biomechanical modelling of the human eye (1)

JO H AN N E S K E P L E RU N I V ER S I TÄ T L IN Z

N e t z w e r k f ü r F o r s c h u n g , L e h r e u n d P r a x i s

Biomechanical Modelling of the Human Eye

Dissertation

zur Erlangung des akademischen Grades

Doktor der Technischen Wissenschaften

im Doktoratsstudium der technischen Wissenschaften

Angefertigt am Institut für Anwendungsorientierte Wissensverarbeitung (FAW)

Eingereicht von:

Dipl.-Ing. (FH) Michael Buchberger

Betreuung:

Univ.-Prof. Dipl.-Ing. Dr. Roland Wagner

Beurteilung:

Univ.-Prof. Dipl.-Ing. Dr. Roland Wagner

Univ.-Doz. Dipl.-Ing. Dr. Thomas Haslwanter

Linz, März 2004

Johannes Kepler UniversitätA-4040 Linz · Altenbergerstraße 69 · Internet: http://www.uni-linz.ac.at · DVR 0093696

Eidesstattliche Erklärung

Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremdeHilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlichoder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe.

Linz, im März 2004 Michael Buchberger

To Bianca and my parents...

Abstract

The goal of this work was the development of a biomechanical model of the human eye. Aninteractive software system was implemented, called „SEE++“ which allows also physicians toobtain a better understanding of the mechanics of eye movements. This software visualizes andsimulates pathologies and eye muscle surgeries, based on the biomechanics of the eye. It canbe used in preoperative planning, medical training and basic research, and shows how Medical-Informatics can improve the diagnosis and treatment of patients.

The interdisciplinary nature of the project required contributions from very different fields.Anatomical studies, in cooperation with researchers as well as practicing physicians, provideddata for defining a mathematical representation of human eye movements. The biomechanicalmodel included a geometrical representation of eye movements, a muscle force prediction model,and a kinematic model that balances muscle forces by using mathematical optimization meth-ods. High-resolution magnetic resonance imaging studies were carried out to visualize eye musclemorphology, and image processing methods used to reconstruct three dimensional approximationmodels of human eye muscles. Modern software engineering methods provided the basis for anextensible object-oriented software design. Three dimensional interactive visualization and a userinterface optimized for medical use were combined into a unique software simulation system forthe clinic and for teaching.

The „SEE++“ software system is currently the most advanced biomechanical representationof the human eye, with respect to simulating eye movements and eye muscle surgeries. Theextensive possibilities for parametrization of the human eye model allow interactive simulationsof pathological cases and surgical corrections, and the predictions correspond well with clinicaldata. This system is used in various clinical facilities as computer aided decision support forstrabismus (squint) surgeries. In medical training and education, it substantially improves thefunctional understanding of human eye movements.

Keywords: Biomechanical Modelling, Eye Surgery, Strabismus, Eye Motility, Medical DecisionSupport Systems

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Kurzfassung

Diese Forschungsarbeit hat das Ziel, ein biomechanisches Modell des menschlichen Auges zuentwickeln. Das implementierte Software System, „SEE++“, soll es Medizinern ermöglichen, einbesseres Verständnis der Mechanik der Augenbewegungen zu bekommen. Augenbewegungsstö-rungen und Augenmuskeloperationen werden dabei auf biomechanische Ursachen und Wirkungenzurückgeführt. Der erfolgreiche klinische Einsatz dieses Systems zeigt, wie computerbasierte Me-thoden der Medizin-Informatik die Diagnose und Behandlung von Patienten verbessern können.

Die interdisziplinären Anforderungen dieses Projekts erforderten Beiträge aus stark unterschied-lichen medizinisch-technischen Forschungsbereichen. Anatomische Studien lieferten Grunddatenfür die Formulierung eines mathematischen Modells der menschlichen Augebewegungen. Biome-chanische Überlegungen führten zu einer geometrischen Beschreibung von Augenbewegungen,einer Muskelkraftsimulation und eines kinematischen Modells. Augenpositionen wurden mit ma-thematischen Optimierungsmethoden aus dem Kräftegleichgewicht der Augenmuskulatur berech-net. Um die Morphologie der Augenmuskulatur besser zu verstehen, wurden umfangreiche Studienmit hochauflösender Magnetresonanztomographie durchgeführt, und mit Bildverarbeitungsme-thoden die dreidimensionalen Rekonstruktionen berechnet. Der Einsatz von modernen Methodendes objektorientierten Software-Engineering bildete die Grundlage für eine flexible Implementie-rung. Dreidimensionale interaktive Visualisierung und optimiertes Benutzerschnittstellen-Designwurden in einem einzigartigen Software System kombiniert.

Das biomechanische Simulationssystem „SEE++“ ist derzeit das weltweit detaillierteste und mo-dernste Softwaresystem für die Modellierung und Simulation von Augenbewegungsstörungen. DasSystem ermöglicht durch umfangreiche Möglichkeiten der Parametrisierung die Simulation vonpathologischen Fällen und deren operative Korrektur. Die Simulationsergebnisse zeigen nach-weislich eine gute Übereinstimmung mit verfügbaren klinischen Vergleichsdaten. Derzeit wirddieses System für die computerbasierte Entscheidungsunterstützung in verschiedenen klinischenEinrichtungen verwendet. Der Einsatz in der medizinischen Ausbildung verbessert das Verständ-nis über Funktion und Wirkungsweise von menschlichen Augenbewegungen. In der medizinisch-physiologischen Grundlagenforschung ermöglicht das System die Auswertung und Evaluierungvon Messergebnissen, und gibt somit einen detaillierteren Einblick in die komplizierte Strukturdes menschlichen Auges.

Schlüsselwörter: Biomechanische Modelle, Augenoperation, Strabismus, Augenmotilität, Klini-sche Entscheidungsunterstützung

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Acknowledgements

Within eight years of research, many people have greatly contributed to this work. First of all, Iwould like to express my deepest gratitude, respect and admiration to Prim. Prof. Dr. SiegfriedPriglinger, head of the ophthalmologic department at the convent hospital of the „BarmherzigenBrüder“ in Linz. He did not only start this project, but also greatly contributed to this work asteacher, mastermind and highly experienced medical expert, always emphasizing improvement inpatient care. His exceptional personality and social engagement inspired me beyond my technicalwork.

I would like to thank Univ.-Prof. Dipl.-Ing. Dr. Roland Wagner, head of the Research Institutefor Applied Knowledge Processing (FAW) at the University of Linz for reviewing and supportingthis work.

From the ETH-Zurich, I would like to thank Univ.-Doz. Dipl.-Ing. Dr. Thomas Haslwanter forexplaining medical details and giving valuable advice and guidance throughout the creation ofthis thesis. From the medical side, many physicians and researchers have been involved in thisresearch work. For valuable cooperation I would like to thank Dr. Joel Miller from the SmithKettlewell Eye Research Institute. Furthermore, Prim. Univ.-Prof. Dr. Erich Salomonowitz,Prim. Univ.-Doz. DDr. Armin Ettl and Dr. Jörg Hildebrandt from the hospital St. Pöltensupported this work by providing clinical patient data. Additionally, the radiologic departmentof the Wagner Jauregg hospital in Linz lead by Prim. Dr. Johannes Trenkler with the help ofUniv.-Doz. Dr. Franz Fellner and Univ.-Doz. DDr. Dipl.-Ing. Mag. Josef Kramer providednecessary equipment and medical expert knowledge to carry out different physiological studies.

For usability testing and evaluation of the software system „SEE++“ I would like to thank Univ.-Prof. Dr. Andrea Langmann from the university in Graz for believing in this work. From theuniversity of Innsbruck I appreciate the help of OA Dr. Eduard Schmid, OA Dr. Ivo Baldisseraand OA Dr. Cornelia Stieldorf. From the hospital of the „Barmherzigen Schwestern“ in Ried Iwould like to thank OA Dr. Robert Hörantner for extensively testing the software and providingvaluable feedback.

From the Upper Austrian University of Applied Sciences in Hagenberg I would especially like tothank Univ.-Prof. Dipl.-Ing. Dr. Witold Jacak for initiating and greatly supporting the „SEE-KID“ project. Moreover, thanks go to FH-Prof. Dipl.-Ing. Dr. Herwig Mayr, who also supportedthe project in its beginnings with his project engineering knowledge.

Many diploma students were involved within this project. First of all, I would like to givemy respect to Dipl.-Ing. (FH) Thomas Kaltofen for extensive implementation work and above-average participation within this work. Additionally, Dipl.-Ing. (FH) Martin Wiesmair, Dipl.-Ing.(FH) Franz Pirklbauer, Dipl.-Ing. (FH) Stefan Satzinger and Dipl.-Ing. (FH) Michael Lacherspent their internship and diploma semester in the field of the „SEE-KID“ project. I would alsolike to thank Dipl.-Ing. (FH) Thomas Kern, Dipl.-Ing. (FH) Johannes Dirnberger, Mag. MichaelGiretzlehner and Dr. Thomas Luckeneder for proof reading this work.

Last but not least, I would like to thank Dipl.-Ing. Dr. Otmar Höglinger from the Upper AustrianResearch GmbH for believing in new technologies and providing a fascinating infrastructure forapplication-oriented research in the field of Medical-Informatics. This work was also supportedby grants from the Austrian Ministry of Science (FFF) and the Upper Austrian government.

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Contents

1 Introduction 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Medical Informatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Clinical Decision Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Eye muscle surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 SEE-KID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Medical Foundations 92.1 Anatomy of the Human Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Eye Muscle „Pulleys“ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Global/Orbital Eye Muscle Layers . . . . . . . . . . . . . . . . . . . . . . 182.1.3 Anatomical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.4 High Resolution MRI-Imaging of the Orbit . . . . . . . . . . . . . . . . . 212.1.5 Human Dissection of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Eye Movement Physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.1 Actions of the Extraocular Muscles . . . . . . . . . . . . . . . . . . . . . . 272.2.2 Kinematic Principles of Eye Movements . . . . . . . . . . . . . . . . . . . 29

2.2.2.1 Donders’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2.2 Listing’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.3 Sensorimotor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.3.1 Innervation of the Eye Muscles . . . . . . . . . . . . . . . . . . . 342.2.3.2 Oculomotor Neurons . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.3.3 Neural Signal Encoding . . . . . . . . . . . . . . . . . . . . . . . 362.2.3.4 The Superior Colliculus . . . . . . . . . . . . . . . . . . . . . . . 372.2.3.5 Brainstem Control of Saccades . . . . . . . . . . . . . . . . . . . 382.2.3.6 Control of Smooth-Pursuit Movements . . . . . . . . . . . . . . . 402.2.3.7 Hering’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.3.8 Sherrington’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 Eye Movement Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.1.1 Electro-Oculography . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.1.2 Infrared-Oculography . . . . . . . . . . . . . . . . . . . . . . . . 432.3.1.3 Scleral Search Coils . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.1.4 Video-Oculography . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.2 Physiologic Muscle Force Measurements . . . . . . . . . . . . . . . . . . . 472.3.3 Measurement of Motion in the Orbit . . . . . . . . . . . . . . . . . . . . . 51

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3 Strabismus 533.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.1 Visual Acuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1.2 Symptoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1.3 Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Binocular Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.1 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.2 Diplopia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Ocular Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4 Clinical Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.1 Corneal Light Reflex Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.2 Cover Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.3 Subjective Clinical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4.4 Hess-Lancaster Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Eye Motility Disorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.5.1 Concomitant Strabismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.2 Incomitant Strabismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5.2.1 Paralytic Strabismus . . . . . . . . . . . . . . . . . . . . . . . . . 733.5.2.2 Duane’s Syndrome . . . . . . . . . . . . . . . . . . . . . . . . . . 773.5.2.3 Fibrosis Syndrome . . . . . . . . . . . . . . . . . . . . . . . . . . 803.5.2.4 Supranuclear Disorders . . . . . . . . . . . . . . . . . . . . . . . 81

3.6 Strabismus Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.6.1 Recession Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.6.2 Resection Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.6.3 Transposition Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.6.4 Amount of Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Biomechanical Modelling 894.1 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Structure of Biomechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3 History of Modelling of the Human Eye . . . . . . . . . . . . . . . . . . . . . . . 924.4 Ocular Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.2 Mathematical Description of Eye Rotations . . . . . . . . . . . . . . . . . 96

4.4.2.1 Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 964.4.2.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4.2.3 Listing’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4.2.4 Definition of Eye Positions . . . . . . . . . . . . . . . . . . . . . 104

4.4.3 Geometrical Abstractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4.3.1 Globe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4.3.2 Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.4.3.3 Evaluation of Muscle Action . . . . . . . . . . . . . . . . . . . . 118

4.4.4 Passive Geometrical Changes . . . . . . . . . . . . . . . . . . . . . . . . . 1214.5 Muscle Force Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.5.1 Length-Tension Relationship . . . . . . . . . . . . . . . . . . . . . . . . . 1234.5.2 Elastic Force Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.5.3 Contractile Force Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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4.5.4 Total Force Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.6 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.6.1 Orbital Restoring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.6.2 Globe Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.6.3 Balancing Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.6.3.1 Solving for Eye Positions . . . . . . . . . . . . . . . . . . . . . . 1404.6.3.2 Solving for Innervations . . . . . . . . . . . . . . . . . . . . . . . 141

4.7 Brainstem Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.7.1 Simulation of Binocular Function . . . . . . . . . . . . . . . . . . . . . . . 143

5 Visualization of Muscle Action 1485.1 Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.1.1 Picture Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.1.2 Generation of Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.2 Surface Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2.1 Calculation of the Muscle Path . . . . . . . . . . . . . . . . . . . . . . . . 156

5.2.1.1 Analyzing Surface Distribution . . . . . . . . . . . . . . . . . . . 1595.2.2 Approximation of Muscle Surface . . . . . . . . . . . . . . . . . . . . . . . 161

5.2.2.1 Optimized Rendering . . . . . . . . . . . . . . . . . . . . . . . . 1625.2.3 Interpolation of Muscle Models . . . . . . . . . . . . . . . . . . . . . . . . 164

5.3 Reconstruction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.3.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6 Software Design and Implementation 1696.1 Design of the Biomechanical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.2 Design of the „SEE++“ Software System . . . . . . . . . . . . . . . . . . . . . . . 1766.3 The „SEE++“ Software System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.3.1 „SEE++“ Simulation Task Flow . . . . . . . . . . . . . . . . . . . . . . . 1816.3.2 Simulation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.3.2.1 Globe Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.3.2.2 Muscle Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.3.2.3 Distribution of Innervation . . . . . . . . . . . . . . . . . . . . . 1856.3.2.4 Gaze Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1866.4.1 Abducens Palsy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.4.1.1 Simulation of the Pathology . . . . . . . . . . . . . . . . . . . . . 1866.4.1.2 Simulation of Surgical Correction . . . . . . . . . . . . . . . . . . 188

6.4.2 Superior Oblique Palsy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.4.2.1 Simulation of the Pathology . . . . . . . . . . . . . . . . . . . . . 1896.4.2.2 Simulation of Surgical Correction . . . . . . . . . . . . . . . . . . 191

6.4.3 Superior Oblique Overaction . . . . . . . . . . . . . . . . . . . . . . . . . 1916.4.3.1 Simulation of the Pathology . . . . . . . . . . . . . . . . . . . . . 1926.4.3.2 Simulation of Surgical Correction . . . . . . . . . . . . . . . . . . 193

6.4.4 Heavy-Eye Syndrome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.4.4.1 Simulation of the Pathology . . . . . . . . . . . . . . . . . . . . . 1966.4.4.2 Simulation of Surgical Correction . . . . . . . . . . . . . . . . . . 199

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7 Conclusion 2017.1 Goals Achieved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Literature 205

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List of Figures

1.1 Example Applications Based on Data from the Visual Human Project r© . . . . . 41.2 Classification of Clinical Decision Support Systems . . . . . . . . . . . . . . . . . 51.3 SEE++ Virtual Eye Muscle Surgery Software . . . . . . . . . . . . . . . . . . . . 8

2.1 Definition of Anatomical Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 View of the Bony Orbita of a Right Eye . . . . . . . . . . . . . . . . . . . . . . . 112.3 Anatomy of the Globe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Anatomy of the Extraocular Eye Muscles . . . . . . . . . . . . . . . . . . . . . . 152.5 Extraocular Muscles of a Right Eye from above . . . . . . . . . . . . . . . . . . . 152.6 Schematic View of the Extraocular Tissue Architecture . . . . . . . . . . . . . . . 172.7 Example of Lateral Rectus Path Influenced by Pulley . . . . . . . . . . . . . . . . 182.8 Different Fiber Layers of Rectus Muscles . . . . . . . . . . . . . . . . . . . . . . . 192.9 Axial MR Scan, showing Lateral and Medial Rectus of both Eyes . . . . . . . . . 232.10 Axial MR Scan using Contrast Agent, showing Lateral and Medial Rectus of both

Eyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.11 Comparison of MR Scans in different Gaze Positions . . . . . . . . . . . . . . . . 242.12 Human Dissection of the Orbit showing Medial Rectus Pulley . . . . . . . . . . . 252.13 Line of Sight, Vertical and Horizontal Axes; Rotations to Other Eye Positions . . 262.14 Binocular Fixation of an Object in Space . . . . . . . . . . . . . . . . . . . . . . 272.15 Rotational Directions for both Eyes . . . . . . . . . . . . . . . . . . . . . . . . . . 282.16 Simple Abstraction of Listing’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 312.17 Definition of Listing’s Law w.r.t. Primary Position . . . . . . . . . . . . . . . . . 322.18 Recording of Saccadic Eye Movements . . . . . . . . . . . . . . . . . . . . . . . . 342.19 Ocular Motor Neurons and Motor Nuclei . . . . . . . . . . . . . . . . . . . . . . . 352.20 Oculomotor Nerve Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.21 Location and Distribution of Oculomotor Nerves . . . . . . . . . . . . . . . . . . 372.22 Motor Circuit for Horizontal Saccades . . . . . . . . . . . . . . . . . . . . . . . . 392.23 EOG Eye Movement Measurement Technique . . . . . . . . . . . . . . . . . . . . 422.24 Schematic Overview of Purkinje Corneal Reflections . . . . . . . . . . . . . . . . 442.25 Purkinje Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.26 Electromagnetic Search Coil Eye Position Measurement . . . . . . . . . . . . . . 452.27 Insertion Procedure for a Scleral Search Coil Lens . . . . . . . . . . . . . . . . . . 462.28 Video-Oculography Pupil Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 462.29 Chronos VOG System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.30 Different Types of Muscle Force Measurement . . . . . . . . . . . . . . . . . . . . 482.31 Example of measuring Force with Forceps . . . . . . . . . . . . . . . . . . . . . . 492.32 Muscle Force Transducer for Intraoperative Measurements . . . . . . . . . . . . . 50

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2.33 Intraconal Tissue Motion around the Optic Nerve . . . . . . . . . . . . . . . . . . 51

3.1 Projection of Objects in Space onto the Retina on an Eye . . . . . . . . . . . . . 573.2 Different Forms of Binocular Fixation . . . . . . . . . . . . . . . . . . . . . . . . 583.3 Example for Inward Squinting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Example for Outward Squinting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.5 Example for Upward Squinting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.6 Example for Downward Squinting . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7 Pseudo-Esotropia due to wide Bridge and Epicanthal Skin Fold . . . . . . . . . . 623.8 Hirschberg Light Reflex Test Method . . . . . . . . . . . . . . . . . . . . . . . . . 633.9 Krimsky Light Reflex Test Method . . . . . . . . . . . . . . . . . . . . . . . . . . 633.10 Prism-Cover Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.11 Cover Tests for Tropias and Phorias . . . . . . . . . . . . . . . . . . . . . . . . . 653.12 Maddox-Wing Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.13 Binocular Fixation in the Hess-Lancaster Test . . . . . . . . . . . . . . . . . . . . 673.14 Hess-Lancaster Diagram for Right Eye (Left Eye Fixing) . . . . . . . . . . . . . . 683.15 Interpretation of Hess-Diagram according to Muscle Actions . . . . . . . . . . . . 683.16 Hess-Diagram Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.17 Example of Disturbance of the Binocular Muscular Team . . . . . . . . . . . . . 723.18 Example of Abnormal Head Posture, Compensating Esotropia . . . . . . . . . . . 733.19 Example of a Right Superior Rectus Palsy . . . . . . . . . . . . . . . . . . . . . . 743.20 Hess-Lancaster Chart for Right Superior Rectus Palsy . . . . . . . . . . . . . . . 753.21 Example of a Right Superior Oblique Palsy . . . . . . . . . . . . . . . . . . . . . 763.22 Hess-Lancaster Chart for Right Superior Oblique Palsy . . . . . . . . . . . . . . . 773.23 Example for Duane’s Retraction Syndrome Type 3 of a Right Eye . . . . . . . . . 783.24 Example for Brown’s Syndrome of a Right Eye . . . . . . . . . . . . . . . . . . . 793.25 Example for an Abducens Gaze Palsy . . . . . . . . . . . . . . . . . . . . . . . . 823.26 Schematic Example of Muscle Recession . . . . . . . . . . . . . . . . . . . . . . . 843.27 Preparation for Medial Rectus Recession . . . . . . . . . . . . . . . . . . . . . . . 843.28 Medial Rectus Recession, continued . . . . . . . . . . . . . . . . . . . . . . . . . . 853.29 Medial Rectus Recession, continued . . . . . . . . . . . . . . . . . . . . . . . . . . 853.30 Schematic example of Muscle Resection . . . . . . . . . . . . . . . . . . . . . . . 863.31 Medial Rectus Resection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.32 Medial Rectus Resection, continued . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1 Halle’s Ophthalmotrope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2 Ruete’s Ophthalmotrope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.3 Coordinate System of a Left Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4 Gimbal Systems for describing 3D Eye Position . . . . . . . . . . . . . . . . . . . 954.5 Geometrical Abstraction of the Globe . . . . . . . . . . . . . . . . . . . . . . . . 1054.6 Geometrical Abstraction of an Eye Muscle . . . . . . . . . . . . . . . . . . . . . . 1064.7 Geometrical „String Model“ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.8 Point of Tangency in the „String Model“ . . . . . . . . . . . . . . . . . . . . . . . 1084.9 Tape Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.10 Geometrical Tape Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.11 Comparison of Conventional vs. Pulley Model . . . . . . . . . . . . . . . . . . . . 1124.12 Primary Position in Pulley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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4.13 Tertiary Position in Pulley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.14 Muscle Action Circles in Pulley Model . . . . . . . . . . . . . . . . . . . . . . . . 1164.15 Muscle Direction Vector in Pulley Model . . . . . . . . . . . . . . . . . . . . . . . 1174.16 Muscle Rotation Axis and Action Circle Center in Pulley Model . . . . . . . . . . 1184.17 Point of Tangency in Pulley Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.18 Muscle Path Comparison using Different Geometrical Models . . . . . . . . . . . 1194.19 Muscle Force Distribution in String and Tape Model . . . . . . . . . . . . . . . . 1214.20 Muscle Force Distribution in the Pulley Model . . . . . . . . . . . . . . . . . . . 1214.21 Elastic Force Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.22 Contractile Force Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.23 Total Force Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.24 Apex Coordinate System for measuring Globe Translation . . . . . . . . . . . . . 1324.25 Torque Error Function for Listing Positions in a Healthy Eye . . . . . . . . . . . 1354.26 Torque Error Function for Pathological Eye . . . . . . . . . . . . . . . . . . . . . 1364.27 Torque Error Minimization in solving for Eye Positions . . . . . . . . . . . . . . . 1414.28 Squint-Angles Diagram for Binocular Fixation . . . . . . . . . . . . . . . . . . . . 1434.29 Simulation Task Flow for the Hess-Lancaster Test . . . . . . . . . . . . . . . . . . 145

5.1 DXF Model Generation Tasks using Marching Cubes . . . . . . . . . . . . . . . . 1495.2 Sorting of Color Index Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.3 Definition of 2D Polygon for MRI Segmentation . . . . . . . . . . . . . . . . . . . 1515.4 Threshold Region for MRI Slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.5 Marching Cube Traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.6 Marching Cubes Standard Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.7 Surface Reconstruction Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.8 DXF Model with Area Centroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.9 Approximation of the Muscle Path . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.10 Angular Measurement of Surface Points . . . . . . . . . . . . . . . . . . . . . . . 1595.11 Example of Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.12 NURBS Approximation of Muscle Cross-Section . . . . . . . . . . . . . . . . . . . 1625.13 Linear Interpolation of NURBS-generated Cross-Section . . . . . . . . . . . . . . 1635.14 Reconstruction of a Left Medial Rectus Muscle . . . . . . . . . . . . . . . . . . . 1665.15 Shaded Reconstruction of a Left Medial Rectus Muscle . . . . . . . . . . . . . . . 1675.16 Morphology of Reconstructed Medial Rectus Muscle . . . . . . . . . . . . . . . . 1675.17 Shaded Reconstruction of a Left Medial Rectus Muscle with MRI Data . . . . . . 168

6.1 Structure of the „SEE++“ Software System . . . . . . . . . . . . . . . . . . . . . 1706.2 Model-View-Controller Structure of the „SEE++“ Software System . . . . . . . . 1716.3 Abstraction of Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.4 Primary Abstractions for Biomechanical Model . . . . . . . . . . . . . . . . . . . 1736.5 Extensible Software Design for the Biomechanical Model . . . . . . . . . . . . . . 1746.6 Optimizer and Related Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1756.7 Structure of the „SEE++“ Package . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.8 Structure of the „SeeMedic“ Package . . . . . . . . . . . . . . . . . . . . . . . . . 1776.9 Structure of the „SeeModel“ Package . . . . . . . . . . . . . . . . . . . . . . . . . 1786.10 Structure of the „SeeView“ Package . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.11 Muscle Force Vector Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

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6.12 Simulation Task Flow for using „SEE++“ . . . . . . . . . . . . . . . . . . . . . . 1816.13 Default View of the „SEE++“ Software System . . . . . . . . . . . . . . . . . . . 1836.14 Globe Data Parameters of the „SEE++“ System . . . . . . . . . . . . . . . . . . 1836.15 Muscle Data Parameters of the „SEE++“ System . . . . . . . . . . . . . . . . . . 1846.16 Gaze Pattern Dialog of the „SEE++“ System . . . . . . . . . . . . . . . . . . . . 1856.17 Muscle Properties for Abducens Palsy Simulation . . . . . . . . . . . . . . . . . . 1876.18 Hess-Lancaster Test for Abducens Palsy . . . . . . . . . . . . . . . . . . . . . . . 1876.19 Simulation of Right Lateral Rectus Resection . . . . . . . . . . . . . . . . . . . . 1886.20 Postoperative Hess-Lancaster Simulation for Abducens Palsy . . . . . . . . . . . 1896.21 Muscle Properties for Superior Oblique Palsy . . . . . . . . . . . . . . . . . . . . 1906.22 Hess-Lancaster Simulation of Superior Oblique Palsy . . . . . . . . . . . . . . . . 1906.23 Postoperative Hess-Lancaster Simulation for Superior Oblique Palsy . . . . . . . 1916.24 Classification of Superior Oblique Overaction . . . . . . . . . . . . . . . . . . . . 1926.25 Hess-Lancaster Simulation of Superior Oblique Overaction . . . . . . . . . . . . . 1936.26 Simulation of Superior Oblique Surgery . . . . . . . . . . . . . . . . . . . . . . . 1946.27 Hess-Lancaster Simulation of Superior Oblique Surgery . . . . . . . . . . . . . . . 1946.28 Muscle Displacement as Hypothesis for Heavy-Eye Syndrome . . . . . . . . . . . 1956.29 Measured Values from Patient with Heavy-Eye Syndrome . . . . . . . . . . . . . 1966.30 Simulation Results for Resized Globes according to Patient Data . . . . . . . . . 1976.31 Simulation Attempt using Data suggested by Schroeder and Krzizok . . . . . . . 1976.32 Superior Oblique Muscle Insertion Transposition in Heavy-Eye Simulation . . . . 1986.33 Hess-Lancaster Simulation Results of the Heavy-Eye Syndrome . . . . . . . . . . 1996.34 Muscle Surgery for Heavy-Eye Simulation . . . . . . . . . . . . . . . . . . . . . . 1996.35 Hess-Lancaster Simulation Results of the Heavy-Eye Surgery . . . . . . . . . . . 200

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Chapter 1

Introduction

The research work presented in this document describes a successful combination and applicationof interdisciplinary research in the fields of computer science and eye muscle strabismus surgery.Basic medical research was incorporated in a mathematical model and subsequently realizedas interactive software system for clinical usage. The main goal of this work was application-oriented research in order to realize a software system that directly affects advancement in patienttreatment.

The goal of this thesis is to prove that software simulation systems can be of valuable assistancein surgical decision making in the field of eye muscle strabismus surgery. The combination ofessential diagnostic data of individual surgical cases, its diagnosis and and its possible treatmentsinto a well designed, clinically applicable computer software system is presented. A new way ofinteractive, virtual eye muscle surgery evaluation an preoperative planning method is proposed,based on a biomechanical model of the human eye in order to predict surgical outcomes onthe basis of objective, anatomically related and measurable data. Additionally, physiologicallymeaningful three-dimensional visualization proves to support detailed evaluation and meaningfulinterpretation of disorders of the human eye movements. However, it is not intended to replacecommon clinical diagnostics, nor surgical expertise, instead, the proposed work should be con-sidered as a clinical decision support tool to process diagnostic data into possible choices andamounts of surgery in an objective manner.

1.1 Overview

Due to the extensive medical implication of this work, the first two Chapters give an overview ofmedical foundations within the field of ophthalmology with special regard to strabismus. In Chap-ter 2, the basic anatomy of the human eye, its muscles and the orbita are explained. Additionally,anatomical measurements provide quantitative data based on available literature. Magnetic res-onance imaging studies and human dissection were performed within the presented research workthat provide detailed data on the anatomy and physiology of the extraocular eye muscles. Resultsfrom magnetic resonance imaging (MRI) studies explained in Sec. 2.1.4 were used to visualizethree dimensional muscle morphology (cf. Ch. 5) which resulted in the most detailed model-basedvisualization of human eye muscles currently available.

1

CHAPTER 1. INTRODUCTION 2

Understanding eye movement physiology is very important in order to build a biomechanicalmodel. Therefore, Chapter 2 contains valuable information on kinematic principles of ocularmovements as well as detailed information on sensorimotor control and neural signal encoding.Especially important for biomechanical modelling are kinematic and sensory principles that havebeen identified as central laws that constrain ocular movements. All these principles need to beincluded in a biomechanical model that gives anatomically related predictions.

In order to relate predictions of a simulation model to clinical data, diagnostic methods are neededto quantify eye motility disorders. These methods are presented in Sec. 2.3 where measurementresults serve as starting point for the simulation of pathological situations. Additionally, physio-logic muscle force measurements that were carried out by a partner institution (Smith KettlewellEye Research Institute, San Francisco) are introduced in Sec. 2.3.2. These measurements wereincluded in the biomechanical model in order to simulate muscle forces.

In Chapter 3 of this thesis, the field of strabismus is introduced. Since the main goal of thebiomechanical model and the software system that are presented in this thesis is the applicationin the field of strabismus, it is essential to understand function and basic medical diagnosis andtreatment in the field of strabismus. Therefore, Chapter 3 gives an overview of the physiologyof binocular vision and ocular dissociation in Sec. 3.2 and Sec. 3.3. In Sec. 3.4 the clinicalassessment of eye motility disorders is described. Without the use of simulation model, thesemostly subjective measurement methods are currently the only diagnostic basis. In Sec. 3.5,major categories and examples of important eye motility disorders are explained, whereas Sec. 3.6describes the most important treatment methods within surgical interventions.

In Chapter 2 as well as in Chapter 3 of this thesis, illustrations of eye muscle anatomy andexamples of major eye muscle disorders were already produced with the biomechanical simulationsystem „SEE++“.

In Chapter 4, the major part of the presented research work is described. In Sec. 4.2, thestructure of a biomechanical model is described which can be split into geometrical, muscle forceand kinematic sub models. In this work, a unique mathematical formulation of ocular geometryusing quaternions is described in Sec. 4.4. Furthermore, the mathematical implementation ofthe muscle force prediction model is explained in Sec. 4.5. Another major achievement thataccounts for the stability of simulation predictions is the formulation of the kinematic model,explained in Sec. 4.6, that connects geometry and muscle forces in order to solve a standard non-linear minimization problem. Currently, this is the only biomechanical eye model that strictlydissociates geometry from muscle force and kinematics. Through the application of standardnon-linear numerical minimization methods, the accuracy of simulation predictions with respectto clinical measurements is unique for this type of medical application.

In Chapter 5 of this thesis, a new approach of three dimensional reconstruction of shape andmorphology of the extraocular muscles is presented. Reconstruction results were incorporated inthe software simulation system „SEE++“ in order to visualize anatomically related data. Imageanalysis and image processing methods were used to generate interpolated muscle models thatwere connected to the muscle force simulation of the biomechanical model.

Finally, Chapter 6 gives an overview of the software system design for the biomechanical model aswell as for the software system „SEE++“ that has been implemented. Modern methods in objectoriented software engineering were used to build a generic extensible and robust software system


that provides flexibility for further advancements. An additional feature is that the biomechan-ical model can be seen as autonomous part within the software simulation system „SEE++“.While the software system provides user interface, most advanced three dimensional interactivevisualization and representation of biomechanical parameters as anatomically meaningful prop-erties, the biomechanical model implements and encapsulates the mathematical formulation forgeometric, muscle force and kinematic models. Therefore, in Sec. 6.1, main parts of the softwaredesign of the biomechanical model are explained and in Sec. 6.2, main aspects of the design modelof the „SEE++“ software system are described.

Additionally, in Chapter 6 the user interface and simulation parameters of the „SEE++“ softwaresystem are explained in Sec. 6.3. This Chapter then concludes with case studies showing differenteye motility disorders that were simulated with the „SEE++“ system. Currently, „SEE++“ isthe only simulation system that is able to also simulate complex eye motility disorders as shownin the „Heavy-Eye“ example in Sec. 6.4.4.

1.2 Medical Informatics

The rapid development of computer systems increasingly enables the use of software systemswithin the medical field. Efficient computers for image processing and 3D-graphics in combinationwith specialized systems offer a practical supplement e.g. in medical diagnostics. In buildingsuch systems, interdisciplinary research in the fields of e.g. medicine, mathematics, physics andinformatics is inevitable. Generally, research activities in the described field can be assigned to thefield of Medical Informatics, which is defined as the application of computers, communications andinformation technology and systems to all fields of medicine - medical care, medical education andmedical research [BM00a]. Therefore, this field combines medical science with several technologiesand disciplines in the information and computer sciences and provides methodologies by whichthese can contribute to better use of the medical knowledge base and ultimately to better medicalcare [BM00a].

Medical Informatics range from computer-based patient records to image processing and fromprimary care practices to hospitals of health care. Processing information plays a vital rolein health care since the field of medical knowledge is growing every day, and there needs tobe a way to properly formalize, store, process, and access that knowledge. Diseases and theirdiagnosis are being explored more and more so health care professionals have to take advantageof the information. The number of administrative and legal requirements dealing with processinginformation is on the rise. Information technology, if utilized correctly, can improve health caretremendously which is why health care professionals have to be educated about health and medicalinformatics.

The presented research work focuses on simulation software systems providing medical decisionsupport by using methods and technologies for the fields of mathematics, physics and softwareengineering. A substantial criterion for the application of such systems in practice is the reliabilityof medically relevant data or results, as well as the scope of interpretation applied to such data.In the application of virtual reality in connection with surgical interventions, the success of anoperation is substantially influenced by data obtained from such a system. Detailed graphicalvisualization enables the surgeon to preoperatively simulate a disease, and afterwards by means of


(a) Segmentation (b) Rendering

Figure 1.1: Example Applications Based on Data from the Visual Human Project r©, from [Ack02]

interactive „virtual surgery“, plan, check and possibly even correct a surgical procedure in order toachieve the best result. Continuous endeavor of current research is associated with the so called„Visual Human Project r©“ (Fig. 1.1), an anatomically detailed, three-dimensional representationof human bodies, in order to use these data in different fields of application (e.g. Visible HumanExplorer, Cross Sectional Anatomy, Voxelman r©, Body Navigator, etc.) [Ack02]. Acquisitionof transverse computed tomography (CT), magnetic resonance (MR) and cryosection images ofrepresentative male and female cadavers has been completed. The male was sectioned at onemillimeter intervals, the female at one-third of a millimeter intervals. The long-term goal of theVisible Human Project r© is to produce a system of knowledge structures that will transparentlylink visual knowledge forms to symbolic knowledge formats such as the names of body parts.Another goal of this work is to represent the function of the human body as realistically as possibleby trying to apply well-known relationships from the mechanics to the anatomy of humans.Complex mathematical models of skeletons, muscles, joints and their graphical, three-dimensionalvisualization form the basis of an interactive system. By means of systematic studying of suchsystems, new insight can be derived, integrated into the model and subsequently be used toextend research.

1.3 Clinical Decision Support

In order to improve a patients prognosis, clinicians continuously make decisions on what di-agnostic procedures they should perform and on what therapeutic actions they should take.Clinical decision support tools may provide predictions on e.g. the diagnosis of a disease andpotential promising treatment suggestions on the basis of clinical information about a patient.However, clinical information can be gathered in many different ways by using existing informa-tion technology infrastructure. Clinical decision support tools often make use of different sourcesof information (Fig. 1.2 [BK03a]) e.g. medical information systems provide electronic medicalpatient records or protocol-based systems offer standard algorithms that define one precise man-ner in which certain classes of patients should be evaluated or treated. Language coding andclassification systems complement these systems by realizing natural, intuitive interfaces. Com-munication systems in health care (e.g. workflow management tools and electronic transmission


of diagnostic findings) properly arrange selective distribution of medically relevant information.

Figure 1.2: Classification of Clinical Decision Support Systems

Intelligent clinical decision support systems overlap the described technologies for the purpose ofcombining medical and administrational data in order to build up clinical useful knowledge withincertain disciplines. Methods and models that operate on this data and are specifically designedfor investigations in certain medical fields enable the formulation of predictive conclusions inorder to evaluate diagnosis and treatment options. By means of systematic analysis of patientdata, such systems may support clinicians in their decision-making processes. The developmentof predictive decision support tools concentrates on five main steps [BM00a]:

1. Analysis: In this step, the clinical problem and its specific characteristics must be exam-ined. This process must be described clearly, including the potential role of the decisionsupport tool.

2. Outcome: The desired clinical outcome that clinicians consider central for decision makingmust be indicated. In most cases, these outcomes are defined in clinically related groups ofdiseases or diagnoses.

3. Predictors: This determines the clinical characteristics that might be used as predictorsof the clinical outcome. Predictors directly influence the way a patient is treated and thustransitively determine the treatment outcome.

4. Quantification: This step defines a relation between the predictors and the clinical out-come. The quantification may be provided by expert clinicians who possess the clinicalknowledge and is captured within a mathematical description (e.g. statistical model).

5. Presentation: The predictive decision support tool needs to be presented to the users inan applicable way. Often, this includes the design and implementation of an interactivesoftware system, incorporating predictors and quantification relationships.

The research work presented here describes the analysis, design, implementation and applicationof a clinical decision support tool in the field of eye muscle strabismus surgery.


1.4 Eye muscle surgery

Surgery of the eye muscles is surgery to weaken, strengthen, or reposition any of the eye musclesthat move the eyeball (the extraocular muscles). The purpose of eye muscle surgery is generallyto align the pair of eyes so that they gaze in the same direction and move together as a team,either to improve or maintain binocular vision, or to improve the orientation of both eyes. Toachieve binocular vision, corresponding images need to be projected onto corresponding areas ofthe retinae of both eyes. In addition, eye muscle surgery can improve eye alignment in peoplewith other e.g. neurologically caused eye disorders (Nystagmus, Duane Syndrome, etc.).

Depth perception (stereopsis) develops around the age of three months. In case of misalignedeyes, successful development of binocular vision and the ability to perceive three-dimensionallyis constricted. Surgery should not be postponed past the age of four since immediate sensoryadaptations occur in the early stage of growth.

The six extraocular muscles attach via tendons to the sclera (the white, opaque, outer protectivecovering of the eyeball) at different places just behind an imaginary equator encircling the top,bottom, left, and right of one eye. The other end of each of these muscles attaches to a partof the orbit (the eye socket in the skull). These muscles enable the eyes to move up, down,to one side or the other, or any angle in between. Modification of one eye muscle also affectscontrol signals of the brain to the other eye. Normally both eyes move together, receive thesame image on corresponding locations on both retinas, and the brain fuses these images into onethree-dimensional image. The exception is in strabismus which is a disorder where one or botheyes deviate out of alignment, most often outwardly (exotropia) or toward the nose (esotropia),sometimes upward (hypertropia) or downward (hypotropia). The brain now receives two differentimages, and either suppresses one or the person sees double (diplopia). This deviation can, inmost cases, be adjusted by weakening or strengthening the appropriate muscles to move the eyestoward the center. For example, if an eye deviates upward, the muscle at the bottom of theeye could be strengthened. Both eyes are controlled by twelve eye muscles (six muscles per eye)including combined bilateral brain signals that form a highly complex mechanical system. In anutshell, the human mechanical eye system seems to work very hard and complicated in order togive a simple impression.

One main problem in treating eye alignment disorders is, that there is no clinical theory availablethat exactly defines surgical treatment consequences based on diagnostic measurements. Up tonow, clinicians use their own practical experience and raw statistical data to derive dose-responserelationships for certain pathological situations [SS00]. This results in repeated surgery, especiallyin case of complicated, combined eye alignment disorders. This leads to the establishment ofdifferent „treatment philosophies“ without verification of underlying correlation to mechanicaland/or neurological properties. Another issue affects incorporation of patient specific data inorder to adjust accuracy of existing statistical approaches. Moreover, the actions of eye musclesvary as a function of eye position, thus, effect of eye muscle surgery must be evaluated within aspecific field of gaze. Realigning both eyes with respect to only one eye position is most oftennot sufficient to achieve a satisfying result. Many clinical approaches have been undertaken todefine certain guidelines for eye muscle surgery, however, all of these assumptions are based ondifferent ideas of the mechanical behavior of the human eye system (e.g. [Pri81]). Cliniciansand teachers still legitimate decisions based upon outdated assumptions of the action of theextraocular muscles. Since the problem of building up fundamental strategies for clinical use, in


this case, lies in the lack of definitions for elementary concepts describing the modes of operationof the human extraocular system, new approaches had to be found.

1.5 SEE-KID

The research work SEE-KID (Software Engineering Environment for Knowledge based Inter-active Eye motility Diagnostics), described in this work, tries to connect aspects of biomechanicalmodelling with methods of modern software engineering [BM00b]. This work is carried out atthe Upper Austrian Research Center at the department for Medical-Informatics in close cooper-ation with the Upper Austrian University of Applied Sciences in Hagenberg. With internationalpartners (e.g. ETH-Zürich and Smith Kettlewell Eye Research Institute in San Francisco), thisproject started in 1995 and was supported by funding of the Austrian Ministry for Science andTechnology (FFF) within the years 2000 and 2002. Additionally, partners from Austrian hos-pitals and research centers in Linz, St. Pölten, Innsbruck and Graz greatly contributed to thiswork as evaluation partners.

Originally, this research project was initiated by Prim. Prof. Dr. Siegfried Priglinger, head ofthe department of ophthalmology at the convent hospital of the „Barmherzigen Brüder“ in Linz,Austria. This department has specialized in correcting eye motility disorders, particularly ininfants, by e.g. recession or resection of certain eye muscles. Most of these surgeries must beperformed at an early age. In order to avoid a permanent misalignment and a sensory adaptationresulting from it, children must be operated according to individual strategies (e.g. fibrosis syn-drome) as soon as possible. Prerequisite for such surgeries is an early diagnosis and a conservativetreatment plan that includes e.g. masking (covering the better eye to stimulate the recovery ofthe pathological eye).

For the success of an eye muscle surgery, an understanding of the disease mechanism and theanatomically functioning mechanisms is necessary, in order to avoid wrong or multiple surgicaltreatments.

Such model-supported eye muscle operations have been performed at the hospital of the „Barm-herzigen Brüder“ in Linz, Austria, since 1978. Also, new operation techniques and treatment op-tions have been developed during this time. Particularly complicated surgeries must be plannedin detail and suitable operation steps must be selected. At present, surgical procedures can beevaluated and improved so far only directly at the patient. In complicated eye motility disorders,even an experienced surgeon will depend on documented empirical values, which often lead tomultiple treatments until the result is satisfying. The result of this research work is a softwaresystem (SEE++, Fig. 1.3), which enables physicians to simulate eye motility disorders on thebasis of measurements from the patient and to perform all possible surgical treatments inter-actively. Using a 3D representation of the anatomy of the human eye, the surgeon can modeldisorders as deviations from a non-pathological „healthy“ eye. Thus the surgeon can determinethe optimal treatment for the patient and plan its proceeding in detail.

The simulated outcome of a virtual surgery is displayed interactively in the 3D visualization, aswell as through measuring parameters and diagrams familiar to clinicians. In addition, referencepoints and measured values are displayed to the surgeon, enabling better orientation while oper-ating. Moreover, the SEE++ system permits to exchange the model base interactively, therefore,


Figure 1.3: SEE++ Virtual Eye Muscle Surgery Software

model predictions can be compared by applying different modelling strategies on the fly. An-other benefit of this system is that it is capable of simulating binocular highly complex diseasesincluding neurologically caused pathologies (e.g. nuclear, inter- or supranuclear lesions).

Chapter 2

Medical Foundations

Understanding visual experience has long challenged the best of human minds, from the AncientGreek’s interest in optics to the study of visual perception by contemporary psychologists andneuroscientists. Today’s scientific study of perception seeks to understand the nature of ourexperience in terms of the underlying mechanisms by which it occurs. Understanding the anatomyand functional behavior of the human eye is causally related to explaining the process of visualperception.

Ongoing new discoveries implying extensive clinical consequences clearly point out, that the basicanatomy and physiology of the human eye is still not explored to the full extent. Especially, newfindings concerning the structure and behavior of eye muscles suggest that the human eye ranksamong the most complex human organs. New radiologic inventions (e.g. MRI, CT, PET and hy-brid technologies [BWM+01]) extend the abilities for diagnostic and scientific exploration. Thisalso explains why today’s fundamental medical research needs interdisciplinary approaches com-prising collaborations of clinicians, technical engineers, mathematicians and physicists. Withoutthe invention and incorporation of new technologies, medical research would not emerge thateffectively.

This chapter will give a basic overview of the human eye’s anatomy which will be required inorder to fully understand principles and properties of the oculomotor plant that will be describedthroughout this thesis. Additionally, physiologic and neurologic principles of oculomotor controlwill be explained providing deeper insight into important aspect of the visual system. Moreover,common measurement techniques are presented, giving new insight into the function of the humanvisual system. Especially in research, these methods are essential for the discovery and validationof various properties of the oculomotor system.

9

CHAPTER 2. MEDICAL FOUNDATIONS 10

Medical professionals and researchers in the medical-technical fields often refer to sections of thebody in terms of anatomical planes (flat surfaces). These planes are imaginary lines - verticalor horizontal - drawn through an upright body. The terms are used to describe a specific bodypart. Fig. 2.1 gives an overview of these anatomical naming conventions.

Figure 2.1: Definition of Anatomical Planes

In Tab. 2.1 general anatomical terms and their meanings are listed. The naming for the anatom-ical planes often differ between coronal plane or frontal plane, sagittal plane or lateral plane andaxial plane or transverse Plane.

Anatomical Terms DirectionMedial Toward the midline of the bodyLateral Away from the midline of the bodyProximal Toward a reference point (extremity)Distal Away from a reference point (extremity)Inferior Lower or belowSuperior Upper or aboveCephalad or Cranial HeadCaudal or Caudad Tail, tail endAnterior or Ventral Toward the frontPosterior or Dorsal Toward the back

Table 2.1: Terms of Medical Directions


2.1 Anatomy of the Human Eye

The eyes, the „sense of perception“ , rank among the most important sensory organs of the humanorganism. They supply us with a constantly updated picture of the environment. The followingexplanations refer to a right eye. The eyeball or globe (lat. bulbus oculi, briefly bulbus, øapprox.24 mm, approximately spherical [PD01]) lies protected in the orbita, recessed in the head.

Fig. 2.2 shows the bony orbita of a right eye. The human orbita is shaped pyramidal and isapprox. 40-50 mm deep. The back end of the orbita (orbital apex) is situated right next to theoptic canal which acts as passage for the optical nerve and the ophthalmic arteries. Except forthe solid orbital boundary, orbital bones are extremely thin.

(a) Orbital bones (b) Orbital apex with nerves and arteries

Figure 2.2: View of the Bony Orbita of a Right Eye [KJCS99]

Referring to Fig. 2.2(a), the following bone structures can be identified:

(1) Os lacrimale - the lacrimal bone, an irregularly rectangular thin plate, forming part of themedial orbital wall of the orbit behind the frontal process of the maxilla (the upper jawbone).

(2) Os ethmoidale - the ethmoid bone, an irregularly shaped bone lying between the orbitalplates of the frontal and anterior to the sphenoid bone (3).

(3) Os sphenoidale - a bone of most irregular shape occupying the base of the skull.

(4) Os zygomaticum - the zygomatic bone, a quadrilateral bone which forms the prominenceof the cheek.

(5) Os frontale - the frontal bone, a large single bone forming the forehead and the uppermargin and roof of the orbit on either side.

(6) Os maxillare - the maxillary bone or the maxilla, an irregularly shaped bone that with itsfellow forms the upper jaw.

In Fig. 2.2(b), a schematic view of the orbital fissures, consisting of a inferior and superior part.The inferior orbital fissure is a cleft between the greater wing of the sphenoid and the orbital plate


of the maxilla, through which pass the maxillary division and the orbital branch of the trigeminalnerve (face nerve) and the infraorbital vessels. The superior orbital fissure is located between thegreater and the lesser wings of the sphenoid establishing a channel of communication betweenthe middle cranial fossa and the orbit, through which pass the oculomotor and trochlear nerves,the ophthalmic division of the trigeminal nerve, the abducens nerve, and the ophthalmic veins.For this work, the area to concentrate on will be the superior orbital fissure, since it provides apassage for all nerves that control the visual organ.

According to Fig. 2.2(b), the following nerves and vessels can be distinguished:

(1) Vena ophthalmica superior - the superior ophthalmic vein

(2) Nervus lacrimalis - the lacrimal nerve, a branch of the ophthalmic nerve supplying sensoryfibres to the lateral part of the upper eyelid, conjunctiva, and lacrimal gland (the glandthat secretes tears).

(3) Nervus frontalis - the frontal nerve, a branch of the ophthalmic nerve which divides withinthe orbit into the supratrochlear and the supraorbital nerves.

(4) Nervus trochlearis - the trochlear nerve that controls the superior oblique eye muscle.

(5) Nervus oculomotoris - the oculomotor nerve is responsible for motor innervation of theupper eyelid muscle, extraocular muscle and pupillary muscle.

(6) Nervus abducens - the abducent nerve, innervates the lateral rectus eye muscle.

(7) Nervus nasociliaris - the nasociliary nerve is a branch of the ophthalmic nerve in the su-perior orbital fissure, passing through the orbit, giving rise to the communicating branch tothe ciliary ganglion, the long ciliary nerves, the posterior and anterior ethmoidal nerves, andterminating as the infratrochlear and nasal branches, which supply the mucous membraneof the nose, the skin of the tip of the nose, and the conjunctiva.

(8) Vena ophthalmica inferior - the inferior ophthalmic vein

(9) Nervus opticus - the optic nerve which is carrying all impulses for the sense of sight.

(10) Arteria ophthalmica - the ophthalmic artery originating from the internal carotid arteryand distributing to the eye, orbit and adjacent facial structures.

The orbita is covered by the periost (or periorbita), a membrane of fibrous connective tissue whichclosely invests all bones except at the articular surfaces. From there, connective tissue and septastabilize and cover intraorbital structures (e.g. globe, muscles and vessels). This anatomicalstructure is also known as Tenon’s capsule.

The globe or bulbus (Fig. 2.3) is built bulb flat-like, composed of three layers of skin [SS98]:

sclera (leather skin) - outer eye skin,

choroidea (vein skin) - middle eye skin, the middle layer of the globe, between retina andsclera


retina (retina) - internal eye skin.

Figure 2.3: Anatomy of the Globe [KJCS99]

Like a high sophisticated camera, the eye has multiple discrete parts which must function togetherproperly to produce a clear vision. To illustrate this, the path of light as it travels through theeye is discussed, and the various ocular structures are identified.

Cornea: The first surface encountered by a ray of light is the tear film. The eye’s surface mustbe kept moist at all times. To achieve this, glands in and near of the eyelids produce bothtears and a special oil which mix together and coat the eye. This tear film coats the corneawhich normally is the crystal clear window to the eye. Behind the cornea, the anteriorchamber is situated, which is filled with aqueous fluid. The aqueous is usually clear likewater and is responsible for maintaining the pressure of the eye.

Iris: Inside the anterior chamber is the iris. This is the part of the eye which is responsible forthe eye-color perceived from an outside viewer. It acts like the diaphragm of a camera,dilating and constricting the pupil to allow more or less light into the eye.

Lens: The next structure encounter is the crystalline lens. The lens is responsible for focusinglight onto the retina. It changes shape slightly to allow adaption of focus between objectsthat are near and those that are far. During the process of aging, the lens becomes lessflexible and able to „accommodate“ or change focus.

Vitreous: This is a jelly-like substance that fills the body of the eye. It is normally clear andin early life, it is firmly attached to the retina behind it. With age, the vitreous becomesmore water-like and may detach from the retina. Often, little clumps or strands of the jellyform and cast shadows which are perceived as „floaters“.

Retina: Finally, light reaches the retina, a thin tissue lining the innermost wall of the eye. Theretina acts much like the film in a camera. The retina responds to light rays hitting it andconverts them to electrical signals carried by the optic nerve to the brain. The outlyingparts of the retina are responsible for peripheral vision while the center area, called the


macula, is used for fine central vision and color vision. The very center of the macula iscalled the fovea. It has a very high concentration of special cells (cones) which make itthe only part of the retina capable of 20/20 vision1.

Retinal Layers: Like film, the retina is composed of several layers with different roles. The firstlayer encountered by light is called the nerve fiber layer. Here, the nerve cells travel fromall the parts of the retina to the optic nerve. Under this layer, most of the retinal bloodvessels are located. They are responsible for supplying the inner parts of the retina. Theoutermost layer is the the photoreceptor layer. The photoreceptor layer, composed of conesfor fine and color vision, and rods for vision in dim light, consists of the cells that actuallyconvert light into nerve impulses. There are approximately 120 million rods and 6 millioncones in a human retina. Most of the cones are located in the macula. The photoreceptorcells lie on top of a layer of cells called the retinal pigment epithelium or RPE. The RPE isresponsible for keeping the photoreceptors healthy and functioning well. Under the RPE isthe retina’s second set of blood vessels which are in a layer called the choroid. The RPE,fed by the blood vessels of the choroid, supply the photoreceptors.

Optic Nerve: The optic nerve is the structure which takes the information from the retina aselectrical signals and delivers it to the brain where this information is interpreted as a visualimage. The optic nerve consists of a bundle of about one million nerve fibers. The positionin the back of the eye where the nerve enters the globe is corresponds to the „blind spot“since there are no rods or cones in these location. Normally, a person does not notice thisblind spot since rapid movements of the eye and processing in the brain compensate forthis absent information.

The globe is tightly suspended within the orbita, surrounded by six extraocular muscles whichare responsible for the movement of the globe. The four straight eye muscles (musculi recti) andthe upper diagonal (superior oblique) eye muscle originate in the posterior part of the orbita.Only the lower diagonal (inferior oblique) muscle originates from the orbital plate of the maxilla(see page 11). All eye muscles insert at the leather skin of the globe. Fig. 2.4 shows the muscleorigins in the posterior orbita and their insertions. Each eye muscle affects the eyeball in threecomponents, whereby the muscle path determines the main direction of pull. The main effect ofeach muscle can be derived from its designation [BKP+03].

Perpendicular arranged to each other, the musculi recti originate in the anulus of Zinn, a pointat the posterior end of the orbita. Their tendons unite to a circular plate (Zinns’ ring) and theirinsertions lie before the equatorial plane of the globe [Gue86]. In contrast, the musculi obliquiinsert behind the globe equator and pull diagonally forward. The m. obl. sup. is the longest ofall eye muscles. Starting at its insertion, it runs above the globe towards the nasal frontal bone,pulls through a cartilaginous hole (the trochlea) and runs from there directly to its origin closeat Zinns’ ring. Obl. inf. originates at the nasal edge of the bony orbita, runs below the globe,crosses the m. rect. inf. and inserts within the rear range of the eyeball. Within the crossingarea, m. obl. inf. and m. rect. inf. are connected by ligamentum Lockwood [Gue86]. Eacheye muscle consists, apart from the purely muscular portion, also of a tendon which connects themuscle at the origin on one side, and at the point of insertion on the other side. The overall

1’20/20 vision’ is a term used to describe normal distance vision. The ’20’ is a distance of 20 feet, which is astandard testing distance for eyesight, used by Optometrists and Doctors.


Figure 2.4: Anatomy of the Extraocular Eye Muscles [KJCS99]

length (muscle and tendon) of eye muscles is very different. The largest differences occur intendon lengths [Kau95]. The m. obl. inf. has the shortest tendon (0-2 mm) and the m.obl.sup.the longest (25-30 mm). The actual muscle length lies between 30 mm (mm obliqui) up to 39mm (m. rect. inf.). Due to the insertion lying before or behind the equator of the globe, eachmuscle partially contacts the eyeball’s surface. At the point of tangency, the muscle loses contactto the globe and pulls toward its origin. With each movement of the globe, the relative positionof a muscles insertion changes with respect to the orbita.

Figure 2.5: Extraocular Muscles of a Right Eye from above [uS98]

In referring to Fig. 2.4 and Fig. 2.5, the names and primary directions of action for each muscleis identified in Tab. 2.2.

Until 1994, eye muscles were assumed to be string like structures that run from the origin straightto the insertion. At the point where a eye muscle touches the globe (point of tangency), themuscles were expected to move freely. If the muscles could move freely between insertion and


Fig.Nr. Muscle name Primary action(s)(1) musculus rectus superior or superior rectus muscle

(upper straight eye muscle)upward

(2) musculus rectus lateralis or lateral rectus muscle(outside straight eye muscle)

sideways outward

(3) musculus rectus inferior or inferior rectus muscle(lower straight eye muscle)

downward

(4) musculus rectus medialis or medial rectus muscle(internal straight eye muscle)

sideways inward

(5) musculus levator palpebrae or levator muscle(upper eyelid muscle)

raises the upper eyelid

(6) musculus obliquus superior or oblique superior muscle(upper diagonal eye muscle)

downward and inside

(7) musculus obliquus inferior or oblique inferior muscle(lower diagonal eye muscle)

upward and outside

Table 2.2: Eye Muscle Names and Primary Actions

origin during an eye movement (shortest path hypothesis), a shift of the muscle path on the globesurface would occur, especially in extreme gaze positions. Thus, the muscle path and thus thedirection of pull would change considerably according to the actual eye position (loss of maindirection of pull).

2.1.1 Eye Muscle „Pulleys“

In order to prevent an eye muscle from slipping away while the globe rotates, connective tissuesurround the globe and stabilizes the muscles within the area of the point of tangency. Thesestabilizers are called Pulleys [BM00b][DMP+95]. Early radiographic studies in monkeys andcomputed tomography studies in human subjects suggested that the bellies of contracted rectiextraocular muscles have paths that are very stable relative to the orbit despite changes ingaze. Miller and Demer [DMP+95] used magnetic resonance imaging (MRI) and manual 3Dimage reconstruction to demonstrate the extreme stability of the recti extraocular muscle pathsthroughout the normal range of ocular rotations. The result was, that only the anterior parts ofthe tendons moved relative to the orbit, as they must, because they are attached to the globe.Histochemical and Immunohistochemical investigations within this study confirmed the findingsof the MRI analysis. Fresh orbital specimens were obtained at autopsy from adult cadavers,dissection and globe enucleation was performed. All specimens were fixed in 10% neutral bufferedformalin, dehydrated and embedded in paraffin. The resulting 7 to 10–µm sections were mountedon glass slides and chemically colored distinguishing collagen, elastin, cartilage, smooth muscle,striated muscle and tendon. Smooth muscle is generally involuntary and differs from striatedmuscle in the much higher actin/myosin ratio, the absence of conspicuous sarcomeres and theability to contract to a much smaller fraction of its resting length.


Figure 2.6: Schematic View of the Extraocular Tissue Architecture from [DMP+95]IR = inferior rectus; LPS = levator palpebrae superioris; LR = lateral rectus; MR = medial rectus; SO = superioroblique; SR = superior rectus.

In Fig. 2.6, a schematic representation of the orbital tissues with respect to eye muscle pulleys isshown. The horizontal section shows the lateral and medial rectus muscles (LR, MR) suspendedby fibroelastic sleeves consisting of collagen and elastin at the posterior part of Tenon’s capsule,approx. 10 mm posterior to the muscle insertion. The sleeves itself are coupled to the orbitby musculofibroelastic septae, extending to the periorbita and to adjacent muscle sleeves. Thecoronal sections of Fig. 2.6 are represented at the level of the superior rectus tendon (SR tndn),and the superior oblique tendon (SO tndn).


Figure 2.7: Example of Lateral Rectus Path Influenced by Pulley

In Fig. 2.7, a left eye elevated by 35 ◦ is shown, focusing the left lateral rectus muscle. The figureshows how the muscle path of the left lateral rectus is inflected by the pulley indicated in red.Since the lateral rectus muscle, in its primary function, pulls the eye outward, away from the nose(see Tab. 2.2), this main action must be retained, also when the globe rotates to other positions.Through the inflection of the muscle path at the location of the pulley, the main function of thelateral rectus muscle is preserved and the globe would be rotated laterally in case contraction ofthe muscle occurs. Comparing the posterior and anterior part of the muscle with respect to thepulley location, the anterior part moves with the rotation of the globe, while the posterior partof the muscle stays stable relative to the point of origin.

2.1.2 Global/Orbital Eye Muscle Layers

Recent anatomic studies of whole orbits confirmed the existence of pulleys (e.g. [Mil89]). Tech-nical improvements have made it possible to reconstruct orbital histology even more detailed andwithout the need to remove the structures from the orbital bones. This leaves the normal spatialrelationships as is, thus providing a better insight in the actions of the different muscles. Formerstudies on mammals (e.g. [MGGN75]) already suggested that the extraocular muscles consist oftwo different layers.

Referring to Fig. 2.8, the global layer is continuous from the origin at the annulus of Zinn to thetendinous scleral insertion on the globe, in contrast to the orbital layer, obviously terminatingposterior to the insertion. Recent studies have shown that the position of the rectus musclepulleys change slightly as a function of gaze [DOP00], thus modifying the mechanical propertiesof muscle action. Pulley positions were identified by the sharp inflections of the muscle path indifferent gaze positions. A study carried out by Demer et.al [KCD02] showed that for each rectusmuscle, a discrete inflection moved significantly posteriorly during contraction and anteriorlyduring relaxation of the muscle. These changes in path inflections have been interpreted asantereoposterior shifts in the rectus muscles pulleys. This so called Active-Pulley hypothesis


Figure 2.8: Different Fiber Layers of Rectus Muscles from [DOP00]

is quantitatively supported by recent studies (e.g. [KCD02][DOP00][CJD00]). The resultingconsequence is that human rectus muscle pulleys move to shift the ocular rotation axis in orderto attain commutative behavior of the oculomotor system.

2.1.3 Anatomical Measurements

Around 1869 by A.W. Volkmann [Vol69], a statistic analysis was carried out with several patients.The results of this study were defined as „standard eye“, an average eye of humans. Volkmann’sspecifications referred to the median lines of eye muscle tendinous insertions (i.e. the approxi-mated center line of a muscle volume), and accuracy was limited due to the seamless transition ofmuscle tendon into neighboring tissue. In 1965, Nakagawa used hand dissected slices of cadaversin order to measure each eye muscles individual cross-section [Nak65]. These clinical results en-dured as basis for many modelling approaches until studies from Mühlendyck and Miller revisedthe data [MR84][MKM84]. With the discovery of Pulleys, new geometrical measurement usingMRI and histologic approaches were defines [CJD00].


Globe Radius: 12.43 mmCornea Radius: 5.50 mm

MR LR SR IR SO IOOrigin

x (mm) -17.00 -13.00 -16.00 -16.00 -15.27 -11.10y (mm) -30.00 -34.00 -31.78 -31.70 8.24 11.34z (mm) 0.60 0.60 3.80 -2.40 12.25 -15.46

Insertionx (mm) -9.15 10.44 0.00 0.00 -0.07 6.73y (mm) 8.42 6.75 7.33 7.65 -8.05 -10.46z (mm) 0.00 0.00 10.05 -9.80 9.48 0.00

Length (mm) 39.96 49.65 44.05 44.80 22.28 34.03L0 (mm) 31.60 36.20 33.90 35.00 32.00 29.10

TendonLength (mm) 3.80 8.40 5.40 4.80 26.50 1.50TendonWidth (mm) 10.30 9.20 10.60 9.80 11.00 9.80

Table 2.3: Muscle Parameters measured by Volkmann from [Gue86]

The geometrical data for muscle length from Tab. 2.3 also includes the tendon of each muscle,whereas the muscle length (L0) represents only the contractile part of the muscle without itstendon. These values are defined on the basis of relaxed denervated muscles (muscles whichare not innervated). All coordinates in this table are defined with respect to an oculocentric,head-fixed coordinate system originating in the center of the globe.

Data from Tab. 2.3 data was refined by Miller et.al. due to the discovery of pulleys and newmeasurement results for further studies [Mil89][CJD00][KCD02].


Globe Radius: 11.99 mmCornea Radius: 5.50 mm

MR LR SR IR SO IOOrigin

x (mm) -17.00 -13.00 -15.00 -17.00 -18.00 -13.00y (mm) -30.00 -34.00 -31.76 -31.76 -31.50 10.00z (mm) 1.00 -1.00 3.60 -2.40 5.00 -15.46

Insertionx (mm) -9.65 10.08 2.76 1.76 2.90 8.00y (mm) 8.84 6.50 6.46 6.85 -8.00 -9.18z (mm) 0.00 0.00 10.25 -10.22 8.82 0.00

Pulleyx (mm) -14.00 12.00 -5.16 -5.16 -15.27 -13.00y (mm) -5.00 -8.00 -10.78 -8.78 11.00 10.00z (mm) 0.14 0.33 10.00 -12.00 11.75 -15.46

Length (mm) 39.96 49.65 44.05 44.80 22.28 34.03L0 (mm) 31.92 37.50 33.82 35.60 34.15 30.55

TendonLength (mm) 4.91 7.71 5.40 4.80 31.90 1.33TendonWidth (mm) 10.50 9.30 10.80 10.00 10.80 8.60

Table 2.4: Revised Muscle Parameters by Miller et.al. [MR84]

In the revised muscle data from from Tab. 2.4, muscle insertions as well as the pulley positionswere measured in three-dimensional space. This means that assuming the globe as geometricalabstraction using a spherical object, each muscle’s insertion does not lie exactly on the globe, dueto the fact that in reality, the globe is shaped ellipsoid. This leads to a compromise in choosingthe globe radius as the smallest distance between coordinate system origin and muscle insertionof all six muscles. The substantial mathematical difference of these data consists of the fact thatnow each muscle rotates its own „virtual globe“, in turn affecting lever arm and force behavior.

2.1.4 High Resolution MRI-Imaging of the Orbit

Imaging techniques have become an indispensable diagnostic tool in the field of ophthalmology.In most hospitals, computed tomography is still the method of choice for orbital imaging becauseof its low costs and excellent depiction of bony details. However, resolution of MRI scanners hasimproved during the past years providing sufficient accuracy to show detailed tiny anatomicalstructures. In contrast to CT, MRI allows for multiplanar imaging without the need for reposi-tioning the patient. Whereas CT uses ionizing radiation, which is known to be harmful to health,MRI utilizes the nuclear magnetic resonance (MR) effect.

Nuclei with a net magnetic moment, such as hydrogen ions (protons) which occur in livingmatter, line up in parallel in a strong magnetic field and change to a higher energy level whena radio frequency (RF) impulse is applied at right angles to the static magnetic field. The


strength of this static magnetic field in MR scanners ranges between 0.5 to 2 Tesla, whereby oneTesla corresponds to a unit of magnetic flux. The tesla (symbolized T) is the standard unit ofmagnetic flux density. Reduced to base units in the International System of Units, 1 T representsone kilogram per second squared per ampere (kg/s2A). In practice, the tesla is a large unit, andis used primarily in industrial electromagnetics. Once the RF impulse is turned off, the nucleirelax to the original energy level and release a RF signal that can be detected by a coil (RFreceiver). Proton density of the examined tissue plays an important role when applying MRIinvestigations to in vivo subjects. The nuclear relaxation process depends on the mobility of theprotons in the substance of interest.

These properties are distinguished by T1 (spin-lattice relaxation time) and T2 (spin-spin relax-ation time) values. T1 is influenced by molecular rotation time and are lowest in the tissues withintermediate-sized molecules such as fat. T1 values are dependent upon magnetic field strength(increasing field strength gives increasing T1). T1 values at 1.5 T are around 800− 1000 ms forbrain tissue while it is considerably higher (3000 ms) for water and lower (300 ms) for fat. Afterthe RF signal is applied, energy of the spin system in the examined tissue will be redistributedand the measurable signal will decay away as the spins. Long T2 relaxation times only occurin molecules which are tumbling rapidly in solution. More immobile species and solids haverapid spin-spin relaxation and often produce MR signals which decay away before they can evenbe detected in MR imaging. There are several distinct mechanisms occurring at the molecularlevel which can contribute to relaxation, but dipole-dipole interactions are the major cause ofT2 relaxation. Additionally, extrinsic factors that are set on the MR scanner influence imageacquisition. Time of repetition (TR) and time of echo (TE) are the most important values. TRis the time between the emission of RF pulses, and TE is the time between the excitation of anRF pulse and the reception of the measured signal. Thus, the relative contribution of any of thetissue specific factors can be varied by choosing different pulse sequences in order to achieve aweighting of a specific parameter. Hence, in MR imaging, tissues with higher T1 values appeardark on T1 weighted images and tissues with higher T2 values appear bright on T2 weightedimages. Cortical structures like bones and fast flowing blood inside arteries and larger veins giveno signal on MRI scans. Protons of fast flowing blood that had been excited by an RF pulse,have left the imaging slice before their signal can be detected.

Following a study of orbital MR imaging [Ett99], MR signal intensities, with respect to ocularand orbital tissues, can be identified from Tab. 2.5.

Tissue type T1w T2wcornea, sclera L Laqueous, vitreous L Hnormal clear lens M Lextraocular muscles M Morbital fat H Mconnective tissue septa M Mnerves M M

Table 2.5: MR Signal Intensities of Ocular and Orbital Tissues on T1-weighted (T1w) and T2-weighted (T2w)images from [Ett99]. H=high signal, M=medium signal, L=low signal.


Figure 2.9: Axial MR Scan, showing Lateral and Medial Rectus of both Eyes, from [FPBK03].T1 weighted scan, 1.5 mm slicing, TR: 602 TE: 16

Figure 2.10: Axial MR Scan, showing Lateral and Medial Rectus of both Eyes, from [FPBK03].T1 weighted scan, 1.5 mm slicing, TR: 602 TE: 16

Within the SEE-KID research work, a MRI study has been carried out in order to visualize theextraocular eye muscles by using newest MRI scanner equipment and special head coil receptors[FPBK03]. In this study, three human subjects were scanned using a 1.5 Tesla Siemens MagnetomSymphony scanner. Coronal and axial scans with T1 and T2 weighting were taken, using a 8channel head phase array as receptor coil. Image series with approx. 19 images and a slicethickness of 1.5 mm, 512× 416 matrix were taken while the subject was fixating reference points


(see Fig. 2.9 and Fig. 2.10). Both eyes were put in local anesthesia using proprietary eye drops.Additionally, one subject was injected Gadolinium Magnivist 0.2ml/kg radiopaque material. Allsubjects were captured focusing different stable gaze positions that were indicated by dots in thescanner tube.

(a) Subject fixating straightahead

(b) Subject fixating to the right (c) Subject fixating to the rightbottom

Figure 2.11: Comparison of MR Scans in different Gaze Positions [FPBK03]

In Fig. 2.11 different gaze positions are compared with respect to the medial rectus muscles ofboth eyes. In Fig. 2.11(a), the subject was looking straight ahead, showing the muscles in a ratherlow innervated state. Compared to Fig. 2.11(b) and Fig. 2.11(c), where the subject is looking tothe right and to the right bottom respectively, deformation of the physiologic cross-sectional area(PCSA) can be noticed. In repeating this described process for every slice in each image series,eye muscle morphology was reconstructed and graphically visualized.

The goal of this study was to show morphology of eye muscles, especially the rectus muscles whilefixating different reference points. These data was analyzed and reconstructed as 3D dynamicrepresentation of extraocular eye muscles in different gaze positions for later use in the SEE-KIDresearch project [BK03b].

High resolution MRI enables exact delineation of space occupying orbital processes in relation tosurrounding anatomical structures, thus facilitating planning of surgical procedures, which willbe essential for computer-assisted surgery and treatment planning [Ett99].

2.1.5 Human Dissection of the Orbit

Since the discovery of pulleys, existing surgical procedures need to be revised and checked for me-chanical consequences that can arise when surgical modification of pulleys and surrounding tissueoccurs. Treatment of strabismus probably affects the action of orbital extraocular muscle layerson their pulleys, can cause unintended effects, especially when altering relationships betweenmuscle insertion and pulley positions and should be better understood and perhaps consideredin surgical planning [DOP00].

Human dissection was performed in order to better understand the functional implications of thenewly discovered anatomical structures within eye muscles (Pulleys, global and orbital layers,etc.) [FAP03]. Compared to striated skeletal muscles, the six extraocular eye muscles showspecific differences concerning structure, distribution of muscle fibers and neural sensitivity to


innervation. Thus, human eye muscles take up a exceptional position among human anatomywith regard to human skeletal muscles.

Figure 2.12: Human Dissection of the Orbit showing Medial Rectus Pulley [FAP03]

All six extraocular muscles consist of 2 distinct muscle portions, a consistently very thin orbitalmuscle layer with a high mitochondrial density in the muscle fibers, encircled by a thick C-shapedglobal layer. This global muscle layer consists of muscle fibers with a variable mitochondrialdensity, additionally, average physiologic cross-sectional areas are larger compared to the orbitalmuscle layer (see Sec. 2.1.2). Concerning muscle fiber types, human extraocular muscles consistof single innervated twitch fibres and multiple innervated non-twitch fibres which are usually onlyfound in skeleton musculature. In Fig. 2.12, the medial rectus muscle of a right eye was dissected,showing the muscle stretched to indicate the connective tissue structure, encircling parts of themuscle in order to inflect the muscle path and therefore directing muscle force, compared tothe anatomic origin, towards a much more anteriorly located point. The Pulley structure can belocated posterior to the globe equator and is comprised of a dense collagen matrix with alternatingbands of collagen fibers precisely arranged at right angles to one another. This three-dimensionalorganization most likely gives high tensile strength to the pulley. Connective tissue and smoothmuscle bundles suspend the pulley from the periorbita. Smooth muscle is distributed in small,discrete bundles attached deeply into the dense pulley tissue. Fine structural observations confirmthe existence and substantial structure of a pulley system in association with the medial rectusextraocular muscle. The presence of pulleys must be considered in models of the oculomotorplant. However, the nature of the connective tissue-smooth muscle suspending the pulley systemto the orbit supports the suggestion that the pulley position, and thus the directional force ofthe eye muscles, may be adjustable [PPBD96].


2.2 Eye Movement Physiology

The movement of the globe approximately corresponds to a rotation of an object in the three-dimensional space around a certain axis. The globe center can be regarded as rotation center.The line of sight is a vector from the globe center through the center of the pupil. Perpendicularto this vector, the vertical and the horizontal axes are defined, whereby the intersection of thesethree axis lies in the globe center Fig. 2.13. Eye positions can be classified by their rotationalproperties [Kau95]:

Figure 2.13: Line of Sight, Vertical and Horizontal Axes; Rotations to Other Eye Positions

Primary position: The eye looks straight forward, with the head fixed and upright (Fig. 2.13(a)). It is accepted that in this position all muscles exhibit minimum force. From primaryposition, all other eye positions can be reached with as small an energy expenditure aspossible.

Secondary position: From the primary position, a rotation around the horizontal or verticalaxis (Fig. 2.13(b)) is performed. The eye looks to the left or right or upward or down.

Tertiary position: From the primary position a rotation around the horizontal and verticalaxis (Fig. 2.13(c)) is performed. The eye looks e.g. to the left and down. The combinationaround two axis can be represented also by a rotation axis, which lies in the plane spannedby the horizontal and vertical coordinate axes (Fig. 2.13(d)).

Both eyes can be moved only in binocular community with one another, i.e. the movement ofonly one eye is usually not possible [Gue86]. Eye muscles are able to reposition the eye accuratelyand very fast, moreover they can hold a certain position without exhaustion. The rotation of aneye around a certain axis results in a certain eye position and thus also realigns the line of sightto a new gaze direction. The gaze direction designates the orientation of the eye, whereby an eyeposition always includes the fixation of a target object. Each muscle is defined through its origin,point of tangency and insertion. The muscle paths from the origin to the insertion at the globeare however additional held by connective tissues, retaining movements, so called „Pulleys“. Forspatial perception, both eyes must exactly fixate an object of interest. The image of this objectis then projected onto the foveae of both eyes, which provides best visual acuity. A cube fixatedwith both eyes is projected upside down (like in a camera) and skewed (due to different distancesof both eyes) onto the retina. In the brain, both images will be merged in order to perceive oneupright three dimensional cube (fusion) (Fig. 2.14(a)).

As an example, a patient suffering from inward squinting (esotropia) of the right eye. Accordingto this, the right eye fails to fixate the cube and misaligns compared to the left eye (Fig. 2.14(b)).Different images are now projected onto the foveae of both eyes, and the brain fails to merge this


(a) (b)

Figure 2.14: Binocular Fixation of an Object in Space [Tho01]

information. The result is, that disturbing (uncrossed) double images are perceived. Especiallyin infants, the brain will compensate on these errors by trying to eliminate those images thatoriginate from the squinting eye. This process leads to a fatal outcome: the squinting eye willlose visual acuity and will suffer from functional deficiency.

2.2.1 Actions of the Extraocular Muscles

Basically, the eye can rotate in three dimensions around fixed axes of a head-fixed coordinatesystem. Each extraocular muscle rotates the globe in specific directions, also dependent of thecurrent position of the eye. Movement of the eye nasally is adduction; temporal movement isabduction. Elevation and depression of the eye are termed sursumduction and deorsumduction,respectively. Torsional eye movements rotate the eye around its visual axis, whereby incycloduc-tion or incyclotorsion (intorsion) is nasal rotation of the vertical meridian and excycloduction orexcyclotorsion (extorsion) is temporal rotation of the vertical meridian.

Referring to Fig. 2.15, the coordinate system used for torsional rotations is different in both eyes.Referring to the figure above, directions of rotation of one eye around the respective axes can bedescribed as follows:

Rotation around Z-axispositive angle = adduction (towards the nose)negative angle = abduction (away from the nose)

Rotation around X-axispositive angle = elevation or sursumduction (upward)negative angle = depression or deorsumduction (downward)


Rotation around Y-axispositive angle = intorsion (inward rolling)negative angle = extorsion (outward rolling)

Figure 2.15: Rotational Directions for both Eyes [BKP+03]

As shown in Fig. 2.15, coordinate axes for ab-/adduction and in-/extorsion point in differentdirections for both eyes. This provides a uniform definition of the rotation direction by usingconsistent designations (e.g. abduction = away from the nose). In both eyes, positive torsion isdefined to rotate the eye inward around the line of sight. If the two axes would not show differentsigns, then intorsion would roll one eye inward and one outward. The axes for elevation anddepression remain the same in both eyes, since up and down rotations specify the same directionsfor both eyes. Additionally, in case of binocular eye movement, abduction of one eye results inadduction of the other eye, thus eye positions are mirrored for ab-/adduction and in-/extorsionmovements.

The primary muscle that moves an eye in a given direction is known as the agonist. A muscle inthe same eye that moves the eye in the same direction as the agonist is known as the synergist,while a muscle in the same eye that moves the eye in the opposite direction of the agonist isthe antagonist. For example, in abduction of the right eye, the right lateral rectus muscle isthe agonist, the right superior and inferior oblique muscles are partly synergists, and the rightmedial rectus is the antagonists. A muscle that moves the opposite eye in the same direction asthe agonist is the contralateral synergist, whereas a muscle in the opposite eye, that moves theeye in the opposite direction as the agonist is called contralateral antagonist.

Yoke muscles are the primary muscles in each eye that accomplish a binocular eye movement(e.g. for right gaze, the right lateral rectus and left medial rectus muscles). Each extraocularmuscle has a yoke muscle in the opposite eye to accomplish movement of both eyes in the samedirection. The primary action of an extraocular muscle is the direction of rotation of the eyewhen that muscle contracts. This term also indicates the gaze position in which the effects of amuscle most easily are demonstrated. Knowledge of primary actions is important as squintingoften increases in the field of action of a weak eye muscle.


Medial and lateral rectus muscles have only horizontal actions. The medial rectus muscle is theprimary adductor of the eye, and the lateral rectus muscle is the primary abductor of the eye.

Superior and inferior rectus muscles are the primary vertical movers of the eye. The superiorrectus acts as the primary elevator and the inferior rectus acts as the primary depressor of theeye. This vertical action is greatest with the eye in the abducted position.

The direction of pull of the vertical recti muscles forms a 23 ◦ angle relative to the visual axis inthe primary position, giving rise to secondary and tertiary functions. The secondary action ofvertical rectus muscles is torsion. The superior rectus is an incyclotorter, and the inferior rectusis an excyclotorter. The tertiary action of both muscles is adduction.

Superior and inferior oblique muscles are the primary muscles of torsion. The superior obliquecreates incyclotorsion, and the inferior oblique creates excyclotorsion. As the direction of pull forboth muscles forms a 51 ◦ angle (relative to the visual axis in the primary position), secondaryand tertiary actions occur.

The secondary action of the oblique muscles is vertical, and it is best demonstrated when the eyeis adducted with the superior oblique acting as a depressor and the inferior oblique acting as anelevator of the eye. The tertiary action for each muscle is abduction.

Within the orbita, the eye is surrounded by flexible fat pads, which, besides rotation, permit globetranslation up to a certain limit. Usually, this translation is negligibly small, however in somepathological situations involving co-contraction of muscles (e.g. Duane syndrome), additionalinformation about globe translation provides an important clue for the estimation and correctionof eye motility disorders. A forward movement of the globe along the Y-axis is denoted asProtrusion, a backward motion is defined as retraction.

2.2.2 Kinematic Principles of Eye Movements

The main task of the eye movement system is to move the eye quickly from the current gazeposition to a new location, in order to fixate an object of interest. Binocular eye movementsare either conjugate (versions) or disconjugate (vergences). Versions are movements of both eyesin the same direction (e.g. right gaze in which both eyes move to the right). Dextroversion ismovement of both eyes to the right, and levoversion is movement of both eyes to the left. Sur-sumversion and deorsumversion are elevation and depression of both eyes, respectively. Vergencemovements are movements where both eyes move in opposite horizontal directions to permit theacquisition of a near or far object. When a subject is Looking at a near object, both eyes movetogether slightly to maintain binocular vision of it, as the object recedes away from the subject,then the two eyes diverge again.

Vision is blurred during an eye movement, so the length of time that the eye is moving must beminimized. In order to minimize the time during which no clear image is captured on the fovea,eye movements that move the fovea from one object/point to another are very rapid. Thesesaccadic eye movements are among the fastest movements the body can make. The eyes canrotate at over 500 deg/sec, and subjects make well over one hundred thousand of these saccadesdaily.

Stabilization movements assure that the image of an object or region in the center of the field-of-


view is kept over the fovea. Sophisticated mechanisms exist to accomplish this goal in the face ofeye, head, body, and object motion. These eye movements are often grouped into four categories:

The vestibular-ocular reflex (VOR) rotates the eyes to compensate for head rotation andtranslation. Rotational and linear acceleration are detected by the semicircular canals andotolith organs in the inner ear. The resultant signals are used to command compensatingeye movements.

The Optokinetic reflex stabilizes the retinal image caused by large-field motion. Retinal slipinduced by field motion is used to initiate eye movements at the appropriate rate to cancelout image motion.

Smooth-pursuit eye movements are similar to the optokinetic reflex, but allow arbitrarilysized targets to be stabilized instead of large-field motion. A moving target is required forsmooth eye movements so that the eyes can move smoothly across a stationary object.

Vergence eye movements counter-rotate the eyes to maintain the images of an object at agiven depth to be maintained at corresponding locations on the two retinae.

In the generation of eye movements, the brain follows certain consistent patterns of behavior(laws). However, it is important to realize, that the brain only adheres to these laws only whenit suits a given purpose [FHMT97].

2.2.2.1 Donders’ Law

Donders’ law states, that the 3D angular position of the eye is always the same for any particulargaze position [Don48]. If the eye adopts a particular torsional orientation when looking forwardat one instance (e.g. after a rotation from the right), it will adopt the same orientation at allother instances (e.g. after a rotation from the left) [FHMT97]. Hence, torsional eye position isnot arbitrary, but uniquely determined by the gaze position.

2.2.2.2 Listing’s Law

Listing’s Law can be considered as an extension to Donders’ law, in that it specifies a specifictorsional value for any particular eye position. It states that, with the head stationary, uprightand the eyes fixating a distant object, all rotation axes of the eye lie in the same plane. Listing’slaw is considered one of the most important principles in eye movement physiology, by which thebrain couples redundant degrees of freedom, in this case by relating the two dimensional spaceof target directions relative to the fovea to a 2D manifold of 3D eye positions [FHMT97].

When the eye leaves one target object and fixates upon another, it rotates about an axis per-pendicular to a plane cutting both, the former and the present lines of sight. Applying thisprinciple in a broader sense, one could say that Listing’s law is probably the consequence of thefacts that the primary position is the average of all eye positions during the day, that most eyemovements are directed radially from or to the primary position and that rotations about a singleaxis are easier to perform than rotation about two axes or about an axis that changes during the


movement. In considering both, horizontal and vertical eye movements, in which case the eye istreated as if it were a pointer (defined by the line of sight). These two dimensional descriptions ofeye rotations specify the orientation of this pointer in space. Since the eye can also make torsionalmovements around the line of sight, a full description of the rotational behavior also requires mea-surement of torsion. Accounting for all three degrees of freedom of eye rotations leads to someunique neurological consequences regarding oculomotor organization. Although this subject canbe explained simply, it still causes a lot of confusion. Some of the difficulty arise from the needto define the kinematics of eye rotations quantitatively, setting up coordinate systems based onrigorously determined sets of spherical angles. Although this is useful for empirical verification,it is a poor way to introduce this subject. Rather than describing eye rotations with respect toa horizontal and vertical angular coordinate system, this explanation will begin intuitively.

(a) (b) (c)

Figure 2.16: Simple Abstraction of Listing’s Law [Nak83]

A simple physical „ball and membrane“ model of the eye is used [Nak83]. The model consistsof a spherical globe attached to a very tightly stretched elastic membrane, carefully secured tothe end of a cylinder, so that the membrane is equally taut in all directions Fig. 2.16. Becauseof the elastic quality of this membrane, there is a natural resting place for the globe. Thestalk that is attached to this globe represents the line of sight. Additionally, a cross is mountedon top of this stalk to reveal the amount of twist (torsion) of the globe Fig. 2.16(a). Theposition of rest is defined as the primary position, and the direction of the line of sight (thedirection the stalk points to) corresponds to the primary gaze direction. The plane spanned bythe membrane of this model in resting position shall correspond to Listing’s plane for the eye.It is the fronto-parallel plane passing through the center of the globe. In Fig. 2.16(b), the globeis rotated with full degrees of rotational freedom. It can be rotated horizontally and verticallyby displacing the stalk and torsionally by twisting the stalk between the fingers. The mostimportant case is shown in Fig. 2.16(c) where the globe is moved by pushing the stalk with asmooth rod. Using this technique, the globe can be moved to any desired gaze position and themodel has sufficient rotational freedom. Compared to Fig. 2.16(b), only two degrees of freedomare available to position the globe, since the twist of the globe around its visual axis (the stalk) isno longer under external control because it cannot be twisted between the fingers. Its orientationis dictated by the elastic membrane, which ensures that it corresponds to the position having the


lowest potential energy. Thus, each gaze position is associated with exactly one orientation ofthe globe.

What is essential to remember is that the behavior of this particular ball and membrane modelis isomorphic with Donders and Listing’s laws. This phenomenon is also described as „pseudoto-stion“, since the eye does not actively rotate around its visual axis, instead, this torsional rotationis intrinsic to the behavior of the mechanical system of the human eye. The orientation of theeye can be predicted by assuming that the eye has made a geodesic (shortest path) rotation fromthe primary position to any other fixation position. The axis of this shortest path rotation isperpendicular to the intended gaze position and thus lies in Listing’s plane. For the case of therubber membrane model, it means that the axis of the shortest path rotation lies in the planespanned by the membrane. The behavior of the model shows a radial or axial symmetry, suchthat there is no net torsion of the eye with respect to an axial reference position (the primaryposition).

Eye movements from tertiary positions to other tertiary positions do occur, but are less frequentand do not occur predominantly in down-, up-, right- or left-gaze. During eye movement from onetertiary to another tertiary position, Listing’s law is fulfilled only if the rotation takes place aboutan axis that is tilted to Listing’s plane by half the angle between the momentary tertiary positionand primary position [Hel63]. Hence, an axis that can change during the movement. Using endpoints of rotation vectors to plot 3D eye positions, all positions will lie closely scattered alonga plane (Listing’s Plane). Recordings of eye movements in humans and monkeys show that thestandard deviation of rotation vectors describing eye positions from this plane is only about 0.5-1.0 deg. The best plane to fit these data is called „displacement plane“ [Has95]. The orientationof this displacement plane also depends on the reference position used to describe eye positions.In shifting the reference position, the plane describing exactly the same eye positions will onlyshift half the angle in the same direction (see Fig. 2.17). In Fig. 2.17(a), the displacement plane isperpendicular to the reference position, whereas in Fig. 2.17(b), the reference position has shifted2α deg. forward. Since the displacement plane tilts only α degree in Fig. 2.17(b), it is now tiltedα deg. backward, with respect to a head fixed coordinate system.

(a) Reference position perpendic-ular to displacement plane

(b) Reference position shifted by2α deg.

(c) Reference position shifted by4α deg.

Figure 2.17: Definition of Listing’s Law w.r.t. Primary Position from [Has95]

For every data-set of arbitrary eye positions, there is one reference position such that the corre-sponding displacement plane is exactly perpendicular to the reference position. Only in this case,this position of the eye is termed primary position and the corresponding displacement plane istermed Listing’s Plane [Has95]. In Literature it is often preferred to use the term Listing’s planein a broader sense, referring to any plane of rotation vectors. Applying this approach, all planes


in Fig. 2.17 would be termed Listing’s plane.

2.2.3 Sensorimotor Control

The visual system can be seen as a combination of a high-acuity detector, the eye, with a high-precision movement system, the oculomotor system [Car00]. Information processing during visualperception is strongly influenced by constraints arising from the human oculomotor system. Theoculomotor system is characterized by the interaction between peripheral reflexes and centralmotor commands of visual origin.

Generally, eye movements fall into three broad classes:

Gaze-stabilization movements - shift the lines-of-sight of the two eyes to precisely compen-sate for self-motion, stabilizing the visual world on the retina. Gaze-stabilization movementsare accomplished by two partially independent brain systems. The vestibulo-ocular systememploys the inertial velocity sensors attached to the skull (the semi-circular canals) to de-termine how quickly and in what direction the head is moving and then rotates the eyes anequal and opposite amount to keep the visual world stable on the retina. The optokineticsystem extracts information from the visual signals of the retina to determine how quicklyand in what direction to rotate the eyes to stabilize the visual world.

Gaze-aligning movements - point a portion of the retina specialized for high resolution (thefovea) at objects of interest in the visual world. Gaze-aligning movements are also dividedinto two broad classes, saccades and smooth pursuit movements. Saccadic eye movementsrapidly shift the lines-of-sight of the two eyes, with regard to the head, from one place inthe visual world to another at rotational velocities up to 500−1000 deg /s. Saccades are themost common eye movements and occur rapidly (e.g. 3 saccades per second). In contrast,smooth pursuit eye movements rotate the eyes at a velocity and in a direction identical tothose of a moving visual target, stabilizing that moving image on the retina.

Gaze-shifting movements - or vergence movements, operate to shift the lines-of-sight of thetwo eyes with regard to each other so that both eyes can remain fixated on a visual stimuliat different distances from the head (see Fig. 2.14).

The neural control of gaze in natural conditions requires the interaction between different strate-gies of oculomotor control. In particular, some eye movements are controlled by visual feedback(e.g. smooth pursuit) whereas others are controlled in open loop with respect to vision (sac-cades). Indeed, visual tracking of moving targets requires the combination of smooth pursuiteye movements with catch-up saccades. Only in a small part of the visual field, i.e. within thecentral 2 deg of the field of vision (the fovea), visual acuity permits the analysis of fine struc-tured objects. In tracking saccades while a subject is looking at a picture (Fig. 2.18(a)), it canbe shown, how the oculomotor control system directs the sense of perception through a visualstimulus (Fig. 2.18(b)). The recorded saccadic eye movements from Fig. 2.18 clearly show howthe picture is sketched by discrete point to point eye movements.

Consequently, the brain needs to sample information of complicated scenes by many fixations,e.g. intervals, during which the eyes perform only microscopic movements on different parts of


an image (see Fig. 2.18). These fixations are separated by rapid eye movements called saccades.During reading, mean fixation durations on words ranges roughly from 200 ms to 300 ms, de-pending on lexical difficulties of the words. These fixations are interrupted by saccades with amean duration between 20 ms and 40 ms. The brain continuously transforms sensory informa-tion into meaningful perceptions. The brain also accomplishes visual transformation task, suchas encoding sequences of line points into curves, curves into shapes, and shapes into recognizableimages. Though these processes are mostly visual, they also make use of more general brainfunctions, such as information chunking, learning and memory, that are employed by other brainregions.

(a) (b)

Figure 2.18: Recording of Saccadic Eye Movements from [KSJ00]

2.2.3.1 Innervation of the Eye Muscles

All eye movements are rotations, accomplished by three antagonistic pairs of muscles (seeSec. 2.2.1). These six muscles are controlled by three brainstem nuclei (Fig. 2.19(b)). Thesenuclei contain the cell bodies for all of the motor neurons (Fig. 2.19) which innervate the oculo-motor muscles and thus serve as a final common path through which all eye movement controlmust be accomplished (Fig. 2.21). Each eye muscle is innervated by approximately 1000 mo-toneurons. The single motoneurons branch out in the eye muscles and innervate approximately4 to 40 muscle fibers respectively. A motor unit is the aggregation of those muscle fibers, whichare connected to one and the same motoneuron (see Fig. 2.19(b)).

The brain uses two different possibilities to increase the tractive force of a muscle:

1. It recruits motor units, which were inactive before and

2. it concentrates the activity of those motor units, which have been active before, but whichhave not been working to full capacity.


(a) Innervation of a Muscle by a Motoneuron [uS98] (b) The Ocular Motor Nuclei [KSJ00]

Figure 2.19: Ocular Motor Neurons and Motor Nuclei

2.2.3.2 Oculomotor Neurons

The eye muscles are innervated by three cerebral nerves. The nervus oculomotoris (III. cerebralnerve) innervates the medial rectus muscle, the superior rectus muscle, the inferior rectus muscleand the inferior oblique muscle (moreover the levator palpebrae muscle as well as - with itsparasympathetic part - the ciliaris muscle and the sphincter pupillae muscle) (see Fig. 2.20).The nervus trochlearis (IV. cerebral nerve) innervates the superior oblique muscle, the nervusabducens (VI. cerebral nerve) innervates the lateral rectus muscle.

The cell bodies of the motoneurons lie together in groups and form the so-called nuclei in thebrainstem (Fig. 2.19(b)). The center of both oculomotor nerves lies in the midbrain (Fig. 2.19(b)).The alignment of those parts of the nuclei, which are associated with the specific eye muscles,is very complicated and most of the details have been discovered in the last few years. The cellbodies of the medial rectus muscle, the inferior rectus muscle and the inferior oblique muscle lieipsilateral (e.g. for the right eye on the right nucleus-side). Only the cell bodies of the superiorrectus muscle lie contralateral (e.g. for the right eye on the left nucleus-side). The fibers of thenerve of the superior rectus muscle cross in the area of the nucleus oculomotoris to the otherside. The cell bodies of the levator palpebrae muscle lie close to the middle line, both ipsilateraland contralateral. The two nuclei of the nervus trochlearis also lie in the midbrain, just belowthe nucleus oculomotoris. The motoneurons of the nervus trochlearis originate contralateral andcross behind the aqueduct, below the quadrigeminal bodies, to the other side. The nuclei of thenervus abducens lie in the bridge and are ipsilateral connected with the respective lateral rectusmuscle (see Fig. 2.20).


(a) (b)

Figure 2.20: Oculomotor Nerve Circuit [KSJ00]

2.2.3.3 Neural Signal Encoding

Engineering models of the eye and its muscles indicate that motor neurons must generate twoclasses of muscle forces to accomplish any eye rotation. A pulsatile burst of force that regulatesthe velocity of an eye movement and a long-lasting increment or decrement in maintained forcethat, after the movement is complete, holds the eye stationary by resisting the elasticity of themuscles which would slowly draw the eye back to a straight-ahead position [Rob64]. Physiologicalexperiments have demonstrated that all motor neurons participate in the generation of both ofthese two types of forces.

These two forces, in turn, appear to be generated by separable neural circuits. In the 1960s itwas suggested that changes to the long-lasting force required after each eye rotation could becomputed from the pulse-, or velocity-signal by the mathematical operation of integration. Inthe 1980s the lesion of a discrete brain area, the nucleus prepositus hypoglossi (Fig. 2.22), wasshown to eliminate from the motor neurons the long-lasting force change required for leftwardand rightward movements without affecting eye velocity during these movements. This, in turn,suggested that most or all eye movements are specified as velocity commands and that brainstemcircuits involving the nucleus prepositus hypoglossi compute, by integration, the long-lastingforce required by a particular velocity command. More recently, a similar circuit has been iden-tified that appears to generate the holding force required for upwards, downwards, and torsional


Figure 2.21: Location and Distribution of Oculomotor Nerves [Tho01]

movements.

2.2.3.4 The Superior Colliculus

The superior colliculus is a major visumotor region, integrating visual and motor informationinto oculomotor signals for the brainstem (Fig. 2.20(b)). It is a multilayered structure in themidbrain and can be divided into two functional regions:

1. the superficial layers

2. the intermediate and deep layers.

The three superficial layers of the superior colliculus receive both direct input from the retinaand a projection from striate cortex for the entire contralateral visual hemifield. Neurons in thesuperficial layers respond to visual stimuli.

In the two intermediate and deep layers cell activity is primarily related to oculomotor actions.The movement-related cells receive visual information from prestriate, middle temporal, andparietal cortices and motor information from the frontal eye field. These layers also containrepresentations of the body surface and of the locations of sound in space.

Individual movement-related neurons in the superior colliculus discharge before saccades of spe-cific amplitudes and directions, just as individual vision-related neurons in the superior colliculusrespond to stimuli at specific distances and direction from the fovea. The movement-related neu-rons form a map of potential eye movements that is in register with visual and auditory receptivemaps, so that the neurons that control eye movements to a certain target are found in the same


region as the cells excited by the sounds and image of that target. The region of the visualfield that contains the targets for the saccades controlled by a given movement-related neuronin the superior colliculus is called the movement field of that neuron. Electrical stimulation ofthe intermediate layers of the superior colliculus evoke saccades into the movement fields of theneurons at the site of the stimulating electrode. The actual eye movement is encoded by theentire ensemble of these broadly tuned cells.

Activity in the superficial and intermediate layers of the superior colliculus can occur indepen-dently. Thus, sensory activity in the superficial layers need not lead to motor output, and motoroutput can occur without sensory activity in the superficial layers. In fact, the neurons in thesuperficial layers do not have a large, direct projection to the intermediate layers. Lesions of asmall part of the colliculus affect the latency, accuracy, and velocity of saccades, whereas lesionsof the entire colliculus render e.g. a monkey unable to make any contralateral saccades, althoughwith time, the ability to make contralateral saccades is recovered.

The saccade-related activity of the superior colliculus neurons is shaped by inputs from the poste-rior parietal cortex, the frontal eye fields, and the substantia nigra pars reticulata (Fig. 2.19(b)).The posterior parietal cortex is involved in the visual guidance of saccades by shaping the visualinputs to the superior colliculus. The posterior parietal cortex contains neurons that are modu-lated by visual attention, i.e., by how behaviorally relevant a visual stimulus is. They respondmore effectively when the stimulus is the target for an eye movement. The frontal eye fields forman executive center that can selectively activate superior colliculus neurons, playing a role in theselection and production of voluntary saccades. The activity of frontal eye fields neurons reflectsthe selection of the visual target for a saccadic eye movement when several potential goals formovements are available. The frontal eye fields is also involved in suppressing reflexive saccadesand generating voluntary, non-visual saccades. The complementary executive control exerted onsaccade generation by the frontal eye fields and the superior colliculus is revealed by the effect ofselective and combined ablation.

Lesions of the superior colliculus prevent the generation of reflexive saccades, whereas the genera-tion of voluntary saccades is disrupted by frontal eye fields lesions. Although saccades can still beproduced after the ablation of either the superior colliculus or the frontal eye fields, a combinedablation of these two structures results in the complete abolition of saccadic eye movements.

2.2.3.5 Brainstem Control of Saccades

The saccadic system, in order to achieve a precise gaze-shift, must supply the brainstem circuitswith a command that controls the amplitude and direction of a movement. Considerable researchnow focusses on how this signal is generated. Current evidence indicates that this command canoriginate in either of two brain structures. The superior colliculus of the midbrain or the frontaleye fields of the neocortex (Fig. 2.19(b)), both containing laminar sheets of neurons that codeall possible saccadic amplitudes and directions in a topographic map-like organization [HM03].Activation of neurons at a particular location in these maps is associated with a particularsaccade and activation of neurons, adjacent to that location associated with saccades havingadjacent coordinates. Lesion experiments indicate that either of these structures can be removedwithout permanently preventing the generation of saccades.

How these signals that topographically encode the amplitude and direction of a saccade are


Figure 2.22: Motor Circuit for Horizontal Saccades [KSJ00]. Excitatory neurons are orange andinhibitory neurons are gray. The dotted line represents the midline of the brainstem.

translated into a form appropriate for the control of the oculomotor brainstem is not known. Onegroup of theories proposes that these signals govern a brainstem feedback loop which acceleratesthe eye to a high velocity and keeps the eye in motion until the desired eye movement is complete[Rob75a]. Other theories place this feedback loop outside the brainstem, or generate saccadiccommands without the explicit use of a feedback loop. In any case, it seems clear that thesuperior colliculus and frontal eye fields are important sources for these signals because if both ofthese structures are removed, no further saccades are possible [STC80]. The superior colliculusand frontal eye fields, in turn, receive input from many areas within the visual processing streamsincluding the visual cortex, as well as the basal ganglia and brain structures involved in auditionand somatosensation. These areas are presumed to participate in the processes that must precedethe decision to make a saccade, processes like „attention“.

As an example, in Fig. 2.22, the motor circuit for horizontal saccades in the brainstem is shown.


The eye velocity component of the motor signal arises from excitatory burst neurons in theparamedian pontine reticular formation (PPRF) that synapse on motor neurons and interneuronsin the abducens nucleus. The abducens motor neurons, project to the lateral rectus muscles whilethe interneurons of the abducens nucleus project to the contralateral medial rectus muscle viafibers that cross the midline and ascend in the medial longitudinal fasciculus (MLF). Excitatoryburst neurons also drive ipsilateral inhibitory burst neurons that inhibit contralateral abducensand excitatory burst neurons. The medial vestibular nucleus also inhibits contralateral abducensneurons. Omnipause neurons inhibit excitatory burst neurons and abducens neurons, preventingunwanted eye movements. The eye position component of the motor signal arises from a „neuralintegrator“ comprised of neurons distributed throughout the medial vestibular nuclei and nucleusprepositus hypoglossi on both sides of the brain stem. These neurons receive velocity signals fromexcitatory burst neurons and integrate its velocity signal into a position signal. The position signalis then transmitted to the ipsilateral abducens neurons.

2.2.3.6 Control of Smooth-Pursuit Movements

In the smooth pursuit system, signals carrying information about target motion are extracted bymotion processing areas in visual cortex and then passed to the dorso-lateral pontine nucleus ofthe brainstem. There, neurons have been identified which code either the direction and velocity ofpursuit eye movements, the direction and velocity of visual target motion, or both. These signalsproceed to the cerebellum where neurons have been shown to specifically encode the velocity ofpursuit eye movements [SK84]. These neurons, in turn, make connections with cells known tobe upstream of the nucleus prepositus hypoglossi (Fig. 2.22), the integrator of the oculomotorsystem for horizontal saccades). As in the saccadic system, the brainstem integrator appears tocompute the long-term holding force from this signal and then to pass the sum of these signalsto the motor neurons.

All eye movement control signals must pass through the ocular motoneurons which serve as a finalcommon path. In all cases these neurons carry signals associated both with the instantaneousvelocity of the eye and the holding force required at the end of the movement. Eye movement sys-tems must provide control signals of this type, presumably by first specifying a velocity commandfrom which changes in holding force can be computed. In the case of saccades, this command isproduced by brain structures that topographically map all permissible saccades in amplitude anddirection coordinates. In the case of pursuit, the brain appears to extract target motion and touse this signal as the oculomotor control input. Together these systems allow humans to redirectthe line-of-sight to stimuli of interest and to stabilize moving objects on the retina for maximumacuity.

2.2.3.7 Hering’s Law

Neural processes of binocular vision involve the control of both eyes at the same time. One veryfundamental function of the visual system is to provide information on the location of objectsin space. The left and the right eyes, positioned about 6 cm apart, each capturing an image ofthe environment from a slightly different point of view, and comparison by the brain of thesetwo retinal images yields information about distance. As long as the difference between theleft and right eye image (the binocular disparity) is not too large, the mechanism of stereopsis


(binocular vision) both interprets the disparity as depth and, by means of sensory fusion, causesthe desperate images to be perceived as single impression. Thus, the two eyes must be wellaligned for normal stereopsis to function properly. This implies that both eyes accurately fixatea single point in space.

According to Hering’s law of equal innervations [Her68], the two eyes are not independentlycontrolled by the brain. Hering observed that many eye movements are conjugate from birth,even when one eye is covered. Hering argued that a common innervation must be sent to theeyes to achieve this conjugacy, implying that the two eyes are controlled as one unit. The means,that changes of innervation are programmed in one of two modes:

1. for a conjugate eye movement, in which both eyes rotate through the same angle and inthe same direction, or

2. for a disjunctive eye movement, in which both eyes rotate through equal angles but inopposite directions.

In positioning both eyes with respect to conjugate eye movements, the brain accepts one eye asthe leading or fixating eye, from where motor commands are received. The other eye will beconsidered as the following eye, and will receive innervations based on those from the fixing eye.By the Hering law, yoke muscles receive equal and simultaneous innervation. The magnitude ofthis innervation is determined by the fixating eye. Since the magnitude of innervation to yokemuscles is determined by the fixating eye, the angle of deviation between eyes (strabismus) mayvary depending on which eye is fixating. The primary deviation is misalignment with the normaleye fixating. If the paretic eye fixates, the ensuing secondary deviation is typically larger thanthe primary deviation.

2.2.3.8 Sherrington’s Law

Rotation of the eyes can be derived from a change in the distribution of active muscle tensionbetween antagonist muscles, although their sum remains constant. From a stable position (e.g.primary position), the change of muscle force distribution occurs in phases in order to begin,carry out, and stop the saccadic or smooth pursuit movement. The distribution is then stabilizedat the new eye position.

Sherrington’s law of reciprocal innervation states that increase of innervation and contraction ofa muscle is associated with a reciprocal decrease in innervation of the antagonist muscle [DE73].

2.3 Measurement Techniques

Measurement techniques in the field of oculomotor physiology can be divided into measurementsof eye movements (positions) and into measurements of the underlying anatomy and physiology(e.g. neural activity, motion and muscle force). These measurements can be accomplished by theuse of different measurement techniques, ranging from minimal invasive methods (e.g. clinical


observations) to methods that need access to the human anatomy. However, in order to under-stand the fundamental workflow of the oculomotor system, all measurement results need to beconsidered.

2.3.1 Eye Movement Measurements

Eye movements not only dictate where a subject looks or what a subject sees, they are indis-pensable for diagnosing ophthalmologic, neurological, and otologic problems. Measurements ofeye movements are used to study how our cortex processes information, what the cerebellum isdoing, and how learning takes place in the brainstem. And they are increasingly used for techni-cal applications, like monitoring the alertness of drivers or controlling the auto-focus of cameras.Measurement techniques are comprised of mostly non invasive techniques that monitor momentof the eyes in two or three dimensions.

2.3.1.1 Electro-Oculography

The electro-oculogram (EOG) is a measurement of bipotentials under two different light intensi-ties (light adapted – non-light adapted eye) produced by changes in two constant eye positions. Itis used as a clinical test for retinal dysfunction. The changes of bipotentials in electro-oculographyis also used under constant light intensity for the measurement of different eye positions. In thenormal eye, there is a steady potential of approximately 6 to 10 mV between the cornea and theretina known as the corneal-retinal potential. The corneal-retinal potential can be measured byplacing a single surface electrode directly lateral to each of the orbits of the eye and a referenceelectrode to the bridge of the nose.

(a) Application of EOG measurement elec-trodes

(b) EOG recording: The subject was in-structed to look to the far left, hold his gaze,look to the far right, hold his gaze and re-peat.

Figure 2.23: EOG Eye Movement Measurement Technique [FFG+02]

By placing electrodes superior and inferior to the orbit of each eye, and a reference electrodelateral to the eye of interest, vertical eye movements can also be measured (see Fig. 2.23(a)).When a test subject is gazing straight ahead, the corneal-retinal dipole is symmetric between thetwo electrodes, and measured EOG output is zero. As the subject gazes to the left, the corneabecomes closer to the left lateral electrode, therefore causing the EOG output to become more


positive (see Fig. 2.23(b)). The inverse of this is true when the subject looks in the right direction.When measuring the EOG output, there is a fairly linear relationship between the horizontal angleof gaze and the EOG output. This relationship remains true up to approximately thirty degreesof arc. Through careful calibration, horizontal eye movements can be measured with a resolutionof approximately one degree of arc. The EOG technique is preferred for recording eye movementsin sleep and dream research.

One specific type of EOG measurements is the electronystagmograhy (ENG). ENG measurementsare used to measure a condition called nystagmus. Measurement of nystagmus, a characteris-tic pattern of eye movement, is invaluable in the diagnosis of various vestibular and balancedysfunctions.

The EOG is not a very stable signal and measurements can vary as a result of varying ambientlight conditions. By having a patient carry out eye movements of constant amplitude in the darkand then in the light, any change in the biopotential would reflect a change in the corneal-retinalpotential. In normal eyes, this potential decreases during dark adaptation, and increases duringlight adaptation. However, the signal can change when there is no eye movement. It is proneto drift and giving spurious signals, the state of the contact between the electrodes and the skinproduces and other source of variability. There have been reports that the velocity of the eye asit moves may itself contribute an extra component to the EOG. It is not a reliable method forquantitative measurement, particularly of medium and large saccades. However, it is a cheap,easy and non-invasive method of recording large eye movements, and is still frequently used byclinicians.

2.3.1.2 Infrared-Oculography

If a fixed light source is directed at the eye, the amount of light reflected back to a fixed detectorwill vary with the eye’s position. This principle enables the determination of the current eyeposition. Infrared light is used as this is „invisible“ to the eye, and doesn’t serve as a distractionto the subject. As infrared detectors are not influenced to any great extent by other light sources,the ambient lighting level does not affect measurements. Spatial resolution reaches up to 0.1 ◦,and temporal resolutions of 1ms can be achieved. It is better for measuring horizontal thanvertical eye movements. Blinks can be a problem, as not only do the lids cover the surface of theeye, but the eye retracts slightly, altering the amount of light reflected for a short time after theblink. The corneal reflection of the light source is measured relative to the location of the pupilcenter. Corneal reflections are known as the Purkinje reflections, or Purkinje images [Cra94].

Due to the construction of the eye, four Purkinje reflections are formed (see Fig. 2.24). Video-based eye trackers typically locate the first Purkinje image. With appropriate calibration proce-dures, these eye trackers are capable of measuring a viewer’s point of regard (POR) on a suitablypositioned (perpendicularly planar) surface on which calibration points are displayed. Two pointsof reference on the eye are needed to separate eye movements from head movements. The posi-tional difference between the pupil center and corneal reflection changes with pure eye rotation,but remains relatively constant with minor head movements.

Since the infra-red light source is usually placed at some fixed position relative to the eye, thePurkinje image is relatively stable while the eyeball, and hence the pupil, rotates in its orbit(see Fig. 2.25). So-called generation-V eye trackers also measure the fourth Purkinje image,


Figure 2.24: Schematic Overview of Purkinje Corneal Reflections, from [Cra94]PR, Purkinje reflections: 1, reflection from front surface of the cornea; 2, reflection from rear surface of thecornea; 3, reflection from front surface of the lens; 4, reflection from rear surface of the lensùalmost the samesize and formed in the same plane as the first Purkinje image, but due to change in index of refraction at rearof lens, intensity is less than 1% of that of the first Purkinje image; IL, incoming light; A, aqueous humor;C, cornea; S, sclera; V, vitreous humor; I, iris; L, lens; CR, center of rotation; EA, eye axis (line of sight);a ≈ 6mm;b ≈ 12.5mm; c ≈ 13.5mm;d ≈ 24mm; r ≈ 7.8mm from [Cra94]

however, due to the anatomical structure of the eye, this reflection gives a very weak signal. Bymeasuring the first and fourth Purkinje reflections, these dual-Purkinje image (DPI) eye trackersseparate translational and rotational eye movements. Both reflections move together throughexactly the same distance upon eye translation but the images move through different distances,thus changing their separation, upon eye rotation. Unfortunately, although the DPI eye trackeris quite precise, head stabilization may be required.

2.3.1.3 Scleral Search Coils

One of the most precise eye movement measurement methods involves attaching a mechanical oroptical reference object mounted on a contact lens which is then worn directly on the eye. Thistechnique evolved to the use of a modern contact lens to which a mounting stalk is attached (seeFig. 2.26 and Fig. 2.27). The contact lens is necessarily large, extending over the cornea and sclera(the lens is subject to slippage if the lens only covers the cornea). Various mechanical or opticaldevices have been placed on the stalk attached to the lens: reflecting phosphors, line diagrams,and wire coils have been the most popular implements in magneto-optical configurations.

The principle method employs a wire coil, which is then measured moving through an electro-magnetic field. When a search coil (see Fig. 2.26(a) is put into an oscillating magnetic field, avoltage is induced in the coil [Has95]. Three orthogonal magnetic fields are emitted by an electro-magnetic field frame (see Fig. 2.26(b)). Usually, one contact lens has 2 search coils mounted, onecoil for the measurement along each axis of a two-dimensional coordinate system. The voltagesinduced by the electromagnetic field frame can directly be related to elements of the rotationmatrix that describes the current eye position relative to a reference position where the coils line


Figure 2.25: Purkinje Corneal Reflections I. and VI. marked with crosses, from [Kan87]

up with the magnetic fields. The coils commonly used for recording 3D eye positions are dualsearch coils (produced by Skalar Instruments, Delft, The Netherlands) which are oriented in sucha way that they are approximately parallel to the axes spanned by the electromagnetic frame.

(a) Scleral Search Coil Lens (b) Electromagnetic Frame for Search CoilMeasurements

Figure 2.26: Electromagnetic Search Coil Eye Position Measurement, from [Ska03]

Problems, like the determination of offsets which are frequently superimposed on the inducedvoltages have been investigated in detail by Hess et al [HVOS+92]. The determination of angularrotations for a measured eye position also involves consideration of „false torsion“ (see Sec. 2.2.2.2)and appropriate correction (cf. [Has95]).

Insertion of the contact lens is shown in Fig. 2.27. Although the scleral search coil is the mostprecise eye movement measurement method (accurate to about 5-10 arc-seconds over a limitedrange of about 5 ◦), it is also the most intrusive method. Insertion of the lens requires care andpractice. Wearing of the lens causes discomfort and therefore is not ideal for clinical application.


Figure 2.27: Insertion Procedure for a Scleral Search Coil Lens, from [Ska03]

2.3.1.4 Video-Oculography

With the development of video and image analysis technology, various methods of automaticallyextracting the eye position from images of the eye have been developed. Tracking the relativemovements of these images gives an eye position signal. 3D video-oculography (3D VOG) systemscommonly work on the following principle: first, the center of the pupil is found. This is achievedby thresholding the gray-level signal of the image of the eye, finding the pupil, and fitting anellipse to its outline (see Fig. 2.28). The center of this ellipse determines the horizontal andvertical eye position. Then, the light-dark distribution of the iris is measured along a circlearound the center of the pupil. Cross-correlating this iris signature with a reference pattern givesthe torsional eye position (cf. [MHCS94] and [Has95]). Up to now, no algorithm exists thatcan distinguish between a translation of the camera with respect to the head, and a shift of thepupil by a rotation of the eye. However, looking at Fig. 2.28, it is obvious that images of theeye contain more information than just the center of the pupil. Corneal reflections, also used ininfrared oculography, patterns of the iris and the shape of the upper and lower eye-lid may giveadditional signals to improve the stability of VOG.

Figure 2.28: Video-Oculography Pupil Detection, [SMI]

However, image based methods tend to have temporal resolutions lower than that achieved withIR techniques. One reason why existing video-oculography (VOG) systems have not filled thisneed is the difficulty of measuring the rotation of the eye around the line of sight. Techniques fortracking the horizontal and vertical position of the pupil are straightforward, and a number ofdifferent algorithms give acceptable results. But measuring the rotation of the eye around the lineof sight is much more difficult, since it requires not only the detection of the pupil, but also relieson the resolution of fine details in the structure of the iris (cf. [Has95], [MHCS96] and [SH03]).Even small displacements between head and camera can cause large measurement artifacts (e.g.


a camera displacement of only 1 mm shifts the center of the pupil by the same amount as a5 ◦-rotation of the eye). Currently, no published algorithms exist for the automated selection ofsuitable landmarks on the iris, which is necessary for automated measurements of ocular torsion.Such algorithms would dramatically improve the applicability of video-oculography, also to otherscientific and medical applications. However, current system accomplish the 3D measurement ofeye positions, provided that the head is upright and not moving rapidly (see Fig. 2.29).

Figure 2.29: Chronos VOG System, from [Ska03]

The VOG eye tracker shown in Fig. 2.29 consists of a head unit, which is individually adjustable,and carries the CMOS cameras for recording eye-in-head images. Additionally, this head unitcarries the system unit, which accommodates the custom-designed architecture for the online,real-time acquisition and pre-processing of image and signal data. This is designed around astandard Windows PC with a PCI plug-in board. The head unit is connected to the systemunit through high-speed digital data links. These provide the necessary data channels for thetransfer of high bandwidth image and signal data from the head unit to the system unit, andthe command sequences from the system unit to the head unit. The image of the eye is reflectedby the dichroic mirror to the optical lens and projected onto the image sensor. An infraredpass filter is fitted in front of the image sensing area in order to exclude sporadic incident lightfrom the environment. These optical elements and the cameras are arranged on the head unit tofacilitate maximal field-of-view for the test subject. A field-of-view approaching ±90 ◦ horizontaland +40/− 60 ◦ vertical is attained.

Nevertheless, there is an increasing requirement in both the clinical and research fields for non-invasive precise measurement of three-dimensional eye movement by using 3D VOG systems.

2.3.2 Physiologic Muscle Force Measurements

During the investigation of muscle force, two types of contraction measurements can be differenti-ated: isotonic contraction and isometric contraction. An isotonic contraction is the measurementof the muscle length and length change due to an activation with constant load (Fig. 2.30(A)). Inthe case of isometric contraction, the strength and the change of force of the muscle is measuredwith the length held constantly (Fig. 2.30(B)). The contraction mechanism takes place on molec-ular level and is described by the so-called filament sliding theory. Actin and myosin filamentsare connected over archings and realize activation-steered shortening, and thus development of


muscle force.

Figure 2.30: Different Types of Muscle Force Measurement, from [BKP+03]

Generally, static and dynamic characteristics of a muscle can be differentiated. Static force be-havior is also called force-length behavior, whereby isometric force measurement is accomplishedas a starting point and, in dependence of the adjusted length, measurement of the produced forceis carried out. Dynamic characteristics of a muscle refers to contraction speed and is analyzedby using isotonic measurements. In the representation of the static characteristics of a muscle,active and passive muscle force can be differentiated. Active muscle force results from activation(innervation) of a muscle initiated by the brain, while passive forces represent the flexible stretchcharacteristics of a muscle, which works opposite to the active force. By applying repeated iso-metric force measurements with differently adjusted lengths, a force length curve of the musclebehavior results. In relating these data to the activation potential of a muscle, a three-dimensionalforce-length-activation function can be defined. This function can be divided again into its activeand passive forces, receiving an active and a passive force curve, which correspond to the totalforce curve of a muscle. The passive force curve describes the flexible forces a muscle exerts, ifit is stretched or contracted accordingly. If a detached muscle is sufficiently stretched, then, ata certain length, the muscle stops to behave like a non-linear spring, but will get stiff very fast,allowing no further flexibility. This is called the „leash region“ and becomes apparent in a drasticrise of the passive strength with increased stretch length. On the other hand, if a muscle getsshorter and loses its stretch, then it will get slack and can not exert force anymore. This is calledthe „slack region“ of the muscle force function.

Meaningful force measurements of the extraocular muscles are extremely difficult to obtain. How-ever, there are difficulties deriving the physiologic behavior of extraocular muscles from suchmeasurements. A clinically useful device to measure force was introduced by Scott et al in 1972(see Fig. 2.31) [SCO72].

Collins described instrumented duction forceps that carried strain gauges to measure force,and an ultrasonic microphone to measure distance, in conjunction with an ultrasonic soundsource [Col78]. The Collins forceps allowed convenient length-tension measurements on an iso-lated, disinserted muscle or an intact eye.Others followed with variations of this approach (e.g.[SCW+84]).


Figure 2.31: Example of measuring Force with Forceps

A common clinical diagnostic test is the forced duction, sometimes called passive duction test.This test is used to identify the cause of the lack of rotation of a muscle from two possible broadcauses. In weakness of the muscle or paresis of the nerve supplying the muscle, the eye does notrotate as it should because, ultimately, the muscle is not contracting properly. In mechanicalcauses of lack of rotation of the eye, the muscle receives input instructing it to contract and rotatethe eye, but mechanical forces are preventing the eye from moving properly. In the exam chair,it is performed by topically anesthetizing the eye with drops, and then further anesthesia maybe given by soaking an applicator with topical anesthetic and applying it to the intended pointof contact with the eye. Classically, a forceps is used to perform the test, by grabbing the eyeand mechanically rotating it while the patient looks in the direction of gaze being tested, andseeing if the examiner can rotate the eye for the patient. Free rotation implies that the eye isnot mechanically resisted, which means the rotation problem is a paresis of the nerve supplyingthe muscle the impulse to contract. If there is mechanical resistance, often not only does the eyenot fully rotate in the direction being tested, but the examiner may feel the resistance. Someexaminers prefer to do the test by pushing the globe in the intended direction with an applicator,rather than grabbing the eye with a forceps, but the overall concept is the same (see Fig. 2.31).

According to Miller [MD96], there exist four problems with all isometric measurements:

Relationship of innervation, length, and tension - is not physiologically meaningful dur-ing isometric measurements, since the eye to be measured is held at a fixed position andmoved from there. Normally, when innervation increases, muscle force increases, and themuscle shortens. When innervation decreases, muscle force also decreases, and the musclelengthens. With forced duction, innervation is held constant, thus, muscle force increasesas the muscle is lengthened, and decreases as it is shortened, resulting in the inverse behav-ior compared to the normal relationship. Isometric measurements also disturb the normalrelationship between length and tension, though not so badly as forced ductions.

Neither forced duction measurements nor isometric measurements can directly predict nor-mal behavior of the extraocular muscles. The force produced in a suddenly stretched muscletends to decay by stress relaxation or yield. Similarly, a sudden increase in load results inan abrupt initial stretch, followed by a gradual increase in stretch, called creep. Seekingto avoid yield and creep by pulling and relaxing the muscle quickly does not help, since


increasing stretch velocity increases viscosity, spuriously increasing measured forces dur-ing the pull and decreasing them during the release. Viscous effects can be minimized bystretching slowly, but this increases yield and creep.

Methods that require the measured muscle to be disinserted - makes it nearly impos-sible to measure force at primary position length. Once the cut end of the muscle tendon isattached to the measuring device, attempts to determine primary position length by abut-ting it to its severed insertion, while holding the eye in primary position and preventingglobe translation, do not inspire confidence. The need to „unwrap“ the muscle from theglobe to take the actual measurement provides another opportunity for error.

Rotating the globe to measure stiffness - may be done in an intact eye [CCSJ81] or withsome muscles disinserted. Here the worry is that translation as well as rotational forces areapplied by the duction forceps. There is no way to avoid translating the eye when pulling atone point on its surface. Translation stretches some muscles and relaxes others, distortingforce measurements in complex ways.

The degree of elastic coupling to surrounding structures - with measurements on disin-serted muscles is seldom clear and definitely not physiologic.

Figure 2.32: Muscle Force Transducer for Intraoperative Measurements [MD96]

To address these problems, Miller et al developed a method that is related to the method of Collins[COS75], in that it leaves the eye free to rotate, preserving the physiologic relationship betweenmuscle tension and length (see Fig. 2.32). The muscle is not disinserted, and its path length(e.g. degree of stretch in a given eye position) is not significantly altered. This device lies flat onthe muscle tendon, causing little modification of ocular mechanics. Since the measured muscleremains attached to the globe, primary position muscle length is easily established. Problems ofglobe translation do not arise, and musculo-fascial couplings are physiologic. However, signalsare slightly distorted as the transducer rotates under the lids. The method is most useful withawake patients, but even with anesthetized patients it has the advantage that primary positionforces can be determined.


2.3.3 Measurement of Motion in the Orbit

It has been shown, that the measurement of the motion of orbital tissue can improve the diagnosisand management of orbital disorders and also give more detailed insight to the kinematics oforbital movements [DMP+95][CJD00][DOP00]. It is easy to measure the motion of the eyes as afunction of gaze, since the eyes are easily observed visually and their motion is readily accessiblefor inspection. However, the motion of the orbital tissues behind the globe is effectively hiddenby the orbit, the eye and the eyelids, so that it is very difficult to apply measurements in vivo.Attempting to measure motion objectively by introducing instruments or devices into the orbitis not yet an option. The risk of damaging the optic nerve and consequently causing blindnessis simply too large. Additionally, such instruments may influence the very motion they aresupposed to measure. As an alternative, Abràmoff suggested to measure the motion first byusing MR imaging and then using image analysis techniques to measure the motion objectively[Abr01]. A suitable image device constructs an image, or a series of images from some regionallyvarying physical properties. In this case, an image can be considered as an ordered set of vectors,where each vector represents the magnitude of one or more of the measured physical properties.Reconstructing optical flow vector fields in these images gives insight into the motion of thecaptured region.

Cine MRI time sequences were obtained using T1-weighted volumes on a 1.5 T MR scannerand a head-coil (TE 6.9, TR 12, matrix 256x256x46) with an acquisition time of 15 sec. Twodimensional image sequences were extracted from these volumes on a transversal axis intersectingboth, horizontal rectus muscles and the optic nerve (cf. Sec. 2.1.4).

Figure 2.33: Intraconal Tissue Motion around the Optic Nerve, from [Abr01]A: flow field displayed over a static MRI of the left orbit. B: Schematic view of motion as expressed by B.

Because the true motion field in the orbit is unknown, simulations and measurements of controlledmotion of an object (i.e. a sirloin steak) was used to compare the computed flow fields with theknown motion fields. A sirloin steak was mounted in a transparent box fitted with an angle ruler.A sirloin steak consists of bands of several millimeters width of alternating muscle and fat tissuethat approximates the alternating fat-muscle-fibrous tissue structure of orbital soft tissue. Thisobject was rotated 5 degrees per captured frame, and a sequence of 21 frames was obtained bythe MR scanner. Pre-filtering using Gaussian smoothing and nonlinear diffusion was performedon all images.


Optical flow algorithms were used to extract the spatio-temporal patterns of image or signalintensity to estimate the optical flow field. In a second stage, the resulting system of equationsis solved to estimate the actual optical flow. The computed flow fields were displayed using amapping technique that shows all flow vectors as colored pixels plotted over the original MR image(see Fig. 2.33). Thus, a multi-modality image is obtained that combines functional (motion) andanatomical information in a single image.

Chapter 3

Strabismus

Squint (strabismus) is the name given to usually persistent or regularly occurring misalignmentof the eyes. Strabismus is a visual defect in which the eyes are misaligned and point in differentdirections. One eye may look straight ahead, while the other eye turns inward, outward, upwardor downward. Strabismus is a common condition among children. About 4 percent of all childrensuffer from strabismus. It can also occur later in life. It occurs in males and females and mayrun in families. However, many people with strabismus have no relatives with this problem.

People that have strabismus suffer not only from the frequently disfiguring externally visibleabnormality, the visual impairment associated with squint is an even greater burden. Squint isnot just a blemish but often a severe visual impairment. The earlier a child develops a squint andthe later it can be treated, the worse the visual impairment will be. By the time a child reachesschool age, the prospects of successful treatment decline dramatically. Babies and small childrenwith strabismus should be treated at the earliest possible moment.

This chapter will give an overview of some important fundamentals in strabismus along with itsvarious pathological classifications. Along with representative examples, basic clinical diagnosisand treatment will be covered, including the effects of surgery, evaluation of eye motility disordersor determination of the amount of surgery. However, more detailed information on strabismuspathology and surgery is well documented and can be found in various literature (e.g. [Kau95],[DE73], [RSS01], [Kan87] etc.). The goal of this chapter is to give insight into current clinicalpractice in the field of strabismus diagnosis and treatment in order to better understand theapplication of a new, computer-aided basis for this medical subject.

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CHAPTER 3. STRABISMUS 54

3.1 Overview

In order to be able to correctly perceive the environment, both eyes must look in the samedirection. This causes almost identical images to be generated within each eye. These twoimages are then fused together in the brain to form a single three-dimensional visual impression.If a squint is present, the difference between the two images caused by the misalignment is toogreat and the brain is unable to converge and to fuse them. The result is irritating double vision.The juvenile brain is able to respond to double images by simply suppressing the image arrivingfrom the deviant eye. This process generally has calamitous consequences as vision in the unusedeye gradually becomes weak (amblyopic). Amblyopia is the term used to describe weak visionin an otherwise organically healthy eye. In the absence of treatment almost 90% of all childrenwho suffer from a squint develop amblyopia on one side. If this squint-related visual weakness isnot detected and treated within early time, it will remain a lifelong affliction. The child will thennever learn to see with both eyes or even have three-dimensional vision and will be at greaterrisk of accidents and restricted in everyday life. Prompt treatment can almost always prevent orcure amblyopia and sometimes also produce good spatial vision.

3.1.1 Visual Acuity

Babies are able to perceive their environment through their eyes quite soon after birth - butonly indistinctly. Among all human senses, the visual sense develops within shortest time, butvisual acuity still has to be developed through constant exercise. Only a limited period in growthis available for this purpose. By the time school age is reached, the eyes’ learning program isvirtually complete. The old adage that „what you don’t learn as a child, you’ll never learn as anadult“ applies to eyesight, too. In the first weeks of life a child is still unable to exactly coordinatethe movements of the two eyes. Brief misalignments during this time are no cause for concern.They may also occur occasionally again in the course of the coming months. The ability to gazealso has to be learned, but if one eye constantly deviates from the direction of the other, thereis no time to loose. The ophthalmologist can diagnose the problem even in infancy and mustinitiate the treatment at the right time.

3.1.2 Symptoms

Children with conspicuous squint have the best prospects because they are taken to the ophthal-mologist within short time by their parents on account of the „blemish“. Unfortunately, thereexists a number of barely visible or invisible deviations. They are only detected when one eye isalready amblyopic - such as during the eye test when starting school, when it is generally too latefor an entirely successful treatment. For this reason alone 4% of all people suffer from seriousone-sided visual deficiency. It is therefore very important to know and to heed all characteristicsthat might indicate an impending or existing squint: sensitivity to light, with tears, squeezing oneeye shut, bad mood or irritability, chronic blepharitis, head held to one side and clumsy motionare alarm signals. Each sign is a valid reason in its own right to obtain an ophthalmologist’sopinion immediately.


3.1.3 Treatment

The primary prerequisite of treatment is to establish an optimum visual acuity for both eyes.Only then, a squint surgery can yield successful and enduring results. Eyes with total loss offixation (e.g. blindness) often re-establish squinting after surgical treatment and cannot holdlong-lasting parallel alignment.

Eyeglasses Many squinting children in Europe are farsighted. Exceeding near fixation can causesquinting with these children due to the convergence impulse. Eyeglasses, determined underfull relaxation, using anesthetic, relaxing eye drops, can minimize or even heal a disorder.

Occlusion treatment Occlusion treatment, in which a light restraining plaster is applied overthe squinting and normal eyes in a specific rhythm as instructed by the ophthalmologist,serves to prevent as well as combat amblyopia. The plaster covering on the normal eyeis intended to have the effect of exercising the squinting eye. Changing over the plasteralternately prevents weak vision in the normal eye caused by the occlusion. The main pre-requisite for the success of amblyopia treatment is strict adherence to the treatment/exercisephases for the squinting eye and the normal eye that have been precisely determined by theophthalmologist in every single case. If glasses, occlusion and eye-drops do not result in animprovement in visual acuity in older pre-school children and younger school children withamblyopia, a training program prescribed by the ophthalmologist can occasionally providefurther help. The amblyopia check-ups and treatment must generally be continued over aperiod of years into the growth phase, in addition to glasses and even after a successfuloperation. The skin plaster can, after improval of vision, often be replaced by an occlusionusing blurring spectacle sheets.

Strabismus surgery and subsequent treatment Half of the children with a squint need cor-rection of the faulty alignment by means of an operation on the extraocular eye muscles.Sometimes, in a mechanical restriction of eye movements, operative positional correction isa prerequisite for all other measures. As a general rule, the operation is only carried outwhen the child wears glasses reliably, can see more or less equally well in both eyes andcan be adequately examined (normally shortly before starting school). In order to improvepreoperative diagnosis, the wearing of prisms, also for longer time, is useful to determinethe true squint angle. The operation does not eliminate the weak vision, neither does itproduce an immediate improvement in spatial vision. Both generally require further oph-thalmic treatment. The operation does not eliminate the need for glasses, because they arethe only means of correcting refractive errors. The type of misalignment and the result ofthe preliminary treatment determine whether a single operation is sufficient. Strabismusoperations are carried out on the children under general anaesthetic by the ophthalmologist.

3.2 Binocular Vision

In vergence disconjugate movement, both eyes move synchronously and symmetrically in oppositedirections, where convergence movements describe the ability to move both eyes inward (to thenose) and divergence defines the movement of both eyes outward (away from the nose). In


convergence, voluntary and in voluntary reflexes occur, in order to fuse the images of both eyesto a single stereoscopic picture.

Tonic convergence - describes the tonus of the extraocular muscles (especially the medial rectimuscles) when awake.

Proximal convergence - is the convergence induced by the knowledge of the proximity of anobject and occurs in optical instruments even though they are set for infinity.

Fusional convergence - is a reflex that ensures the bilateral projection of corresponding areasonto the retinae of both eyes. This reflex occurs without a change in optical refraction andis stimulated by the perception of double images (diplopia). Thus, fusional convergencegenerates compensatory eye movements in order to overcome disparity of the retinal images.The amplitude of fusion denotes the maximum magnitude of eye movement that residesfrom fusional convergence. Fusional amplitudes may be corrected through the applicationof prisms and can be measured with a synoptophore. The usual fusional amplitude ofconvergence that is measured to the breaking point of diplopia for fixation of far targets isapproximately 30 prism dioptres and for near fixation approximately more than 35 prismdioptres, whereas one dioptry corresponds to approximately 0.5 degrees. Generally, fusionalconvergence supports to control exophoria (latent divergent strabismus), whereas fusionaldivergence helps to compensate for esophoria (latent convergent strabismus).

Accommodative convergence - is convergence induced by accommodation. For each dioptreis associated with a near linear increasing relationship to the angle of accommodative con-vergence and results in the so called accommodative convergence/accomodation (AC/A)ratio. Normally 3 to 5 prism dioptres per dioptre of accommodation. Anomalies in theAC/A ratio most often indicate a cause for strabismus. A high AC/A ratio due to ac-commodation for the fixation of near objects may induce excessive convergence and resultin esotropia (inward squinting). Thus, a low AC/A ratio may lead to divergence that iscausally related to exotropia (outward squinting) while a subject fixates a near object.

Voluntary convergence - is convergence that can be produced at will.

Binocular vision develops within the first years of life and is additionally accomplished by theability to three-dimensionally perceive an image due to stereopsis. Three essential factors arerequired for the successful development of binocular vision and stereopsis.

• Clear and undisturbed refraction in both eyes,

• the ability of different areas in the brain to fuse slightly different images from both eyes,and

• the ability of exact coordination of eye movements in all possible gaze positions within thephysiologic field of gaze.

Thus, the ability of fusion strongly depends on the relationship between both retinae.


3.2.1 Projection

Projection is defined as the interpretation of the position of an object in space on the basisof corresponding areas that are stimulated on the retina (see Fig. 3.1). Following Fig. 3.1(a),an object (F’) and an additional element (T’) are stimulating the right fovea (F) and the righttemporal area of the retina (T), respectively, the brain will perceive the red object (F’) as straightahead, whereas the black object (T’) will be located in the nasal visual field of the right eye. Thissituation corresponds to the normal projection of images onto the retina of an eye. According tothis definition, nasally located elements will project onto temporal areas on the retina, whereaselements that are located in the upper part of the physiologic field of gaze are projected ontolower areas onto the retina, and vice versa.

(a) Right Eye (b) Both Eyes

Figure 3.1: Projection of Objects in Space onto the Retina on an Eye, adapted from [Kan87]

When both eyes are kept open, the red object (F’) in Fig. 3.1(b) stimulates the retinae of botheyes (F), and the black object (T’) stimulates temporal areas of the retina in the right eye (T)as well as nasal areas of the retina in the left eye (N). Accordingly, the right eye projects theobject into temporal areas of the visual field, and the left eye projects into nasal areas of thevisual field. Since the stimulated retinal elements both represent the same object, both retinalpoints will project to the same location in space and no double images will occur.

According to the implications of retinal projection, a horopter (see Fig. 3.1(b)) represents avirtual spherical arc that captures all points in space that correspond to diplopia free binocularvision. Points that are fixated beyond or before this horopter will be accompanied by doubleimages (diplopia) and are the foundation for physiological double vision.

The line of sight or visual axis, corresponds to the line that is virtually drawn from the fixationpoint in space to the retinal point on the fovea and additionally almost intersects the center ofthe pupil. From Fig. 3.1(b) it is easy to see that both red lines of sight intersect at the desiredfixation point (F’) in order project a single, sharp image onto the foveae. The retinal areas (F)in both eyes are denoted as corresponding retinal areas.


(a) (b) (c)

Figure 3.2: Different Forms of Binocular Fixation, from [Kan87]

3.2.2 Diplopia

In strabismus, a dissociation of the lines of sight of both eye occurs which can be latent (phoria) ormanifest (tropia). Concerning manifest forms of strabismus there can be two different problemsin the alignment of the visual axes, namely confusion and diplopia. In Fig. 3.2(b), a convergentmanifest strabismus of the right eye is shown, so that the fovea of the right eye is stimulatedby a black triangle and the red object is projected onto the left fovea. Confusion results in theoverlapping of these different images into the same vertical axis in space.

After that, diplopia results from the stimulation of an excentric retinal area in the fovea ofone eye. In Fig. 3.2(c), the red object in space does not stimulate corresponding retinal areas,instead, a nasally shifted position on the retina is stimulated in the left eye, whereas the objectis on the fovea of the right eye. In case of convergent strabismus, uncrossed double imagesoccur (see Fig. 3.2(c)), while in divergent strabismus diplopia is perceived with crossed doubleimages. Mainly in young children, a mechanism of alternating suppression leads to a temporaryphenomenon in binocular vision, where the misaligned image is actively masked out by the brain.When the fixating eye is covered, the squinting eye takes up the fixation and suppression stopsimmediately. But in not alternating (monocular) strabismus, amblyopia can be seen as a resultof continuous monocular suppression of the image of the affected eye.

3.3 Ocular Dissociation

The term orthophoria defines the ideal condition of ocular balance wherein the oculomotor systemis in perfect equilibrium so that both eyes retain their normal positional relationship (i.e. botheyes remain directed upon a fixation point or remain parallel). In such a condition, the positionsof both eyes are identical and a correct posture of the eyes is maintained without effort forall directions of gaze and all physiological distances of a fixation point. However, this idealequilibrium is rare for distant and seldom existent, especially when fixating near objects. Usually,both eyes are maintained on the fixation point only under stress with the aid of corrective


fusion reflexes, originating from the brain. This can be classified as a tendency for deviationof the eyes from parallelism, which can be prevented by the mechanism of binocular fusion.This results in a typical deviation of both eyes, when they are dissociated (e.g. one eye iscovered), and binocular vision is mutually prevented on purpose. This „common“ deviation iscalled heterophoria and better represents the practical reality compared to the idealized caseof orthophoria. However, functional heterophoria is clinically referred to as a significant, latentdeviation of the eyes, compared to the unrealized ideal of orthophoria or the physiologicallynormal deviations of heterophoria.

Additionally, the time-incidence of the deviation is denoted as follows:

Latent strabismus (Heterophoria) - is referred to as „hidden“ strabismus, which only occurswhen binocular vision is prevented by dissociation of the eyes, usually achieved by coveringand subsequent uncovering of one eye.

Manifest strabismus (constant strabismus, Heterotropia) - is defined as deviation thatoccurs all the time, also without dissociation of the eyes.

Variations in the characteristics of deviation of movements of the eye are also distinguished usingthe following terms:

Concomitant strabismus - occurs when the deviation remains the same, or approximatelythe same, in all position of the eyes and additionally is unaltered no matter which eye isused for fixation of a target object. Thus, both eyes move together in coordination, andthe visual axes, although abnormally directed, retain the same abnormal relationship toeach other throughout the complete physiologic field of gaze. It is now well known thatso-called concomitant deviations are in fact variable by nature (cf. [RSS01]), and that thesevariations can be measured. Therefore the term concomitant is too unspecific, if it has tosignify that the angle of deviation is invariable. Presently, this definition is used if there isno limitation on duction movements, in spite of an oculomotor disorder, and that there isan accompanying sensorial disorder to a greater or lesser degree.

Incomitant strabismus - means that the deviation alters with changing the position of theeyes and varies depending on whether the non-pathological or pathological eye is used forfixation. This usually originates in paretic or paralytic, sometimes spastic problems. Inparalytic strabismus, when the eyes are turned away from a paralyzed muscle, which is thusnot contracted, eye positions may be relatively normal. However, when the eyes move inthe direction of action of the paralyzed muscle, movement becomes limited or even absentand deviation grows with the angle of excitation.

According to the direction, time-incidence and degree of deviation, heterophoria and heterotropiaare described by using the following terminology [DE73]:

Esophoria and Esotropia (latent and manifest convergent strabismus) - is apparent,when the deviating eye turns inward, towards the nose (see Fig. 3.3).


Figure 3.3: Example for Inward Squinting

Exophoria and Exotropia (latent and manifest divergent strabismus) - designates aoutward (away from the nose) deviating eye (see Fig. 3.4).

Figure 3.4: Example for Outward Squinting

Hyperphoria and Hypertropia (latent and manifest vertical strabismus) - occurswhen the eye is deviating upward. Vertical upward deviating strabismus is also oftentermed strabismus sursumvergence (see Fig. 3.5).

Figure 3.5: Example for Upward Squinting

Hypophoria and Hypotropia (latent and manifest vertical strabismus) - occurs whenthe eye is deviating downward. Vertical downward deviating strabismus is also often termedstrabismus deorsumvergence (see Fig. 3.6).

Figure 3.6: Example for Downward Squinting

In order to denote the affected pathological eye within this terminology, the following terms areused commonly:

Monocular strabismus - can be denoted as right or left uniocular or monocular strabismus


that always affects one eye so that the other eye is preferred as the leading fixating eye. Theaffected eye will then follow the fixating eye and deviate according to the actual pathology.

Binocular or alternating strabismus - sometimes affects one eye and sometimes the other,so that fixation can be assumed and maintained by each eye in turn, and when both eyesare open, each is able to hold fixation freely, without any obvious preference for fixing witheither eye. Sometimes, however, it is legitimate to retain the term when there is a preferenceto take up fixation with one eye that is distinctly dominant, despite the absence of an equallevel of vision and the ability of each eye to retain fixation of the non-dominating eye aftercovering the other eye.

It has to be noted that the mechanism controlling movements of the eyes is variable and struc-turally determined, but is non-obligatory, non-rigid and physiological in nature although theoculomotor control system conforms to definite laws within the limits of which it must remain(see Sec. 2.2.2). It is obvious, however, that a mechanism of such complexity and precise controlmust possess the weakness of vulnerability and it is not surprising that disruptions frequentlyoccur from structural and functional causes, both in its peripheral and central parts. Abnormal-ities in ocular motility are therefore frequent and, indeed, constitute one of the most commonof ocular disabilities. The essential factor determining the efficiency of the oculomotor controlsystem is the early (in childhood) development of the ability of fixation and the fusional reflexesaccomplished by the brain.

3.4 Clinical Assessment

In evaluation of ocular alignment a first decision about the information that is required mustbe taken. Measurements can give information on the eye alignment during everyday binocularviewing, or the maximal deviation of the visual axes under conditions of disrupted binocularvision, or both of these. Subjective methods are useful for cooperative, communicative olderpatients, but objective methods must be used in younger patients or those less cooperative.Finally, some testing methods are useful only under research laboratory conditions.

Most laboratory tests are objective but depend on the measurement instrument that is used.The absolute position of the eye in space may be determined by measurement of the quantityof light reflected by the cornea from a deviated eye. The electro-oculogram (see Sec. 2.3.1.1)is generated by alterations of eye positions and electrodes capture the imbalance of electricalpotentials. Insulated wire placed in a silicone rubber (eye coil) generates response to a magneticfield (see Sec. 2.3.1.3) and can be used to exactly determine eye position.

Clinical investigations imply the analysis of atypical head positions that may indicate restrictiveor paralytic strabismus, the presence of a null point in a patient who has nystagmus, or alphabetpattern strabismus. Usually, a patient places the head in a position that provides comfortablesingle binocular vision for the straight ahead view, but occasionally the head is placed to separatediplopic images maximally. The examiner must differentiate between head turns, tilts, andvertical head positions and attempt to quantify these.

In patients who have vertical strabismus, lid asymmetry is often found. If hypotropia is presentand the non fixing eye is lower than the fixing eye, then the lid position is lower in this non fixing


Figure 3.7: Pseudo-Esotropia due to wide Bridge and Epicanthal Skin Fold, from [Kan87]

eye. This is termed pseudoptosis, if the lid regains normal position when the previously hypotropiceye fixates in primary position. Epicanthal skin folds that extend over the nasal sclera in a smallchild may simulate esotropia (see Fig. 3.7). Vertical displacement of an orbit may simulate verticalstrabismus, and an abnormal increase in the interorbital distance may simulate exotropia. Theexaminer can declare a patient to be strabismic only after the appropriate alignment testing hasbeen performed.

Objective clinical methods to determine and measure deviations of the visual axes include corneallight reflex tests, cover tests, and haploscopic tests. These tests do not require any response bythe patient and thus are independent of the patient’s ability to interpret the testing environment.

3.4.1 Corneal Light Reflex Tests

Corneal light reflex tests, the oldest testing methods, are suitable for all patients. The anglekappa (i.e. the angle formed between two imaginary lines: the visual axis and the pupillary axis)has to be taken into account, and the fixation of one eye is required. The Hirschberg methodrelies on a pupil size of 4mm and assumes each millimeter of light displacement across the corneais equivalent to approximately 7 ◦ of deviance. A light reflection at the pupillary border signifiesa 15 ◦ deviation (see Fig. 3.8), at the mid-iris a deviation of 30 ◦ and at the limbus a deviation of45 ◦. The patient shown in Fig. 3.8 has a left esotropia, thus, the corneal light reflex of the lefteye is displaced temporally with respect to the pupillary border of the left eye, while the reflexis centered in the pupil of the right eye.


Figure 3.8: Hirschberg Light Reflex Test Method, from [Kan87]

The test should be performed with the light centered in each eye’s pupil to detect the presenceof secondary deviations. The disadvantages of this test are the estimations necessary to measurethe eye deviation and the inability to control accommodation when testing at near fixation, asthe measuring light serves as the fixation target. Distance testing is difficult because the dimnessof the target light is reflected in the corneas.

The Krimsky test quantifies the light reflex displacement using appropriately held prisms. Theoriginal description suggested to place the prism before the aligned, fixating eye (see Fig. 3.9),but most users today find it easier to hold the prism before the deviating eye. The strength ofa base-out prism over the fixing right eye to center the pupillary light reflex in the esotropic lefteye is defined as the amount of left esotropia (see Fig. 3.9).

Figure 3.9: Krimsky Light Reflex Test Method, from [Kan87]

Prisms must be appropriately handled to yield accurate measurement of strabismus. They deflectlight toward their base, but the patient views the light as deflected toward the prism apex. Theprism diopter is defined as the strength of prism necessary to deflect a light beam 1cm at 1mdistance. Glass prisms are calibrated when positioned with the back surface perpendicular tothe visual axes. Plastic prisms, whether loose or in bar form must be held with the rear surfacein the frontal plane, to approximate closely the position of minimal deviation of light throughthe prism. Prisms cannot be stacked base to base as the sum prism strength is much greaterthan the sum of each individual prism strength, but they may be stacked with bases 90 ◦ apart.


Large deviations are best neutralized when the prisms are divided between the two eyes. Formeasurements when patients view in eccentric gaze positions and for those in the head tilt test,care is required to ensure the prisms are held in the frontal plane.

3.4.2 Cover Tests

These objective tests detect and measure horizontal and vertical strabismus, but they cannotmeasure torsional deviations and detect only some and not all torsional deviations. All covertests demand the ability of each eye to look at a fixation target at near and distance, and tomove to take up fixation upon that target.

Figure 3.10: Prism-Cover Test, from [Kan87]

The monocular cover test detects constant visual axis deviations. The examiner observes theuncovered eye for movement as its fellow eye is covered with a paddle, the hand or the thumb. Anasal movement implies exotropia, temporal movement esotropia, upward movement hypotropia,and downward movement hypertropia of the uncovered eye (see Fig. 3.11). Each eye is coveredin turn. An accommodation-controlling fixation target is presented to the patient, who ideallydescribes the target. Small toys are suitable for young children, but bright white lights are toobe avoided as the patient cannot accommodate on the contours of a light. Tropias establishedby the cover test may be measured using the simultaneous prism and cover test. A prism ofappropriate strength held in the appropriate direction is introduced before one eye as its fellowis covered (see Fig. 3.10). Prism strength is increased until eye movement ceases and the prismstrength corresponds to the size of the strabismus. The test is then repeated with the prismbefore the other eye.

The uncover test requires observation of the covered eye as the cover is removed. If that eyedeviated under cover it may regain fixation or may remain deviated. The former implies the


presence of a phoria, a latent deviation held in check by sensory fusion, or an intermittent tropia;the latter implies a tropia with fixation preference for the fellow eye.

Figure 3.11: Cover Tests for Tropias and Phorias, from [DE93].(a) For exotropia, covering the right eye drives inward movement of the left eye to take up fixation; uncovering the right eye shows recoveryof fixation by the right eye and leftward movement of both eyes; covering the left eye discloses no shift of the preferred right eye. (b) Foresotropia, covering the right eye drives outward movement of the left eye to take up fixation; uncovering the right eye shows recovery of fixationby the right eye and rightward movement of both eyes; covering the left eye discloses no shift of the preferred right eye. (c) For hypertropia,covering the right eye drives downward movement of the left eye to take up fixation; uncovering the right eye shows recovery of fixation bythe right eye and upward movement of both eyes; covering the left eye shows no shift of the preferred right eye. (d) For exophoria, the lefteye deviates outward behind a cover and returns to primary position when the cover is removed. An immediate inward movement denotes aphoria, a delayed inward movement denotes an intermittent exotropia.

Phorias may be detected more directly using the alternate cover test, in which each eye is occludedalternately to dissociate the visual axes maximally. Care must be taken to permit time for eacheye to reside behind the cover (the cover must not be „fanned“ before the eyes). Appropriatelyheld prisms enable quantification of the phoria (see Fig. 3.10). Some patients have poorly definedend points and a range over which eye movements shift from one direction to the opposite asprism strength is increased, thus, the strabismus measurement may be estimated as the midpointbetween clearly defined movements in each direction.

If the cause of strabismus is paralytic or restrictive, patients may have greater cover test mea-surements when the paretic or restricted eye fixes in a given gaze position (secondary deviation)than when the unaffected eye fixates (primary deviation). This phenomenon arises from Hering’sLaw (see Sec. 2.2.3.7), which demands equal innervation to yoke muscles, thus, the yoke of aparalyzed or restricted muscle receives excess innervation when the pathologic eye is fixing.

Strabismus should be detected and measured in primary position at distance and near fixation,and in gaze up, down, right, and left 30 ◦ from primary position. These nine diagnostic gazepositions include the above plus up and right, up and left, down and right, and down and leftand are useful to measure cyclovertical muscle palsies. For patients who have oblique muscledysfunction, measurements are taken with the head tilted right and left at distance fixation.

3.4.3 Subjective Clinical Tests

Subjective clinical methods include diplopia tests and haploscopic tests, which require cooper-ation, intelligence, and the ability of the patient to communicate the sensory percept to the


examiner.

Figure 3.12: Maddox-Wing Test, from [YD03]

The Maddox-Rod test dissociates both eyes for near and distance fixation and measures het-erophoria (see Fig. 3.12). A red Maddox rod in a trial frame may be used to evaluate subjectiveocular torsion. The grooves must be aligned with the mark on the rim, as they tend to rotatewithin. The red glass requires the patient to alert the examiner when the red light viewed behinda red filter before the right eye and a white light viewed with the left eye are superimposed ordisplaced one from the other. Fusion is disrupted by the red glass and thus horizontal and verticalphorias are uncovered and measured. The gaze position of maximal image separation is a clueto the identity of paretic or restricted muscles. This is a useful bedside test, but accommodationis not controlled.

The Maddox rod consists of closely aligned, powerful glass or plastic cylinders. When illuminated,these cylinders project a line upon the patient’s retina perpendicular to the groove orientation.The line is aligned horizontally to detect and measure horizontal phorias (accommodation cannotbe controlled with this test). Torsion may be detected and quantified using the Maddox rod whichis placed in a trial frame scaled in degrees. It is common to place two Maddox rods of differingcolor in each trial frame cell and permit the patient to rotate each to his or her perception of thehorizontal. The torsional position of each eye may be read directly in degrees from the angularscale used for cylinder axes.

3.4.4 Hess-Lancaster Test

The Hess-Lancaster test is a test for binocular alignment using separated images for both eyes.Generally, there are many similar testing methods (e.g. Less-Screen, Helmholtz-Test, etc.) thatuse the same testing principle as the Hess-Lancaster test. The also exists the so called Hess test,which uses a different coordinate system for measurement, but the basic interpretation of theresults is identical.

During the clinical Hess-Lancaster test the following steps are carried out:

1. The patient wears red-green-glasses with the red filter in front of the e.g. right eye (fixingeye) initially (see Fig. 3.13).

2. The patient gets a green light pointer, the examiner gets a red light pointer.


3. The examiner projects a red light spot onto the so-called Hess screen and asks the patient tobring the green light spot (following eye) over the red light spot. Under normal conditionsboth light spots overlay in all nine main gaze directions (see Fig. 3.13(a)).

4. Now the red filter is put in front of the other eye and the previous steps are repeated withthe other eye, which is now the fixing eye.

If the patient fixates with the „normal“ unaffected left eye, the fixation can be reached withnormal innervation. However, if the right medial rectus muscle is palsied, the patient’s greenlight pointer will point into a direction, which does not correspond to the real location of thefixation point in space (see Fig. 3.13(c)). After the test is finished, the relative positions areconnected with straight lines.

(a) (b) (c)

Figure 3.13: Binocular Fixation in the Hess-Lancaster Test, from [Kan87]

If e.g. the patient suffers from a palsy of the lateral rectus muscle on the right eye and alsofixates with the right eye (red filter), the „normal“ left medial rectus muscle receives an excessiveinnervation (Hering’s law). As a result, the patient’s green light pointer will point to a position,which lies far beyond the correct one (see Fig. 3.13(b)). The results of the Hess-Lancaster testare in general two diagrams (left eye and right eye fixing) with the corresponding gaze positions,which in turn show the deviation and the squint angle (see Fig. 3.14).

In the Hess-Lancaster diagram from Fig. 3.14, the blue points represent those gaze positions,which the patient should fixate (intended gaze positions) and the red points represent those gazepositions, which the patient was able to reach with the following eye. The difference betweenthe blue and the red points shows the respective deviation of binocular coordination. At thesame time next to each red point (following gaze position) the torsion of the following eye isshown (as text) in order to get the torsional position of the following eye. The following list givesan overview of the interpretation of Hess or Hess-Lancaster diagrams for a right medial rectusmuscle palsy:


Figure 3.14: Hess-Lancaster Diagram for Right Eye (Left Eye Fixing)

1. The two diagrams from Fig. 3.15 (left eyeand right eye fixing) are compared.

2. The smaller diagram shows the eye withthe palsied muscle.

3. The larger diagram corresponds to theeye with the overacting muscle.

4. The smaller diagram shows the biggestrestriction in the main functional direc-tion of the palsied muscle (in Fig. 3.15(b),the right medial rectus muscle).

5. The larger diagram shows the biggest ex-pansion in the main functional directionof the synergistic muscle (in Fig. 3.15(a),the left lateral rectus muscle).

(a)

(b)

Figure 3.15: Interpretation of Hess-Diagramaccording to Muscle Actions, from [Kan87]


Figure 3.16: Hess-DiagramInterpretation, from[Kan87]

Changes in the Hess diagrams are a prognostic help. A palsy ofthe right superior rectus muscle for example will show a restric-tion of the affected muscle and an overfunction of the synergist(left inferior oblique muscle) (see Fig. 3.16(a)). As a result of thissignificant incomitant reaction shown in both diagrams, the diag-nosis can be made directly out of the diagrams. However, when thepalsied muscle has recovered, both diagrams show approximatelynormal values again. If the palsy is not corrected, the shapes ofboth diagrams change and a secondary contractor of the ipsilat-eral antagonist (right inferior rectus muscle) develops, which canbe seen as an overfunction in the diagram. This in turn can leadto a secondary (inhibition) palsy of the left superior oblique mus-cle, which is represented in the diagram through reduced activity(see Fig. 3.16(b)) and which gives the false impression that theleft superior oblique muscle is the real cause for the pathologicalsituation. By-and-by the two diagrams get even more concomitantup to the point were it is impossible to determine which musclewas the primary palsied muscle (see Fig. 3.16(c)).

3.5 Eye Motility Disorders

Pathological anomalies in ocular motility are mainly differentiated in two fundamental types.The reasons for motility disorders can therefore be a failure in the development of the fixationreflexes or a disruption of fixation reflexes from structural or functional causes. Each of thesecategoric disorders will produce different symptoms and therefore demand also different therapy.

Anomalies in the conjugate fixation reflex make it impossible to fixate objects, since both eyescannot move in binocular community. This will result in heterophoria (latent strabismus) whenthe fixation can be established with stress, and in concomitant, manifest strabismus when fixa-tion is not possible at all. Similarly, the near fixation reflex may be disrupted, resulting in ananomaly in convergence. Some structural or neuro-muscular lesion may prohibit the develop-ment of adequate movements, already from birth on, so that congenital incomitant strabismus isdeveloped. On the other hand, if a lesion occurs in the peripheral (infranuclear) neuro-muscularmechanism after the reflexes have been established, an acquired incomitant strabismus, resolvablein terms of individual eye movements results due to some pathological accident. A lesion in thecentral (supranuclear) mechanism results in binocular deviation, resolvable in terms of associatedbinocular eye movements.

Thus, a supranuclear lesion indicates disorders of movement caused by the destruction or func-tional impairment of brain structures above the level of the motor neurons, such as the motorcortex (see Sec. 2.2.3). Furthermore, infranuclear lesions refer to disruptions below a nucleus ofa nerve (e.g. eye muscle palsies), whereas intranuclear lesions affect connections between otheroculomotor nuclei (palsies that affect interneuronal connections, e.g. the medial longitudinalfasciculus, see Sec. 2.2.3.5).

The fundamental fact about the development of pathological dissociation is, that if the normal


fixation reflexes are not attained and to some extent fixed by usage in the early months and yearsof life and certainly before the age of five years, while the nervous system is still flexible enoughfor adaptations, these reflexes will never develop.

3.5.1 Concomitant Strabismus

Concomitant strabismus can be classified as congenital, sometimes also acquired, functional eyemotility disorder, caused by common predisposition that lead to a pathological development of eyemovement. Functional eye motility disorders are influenced by many different factors, wherebythree groups of factors can be distinguished and are particularly important [DE73]:

Static anatomical conditions - that produce a faulty position of the eyes relative to eachother,

excessive or abnormal innervations - particularly important in a disharmony of the rela-tionship between accommodation and convergence, and

deficient development of the ability to fuse - due to a unilateral sensory failure or a lackof central organization.

It is important to notice that it is extremely rare that strabismus is caused by one of these factorsalone. However, generally, all these factors depend on some obstruction to the development ofthe binocular fixation reflexes which coordinate eye movements.

Generally, the following reasons can be responsible for the development of concomitant strabis-mus:

Hyperopia (long sightedness) - is a very common cause for Esophoria, since a young childmust accommodate in order to get a sharp image, when fixating far objects. Unfortu-nately, accommodation also implies convergence of both eyes and can subsequently lead tostrabismus, when not cured within certain time.

Fusional dysfunction - that can be congenital or acquired (e.g. caused by scarlet fever ormeasles). Additionally, concussion or accidents can cause acquired concomitant strabismus.

Unbalanced refraction in both eyes - can hinder fusion due to differently sized retinal im-ages that occur when fixating objects.

Developmental disorders of eye muscles - can additionally contribute to mechanicallycaused concomitant strabismus.

In the great majority of the cases, good fusional reflexes can compensate for a mechanical mis-alignment, and the result is latent strabismus. The common occurrence of divergent strabismus,wherein, except in most young children, the eyes revert to their divergent resting position afterthe development of impaired vision or blindness in one eye. Generally, children tend to convergentresting positions, whereas adults tend to divergent resting positions of the eyes. This illustratesthe tendency of anatomical conditions to make their influence felt in the absence of fusion.


Innervational, particularly accommodational influences similarly tend to the development of con-comitant strabismus, but permanent manifest strabismus does not occur if the binocular fixationreflexes have not yet been developed and the fusional vergence is good. If the fusional ability istoo weak, concomitant strabismus develops in childhood, before fusion has become stabilized.

A deficiency of fusion can also be responsible for the development of concomitant strabismus, espe-cially when complex neural control mechanisms are labile or unstable. Exhaustion or excitabilityand an unstable neuropathic constitution (i.e. a sudden shock or a psychological trauma) leadsto the disruption of the visual system, and strabismus occurs.

Heredity is most frequently an evidence for concomitant strabismus, as relatives of a family inthe same generation show a higher incidence. However, concomitant strabismus does not takethe same form throughout a family, instead it is varying considerably in being uniocular oralternating. Concerning the incidence and type of amblyopia, variations have even been seen inmonozygotic twins.

3.5.2 Incomitant Strabismus

Incomitant Strabismus can be seen as a dissociation of the ocular movements, wherein the devi-ation is irregular, varying in an uncoordinated manner, in different directions of gaze. This typeof strabismus results of defects in the final motor path of the binocular reflexes. The essential dif-ference between concomitant and incomitant strabismus is, that in concomitant strabismus, thecentral organization of binocular vision functions inadequately or not at all, whereas in incomi-tant strabismus, the brain is usually able to command movements, but the motor apparatus isrestricted in carrying out these instructions with adequacy. However, this distinction is variable,since in many forms of incomitant strabismus in early life, disruptions of the sensory stimuli areestablished, similar to those typically occurring in concomitant strabismus. Conversely, certainforms of concomitant strabismus (e.g. accommodative strabismus or hyperopia) do not show anydisruptions of the sensory stimuli, despite the failure to maintain binocular single vision on nearfixation. In incomitant strabismus, lesions occur in the lower motor neuron level (the nuclei,nerves or muscles), and consequently deviations are not resolvable in terms of eye movements,but in terms of individual muscles. Herein also lies the distinction from conjugate deviations dueto lesions above the motor neuron level, affecting the supranuclear mechanism which concernsitself with the control of movements and does not take the contraction of individual muscles intoaccount. Concomitant strabismus, therefore, gives relatively constant deviations in all directionsof gaze, whereas incomitant strabismus may become evident only when the eye is turned intothe field of action of the affected muscle where deviation always grows proportional to the gazeposition.

Incomitant strabismus may be either paretic or spastic in nature, but most cases show clinicallya simple under- or over-action of a muscle, although this is mostly associated with changes inactivity of other muscles, so that the resulting deviation may be very complicated and mayeventually rapidly assume entirely different characteristics.

One way to simplify diagnostics is to divide the symptoms into four distinctive categories [DE73]:

Overaction of the ipsilateral antagonist which, if maintained for longer time, may result in


permanent contractual changes.

Overaction of the contralateral synergist - which is due to the demand of equal innervation(Hering’s law, see Sec. 2.2.3.7) which requires an equal distribution of innervational impulsesbetween both eyes. The overaction of the contralateral synergist (yoke muscle) becomesapparent when an attempt is made to move the paretic muscle in the other eye, and mayalso result in contractual changes.

Underaction of the antagonist of the contralateral synergist - becomes noticeablethrough Sherrington’s law (see Sec. 2.2.3.8) of reciprocal innervation. This is due to ancompensatory overaction of the contralateral synergist and results in an inhibitory under-action of its antagonist, predetermined by the law of reciprocal innervation.

Overaction of the ipsilateral synergist(s) - describes a compensatory reaction that is car-ried out in an attempt to accomplish the deficient movement. This type of overactionis seldom noticeable, since ipsilateral synergists always have very limited compensatoryactions with respect to an affected muscle.

Figure 3.17: Example of Disturbance of the Binocular Muscular Team

In Fig. 3.17, one example of a disturbance of the binocular muscular team is illustrated. A palsyof the right lateral rectus muscle results in an overaction of the right medial rectus (the ipsilateralantagonist). Additionally, this pathological situation produces a compensatory overaction of theleft medial rectus muscle (the contralateral synergist) and an inhibitory underaction of the leftlateral rectus muscle (the ipsilateral antagonist of the contralateral synergist).

If the paretic eye is habitually used for fixation, the secondary deviation (deviation when fixingwith a pathologic eye) produced by the contralateral synergist becomes accustomed. Conversely,if the non-paretic eye is constantly used for fixation, the primary deviation (deviation when fixing


with a normal, healthy eye) will function against the recovery of the affected, palsied muscle ofthe paretic eye, by facilitating the overaction of the ipsilateral antagonist.

It is also possible that a pathological situation (e.g. described in Fig. 3.17) disappears so that theoriginally paretic strabismus changes its characteristics and subsequently shows a purely spasticdeviation. However, if the muscle palsy disappears, it leaves permanent changes of the contractureor stretching in the various muscles involved. Thus, the original incomitant strabismus maybecome virtually concomitant.

3.5.2.1 Paralytic Strabismus

Paralytic strabismus means, that one of the muscles attached to the globe is paralyzed and the eyeaffected may turn in, out, up or down depending on the muscle involved i.e. the eye movement isrestricted in the direction of the action of the paralyzed muscle. The reasons for paralytic squintcan originate from a breakdown of motor control before or after the binocular fixation reflexeshave been developed. It may also be caused by certain nerve palsies, which in turn may be causedby peripheral diseases of the cranial nerves (e.g. meningitis, encephalitis etc.). Clinical progressand treatment of paretic strabismus therefore differs considerably with respect to its differentcauses. The following objective signs of paralytic strabismus can be mentioned:

• Abnormal deviation of the eyes,

• deviation of movements, and

• the adoption of compensating postural attitudes (abnormal head posture).

A common symptom for paralytic strabismus is binocular diplopia (double images) that disappearwhen either eye is closed. This is caused by a misalignment of the eyes, which can be secondaryto nerve or muscle related disorders. Diplopia can also cause a reduction of reading, driving, andvocational skills.

Figure 3.18: Example of Abnormal Head Posture compensating Esotropia, from [Kau95]


Especially abnormal head postures are a noticeable sign for paralytic strabismus. Patients withearly manifest strabismus show abnormal head posture in that the head is tilted to the shoulderso that fixation in excessive adduction of the leading eye can be accomplished at the same time.The tilt of the head occurs mostly in the direction where the leading eye is situated (see Fig. 3.18).The head is turned towards the field of action of the paralyzed muscle, whereas in paralysis ofany of the recti muscles the chin is elevated in paralysis of the elevators of the eye - superior rectiand inferior obliques. It is depressed in paralysis of the depressors of the eye - inferior recti andsuperior oblique. The head is tilted towards the normal side in paralysis of the superior oblique.It is tilted towards the side of the paralyzed muscle in paralysis of the superior and inferior recti,and the inferior oblique muscle.

In the following, some examples of individual muscle palsies are given along with the respectiveclinical signs. In contrast to neurogenic palsies that occur due to lesions in the neural oculomotorpathway, muscle palsies are restricted to limitations of the function of specific muscles.

Palsy of the Right Superior Rectus Muscle

In primary position, the right superior rectus muscle is mainly an elevator so that the primarydeviation of the affected eye is downwards (see Fig. 3.19(e)), due to an overaction of the rightinferior rectus muscle (the ipsilateral antagonist). The secondary deviation of the unaffected eyeduring fixation with the affected eye is upwards (see Fig. 3.19(f)), due to overaction of the leftinferior oblique and the left superior rectus (the contralateral synergists). Usually, the secondarydeviation is greater than the primary, which complies to the definition of incomitant deviation.However, if contraction develops in the ipsilateral antagonist (the right inferior rectus muscle), theprimary deviation increases to become approximately equal to the secondary deviation, assumingconcomitant features. Since the right superior rectus muscle is, in its secondary and tertiaryfunctions also adductor and intorter, the loss of the adducting influence is of little significance aslong as the right medial rectus is intact, but there is noticeable extorsion due to an overaction ofthe right inferior rectus and the right inferior oblique muscles.

(a) (b) (c)

(d)

(e) Primary Deviation

(f) Secondary Deviation

(g)

(h) (i) (j)

Figure 3.19: Example of a Right Superior Rectus Palsy


(a) Primary Deviation, Left Eye Fixing (b) Secondary Deviation, Right Eye Fix-ing

Figure 3.20: Hess-Lancaster Chart for Right Superior Rectus Palsy

Ocular movements show limitations of the palsied eye in an upward and outward direction (seeFig. 3.19(a)) so that there is an increase of deviation when looking up and to the right using theunaffected eye. Using the right eye to fixate, the secondary deviation results in an upshoot ofthe unaffected left eye, due to an overaction the left inferior oblique (the contralateral synergist)which is also apparent in right lateral gaze (see Fig. 3.19(d)). A downshoot of the affected righteye occurs due to an overaction of the right inferior rectus muscle (the ipsilateral antagonist),or otherwise, a defective downward movement of the unaffected left eye, when fixating with theright eye, due to a weakness of the left superior oblique (the antagonist of the contralateralsynergist) which is evident particularly on looking down and to the right (see Fig. 3.19(h)). Thedeviation is also evident on upward movement from primary position (see Fig. 3.19(b)), but thereis seldom any defect on downward movement (see Fig. 3.19(i)). The vertical ocular movementsare relatively normal in abduction since the right inferior oblique compensates for the weak rightsuperior rectus muscle in these positions (see Fig. 3.19(c), Fig. 3.19(g)), although there may be aslight overaction of the ipsilateral synergist when looking to the left and down (see Fig. 3.19(j)).There is an increase in extorsion in the affected eye during right gaze, due to overaction of theright inferior oblique, also in left gaze, due to an overaction of the right inferior rectus muscle.

The compensatory (abnormal) head posture in a right superior rectus palsy is usually a rotationof the head into the field of action of the affected muscle. In this case, the head would be turnedup and to the right, and also tilted to the side of the affected eye.

The Hess-charts in Fig. 3.20 show a shrinkage, away from the direction of action of the rightsuperior rectus muscle with enlargement towards the direction of the right inferior rectus andthe left inferior oblique accompanied by a slight shrinkage away from the action of the leftsuperior oblique. When fixating with the left, unaffected eye, the resulting primary deviationin Fig. 3.20(a) shows the overaction of the ipsilateral antagonist in the right eye through a


downward movement of the diagram towards the direction of action of the right inferior rectusmuscle. When fixating with the affected, right eye, the secondary deviation in Fig. 3.20(b) showsthe overaction of the left superior rectus muscle, which results in a movement of the diagramtowards the direction of action of the left superior rectus muscle.

Palsy of the Right Superior Oblique Muscle

In primary position, the right superior oblique muscle is primarily a depressor, so that the primarydeviation of the affected right eye in this particular situation is an upward movement, due toan overaction of the right inferior oblique muscle (see Fig. 3.21(e)). The secondary deviationof the unaffected eye during fixation with the affected right eye is downward movement due tooveraction of the left inferior rectus muscle (see Fig. 3.21(f)).

(a) (b) (c)

(d)

(e) Primary Deviation

(f) Secondary Deviation

(g)

(h) (i) (j)

Figure 3.21: Example of a Right Superior Oblique Palsy

Since the right superior oblique muscle is also an abductor and intorter, the loss of the abductinginfluence is of little significance, as long as there is an intact right lateral rectus muscle, but thereis some extorsion due to overactions of the right inferior oblique and the right inferior rectusmuscles, which can easily be seen in Fig. 3.21(f) and Fig. 3.21(e) when looking at the dark (blue)cross in the pupil.

Ocular movements show limitations of the palsied eye when looking downwards and inwards,since this is the direction of action of the affected muscle (see Fig. 3.21(j)). This situation is ac-centuated by an upshoot of the affected right eye, due to an overaction of the right inferior obliquemuscle, which is evident in all positions of left gaze (Fig. 3.21(c), Fig. 3.21(g) and Fig. 3.21(j)).Particularly, when looking up and to the left, a downshoot of the unaffected eye, due to over-action of the left inferior rectus muscle can be noticed. Restricted downward movement of theaffected right eye is also evident on downward movement (see Fig. 3.21(i)), and to some extentin upward movement from primary position (see Fig. 3.21(b)). The ocular movements are rathernormal on looking to the palsied side (see Fig. 3.21(a), Fig. 3.21(d) and Fig. 3.21(h)).

The compensatory (abnormal) head posture is usually a rotation of the face to the left and


(a) Primary Deviation, Left Eye Fixing (b) Secondary Deviation, Right Eye Fix-ing

Figure 3.22: Hess-Lancaster Chart for Right Superior Oblique Palsy

downwards, accompanied by a tilting of the head to the left. Both, the turning of the face, aswell as the tilting of the head are oriented in the direction of the unaffected eye.

The Hess-Lancaster charts shown in Fig. 3.22 show a shrinkage away from the direction of actionof the right superior oblique muscle with a slight enlargement towards the direction of action ofthe right inferior oblique (see Fig. 3.22(a)), and an enlargement in the direction of action of theleft inferior rectus muscle with a shrinkage away from the direction of action of the left superiorrectus muscle (see Fig. 3.22(b)).

3.5.2.2 Duane’s Syndrome

Duane’s syndrome is a congenital ocular motility disorder characterized by limited abductionand/or limited adduction. The palpebral fissure narrows (the globe retracts) on attempted ad-duction. Upward or downward deviation may occur with attempted adduction due to a leasheffect. Often associated with this condition is a „tether“ phenomenon consisting of overelevation,overdepression, or both, in adduction as the retracted globe escapes from its horizontal rectusrestrictions. It is a condition of aberrant innervation that results in co-contraction of the medialand lateral recti in the affected eye. Thus, Duane’s syndrome can be considered to be a congen-ital miswiring of the medial and the lateral rectus muscles such that globe retraction occurs onadduction.

Duane’s syndrome is often clinically subdivided into three types (1-3). Different clinical typesmay be present within the same family suggesting that the same genetic defect may produce arange of clinical presentations.

Duane’s syndrome type 1: The ability to move the affected eye outward towards the ear(abduction) is limited, but the ability to move the affected eye inward towards the nose


(a) Duane’s Retraction Syndrome Type 3 (b) Hess-Lancaster Chart for Duane’s Retraction Syn-drome

Figure 3.23: Example for Duane’s Retraction Syndrome Type 3 of a Right Eye

(adduction) is normal or nearly so. The eye opening (palpebral fissure) narrows and theeyeball retracts into the orbit when looking inward towards the nose (adduction). Whenlooking outward towards the ear (abduction) the reverse occurs.

Duane’s syndrome type 2: The ability to move the affected eye inward towards the nose(adduction) is limited, whereas the ability to move the eye outward (abduction) is normalor only slightly limited. The eye opening (palpebral fissure) narrows and the eyeball retractsinto the globe when the affected eye attempts to look inward towards the nose (adduction).

Duane’s syndrome type 3: The ability to move the affected eye both inward towards thenose (adduction) and outward towards the ear (abduction) is limited. The eye opening(palpebral fissure) narrows and the eyeball retracts when the affected eye attempts to lookinward towards the nose (adduction).

Each of these three types can be further classified into three subgroups designated A, B, and Cto describe the eyes when looking straight (in primary position). In subgroup A the affected eyeis turned inward towards the nose (esotropia). In subgroup B the affected eye is turned outwardtowards the ear (exotropia), and in subgroup C the eyes are in a straight primary position.

In Fig. 3.23, Duane’s Retraction Syndrome Type 3 for a right eye is illustrated. Due to a co-contraction of the right medial and right lateral rectus muscles, the affected right eye will showretraction as indicated in Fig. 3.23(a) when looking to the right. The Hess-Lancaster chart forthe right eye in Fig. 3.23(b) shows the typical restriction in ab/adduction of the right eye, dueto co-innervation of both recti muscles. This results in a shrinkage of the diagram for both sides,away from the fields of action of the right lateral rectus and the right medial rectus muscles.


Brown’s Syndrome

This ocular motility disorder, characterized by an inability to elevate the adducted eye actively orpassively, was first described by Brown. It has since become recognized that there is a variety ofcauses, that the condition may be congenital or acquired, and that the defect can be permanent,transient, or intermittent.

Brown’s syndrome is characterized by a deficiency of elevation in the adducting position. Im-proved elevation is usually apparent in the midline, with normal or near-normal elevation inabduction (see Fig. 3.24). There is occasional widening of the palpebral fissure on attemptedelevation in adduction. With lateral gaze in the opposite direction, the affected eye may depressin adduction, although no overdepression simulating overaction of the superior oblique muscleoccurs on duction testing. Exodeviation (V pattern) often occurs as the eyes are moved upwardin the midline. Many patients are orthophoric and experience diplopia in the primary position,although with time hypotropia may develop with a compensatory head posture turn towards theopposite eye. In some cases, there is discomfort on attempted elevation in adduction, the patientmay feel or even hear a click under the same circumstances, and there may be a palpable massor tenderness in the trochlear region. A positive forced duction test is the hallmark of Brown’ssyndrome.

(a) Primary Deviation, Left Eye fixing (b) Secondary Deviation, Right Eye Fix-ing

Figure 3.24: Example for Brown’s Syndrome of a Right Eye

From the Hess-Lancaster charts of Brown’s syndrome in Fig. 3.24, the characteristic V-Patternof the primary and secondary deviations can be recognized. The Hess chart for the primarydeviations (fixing with the unaffected eye) in Fig. 3.24(a) shows a deficiency in elevation whenthe right eye is in adduction and near normal elevation in abduction. When fixing with theaffected eye (see Fig. 3.24(b)), improved elevation of the unaffected eye is evident when theaffected right eye looks in adduction.

The anatomical cause of the syndrome is a tight superior oblique tendon. Acquired Brown


syndrome has been attributed to a variety of causes, including superior oblique surgery, scleralbuckling bands, trauma, focal metastasis to the superior oblique, and following sinus surgeryand inflammation in the trochlear region. An identical motility pattern, as seen in Brown’ssyndrome, can be acquired by patients with juvenile or adult rheumatoid arthritis. It appearsthat this form of Brown’s syndrome shares similar characteristics to inflammatory disorders thataffect the tendons of the fingers.

3.5.2.3 Fibrosis Syndrome

This syndrome is characterized by replacement of normal muscle tissue by fibrous tissue in varyingdegrees. The various clinical presentations depend on the number of muscles affected, the degreeof fibrosis, and whether the involvement is unilateral or bilateral. The condition is congenital,with males and females equally affected.

Fibrosis Syndrome is characterized encompassing

• fibrosis of the extraocular muscles,

• fibrosis of Tenon’s capsule,

• adhesions between muscles, Tenon’s capsule, and globe,

• inelasticity and fragility of the conjunctiva,

• absence of elevation or depression of the eyes,

• little or no horizontal movement,

• eyes fixed 20 to 30 degrees below the horizontal,

• chin elevation and

• the condition being present at birth.

Congenital fibrosis of the inferior rectus is probably a variant of the general fibrosis syndrome.The inferior rectus alone or together with a levator may be involved, with little or no involvementof the other extraocular muscles. The condition may be unilateral or bilateral and is commonlyasymmetric. Because patients cannot typically elevate their eyes even to the midline, they adapta compensating head posture with their chin up to maintain binocular vision.

Another variant of the general fibrosis syndrome is strabismus fixus, in which the eyes are in amarkedly fixed position of esotropia or exotropia. The eyes are so firmly fixed that they cannotbe actively or passively moved in a horizontal direction, although vertical movement is usuallypossible.

Another possible variation of the general fibrosis syndrome is the vertical retraction syndrome.In this condition, horizontal movements are normal, but elevation and depression are reduced,while the eye is abducted. In addition to the vertical limitation, there is retraction of each eyeduring attempted depression, with the eye in the abducted position.


Congenital fibrosis of the extraocular muscles (CFEOM) describes a group of rare congenital(present at birth) eye movement disorders that result from the dysfunction of all or part of theoculomotor nerve (cranial nerve III) and/or the muscles this cranial nerve innervates. Patientsaffected with CFEOM are typically born with ophthalmoplegia (an inability to move the eyesin certain directions) and ptosis (droopy eyelids). In addition, the eyes are usually fixed in anabnormal position.

3.5.2.4 Supranuclear Disorders

A supranuclear gaze palsy is an inability to look in a particular direction as a result of cerebralimpairment. There is a loss of the voluntary aspect of eye movements, but, as the brainstem isstill intact, all the reflex conjugate eye movements are normal.

The type of gaze problem is dependent upon the lesion - thus a right hemisphere lesion, particu-larly the frontal lobes, leads to a contralateral gaze palsy, i.e. an inability to look away from thelesion.

As an example for a supranuclear gaze palsy, the abducens gaze palsy will be described in moredetail.

Abducens Gaze Palsy

Cranial nerve VI, also known as the abducens nerve, innervates the ipsilateral lateral rectus (LR),which functions to abduct the ipsilateral eye.

Patients usually present with horizontal diplopia and an esotropia in primary gaze. The deviation,as would be expected, is noted to be greater when the patient fixates with the paretic eye. Patientsalso may present with an abnormal head posture to maintain binocularity and binocular fusionand to minimize diplopia.

It is rare to find true congenital sixth nerve palsy. A typical workup of a sixth nerve palsyinvolves excluding paresis of other cranial nerves (including VII and VIII), a check of ocularmuscle motility and evaluating pupillary responsiveness. Only the ipsilateral lateral rectus thatis solely innervated by the involved peripheral sixth cranial nerve is affected, therefore, onlydeviations in the horizontal plane are produced. In isolated cases of peripheral nerve lesions, novertical or torsional deviations are present.

Central nervous system lesions of the abducens nerve tract are localized easily secondary to thetypical findings associated with each kind of lesion. Damage to the sixth nerve nucleus resultsin an ipsilateral gaze palsy. The lack of contralateral adduction defects (see Fig. 3.25) makes iteasy to differentiate nuclear from a fascicular or non-nuclear lesion (see Sec. 3.5.2.1).

Abducens palsy frequently is seen as a postviral syndrome in younger patients.


(a) Primary Deviation, Left Eye fixing (b) Secondary Deviation, Right Eye Fixing

Figure 3.25: Example for an Abducens Gaze Palsy

Nystagmus

The most primitive fault is a failure in fixation, due to anomalies in central vision, in whichcase innate attempts to fixate cause rapid jumps known as nystagmus. Nystagmus is generallydescribed as an involuntary movement of the eyes, which reduces vision. The movement is usuallyside to side (but can be up and down or circular motion) and can be either jerk or pendular.Normal (physiological) nystagmus occurs for example when a passenger of a train watches astelegraph poles pass the window. The eyes will travel one way, and then jump in the oppositedirection to begin watching the next pole. There are over 40 different types of nystagmus butthe main division is between congenital and acquired Nystagmus.

Congenital nystagmus - is thought to be present at birth, but is usually not apparent untilthe age of five months.

Acquired nystagmus - occurs later than 6 months of age, and can be caused by a stroke, diseasesuch as multiple sclerosis, or even a heavy strike to the head. Patients with nystagmus maysuffer from the perception of a moving world, known as oscillopsia. The number of peoplethat suffer from congenital nystagmus is much less than those with acquired nystagmus.This is thought to be because an infant can adapt to the perception of motion better thatan adult that previously had normal vision.

It is not clear whether congenital nystagmus is actually present at birth, or whether it occursearly in the child’s vision development. It is therefore also referred to as early-onset or infantilenystagmus. Many children with nystagmus have no other vision or brain problems. This is known


as idiopathic, which means of unknown cause. However, nystagmus is often a symptom of otherconditions such as albinism, aniridia, cataracts, cone dysfunction and many others. Nystagmuscan be present with cerebral palsy, Down’s Syndrome and many motor system diseases. A typeof congenital nystagmus is latent nystagmus, and is only present when one eye is covered. Thisis not usually noticed until the first visit to the physician.

3.6 Strabismus Surgery

The goal of surgery of the extraocular muscles is to balance an ocular misalignment in order torestore binocular alignment. Generally, it is also desirable to reestablish single binocular vision.Generally, the attainment of peripheral fusion with fusional vergence amplitudes sufficient tomaintain alignment of the eyes, comfortable single binocular vision to enable the patient toperform visual tasks without asthenopia and improved esthetic appearance are the most commonobjectives.

Ideal preoperative evaluation of the strabismus surgical patient includes quantification of themisalignment in primary positions at distance and at near, as well as in the nine diagnostic gazepositions (see Sec. 3.4). In most patients, the maximal deviation under conditions of completedissociation of the visual axes is the deviation for which surgery is to be designed. As indicated,measurements are taken both with and without the appropriate optical correction. Finally,duction and version testing and, when appropriate, forced duction testing may be performed.

All surgical methods are accomplished through the modification of the eye muscles since this is themost effective way to influence the mechanical properties of the oculomotor system. Basically,there are three different options to modify the mechanical functions of eye movements, whenconsidering extraocular muscle surgery:

• Weakening surgery (recession), in order to reduce the traction of a muscle,

• strengthening surgery (resection), in order to raise traction of a muscle,

• transposition surgery, in order to influence the pulling direction of a specific muscle.

Strabismus surgery demands careful planning. Barring unusual circumstances or unusualanatomy, the surgical plan is prepared before anesthetic is given. It is helpful to keep a de-scription of the surgical plan, for ready reference before and during surgery. The exact locationand incision technique depends upon the muscles to be operated, pre-existing scarring, and thepatient’s previous surgical history. Absorbable sutures generally are utilized when vascular heal-ing of tissues occurs, such as in typical recession and resection techniques. These sutures generallyabsorb within 7-10 days if covered with conjunctiva, slightly longer if exposed. Permanent suturesare utilized if avascular tissue such as the superior oblique tendon is harnessed, as in the superioroblique tuck or silicone band-lengthening procedure.


3.6.1 Recession Surgery

Recession surgery aims a transposition of the muscle insertion posterior along the muscles primarydirection of action. This surgical technique can be applied to all six extraocular muscles. In thisprocedure (see Fig. 3.26), the surgeon must detach the muscle from the eye and reattach it furtherback on the eye, thereby reducing the relative strength of the muscle.

Figure 3.26: Schematic Example of Muscle Recession, from [Kau95]

As an example for the weakening of a extraocular muscle, a medial rectus recession is describedin detail (cf. [YD03]).

To perform a medial rectus recession (Fig. 3.27), the surgeon grasps the eye at the conjunctiva-Tenon’s capsule junction with a 0.3mm forceps and rotates the eye into elevation and abduction(Fig. 3.27(a)). The surgeon then elevates the conjunctiva at the base of the fornix, and incises theconjunctiva 8mm from the limbus (Fig. 3.27(b)). At this point, all visible conjunctival vessels areprepared lightly to ensure good visibility during localization of the tendon. The surgeon graspsthe fascia within the conjunctival incision with gentle pressure against the sclera, and elevates itfrom the globe (Fig. 3.27(c)). The scissors are then used to incise Tenon’s capsule at this point,and expose the sclera (Fig. 3.27(d)). Visualization of the sclera is maintained using the posteriorforceps.

(a) (b) (c) (d) (e)

Figure 3.27: Preparation for Medial Rectus Recession, from [YD03]

A self-retaining muscle hook is then passed behind the medial rectus muscle with no posteriormovement of the hook further than the site of the incision itself (Fig. 3.27(e)). When the medialrectus is on the hook, the surgeon confirms that the entire tendon is engaged. The posteriorarm of the 0.3mm forceps may be used as a probe to locate the superior pole of the muscle(Fig. 3.28(a)). The pole is secured using the forceps, and the muscle hook is withdrawn to theinferior aspect of the tendon (Fig. 3.28(b)). Then, the hook is passed beyond the superior pole


held by the forceps (Fig. 3.28(c)). This way, it is certain that the full length of the tendon issecured completely.

A small hook is introduced between the insertion of the tendon and Tenon’s capsule anteriorand used to dissect carefully the fascia from the surface of the tendon, as it is moved posteriorlyalong its long axis (Fig. 3.28(d)). Afterwards, the scissors are used to incise the superior aspectof Tenon’s fascia (Fig. 3.28(e)).

(a) (b) (c) (d) (e)

Figure 3.28: Medial Rectus Recession, continued, from [YD03]

Gentle traction, away from the operated muscle, allows the surgeon to control the hook beneaththe tendon and accomplish suture passage at the insertion. The surgeon may then control theforceps to position optimally and to control the globe. A lock to the edge of the tendon byapplying a suture (Fig. 3.29(a)) is performed. A scissors is used for removal of the tendon(Fig. 3.29(b)). Visualization of the tendon may be preserved if a dry cotton pledget is passedbetween the tendon and globe.

The sutures are then drawn in the direction in which they were passed, to avoid breaking thesutures out of the sclera. The receded muscle is demonstrated where an attempt is made topreserve the original orientation of the tendon to the globe at its normal width (Fig. 3.29(c)).After the tendon has been secured to the globe, a hook is introduced above the superior pole ofthe earlier insertion and the fascia is rotated inferiorly (Fig. 3.29(d)) over the operative site. Theincision is closed with a single plain suture (Fig. 3.29(e)).

(a) (b) (c) (d) (e)

Figure 3.29: Medial Rectus Recession, continued, from [YD03]

The principles of recession for the other recti muscles are basically the same as described for themedial rectus. However, recession of the lateral rectus and vertical recti includes visualizationand preservation of the neighboring oblique muscles before the procedure is performed.


3.6.2 Resection Surgery

Resection of a muscle is performed in order to strengthen its force. In a muscle resection procedureshown in Fig. 3.30, the surgeon must detach the muscle from the eye (Fig. 3.30(a)), excise aportion of the distal end, and reattach the muscle to the eye on the same position (Fig. 3.30(b)).The shortening of the muscle provides greater pull in the field of action that the muscle functions.

(a) Resection of Detached Muscle (b) Reinsertion of Muscle

Figure 3.30: Schematic Example of Muscle Resection, from [Kau95]

The preparation procedure in resection surgery is related to the recession surgery in that theconjunctival vessels are prepared and Tenon’s fascia is entered. A large muscle hook is passedbeneath the tendon, going no further posteriorly than the insertion itself. The Tenon’s capsulebeneath the olive tip of the large muscle hook is incised (Fig. 3.31(a)) in order to visualize thesclera at both poles of the tendon. Two small hooks are passed along the long axis of the tendonso that the muscular fascia can be exposed for incision (Fig. 3.31(b)). A second large musclehook is passed beneath the tendon, and traction is applied between the two muscle hooks, withthe insertion and both hook tips kept parallel. The anterior arm of the caliper is placed on themidportion of the anterior hook, and the posterior portion delineates the site for needle passage(Fig. 3.31(c)). A marking pen may be used if it is not desirable to remeasure during suturepassage.

(a) (b) (c) (d) (e)

Figure 3.31: Medial Rectus Resection, from [YD03]

The same technique of suture passage is employed as described earlier for recession. The needle ispassed tangentially to the tendon and globe, and woven through the tendon from its midportionto the superior pole and then locked upon itself. The needle at the opposite end of the sutureis then passed from the midportion to the inferior pole and once again locked, and the tendon


secured at the point of desired resection (Fig. 3.31(d)). The tendon is cut ahead of the clamp(Fig. 3.31(e)), and the resection of the tendon is completed at the original insertion.

(a) (b) (c)

Figure 3.32: Medial Rectus Resection, continued, from [YD03]

Once the surgeon is satisfied with the location and depth of the scleral pass, the first suturethrow is placed before the muscle is drawn forward and knotted (Fig. 3.32(a) & Fig. 3.32(b)).The conjunctiva can be closed with two interrupted plain sutures (Fig. 3.32(c)).

3.6.3 Transposition Surgery

Transposition surgery has the goal to modify that direction of action of the extraocular musclesin order to correct anomalies of ocular alignment. The vertical transposition surgery of thehorizontal recti muscles is applied to correct symptomatic „A“- and „V“-patterns, that arise fromeso- and exotropia, in absence of significant overaction of the oblique muscles. The effects ofmuscle transposition are not completely predictable, and only a few of the many transpositionswith therapeutic potential are attempted by clinicians (cf. [MDR93]).

An „A“-symptomatic esotropia is treated with a bilateral recession of both horizontal medialrecti muscles in combination with an upward transposition.

An „A“-symptomatic exotropia is treated with a bilateral recession of both horizontal lateralrecti muscles in combination with a downward transposition.

A „V“-symptomatic esotropia is treated with a bilateral recession of both horizontal medialrecti muscles in combination with a downward transposition.

A „V“-symptomatic exotropia is treated with a bilateral recession of both horizontal medialrecti muscles in combination with an upward transposition.

In applying these combined surgical methods, careful preoperative planning is necessary, es-pecially the consideration of effects that modify lever-arm and unreel-strain of a muscle givesinformation about the possible results or complications that may arise and need to be avoided.

3.6.4 Amount of Surgery

The amount of surgery is the main criteria that is to be decided preoperatively on the basis ofdiagnosis and measurements. Since the oculomotor systems forms a relatively complex mechanical


system, these decisions are not trivial. However, it is now generally accepted, that the choiceand amount of surgery are matters of importance on which the end result closely depends (cf.[RSS01]). From the purely geometric point of view, a 1mm rectus muscle recession, combinedwith a 1mm advancement of the ipsilateral antagonist, should result in a 4.7 ◦ change of anglefor a globe of 24.5mm diameter, 5 ◦ for a globe of 23mm diameter, using the simple sphericalcircumference. In reality, however, the result is reduced by the passive forces of the other musclesand anatomical influences like pulleys and other movement retaining components. In single-muscle surgery, the effect is further reduced by the fact that the counter-tension of the antagonisticmuscle remains unchanged. The coefficient of „surgical effectiveness“, or the ratio between thetheoretical geometric effect and the actual result, is close to 1 only when the counter-tensionof the antagonistic muscle is adjusted by an equivalent amount in the opposite direction (interms of tension and not of distance). The effect also depends on the muscle that is operated onand additionally on factors like age, the manner in which surgery is carried out, the eye that isoperated on (affected or unaffected), the binocular status and type of the strabismus, concomitantor incomitant features and a lot of different other relations. Finally, there is no linear relationshipbetween the amount of surgery and the effect of a surgical procedure. All these factors have ledto a wide range of figures being proposed, giving little credibility to establishing the amount ofsurgery [RSS01].

„Although for some authors the precise amount of surgery is unimportant, or purelyindicative, this attitude is currently unacceptable [RSS01].“

From the analysis of their results, some authors have shown that, in spite of variables, it is possibleto measure the amount of surgery to be performed. Postoperative statistical analysis of resultstogether with known, and in particular mechanical parameters, have, at least for some, providedthe basis for a mean surgical effectiveness chart. Figures and formulae proposed for calculatingthe amount of surgery necessary to obtain a certain surgical effect are only valid from 2 to 7 or8 mm, and are only applicable from the age of 2

12 to 3 years. However, diagrams and formulae

provide average values, applicable only when patient data is in this certain field of tolerance.

Thus, the preoperative process of measurement, diagnosis and planning deals with a certainamount of tolerance that is intrinsic to every part of this process, since it is unlikely possible tocarry out a surgical procedure with an accuracy of e.g. 1 mm. Additionally, the terminology forpathologies in the field of strabismus is different in each language and also not clearly separable,since in some pathological cases, the exact cause is still unknown. Dealing with preoperativeplanning and the determination of the amount for eye muscle surgery is somehow the key factin this thesis, which proposes an entirely different way to describe ocular motility defects. Com-puter programs that realize and calculate results of formulae that have been based on statisticalresult are already used, but do not sufficiently solve the problem. Instead, this thesis proposesbiomechanical modelling of pathologies in terms of simulation parameters that are confined toan underlying model of geometry, muscle force and kinematics of the human eye. Essentiallysupported by an interactive and clinically applicable computer simulation program, eye motilitydisorders can be expressed in their impact on a „virtual“ patient that represents all measuredvalues from the real patient that should be treated.

Chapter 4

Biomechanical Modelling

Different strategies in modelling diagnosis and treatment of strabismus in the field of medicinehave been proposed over the last years (cf. [C7̈2], [Jam22], [SS00]). These models had the primarygoal to suggest or predict possible outcomes of a treatment procedure based on measurementsthat were acquired from patients. Since many authors proposed new standard values based uponclinical results, these values differ considerably. It is probable, that these authors did not take thesame amounts of surgery, or even comparable or representative types of strabismus into account.The problem becomes evident when considering natural variations in the anatomy and physiologyof different patients, which makes it nearly impossible to drive a model that is based on clinicalresults to exactly the same results, when predicting surgery for comparable patients. This alsoexpresses the non-linearity of the relationship between surgical dose and postoperative response.However, a model-driven approach, able to coordinate broad ranges of laboratory research andclinical experience, can accelerate progress of diagnosis and treatment of strabismus.

This chapter proposes a new approach of predicting surgical results for strabismus surgeries inthat the the human oculomotor system is described by a biomechanical model that incorpo-rates anatomically related parameters in order to enable the simulation of pathological situationsthrough the modification of these parameters. Biomechanics is a branch of mechanics and me-chanics again is a branch of physics. Therefore, biomechanics deals with the mechanical lawsand rules of biological structures and the interaction of these biological structures. Since suchmodels already existed before (cf. [MR84]), but were used rarely in clinical application, thisthesis proposes a different mathematical approach in combination with an interactive, easy touse software simulation system that enables interactive eye motility simulation and preoperativeplanning of surgery.

89

CHAPTER 4. BIOMECHANICAL MODELLING 90

4.1 Analytical Models

Analytical models are mathematical models that have a closed form solution, i.e. the solution tothe equations used to describe changes in a system can be expressed as a mathematical analyticfunction. Scientific practice involves the construction, validation and application of scientificmodels, thus, an analytical approach implies the invention of hypotheses and theories and thesubsequent effort of validation within the scientific model space. Reapplication of model resultsinto the real world may support or reject the proposed theories. Further more, models providean environment for interactive engagement. Evidence from science education research shows thatsignificant gains in knowledge and understanding are achieved within interactive activities. Thus,it is important that the environment created around a model provides an interactive experience inorder to be valuable for practical use. Working with models can enhance systems thinking abilitiesby allowing sensitivity studies to assess how changes in key system variables alter the system’sbehavior. Such sensitivity studies can help to identify leverage points of a system to either help toaffect a desired change with a minimum effort, or to help estimate the risks or benefits associatedwith proposed or accidental changes in a system. Concerning medical application, especially inthe field of strabismus surgery, there exist different model types that aim clinical improvementon the basis of analytical models.

Medical expert systems model the relationships between symptoms and diseases by usingexplicit domain knowledge that is embedded into a model. Such systems use inferencestrategies to apply knowledge to the available data (which is often noisy and incomplete)through heuristic reasoning. As with all expert systems, the design of medical diagnosticsystems raises essential questions:

• How to find a suitable representation of the observed data?• How to define a suitable representation of the knowledge of the medical domain the

system should work in?• How to obtain measurements that could validate a diagnostic hypothesis?• How to handle inaccuracy and uncertainty of the observed data?

Early systems adopted a sequential data-driven strategy to perform heuristic classifi-cation. Knowledge was encapsulated in production rules in the form of „if ... then“statements [CHG00]. Rules are a simple and modular mechanism, but their formalismis restrictive, and a huge amount of rules are needed to deal with real-life problems.

The application of expert systems can be seen as an effective way to distribute extantprofessional expertise, which is specific and also very limited to the domain of appliance.Conversely, expert systems don’t seem to be an adequate model to prove hypothesis ortheories due to their limited generality.

Empirical models are such types of models that generalize empirical knowledge in terms ofsurgical „dose-response“ tabular relationship (cf. [MD99]). Whether developed informallyor with computerized databases and statistical techniques, empirical generalizations sum-marize experience and are therefore models of observations. Empirical generalizations areprobably the basis of professional competence in most fields. However, because these mod-els are so closely related to experience, they prevent fundamental insight into the causes ofobserved patterns.


Homeomorphic models are organized by following the human structure in that these modelsare built of comparable physiologic parts found in the domain of interest. All interactionswith a homeomorphic model reflect underlying physiological processes. Thus, a homeomor-phic model can treat arbitrary new situations only so long as they can be expressed usingthe model’s terms.

Biomechanical models are homeomorphic models that try to simulate physiologic functionsof the human body by using parts, properties and parameters that are comparable to thehuman example. Moreover, biomechanical models also include constrains and relationshipsbetween the model parts and therefore define the essential functions that can be evaluatedand modified.

In the following sections, this work presents the formulation of a biomechanical model of theoculomotor system, based on geometric, force and kinematic principles that have been identifiedby studying human subjects (cf. [BKPH03]).

4.2 Structure of Biomechanical Models

Biomechanics, as the name implies, is a branch of mechanics that examines forces acting uponbiological structures and the effects induced by these forces. Thus, for successfully develop-ing biomechanical models, a good understanding of three different areas is required: Biologicalstructures, mechanical analysis and an understanding of movements [Mil96].

Initially important for creating a biomechanical model is to extract the relevant anatomic struc-tures of the oculomuscular system (see Sec. 2.1.4) and form proper abstractions that can bemodelled using mathematical methods. Therefore, abstract representations of all six extraoculareye muscles and the globe need to be defined and used as parts of the overall biomechanical model.A muscle itself is composed of several other abstract representations. These are for example theinsertion and origin of a muscle which are decomposed again.

However, representing muscles and the globe does not suffice in order to form a complete biome-chanical model. A description of the geometrical interaction between all abstract representationsis also required. This part of a biomechanical model is defined as the geometrical model. It isdesirable to form the geometrical model as autonomous, exchangeable part that interacts withother biomechanical components in order to preserve system variability. A well formed biome-chanical model consist of exchangeable components that provide flexibility and compatibility forlater modifications.

Apart from a geometrical part of the system model, there is also need for a model describingmuscle forces and the influence of these forces on the ocular geometry. Therefore, a biomechanicalmodel also requires a muscle force model which simulates and predicts the transformation ofinnervations into muscle force and subsequently supplies this muscle force to the ocular geometry,which resolves forces in terms of eye rotations.

Finally, a biomechanical model also needs to consider kinematics of ocular movements. Adoptedto a kinematic model of the human eye, forward and inverse kinematics can be identified as


predicting eye position on the basis of muscle innervations and predicting muscle innervation onthe basis of eye positions.

Thus, a biomechanical model consists of several abstract representations and sub-models. Ab-stract representations refer to different anatomical parts of the human eye, whereas sub-modelsimply the mathematical descriptions of geometrical properties, muscle forces and kinematics[BKPH03].

4.3 History of Modelling of the Human Eye

One of the first modelling attempts to get a better understanding about the oculomotor systemwas the so called „ophthalmotrope“, which has been designed and constructed by C.G.T. Ruetein 1845 [ST90]. An ophthalmotrope is a mechanical model of the eye, usually built out of copper,and is used to gain better understanding about eye rotations around different axes in 3D-space(see Fig. 4.1).

Figure 4.1: Halle’s Ophthalmotrope, from [ST90]

In 1848, F.C. Donders made an interesting discovery. If the eye fixates an object somewhere inspace, the position of this object also determines the gaze position of the eye. But the positionof the object in space does not specify the amount of torsional rotation, nor is this amount oftorsion arbitrary. In fact, the amount of torsion is clearly specified through the gaze position ofthe eye itself.

When the German edition of Donders’ work appeared, it drew the attention of H. von Helmholtz,who noticed that Donders had found the existence of pseudotorsion and proposed to call Donders’discovery „Donders’ Law“ [Hel63]. Pseudotorsion is a problem as it is a measured value of torsionwhich does not really exist, but is evoked by the measuring procedure. As Donders did not givea reason for pseudotorsion, it was von Helmholtz, who gave an explanation for its occurrence.Pseudotorsion is caused by the fact that, in tertiary positions of gaze, the vertical meridian


through the eye does not coincide with a vertical line in space, nor does the horizontal meridiancoincide with a horizontal line in space. „The reason for this discrepancy is that ’horizontal’ and’vertical’ are defined according to the coordinate system used“ [ST90]. Thus, a solution had tobe found to eliminate the problem of pseudotorsion.

Since pseudotorsion is caused by an unsuitably defined coordinate system, J. B. Listing had theidea to use polar coordinate systems to overcome this problem. Listing also noticed that inthis coordinate system all tertiary positions of gaze can be reached by a single rotation aroundone particular axis. Therefore, when Ruete read about Listing’s discovery in 1853, he suggestedto call it „Listing’s Law“. He also noticed that his first ophthalmotrope was not correct, as itviolated Listing’s law, so he developed a new version of the ophthalmotrope, which complied withListing’s law and already included parts for simulating muscles (see Fig. 4.2).

Figure 4.2: Ruete’s Ophthalmotrope, from [ST90]

As Donders’ law and Listing’s law were published, a consistent sequence of rotations for describinggaze rotations had to be found. As a consequence, in 1854 A. Fick introduced the so called Fick-sequence of rotations, where the position of the eye is characterized by rotations around thevertical, the horizontal and then the torsional axis. Later, in 1863, von Helmholtz suggested adifferent sequence by exchanging the first two axes. Therefore, the Helmholtz-sequence describesfirst a rotation around the horizontal, then around the vertical and finally around the torsionalaxis [Has95]. Such gimbal systems are a convenient way to describe the sequence of rotations.

In 1869 A. W. Volkmann provided the basis for all future developments in the field of extraocularmechanical research, especially for the development of improved models based upon ophthal-motropes. Volkmann made a statistical analysis of the characteristics of the eyes of numerouspatients and published the data of an average human eye [Vol69].

Using Volkmann’s eye data, Krewson [Kre50] published the first geometrical model of the human


eye called „string model“. The „string model“ defines muscles as strings which take the shortestpath from the insertion to the origin without considering any orbital connective tissues or muscleforces. A second attempt for building a geometrical model using Volkmann’s data was made byRobinson [Rob75b]. Robinson named his model „tape model“ and tried to include some sort offixing structure to reduce muscle side-slip during tertiary positions of gaze. Although this was afirst step in the right direction, the muscles in the „tape model“ still showed too much side-slipcompared to actual observations of the oculomuscular apparatus. Later, Kusel and Haase [KH77]tried to improve the „tape model“ and introduced the restricted „tape model“, but the predictionsof their model were still very close to the original „tape model“.

Robinson was the first who noticed that a geometrical model alone is not sufficient to successfullybuild a realistic model of the human eye. So he also took muscle forces into account and builtthe first biomechanical model. Following Robinson’s observation, Günther [Gue86] extended therestricted „tape model“ of Kusel and Haase into a biomechanical model. Although the predictionsof these models were better than those of the previous ones, they still were not perfect.

The latest discovery in the field of extraocular research was the analysis of muscle pulley structuresin 1995. Based on this new research results, Miller and Demer [MD99] started another attemptto build a biomechanical model of the human eye. Their so called orbit model represents thelatest development in the field of biomechanical eye motility research.

4.4 Ocular Geometry

The mathematical derivation for the modelling of ocular geometry is comprised of the geometri-cal definition of eye rotations and the geometrical representation of the extraocular muscles thatcause these rotations. In order to describe the transformation of eye muscle action into respec-tive angular rotations, the lever-arm of each eye muscle must be defined. This is accomplishedby the geometric interpretation of muscle action, that assumes that each muscle, in its staticbehavior, rotates the eye around a specific axis in space. Additionally, geometrical constrains likethe sequence of rotation and conformance to Listing’s law needs to be taken into account (seeSec. 2.2.1 and Sec. 2.2.2.2).

4.4.1 Coordinate Systems

In order to describe or measure 3D eye positions, coordinate systems need to be defined, allowingthe exact definition of the orientation the eye. Let {X,Y, Z} denote a cartesian coordinate systemof a left eye (see Fig. 4.3), so that the positive X-axis points to the right, the positive Z-axis pointsupward and the negative Y-axis coincides with the line of sight through the center of the pupilwhen the eye is in primary position.

For 3D orientation of the eye, Euler’s theorem can by applied, which states that for every twoorientations of an object, the object can always move from one position to the other by a singlerotation around a fixed axis. This combined rotation is usually decomposed into three consecutiverotations around well defined, hierarchically nested axes. The sequence of rotation plays animportant role, since the execution of rotations specifying the same angles but in different order,leads to a different final orientation of the rotated object. One important distinction within


Figure 4.3: Coordinate System of a Left Eye

this definition is to consider active and passive rotation behavior. Active rotations are definedas rotations where the coordinate system is constant and not affected by preceding rotations,whereas passive rotations change the coordinate system, reorienting the coordinate axes with eachrotation that is applied. Thus, in passive rotations, each rotational modification also changes thecoordinate axes around which successive rotations will be performed. In adopting these propertiesto eye rotations, active rotations correspond to rotations in a head-fixed coordinate system, andpassive rotations are performed in an eye-fixed coordinate system.

(a) Fick Gimbal (b) Helmholtz Gimbal

Figure 4.4: Gimbal Systems for describing 3D Eye Position, adapted from [Has95]

A combination of a horizontal and a vertical rotation of the eye is a well defined sequence,uniquely characterizing the direction of the line of sight. However, this does not completelydetermine the 3D eye position, since the rotation around the line of sight is still unspecified. Athird rotation is needed to completely determine the orientation of the eye. Systems that usesuch a combination of three rotations for the description of eye positions generally use passiverotations, or rotations of the coordinate system. Due to non-commutativity of rotations in 3D


space, it is necessary to use a uniform order of rotations to describe eye positions and to denotethis rotation order along with the definition of eye positions. Such rotations of the coordinatesystem can effectively be demonstrated by using gimbal systems, in which the hierarchy of passiverotations is automatically implemented as intrinsic system property. A coordinate system wasdefined by Fick [Fic54], whereby rotations have to be executed first around the vertical axis, thenaround the horizontal and around the torsional axes (see Fig. 4.4(a)). In contrast, Fig. 4.4(b)shows a rotation sequence defined by Helmholtz, first around the horizontal, afterwards aroundthe vertical and torsional axes [Hel63]. Whereas the Fick gimbal system rotates three angularcomponents {α, β, γ} around the coordinate axes {Z,X, Y } respectively, the Helmholtz gimbalwill rotate around the axes {X,Z, Y }. Both systems of rotation describe a gimballed suspensionof the globe with consideration of the respective rotation sequence, where rotations are executedfrom the outside to the inside. From the representation of the two systems in Fig. 4.4 it is easy torecognize that some specified angles of an eye position lead to different orientations of the globeand thus to different eye positions when comparing both systems.

4.4.2 Mathematical Description of Eye Rotations

Based on the coordinate system for one eye, depicted in Fig. 4.3, the angular rotations around themain coordinate axis are described by the Fick sequence of rotations (cf. Sec. 4.4.1). This ensuresthat each eye position is expressed by three angular rotations {α, β, γ} around the coordinateaxes {Z,X, Y } respectively. There are many different ways to represent rotations in three-dimensional space. In this work, matrices and quaternions will be used to define rotations andeye positions for the derivation of the ocular geometry. Eye rotations in three-dimensional spaceconsist of rotations as well as translations. The discussion here will be restricted to the rotationalcomponents of the total eye movement. Translations of the eye will be treated later in thisdocument.

First, let a vector ~SC(sx, sy, sz) represent a line originating from the center of the eye fixedcoordinate system. Next, it will be described how this vector can be rotated according to a giveneye position.

4.4.2.1 Rotation Matrices

In order to specify a rotation matrix that describes an eye position in Fick rotational order, thestandard definition of rotations around the three principle axes are used. The rotation matricesused in this derivation use passive rotational order.

Generally, elements of a rotation matrix can be indexed in the following row-based form,

R =

R11 R12 R13

R21 R22 R23

R31 R32 R33

.

(4.1)


A rotation of γ radians around the Y-Axis is defined as,

RY (γ) =

cos(γ) 0 − sin(γ)0 1 0

sin(γ) 0 cos(γ)

,

(4.2)

a rotation of β radians around the X-Axis is defined as,

RX(β) =

1 0 00 cos(β) sin(β)0 − sin(β) cos(β)

,

(4.3)

finally, a rotation of α radians around the Z-Axis is defined as,

RZ(α) =

cos(α) sin(α) 0− sin(α) cos(α) 0

0 0 1

.

(4.4)

Thus, a generalized rotation matrix that can rotate the eye around each of the principle coordinateaxis in the Fick rotational sequence (Z-X-Y) can be defined using,

R′Fick(α, β, γ) = RT

Y RTXRT

Z , (4.5)

which conforms to an active rotation (i.e. a rotation of the object), where T denotes matrixinversion. If rotated in the object space (eye fixed coordinate system), the rotation sequenceneeds to be inverted to,

RFick(α, β, γ) = RZRXRY , (4.6)

and solved for the matrix product, this results in

RFick(α, β, γ) =

(cos(α) cos(γ) + sin(α) sin(β) sin(γ) sin(α) cos(β) sin(α) sin(β) cos(γ)− cos(α) sin(γ)

sin(α)− cos(γ) + cos(α) sin(β) sin(γ) cos(α) cos(β) cos(α) sin(β) cos(γ) + sin(α) sin(γ)cos(β) sin(γ) − sin(β) cos(β) cos(γ)

).

Please note that the sequence of rotations from Eqn. 4.5 around the principle axes in worldcoordinate space (head-fixed) can be rewritten to perform rotation in the object’s coordinatespace (eye-fixed) by inverting Eqn. 4.5 to Eqn. 4.6.

Given the rotation matrix from Eqn. 4.6, the vector ~S can be reoriented to ~S′ according to theangular Fick coordinates (α, β, γ) using,

~S′ = RFick(α, β, γ) ~S. (4.7)

Thus, an eye position can be described by reorienting the geometry of the eye according toEqn. 4.7. The eye movement can be described by linearly interpolating the rotational angles(α, β, γ) between two positions, which then describe the shortest path rotation between thesetwo positions. Moreover, the product of multiple Fick rotation matrices (CFick) describes thefinal eye position that is reached by subsequent execution of each rotation. The final eye positionof a rotation (α, β, γ) from primary position, followed by a rotation (α′, β′, γ′) can be describedby,

CFick = RFick(α′, β′, γ′) RFick(α, β, γ). (4.8)


However, this construction assumes that the eye initially always starts its movement from primaryposition (i.e. the position where the line of sight is equal to the Y-Axis of the eye fixed coordinatesystem). If the shortest path rotation from primary position to a position that is composed oftwo or more Fick rotations of the form (4.6) needs to be known, Fick angular coordinates can beresolved from the compound rotation matrix of the form (4.8).

In order to regain angular coordinates from a Fick rotation matrix of the form (4.1), the followingentities can be used:

tan(α) =cos(β) sin(α)cos(α) cos(β)

=R12

R22,

sin(β) = −R32, (4.9)

tan(γ) =cos(β) sin(γ)cos(β) cos(γ)

=R31

R33.

The presented derivation enables the definition of eye position by using rotation matrices ofthe form (4.6) and to determine eye position from given rotation matrices using Eqn. 4.9. Whenspecifying all entities of the ocular geometry, a new eye position can be described by transformingeach geometric entity using Eqn. 4.7. However, rotation matrices are complicated to handle,especially when defining rotation around arbitrary axis.

Using the presented method of Euler angles to represent a set of three rotations specified insuccessive order may suffer from a problem known as „gimbal-lock“. The problem is that nomatter what the order in which the three rotations are carried out, there will always be a valuefor one angle of rotation that yields infinite values of the other two angles. For example, Eulerangles are normally represented as yaw, pitch, and roll, in that order. Given that order of rotation,it is easy to show that when the pitch angle is headed straight up or straight down, the values ofyaw and roll are undefined. In a physical gimbal system, this is known as gimbal-lock and refersto the situation in that two or more principle axis of a coordinate system align, resulting in aloss of rotational degree of freedom.

The notion of axis-angle definition for rotation represents a more convenient way with respect todefining eye positions and eye movements, and shall be introduced by using quaternion algebra.

4.4.2.2 Quaternions

Each rotation may be parameterized by a unit vector along the axis of rotation, and the angleof rotation. Rotations collectively form a three-dimensional space which can be pictured as asolid 3-sphere with diametrically opposite points of its surface identified. Rotations may also beparameterized by Euler angles via various combinations of rotations around the X, Y, Z axes (seeSec. 4.4.2.1). Unfortunately, there are 12 such parameterizations possible, and the use of Eulerangles can give rise to gimbal-lock problems (see Sec. 4.4.1). Using an axis and angle represen-tation in specifying an arbitrary axis and an angle (positive if in a counterclockwise direction),this is an efficient way to avoid gimbal-lock. Additionally, axis-angle representations are a moreintuitive and practical oriented method of representing rotations of objects. Quaternion algebrais a powerful mathematical tool that combines vector notation with rotational operations.

Basically, quaternions are hypercomplex numbers, just as a single complex number, z = x + iy,


can be used to specify a point or vector in a two dimensional space, a single quaternion,

q = a+ bI + cJ + dK, (4.10)

can be used to specify a point in a four dimensional space, and a quaternion with a = 0 can beused to describe vectors in Euclidean 3-space. A quaternion can be rewritten as,

q = (s, ~V ), (4.11)

where s = a, and ~V = bI+cJ+dK. In this notation, s is called the scalar part of the quaternion,and ~V is the vector part. Additionally,

Scal(q) = s, (4.12)V ect(q) = ~V , (4.13)

extract scalar and vector parts from a quaternion so that Scal(q) results in a real number andV ect(q) gives the original cartesian three-dimensional vector.

This definition can be extended to include cartesian vector spaces: Let { ~E1, ~E2, ~E3} be a set ofbase vectors in a cartesian vector space. Then, a vector ~V = (x1, x2, x3) can be represented as:

~V = x1~E1 + x2

~E2 + x3~E3.

There exists an isomorphism of a three-dimensional vector space into the four-dimensional vectorspace of quaternions.

Let~V = a1

~E1 + a2~E2 + a3

~E3,

andw = a0 + a1I + a2J + a3K.

w could also be written as: w = a0 + ~V , where a0 is called the scalar part of the quaternion,and ~V , the vector part. The quantities {I, J , K}, the unit quaternions, stand in relation toquaternions in the same way that { ~E1, ~E2, ~E3} unit vectors relate to vectors. However, {I, J,K}do not combine in exactly the same way as the unit vectors, { ~E1, ~E2, ~E3}.

This leads to the definition that a vector ~S = (x1, x2, x3) can be represented by a quaternion ofthe form,

q = [0, (x1, x2, x3)], (4.14)

where the scalar part is zero and the vector part of the quaternion contains the identical cartesianvector. The general form of a quaternion is denoted as,

qs = [s, (x1, x2, x3)], (4.15)

Given two quaternions according to Eqn. 4.15,

q1 = [a0, (a1, a2, a3)],q2 = [b0, (b1, b2, b3)],


the following algebraic operations can be defined:

q1 + q2 = [a0 + b0, (a1 + b1, a2 + b2, a3 + b3)], (4.16)q1− q2 = [a0 − b0, (a1 − b1, a2 − b2, a3 − b3)]. (4.17)

According to Eqn. 4.11, q1 and q2 can also be written as,

q1 = [s1, ~V1], (4.18)q2 = [s2, ~V2],

where s1 and s2 are the scalar parts and ~V1 and ~V2 are the vector parts of the quaternions q1 andq2 respectively. Additionally, a „pure“ quaternion is a quaternion whose sca1ar part is zero,

qp = [0, ~V ]. (4.19)

Let q1 and q2 be in the form of (4.18), then, multiplication of two quaternions can be accomplishedby,

q1q2 = [s1s2 − ~V1 · ~V2, (s1 ~V2 + s2 ~V1 + ~V1 × ~V2)], (4.20)

where ~V1 × ~V2 stands for the vector-cross-product, and~V1 · ~V2 stands for the vector-scalar-product.

Two „pure“ quaternions in the form (4.19) can be multiplied using,

qp1qp2 = [− ~V1 · ~V2, ( ~V1 × ~V2)]. (4.21)

Multiplication of a quaternion in the form (4.15) or (4.19) by a scalar number n is defined as,

qn = nq = [ns, n~V ] = [nx0, (nx1, nx2, nx3)]. (4.22)

In relation to vector operations in cartesian three-dimensional space, the following entities canbe identified when using quaternion algebra:

The conjugate of a quaternion q in the form of (4.11) inverts the vector part in that,

q′ = [s,−~V ], (4.23)

the absolute value, or magnitude of a quaternion the form of (4.11) can be calculated using,

|q| =√s2 + ~V · ~V , (4.24)

and the norm n of a quaternion is denoted as,

n(q) = (s2 + ~V · ~V ) = |q|2. (4.25)


A very important operation with quaternions, particularly with regard to representing rotations,is the inverse function,

q−1 = (1|q|2

)[s,−~V ] = 1/q =1

n(q)q′, (4.26)

whereby the division of two quaternions can be expressed by,

q3 =q1q2

= q1(q−12 ) = q1q

−12 . (4.27)

In order to rotate a vector by a quaternion, the vector is multiplied on the right by the quaternion,and on the left by the inverse of the quaternion.

From (4.19) it follows that a „pure“ quaternion vp can be rotated using,

v′p = Rot(vp, q) = q−1vpq. (4.28)

The result v′p will always be a quaternion with zero scalar component, [0, V ect((v′p))].This guarantees that:

Rot(vp1, q)Rot(vp2, q) = Rot(vp1vp2, q) (4.29)

which implies that dot and cross products are preserved. This effect is embedded in the quaternionproduct.

By using the inverse of a quaternion q = [s, ~V ] as q−1[s,−~V ]/ |q|2, the effects of magnitude aredivided out so that any scalar multiple of a quaternion gives the same amount of rotation.

When the magnitude of q = 1, these unit-quaternions lie on a sphere of radius 1, q = [w, (x, y, z)]such that w2 + x2 + y2 + z2 = 1.These unit quaternions carry the amount of rotation in w, as cos(θ/2), while the vector part(x, y, z) points along the axis of rotation with magnitude sin(θ/2) and, the axis of rotation isthat line in space which remains unmoved during the rotation.

In referring to Eqn. 4.24, the magnitude of,

q = [cos(θ/2), sin(θ/2)(~u)] = 1, (4.30)

since cos2 +sin2 = 1 and where ~u denotes the unit-vector of the axis of rotation.

This sphere of unit quaternions spans a 4-dimensional vector space (S4) and forms a sub-groupof the 3-dimensional vector space (S3). This spherical metric of S3 is the same as the angularmetric of S3.

Thus, a quaternion for rotating is stored as,

qr = [cos(θ/2), sin(θ/2)(~u)], (4.31)

where θ refers to the radian angle of rotation around the unit-vector of the axis of rotation ~u. Ashort (axis-angle) notation of a rotation quaternion can therefor be denoted as:

qr = [θ, (~u)]. (4.32)


Rotation quaternions of the form (4.31) can be combined by quaternion multiplication, usingEqn. 4.29.

An eye position can therefore be represented by a unit-quaternion that describes a rotation fromprimary position around a specified rotation axis ~u with an appropriate angle θ using Eqn. 4.31.

In order to represent eye movement, interpolation of rotations is conveniently performed usingquaternion algebra.Let qa and qb denote two distinct quaternions that describe the start and end position of an eyemovement respectively.Then the rotation quaternion,

q̄ = qa(q−1a qb)t, (4.33)

where the quaternion power operator is defined as,

qt = [st, ~V ]. (4.34)

This will describe any desired intermediate rotation that lies on the shortest path between thetwo eye positions. The time parameter t can be introduced into the angle so that the adjustmentof q varies uniformly over the great arc between qa and qb.

Eqn. 4.7 can now be rewritten using a rotation quaternion q in the form (4.28) that representsthe current eye position and

~S′ = V ect(Rot(~S, q)), (4.35)

represents the rotation of a vector ~S into a new position ~S′.

Finally, a quaternion can be transformed into a rotation matrix.Let q be a rotation quaternion in the form of (4.15), then, the corresponding rotation matrix isdefined by,

R =

1− 2x22 − 2x3

2 2x1x2 − 2x3S 2x1x3 + 2x2S2x1x2 + 2x3S 1− 2x1

2 − 2x32 2x2x3 − 2x1S

2x1x3 − 2x2S 2x2x3 + 2x1S 1− 2x12 − 2x2

2

. (4.36)

Further information on quaternions, derivations and proofs can be found in [PW82] or [WP03].

4.4.2.3 Listing’s Law

In order to generate 3D eye positions based on 2D reference positions, torsional rotation anglesneed to be calculated in order to fulfill Listing’s law. Referring to Sec. 2.2.2.2, Listing’s law can bederived geometrically. Based on the coordinate system definition from Sec. 2.2.1 and Sec. 4.4.1,an eye position can be represented by three successive rotations around the coordinate systemaxes.

Let {α, β, γ} denote three angular rotations around the coordinate axes X, Z and Y respectively,whereas α represents ab-/adduction, β elevation or depression and γ torsional eye movements.This sequence of rotation corresponds to the definition of the Fick rotation order from Sec. 4.4.1.Since the Y-Axis represents the line of sight, torsional rotation in accordance with Listing’s lawneeds to be determined as a function of the angles α and β only.


The torsional rotation of a Point P (x, y, z) around the line of sight results in a new referenceposition P ′(x′, y′, z′) and can be described using Eqn. 4.37,

x′ = x cos(γ)− z sin(γ), (4.37)y′ = y,

z′ = z cos(γ) + x sin(γ).

Additionally, P ′ is rotated around the horizonal X-Axis into position P ′′(x′′, y′′, z′′) in order torepresent elevation or depression movements using Eqn. 4.38,

x′′ = x′, (4.38)y′′ = y′ cos(β) + z sin(β),z′′ = z′ cos(β) + y′ sin(β),

and finally ab-/adduction movement is represented by a rotation of P ′′ around the Z-Axis usingEqn. 4.39, defining the final position of P ′′′(x′′′, y′′′, z′′′),

x′′′ = x′′ cos(α) + y′′ sin(α), (4.39)y′′′ = y′′ cos(α)− x′′ sin(α),z′′′ = z′′.

Substitution of Eqn. 4.37 and Eqn. 4.38 in Eqn. 4.39 gives Eqn. 4.40, a combined rotation aroundall three axes of an eye fixed coordinate system, obeying the Fick sequence of rotation,

x′′′ = y cos(γ) sin(α)− (z cos(β) + x sin(α) sin(β)) sin(γ) + (4.40)cos(α)(x cos(γ) + y sin(β) sin(γ)),

y′′′ = y cos(α) cos(β)− x cos(β) sin(α) + z sin(β),z′′′ = z cos(β) cos(γ) + cos(α)(−y cos(γ) sin(β) + x sin(γ)) +

sin(α)(x cos(γ) sin(β) + y sin(γ)).

According to Listing’s law, Eqn. 4.40 defines a rotation axis in 3D-space, so that any eye positiondefined through the angles {α, β, γ} can be reached by a single rotation around this axis. Sincethis axis stays constant during the rotational movement, it can be found by setting P ′′′ = P inEqn. 4.40 and subsequently solving for x, y and z. Thus, all points of the rotation axis satisfyEqn. 4.41,

x =y cos(γ) sin(α)− z cos(β) sin(γ) + y cos(α) sin(β) sin(γ)

1− cos(α) cos(γ) + sin(α) sin(β) sin(γ), (4.41)

y =x cos(β) sin(α)− z sin(β)−1 + cos(α) cos(β)

,

z = −cos(α)(−y cos(γ) sin(β) + x sin(γ)) + sin(α)(x cos(γ) sin(β) + y sin(γ)))−1 + cos(β) cos(γ)

. (4.42)

Since Listing’s law also states, that all rotation axes of the eye lie in the fronto-parallel plane(Listing’s plane), this constraint needs to be added by using the equation for the X − Z plane,

y = 0, (4.43)


constraining all coordinates of the rotation axes from Eqn. 4.41. By replacing Eqn. 4.43 intoEqn. 4.41, the system of equations can be solved by forward substitution of Eqn. 4.41 in Eqn. 4.42,resulting in Eqn. 4.44,

z = − z cos(β) sin(γ)(cos(γ) sin(α) sin(β) + cos(α) sin(γ))(−1 + cos(β) cos(γ))(−1 + cos(α) cos(γ)− sin(α) sin(β) sin(γ))

. (4.44)

Solving Eqn. 4.44 for γ eliminates the last free variable z and expresses the torsional angle ofrotation as a function of α and β, ensuring that each eye position is assigned a unique torsionalvalue according to Eqn. 4.45,

⊗ (α, β) = γ = cos−1

(cos(α) + cos(β)

1 + cos(α) cos(β)

). (4.45)

4.4.2.4 Definition of Eye Positions

In describing eye positions with the methods presented in Sec. 4.4.2.1 and Sec. 4.4.2.2, there arebasically two possibilities:

3D eye positions are used to fully specify all three degrees of freedom for a given eye positionin space. This corresponds to the definition of three angular coordinates in Fick rotationalorder that specify add-/abduction, elevation/depression and torsion respectively.

2D eye positions are used when specifying ab-/adduction and elevation/depression angles, ex-cept the torsional rotation angle is calculated as suggested by Listing’s law and thereforeconstrains eye rotation axes to lie in Listing’s plane (see Sec. 2.2.2.2).

Eye positions that fulfill Listing’s law can therefore be defined by using Eqn. 4.6 in the followingway:

RListing(α, β) = RFick(α, β,⊗(α, β)), (4.46)

where γ is substituted by Eqn. 4.45.

Using quaternion notation, 3D eye positions can be defined by considering passive rotation historyof every coordinate system axis. In Fick rotation sequence, first, the X-axis of the coordinatesystem is rotated around the Z-axis by α degrees using a quaternion of the form (4.32). Let~XA(1, 0, 0), ~Y A(0, 1, 0) and ~ZA(0, 0, 1) denote the base vectors of the eye fixed coordinate system.

Then, the rotation quaternion,

qrz = [α, ( ~ZA)],

defines a rotation of α degrees around the base axis ~ZA, and,

~XA′ = V ect(Rot( ~XA, qrx)),

reorients the original X-axis to a new vector ~XA′. Accordingly, the y-Axis of the eye fixedcoordinate system is rotated around the Z- and X-axis respectively by using,

qry = [β, ( ~XA′)]qrz,

~Y A′ = V ect(Rot( ~Y A, qry)),


resulting in a new Y-Axis ~Y A′. The rotation quaternions that rotate around each of the pre-oriented eye fixed axes can now be defined as,

qry = [γ, ( ~Y A′)],

qrx = [β, ( ~XA′)], (4.47)qrz,

where γ is given by Eqn. 4.45.

The complete rotation quaternion for defining eye positions according to the Fick sequence ofrotation is given as

qListing = qryqrxqrz. (4.48)

4.4.3 Geometrical Abstractions

In the derivation of the ocular geometry, geometrical abstractions will serve as modelling ele-ments that mathematically define constrains and conditions that represent its human counter-part. Additionally certain equations can be identified as interfaces to other elements of the overallbiomechanical model.

4.4.3.1 Globe

The first essential geometrical abstraction is the modelling of the globe. The geometrical objectof a sphere is used to represent the human eyeball and is defined by using the spherical equation,

x2 + y2 + z2 = r2, (4.49)

where x, y, z are coordinates that lie on the sphere with radius r.

Figure 4.5: Geometrical Abstraction of the Globe

The illustration in Fig. 4.5 shows that the globe is placed in the center of the eye fixed coordinatesystem which is identical with the rotation center of the sphere. The line of sight originates from


the center of rotation (C) and intersects the center of the pupil. In primary position, the line ofsight coincides with the Y-axis of the head-fixed coordinate system. The orientation of the globe(i.e. the eye position) is changed according to the rotation of the line of sight using Eqn. 4.35.

Although, this geometrical abstraction assumes that the globe is spherical, further derivationswill show, that the „natrual“ ellipsoid shape of the human eyeball will be considered when definingthe lever-arm of each muscle. Thus, the spherical approximation of the globe was chosen for easiervisualization and can be replaced by a more realistic (i.e. more complex) geometrical structure,without affecting the models behavior.

4.4.3.2 Muscles

In order to geometrically define the action of the extraocular muscles, specific landmark-pointsthat describe the muscle path and the direction of pull are chosen for each muscle. Derivationswill be given starting from older, historic modelling approaches up to the geometrical model usedin this thesis, the so called „Pulley“ model.

Figure 4.6: Geometrical Abstraction of an Eye Muscle

An extraocular muscle can geometrically be approximated by a straight line. Muscle force anddeformation due to contraction or relaxation are not of primary interest in the geometrical model.These properties will be added by additional models of muscle force and dynamic 3D-visualization.The geometrical model itself is only responsible for predicting how a muscle can transfer its forceinto a rotation axis that affects eye position. Following Fig. 4.6, the requirements for a specificgeometrical model of the extraocular muscles are the calculation of the following entities (cf.[BKPH03] and [Buc02]):

The origin of each muscle represents the point where a muscle originates in the posterior areaof the orbit (i.e. the anulus of Zinn).

The point of tangency of each extraocular eye muscle is defined as that point where the musclefirst contacts the globe. This special point is intrinsic to the geometrical definition, since


it changes as a function of gaze and is defined using different mathematical formulations indifferent geometrical models.

The insertion point is that point on the globe where the tendon of the muscle finally insertson the globe.

The muscle action circle is defined as that circle that lies on the globe and is defined by themuscle’s plane of action.

The arc of contact of a muscle is defined as the circular path between the insertion and thepoint of tangency. This path is always part of the muscle action circle.

The muscle rotation axis is defined as a vector that represents an axis around which a specificmuscle rotates the eye. Thus axis is perpendicular to the muscle action circle or the muscle’splane of action.

String Model

The simplest geometrical representation of extraocular muscle action is the „string model“ (seeFig. 4.7). This model is also known as the „shortest path hypothesis“ (cf. [Kre50]), which meansthat muscle strings are assumed to be always tight and therefore take the shortest possibleconnection between the insertion and the origin (see Fig. 4.7(a)).

(a) Muscle Path in Primary Position (b) Muscle Path in Secondary Position

Figure 4.7: Geometrical „String Model“, from [BKPH03]

The definition of the „string model“ can be reduced to specifying the point of tangency as afunction of gaze position (see Fig. 4.8). Let ~I be the vector from the center of the coordinatesystem C to the insertion point of a muscle from Sec. 2.1.3 and ~I ′ the vector to the insertionpoint in the actual eye position, such that,

~I ′ = V ect(Rot(~I, q)), (4.50)

where the rotation quaternion q in the form (4.32) specifies the current eye position. Let ~Odenote the origin of an eye muscle, according to Sec. 2.1.3 and ~R denote a vector with lengthr =

∣∣∣~R∣∣∣ that is perpendicular to the vector ~O. Then, the right triangle O −R−C gives l as the


shortest path from origin to insertion by using,

l =

√∣∣∣ ~O∣∣∣2 − ∣∣∣~R∣∣∣2. (4.51)

From the definition of the right triangle, the angle between ~O and ~R is defined by,

δ = cos−1(l/r). (4.52)

The rotation axis ~RM of the muscle can be defined as vector that is perpendicular to the muscleaction plane spanned by the vectors ~O and ~I ′ using,

~RM =~O × ~I ′∣∣∣ ~O × ~I ′

∣∣∣ . (4.53)

Finally, the vector ~QN , which points along the vector ~O with length r needs to be rotated aroundthe muscle rotation axis ~RM by the angle δ. The vector ~QN is denoted as,

~QN =~O∣∣∣ ~O∣∣∣r. (4.54)

Using a rotation quaternion of the form (4.32),

rq = [δ, ( ~RM)], (4.55)

the vector ~QN can be rotated into the position of the point of tangency T by,

~T = V ect(Rot( ~QN, rq)). (4.56)

Figure 4.8: Point of Tangency in the „String Model“

In secondary gaze positions (see Fig. 4.7(b)), the shortest path hypothesis in this model leads toa displacement of the point of tangency which is known as „side-slip“. In this model, the muscle


action circle is always a great circle on the globe and its center coincides with the center of theglobe.

However, the predictions of the „string model“ do not correspond to clinical expectations. Par-ticularly in secondary and tertiary gaze positions, this model gives inadequate results. Only inprimary position, predictions for muscle path and muscle match clinical expectations. Fig. 4.7(a)shows a left eye with the lateral rectus muscle in primary position, whereas Fig. 4.7(b) showsan adduction of about 34 ◦ (secondary gaze position). It is easy to recognize that the path ofthe lateral rectus muscle shifts upwards with a change in gaze position (the lateral rectus muscle„shifts its muscle path with the line of sight“), which does not correspond to clinical observations.This shift results in a drastic change of the muscle path and thus in an abnormal rotation ofthe eye (i.e. the lateral rectus now mainly elevates the eye!). The lateral rectus muscle cannotsustain its mainly abducting effect on the globe, it „slips“ away as the eye turns towards the nose.

Tape Model

Based on the „string model“, Robinson ([Rob75a]) developed the so called „tape model“ which triesto reduce the side-slip of the muscles in secondary and tertiary positions. Therefore, Robinsonlimited the side-slip of the point of tangency with an empirical value depending on the extent ofthe rotation of the eye. This side-slip retaining component is introduced in the „tape model“ viaa reduction of the maximum side-slip angle that is calculated with respect to the „string model“.

The maximum side-slip angle is determined by measuring the angular displacement of two per-pendicular vectors with respect to the current eye position in a head-fixed coordinate system (seeFig. 4.9). Let ~MP be the vector perpendicular to ~I and ~O, and let ~TP be the „tangential“ vectorwith respect to the arc of contact of the muscle such that,

~MP =~I × ~O∣∣∣~I × ~O

∣∣∣ , (4.57)

~TP =~MP × ~I∣∣∣ ~MP × ~I

∣∣∣ . (4.58)

These two base vectors are defined with respect to I, the insertion point in primary position.Now, these two base vectors are also defined for the current gaze position, using the vector ~I ′from Eqn. 4.50,

~MP ′ =~I ′ × ~O∣∣∣~I ′ × ~O

∣∣∣ , (4.59)

~TP ′ =~MP × ~I ′∣∣∣ ~MP × ~I ′

∣∣∣ . (4.60)

The maximum angle of deviance is measured between the vectors ~MP and ~MP ′. However,since the vector ~MP is defined in primary position, it needs to be transformed in the head-fixedcoordinate system in order to measure the true angle of deviance. Let q be the quaternion that


Figure 4.9: Tape Model Calculations

defines the current eye position in the form (4.32). Then, q−1 will represent a backward rotationinto the head-fixed coordinate space, and,

~MPh = V ect(Rot( ~MP, q−1)), (4.61)~TPh = V ect(Rot( ~TP , q−1))

will transform the vectors ~MP and ~TP into head-fixed vectors ~MPh and ~MPh respectively. Now,the maximum angle of deviation vmax of the muscle action circles between the primary positionand the current gaze position with respect to the „string model“ is defined as,

vmax = sgn( ~MP ′ · ~TPh) cos−1( ~MP ′ · ~MPh), (4.62)

where the first part of this equation specifies the angular direction in terms of the sign of thedeviation angle. The actual deviation angle e with respect to the vector ~MPh can be calculatedusing,

e = cos−1( ~MPh · ~O). (4.63)

Finally, the reduced side-slip angle v can be defined as a fraction of the maximum deviation anglevmax with respect to the actual excitation angle e using,

v = vmax |cos(e)| . (4.64)

The angle v now defines the the actual reduced displacement angle for the „tape model“ that isapplied by using a rotation around the axis that is defined by the vector ~O based on the point oftangency from the „string model“. Thus, the point of tangency T from the „string model“ (4.56)in primary position is rotated using the following rotation quaternion qt of the form (4.32),

qt = [v, ( ~O)], (4.65)

resulting in the definition of the point of tangency TT in the „tape model“,

~TT = V ect(Rot(~T , qt)). (4.66)


Because of the modification to the movement of the point of tangency, the center of the muscleaction circle of a specific muscle does no longer coincide with the globe center in secondary andtertiary positions. Instead, the center of a specific muscle action circle is now defined through amuscle’s point of tangency vector ~TT and origin vector ~O, just like the muscle rotation axis. Asin the „string model“, the vector perpendicular to the plane spanned through these three pointsrepresents the muscle rotation axis ~RM , such that,

~RM =~TT × ~O∣∣∣ ~TT × ~O

∣∣∣ . (4.67)

The center of the muscle action circle can now be calculated as the intersection point of therotation axis with the muscle action plane. Since the muscle rotation axis is perpendicular to themuscle action plane, and the muscle action plane contains the point of tangency and origin ofa muscle, the center of the muscle action circle lies in the extension of the muscle rotation axis.Thus, the center of the muscle action circle can be calculated using,

~Cc = (~C − ~RM)~C − ~TT

~RM − ~C. (4.68)

(a) Muscle Path in Primary Position (b) Muscle Path in Secondary Position

Figure 4.10: Geometrical Tape Model, from [BKPH03]

The limitation of the movements of the point of tangency results in a reduced muscle side-slipin secondary and tertiary positions. Fig. 4.10 shows this effect by comparing an eye in primaryposition (Fig. 4.10(a)) and after an adduction of 34 ◦ (Fig. 4.10(b)). Although the side-slip isreduced in the „tape model“, the model predictions still differ from anatomical findings. In thefollowing part of this thesis, a new geometrical pulley model will be described which tries toovercome this problem and also takes pulleys into account.


Pulley Model

Results of investigations of muscle pulley structures provided the reason why all the conventionalmodels like the string or „tape model“ where not exact and the predictions of these models did notcompare to clinical findings (see Sec. 2.1.1). But pulleys did not only influence existing models.They also considerably affected existing surgery techniques. Before the discovery of pulleys mostsurgeons tended to remove portions of the orbital connective tissues covering the eye muscles,however, it has been noticed that if too much of the connective tissue is removed, unpredictableand unwanted results occurred after surgery. With the discovery of pulleys, the reason for thisbehavior became clear. When surgeons damaged or even removed pulley structures, they de-stroyed the functional origin of a muscle and therefore destabilized the muscle and sometimeseven the whole oculomuscular system (see [CJD00]).

In Fig. 4.11, the difference between conventional models and the pulley model is illustrated.While conventional models assume that the muscle tendon is coupled tightly to the globe (seeFig. 4.11(A)), the pulley model introduces an new anatomical structure that lets the muscle slidethrough a fascial pulley which is elastically coupled to the orbital wall (see Fig. 4.11(B)). Thefunctional consequence is obvious in comparing both models when the eye is elevated. Whileconventional models tend to define a constant axis of rotation with respect to different gazepositions (compare Fig. 4.11(A) and Fig. 4.11(B)), the pulley model adapts the axis of rotationsuch that the primary direction of action of the muscle is preserved with respect to the currentgaze position (compare Fig. 4.11(C) and Fig. 4.11(D)). This means, that the axis of rotation ofa muscle is expressed according to an eye-fixed coordinate system in the pulley model, since thecoordinate system axes are moved as a function of gaze position.

Figure 4.11: Comparison of Conventional vs. Pulley Model, from [MD99]


For the mathematical derivation of the pulley model, a new geometrical pulley point is introduced,which also serves as functional origin of a muscle. In contrast to other models, the pulley modelredirects a muscle’s main direction of pull towards the functional pulley point instead of usingthe anatomical origin (o) as force directing reference position. This results in a stabilization ofthe muscle path in the posterior area of the orbit as well as an adaption of the muscle rotationaxis with respect to the current eye position.

For the calculation of the point of tangency, the muscle action circle must be defined. In primaryposition, the muscle action circle of each muscle is a great circle. Therefore, in primary position,the center of a muscle action circle coincides with C as shown in Fig. 4.12. In secondary andtertiary positions, the muscle action circle of a specific muscle is no longer a great circle and forthis reason, the center has to be calculated in a different way.

Thus, to calculate the center of a muscle action circle in all different eye positions, the first stepis the definition of three vectors { ~SX, ~SY , ~SZ} based on the insertion in primary position I, thepulley location P and the center of the globe C.

According to Fig. 4.12, let ~SZ denote the unit vector from C to I,

~SZ =~I∣∣∣~I∣∣∣ , (4.69)

~SY the vector perpendicular to the plane spanned by the three points C, P and I,

~SY =~P × ~I∣∣∣~P × ~I

∣∣∣ , (4.70)

and ~SX be the cross product of ~SY and ~SZ,

~SX = ~SY × ~SZ. (4.71)

In order to calculate the center of a muscle action circle in a secondary or tertiary position the eyeand, consequently, the vectors { ~SX, ~SY , ~SZ} have to be rotated out of the primary position intoa secondary or tertiary position. Since the orientation of the eye is defined through a rotationquaternion q of the form (4.32), q is also used for reorienting three vectors { ~SX, ~SY , ~SZ}, suchthat,

~TX = V ect(Rot( ~SX, q)), (4.72)~TY = V ect(Rot( ~SY , q)),~TZ = V ect(Rot( ~SZ, q)).

Thus, the resulting vectors { ~TX, ~TY , ~TZ} define the three base-vectors { ~SX, ~SY , ~SZ} in thecurrent eye position (see Fig. 4.13). Additionally, the insertion point I ′ is defined as the rotatedinsertion point in primary position I according to (4.50).

From Fig. 4.13, it can be seen that the rotated primary position muscle action circle that is definedby the pulley model is no longer the shortest possible path between the insertion and the origin


Figure 4.12: Primary Position in Pulley Model, from [BKPH03]

on the surface of the bulbus. Therefore, three new vectors { ~GX, ~GY , ~GZ} are calculated whichform the basis of a new muscle action circle representing the shortest path between the insertionand the pulley. These vectors are calculated with the same equations shown in (4.69)-(4.71), butwith I replaced by I ′, using,

~GZ =~I ′∣∣∣~I ′∣∣∣ , (4.73)

~GY =~P × ~I ′∣∣∣~P × ~I ′

∣∣∣ ,~GX = ~GY × ~GZ.

There are three different muscle action circles which can be determined with a set of givenequations. These circles are the primary position muscle action circle, the rotated primaryposition muscle action circle in tertiary and secondary positions (zero side-slip) and the shortestpath muscle action circle in tertiary and secondary positions (full side-slip).

The two different muscle action circles in secondary and tertiary positions are important, becausethe „correct“ muscle action circle of a specific muscle in secondary and tertiary positions liessomewhere between these two (see Fig. 4.14). In order to determine the actual muscle actioncircle, the full side-slip angle Ψ is calculated, which lies between the rotated primary positionmuscle action circle and the shortest path muscle action circle measured at the insertion point.The angle between the two muscle action circles is calculated using,

Ψ = tan−1

(− ~GX · ~TY− ~GX · − ~TX

). (4.74)


Figure 4.13: Tertiary Position in Pulley Model, from [BKPH03]

However, as already mentioned before, the „correct“ muscle action circle lies somewhere betweenthe rotated primary position muscle action circle and the shortest path muscle action circle.Therefore, the angle Ψ has to be reduced by an empirical side-slip scaling parameter defined asα. This scaling parameter was fitted to comply to other models (i.e. Orbit [MR84]) and describesthe rotational force due to muscle width that tends to rotate the eye around the axis ~I ′. Thedifferent values for α of each muscle have been taken from the EyeLab model (cf. [PWD00]) andare listed in Tab. 4.1.

Muscle Side-slip Scalar (α)medial rectus 0.3443lateral rectus 0.2909inferior rectus 0.2954superior rectus 0.2850inferior oblique 0.0722superior oblique 0.1240

Table 4.1: Side-slip Scaling Values for Pulley Model

Using Eqn. 4.74 for calculating Ψ and the side-slip scaling parameter α, the side-slip angle θ canbe calculated using,

θ = −α tan−1

(~GX · ~TY~GX · ~TX

). (4.75)

With the side-slip angle θ and the vectors ~TX and ~TY , a vector ~D can be defined which describesthe direction in which the muscle departs from its insertion (see Fig. 4.15). This vector ~D can


Figure 4.14: Muscle Action Circles in Pulley Model, from [Kal02]

be defined as a linear combination of ~TX and ~TY such that,

~D = ~TY sin(θ)− ~TX cos(θ). (4.76)

To determine the center of the actual muscle action circle in secondary and tertiary gaze positionsa vector ~N , which is perpendicular to the actual muscle action circle and represents the axis ofrotation of the globe is defined. However, calculating ~N is easy, since I ′, P and ~D are all lying inthe plane of the muscle action circle and therefore can be used for calculating the perpendicularvector of the plane spanned by the vector ~D and by the vector defined through the points P andI ′ (see Fig. 4.16). Consequently, the equation for calculating ~N is given by,

~N =~D × ~I ′P∣∣∣ ~D × ~I ′P

∣∣∣ . (4.77)

Using the vector ~N , the calculation of Cc, the center of the pulley model muscle action circle, isnow possible by using,

Cc = (~I ′ ~N) ~N. (4.78)

With the calculation of Cc, the center of the muscle action circle of a specific muscle in a primary,secondary or tertiary position is defined. Moreover, the vector ~N corresponds to the rotationaxis of a specific muscle. Therefore, the only aspect remaining is the determination of the pointof tangency.

In order to calculate the point of tangency, two vectors { ~EX, ~EY } are defined, which form thebasis of the pulley model muscle action circle and are shown in Fig. 4.17. These two vectors are


Figure 4.15: Muscle Direction Vector in Pulley Model, from [Kal02]

formed using P , Cc and ~N according to,

~EX =~P − ~Cc∣∣∣~P − ~Cc

∣∣∣ , (4.79)

~EY =~N × ~EX∣∣∣ ~N × ~EX

∣∣∣ .With the help of ~EX and ~EY , the angle β which describes the angle between the vector fromCc to I ′ and the vector from Cc to P , can be defined. The angle β is also shown in Fig. 4.17 andis calculated using,

β = tan−1

(~I ′ · ~EY~I ′ · ~EX

). (4.80)

However, in order to determine the point of tangency, the calculation of the angle γ is necessary(see Fig. 4.17). Therefore, the radius of the pulley model muscle action circle is required and thelength of the vector from Cc to P . The radius r of the muscle action circle and the length l ofthe vector from the pulley to the center of the muscle action circle can be calculated using,

l =∣∣∣~P − ~Cc

∣∣∣ , (4.81)

r =∣∣∣~I ′ − ~Cc

∣∣∣ .Since the angle between the vector from the point of tangency to Cc and the vector from thepoint of tangency to the pulley P must be exactly 90 ◦, implied by the definition of the tangent


Figure 4.16: Muscle Rotation Axis and Action Circle Center in Pulley Model, from [Kal02]

function, the angle γ can be calculated by,

γ = 90 ◦ − sin−1(rl

). (4.82)

Thus, the angle between the insertion and the point of tangency at the center of the muscleaction circle is (β − γ). With this angle, the point of tangency t can be calculated by rotatingthe point I ′ around an axis defined through the vector ~N using a rotation quaternion rq of theform (4.32),

rq = [β − γ, ( ~N)]. (4.83)

Next, the rotation quaternion rq is used to rotate the point I ′ in the muscle’s direction of pullby (β − γ) degrees and the resulting quaternion is assigned to the point T ,

T = V ect(Rot(~I ′, rq)). (4.84)

4.4.3.3 Evaluation of Muscle Action

In order to complete the definition of extraocular geometry, one needs also to consider musclesand their directions of pull. A muscle’s geometric description is based on definition of importantreference points, while the muscle’s direction of pull is determined by the definition of a rotationaxis, around which the muscle would rotate the eye. The geometric description of a muscle isdefined by a muscle path from its origin to its insertion and how this muscle path changes withdifferent gaze positions. The illustration in Fig. 4.18(c) shows the medial rectus muscle definedby muscle origin, pulley, point of tangency and insertion. The pulley stabilizes the muscle path in


Figure 4.17: Point of Tangency in Pulley Model, from [BKPH03]

the rear orbit. The point of tangency marks that range, at which the muscle path first contactsthe globe. Reactions of the muscle path to changes in gaze position are differently represented bydifferent models. Thus, substantial differences exist in the definition of the muscle path betweenstring, tape and pulley models. The „string model“ and the „tape model“ use origin, point oftangency and insertion to define the muscle path, whereas the pulley model additionally considerthe pulley point.

(a) „String Model“ Path (b) Tape Model Path (c) Pulley Model Path

Figure 4.18: Muscle Path Comparison using Different Geometrical Models from [BKP+03]

In comparing the muscle path representation of these models (see Fig. 4.18), noticeable differencesin the movement of the point of tangency occur. In the „string model“ (Fig. 4.18(a)) as well as inthe „tape model“ (Fig. 4.18(b)), the point of tangency and thus also the entire rear muscle pathis pulled downward. This also substantially affects the direction of pull in other gaze positions.Using the pulley model (Fig. 4.18(c)), stabilization of the direction of pull of the muscle is reachedby introducing a new reference point (pulley), and thus a substantially more realistic result occurs.

To geometrically evaluate the rotational behavior of a muscle according to clinical methods, therotation axis can be split into its respective components of action. This results in specifyingnormalized rotational magnitudes of rotational action for ab-/adduction, elevation/depressionand in-/extrorsion. These rotational actions are measured within an oblique coordinate system


{ ~D1, ~D2, ~D3} where ~D3 denotes the head-fixed vertical axis, ~D1 describes the head-fixed horizon-tal axis that is rotated around ~D3 according to ab-/adduction movements and ~D2 correspondsto the line of sight. Thus, the orientation of this new coordinate system depends only on ab-/adduction and elevation/depression movements of the globe. In order to measure muscle actionin a certain eye position, the rotation axis of a muscle, defined using string, tape or pulley modelneeds to be transformed into the new coordinate system.

Let {α, β, γ} denote an eye position according to the Fick rotational sequence and ~RA be anormalized muscle rotation axis according to the string, tape or pulley model (4.53, 4.67 or 4.77).Using the rotation quaternions qrx and qrz from (4.47), the rotation quaternion,

qt = qrxqrz, (4.85)

specifies the rotational transformation of coordinates from head-fixed space into the new mea-surement coordinate system. Therefore, the rotation axis ~RA(x, y, z) can be transformed into anaxis ~RA′(x′, y′, z′) using,

~RA′ = V ect(Rot( ~RA, qt)), (4.86)

where z′ represents the ab-/adducting rotational component, x′ represents theelevating/depressing component and y′ specifies the torsional rotation. Since ~RA′ is a unitvector, all rotational components lie between −1 and 1.

The muscle force distribution shows this relative rotational components for selected eye musclesalong a horizontal view range (see Fig. 4.19) in a certain elevation/depression level. The ro-tational components are indicated in standardized values between -1 and 1 for ab-/adduction,elevation/depression and in-/extorsion. The illustration in Fig. 4.19 show the force distributionsof the lateral rectus muscle in the „string“ and „tape“ models with an elevation of 15 ◦ along thehorizontal field of vision with up to 60 ◦ of abduction/adduction of a left eye.

Here, the physiologically incorrect prediction of the „string model“ becomes clearly evident at 36 ◦

of adduction. The lateral rectus muscle drastically changes its abducting effect into adduction(see Fig. 4.19(a)). The comparison with the „tape model“ in Fig. 4.19(b) shows better behaviorfor accurately the same scenario due to retention of the main direction of pull of the lateralrectus. These differences result from the differentiated mathematical modelling of the anatom-ical structures. While in the „string model“ the muscle path within its contact range with theglobe was defined by the shortest path between insertion and origin, the „tape model“ containsan angle-reducing component, which describes the muscle path by the relative motion of thepoint of tangency as a function of gaze position. This allows the simulation of stabilizing connec-tive tissues, in order to limit the movement of muscles in extreme gaze positions. Graphically,this simulation results in a bent muscle path between insertion and point of tangency, whichsubstantially better corresponds to anatomical conditions.


(a) „String Model“ Distribution

(b) Tape Model Distribution

Figure 4.19: Muscle Force Distribution in String and Tape Model from [BKP+03]

Figure 4.20: Muscle Force Distribution in the Pulley Model

Only through the introduction of a model that includes pulleys, more detailed simulations andpathological case studies can be accomplished (see Fig. 4.20). The simulation of pulleys showsan absolute retention of a muscle’s main action in the entire physiological field of vision and isthus best suitable for being used as geometrical representation.

4.4.4 Passive Geometrical Changes

Passive changes in ocular geometry occur when some muscles move the globe into a differentgaze position. Due to the rotation of the globe, muscle length changes in every eye muscle,since a contracting muscle shortens and an antagonistic muscle lengthens during eye movement.These geometric variations are very essential to the biomechanical model because muscle length


and length changes are major parameters that affect muscle force prediction. The mathematicaldescription of muscle length can be split into the sum of the contact arc length and the lengthof the posterior muscle part. The length of the contact arc can be calculated as the sphericaldistance along the muscle action circle between the insertion point and the point of tangency.Let ~T and ~I ′ be the vectors from the center of the muscle action circle to the insertion and originpoint. Then, the radius of the muscle action circle is defined as,

rad =∣∣∣~I ′∣∣∣ , (4.87)

and the angle λ between the two vectors can be calculated using,

λ = cos−1(~I ′ · ~T ). (4.88)

Using the radius and the angle λ, the spherical distance on the muscle action circle betweeninsertion and point of tangency is,

darc = rad · λ. (4.89)

For the pulley model, the length of the posterior part of the muscle is also calculated in two steps.First, the length from the origin point of the muscle to the pulley point is given by the length ofthe vector ~PO,

l1 =∣∣∣ ~PO∣∣∣ . (4.90)

Then, for the length from the pulley point to the point of tangency, an adapted formula from(4.51) for the shortest path distance is used,

l2 =√~P · ~P − ~I ′ · ~I ′ − Tl, (4.91)

where ~P is the vector from the center of the globe to the pulley point, ~I ′ is the vector from thecenter of the globe to the insertion point and Tl denotes the length of the tendon which is givenin Sec. 2.1.3. The subtraction of Tl is important, since the net muscle length is measured withoutthe muscle tendon.

For geometrical models that do not use pulleys, the posterior part of the muscle length is simplycalculated using the shortest path distance between origin and point of tangency using,

l =∣∣∣ ~O − ~T

∣∣∣− Tl. (4.92)

Finally, the overall muscle length ml can be computed using,

ml = darc + l1 + l2, (4.93)

for the pulley model, and,ml = darc + l, (4.94)

for any other models that do not use pulleys.

Based upon these calculations, the passive length change with respect to geometrical measure-ments from Sec. 2.1.3 can be computed. The passive length change dl is always measured inpercent, based on the geometrical muscle length in primary position, using,

dl =100 · (ml − L0)

L0. (4.95)


4.5 Muscle Force Prediction

Besides ocular geometry, muscle force prediction is another essential part of the overall biome-chanical model. The muscle force prediction model has the function of providing forces for eachmuscle, based on muscle length, tension and actual innervation.

For skeletal muscles, many different modelling approaches can be found in literature. Mostmodels are devoted to muscle force prediction in that they only provide models for estimatingthe global uniaxial output force of given muscles in defined conditions and experiments (cf.[MWTT98] and [KS98]). Conversely, other models are concerned with the understanding of thecontractile mechanism, and describe the chemico-mechanical aspects of the contraction process,but are hardly related to a realistic global output force involving the 3D anatomical and passiveproperties of a specific muscle. Only few studies attempt to provide a model of muscles includinganatomical and mechanical, active and passive properties, allowing for a realistic simulation ofits contractile behavior in relation with its deformation and its global output force (cf. [BL99]and [BCL99]).

However, since all of these studies primarily investigated skeletal muscles, results cannot directlybe related to eye muscles. Further, since the anatomy of human eye muscles is still not fullyexplored, models that take fiber-types and contraction velocities into account cannot be adaptedto produce accurate results due to currently many unknown properties of human eye muscles.Therefore, for current proposes, a static model of eye muscle force prediction using the bestavailable data is preferable. The static muscle force prediction model presented in this thesisis adapted from studies conducted by Miller and Robinson (cf. [MR84] and [Rob75b]), takinglength-tension-innervation relationships into account.

4.5.1 Length-Tension Relationship

As mentioned before, muscle force depends on muscle length and tension. Tension is a function oftwo variables, namely muscle stretch and muscle innervation. The more a muscle is stretched thehigher the tension, the second variable is motor command or innervation. High motor commandslead to muscle contraction which also increases tension. Based upon measurements that werecarried out by Robinson et al. [ROS69], Collins et al. [CSO69] and Miller (see Sec. 2.3.2), thelength-tension relationship was modelled for a given stretch and motor command by a hyperbola.This length-tension relationship was modelled according to the following equation,

Fi(dli, ei) = λi

(k

2(dli + ei) +

√k2

4(dli + ei)

2 + a2

). (4.96)

Here k is the asymptotic slope (equal to stiffness for large extensions) and a determines thesharpness of curvature of the transition to the „slack“ state. Miller and Robinson approximatedthe values of k = 1.8g and a = 6.24g to fit to the measured data. The other parameters in (4.96)are indexed by muscle number. Changes in muscle length dl is expressed as a percentage of thelength of the muscle in primary position using Eqn. 4.95. Thus, Eqn. 4.96 can be applied to allmuscles with respect to their differing reference lengths (L0). The parameter ei translates thewhole curve left or right along the length axis. It thus simulates a change in innervation. It


is thought that muscle force is linearly proportional to the average muscle cross sectional area,accordingly, the parameter λi is used to scale the lateral rectus muscle length-tension hyperbola.Therefore, λ is 1 for the lateral rectus muscle and the scaling parameter for each of the otherfive muscles is taken to be its cross sectional area relative to that of the lateral rectus, shown inTab. 4.2.

Muscle λ

Lateral Rectus 1.00Medial Rectus 1.07Superior Rectus 0.80Inferior Rectus 0.97Superior Oblique 0.41Inferior Oblique 0.38

Table 4.2: Cross Sectional Muscle scaling Parameter Values

The remaining parameters dli and ei are now used to formulate the muscle force prediction basedupon experimental data. The parameter ei slides the force curve along the length axis andtherefore influences the output force when the length of a muscle does not change. This is relatedto a simulation of changing innervation, thus, a mathematical formulation for innervation can bederived from the parameter ei by using,

ivi(ei) = Fi(0, ei)− F (0, e0), (4.97)e0 = −21.7264.

The first term in (4.97) describes the total isometric force for zero length change and the secondterm describes the passive isometric muscle force for zero length change. The net change in force,and therefore also innervation can be determined in subtracting the passive isometric force fromthe total isometric force. In Eqn. 4.97, e0 is that constant value of e that fits the passive forcecurve from the force data. Thus, changes in e0 modifies default passive force behavior for allmuscles as initial force displacement ratio in units of percent length changes with respect to L0

from (4.95). Since this mathematical derivation of innervation is based upon the parameter e ofEqn. 4.96, the units for iv are arbitrary. However, it can be seen that a change in innervationalters the total force output by the same amount as a numerically equal change of dl.

In using this muscle force prediction model, two fundamental problems arise. When a detachedmuscle is stretched sufficiently it stops behaving like a non linear spring and becomes stiff veryquickly. This so called leash region is not incorporated in the length-tension relationship fromEqn. 4.96. Conversely, if the muscle is shortened sufficiently it will become slack and exertno force. However, the hyperbolic nature of Eqn. 4.96 ensures that the tension for muscleswill never become zero. Miller and Robinson addressed these problems by creating three forcesurfaces corresponding to elastic (passive), contractile (developed) and their sum total force thatwere modified by hand to fit experimental data. The relationship between elastic force FE andcontractile force FC can be defined as,

FT (dl, iv) = FE(dl) + FC(dl, iv), (4.98)

where F represents the total force curve and iv defines innervation according to Eqn. 4.97.


4.5.2 Elastic Force Data

Elastic force is that force that a muscle exhibits when it is stretched or compressed due to itsmaterial properties. Thus, the elastic force data is that part of Eqn. 4.96 that is only sensitive tolength changes dl. The curve consists of static values for passive isometric tension values givenat different muscle lengths (cf. Sec. 2.3.2).

Figure 4.21: Elastic Force Data

It is easy to see from Fig. 4.21 that the elastic force function is constant throughout the innervationaxis (iv), thus only depends on the length change of the muscle (dl axis). To incorporate theleash and slack regions, the dl axis was split into three regions. In the region 17% ≤ dl ≤ 40%,elastic force was simply calculated using Eqn. 4.96,

FE(dl) = F (dl, e0), 17% ≤ dl ≤ 40%. (4.99)

In the region dl < 17%, muscle force was smoothly tapered by hand to zero in order to reflectthe slack region. Similarly for dl > 40%, force was extrapolated to an arbitrarily large value tomodel the rapid increase in stiffness exhibited in the leash region. From Fig. 4.21 it can be seenthat at a value of dl >= 54, force data stays constant at 82g, which simulates rigid behavior ofthe muscle.

Muscle data for the leash and the slack regions are stored as tables that are used for interpolation.Let,

TSd[x] = (x1, x2, x3, . . . , xn), (4.100)

denote the table that stores the dl values of the slack region and,

TSf [y] = (y1, y2, y3, . . . , yn), (4.101)


denote the table that stores the force value for each xi. Then, any real value x2 ≤ u ≤ xn−2 canbe found by using a cubic interpolation for an index i, such that,

xa ≤ TSd[xi] ≤ xb, 2 ≤ a, b ≤ n− 2. (4.102)

Based on 4 sample points (v1, v2, v3, v4) = (TSf [xi−1], TSf [yi], TSf [yi+1], TSf [yi+2]), the approxi-mated force value FSa can be calculated using,

a0 = v4 − v3 − v1 + v2,

a1 = v1 − v2 − a0,

a2 = v3 − v1, (4.103)a3 = v2,

u′ =1

(xb − xa)(u− xa)

FSa(u) = u′3a0 + u′

2a1 + u′a2 + a3.

For the leash region of the elastic muscle data, Eqn. 4.100 - Eqn. 4.103 is used with the respectivetable data that extrapolates force values up to 82g, and according to the cubic interpolation, afunction FLa(u) can be defined in the same manner as Eqn. 4.103.

Using these interpolation values, the elastic force function can be completed in addition toEqn. 4.99 as,

FE(dl) =

FSa(dl), dl < 17%,FLa(dl), dl > 40%,

82, dl ≥ 54%.(4.104)

In order to simulate muscle dysfunction concerning elastic force, FE is additionally scaled by anelastic force scaling factor es such that,

FE(dl, es) = FE(dl) · es, 0 ≤ es ≤ 2. (4.105)

4.5.3 Contractile Force Data

Contractile muscle force can be defined as a function of innervation iv and length change dl.Based on Eqn. 4.98, the contractile force function FC can also be extracted from Eqn. 4.96with additional manual changes. Compared to force data analysis of human skeletal muscles,contractile force in eye muscles does not fall off when muscle length increases above normal. Thiscould be due to the sudden rise of FE at large dl. However, it is obvious, that for large dl values,the total output force is dominated mainly by the elastic force function FE . On the other hand,contractile force FC needs to be zero when innervation iv is zero. Thus, the contractile muscleforce function, shown in Fig. 4.22, is calculated in three regions.

In the first region, where −20% ≤ dl ≤ 45%, 0g ≤ iv ≤ 100g, the contractile force is calculatedusing Eqn. 4.96 by,

FC(dl, iv) = F (dl, iv)− FE(dl), −20% ≤ dl ≤ 45%, 0g ≤ iv ≤ 100g. (4.106)


Figure 4.22: Contractile Force Data

Here, the first term evaluates the total force using Eqn. 4.96 and the second term uses Eqn. 4.99to subtract the elastic force to get the contractile output force based on Eqn. 4.98. The secondregion is defined as −20% ≤ dl ≤ 45%, 0g ≤ iv > 100g and is interpolated in a manuallygenerated force data table. To reflect saturation of innervation, contractile force is restrainedat a maximum value of approximately 100g for high iv. Bicubic interpolation is used as tablelookup strategy. Let,

TCd[x] = (x1, x2, x3, . . . , xn), (4.107)

denote the table that stores the dl values of the interpolation region and,

TCi[y] = (y1, y2, y3, . . . , yn), (4.108)

denote the table that stores the innervation values of this region. Then, a two dimensional tableof the form,

TCf [x, y] = (z1, z2, z3, . . . , zn), (4.109)

can be defined which stores the force values for each combination of xi and yi such that for anyreal values x4 < u < xn−4 and y4 < v < yn−4, an interpolated force value in the table TCf can befound by using an average of 16 points, surrounding the closest corresponding value. Therefore,two indices xi and yi for Eqn. 4.109 can be found for u and v respectively, such that,

xa ≤ TCd[xi] ≤ xb,ya ≤ TCi[yi] ≤ yb, 4 ≤ a < b ≤ n− 4.

(4.110)

Using xi and yi as reference point, a weighted sum of force values of 16 surrounding points canbe calculated using,

FCa(u, v) =2∑

m=−1

2∑n=−1

TCf [xi+m, yi+ n] ·R(m− dx) ·R(dy − n), (4.111)


where dx = TCd[xi] − u and dy = TCd[yi] − v denote the real differences between table valuesand target coordinates u and v. Finally, R is the cubic weighting function that defines how mucheach of the 16 points that are used for interpolation contributes to the final value,

R(x) = 16 · [P (x+ 2)3 − 4P (x+ 1)3 + 6P (x)3 − 4P (x− 1)3],

P (x) ={x, x > 0,0, x ≤ 0.

(4.112)

In the remaining section, where dl > 45%, force in the muscle is taken as that value for dl = 45%,which has already been defined. Hence, the definition for the contractile force function, can becompleted from Eqn. 4.106 in adding,

FC(dl, iv) ={FCa(dl, iv), −20% ≤ dl ≤ 45%, 0g ≤ iv > 100g,FC(45, iv), dl > 45%.

(4.113)

In order to simulate muscle dysfunction concerning contractile force, FC is additionally scaled byan contractile force scaling factor cs such that,

FC(dl, iv, cs) = FC(dl, iv) · cs, 0 ≤ cs ≤ 2. (4.114)

4.5.4 Total Force Data

The total force of a muscle can be calculated by summing elastic and contractile force data. Thisresults in a total force function FT that depends on innervation iv and length change dl, usingEqn. 4.105 and Eqn. 4.114 such that,

FT (dl, iv, es, cs, ts) = ts · (FE(dl, es) + FC(dl, iv, cs)), 0 ≤ ts ≤ 2. (4.115)

In Eqn. 4.115, the parameter ts indicates the total force scaling parameter, which provides thesimulation of muscle dysfunction that affects all properties of muscle force.

(a) Elastic Force

+

(b) Contractile Force

=

(c) Total Force

Figure 4.23: Total Force Data

In Fig. 4.23, the addition of elastic and contractile force is shown. The resulting total force curve(Fig. 4.23(c)) shows the dominating elastic leash region (Fig. 4.23(a)) for high values of dl and


innervation dependent muscle force (Fig. 4.23(b)) that is generated with rising iv. In using thisforce prediction model, Eqn. 4.115 provides different scaling parameters to modify the behaviorof a muscle in pathological situation.

Elastic strength (es) scales the passive force function in order to influence the elastic propertiesof a muscle. A reduction of this value, for example, results in a muscle with a reduced elasticforce when the muscle stretches.

Contractile strength (cs) scales the active force function and simulates either overaction or amuscle palsy in relation to the cerebral innervation. A value of zero would correspond to acomplete muscle palsy (complete loss of the contractile strength).

Muscle strength (λ) scales the total force function (passive and active force). For this value,default values are used which describe the different scaling of all muscles in relation to thelateral rectus muscle (see Tab. 4.2). If this value is changed, the muscle is strengthened orweakened relative to the lateral rectus muscle. In order to scale the total force in relationto the current muscle only, one should change the muscle’s total strength.

Total strength (ts) scales the total force function in relation to the current muscle, but onlyafter the total muscle force has already been scaled by the factor λ.

For the simulation of surgeries, the length of a muscle is very important. The length of the muscleand the tendon also changes the way how the simulation interprets the muscle force curves. Asan example, a rubber band with a length of 10mm (relaxed length L0 = 10mm) is stretched by10mm, which results in a path length of 20mm and a relative stretch of 100%. Now, if the bandis shortened to 5mm relaxed length, and once again stretched by 10mm, the relative stretch isnow 200%. This would lead to a totally different resulting force in the force model and shows,how the modification of the muscle length can influence force behavior of a muscle. Referring toSec. 4.4.4, the geometrical model offers different parameters to modify muscle length.

Muscle length (L0) defines the length of a relaxed, denervated muscle without tendon in mm.

Tendon length (Tl) defines the additional length of the tendon in mm.

To simulate a resection of the lateral rectus muscle by 5mm, the length of the muscle and/or thetendon has to be changed. When the insertion of the lateral rectus muscle is disinserted from theglobe, the muscle is shortened by 5mm and afterwards reattached again at the same position. Incarrying out this procedure, 1mm of muscle length is lost during the separation and 1mm duringthe fixation of the length of the muscle tendon. Since the model cannot predict this conditionautomatically, it is necessary to shorten the muscle by 7mm in this situation. Therefore, thetendon length of the lateral rectus muscle is changed from 8.4 to 1.4mm. If a larger resectionis carried out, it is possible that the muscle loses its tendon (Tl = 0) and in such a case, theremaining shortening has to be applied to the muscle length parameter (L0), consequently bothparameters have to be changed.


4.6 Kinematics

Using the orbital geometry from Sec. 4.4.3 and the muscle force prediction from Sec. 4.5, thekinematics of the biomechanical model can be defined. The goal of the kinematic model is torelate geometry and muscle force prediction in such a way that eye positions and innervations canbe derived from given parameterizations of one eye. Thus, the kinematic model is responsible forthe correct transformation between muscle simulation and geometrical representation. Startingpoint for the derivation of the kinematic model is the complete definition of data for one eye.This reference data set is given by a geometrical model and the muscle force prediction model forall 6 eye muscles together with initial values for all parameters. Then, two essential operationsthat conform to forward and inverse kinematics, need to be defined.

Forward Kinematics defines the operation that resolves an eye position based upon a givenset of innervations for all eye muscles.

Inverse Kinematics defines the operation that resolves a set of innervations for all eye musclesthat are required to drive the eye into a given position.

To solve the kinematic operations, a mathematical definition of a stable eye position is required.Such a stable eye position is only reached, when all forces that act on the globe are in anequilibrium. To measure if an equilibrium is reached, a force balance equation is defined, thatderives a torque imbalance vector based on the current eye position and muscle forces. Thetorque imbalance vector ~t can therefore be derived using,

~t =6∑

i=1

FTi(dli, ivi, esi, csi, tsi) · ~ni, (4.116)

where FTi denotes the total output force of a muscle i based on Eqn. 4.115, and the vector ~ni

denotes the unit moment vector from Eqn. 4.77. A stable eye position can therefore be definedas the constraint, ∣∣~t∣∣ ≈ 0. (4.117)

It is however not enough to just balance force of the eye muscles to describe a stable eye position.Additional moments that arise from orbital restoring forces and globe translation that alter forcesand rotational behavior must also be considered.

4.6.1 Orbital Restoring Force

In addition to the torque that is exerted by the muscles as the eye moves further away fromprimary position, it encounters resistance from non-muscular elastic tissues in the orbit such asTenon’s Capsule which acts to restore the eye to primary position. This resistance can be definedby a passive moment vector ~P (Pα, Pβ , Pγ), where the components of the vector correspond toangular coordinates (α, β, γ) based on the angular coordinates of the current eye position usingEqn. 4.9. In modelling the orbital restoring force, a simple nonlinear spring model of the form,

K(w,w′) = K1 · w +K2 · w′3. (4.118)


is used, where K1 and K2 denote the spring constants. The torsional restoring force that acts inthe opposite direction of γ can therefore be denoted as,

Pγ = Kt · γ +K ′t · γ3. (4.119)

In Eqn. 4.119, Kt and K ′t are the torsional stiffness constants, taken from [MR84], as Kt = 0.5g/ ◦

and K ′t = 324µg/ ◦3.

Similarly, the orbital restoring force acting in the opposite direction of α and β can also beformulated with respect to Eqn. 4.118,(

Pα

Pβ

)= K ·

(α

β

)+K ′ · (α2 + β2) ·

(α

β

), (4.120)

where the spring constants K and K ′ were also taken from [MR84], as K = 0.25g/ ◦ and K ′ =81µg/ ◦.

The orbital restoring torque vector ~P needs to be considered when balancing forcesusing Eqn. 4.116.

4.6.2 Globe Translation

Globe translation is the anteroposterior movement of the globe during eye movements (see 2.2.1).Usually the eye’s center of rotation remains fixed within the head-fixed coordinate system. How-ever, in pathological situations, globe translation can be of great value in diagnosis and treatment,and therefore it is of interest to the biomechanical model. When the globe translates during ro-tation, the center of rotation is shifted and thus, the axis of rotation is modified. Additionally,globe translation alters the relative position to the muscle origin and therefore also the stretchof each muscle, which in turn modifies output force and force balance in Eqn. 4.116.

In order to measure globe translation, a new coordinate system is defined that describes theorbital cone through the definition of an apex point that lies midway between the origin pointof the superior rectus (Osr) and inferior rectus (Oir) muscles (see Fig. 4.24). Let { ~Hx, ~Hy, ~Hz}denote a head-fixed coordinate system, and let ~V = ~Osr+ ~Oir

2 , denote the apex point in theposterior region of the orbit. The new apex coordinate system can then be defined by three basevectors { ~Ax, ~Ay, ~Az},

~Ay = ~V ,

~Ax = ~Hz × ~Ay

~Az = ~Ax × ~Ay,

where the base vector ~Ay represents the vector from the origin to the apex point, the vector~Ax is perpendicular to the head-fixed vertical axis and the vector ~Ay and the vector ~Az isperpendicular to ~Ax and ~Ay. Then, the rotation quaternions that transform between head-fixedand apex coordinate systems can be formulated using two rotation angles ψ and ω, such that,

ψ = cos−1( ~Ay · ~Hy),

ω = cos−1( ~Az · ~Hz),


Figure 4.24: Apex Coordinate System for measuring Globe Translation

where ψ denotes the angle between the head-fixed axis ~Ey and the apex axis ~Ay, and ω denotesthe angle between the two vertical axes ~Ez and ~Az. The forward transformation to the apexcoordinate system can therefore be represented as combined rotation in the form of (4.32), aroundthe head-fixed axes ~Hx and ~Hz,

qapex = [ω, ( ~Hx)] · [ψ, ( ~Hz)]. (4.121)

The reverse transformation into the head fixed coordinate system can be formulated using theinverse rotation of (4.121),

qhead = qapex−1. (4.122)

To measure globe translation, a relationship between the torque imbalance equation (4.116) andthe amount of translation must be defined. This is reached by the introduction of a stiffness vector~Fa = (54, 27, 54), that is linearly related to translation values. Thus, the translation vector ~Gtrans

can be found by first transforming the torque imbalance vector ~t to the apex coordinate system,and then restrict the length of this vector using the stiffness values ~Fa, which leads to,

~Gtrans =V ect(Rot(~t, qapex))

2 ~Fa

. (4.123)

The amount of globe translation gt along the axis ~Ay for a given eye position can then be definedas,

gt =∣∣∣ ~Gtrans

∣∣∣ , (4.124)


and the translation that affects ocular geometry can be represented by a vector ~Trans that isdefined with respect to the head-fixed coordinate system using,

~Trans = V ect(Rot( ~Gtrans, qhead)). (4.125)

This translation vector ~Trans can now be used to modify ocular geometry in order to reflectrotational changes. Therefore, the vector ~−Trans is used to translate the origin point O and thepulley point P of each muscle (see Sec. 4.4.3.2) in the opposite direction, prior to calculating thetorque imbalance vector ~t, using Eqn. 4.116. Instead of translating the whole ocular geometry,the pulley and the origin point are translated in the opposite direction, which results in the samerelative effect. In order to formulate this modification, the calculation of the muscle rotationaxis within the ocular geometry is necessary. Therefore, Eqn. 4.53, Eqn. 4.67 and Eqn. 4.77 thatdefine the muscle rotation axes in the string, tape and pulley model respectively, need to bemodified, so that each rotation axis is calculated with respect to a displaced pulley and muscleorigin such that,

Gt( ~Trans, ~RA) := ~RA→ ~−Trans, (4.126)

where ~RA denotes any muscle rotation axis from Eqn. 4.53, Eqn. 4.67 or Eqn. 4.77, and theoperator → ensures that these rotation axes are calculated with modified pulley and origin data.

4.6.3 Balancing Forces

In order to find stable eye positions, it is necessary to refine the torque imbalance equation (4.116)to additionally consider orbital restoring forces and globe translation, since eye muscle are not theonly actors that exert force onto the globe. This leads to the refined torque imbalance equation,

~T = ~P +6∑

i=1

FTi(dli, ivi, esi, csi, tsi) ·Gt( ~trans, ~rai), (4.127)

where the force of each eye muscle is multiplied by the unit moment vector which is calculatedwith translated origin and pulley points (Gt), and ~P denotes the orbital restoring force, thatmodifies the global rotational balance using Eqn. 4.119 and Eqn. 4.120. Each value for the torqueimbalance vector ~T in Eqn. 4.127 is mainly determined by six innervations (~iv) and six lengthchanges (~dl). Each modification to the current eye position results in different length changevalues and thus also in a different torque imbalance vector. Conversely, changing innervationsgive different force values and therefore also affect torque balance. It can therefore be seenthat different eye positions lead to different dl values by applying Eqn. 4.95 and that differentinnervation values lead to different values for e, using Eqn. 4.97. Thus, an adequate way tocontrol the value of ~T is to define an innervation vector ~Iv and an eye position vector ~Ep suchthat,

~Iv = {iv1, iv2, iv3, iv4, iv5, iv6}, (4.128)~Ep = {ex, ey, ez},

where the values {iv1, iv2, . . . , iv6} contain the innervation values for each eye muscle and thevalues {ex, ey, ez} contains a rotation vector that describes an eye position based on a rotationquaternion of the form (4.32),

~Ep =(

12· tan(θ)

)· ~U, (4.129)


where the orientation vector ~U is parallel to the axis of rotation and the length is given by halfthe tangens of the rotation angle in radians. In order to transform an eye position ~Ep into a6-vector of dl values, passive length changes need to be calculated, based on ~Ep by applying themethod described in Sec. 4.4.4.

A rotation vector ~Ep of the form (4.129) can be transformed into a quaternion using,

Q( ~Ep) = [cos(tan−1(| ~Ep|)), sin(tan−1( ~Ep))], (4.130)

and,

R(q) = tan(cos−1(Scal(q)) · V ect(q), (4.131)

gives the rotation vector from the quaternion q using Eqn. 4.12.

However, before the minimization can be applied, the degrees of freedom for the innervationaldependency of the objective function needs to be reduced, since the muscles are always working inpairs according to Sherrington’s law described in Sec. 2.2.3.8, Sec. 2.2.1 and Sec. 3.5.2. Assumingthat the innervation vector Iv has the following form,

~I ′v = {ivRL, ivRM , ivRS , ivRI , ivOS , ivOI}, (4.132)

where agonist muscle innervations correspond to odd, and antagonistic muscle innervations cor-respond to even vector indices, then, only agonist muscle innervations need to be specified, sincethe antagonist muscle innervations can be found by using Sherrington’s law of reciprocal innerva-tions. Thus, the degrees of freedom are reduced from 6 to 3, where antagonist muscle innervationsare dictated by innervations specified for agonist muscles.

Sherrington’s law of reciprocal innervations can be applied over ~I ′v by,

Ive(Ivo) =(h+ w)2

Ivo + w− w, (4.133)

where Ive and Ivo are 3-vectors of even and odd innervations from ~I ′v respectively, and w = 7.6, h =14.3 are constants taken from [Rob75b] and fit experimental data. Thus, a new innervation vector~Iv can be defined as,

~Iv = {Ivo(1), Ive(1), Ivo(2), Ive(2), Ivo(3), Ive(3)}, (4.134)

resulting in a 6-vector of innervations that is determined by Ivo only.

Thus, a stable eye position depends on the parameter vectors ~Iv and ~Ep where the model param-eters ix and ex dictate the model function defined in Eqn. 4.127. This can easily be seen whenlooking at the muscle force prediction model in Sec. 4.5, where Eqn. 4.96 is a function where theparameters are non linearly dependent on the function values.

To quantify how well model parameter values in ~Iv and ~Ep approximate a stable eye position,Eqn. 4.127 can be used such that the squared length LT of the vector ~T gives information aboutkinematic instability,

LT (~Iv, ~Ep) =

∣∣∣∣∣~P +6∑

i=1

FTi(dli, ivi, esi, csi, tsi) ·Gt( ~Trans, ~RAi)

∣∣∣∣∣2

, (4.135)


Figure 4.25: Torque Error Function for Listing Positions in a Healthy Eye

where the dli values are calculated from ~Ep using the equations from Sec. 4.4.4.

When plotting 2D eye positions of the form (4.46), the error function (4.135) can be visualized.

Fig. 4.25 shows the torque error function for eye positions between −30 ◦ and 30 ◦ ab-/adductionand elevation/depression, where the torsional position is implicitly specified using Listing’s tor-sion from Eqn. 4.45. It can be seen that the minimum of this function also specifies the bestapproximation to a stable eye position, thus, the desired goal is a minimization of Eqn. 4.135which can be reached by finding values for ~Iv and ~Ep such that,

minLT (~Iv, ~Ep). (4.136)

The non linear manner of the error function (4.135) becomes evident when the ocular geometry ischanged to some pathological situation. The curve that is shown in Fig. 4.26 depicts the torqueimbalance function when the lateral rectus muscle has a pathological muscle path (i.e. the muscleinsertion is transposed towards the superior rectus muscle).

In comparing Fig. 4.25 and Fig. 4.26, it can be seen that the shape of the curve changes drasticallywith different geometrical or muscle force prediction values. Thus, the procedure for finding theminimum in Eqn. 4.135 needs to be as stable as possible. Additionally, the structure of themathematical formulation also guarantees that all components of the biomechanical model canindependently be exchanged by other models without invalidating the model itself. It is thereforealso desirable to use a stable minimization approach in order to provide flexibility with respectto the mathematical model of ocular geometry and muscle force prediction.


Figure 4.26: Torque Error Function for Pathological Eye

However, the minimization algorithm that is used to find a stable eye position within the torqueimbalance equation is based upon a nonlinear least-squares problem, where Eqn. 4.135 can beused as objective function. The model is known to have the form LT (~Iv, ~Ep) and the goal is tochoose values for ~Iv or ~Ep that give the best fit in order to minimize LT . The nonlinear least-squares problem can be regarded as a general unconstrained minimization problem. In leastsquares problems, the objective function f has usually the form,

f(x) =12

m∑j=1

rj(x)2, (4.137)

where each rj is a residual and x is a column vector of the free model parameters (x1, x2, · · · , xn).Using Eqn. 4.137, a residual vector r can be defined such that,

r(x) = (r1(x), r2(x), · · · , rm(x))T , (4.138)

where T transforms the column vector r(x) into a row-based vector. The first derivatives of f(x)is denoted as the Jacobian matrix J of r,

J(x) =

∂r1∂x1

∂r2∂x1

· · · ∂rm∂x1

......

. . ....

∂r1∂xn

∂r2∂xn

· · · ∂rm∂xn

. (4.139)

The gradient (∇f(x)) is given by,

∇f(x) =m∑

j=1

rj(x)∇rj(x) = J(x)T r(x). (4.140)


The second derivatives of the objective function can be computed in the form of the Hessianmatrix and is described as,

∇2f(x) =m∑

j=1

∇rj(x)∇rj(x)T +m∑

j=1

rj(x)∇2rj(x) = (4.141)

= J(x)TJ(x) +m∑

j=1

rj(x)∇2rj(x).

The Jacobian matrix can be used to determine the direction of decent for the objective function,whereas the Hessian matrix gives information whether a current residual vector is a local min-imum. The goal of minimization is to calculate the gradient in Eqn. 4.140 and the Hessian inEqn. 4.141 and use this information to iteratively converge to a minimum of the objective func-tion. Therefore, the first part of the Hessian is already known through Eqn. 4.139 as J(x)TJ(x),and the second part of the Hessian needs to be calculated as the summation of the second partialderivatives and the residual values. However, the distinctive feature of least squares problems isthat by knowing the Jacobian and therefore being able to compute the first part of the Hessianfor free. Moreover, the term J(x)TJ(x) in Eqn. 4.141 is often more important than the secondsummation term in Eqn. 4.141, either because of nearly linear compliance of the model near thesolution when ∇2rj is very small or because of small residuals rj . Most algorithms for minimizingnon linear least squares using equations of the form (4.137) exploit these structural propertiesof the Hessian matrix. The simplest of these algorithmic methods is the Gauss-Newton method(cf. [NW99]) which excludes the second order term in Eqn. 4.141 and therefore approximatesthe Hessian matrix and uses a line search strategy to find a local minimum. The idea of a linesearch method is to use the direction of the chosen step, but to control the length, by solving aone-dimensional problem of minimizing,

φ(α) = f(αpk + xk), (4.142)

where α is the step size and pk is the search direction chosen from the position xk. The property,

φ′(α) = ∇f(αpk + xk)pk,

gives the information that if the gradient can be computed, an efficient one-dimensional searchwith derivative can be used. The search direction can be evaluated by solving,

JTk Jkpk = −JT

k rk. (4.143)

Typically, an effective line search only looks towards α > 0 since a reasonable method shouldguarantee that the search direction is a descent direction, which can be expressed as φ′(α) < 0.It is typically not worth the effort to find an exact minimum of Eqn. 4.142, since the searchdirection is rarely exactly the right direction, thus it is enough to move closer. However, eachiterative step in the minimization can be computed using,

xk+1 = xk + αkpk. (4.144)

Most rules for terminating the iterative solver are based on the residual,

rk = ∇2f(xk)pk +∇f(xk), (4.145)


and evaluate the relative change of rk with respect to each iteration in the minimization procedure.Therefore, the iterative approach can be terminated by using,

|rk| ≤ ηk |∇f(xk)| , (4.146)

where ηk could be chosen to be min(0.5, |∇f(xk)|), or any other problem dependent value withinthe interval [0; 1]. Additionally, a maximum number of iterations kmax is often used to limitnon-converging situations.

Using the Gauss-Newton approach, if J(xk) is rank-deficient for some k, the coefficient matrixin Eqn. 4.143 will become singular. However, the system in Eqn. 4.143 will still have a solution,in fact, there are infinitely many solution for pk in this case. To overcome this weakness, theline search strategy of the Gauss-Newton method is replaced by a trust-region search strategy.This results in the realization of the Levenberg Marquardt method, which still uses the Hessianapproximation in the same manner as the Gauss-Newton method, but improves convergence byusing a more sophisticated directional search method. Line search and trust-region methodsboth generate steps with the help of a quadratic model of the objective function, but they usethis model in different ways. The main difference is that line search methods use the model forgenerating a search direction, and subsequently focus on finding a suitable value for the steplength α along this direction, whereas trust-region methods define a region around the currentiterate within which they trust the model to be an adequate representation of the objectivefunction. The step within trust-region methods is then evaluated as an approximate minimumwithin the defined region. In case a step is not acceptable, the size of the region is reduced.In general, the step direction changes whenever the size of the trust-region is altered. However,choosing the size of the trust region is crucial with resect to finding a good approximation for alocal minimum. The model function for the trust-region approach can be denoted as,

mk(p) = fk +∇fTk p+

12pTBkp, (4.147)

where fk = f(xk),∇fk = ∇fk(xk) and Bk = ∇2f(xk). The derivation of Eqn. 4.147 is identicalto the first two terms of the Taylor-series expansion of f around xk due to the fact that,

f(x+ p) = f(x) +∇f(x)T p+12pT∇2f(x+ tp)p, (4.148)

can be solved for some t ∈ [0; 1], and is used to study all the points in the immediate vicinityof f(xk) in order to identify a local minimum. In Eqn. 4.147, m(p) approximates the functionvalues around the current iteration step, thus, the goal is to find a minimum for p ≤ ∆k, where∆k is the trust-region radius. Consequently, to obtain each step, a solution of the subproblem,

minp∈Rn

mk(p) = fk +∇fTk p+

12pTBkp, (4.149)

is required. Adapted to the current situation, the model function in Eqn. 4.147 can be rewrittenas,

minp∈Rn

mk(p) =12|rk|2 + pTJT

k rk +12pTJT

k Jkp. (4.150)

and the subproblem to be solved according to Eqn. 4.149 is stated as,

minp

12|Jkp+ rk|2 , where |p| ≤ ∆k. (4.151)


The solution to the subproblem (4.151) can be characterized with respect to comparing the stepdirection with a Gauss-Newton approach. When a Gauss-Newton solution (pGN

k ) of Eqn. 4.143lies strictly inside the trust-region (that is,

∣∣pGNk

∣∣ < ∆k), then this step pGNk also solves the

subproblem (4.151). Otherwise, there is a λ > 0 such that the solution p = pLM of Eqn. 4.151satisfies |pk| = ∆k (cf. [NW99]) and,

(JTk Jk + λI)pLM

k = −JTk rk. (4.152)

The vector pLMk is a solution to the trust-region problem (4.151) for some ∆k > 0 if and only if

there is a scalar λ ≥ 0 such that,

(JTJ + λI)pLMk = −JT r,

λ(∆k −∣∣pLM

k

∣∣) = 0.

Analytically, the minimization problem of (4.152) becomes,

(JTk Jk + λD2

k)pLMk = −JT

k rk, (4.153)

and equivalently solves the linear least squares problem of the form,

minpk

∣∣∣∣[ Jk√λDk

]pk +

[rk0

]∣∣∣∣ , (4.154)

where the matrix Dk is a diagonal matrix that is used to scale the step direction p and can changefrom iteration to iteration as new information about the typical range of values is gathered.The trust-region radius ∆k is adjusted between iterations according to the agreement betweenpredicted and actual reduction in the objective function such that,

ρk =f(xk)− f(xk + pk)mk(0)−mk(pk)

. (4.155)

For a step to be accepted, the ratio ρ must exceed a small positive number (typically 0.0001). Ifthis test fails, the trust region is decreased and the step is recalculated. When the ratio is close toone, the trust region for the next iteration is expanded. The solution of Eqn. 4.154 can be foundby applying matrix factorization methods (e.g. QR factorization) to solve a linear system ofequations. The local convergence behavior of the Levenberg Marquardt method is similar to theGauss-Newton method. Near a solution with small residuals, the model function (4.151) will givean accurate picture of the objective function (4.137). The trust region will eventually becomeinactive, and the algorithm will take unit Gauss-Newton steps, giving the rapid linear convergencerate that is provided by the Gauss-Newton method. Generally, with the determination of λ inEqn. 4.154 the Levenberg Marquardt method changes is preference between a line search andtrust-region strategy with respect to the parameter λ. However, the introduction of the trust-region approach for determining λ is just one possibility among many other strategies includingthe direct modification of this value using heuristic approaches (see [NW99]).

The Levenberg Marquardt algorithm for minimizing a n-dimensional non-linear function can beformulated as follows:


First initial values for x0 need to be defined and the objective function (4.137) f(x0) is evaluated,the iteration counter k = 0, ∆0 = ∆̄ where ∆̄ is an overall bound on step lengths and λ = λ0.

(1) Test for convergence: If the condition defined in Eqn. 4.146 is satisfied, or the maximumnumber of iteration is exceeded, the algorithm terminates with xk as the solution.

(2) Solve trust-region subproblem This step solves the subproblem Eqn. 4.154 using a trust-region algorithm. According to Eqn. 4.154, a new vector pk is determined that gives the newsearch direction for the current iteration. The multiplier λk is set to zero if the minimum-norm Gauss-Newton step pGN

k is smaller than ∆k, otherwise, λk is chosen such that,

∆k = |Dkpk| . (4.156)

(2.1) Evaluate ρk: Using Eqn. 4.155, the ratio ρk gives information on the alternation of thetrust-region radius ∆k.

(3) Improve step The next iteration point is updated by,

xk+1 = xk + pk. (4.157)

Then, k is increased by 1 and the procedure proceeds by starting again from (1).

A complete implementation and analysis of the Levenberg Marquardt algorithm can be found in[Fin96]. Details information on different methods of the trust-region implementation, proofs andanalysis of convergence can also be found in [PTVF97] and [NW99].

Recalling the torque imbalance equation (4.135), the Levenberg Marquardt method is used toeither find a stable eye position for constant innervations, or innervations for a constant eyeposition.

By using Eqn. 4.135 as objective function for the Levenberg Marquardt minimization method,stable eye positions can be approximated by minimizing the torque imbalance vector in thateither ~Iv or ~Ep is set constant and Eqn. 4.135 is minimized over the free parameters. Thus, thesolution to the minimization problem,

minLMLT (~Iv, ~Ep), (4.158)

gives a stable eye position in terms of balanced muscle and orbital forces.

4.6.3.1 Solving for Eye Positions

Eqn. 4.158 can be used to solve the forward kinematics in that a stable eye position is foundfor a given set of innervations. According to Eqn. 4.135, this results in setting Iv constant andminimize over Ep such that,

Epmin( ~Ep) = min~Ep

LMLT (~Iv, ~Ep), ~Iv constant. (4.159)

The resulting eye position Epmin in the form of (4.129) can then be regarded as one solution thatsatisfies Eqn. 4.117 for a given constant set of 6 innervations ~Iv.


Figure 4.27: Torque Error Minimization in solving for Eye Positions

In Fig. 4.27, the optimization steps for a pathological situation was rendered. Constant innerva-tions ~Iv that would hold a „normal “ healthy eye in primary position are given. The indicatedminimum of this function corresponds to a modified primary position, due to the different me-chanical properties of the pathological situation that was modelled according to Fig. 4.26.

However, it must be noticed that each step in the minimization of Eqn. 4.159 modifies the vector~Ep in all 3 components, whereas Fig. 4.27 only shows ab-/adduction and elevation/depressionaxis for visualizing eye positions. Thus, the torque error function will change its shape within this3-dimensional representation, whenever the torsional eye position (that is ey of ~Ep) is adjusted bythe minimization procedure. This also explains, why the minimization steps that were renderedas vectors in Fig. 4.27 do not align with the shape of the function unless the minimum is reached.

4.6.3.2 Solving for Innervations

Using Eqn. 4.158 to solve for innervations is denoted as solving the inverse kinematics problem.In this case, a given eye position ~Ep will be constant in Eqn. 4.158, and innervations ~Iv need tobe determined. Here, the odd innervation vector Ivo from Eqn. 4.133 is used in order to reducethe degrees of freedom, and all 6 innervations are regained by applying Eqn. 4.134. This leads to


a minimization of Eqn. 4.158 over ~Iv such that,

Ivmin(~Iv) = min~Iv

LMLT (Ivo(~Iv), ~Ep), ~Ep constant. (4.160)

By using Eqn. 4.133, it is possible to reduce the inverse kinematics problem from 6 to 3 degrees offreedom, thus allowing the optimizer to converge to values for ~Iv that are confined to analyticallyformulated laws that are supposed to represent anatomical behavior. However, the goal of thisprocedure is to find meaningful values in terms of innervations that correspond to anatomicalfindings in that innervations of antagonist muscles are dictated by agonist muscle innervations.

4.7 Brainstem Simulation

According to Sec. 2.2.3.1, innervations of the oculomotor system originate in the brainstem nucleiwhich are in turn connected to supranuclear brain structures. In order to build a simple inter-face to controlling these structures in a simulation model, the mathematical representation ofinnervations will be used. The presented biomechanical model does not provide an independentsimulation of the brainstem or the supranuclear structures. However, it provides a model of thedistribution of innervations for each eye, which makes it possible to change the percentage ofinnervation of the oculomotor nuclei to the respective muscles. If such a distribution of innerva-tion is changed, all innervations in relation to the affected muscle for the affected eye are scaled.The maximum allowable innervation of 250% corresponds to the maximum allowable stimulationwhich the nucleus shall be able to generate.

If the distribution of innervation is modified, it is possible to simulate either inter- or supranucleargaze palsies or even retraction syndromes (see Sec. 3.5.2.2). For simulating a nuclear gaze palsy,changes to the distribution of innervation of the nervus abducens to the lateral rectus muscleon the right eye and to the nervus oculomotorius to the medial rectus muscle of the left eyeare required. During an internuclear gaze palsy only one eye would be affected and thereforealso only one eye would require changes to the distribution of innervation. Since the presentedmodel uses a so-called invisible reference eye (see Sec. 4.7.1), which forms the basis for most ofthe calculations, and if the distribution of innervation of this reference eye is changed, only thefollowing eye would reflect these changes in the Hess-Lancaster test. This enables for examplethe simulation of a lesion, which is only visible through the following eye alternately dependingwhether the left or the right eye is fixing.

The mathematical formulation for simulating brainstem control is straight forward, by supplyinga distribution matrix to each eye that scales innervation prior to the application of the muscleforce prediction model (see Sec. 4.5). Therefore, each eye is associated with a matrix Dn of theform,

RL RM RS RI OS OI

Dn =

s11 s12 · · · s16...

.... . .

...s61 s62 · · · s66

AbducensOculo/RMOculo/RSOculo/RIOculo/OSOculo/OI

, 0 ≤ sij ≤ 2.5,(4.161)

where n denotes the corresponding eye (i.e. left, right or reference eye) and the matrix elementssij correspond to the scaling parameter of a oculomotor nerve i with respect to a muscle j.


Each row in the Matrix of Eqn. 4.161 has the form of (4.132) and each column is ordered bythe respective oculomotor nucleus for each muscle (see Sec. 2.2.3.2). The initial innervationdistribution matrix for an eye is chosen as the identity matrix Dn = I. In order to scale specificinnervations for one eye, an innervation vector ~I ′v of the form (4.132) is multiplied by Dn suchthat,

~Iv = ( ~Iv ′ ·Dn)T , (4.162)

where T denotes matrix transposition.

4.7.1 Simulation of Binocular Function

The forward and inverse kinematics solutions from Sec. 4.6.3.1 and Sec. 4.6.3.2 provide twoessential operations to assemble a procedure that aims to simulate behavior of binocular eyemovements. The presented system uses a model for simulating the binocular Hess-Lancastertest (see Sec. 3.13). The results of the test can be visualized in the form of Hess-diagrams (lefteye and right eye fixing) and in the form of a text-based view (squint-angles diagram). Themain difference between the two visualizations is the declaration of the deviations. While theHess-diagram (see Sec. 3.13) offers some visual impression of the pathological situation and canbe calculated for arbitrarily chosen gaze positions, the squint-angles diagram offers a text-basedillustration of deviation values in the nine main gaze directions.

(a) Left Eye (b) Right Eye

Figure 4.28: Squint-Angles Diagram for Binocular Fixation

In the squint-angles diagram from Fig. 4.28, the deviations for each gaze direction for the left andthe right eye fixing are shown, whereby HD denotes horizontal deviation and V D denotes verticaldeviation. In the white fields of the diagram, the torsional deviation is given as excyclo or incyclo(EX or IN) and the protrusion is denoted in mm (PR for negative protrusion = retraction). Allvalues except for protrusion are given in degrees.

The signs for the horizontal deviations are interpreted as follows:

Positive HD: Towards the nose (adduction)

Negative HD: Away from the nose (abduction)


The signs for the vertical deviations are defined differently for right eye and left eye fixing:

Right eye fixing

Positive VD: Deviation downwards (depression)

Negative VD: Deviation upwards (elevation)

Left eye fixing

Positive VD: Deviation upwards (elevation)

Negative VD: Deviation downwards (depression)

A gaze pattern is the basis for simulation of the Hess-Lancaster test. At the execution of thesimulation, for every eye position in the gaze pattern, a fixation of the particular fixing eye iscarried out. The other eye will then, by the description of the simulation, be regarded as followingeye and will deviate from the fixing eye, according to some current pathological situation. Thepositions of fixation specified in the gaze pattern can be assigned to the blue points in a Hess-Lancaster diagram. The red points depict the resulting deviation points of the following eye (seeFig. 3.14).

Fig. 6.12 shows a schematic simulation task flow for the Hess-Lancaster test, which is performedfor each gaze position of the fixing eye gaze pattern. The goal is the determination of the deviationof the following eye based on positions of the fixing eye. Thereby, the fixing and following eyecan be exchanged, i.e. first the right eye is the fixing eye and the left eye the following and viceversa. This makes it possible to calculate the right resp. left fixation. For simplicity, let FP andFI denote two operations that find eye positions and innervation respectively of a specific eye,according to each eye’s specific parameters, using Eqn. 4.159 and Eqn. 4.160 such that,

FPFix(~Iv) = Epmin(~Iv), for the fixing eye,

FPFol(~Iv) = Epmin(~Iv), for the following eye, (4.163)

FPRef (~Iv) = Epmin(~Iv), for the reference eye,

and,

FIFix( ~Ep) = Ivmin( ~Ep), for the fixing eye,

FIFol( ~Ep) = Ivmin( ~Ep), for the following eye, (4.164)

FIRef ( ~Ep) = Ivmin( ~Ep), for the reference eye.

Thus, if the left fixing eye e.g. looks to an eye position ~A(ax, ay, az) denoted in the form of(4.129), then,

~B = FPFol(FIFix( ~A)),

will describe the position of the right, following eye that results from the innervations that arerequired to hold the left, fixing eye in position ~A. Note that the right eye will turn against theleft eye in this case, since the coordinate systems of both eyes are not identical (see Sec. 2.2.1),and the innervations are applied to the right eye without any modification.


Figure 4.29: Simulation Task Flow for the Hess-Lancaster Test

(A) Starting from a „healthy“ reference eye and a predetermined fixation position a complete 3Deye position is calculated. If a fixation position is specified, only the abduction/adductionand elevation/depression can be chosen freely. The specification of a torsional value is notpossible, since a patient does not explicitly rotate the eye around a specified torsional angleon command. As a result, the complete position of the fixing eye has to be calculated bygenerating innervations for eye positions with listing torsion with the help of the referenceeye until the position of the fixing eye matches with these innervations and the desired2D eye position. This is similar to a minimization problem of the form (4.143) using theobjective function,

~Ivp = FIRef (~P ),

f(~P , ~Ivp) =∣∣∣(ZAng(FPFix(~P )), XAng(FPFix(~P )))− (ZAng(~P ), XAng(~P ))

∣∣∣2 ,torsion = YAng(f(~P , ~Iv)),

(4.165)

where ~Ivp are starting innervations of the reference eye that correspond to the rotationvector ~P of the form (4.129) that describes the 2D eye position in that the torsional rotation


is always 0. f is the objective function that describes the length of the angular differencevector between reference eye and fixing eye, where XAng, YAng and ZAng extract the radianangle from a rotation vector of the form (4.129). A valid 3D eye position is then found,when the innervations of the reference eye drive the following eye to a 2D position thatmatches ~P , and f(~P , ~Ivp) ≈ 0 holds. The value for torsion will then be the torsionalrotation that expands the 2D eye position ~P into a 3D eye position vector ~PFick such that,

~PFick = (ZAng(~P ), XAng(~P ), torsion), (4.166)

where ~PFick is a 3D eye position of the form (4.6), and the corresponding rotation quaternionqp can be computed by using Eqn. 4.48,

qp = qryqrxqrz, (4.167)

where qrz, qrx and qry correspond to the quaternions that describe rotation ofangles ZAng( ~PFick), XAng( ~PFick) and YAng( ~PFick) respectively around each coordinate axis.

(B) Using the fixing eye parameters and the 3D eye position pq from (A), the innervations for allsix eye muscles are calculated to bring the fixing eye in the desired position using Eqn. 4.164,

~IvF ix = (FIFix(R(pq)) ·DFix)T , (4.168)

where R transforms the quaternion pq into a rotation vector using Eqn. 4.131, and DFix

scales the output innervation vector according to the innervation distribution matrix forthe fixing eye using Eqn. 4.161.

(C) These innervations are now passed into the reference eye in order to calculate its eye position.If the fixing eye is pathological, the eye position calculated with the reference eye differs fromthe previously calculated eye position. The result of this calculation is now the intendedfixation position of the fixing eye ~pFix such that,

~PFix = FPRef (( ~IvF ix ·DRef )T ), (4.169)

where the innervation vector ~IvF ix from (B) is scaled by the brainstem matrix DRef fromEqn. 4.161 with the addition that the columns for the medial rectus and the lateral rectusmuscles are exchanged prior to multiplication using,

RM RL RS RI OS OI

DRef =

D22 D21 · · · D16

D12 D11 · · · D16...

.... . .

...D62 D61 · · · D66

AbducensOculo/RMOculo/RSOculo/RIOculo/OSOculo/OI

, 0 ≤ Dij ≤ 2.5.(4.170)

Note that the values D11, D22 and D21, D12 are additionally transposed within the matrixin order to reflect the downward order of the rows that correspond to the oculomotor nuclei.

(D) During this step, the intended fixation position of the fixing eye is mirrored in order toget the intended position of the following eye. Since innervations cannot be mirrored (thecontralateral synergist of the right lateral rectus muscle is the left medial rectus muscle,which may not act in exactly the same way) this „indirection“ of mirroring the eye position


is necessary. Thus, the rotation vector ~PFix from (C) is converted into mirrored Fickrotational angles ~PFick such that,

~P ′Fick = (−ZAng( ~PFix), XAng( ~PFix),−YAng( ~PFix)), (4.171)

and the corresponding rotation quaternion qf can be computed by using Eqn. 4.48.

The mirroring of an eye position is necessary, since abduction/adduction and torsion ofboth eyes are different for describing the same direction (see Sec. 2.2.1).

(E) Now, innervations are determined, using the reference eye with the mirrored eye position qfso that it will be possible to calculate the position of the following eye. These innervationscan be found by applying,

~IvFol = (FIRef (R(qf )). (4.172)

(F) The innervations of the reference eye ~IvFol now determine the position of the following eye.This position is one of the red points in the Hess-diagram and can be found by using,

~PFol = FPFol(( ~IvFol ·DFol)T ), (4.173)

where DFol denotes the innervation distribution matrix for the following eye.

Chapter 5

Visualization of Muscle Action

Besides the path of a muscle, which can be calculated using the methods from Sec. 4.4, the mostimportant information about a muscle is its volume and the distribution of its volume. In fact,the distribution of the volume changes with different muscle activations. In order to simulatethe deformation process of a given muscle, the initial shape of the muscle at different activationsneeds to be acquired. To provide this information, a three-dimensional representation of MRimages is used for calculating the basic parameters of the muscle. This data, constructed from aset of MR images, each representing a slice of the muscle, defines the surface and thus the volumeof the model (see Sec. 2.1.4).

The goal of this chapter is to outline the image processing techniques that were used to reconstructa three dimensional representation of the extraocular muscles and additionally make the resultingmodel deformable according to underlying MR data and biomechanical predictions. The basicdata was acquired within a study [FPBK03], where MR images were captured in different gazepositions in order to analyze the shape and deformation of the extraocular muscles (see Sec. 2.1.4).A 3D polygon model was built, and different states of innervation of a muscle can interactivelybe visualized by interpolating between static reconstructions of MRI data.

The reconstruction procedure consists of two main parts. First, MRI data is segmented bydefining selection polygons on each image slice. These selections are used to scan each sliceand detect muscle boundaries in order to define a first basic polygon model. The second stepinvolves smoothing and interpolation using spline approximation and data variance analysis inorder to render different states of innervation. The resulting representation of a muscle is thenincorporated into the biomechanical model and connected to the muscle force prediction model(see Sec. 4.5) in order to visualize model predictions in terms of muscle deformation.

148

CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 149

5.1 Image Analysis

One basic method used for constructing three-dimensional models is the marching cube algorithm.The idea of this algorithm is to generate a continuous surface by analyzing the relationship ofneighboring points in space. Each point is classified as either inside or outside the volume.Surfaces are created between points, situated on opposite sides. This process fills an area in3D-space with a large number of small cubes, determining which corners of the cubes lie insideor outside of the volume. According to a predefined set of rules, this information can be usedto generate a continuous surface, which approximates the surface of the original volume withan accuracy inversely proportional to the size of the cubes. To achieve this goal, the marchingcube algorithm creates a series of planes, intersecting MRI data. These planes are all aligned inparallel and are evenly spaced. Then, for each plane, the parts that lie inside and outside of thetraced object are determined. In this case, this is done by assigning one MR image to each plane.Image pixels of a certain color are defined to be inside, while others are outside. The value of thepoints between the planes is assumed to be of the same value as the nearest pixel. The resultof this step is a volume representation, where each pixel represents a small cube (i.e. a voxel),either (partially) inside or outside of the original object.

set threshold

set threshold

define data volume

sort color table

MR data from imaging device

define polygons

picture setup

generate polyhedron

use scan-line/marchingcubes algorithms

generate surface

storage

DXF file

Figure 5.1: DXF Model Generation Tasks using Marching Cubes, from [Sat03]

The overall process of image analysis is depicted in Fig. 5.1. Starting from raw MRI data, eachMR image is modified by the „picture“ setup procedure, where the color index table is sorted, asegmentation polygon is defined and a threshold for analyzing muscle boundaries is selected.


5.1.1 Picture Setup

The color index table of each MR image file is used to reorder the color values, whereby individualpixels in the picture refer to the appropriate entry in the color index table. The color index tableis encoded in RGB color scheme, but, due to the grayscale MR images, the respective RGB entriesare always identical (A grayscale color 50 is expressed as RGB(50,50,50)). In MR images, usuallynot all of the 256 possible colors are used. Thus, the first step in this process is to investigatethe picture with respect to the used colors, mark these colors for the subsequent sort procedureand shift these entries to the end of the color index table. Additionally, unused color entries arezeroed after sorting. The sorting procedure then orders all used color entries according to theirgrayscale values (see Fig. 5.2). The changed color index table is than reapplied to the actualimage in that each pixel reference value is exchanged in order to reference the same color as inthe unsorted color index table.

Figure 5.2: Sorting of Color Index Table

This ensures, that the appearance of the picture is not affected by this action. The reason forthis preprocessing step is that color comparisons can be efficiently implemented within a sortedcolor index table, in that only the color index of the pixels need to be compared, since highercolor index values directly correspond to higher grayscale colors. This comparison operation willbe used within the next procedure to determine the boundaries of the muscle by thresholding thecolor values of the image.

The next step in the picture setup procedure is the definition of 2D polygons to surround theregion of interest for each MRI slice.

In Fig. 5.3, the definition of a image segmentation polygon is shown. The region of interest (thesuperior rectus muscle in Fig. 5.3(a)) is surround by a manually defined area in order to use ascanline approach to rasterize every point within this area (see Fig. 5.3(b)) until the completearea has been processes, as shown in Fig. 5.3(c). Within this rasterization process, the marchingcubes algorithm, together with color threshold analysis is used to identify boundaries of themuscle within the region of interest.

In order to decide whether an image pixel lies inside or outside the muscle area, a threshold regionis additionally defined for the color index table of each MRI slice. Fig. 5.4 depicts a thresholdregion that is defined within the 256 possible grayscale values and classifies an image pixel aspart of the muscle when the pixel index value for the color table is within the interval [8; 134].


(a) Definition of 2DPolygon

(b) Scanline Rasteri-zation

(c) Rasterized Poly-gon

Figure 5.3: Definition of 2D Polygon for MRI Segmentation, from [Pri01]

Figure 5.4: Threshold Region for MRI Slice, from [Pri01]

Thus, each image of size SIZE(I) = m× n can be represented as a table of color index values,

I[i] = {p1, p2, · · · , pm×n}, (5.1)

and an index function PI(x, y) can be defined such that,

PI(x, y) = I[x+ (y · n)], 0 ≤ x < m, 0 ≤ y < n. (5.2)

Additionally, an image stack I[i] is defined, such that,

I[i] = {PI1, P I2, · · · , P Ij} (5.3)

where j MR images are collected, each associated with a 2D polygon definition and a thresholdarea such that,

P [i] = (p1, p2, · · · , pj),T [i] = (t1, t2, · · · , tj),

where P [i] holds the polygon definitions as lists of 2D points pi = (P1, P2, · · · , Pk) with individuallengths ki, and T [i] stored the threshold regions in the form of a list of intervals ti = [ta; tb] where0 ≤ ta ≤ tb ≤ 255.


5.1.2 Generation of Polyhedron

In the second stage of the image analysis procedure, a polygon mesh is generated by apply-ing the surface reconstruction method of marching cubes. Therefore, a series of MR images isincorporated into a volume data set, based on the definition from (5.3) such that,

V [x, y, z] = {PIz1(x, y), P Iz2(x, y), · · · , P Ij(x, y)}, (5.4)

where the size of all images is reduced to the size of the largest rectangular boundary of all poly-gons P [i] for all MRI slices. The scanline algorithm (cf. [Pav94]) is used to generate V accordingto Fig. 5.3. This algorithm takes a list of the polygon edges P [i] and performs topological sortingso that the inner polygon area can be scanned along a horizontal line that is moved from the topmost y-value of the polygon points to the bottom.

Figure 5.5: Marching Cube Traversal, from [BK03b]

Thus, each pixel value within the area of the polygon is visited once. The resulting volume Vthen has the following properties:

• The data volume is comparable to a cuboid that contains voxels (i.e. 3D pixels) thatencompasses all pixels that are defined by each polygon.

• Each image slice contained in V only contains pixels that are situated within the respectivepolygon area. Pixels that are outside the polygon edges are set inactive.

• All inactive pixels are assigned a color index value that is outside the respective thresholdrange (i.e. V [xo, yo, zo] /∈ T [i], where (xo, yo, zo) represents a pixel that is outside a polygon).

• All pixel values that are inside the polygon edges are transferred to V without any changes.

To determine the surface of the muscle, the algorithm assigns flags to each possible group of eightneighboring pixels within two image slices to the corners of a cube (see Fig. 5.5). Every cube canbe generated using,

Q(x, y, z) = (V [x, y, z], V [x+ 1, y, z], V [x+ 1, y + 1, z], V [x, y + 1, z], (5.5)V [x, y, z + 1], V [x+ 1, y, z + 1], V [x+ 1, y + 1, z + 1], V [x, y + 1, z + 1]),

1 ≤ x ≤ m− 2, 1 ≤ y ≤ n− 2, 1 ≤ z ≤ j − 2.


Then, for each cube defined by (5.5), a classification function is defined that can determine if acube corner is inside or outside the threshold region such that,

C(c, t) ={

True, ta ≤ c ≤ tb,False, otherwise.

(5.6)

To generate the surface, the algorithm performs the following steps:

1. Determine the state of the corners from the pixel values.

2. Find a transformation T that transforms the actual cube, matching one of the 15 predefinedclasses (see Fig. 5.6).

3. Build a set of triangles, representing the surface passing through this cube.

4. Transform the triangles using the inverse transformation T−1 from step 2.

5. Add triangles to the set that contains the already constructed surface.

Figure 5.6: Marching Cubes Standard Classes, from [BK03b]

In the first step, the corners of each marching cube are set such that, 0 = False when thecorner of the cube is outside and 1 = True when the corresponding corner of the cube is insidethe muscle surface according to Eqn. 5.6. Because the cube has eight corners and each corner


has to be in one of the two states, 28 = 256 possible cube configurations exist. Each of theseconfigurations defines if and how the surface passes through the cube.

Using the cube corner values that define the points as inside or outside the muscle surface, itcan be shown that by rotation and inversion of the corners, each of the different cubes can bemapped to one out of standard classes (cf. [LC87]), as shown in Fig. 5.6.

The implementation of the marching cubes algorithm performs additional optimization to producea smoother surface. It is not only classifying the corners of a cube as inside or outside, but alsodetermines how far inside or outside each corner is by evaluating the specific color value of eachpixel. Once a specific standard configuration is identified according to Fig. 5.6, the extension ofthe triangle along each edge of the cube can be calculated by linear interpolation of the colorvalues. Let A and B be two corner points of a marching cube, ca and cb their respective colorvalues and T [i] = [ta; tb] the respective threshold interval for a specific slice i. The marching cubestandard classes from Fig. 5.6 define triangle corners between cube edges where one cube cornerlies inside the muscle area (V [xo, yo, zo] ∈ T [i]) and one cube corner lies outside the muscle area(V [xo, yo, zo] /∈ T [i]). Then, the distance d of one corner point of the triangle within the cubecan be calculated using,

d =

1− t−cb

ca−cb, ca > cb,

t−cacb−ca

, ca < cb,

0.5, ca = cb,

(5.7)

where t is the lower or upper bound of the threshold interval such that,

t ={ta, ca ≤ a ≤ cb, ca ≤ cb,tb, ca ≤ b ≤ cb, ca ≤ cb.

(5.8)

After performing these steps for all cubes in the area of interest, the result is a set of triangles,describing the surface of the object. It is obvious from the above layout, that for j images of sizem× n pixels, the asymptotic runtime of this algorithm is O(j ×m× n). It is therefore desirableto keep the images as small as possible.

Pixels with color values near a given threshold are assumed to be nearer to the surface then pixelvalues far from the threshold. The algorithm is then moving the corner points of the surfaceparts inside or outside, depending on the ratio of the distances of the corners of the cube. Thisis especially important when the area of the muscle cross section shrinks rapidly from one MRimage to the next.

The resulting list of triangles that is generated by the marching cubes algorithm can be definedas,

L = {TR1, TR2, · · · , TRn}, (5.9)TRi = {X1, X2, X3},

where L denotes a list of n triangles TRi, each consisting of a list of 3 points X1 to X3.

The resulting resolution of the triangle mesh depends on the size of the cubes where image pixelsthat would lie between two MR images are linearly interpolated. Output data of this process isrepresented using DXF files, storing each triangle as single primitive within the file.


5.2 Surface Reconstruction

Based on the output of the first stage of the reconstruction process, described in Sec. 5.1, thepolygon model is analyzed and interpolation parameters are defined. The goal of the surfacereconstruction process is to define a mathematical interpolation model based on the data fromSec. 5.1.

While output data from the marching cubes algorithm is interchangeable and easy to generate, itis very insufficient in terms of memory usage and rendering speed. Moreover, this data structuredoes not contain any connectivity or volume information. Since the last two points are veryimportant, input data needs to be transformed into a more suitable representation. In Fig. 5.7,an overview of the main processing steps is depicted.

interpolation between the two EOM models

spline interpolation

calculate volume and display surface

import DXF

DXF file from MR analysis

calculation of gravity line

transform DXF format

calculate area centroids

Figure 5.7: Surface Reconstruction Tasks, from [Sat03]

The first step is to find connectivity information. For this purpose a simplified boundary repre-sentation method is used. Because the surface is built from triangles only, there is no need tostore edge information but only triangles and corners. In order to build a usable data structure,the input file is transformed into a list of triangles. Corner points of two or more triangles, whichare within a certain limit of spatial proximity, are welded together. This replaces all points thatare to be welded with a new point, which is the geometric average of these points. It can beshown that, because of the way the marching cube algorithm works, points are either equal toeach other, or have a spatial distance which is higher or equal to 1

2σ where σ is the smallestdistance between two pixels in the stack of images used. Due to this, as long as the threshold forthe welding is smaller than 1

2σ, the resulting set of triangles will describe the same (intended)volume as the triangles in the DXF file.

The boundary representation is then transformed into an indexed triangle list. In this form,connectivity information can still be gained from the list, but at higher cost. However, thisrepresentation is ideal for current rendering hardware. This list eliminates the references from


the points to the triangles. Neighboring triangles can still be identified by searching through thelist of triangles and looking for references to the same corner points.

Based on the definition from Eqn. 5.9, the data structure for the list of triangles is modified suchthat,

L = {(p1, q1, r1), (p2, q2, r2), · · · , (pn, qn, rn)}, (5.10)PTS = {PT 1, PT 2, · · · , PTm},PT i = (xp, yp, zp),

where L denotes a list of n triangles where each list element contains 3 indices p, q, r into a vertexlist PTS. The vertex list holds all corner points PT i, where each point contains the coordinates(xp, yp, zp).

Because the model can be stored using a small amount of memory, and a linear traversal overthe primitives can be used to display them all, this method of storing geometry is widely used intoday’s rendering applications. When the input data has been transformed into a data structurethat can efficiently be rendered, restoration of the volume information is still needed. Before de-ciding on how to do this, the transformation of the volume should be taken into account. Fig. 5.7gives a brief overview of the generation tasks involved in constructing the surface representationmodel.

Given two volume models of the same muscle in two different activation states, the actual goalis to generate a new model in an intermediary state, by interpolating between these states. Thetwo surface representations do however not contain any topological information. The positions ofthe triangles of one surface model are not connected in any way with the position of the triangleson the other surface model. To overcome this flaw, the final model needs to contain topologicaldata as well as volume information. The following information can be identified as vital forinterpolation:

• position and orientation of the muscle in space,

• length of the muscle,

• activation of the muscle,

• volume of the muscle.

It is important to distinguish muscle movement from muscle volume deformation in a set of inputmodels. This distinction is especially important since the interpolation of position and volumeworks in an entirely different way. Activation cannot be extracted from the input model itselfbut has to be supplied from some external source (e.g. from the physician who acquired the MRdata). The model does not only contain the current volume of a muscle, but also information onhow the volume is distributed along the muscle.

5.2.1 Calculation of the Muscle Path

Based on the data definition from Eqn. 5.10, the approximation of the muscle path (see Fig. 5.28)can be calculated. This can be done by defining the area centroids on each image slice using all


triangle corner points that belong to one MR image and were generated by the marching cubesalgorithm.

Let P̃ (d) be the function that describes the muscle path as approximated curve that connectsall area centroids. To get the path, the muscle is divided into a series of slices along the z-axis.Because the images PIi(x, y) used for generating input data are coronal cuts and the image axesx and y are mapped to x and y in 3D space respectively, the muscle always has its longest extentsalong the z-axis.

Figure 5.8: DXF Model with Area Centroids

A bounding parallelepiped is defined to exactly fit the muscle and to find the longest dimensionof extension. The muscle is then cut into equally spaced slices along this dimension. The areacentroid of each slice is calculated (see Fig. 5.8) and the resulting set of points is approximatedby a series of Hermite splines. An area centroid is calculated such that,

A =12

n−1∑i=0

(xiyi+1 − yi+1yi), (5.11)

cx =1nA

n−1∑i=0

(xi + xi+1)(xiyi+1 − xi+1yi),

cy =1nA

n−1∑i=0

(yi + yi+1)(xiyi+1 − xi+1yi),

where A denotes the area defined by the polygon of all n points (xi, yi) on one slice and (cx, cy)denotes the area centroid.

The algorithm is generating splines by using cubic regression on the surface points of each slice,starting with one spline and recursively splitting and adding splines until the whole set of splinessatisfies a least mean square threshold. The identification of the slices is easy, as a cut can beperformed every δ units, starting at −0.5 ∗ δ, where delta is the resolution of the marching cubes(e.g. δ = 1). This way, each slice contains exactly the points which were defined by one imagein the marching cube algorithm. In the ideal case (i.e. the model contains no noise) these pointsform a ring whose borders are two star shaped polygons around the center of gravity.


The best fitting spline is calculated using normal cubic regression. The control points are definedto be equally spaced along the spline. When Ci, i ∈ {0..n} are the area centroids and ui, i ∈ {0..n}define the corresponding locations along the hermite spline curve,

S(u) = au3 + bu2 + cu+ d, (5.12)

then the error ε,

ε =n∑

i=0

(S(ui)− Ci)2, (5.13)

needs to be minimized. After the regression, the average error εn is compared to a previously

defined threshold. If the criterion is met, the process stops, otherwise the list of points is split intotwo halves and regression is performed on each one. This process is repeated recursively until thecriterion is met on each small spline segment. The process can be proven to stop because whenthe amount of points for one spline segment is 4 or less, then the regression produces a splinewhere ε is zero. The complete curve is made continuous by setting the start and end points aswell as the tangents of two consecutive splines equal. In Fig. 5.28 an example of the calculationof the muscle path is depicted, showing the interpolated curve P (d) with respect to the areacentroids.

Figure 5.9: Approximation of the Muscle Path

Thus, the muscle path can be denoted as function of a set of connected Hermite splines,

P̃ (d) : R → R3, (5.14)

in order to represent position, orientation and length of the muscle, where the parameter d denotesthe position along the path.

In fact, most of the defined slices consist of too few points to represent a muscle cross-section.This leads to falsely calculated area centroids, thus resulting in an incorrect muscle path. Hence,


the slices including too few points have to be identified and filled up with additional points torepresent a better approximation of muscle cross-section. The reason for the existence of suchmuscle slices are artifacts in the MRI data and the arbitrarily chosen resolution when applyingthe marching cubes algorithm.

5.2.1.1 Analyzing Surface Distribution

The purpose of analyzing the distribution of the surface along the muscle path is to identify cross-sections which include either too few points or false points. Once identified, the cross-sectionsare corrected by using points from the cross-section nearest to the ones with not enough points.

By applying an analysis of variance such that each muscle slice is treated as group of values,

Xi = {l1, l2, · · · , ln}, (5.15)

where each value li is defined as the angle α that is enclosed by a vector from the area centroidto a surface point and a vector that is parallel to the x-axis of the muscle plane as shown inFig. 5.10.

Figure 5.10: Angular Measurement of Surface Points

Then, the arithmetic mean of the angular distribution of the points around the area centroid isdefined as,

x̄ =∑n

i=1 lin− 1

. (5.16)

The variation with respect to the arithmetic mean can then be calculated using,

s2 =∑n

i=1(li − x̄)2

n− 1, (5.17)

where s2 is defined as variation of angular displacements li and,

v =√s

x̄· 100, (5.18)


is the variation coefficient that relates variation s to the arithmetic mean and provides a bettercomparison between different groups of data.

An example result of the complete identification process is shown in Fig. 5.11, where all de-fined cross-sections were analyzed and visualized in a diagram. The x-axis shows the numberof arbitrary chosen cross-sections of one muscle and the y-axis shows the value assigned to eachcross-section by means of analysis of variance. It can be seen that there are sections with low andhigh values, but only six of them reach a value which is higher than 80. These six sections withhigh values in variance are the cross-sections that consist of points which were derived directlyfrom the MR images, all others sections lie between the MR images and were approximated bythe marching cubes algorithm, thus not including enough information for further processing.

Hence, the diagram shows which cross-sections have to be edited by adding additional points inorder to smoothen the calculation of the gravity line.

0

20

40

60

80

100

120

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73

number of cross-sections

vari

ance

Figure 5.11: Example of Analysis of Variance, from [Sat03]

The following steps are performed in order to smoothen the distribution of surface points aroundthe muscle path:

• Iterate over the values received from the analysis of variance.

• If a value is below a certain threshold, start searching for a slice which has a better variance.

• Find the best variance by searching left and right, take the first slice found which hascorrect points and the shortest distance to the faulty slice.

• Add surface points to the faulty slice in order to produce a better distribution around thearea centroid.

Finally, after all slices are corrected, the muscle path P̃ (d) is recalculated in order to reflectsurface modifications.


5.2.2 Approximation of Muscle Surface

After the complete path is established, a radius function is defined that describes a NURBS (cf.[PT97]) interpolated curve that approximates all surface points around the muscle path withina muscle slice.

Before describing the interpolation itself, it is necessary to determine the input points over whichthe approximation should take place. All surface points of one cross-section are split accordingto their angular displacement from Eqn. 5.15 into two parts, those having an angle which islower than 180 degrees and those whose angle is greater than 180 degrees (see Fig. 5.12). Now,the partitioned points of each cross-section have to be processed in order to contain only uniquepoints. The reason why all points of one slice have to be different from each other is that in somecases it is possible that a spline curve is created which has a cusp (visual discontinuity), althoughthe resulting spline curve is always c1 continuous1.

All slices are processed by calculating a NURBS curve for each cross-section, so that this curvefits exactly between the points used as control points. Thus a control point vector,

Pc = {Pc1, P c2, · · · , P cn}, (5.19)

includes all surface points on one cross-section in order to approximate a curve. The NURBSapproximation can be denoted as,

C(u) =

n∑i=0

Ni,p(u)wiPci

n∑i=0

Ni,p(u)wi

, a ≤ u ≤ b, (5.20)

where the points Pci are the control points, wi are the weights that influence the control pointsand Ni,p are p-degree B-spline functions defined recursively as,

Ni,0(u) ={

1 : if ui ≤ u < ui+1

0 : otherwise

Ni,p(u) =u− ui

ui+pNi,p(u) +

ui+p − u

ui+p+1 − ui+1Ni+1,p−1(u), (5.21)

where U denotes a knot-vector,

U = {a, . . . , a︸︷︷︸p+1

, up+1, . . . , um−p−1, b, . . . , b︸︷︷︸p+1

}. (5.22)

containing values for u at which pieces of the curve join continuously. The knot vector usuallycontains internal and external knots, in the form that the values a and b are repeatedly occurringp+ 1 times at the beginning and at the end of the vector. Therefore, any value u for Eqn. 5.20needs to be within a and b. The spline basis functions Ni,p only depend on the value of p and thevalues of the knot vector, where p is the order of the curve. Increasing the order p also increasesthe continuity and smoothness of the curve at the knots, but tends to move the curve away fromits control points. Detailed information on NURBS and B-splines can be found in [PT97]. Within

1“c1 continuous” means that the first derivative is continuous.


the presented research work, partly based on an MRI study described in Sec. 2.1.4, a softwaresystem „Visu“ was implemented which is described in [Pri01], [Lac01] and [Sat03].

In using the NURBS approximation for the muscle surface around the muscle path, two continuouscurves can be approximated for the upper and lower half of the muscle cross-section as shown inFig. 5.12.

Figure 5.12: NURBS Approximation of Muscle Cross-Section, from [Sat03]

The next step is to improve the surface also in the length of the model. This has to be done,because the already calculated shapes of the cross-sections have slightly different area centroidswhich result in a shift of the sections to each other and thus in a „bumpy“ surface.

5.2.2.1 Optimized Rendering

In order to display the muscle surface and to realize a correct mapping of a texture to the surfaceof the muscle, it is vital that all spline points have a constant distance to each other. This leadsto point locations of each cross-section corresponding to points of a consecutive cross-section bylying in the same plane. This property is, however, not realized after the NURBS interpolation.The previously calculated splines have to be processed in order to obtain equally spaced points.

Since for all points of a spline the angle α from Eqn. 5.15 can be calculated, the idea is to have avector ~P = P (i) that samples spline points along a circular path, centered at the area centroid.According to Fig. 5.13, the area centroid of a muscle slice is denoted as M and the vector ~P canbe defined as,

P (i) = ~M +(

cos(θ · i)sin(θ · i)

), (5.23)

where θ = 360 ◦

m and m is the number of points to be generated. Consequently, i needs to bewithin the interval [0;m].

The angle θ · i can be used to find points of the spline curve that fit the criteria β ≤ θ · i < γ,where β and γ define two vectors ~V1 and ~V2 that point to two consecutive points of the splinecurve. The intermediate spline point Xi can then be calculated as the intersection point between


~P ′ and ~V12 = ~V2 − ~V1, where ~P ′ is defined as elongated vector ~P such that,

~P ′ = ~P · (| ~V1|+ | ~M |). (5.24)

Thus, the intersection point can be found by solving,

~Xi = ~M + s · ~P ′ (5.25)~Xi = ~V1 + t · ~V12,

for the free variables s and t, resulting in,

t =~My + s ∗ ~Py − ~V1y

~V12y

(5.26)

s =~Mx ∗ ~V12y − ~My ∗ ~V12x + ~V12x ∗ ~V1y − ~V12y ∗ ~V1x

~V12x ∗ ~Py − ~V12y ∗ ~Px

.

The resulting vectors ~Xi (drawn in dashed lines in Figure 5.13) now define equally spaced, splineapproximated muscle surface points.

Figure 5.13: Linear Interpolation of NURBS-generated Cross-Section, from [Sat03]


By using the same spline interpolation again on the sum of all spline interpolated cross-sections,a homogenous surface without disturbances is obtained.

5.2.3 Interpolation of Muscle Models

Once all of the input models are processed, they are assigned values for their activation accordingto Sec. 2.1.4. Currently this assignment has to be set manually, as there is no direct way to identifythe activation of a muscle based on pure MRI data. Then, the complete set of input models,along with their activation values can be used to interpolate the muscle at any given activationin between.

Both activation and length change whenever interpolation is applied. Due to activation, thelength of a muscle changes. Passive length change is imposed on all other muscles when one isactivated, because of the mechanical properties of the eye. Eye muscles are always working inpairs, when one muscle contracts, the other muscle is extended and vice versa (see Sec. 2.2.1).Both, activation and length change of a muscle can be obtained from the biomechanical model(see Sec. 4.4.4 and Sec. 4.5). The length change and the activation can be handled as separateproblems. First the muscle is interpolated from the input images and the length change isdisregarded. Afterwards the muscle length is adapted and the muscle is scaled with the constraintto keep its volume constant. The interpolation is done purely on parameter basis.

The process of interpolating the muscle volume without length changes can be split into thefollowing steps:

1. Identify all input models used for interpolation.

2. Interpolate the muscle path.

3. Interpolate all muscle surface points on each muscle cross-section.

The first step consists of defining weighting factors for the different input models, depending ontheir activation relative to the a desired activation A. Let wM be the weight associated with aninput model M . Interpolation of nearest neighbors is applied, so the process starts by findingtwo models M and N with activations AM and AN respectively, for which AM ≤ A ≤ AN andthere is no model O for which AM < AO < AN . Then a weight is assigned to each of the twoneighbors that depends inversely linear on the distance from the desired activation. The weightsare defined to be,

wM = 1− |AM −A||AM −AN |

(5.27)

wN = 1− |AN −A||AM −AN |

.

The next step is to interpolate the muscle path. Linear interpolation is used for the path. So ifPM (d) is the muscle path of model M , according to Eqn. 5.14, and PN (d) is the path of modelN , then the resulting interpolated path Pi(d) is can be denoted as,

Pi(d) = PM (d) · wM + PN (d) · wN . (5.28)


Since two muscle paths may not have an equal number of spline segments, splitting of musclepaths will be necessary. For this, a splitting operation is defined, such that it is possible to cutout a new spline from an existing one such that all points of the new spline are also points ofthe old spline. This configuration can be achieved by recursively splitting larger splines into twohalves when the corresponding part of the other spline set also consists of multiple splines (cf.[Lac01]).

The method for interpolating the muscle surface modifies the distribution of the volume alongthe muscle path. The change of the distribution is given by the input models. Provided thatall input models use the same resolution when defining the muscle surface points on each cross-section (i.e. parameter m in see Eqn. 5.23), all surface point on a cross-section can directly berelated to surface points within the same cross-section in another model.

From Eqn. 5.25 let,

XM = {XM1, XM2, · · · , XMm}, (5.29)XN = {XN1, XN2, · · · , XNm},

be the surface points belonging to one cross-section of the model M and N respectively. Then,for a given activation A, according to Eqn. 5.27, the intermediate muscle surface distribution canbe found by defining a weighting factor w such that,

w =|AM −A|AM −AN

, (5.30)

and use this factor to scale the vector that connects source and destination surface points byusing,

XIi = (XN i −XM i) · w, (5.31)

where XM i and XN i denote the ith surface points on a muscle cross-section and XIi gives theinterpolated surface point according to the weighting factor w.

When repeating the conversion process described in Eqn. 5.31, a new list of surface points XIresults that describes an interpolated distribution of the muscle surface according to a givenactivation and an interpolated muscle path from Eqn. 5.28.


5.3 Reconstruction Results

In order to reach good matching results, high resolution MR images were taken from a study thatwas carried out within this research work, described in Sec. 2.1.4. Therefore, DICOM outputimages from the MR scanner were used directly for the reconstruction of the muscle model. Thereconstructed model was rendered using the OpenGL graphics platform. Each of the followingscreenshots shows one MR image representing one cross-section of a muscle and is overlappedwith the model at the corresponding location.

In Fig. 5.14, a medial rectus muscle of a left eye was reconstructed, while the patient was lookingin primary position. Underlying MRI data was transparently visualized in order to evaluateaccuracy of the algorithm. The wireframe muscle shown in Fig. 5.14 is the result of the NURBSapproximation described in Sec. 5.2.2.

Figure 5.14: Reconstruction of a Left Medial Rectus Muscle

Additionally, texturing was applied to the wireframe model in order to improve visual quality.Fig. 5.15 shows a shaded, textured representation of a medial rectus muscle in a left eye.

The screenshots from Fig. 5.16 show a medial rectus muscle in different innervation states.Fig. 5.16(a) shows the muscle model with an innervation of 0, thus corresponding to an initial re-construction when the patient was looking in primary position. The second static reconstructionshown in Fig. 5.16(c), which depicts the muscle model that was reconstructed from MRI data,when the subject was looking in secondary position. An arbitrary innervation of 1 is associatedwith Fig. 5.16(c). By using the interpolation methods described in this chapter, it is now possi-ble to interpolate muscle surface between the two states shown in Fig. 5.16(a) and Fig. 5.16(c).Therefore, an innervation value of 0.5 is assumed and the resulting interpolated muscle model isshown in Fig. 5.16(b).

In the case depicted in Fig. 5.16(b), an innervation of 0.5 means that the interpolated mus-cle model shows a state that is exactly in the middle between the primary and the secondarypositions.


Figure 5.15: Shaded Reconstruction of a Left Medial Rectus Muscle

(a) Primary Position (b) Interpolated Position (c) Secondary Position

Figure 5.16: Morphology of Reconstructed Medial Rectus Muscle

5.3.1 Validation

Validation of the resulting reconstructions of MRI data concentrates on the evaluation of recon-struction results with respect to the underlying MRI data. However, validation of the reconstruc-tion process cannot be realized by simply comparing pictures. Instead, the extracted data has tobe compared with real-life dimensions, which means to use data measured on within data thatis provided by the MR scanner. A possible way is to compare muscle dimension measurementsfrom the extracted data with muscle dimensions data that can be measured in the MR images.

The standard software Adobe Photoshop was used to measure the pixel dimensions of musclesin MR images and real dimensions in mm were calculated from DICOM parameters that areincluded with each MR image. Since the reconstructed muscle model is defined in a worldcoordinate system with units in mm, the measured values from underlying MR data can directlybe compared.

The measured lengths and heights from eleven coronal MRI cross-sections are listed in Tab. 5.1,whereby the height of the muscle was taken as the longest dimension of the muscle on the current


Figure 5.17: Shaded Reconstruction of a Left Medial Rectus Muscle with MRI Data

MR image in vertical direction (the axis that develops vertically within the sagittal plane), andthe width was taken as the longest extension of the muscle in horizontal direction (the axis thatdevelops horizontally within the coronal plane).

Muscle Image Height Width LengthCross-Section MR Model MR Model MR Model

1 9.4 8.44 5.1 4.94 0.0 0.02 9.4 8.48 4.7 4.14 1.8 1.413 9.4 9.07 4.3 4.43 3.6 3.194 9.8 9.17 4.7 4.46 5.4 5.435 9.8 9.51 4.3 4.35 7.2 6.996 10.2 9.70 4.3 4.27 9.0 8.777 9.8 9.87 4.3 4.12 10.8 10.848 9.8 9.83 3.9 4.00 12.6 12.209 9.4 9.77 3.5 3.83 14.4 14.4010 9.4 9.63 3.9 3.78 16.2 16.1711 9.4 9.59 3.9 3.96 18.0 17.92

Table 5.1: Comparison of Medial Rectus Muscle Dimension Data in Millimeters, measured formMR Images and the Model

The inter-slice distance was given by DICOM parameters as 1.8 mm, thus the length of themuscle that can be displayed according to MRI data is 10 · 1.8mm = 18mm.

Chapter 6

Software Design and Implementation

The biomechanical model as well as the eye muscle surface reconstruction described in Ch. 4and Ch. 5 were implemented and incorporated into an interactive simulation software system„SEE++“. This system enables the simulation of pathological situations as well as the predic-tion of surgical interventions by means of graphical three-dimensional visualization of both eyesincluding the extraocular muscles.

This chapter gives a short overview of the software design model that was used to implement thebiomechanical model. Additionally, the integration of the biomechanical model into a softwaresystem „SEE++“ is described. The software design model is structured in a way such that thebiomechanical model is represented as standalone software component that can be integrated intodifferent applications. The software system „SEE++“ is therefore one possible application thatuses the biomechanical model to simulate eye muscle pathologies and surgeries. Moreover, thebiomechanical model contains exchangeable sub-models (i.e. geometry, muscle force predictionand kinematics) that provide flexibility and scalability for future extensions.

The software system described in this chapter was implemented using the C++ programminglanguage and the MFC class library framework. Computer-aided Software-Engineering tools (i.e.Rational Rose, Rational Unified Process) were used to build a design model using the UnifiedModelling Language (UML) within an integrated round-trip engineering approach (cf. [Buc01]).

169

CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 170

The basic structure of the software system „SEE++“ that was implemented within this researchwork is depicted in Fig. 6.1. The research work described in this thesis concentrates on resultsof the development of a biomechanical model for the human eye within the „SEE-KID“ project,which subsequently can be split into the development of a mathematical model and a softwaresystem. From Fig. 6.1 it can be seen that the implementation of the biomechanical model is anautonomous part within the „SEE++“ software system. The structure of the software systemitself, besides the biomechanical model, consists of a „SEE++“ and a GUI package. The „SEE++“package acts as pure interface to the underlying biomechanical model, whereas the GUI packagecontains the application and the user interface.

Figure 6.1: Structure of the „SEE++“ Software System

Because of the strict division of the software system into biomechanical model and applicationcomponents, the „SEE++“ system provides exchangeable structure such that any other biome-chanical model could be incorporated as long as it conforms to the „SEE++“ interface package.

Several tools were used to manage the high complexity of the system design and implementation.For realizing the software design, Rational Rose 2002 was used (cf. [Cor02b]) and implementa-tion is based on Microsoft Visual Studio .NET using the MFC (Microsoft Foundation Classes)framework (cf. [Cor02a]). The graphical three-dimensional visualization within the „SEE++“software system was implemented using the OpenGL graphics library. The design is described inUML notation and consists mainly of class diagrams which are structured in packages.

However, Fig. 6.2 illustrates that the design of the „SEE++“ system is based on a slightlymodified form of the Model-View-Controller (MVC) concept. The idea behind the MVC conceptis to separate the model, the views and the controller to make the model as independent aspossible. Moreover, the MVC concept enables the use of multiple views onto one shared model(cf. [GHJV94]).

In the „SEE++“ system, the See++ package is used by the instancer to provide an interfacebetween the GUI package and the biomechanical model. Consequently, the GUI package onlyuses the classes of the See++ package and the See++ package only uses the instancer to accessdata and functionality of the biomechanical model (cf. Fig. 6.2). Since the „SEE++“ system usesthe MVC concept, all data is stored within the model, thus the views and the controller do not


Figure 6.2: Model-View-Controller Structure of the „SEE++“ Software System

store any data on their own.

One of the main goals of the „SEE++“ system is flexibility and reusability. Therefore, differentdesign patterns have been used. The instancer uses the Singleton design pattern, which meansthat a global point of access is provided to the other classes and only one instance of the instancerclass can exist at the same time. The instancer also uses the Proxy design pattern. Since mostof the objects required during runtime are managed by the instancer and returned to the See++package and the GUI package when needed, the usage of the Proxy design pattern has theconsequence that an object is not allocated in memory until it is requested for the first time. Theadvantage of this procedure is that it reduces the start-up time and memory consumption of thesystem.

The reason, why the GUI package is placed between the view and the controller section inFig. 6.2 is that the GUI package implements the user interface parts of the different views of the„SEE++“ system, but also handles the mouse and keyboard input, which, according to the MVCconcept, is realized by the controller. Nevertheless, the actual handling of the user input and theimplementation of the views resides in the See++ package, since the GUI package only delegatesthese tasks.

6.1 Design of the Biomechanical Model

The structure of the biomechanical model should be easy to extend in order to permit integrationof latest research results. As a consequence, the model design should be able to handle modelextensions and modifications to existing parts of the model that do not completely invalidatecurrent model predictions.

In order to satisfy these requirements, the following design guidelines need to be applied:


• Not every part of the model should be a direct sub-model of the biomechanical model.Instead, sub-models should use other sub-models, so that the biomechanical model usesonly those parts of the model which are directly required for calculating model predictions.

• Sub-models which have a lot in common should not duplicate this shared functionality ineach model. A better approach is to base these models on an abstract base model whichcontains all the common functionality, but cannot be instantiated on its own. As a result,the addition of new models is much easier, since only the model-specific parts have to beimplemented in the new model.

• All model data should be stored once in order to avoid redundancies. The biomechanicalmodel should therefore provide an abstract representation of anatomical parts.

The software design for the biomechanical model starts with finding proper abstractions for themedical relevant entities of the oculomotor structures. First, all muscles are modelled using anabstract base class that unites all common operations and additionally aggregates anatomicalsubparts like insertion, origin and pulley points (see Fig. 6.3).

Figure 6.3: Abstraction of Muscles

The class diagram shown in Fig. 6.3 treats each geometrical point of a muscle as own class andtherefore provides different operations when accessing insertion, origin or pulley of a muscle. Thisfor example would mean that, when changing the radius of the bulbus, the muscle insertions andthe pulleys are extended, whereas the origin points stays the same.

It is now desirable to incorporate these geometrical abstractions into a design for a completebiomechanical model, consisting of geometrical, muscle force and kinematic model according toSec. 4.2. One possibility is illustrated in Fig. 6.4, which defines one superclass „BiomechanicalEye Model“ that references all sub-models for modelling geometry, muscle force and kinematics.Moreover, this superclass also directly accesses the six different eye muscle classes and the bulbusclass which hold all data for one eye.

The model structure shown in Fig. 6.4 does however lead to some implementation difficulties.Since the superclass „Biomechanical Eye Model“ directly references different geometrical sub-models like String- or Tape model, common functionality for geometrical calculations cannot beincorporated into this design. When for example, an additional pulley model needs to be added,


Figure 6.4: Primary Abstractions for Biomechanical Model

implementation redundancies will occur in that all basic functionality needs to be duplicatedwithin a new geometrical model. Moreover, external usage of the „Biomechanical Eye Model“ de-pends on knowing which geometrical models are available and also needs to differentiate betweendifferent access modes of these models. Identical considerations hold for the muscle force modeland the kinematic model.

A conceptual description of a biomechanical model that is very similar to that shown in Fig. 6.4is suggested in [MD99] and [Gue86]. However, the previously mentioned flaws are also present inthese models making them hard to modify and to extend.

In order to provide the necessary flexibility and extensibility of the design model, new abstractionsneed to be incorporated into the structure from Fig. 6.4. This implies the introduction of abstractbase classes for each of the sub-models for geometry, muscle force and kinematics. This way,superclasses need to access only one interface in order to be able to use different sub-modelswithin the same functional responsibilities.

Fig. 6.5 shows an extended version of the model structure, where the biomechanical model nowuses only one sub-model, namely the kinematic model. The kinematic model itself uses only themuscle force model. The muscle force model is an abstract base class with currently one derivedsubclass. Derived models are concrete realizations of an abstract base model, where the abstractbase class holds all common functionality. Finally, the muscle force model uses the geometricalmodel which is also an abstract base model. The geometrical model has three derived models,the String-, Tape- and Pulley model, and uses the models of the six extraocular eye muscles andthe bulbus.

The structure for modelling the eye muscles, previously shown in Fig. 6.3 has also been extendedin Fig. 6.5. All muscles are now derivations of one abstract base class and the different parts ofa muscle, like insertion, pulley and origin, are now parts of the abstract base model only, so theyneed not be added to each individual muscle.

The following properties can be derived from the system design in Fig. 6.5:


Figure 6.5: Extensible Software Design for the Biomechanical Model

• New sub-models can be added to the biomechanical model more easily.

• Only parts affected directly by additions have to be validated for their correctness onceagain.

• Extensions to existing models only influence parts of the model and not the whole model.

According to the desired structure of a biomechanical model (see Sec. 4.2), the system designmodel exposes the „Biomechanical Eye Model„ class on the top of the class hierarchy. This classreferences the kinematic model which is responsible to connect geometry and muscle force model,but the geometrical model is only accessible through the muscle force model.

In the „SEE-KID“ model (cf. Fig. 6.5), the requirements for a specific geometrical model thatare enforced by the abstract base class „Geometrical Model“ are the calculation of the followingentities:

• The point of tangency of each extraocular eye muscle,

• the rotation axis around which a specific muscle rotates the eye (muscle rotation axis),

• the center of the muscle action circle,

• and the length of a specific muscle.

The listed requirements are specific to each geometrical model. In the „string model“, for example,the center of the muscle action circle is also the center of the bulbus (cf. Sec. 4.8). The muscle


action circle is defined by the muscle rotation axis and since the muscle rotation axis changeswith nearly every movement of the bulbus, the muscle action circle reacts in the same way.

Apart from containing the bulbus and the muscles, the class „GeometricalModel“ also providesmethods for specifying eye positions, performing eye rotations (using Fick or Helmholtz rota-tion sequences) and manipulating the properties of the contained bulbus and muscles. Thesefunctionalities are independent of a specific geometrical model and are therefore implemented inthe class „GeometricalModel“. Thus, the derived models of „GeometricalModel“ do not have toreimplement this functionality.

Similar conditions can be applied to the muscle force model. The abstract base class „Muscle-Model“ defines common functionality to load and store muscle force tables. Additionally, methodsfor cubic and bicubic data interpolation are provided (cf. Sec. 4.5). Each specific muscle forceprediction model then only implements functionality that is specific to a certain model.

For the kinematic model, the calculation of the forward and inverse kinematics are non-linearoptimization problems and optimization methods have to be used (see Sec. 4.6.3). The „SEE-KID“ model contains two different algorithms for solving kinematics. The Levenberg Marquardtalgorithm, which uses gradients, and the Downhill Simplex algorithm, which only needs functionevaluations to perform function minimization (cf. Sec. 4.6). Concerning convergence speed,the Levenberg Marquardt algorithm performs much faster since it uses gradients, whereas thedownhill simplex method is slower but much more accurate in finding minimums.

Figure 6.6: Optimizer and Related Classes

In the „SEE-KID“ model, the object-oriented design shown in Fig. 6.6 enables the use of bothalgorithms (see Fig. 6.6). Therefore, an abstract base class called „Optimizer“ is introducedwhich provides a common interface for solving the non-linear optimization problems. The classes„LevenbergMarquardt“ and „DownhillSimplex“ are derived from the class „Optimizer“ and imple-ment the respective algorithms. Moreover, the class „Optimizer“ uses the classes „ParamObject“and „FunctionObject“. The class „ParamObject“ enables the parametrization of the algorithms,whereas the class „FunctionObject“ is an abstract base class used for defining the objective func-tion (cf. Eqn. 4.135). Thus, a derived class of „FunctionObject“ implements the specific functionfor the optimization and since the forward and inverse kinematics are the problems to be solved,the kinematic model is derived from „FunctionObject“.


6.2 Design of the „SEE++“ Software System

According to Fig. 6.2, the design model for the „SEE++“ system consists of the GUI and the„SEE++“package. These packages hold functionality that provides the user interface and theinterface to the biomechanical model, described in Sec. 6.1. The content of the „SEE++“ packageis visualized in Fig. 6.7. The main functionality is contained within the packages „SeeMedic“ and„SeeModel“.

Figure 6.7: Structure of the „SEE++“ Package

The packages that are contained within the „SEE++“ package, according to Fig. 6.7 can bedescribed as follows:

• The SeePoint package provides classes for the different kinds of points like insertion pointsor points of tangency.

• The SeeGeneral package contains the base class SeeObject from which nearly all otherclasses of the See++ package are derived.

• The SeePatientData package is used for storing and loading data of patients.

• The SeeConstraint package describes limits which are applied to most of the user-controlledvalues in the GUI package.

• The SeeHelp package implements the context sensitive help.

• The SeeSurgery package provides different surgery techniques in connection with classes ofthe SeeView package.


• The SeeProjection package prepares data returned by the SeeModel package for use in theSeeView package.

The basic process of data flow within the „SEE++“ system starts with loading patient data(SeePatient package) and distributing these data into classes contained in the SeeMedic package.Next, simulations are performed using the biomechanical model through the interface of theSeeModel package, where modifications of patient specific data during simulation are checkedfor validity by classes that are contained in the SeeConstraint package. Simulation results aregraphically visualized by functionality that is assigned to the SeeProjection and SeeView packages.Surgical interventions that define specific methods of how the patient data can be modified inorder to simulate or correct pathological situations are realized within the SeeSurgery package.

Figure 6.8: Structure of the „SeeMedic“ Package

The SeeMedic and the SeeModel packages utilize the Adapter design pattern, which means thateach of the classes of these packages adapts the interface of another class (cf. [GHJV94]). Thus,the SeeMedic package provides access to the six extraocular eye muscles and their differentcomponents, although the classes of the SeeMedic package do not store any values (see Fig. 6.8).Instead, these values are retrieved from the class SeeDataModel, which is located in the SeeModelpackage (see Fig. 6.9). Since the SeeModel package also uses the Adapter design pattern, theSeeDataModel in turn retrieves the data from the biomechanical model through the instancer.The other classes in the SeeModel package are adapters of the different models contained in thebiomechanical model „SEE-KID“.

The class SeeMedic is the base class for all medical objects and the class SeeOrbita is used as acontainer for the six extraocular eye muscles and the bulbus. Moreover, for the four straight and


the two oblique muscles, two different abstract base classes called SeeRectus and SeeObliquushave been introduced, both derived from SeeMuscle.

The approach of using the Adapter design pattern in the SeeMedic and the SeeModel packagesmay seem too much effort. However, the advantage of this design is that the model stays inde-pendent from the views, since the views implemented in the SeeView package only use the classesof the SeeMedic package. On the other hand, the GUI package and the views are independent ofthe biomechanical model, which makes it quite easy to exchange either the biomechanical modelor the GUI.

Figure 6.9: Structure of the „SeeModel“ Package

The class SeeModel is the base class for all models in the SeeModel package. The class SeeData-Model provides access to the data stored in the biomechanical model and the class SeeStateModelserializes the internal state of the See++ package and the GUI package between different execu-tions of the „SEE++“ software system.

In order to visualize output data of the „SEE++“ software system, three different types of di-agrams (views) are supported. The muscle force distribution (MFD) diagram, the muscle forcevector (MFV) diagram and the Hess diagram. The MFD and the MFV diagram represent differ-ent visualizations of the output data of a geometrical model, whereas the Hess diagram is basedon the output data of the biomechanical model (see Sec. 4.7.1). The implementation of thesediagrams is not part of the „SEE-KID“ model, as they are implemented in the SeeView package,shown in Fig. 6.10, which is part of the See++ package.

Fig. 6.10 shows the SeeView package which contains the classes of the different views that the„SEE++“ software system supports. All views are derived from the abstract base class SeeViewand all diagrams are additionally derived from the abstract base class SeeDiagramView. Theclasses SeeVText and SeeV3D visualize the different properties of the muscles and the bulbus,either in a textual or in a three-dimensional form. The three diagrams which are all derived fromSeeDiagramView are visualizations of the output data of the biomechanical model.

For the calculation of the MFD diagram, implemented in the class SeeVMFDD, the musclerotation axis of a specific muscle is used. It is also possible to calculate one MFD diagram forseveral muscles by adding up the diagrams for each muscle. Thus, the MFD diagram visualizes theforce distribution of one or several muscles of one eye in a specific elevation plane (cf. Sec. 4.4.3.3).Each curve of the diagram shown in Fig. 4.19, represents a different rotational component of thevisualized muscle(s) in an arbitrary but fixed elevation plane within a specific range.

The MFV diagram and the implementation of this diagram can be found in the class SeeVMFVD.


Figure 6.10: Structure of the „SeeView“ Package

For the calculation of the MFV diagram, shown in Fig. 6.11, arbitrary gaze positions are chosenwhich are only constrained by anatomical boundaries. These gaze positions are drawn in a 2D-diagram where the x-axis represents adduction and abduction and the y-axis represents elevationand depression in a specific range. Each of the gaze positions is then used for drawing a vector,whereas the length and the direction of the vector are specified by the force distribution of aspecific muscle along with its particular pulling direction. The MFV diagram, like the MFDdiagram, also supports the visualization of several muscles by adding up the diagrams of eachmuscle.

Figure 6.11: Muscle Force Vector Diagram

Fig. 6.11 illustrates a MFV diagram of the lateral rectus muscle of a left eye with a set of gaze


positions within the 30 degree physiologic field of vision. These gaze positions are considered asdefault gaze positions for a „normal“ human subject.

However, MFD and MFV diagrams themselves cannot be gathered through clinical measure-ments. For this reason the “SEE++„ system supports the Hess diagram which can also bemeasured clinically (cf. Sec. 3.13).

The Hess diagram is a visualization of combined output data of the kinematic model and isimplemented in the class SeeVHess (see Fig. 6.10). The Hess diagram also depends on arbitrarygaze positions which must not exceed anatomic boundaries and are plotted in the diagram aspoints. These gaze positions are then used as an input for the calculation of the simulation (cf.Sec. 4.7.1). The axes of the Hess diagram are defined according to the MFV diagram, where thex-axis represents ab-/adduction and the y-axis represents elevation/depression within a specificrange (see Fig. 3.14). A Hess diagram contains different points which represent the specified andthe calculated gaze positions. To improve the visibility of the diagram, points of specified andcalculated gaze positions are drawn using different colors and are connected through thin lines.

In addition to the described packages, different helper classes for matrix calculations, quaternionalgebra and OpenGL are used. For matrix calculations the „NewMat“ framework was used (see[Dav03]) and for quaternion and 3D calculations the „3DMath“ classes (see [Fal99]) were adapted.

6.3 The „SEE++“ Software System

The „SEE++“ software is a new simulation system that aims at the forecast of clinical oper-ation results, as well as the representation of pathological situations in the field of strabismussurgery. The system is based on a highly developed mathematical simulation model (biome-chanical model), which copies the behavior of the human eye realistically and thus provides anexperimental platform for the simulation of pathologies and the evaluation of possible treatments.

„SEE++“ is a biomechanical system for the interactive three-dimensional simulation and visual-ization of eye motility disorders and their surgical correction.

The „SEE++“ software system offers the following functions:

• Compact, descriptive and thus well understandable knowledge transfer in teaching andtraining,

• scientifically oriented procedures for practice,

• fundamental references and numerous examples,

• a basis for individual considerations of diagnostics and operational correction of eye motilitydisorders.

The „SEE++“ software system aims the following target user group:

• Ophthalmologists, specialists as well as orthoptics can use „SEE++“ to support measure-ments and to archive pathologies and treatment methods.


• Researchers in the field of ophthalmology, strabismology and neurology, pediatrists as wellas researchers in the field of biophysics use „SEE++“ as an extensive scientific tool for theinvestigation of the mechanics of eye movements.

• „SEE++“ offers a substantial support to teachers through descriptive representation of thefundamentals for understanding eye movements.

• Students have the possibility to interactively deepen their knowledge and to recall anddescribe previously studied fundamentals of eye movement and strabismus.

6.3.1 „SEE++“ Simulation Task Flow

When using „SEE++“ to model and simulate eye motility disorders, a certain simulation task flowcan be defined in order to perform simulation and evaluation of model predictions. In Fig. 6.12,this task flow is illustrated as a sequence of simulation steps that can be performed iterativelywithin the software system.

Figure 6.12: Simulation Task Flow for using „SEE++“

1. Parametrization with patient data: One main goal of „SEE++“ is to give close-to-realityrelated predictions of a patient-specific situation. In this first step of the simulation taskflow, model values will be based directly on the patient. Parameters can be modified suchas globe radius, cornea radius, muscle lengths, insertions, tendons etc. At the same time,general data are entered like name or description.

2. Simulation of a pathology: During the simulation of a pathology, model parameters arechanged in such a way that the resulting model predictions correspond to measured valuesof the patient, as closely as possible. A model prediction is done in „SEE++“ by thesimulation of a clinical Hess-Lancaster test, whereby the representation for right or leftfixation is used. By determination of the Hess-Lancaster data of the patient, these valuescan be compared to simulated data to find whether the simulation corresponds to thepathology of the patient. Also the 3D representation of the patient offers an additionalsupport regarding the evaluation of a simulation.


3. Comparison of simulation results with patient data: This comparison refers to theHess-Lancaster investigation already mentioned, whereby the process of comparing canalso serve as a verification of the diagnosis posed before. Thus, in this step it is to de-termine whether the simulation result agrees sufficiently with the measured patient data.This, at the same time, offers a basis for a later simulated treatment by interactive virtualsurgery of the modelled pathology.

4. Simulation of surgery: Here the actual operation is simulated by interactively modifyingdifferent model parameters using the mouse within the 3D representation. Points of refer-ence support orientation and the dosage of the surgery performed. Furthermore, differentoperation techniques are available such as transposition and tangential repositioning of amuscles insertion. Muscle force and innervation parameters are changed manually in theprogram so that they correspond to a comparable surgical procedure. For example, a mus-cle resection can be accomplished by changing the value of the parameter for muscle length.The 3D model visualizes these changes immediately after confirmation of entered values.

5. Evaluation of results: According to step 3, a comparison of the simulation results is carriedout again. On the basis of the binocular Hess-Lancaster test, the outcome of a surgery canbe judged regarding to the correction of a pathological situation, and whether it is stillnecessary to apply additional changes (simulation trials).

6. Simulation result: The simulation result represents the last condition of all model param-eters in the task flow of the simulation of a pathology with „SEE++“. The system enablesthe user to assign and archive scenarios to a patient. Thereby the results of different sim-ulations may be compared and e.g. simulation strategies can be developed. Each scenariostores any step of a treatment of a patient and can later be recalled in textual or graphicways.

6.3.2 Simulation properties

When starting the program, the default view depicted in Fig. 6.13 appears. All functions of theprogram are accessible through the main menu in a structured manner. At the same time, atree representation in the Treeview provides the same functions for direct access. The „SEE++“system features four different diagram windows, where each diagram can be displayed arbitrarilywithin each view window. A fixed component of the system is the 3D view, in which the currentsimulation is illustrated by a „virtual patient“. The „Toolbar for 3D-View“ is used in order toconfigure this view, i.e. to show or hide muscles, the globe, points of reference, etc.. The „Toolbarfor Main Functions“ allows quick and direct access to the most important features of the mainmenu or the Treeview, as well as it allows to switch the current model. Depending on the currentposition of the mouse cursor, additional information is displayed in the Status Bar at the bottomof the main window.

Medical data includes all data necessary for modelling and simulating a „virtual patient“. Whenreferring to a patient file, patient data and interactions can be distinguished. Interactions mapsimulation and surgeries of the „virtual patient“ through appropriate choice of the simulationparameters. All values of these simulation parameters, which describe a pathological situationor simulate a surgical intervention, are centralized into medical data.


Figure 6.13: Default View of the „SEE++“ Software System

Thus, Medical data includes data for left eye, right eye and a so-called „reference eye“.

6.3.2.1 Globe Data

Every eye simulated by „SEE++“ has its own globe data parameters. In this explanation, theleft eye is used as an example. Of course, all details apply similarly to the right eye and thereference eye.

Figure 6.14: Globe Data Parameters of the „SEE++“ System

This dialog shown in Fig. 6.14 displays the current patient’s name if previously entered in thepatient’s data. The values to be adjusted here are „Globe Radius“ and „Cornea Radius“, wherebypreset default values depend on the geometric model selected. The globe affects the result of


the simulation fundamentally, since an enlarged or reduced radius leads to different forces of themuscles acting on the globe. When this value is modified, additionally all points of insertion andpulleys (functional origins) of all muscles are adapted automatically. Changing the cornea radiusaffects only the geometric shape of the globe, but not the result of the simulation.

6.3.2.2 Muscle Data

The muscle data dialog shown in Fig. 6.15 is one of the most important elements of the „SEE++“system. Here, the muscle force model can be adapted to pathological conditions with regard toparticular eye muscles. All data in this dialog influences the development of force of all or ofcertain muscles only. Consequently, muscle palsies, overactions or fibroses can be simulated (cf.Sec. 3.5). Again, muscle data is present for all eyes simulated by „SEE++“ (left eye, right eyeand reference eye).

Figure 6.15: Muscle Data Parameters of the „SEE++“ System

The muscle data dialog also offers the possibility to manually modify some geometric properties(i.e. origin, pulley and insertion). Modifying these parameters causes alteration of muscle path,and thus a different result of the simulation. All values are defined in primary position, even ifthe 3D view depicts a different eye position, the muscle dialog, in its geometric values, alwaysrefers to primary position.

Modification of muscle force parameters applies to the muscle force curves (i.e. elastic, contractileand total force) situated on the right side of the dialog. These curves can be adjusted accordingto the simulation parameters defined in Sec. 4.5.


6.3.2.3 Distribution of Innervation

The dialog for modifying the distribution of innervation controls the activation potential of themuscles on the basis of stimuli generated by motor nuclei of the cranial nerves (cf. Sec. 2.2.3).Since this distribution is connected to every eye, it represents an abstraction from the actualanatomical structure (cf. Sec. 2.2.3). Different nuclear or supranuclear lesions can be modelledby appropriate adaptation of the distribution of innervation for the left and/or the right eye. Thedistribution of innervation is available for all eyes simulated by „SEE++“ (left eye, right eye andreference eye).

6.3.2.4 Gaze Patterns

Additionally, a gaze pattern can be defined for the left and the right eye, each specifying fixationpositions for the simulation of the Hess-Lancaster test (cf. Sec. 3.13). The dialog shown inFig. 6.16 enables modification and storage of gaze patterns, as well as the possibility to manuallyenter measured patient values in order to compare simulation predictions with patient measureddata.

Figure 6.16: Gaze Pattern Dialog of the „SEE++“ System

Changes concerning medical data are saved as interactions in scenarios that are assigned to apatient. They are saved along with the patient file as well. By using this scenario concept,simulation tasks can be retrieved at a later time without loosing the order in which they wereperformed.


6.4 Evaluation

In the following sections, different examples for the simulation and correction of pathological casesare described. The first two examples show common basic cases of pathologies that affect only asingle muscle. In further examples, simulation results are also compared to clinical measurementsacquired pre- and postoperatively from patients that had surgery.

6.4.1 Abducens Palsy

In the first example, an abducens palsy of the right eye will be simulated. Abducens palsy is anincomitant form of squinting, i.e. the squint angle increases in the main functional direction ofthe muscle concerned - namely the lateral rectus muscle - towards abduction. Clinically, a patientincreasingly shows double images (uncrossed) towards abduction, thus in right gaze positions.Possible causes for an abducens palsy are, among other things, traumata of the peripheral nerve(that is the entire nerve without the nucleus) caused for example by a basal skull fracture. As aresult, a damage to the nerve from the nucleus (in the brainstem) up to the insertion can takeplace. In case of a damage directly within the area of the nucleus, there is a possibility that alsothe interneurons (the connections between the nucleus of the abducens nerve and the conjugatedmuscle on the other side, the left medial rectus muscle in this case), which are situated in theneighborhood, are hurt. This case is not assumed in this example.

6.4.1.1 Simulation of the Pathology

As a consequence of a lesion to the nerve, the contractile strength of the muscle has to be reducedand furthermore its elastic parts have to be modified. This is the basis for the simulation. In the„SEE++“ software system, this force reduction is carried out in the muscle data dialog, shownin Fig. 6.17.


Figure 6.17: Muscle Properties for Abducens Palsy Simulation

In the field „Total Strength“ the strength of the muscle can be reduced by changing the valuefrom 1(%/100) to 0.5(%/100). When the Hess-diagram calculation is displayed, the modificationsfrom the muscle data dialog are immediately taken into account (see Fig. 6.18).

Figure 6.18: Hess-Lancaster Test for Abducens Palsy

In the calculated Hess-diagram shown in Fig. 6.18, for the left eye (right eye fixing) the exceeding


reaction of the following left eye into adduction can be seen. Conversely, in the second Hess-diagram the restriction of the right eye (left eye fixing) in abduction is evident. On the basisof the results of the simulation, which are represented by the Hess-diagrams in Fig. 6.18, thesimulation of the abducens palsy can be considered as sufficient.

6.4.1.2 Simulation of Surgical Correction

For the surgical correction of the simulated abducens palsy, a strengthening of the paretic muscleis necessary. The surgery is carried out by shortening the affected lateral rectus muscle (resection).During a resection surgery, the insertion of a muscle is separated from the globe, a piece of themuscle is cut off or folded in and afterwards, the muscle is fixed again at the same position on theglobe. For shortening a muscle in „SEE++“, the muscle data dialog is used again. The relaxed,denervated muscle length (L0) of the right lateral rectus muscle is reduced by 4 mm from 37.5mm to 33.5 mm (see Fig. 6.19).

Figure 6.19: Simulation of Right Lateral Rectus Resection

Now the two Hess-diagrams in Fig. 6.20 show that the target of the surgical correction has beenachieved, namely to get the double image-free zone into the primary position without substantiallyweakening adduction.


Figure 6.20: Postoperative Hess-Lancaster Simulation for Abducens Palsy

However, a complete „healing“ of the palsy by modifying the innervational component in themodel is clinically not possible, since a surgical modification of innervation is impossible.

6.4.2 Superior Oblique Palsy

In the second example, a palsy of the superior oblique muscle of the right eye will be simulated.This palsy is, similar to the abducens palsy, an incomitant form of squint. The vertical deviationof the palsied eye increases towards adduction and depression, equally, extorsion increases inabduction. The horizontal component is affected in the sense of a convergent deviation. Again,similar to the abducens palsy, a possible cause can be a basal skull fracture, since the trochlearnerve, like the abducens nerve, is vulnerable to traumatic injuries due to its length. Clinically,the patient usually takes an abnormal head posture (tilt to the left) for balancing extorsion andfor maintenance of binocular vision. Vertically shifted and outward-tilted double images increasein depression and convergence (in the main functional range of the concerned muscle).


Similar to the simulation of the abducens palsy (see Sec. 6.4.1), the contractile strength of themuscle has to be reduced and furthermore, its elastic parts have to be changed (see Fig. 6.21).


Figure 6.21: Muscle Properties for Superior Oblique Palsy

For successfully simulating a superior oblique palsy, the strength of the superior oblique muscle ofthe right eye has to be reduced. In the „SEE++“ software system, this force reduction is carriedout in the muscle data dialog shown in Fig. 6.21.

In the field „Total Strength“ the strength of the muscle can be reduced by changing the valuefrom 1(%/100) to 0.3(%/100).

Figure 6.22: Hess-Lancaster Simulation of Superior Oblique Palsy


In the calculated Hess-diagram shown in Fig. 6.22, the exceeding reaction of the inferior rectusmuscle of the left eye (right eye fixing) can be seen. In the second Hess-diagram, the increasingincomitant restriction of the superior oblique muscle of the right eye (left eye fixing) in adductioncan be clearly seen. On the basis of the results of this simulation, the simulation of the superioroblique palsy can be considered as sufficient.


For the surgical correction of the simulated superior oblique palsy, a strengthening of the pareticmuscle is necessary. The surgery is carried out by shortening the affected superior oblique muscle(resection). According to Sec. 6.4.1, the relaxed, denervated length (L0) of the muscle (superioroblique muscle) is reduced by 5 mm from 34.15 mm to 29.15 mm and the simulation results areshown in Fig. 6.23.

Figure 6.23: Postoperative Hess-Lancaster Simulation for Superior Oblique Palsy

Now, the two Hess-diagrams in Fig. 6.23 show that the target of the surgical correction hasbeen achieved, namely to get the double image-free zone into the primary position. However,as explained in Sec. 6.4.1, a complete „healing“ of the palsy, even in extreme adduction anddepression, is clinically not possible, since a surgical modification of the innervation is impossible.

6.4.3 Superior Oblique Overaction

Vertically incomitant squinting is characterized as horizontal misalignment of the eyes in whichthe magnitude of the horizontal deviation differs in upgaze when compared to downgaze. Com-monly, so called A-patterns and V-patterns are seen with respect to the Hess-Lancaster test.These patterns are named using letters of the alphabet whose shapes have visual similarities tothe ocular motility patterns that they describe.


(a) Increased Divergence in Downgaze (b) Increased Convergence in Upgaze

Figure 6.24: Classification of Superior Oblique Overaction

The term A-pattern designates a vertically incomitant horizontal deviation, shown in Fig. 6.24,in which there is more convergence in midline upgaze (see Fig. 6.24(b)) and less convergence(increased divergence) in midline downgaze (see Fig. 6.24(a)). An A-pattern esotropia is aninward deviation of the visual axes in which there is more inward deviation of the eyes in midlineupgaze than in midline downgaze. An A-pattern exotropia is an outward deviation of the visualaxes in which there is more divergence of the eyes in midline downgaze than in midline upgaze.

With significant A-patterns, version testing usually reveals superior oblique muscle overaction.The tertiary abduction effect of the superior oblique muscle is believed to produce the A-pattern.The abducting force is greatest in downgaze within the superior oblique’s primary field of action,causing an increased relative divergence of the eyes in downgaze.


In order to simulate superior oblique overaction, clinically measured patient data was used tocompare predictions of the „SEE++“ software system when adjusting simulation parameters.The simulated pathology is shown in Fig. 6.25, where the measured patient data is displayed asgreen Hess-Lancaster diagram and the simulation results are shown in red.

In order to simulate the pathology, the path of the superior oblique muscle from the trochlea tothe insertion was modified. In the „SEE++“ software system, this can be done by displacing thevirtual pulley of the superior oblique muscle away from the nose. Thus, the pulley position ofthe superior oblique was changed from −15.270/11.000/11.750 to −13.250/11.000/11.750.


Figure 6.25: Hess-Lancaster Simulation of Superior Oblique Overaction

In order to show an overacting superior oblique muscle, the total muscle strength was changedfrom 1.00(%/100) to 2.10(%/100), additionally the relaxed, denervated muscle length (L0) wasreduced from 34.150 mm to 32.150 mm. Conversely, the total muscle strength of the inferioroblique muscle was reduced from 1.00(%/100) to 0.20(%/100) and the muscle length (L0) waschanged from 30.55 mm to 34.55 mm. Thus, the inferior oblique muscle was lengthened, whereasthe superior oblique muscle was shortened. The result of the simulation can be seen in Fig. 6.25.


In order to correct the pathological situation shown in Fig. 6.25, a weakening of the concernedmuscle needs to be carried out. Therefore, the right superior rectus muscle was transposed alongits main direction of action, as shown in Fig. 6.26. This leads to a sufficient correction withinhorizontal gaze positions.


(a) Recession and Transposition of the SuperiorOblique Muscle

(b) Final Position of the Superior Oblique Muscleafter Surgery

Figure 6.26: Simulation of Superior Oblique Surgery

The complete surgical treatment is characterized by a recession of the right superior obliquemuscle of 8.6 mm followed by a tangential transposition of 4.5 mm (see Fig. 6.26(a)). Theresulting position of the superior oblique muscle can be seen in Fig. 6.26(b).

The simulation results of this operation are shown in Fig. 6.27.

Figure 6.27: Hess-Lancaster Simulation of Superior Oblique Surgery

It can be seen that the results of the simulation predictions correspond to postoperative treatment


prospects in that within the normal physiologic field of gaze, binocular function has been restoredto near normal conditions.

6.4.4 Heavy-Eye Syndrome

Motility disorders caused by a myopic globe with high axial length were reported in literaturelong ago. Donder reported in 1864 [Don64] frequent divergence in high myope patients whoseelongated globes find difficulty in orientation. The same phenomenon is described in a text bookby Duke Elder (1968) [DE73] in patients with only one myopic eye. A deviation of this type wascalled the heavy eye syndrome by Ward (1967) (cf. [Kau95]). Such an eye is frequently limitedin vertical excursions.

For this reason R. Hugonnier and Magnard (1960) suggested the designation „the nervous syn-drome of high myopia“, which, because of the muscular aetiology, was changed to „myopic myosi-tis“ by Hugonnier (1965) [Hug65].

The term heavy eye syndrome is descriptive for hypotrophia. Anatomic studies showed an irreg-ular path of the extraocular muscle from the origin through the orbit to the point of insertion.Kaufmann [Kau95] interpreted the heavy eye syndrome with a dislocation of the medial andlateral rectus muscles into a caudal position. These translocations should cause an overweightof downward rotating force. Furthermore, Kaufmann reports a shift of the superior and inferiorrectus muscle in nasal direction with the result of adduction overweight. Detailed informationwas gathered using MRI analysis of myope patients.

Several publications analyzed the exact muscle path and quantified the translocation of theaffected muscle. Krzizok [KKT97] reported a dislocation of the lateral rectus muscle into thetemporocaudal quadrant by 3.4 mm based on MRI findings. Furthermore, Krzizok suggested afixation of the dislocated muscle in the physiological meridian as causal therapy.

In a publication by Schroeder [SKT98], the dislocation of all four rectus muscles is quantified.Schroeder reported a dislocation of 2.9 mm of the lateral rectus muscle into the lower temporalquadrant, the path of the superior rectus muscle was altered 1.5 mm medially and the path ofthe inferior rectus muscle was shifted 1.3 mm medially, the medial rectus muscle was dislocated1.3 mm downwards. Before these MRI studies, aetiology of the heavy eye syndrome was unclear.

(a) Lateral Rectus Muscle with Normal Path (b) Displacement of the Lateral Rectus Muscleinto Temporocaudal Quadrant (Base Hypothesisfor Hypotrophy in Heavy-Eye Syndrome)

Figure 6.28: Muscle Displacement as Hypothesis for Heavy-Eye Syndrome


Fig. 6.28 illustrates the effect of dislocation of the pulley in temporocaudal position. The pullingdirection of the muscle shown in Fig. 6.28(a) is shifted in the same way. Contraction of themuscle shown in Fig. 6.28(b) causes mainly abduction but also a downward movement. Thebiomechanical relevance of these dislocations with the result of downward movement of the eyewas not analyzed up to now.

Dislocations of rectus muscles are reported in a range from 1.3 mm up to 3.4 mm. The operationis usually performed with permanent sutures or silicon loops. This technique is also known as„guide pulleys“, which aims to establish an artificial fixation and avoid a further side-slip of themuscle. These „guide pulleys“ fixate the muscles to the globe and therefore influence the „natural“pulleys. MRI analysis was not able to clarify mechanical effects which may be responsible for theheavy eye syndrome.


The primary aspect for the simulation of the Heavy-Eye Syndrome is the enlargement of the globeof the affected eye. Due to an oversized globe, the rotational effect of all muscles that act on theaffected eye is much lower compared to a normal eye. Moreover, muscle tension increases, sincethe distance from the pulley to the insertion also extents, while muscle length stays constant.

For this simulation, a 48 years old patient with Heavy-Eye Syndrome was measured and comparedwith predictions of the „SEE++“ biomechanical simulation system. The measured values inFig. 6.29 are shown in green color.

Figure 6.29: Measured Values from Patient with Heavy-Eye Syndrome

The first step in the simulation is the modification of the globe radius for both eyes according toultrasonic globe measurements of the patient. Thus, the globe radius of the left eye was changedto 14.00 mm and the globe radius of the right eye was changed to 16.50 mm. The results of thesimulation for these changes can be seen in Fig. 6.30.


Figure 6.30: Simulation Results for Resized Globes according to Patient Data

The second step is the simulation of the muscle dislocations in order to reproduce the Heavy-EyeSyndrome according to suggestions from Schroeder and Krzizok. By dislocating the pulleys of allfour rectus muscles, the typical downward overaction of the affected eye could not be reproducedin the „SEE++“ system, using the suggested approach, as illustrated by the simulation results inFig. 6.31.

Figure 6.31: Simulation Attempt using Data suggested by Schroeder and Krzizok

In contrast to suggestions found in literature, biomechanical simulation of pulley dislocationsusing the „SEE++“ software system barely affects horizontal gaze positions in the Heavy-EyeSyndrome. Therefore, the oblique muscles obviously contribute to this pathology due to supe-rior oblique overaction and inferior oblique underaction which in turn influences horizontal gazepositions.

In order to simulate the Heavy-Eye Syndrome according to measured patient data, modificationof the oblique muscles was performed as shown in Fig. 6.34, in that the insertion of the superior


oblique was moved so that both oblique muscle form a steeper angle to the median-sagittalplane (compare Fig. 6.32(a) and Fig. 6.32(b)). Additionally, the oblique superior muscle wasstrengthened, whereas the inferior oblique muscle was weakened. This results in amplification ofthe downward movement of the globe.

(a) Normal Insertion Location ofthe Superior Oblique Muscle

(b) Transposed Superior ObliqueInsertion

Figure 6.32: Superior Oblique Muscle Insertion Transposition in Heavy-Eye Simulation

The complete values for the simulation parameters that lead to the simulation results shown inFig. 6.33 are denoted as follows:

• Globe radius of the right eye was changed from 11.90 mm to 16.50 mm.

• Globe radius of the left eye was changed from 11.90 mm to 14.00 mm.

• The right lateral rectus pulley was displaced 2.90 mm inferiorly.

• The right medial rectus pulley was displaced 1.30 mm inferiorly.

• The right superior rectus pulley was displaced 1.50 mm medially.

• The right inferior rectus pulley was displaced 1.30 mm medially.

• The right medial rectus contractile strength was changed from 1.00(%/100) to 2.00(%/100).

• The right lateral rectus contractile strength was changed from 1.00(%/100) to 0.50(%/100),the elastic strength was changed from 1.00(%/100) to 0.20(%/100).

• The right superior rectus contractile strength was changed from 1.00(%/100) to 1.10(%/100),the elastic strength was changed from 1.00(%/100) to 0.50(%/100).

• The right inferior rectus total strength was changed from 1.00(%/100) to 1.80(%/100).


Figure 6.33: Hess-Lancaster Simulation Results of the Heavy-Eye Syndrome


The patient with Heavy-Eye syndrome simulated in Sec. 6.4.4.1 was treated with superior oblique,medial and lateral rectus surgery. The insertion of the superior oblique muscle was transposedto the vertical pole (see Fig. 6.34(b). The medial rectus muscle was recessed by 4 mm andtangentially transposed downward by 4 mm, whereas plication surgery was performed at thelateral rectus muscle by 7 mm followed by a 4 mm upward tangential transposition.

(a) Directions of Muscle Trans-positions(1) Anterior/Posterior Transpo-sition (2) Tangential Transposi-tion

(b) Fixation of Muscle to theVertical Pole

Figure 6.34: Muscle Surgery for Heavy-Eye Simulation

The same surgery was simulated using the „SEE++“ software system. The following modifications


to simulation parameters were performed:

• The right superior oblique muscle was positioned at the vertical pole (see Fig. 6.34(b)),in that the insertion was moved 13.20 mm horizontally posterior and subsequently moved-2.90 mm tangentially (see Fig. 6.34(a)).

• The insertion of the right medial rectus muscle was moved 4.00 mm horizontal posteriorand subsequently moved -4.00 mm tangentially (see Fig. 6.34(a)).

• The insertion of the right lateral rectus muscle was moved 5.00 mm tangentially (seeFig. 6.34(a)).

• The right lateral rectus muscle length (L0) was changed from 37.50 mm to 32.00 mm.

The simulation result is shown in the Hess-Lancaster diagram in Fig. 6.35. The results showsatisfactory reorientation of the lines of sight in primary position. However, full restoration ofocular motility in case of Heavy-Eye Syndrome is most often not possible due to the disease’sextensive pathological impact.

Figure 6.35: Hess-Lancaster Simulation Results of the Heavy-Eye Surgery

The simulation results from Fig. 6.35 show that the biomechanical model described in this thesis isalso capable of simulating complicated pathological situations as well as their surgical treatment.Amounts of surgery that were performed on the patient can be used identically or with only smalladjustments within the biomechanical model, which provides proper anatomical conformance inmost cases. This way, the „SEE++“ software system provides a good approximation for clinicalpre- and postoperative simulation.

Chapter 7

Conclusion

The research work described in this thesis has been inspired by the idea that modern technologiescan also contribute and even advance medical processes, in this case, the field of strabismussurgeries. Besides the technical skills in informatics, software engineering and biomechanics,extensive medical knowledge was needed to create a modern, interactive software system thatsupports surgeons in complicated medical decision making and explains students fundamentalsabout the way how the oculomotor system works.

One of the most challenging task within this research work was communication with physiciansand medical personnel. Since this research work already started in 1996, almost the first twoyears were spent with gathering medical knowledge and finding common ways to communicateand talk about medical and technical topics. This common language can be identified as vitalfor all further work that was accomplished within this research project.

From the technical point of view, a complex software system needed to be designed that pro-vides flexibility and extensibility, since many state of the art research results within the field ofextraocular physiology needed to be incorporated successively. Moreover, it was even necessaryto carry out medical studies with partner institutions in order to gain deeper insight in specifictopics of extraocular muscle function and provide the possibility to study pathological situations.One of these studies is described in Sec. 2.1.4 and Sec. 5, where high resolution MRI studieswere carried out in order to visualize 3D reconstructions of the morphology of the extraocularmuscles in normal human subjects. However, many other research data was needed to developa biomechanical model of the human eye. Many other partners provided data about extraocularmuscle force measurements, extraocular geometry and eye motility diagnostics. Without thesemedical basis, a mathematical model would never show any relation in its predictions comparedto the human eye.

While developing this model and the software system, it was early realized that the basic un-derstanding of the function of the oculomotor system is still under heavy discussion. Afterimplementing and simulating ocular geometry based on the string and tape models (cf. Sec. 4.4),the pulley hypothesis proposed a very different operation mode of the oculomotor system. Inte-grating pulleys into the biomechanical model also clarified how muscle force predictions can berelated to eye positions in a way such that muscle force equilibrium defines a stable eye position.Nevertheless, these new findings were incorporated into other biomechanical models before, but

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one of the main achievements of this work was that the problem of finding stable eye positions wasreduced to a common non-linear optimization problem, without invalidating model predictions.

Another main feature of the developed software system „SEE++“ is, that it provides a simpleand easy to use interface which controls biomechanical model parameters in way that is familiarto medical personnel and physicians. Moreover, one of the goals was that a physician workingwith this software system does not need extensive mathematical background knowledge in orderto simulate complex pathological eye motility disorder. In contrast, the user is able to identifyanatomically related parameters and surgical techniques that closely correspond to clinical ex-perience. Thus, the biomechanical model can be parameterized, using measured patient datawithout additional modification.

Concerning the mathematical background of the biomechanical model, implementation was splitinto pure modelling of the biomechanics of the oculomotor system and representation of anatomi-cal abstraction that form an interface to the biomechanical model. This way, maximum flexibilityon both ends of the software system was reached. The user interface can be modified without theneed to adapt the biomechanical model. Conversely, the biomechanical model can be modified inits workflow or its behavior that controls model predictions, without modifying the user interfacethat controls anatomically related parameters that influence the model.

One of the most important parts of the system is the interactive 3D representation of a „virtual“patient. To provide anatomically relevant models in three dimensions, extensive work has beencarried out in the field of image analysis and image processing (cf. Sec. 5). Therefore, knowledgein 3D computer graphics and geometry needed to be incorporated into the software system.Modern methods for user interface design and interactive control of 3D scene rendering wereimplemented to give the user most intuitive control (cf. [Fal99]).

Numerical mathematics, especially algorithms for solving non-linear optimization problems wereused to connect ocular geometry with muscle force simulation and solve forward and inversekinematics. Based on these formulations, clinical test methods were investigated and checked forsuitability with respect to integration into the software system. Especially the Hess-Lancastertest of binocular function (cf. Sec. 3.13) turned out to give valuable information due to itsextensive clinical usage and its relation to other similar measurement methods (cf. Sec. 3.4)within clinical assessment of eye motility disorders. Therefore, the clinical Hess-Lancaster testwas implemented based on the biomechanical model (cf. Sec. 4.7.1). This abstraction of aclinical measurement technique also solves the crucial problem of connecting both eyes in orderto transform innervations of one eye to innervations of the fellow eye. Since the horizontal rectusmuscles are mirrored with respect to their locations between both eyes, innervations can betransformed by using a virtual reference eye that transforms innervations based on mirrored eyepositions. This also provided a basis for simulating neural control of the oculomotor nuclei inorder to also simulate neurological disorders.

Within the last four years, research within this project has been concentrated on refining thebiomechanical model by evaluating different pathological cases and comparing model predictionsto clinically measured patient data. Extensive coordination between clinical studies and technicalrealization was performed during this time. Different partner institutions have been engagedwithin this process. The university hospitals of Graz and Innsbruck have contributed to thiswork in providing clinical data and studies. The hospital St. Pölten and the Wagner Jauregghospital in Linz assisted in carrying out anatomical and physiological studies using MRI and


human dissections. The convent hospital of the Barmherzigen Brüder is the primary cooperationpartner within this research work and has been the first institution to use the „SEE++“ softwaresystem clinically. Currently, the university hospital of Vienna evaluates the system for educationalpurposes. International cooperations supported this work by providing in depth knowledge aboutocular physiology and anatomy. The Smith Kettlewell Eye Research Institute in San Franciscoprovided muscle force measurement data (cf. Sec. 2.3.2) and the university hospital of the ETHZurich provided measurement data of eye movements.

Besides the efforts of research and software implementation, the most challenging task was theintegration of the software system into the clinical environment. During the last four years, pub-lications and conference presentations in the field of strabismus research have been published andworldwide attention has been gained. In order to convince physicians of the reasonable practicalrelevance of such a software system, pure mathematical proofs do not suffice. Instead, manycase studies have additionally been presented in order to show clinical compliance of simulationpredictions. The idea was to propose a new way of planning and simulating eye muscle surgeriesand to provide methods to study the functions of the oculomotor system. Currently, a book oneye motility disorders and computer aided simulation and treatment is being published to givedeeper insight into possibilities and future prospects in using this new methodology.

7.1 Goals Achieved

Within the medical field, the main goals were to advance the clinical treatment of eye motilitydisorders and to provide computer aided teaching support. Within the clinical integration processof the software system „SEE++“ these goals were achieved by carrying out clinical trials andcomparisons of patient data. Thus, clinical application of the „SEE++“ improved patient careby supporting surgeons in diagnosis and preoperative planning. Often, up to three successiveoperations are performed in order to achieve a successful treatment result. Especially in suchcases, the application of computer aided surgical treatment contributes to minimize repeatedsurgical treatment which results in benefits for the patients and directly reduces treatment costs.

The integration of the biomechanical eye model into the field of medical training and educationenables students and teachers to interactively explain and study basic functional aspects aswell as surgical methods. Educational trial lessons have shown that the presented softwaresystem supports students in self-studying the function of the oculomotor system and considerablyenhance basic understanding of functional implications of different surgical treatment methods.

In basic research, this biomechanical model provides a way of gaining a more detailed under-standing in principles and processes that affect oculomotor control. In this case, biomechanicalmodels provides an efficient method for checking hypotheses and verifying experimental data.Moreover, state of the art research results in anatomy and physiology can be incorporated intothe biomechanical model and subsequently improve simulation predictions.

The „SEE++“ software system currently implements the worldwide most accurate and up todate biomechanical model of the human eye. Due to a well structured object oriented design,this system provides adequate flexibility for further improvements. Integration of interactive threedimensional visualization methods and a user interface that corresponds to anatomical notionsprovide an efficient new way for physicians to intuitively handle biomechanical simulations.


7.2 Future Work

During research, many additional topics and ideas for enhancements and future investigationshave been discovered. Most of the future work that is planned will concentrate on further im-provements of the biomechanical model and the software system „SEE++“. However, additionalwork will try to use parts of the „SEE++“ software for the development of new measurement andclinical diagnostic methods.

One of the next improvements of the biomechanical model will introduce the active pulley hypoth-esis (cf. Sec. 2.1.2) in that a muscle consists of two distinct layers, one inserting at the pulleyand one on the globe. Additionally, data from Demer et. al. [KCD02] suggests that pulleysare dislocated as a function of eye position or muscle innervation. Integration of active pulleysmay lead to even more accurate simulation predictions, especially when simulating pathologicalsituations.

Up to now, a physician that is interested in exploring surgical methods needs to reproduce apathological situation in terms of model parameter values before simulation of surgery can beaccomplished. One major extension to the biomechanical model will deal with the automated gen-eration of pathological simulations based upon measured patient data. Therefore, biomechanicalmodel parameter values need to be fit to actual clinical measurements. Using such functionalitywould greatly improve clinical usage.

Currently, the extension of the „SEE++“ system into a scalable component architecture, realizinga biomechanical „construction kit“ is evaluated. This construction kit should provide standardtypes of elements in order to aggregate and combine new biomechanical models based on agraphical interactive way of programming.

In order to assign standard cases to specific classes of pathologies, patient data will be - ad-ditionally to the computation of functional interpretations - stored in a knowledge base. Thisenables the system to suggest a suitable surgery for a new pathological case that fits into one ofalready stored pathological classes. This practice of evidence-based medicine means integratingindividual clinical expertise with the best available external clinical evidence from systematicresearch.

One of the most exciting future development will aim improvements of eye position measurementmethods (cf. Sec. 2.3), especially within efficient objective measurements using video oculography(cf. Sec. 2.3.1.4). Currently, two techniques exist for the recording of eye positions: scleral searchcoils (cf. Sec. 2.3.1.3), and video-oculography (VOG). While scleral search coils have a hightemporal and spatial resolution, they are too expensive and invasive to be used outside theresearch laboratory. One part of the future work is the development of new VOG algorithmsthat compensate for translations of the camera with respect to the head. Additionally, the„SEE++“ software system shall serve as front end for VOG measurements in order to show howeye movements change after surgery on the extraocular muscles.

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[MHCS96] S.T. Moore, T. Haslwanter, I.S. Curthoys, and S.T. Smith. A geometric basis for measurement ofthree-dimensional eye position using image processing. Vision Res., (36):445–459, 1996.

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[MR84] J.M. Miller and D.A. Robinson. A Model of the Mechanics of Binocular Alignment. Computers andBiomedical Research, (17):436–470, January 1984.

[MWTT98] W. Maurel, W. Wu, N.M. Thalmann, and D. Thalmann. Biomechanical Models of Soft TissueSimulation. ESPRIT basic research series. Springer, Germany, 1998. ISBN 3-540-63742-7.

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[PPBD96] J.D. Porter, V. Poukens, R.S. Baker, and J.L. Demer. Structure-function correlations in the humanmedial rectus extraocular muscle pulleys. Investigative Ophthalmology & Visual Science, 37(2):468–472, February 1996.

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Curriculum Vitae

Michael Buchberger

Address

Bahnhofstr. 63A-4230 PregartenAustriaPhone: +43 664 1651980Email: [email protected]: www.uar-mi.at/staff/mbuchber

Personal Details

Gender: MaleDate of birth: 12th of April, 1974Place of birth: Linz, AustriaCitizenship: Austrian

Education

2000–2003 University of Linz, Austria

Doctorate studyResearch Project „SEE-KID“ (Software Engineering Environment for Knowl-edge Based, Interactive Eye Motility Diagnostics)

1994–1998 Upper Austria University of Applied Sciences, Hagenberg

Study of Software-EngineeringGraduation with honors

Diploma Thesis: Design and Implementation of a Client/Server System for aDigital Video Archive


Study of Informatics for 4 semesters

1988–1992 Bundes- Oberstufenrealgymnasium Linz/Honauerstr., Austria

High-School with special emphasis on Informatics.

Working Experience

since 01/2003 Upper Austrian Research GmbH, Hagenberg, Austria

Head of the Research Department for Medical-Informatics

2000–2003 Upper Austrian University of Applied Sciences, Hagenberg, Austria

Research Assistant at the Department for Software-Engineering for Medicine

• Project advisor research project „SEE-KID“ (www.see-kid.at), funded bythe Austrian Ministry of Science and Technology (FFF)

• Collaboration within the establishment of the research department of theUpper Austrian University of Applied Sciences (FORTE), Hagenberg

• Collaboration within the establishment of an „Information MarketplaceHagenberg“, funded by the Austrian Ministry of Science and Technology(FFF)

03/2002 Consulting for SKI-DATA AG, Salzburg, Austria

Consulting activities in the fields of project engineering and object-orientedsoftware design using modelling languages (UML).

09/2001 Consulting for Posimis Internet GmbH, Linz, Austria

Consulting in the field of computer graphics and animation for Internet appli-cations.

2000–2001 Upper Austrian University of Applied Sciences, Hagenberg, Austria

Design and Implementation of the Internet presentation of the Polytechnic Uni-versity in Hagenberg. Implementation of a dynamic scripting language for thegeneration of web pages.

1998–2000 Research Institute for Symbolic Computation (RISC), Hagenberg, Austria

• Collaboration within a project for controlling automated transport systemfor TMS Voest Alpine, Linz, Austria.

• Project manager for the re-engineering of the accounting system for the„Rinderbörse“, Linz, Austria.

1994–1998 AMS-Engineering GmbH, Hagenberg, Austria

• Project manager for different project in the field of industry automationand data analysis.

• Project manager for the project Video X-pert, a Client/Server system fordigital video archiving.

1992–1994 Landesbildstelle Oberösterreich, Linz, Austria

Programmer in the field of computer aided teaching for Geography and English.

Teaching

2002–2003 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg,Austria

Lecturer at the Department for Hardware/Software Systems Engineering, Systems-Engineering IV.

01/2002 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg,Austria

Seminar for object-oriented software design using modelling languages (UML).

2001–2002 Trainer at WIFI Linz, Austria

WIFI trainer in Linz und Steyr at the WIFI Fachakademie for Informatics.

• Courses in C/C++ programming

• WEB-Designer course for programming in Javascript


Lecturer at the Department for Software-Engineering for Medicine, Project-Engineering Systems-Engineering: Virtual Surgery.


Lecturer at the Department for Software-Engineering, Project-Engineering: Datavisualization of automation systems using XML.


Lecturer at the Department for Software-Engineering for Medicine, Project-Engineering: Simulation Expert for Eyes + diagnosis + transposition surgery(SEE++).


Lecturer at the Department for Software-Engineering. Lecture: Introductionto object-oriented design using modelling languages.

1999 Guest lecturer at the Upper Austrian University of Applied Sciences

Guest lectures in Software-Engineering and C++ programming.


Student Tutor for Algorithms and data-structures for technical Mathematics.

Language Knowledge

German nativeEnglish near nativeFrench fair

Programming Skills

Visual Studio, MFC, Borland/Inprise C++ Builder, SNIFF++, Oracle De-signer Tools, Matlab, Mathematica, Labview, GUPTA SQL, ERWin, RationalSoftware Suite, Select OMT/UML, Objectdomain, Visual J++, J-Builder, Del-phi, Borland Turbo Pascal, Borland C++.

.NET, C/C++, Pascal, Java/Java-Script, Modula-2, Oberon, Python, Perl,SQL, Basic.

Consultations

01/2004– Consultant in the development of the strategic program of the Upper Austriangovernment for research and technology transfer „Innovatives Oberösterreich2010+“.

Listed in Who’s Who in Science and Engineering, 7th Edition, 2003-2004.

Publications

Reviewed Publications

M. Buchberger, T. Kaltofen, S. Priglinger, R. Hörantner, Construction andApplication of an Object-Oriented Computer Model for Simulating Ocular Po-sitioning Defects, Journal of the Austrian Ophthalmologic Society Nr. 17/4,Springer-Verlag, pp. 151-157, Vienna, Austria, 2003.

R. Hörantner, M. Buchberger, T. Kaltofen, S. Priglinger, Differentialdiagnosevertikaler Schielformen bedingt durch schräge Augenmuskeln und Pulleys, Jour-nal of the Austrian Ophthalmologic Society Nr. 17/4, Springer-Verlag, pp.158-163, Vienna, Austria, 2003.

M. Buchberger, Ein biomechanisches Modell der Augenmotilitõt, Journal of theAustrian Ophthalmologic Society Nr. 16/4, Springer-Verlag, pp. 176-182, Vi-enna, Austria, 2002.

Conference Papers

M. Buchberger, T. Kaltofen, An Ophthalmologic Diagnostic Tool Using MR Im-ages for Biomechanically-Based Muscle Volume Deformation, Proc. of MedicalImaging 2003, Image Processing; Milan Sonka, Univ. of Iowa (USA); J. MichaelFitzpatrick, Vanderbilt Univ. (USA), San Diego, USA, 2003.

S. Priglinger, R. Hörantner, M. Buchberger, T. Kaltofen, Functional Topogra-phy in an Eye Model, Proceedings of Development and Perspectives in VisualProcessing and Eye Movements, The Heidelberg Meeting on Eye Movements,International Symposium of the German Ophthalmological Society, Heidelberg,Germany, 2002.

M. Buchberger, Ein dynamisches, volumserhaltendes 3D-Deformationsmodellvon extraokularen Muskeln, basierend auf der Analyse von MRI-Daten, Proc. 1.Jahrestagung der Österreichischen Wissenschaftlichen Gesellschaft für Telemedi-zin, pp. 37, Innsbruck, Austria, 2001.

M. Buchberger, H. Mayr, SEE-KID: Software Engineering Environment forKnowledge-based Interactive Eye Motility Diagnostics, Proc. International Sym-posium on Telemedicine, EU Human Potential Programme, pp. 67-79, Gothen-burg, Sweden, 2000.

Conference Presentations

R. Hörantner, M. Buchberger, T. Kaltofen, S. Priglinger, Myopia Alta, a SEE++case study, Consilium Strabologicum Austriacum, Innsbruck, Austria, 2003.

M. Buchberger, T. Kaltofen, Ein computerunterstütztes biomechanisches Mod-ell zur Vorbereitung, Planung und Simulation von Strabismus-Operationen, 44.Jahrestagung der österreichischen Ophthalmologischen Gesellschaft, Salzburg,Austria, 2003.

M. Buchberger, T. Klatofen Computer-based Simulation in Medical Informatics,Talk at the Brown-Bag Symposium, Smith Kettlewell Eye Research Institute,San Francisco, USA, 2003.

M. Buchberger, T.Kaltofen, S. Priglinger, R. Hörantner, SEE-KID and EOM-Modelling, The Heidelberg Meeting on Eye Movements, International Sympo-sium of the German Ophthalmological Society, Heidelberg, Germany, 2002.

M.Buchberger, J. Hildebrandt, T. Kaltofen MR-Cinematographie der Augen-motilität, Consilium Strabologicum Austriacum, St. Pölten, Austria, 2001.

References

These persons are familiar with my professional qualifications and my character:

Univ.-Prof. Dipl.-Ing. Dr. Roland WagnerThesis supervisor Phone: +43 732 2468 8791FAW-University of Linz Fax: +43 732 2468 9308Altenberger Straße 69 Email: [email protected] Linz, Austria

Univ.-Doz. Dipl.-Ing. Dr. Thomas HaslwanterThesis advisor Phone: +41 1 255 3996Institute for Theoretical Physics, Fax: +41 1 255 4507ETHZ and Dept. of NeurologyUniversity Hospital Zurich Email: [email protected]. 268091 Zurich, Switzerland

Univ.-Prof. Dipl.-Ing. Dr. Witold JacakThesis advisor Phone: +43 7236 3888 2000Upper Austrian University of Applied Sciences Fax: +43 7236 3888 99Hauptstr. 117 Email: [email protected] Hagenberg, Austria

Prim. Prof. Dr. Siegfried PriglingerThesis advisor Phone: +43 732 7897 1300Convent Hospital of the „Barmherzigen Brüder“ Fax: +43 732 7897 1099Seilerstätte 2 Email: [email protected] Linz, Austria

Prim. Univ.-Doz. DDr. Armin EttlCollaborator Phone: +43 2742 300 2869Hospital St. Pölten Fax: +43 2742 300 3285Propst Führer Str. 4 Email: [email protected] St. Pölten, Austria

Univ.-Doz. Dr. Franz FellnerCollaborator Phone: +43 732 6921 26701Radiologic Institute of the Wagner Jauregg Hospital Fax: +43 732 6921 26704Wagner-Jauregg-Weg 15 Email: [email protected] Linz, Austria

biomechanical modelling of the human eye (1)

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