new determination of muscle forces acting on the femur and stress...

Chair of Structural Analysis In Cooperation with: Int. Master’s Program on Computational Mechanics “Klinikum Großhadern”

Title:

Determination of muscle forces acting on the femur and stress analysis

Masterthesis

Author: Silke Renner, Technische Universität München Student number: 2840064 Date: November 2007

Supervisors: Univ. Prof. Dr.-Ing Kai Uwe Bletzinger

Prof. Dr. med. Bernhard Heimkes

M.Sc. Christoph Müller

Abstract

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Abstract

.A two-dimensional biomechanical model of the juvenile hip joint, called the “Stemmkörpermodel”, was developed in the one-legged stance. The one-legged stance is the relevant phase during walking considering the loading of the femur. Forces acting on the apophysis of the greater trochanter in the one-legged stance were computed based on anatomical and radiological investigations and the information was integrated into Pauwels’ biomechanical hip model. A basic assumption was the validity of the Pauwels’ bone remodeling law, which states that growth plates are oriented at a right angle to the acting resultant forces. According to the developed biomechanical “Stemmkörpermodel“, the greater trochanter apohysis has to absorb a considerable force, equivalent to almost twice the body weight, and is subject to pressure stress from a craniolateral direction. Thus, the apophysis of the greater trochanter is a “pressure apophysis”, which was still not entirely clear. Using the musculoskeletal modelling system “AnyBody”, the muscle force magnitudes and the inclination of the muscle force vectors included in the “Stemmkörpermodel” during the gait cycle were determined. Comparing these determined values with the expected values, the lightly modified “Stemmkörpermodel” showed good agreement with the expected values. Thus, the “Stemmkörpermodel” can be said to be verified by AnyBody. In addition, a stress analysis of the femur using the finite element software Ansys was performed, comparing the biomechanical Pauwels’ model and the “Stemmkörpermodel” in the one-legged stance.

Acknowledgement

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Acknowledgement

This thesis could not have been completed without the help and support of many

people to whom I am very grateful.

My gratitude is sincerely expressed to Prof. Dr. med. Bernhard Heimkes from

Klinikum Großhadern for his guidance, medical material support and expert knowledge.

Thanks to my supervisor Univ. Prof. Dr.-Ing Kai Uwe Bletzinger from the Technical

University of Munich for his encouragement and long and useful discussion about my work.

Special thanks to M.Sc. Christoph Müller from CADFEM for the assistance in the

generation of the FE model of the femur and the supply of the softwares: Simpleware,

AnyBody and Ansys.

Munich, November 2007

Silke Renner

Table of Contents

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Table of Contents

1 INTRODUCTION.............................................................................................................................................. 1

1.1 MOTIVATION FOR THE PROJECT..................................................................................................................... 1 1.2 STRUCTURE OF THE THESIS............................................................................................................................ 1

2 BACKGROUND ................................................................................................................................................ 3

2.1 TERMINOLOGY DESCRIBING THE MUSCULOSKELETAL SYSTEM...................................................................... 3 2.2 HIP ANATOMY ............................................................................................................................................... 4 2.3 THE FEMUR: A HOLLOW BONE ....................................................................................................................... 5

2.3.1 The inner architecture .......................................................................................................................... 6 2.3.2 Classification according to the CCD angle .......................................................................................... 6 2.3.2 Antetorsion and Retrotorsion ............................................................................................................... 7

2.4 MUSCLES....................................................................................................................................................... 7 2.4.1 Structure of skeletal muscle .................................................................................................................. 7 2.4.2 Muscles of the lower extremity ............................................................................................................. 8

2.5 DIFFERENT KINDS OF JOINTS IN THE HUMAN BODY ........................................................................................ 9 2.5.1 Joints of the lower extremity................................................................................................................. 9

2.6 GAIT CYCLE................................................................................................................................................. 10 2.6.1 Stance and Swing phase...................................................................................................................... 10 2.6.2 Alternative Nomenclature................................................................................................................... 11

3 THEORY .......................................................................................................................................................... 13

3.1 MATHEMATICAL MODEL ............................................................................................................................. 13 3.1.1 “Pauwels’ model” .............................................................................................................................. 13 3.1.2 Modification aspects of the “Pauwels” model ................................................................................... 15 3.1.3 Heimkes “Stemmkörpermodel” .......................................................................................................... 17

3.1.3.1 Forces acting on the juvenile hip in the one-legged stance........................................................................... 19

4 SUBJECT OF STUDY IN RESEARCH: FEMUR ....................................................................................... 20

4.1 BIOMATERIAL BONE.................................................................................................................................... 20 4.2 THE EARLY STEPS ........................................................................................................................................ 22 4.3 LOADING CONDITIONS SIMULATED IN THE LITERATURE.............................................................................. 24 4.4 CONSTRAINTS SIMULATED IN THE LITERATURE ........................................................................................... 25

5 METHODOLOGY........................................................................................................................................... 26

6 ANALYSIS USING ANYBODY..................................................................................................................... 28

6.1 BACKGROUND ANYBODY ........................................................................................................................... 28 6.1.1 Kinematical Analysis .......................................................................................................................... 28 6.2.1 Inverse Dynamics ............................................................................................................................... 29 6.1.2 Redundancy problem .......................................................................................................................... 31 6.1.4 Optimization in ANYBODY................................................................................................................. 32

6.1.4.1 Min/Max criterion......................................................................................................................................... 32 6.1.5 Programming language: AnyScript .................................................................................................... 35 6.1.6 ANYBODY Model Repository ............................................................................................................. 35

6.2 “ANYGAIT” MODEL..................................................................................................................................... 36 6.2.1 Degrees of freedom............................................................................................................................. 36 6.2.2 Marker driven model .......................................................................................................................... 37 6.2.3 Force plates: Ground reaction forces ................................................................................................ 38 6.2.4 Boundary condition ............................................................................................................................ 39

6.3 EXPECTED DISPLACEMENT OF THE BODY CENTRE OF MASS DURING GAIT.................................................... 39 6.3.1 Calculation of the body centre of mass............................................................................................... 41

6.3.1.1 Force platform method ................................................................................................................................. 41 6.3.1.2 Full body model............................................................................................................................................ 43 6.3.1.3 Visualization of COM position during gait cycle ......................................................................................... 45

6.4 MUSCULOSKELETAL MODELLING IN ANYBODY ...................................................................................... 46 6.4.1 Direction of “AnyViaPointMuscle” pull ............................................................................................ 47 6.4.2 “ViaPointMuscles” of the “Stemmkörpermodel” .............................................................................. 48

Table of Contents

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6.5 DETERMINATION OF THE HIP JOINT FORCE................................................................................................... 50 6.5.1 Comparison of the hip joint force: Heimkes/AnyBody........................................................................ 50 6.5.2 Correction of the hip joint force ......................................................................................................... 50

6.5.2.1 New nomenclature: “Stemmkörpermodel” ................................................................................................... 52 6.6 VERIFICATION OF THE “STEMMKÖRPERMODEL” ......................................................................................... 53

6.6.1 Verification of the force vectors ......................................................................................................... 54 6.6.2 Verification of the angles.................................................................................................................... 54 6.6.3 Visualization of the force parallelogram at the greater trochanter.................................................... 56

6.7 COORDINATE TRANSFORMATION ................................................................................................................ 57

7 GENERATION OF THE FE MODEL OF THE FEMUR........................................................................... 58

7.1 SIMPLEWARE SOFTWARE ............................................................................................................................. 58 7.1.1 ScanIP: image processing software.................................................................................................... 58 7.1.2 ScanFE: mesh generation / material assignment module................................................................... 61

7.1.2.1 Correction of the material properties ............................................................................................................ 62

8 STRESS ANALYSIS USING ANSYS ............................................................................................................ 66

8.1 APPLIED LOADS ........................................................................................................................................... 66 8.1.1 Hip joint force..................................................................................................................................... 66

8.1.1.1 Determination of the hip joint load application point ................................................................................... 67 8.2 CONSTRAINTS.............................................................................................................................................. 67 8.3 POSTPROCESSING IN ANSYS ........................................................................................................................ 67

8.3.1 Stress tensor........................................................................................................................................ 68 8.3.2 Stress vector........................................................................................................................................ 68 8.3.3 Calculation of normal and shear stresses of a defined cutting plane ................................................. 69

8.4 ANALYTICAL SOLUTION .............................................................................................................................. 69 8.4.1 Bending stress of a cutting section ..................................................................................................... 69 8.4.2 Shear stress of a cutting section ......................................................................................................... 72 8.4.3 Visualization of the results in GiD...................................................................................................... 73

8.4.3.1 Total normal stress........................................................................................................................................ 73 8.4.3.2 Total shear stress .......................................................................................................................................... 74

8.5 EQUILIBRIUM CHECK ................................................................................................................................... 75 8.5.1 Internal equilibrium............................................................................................................................ 75

8.6 RESULTS: COMPARISON PAUWELS MODEL / “STEMMKÖRPERMODEL” ........................................................ 78 8.6.1. Transversal cut: FEM and analytical solution .................................................................................. 78 8.6.2. Cut through the femoral head/neck ................................................................................................... 81

9 SUMMARY AND DISCUSSION ................................................................................................................... 83

10 REFERENCES............................................................................................................................................... 84

Table of Figures

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Table of Figures

FIGURE 2.1: Anatomical body planes 3 FIGURE 2.2: Hip anatomy 4 FIGURE 2.3: Femur in ventral view and in dorsal view 5 FIGURE 2.4: Trabecular and cortical bone of the femur 6 FIGURE 2.5: Coxa normala (left); coxa vara (middle); coxa valga (right) 6 FIGURE 2.6: Antetorsion, normal torsion and retrotorsion 7 FIGURE 2.7: Structure of an arbitrary skeletal muscle 8 FIGURE 2.8: Different kind of joints in the human body 9 FIGURE 2.9: One complete gait cycle 10 FIGURE 2.10: All events of the stance / swing phase 11 FIGURE 2.11: Alternative nomenclature for the gait cycle 11 FIGURE 3.1: 16. Phase of walking after Fischer and Braune 13 FIGURE 3.2: Pauwels mathematical model 14 FIGURE 3.3: Femur end from the lateral view 15 FIGURE 3.4: Increase of the angle EY during growth (left); Angle AY remains nearly constant during growth (right) 16 FIGURE 3.5: Scheme of the two force parallelograms (Pauwels, Heimkes) 18 FIGURE 3.6: Computed force vectors dependent on the body weight 19 FIGURE 4.1: Force-deformation curve of bones with an elastic and plastic deformation region 20 FIGURE 4.2: Ultimate strength of the cortical and compact bone 21 FIGURE 4.3: Direction/magnitude of the hip joint force in dependency of the CCD-angle 21 FIGURE 4.4: Stress trajectories in a crane design compared with trabeculae in a femur 22 FIGURE 4.5: Lines of stress in the upper femur (Koch) 23 FIGURE 4.6: Percent of the literature according to the applied load 24 FIGURE 5.1: Flow chart of the followed steps within the project 26

Table of Figures

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FIGURE 6.1: Mechanical model representing the musculosketal system 29 FIGURE 6.2: Movement chain of gait analysis 30 FIGURE 6.3: Simple arm model with three muscles 31 FIGURE 6.4: Simple arm model 33 FIGURE 6.5: Table showing the relative muscles activities 34 FIGURE 6.6: model with applied forces (F1=2F2) (left); relative activities computed by the iterative method (right) 34 FIGURE 6.7: Two parts of the Repository 35 FIGURE 6.8: DOF’s per bone and constraints per joint 36 FIGURE 6.9: Infrared reflecting markers on a human body 37 FIGURE 6.10: Visualized markers 37 FIGURE 6.11: Typical force plate 38 FIGURE 6.12: Scheme of vertical / lateral displacement during one gait cycle 40 FIGURE 6.13: Displacements of COM: saggital and horizontal view 40 FIGURE 6.14: Vertical ground reaction forces given by textfiles 42 FIGURE 6.15: Start and end position of the gait cycle in AnyBody 42 FIGURE 6.16: Definition of one step interval of the force plate method 43 FIGURE 6.17: Full body model (no muscles) 43 FIGURE 6.18: Vertical/Lateral displacement of the COM (AnyBody) 44 FIGURE 6.19: The generic three component Hill-type muscle-tendon model described by Zajac 47 FIGURE 6.20: Muscle from origin to insertion 47 FIGURE 6.21: Via-point muscles 48 FIGURE 6.22: gluteus minimus (green) and gluteus medius (blue) 48 FIGURE 6.23: gluteus maximus (second part): insertion / origin / three via-points 49 FIGURE 6.24: Comparison of the hip joint force Heimkes/AnyBody 50

Table of Figures

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FIGURE 6.25: Activity patter of ilipsoas and rectus femoris 51 FIGURE 6.26: Comparison of the corrected hip joint force Heimkes/AnyBody 51 FIGURE 6.27: New nomenclature “Stemmkörpermodel” 52 FIGURE 6.28: Computed force vectors dependent on the body weight 53 FIGURE 6.29: Expected force values according to the body weight of 64.9 kg 53 FIGURE 6.30: Loading response: Computed muscle force vector values in kg 54 FIGURE 6.31: Global coordinate system of AnyBody and resultant force vector Rh 54 FIGURE 6.32: Loading response: Angles to the perpendicular of the computed force vectors 55 FIGURE 6.33: Loading response: planar angle to the perpendicular in the frontal plane 56 FIGURE 6.34: Trochanter resultant in the frontal and lateral view at 9% 56 FIGURE 6.35: Global and local coordinate system of the femur 57 FIGURE 7.1: Simpleware software products 58 FIGURE 7.2: Slice of the femur in the x-z plane 58 FIGURE 7.3: cavities in mask1 59 FIGURE 7.4: The marrow of the femur of one slice 59 FIGURE 7.5: Three defined masks of the femur 60 FIGURE 7.6: Comparison of the femur before and after applying the Recursive Gaussian Filter 60 FIGURE 7.7: Voxel model of the proximal femur 61 FIGURE 7.8: Displacement of the femur at 9% of the gait cycle (loading according Heimkes) 62 FIGURE 7.9: assigned material properties from “ScanFE”; E-modulus versus density 62 FIGURE 7.10: Comparison of the conversion formula 63 FIGURE 7.11: E-modulus versus density (VAKHUM: femur) 64 FIGURE 7.12: blue elements (cortical bone); red elements (spongiosa bone) 65 FIGURE 7.13: Displacement of the femur at 9% of the gait cycle (loading according Heimkes) 65

Table of Figures

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FIGURE 8.1: Hip joint force in the frontal plane 66 FIGURE 8.2: Link elements for determination of load application point 67 FIGURE 8.3: illustration of the stress vector defined on a cutting plane 68 FIGURE 8.4: shear/normal part of the stress vector 69 FIGURE 8.5: scheme of delaunay triangulation in a plane 70 FIGURE 8.6: point cloud (left); triangulated point cloud (middle); total area (right) 71 FIGURE 8.7: illustration of the procedure of the calculation of the bending moment 71 FIGURE 8.8: approximated inner and outer diameter of the cross section 72 FIGURE 8.9: total normal stress [Pa] in the hollow circle cross section 73 FIGURE 8.10: total normal stress in the real complex cross section 73 FIGURE 8.11: comparison compression/tension values over the cross-section 74 FIGURE 8.12: Heimkes: total shear stress (left); tau_yz (middle); tau_xz (right); unit [Pa] 74 FIGURE 8.13: Pauwels: total shear stress (left); tau_yz (middle); tau_xz (right); unit [Pa] 75 FIGURE 8.14: illustration of the interpolation normal stress values 76 FIGURE 8.15: internal equilibrium 77 FIGURE 8.16: stress in z-direction [MPa] of the Pauwels model 78 FIGURE 8.17: stress in z-direction [MPa] of the “Stemmkörpermodel” 78 FIGURE 8.18: Pauwels model: view from above 79 FIGURE 8.19: “Stemmkörpermodel”: view from above 79 FIGURE 8.20: comparison pure moment result 80 FIGURE 8.21: Ansys total shear force 80 FIGURE 8.22: defined cutting planes of the proximal femur 81 FIGURE 8.23: Heimkes: normal and shear stresses of the femoral head 81 FIGURE 8.24: Normal stresses [Pa] of the cut through the femoral neck 82

1 Introduction

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1 Introduction

1.1 Motivation for the project

Physiological loading of the bone or implant is of great importance for investigations involving micro-motion, fracture fixation/healing and for implant design and its primary stability. Considering hip joint replacement operations, the question may arise to what degree a certain surgical approach modifies the load distribution within the femur and the maybe resulting consequences of this decision. Especially in biological processes such as fracture healing the fact that muscles are major contributors to femoral loading plays an important role. Nevertheless, the knowledge of musculoskeletal forces is still limited. The existing possibilities to measure muscle forces in humans by invasive methods are excluded due to ethical considerations. However, the non-invasive methods such as electromyography are used to get an insight of the activation patter of muscles, but for the prediction of the magnitudes of muscle forces they cannot be utilized. Therefore, the only opportunity to estimate the complex distribution of muscle forces is offered by computer analysis on the basis of optimization algorithms. Pauwels was one of the first researchers, who included the effect of muscles on femoral loading in his analytical analysis for the one-legged stance. His determined mathematical model of the hip joint force can be found in nearly all biomechanical books and is accepted and valid until today. Prof. Dr. med. Heimkes from Klinikum Großhadern in Munich postulates in his developed “Stemmkörpermodel” based on anatomical, radiological and computational results, that the Pauwels model has to be modified. If muscle activity is considered, there is a general consensus that the muscle forces tend to reduce the load acting within the bone. Thus, the bending of the femur compared with the Pauwels model is expected to be reduced. In this project, the musculoskeletal modelling system ANYBODY will be used to determine the muscle forces of the so-called “Stemmkörpermodel” of Prof. Dr. med. Heimkes. The aim is to verify the found magnitudes of muscle forces and their inclination angle to the vertical of the “Stemmkörpermodel”. Furthermore, a Finite Element Analysis of the loaded femur will be performed, comparing the biomechanical model of Pauwels with the “Stemmkörpermodel”of Prof. Dr. med. Heimkes.

1.2 Structure of the thesis

In this thesis the main point of investigation is the femoral loading in the one-legged during walking, but also the complete gait cycle will be investigated. Therefore, in chapter 2, some general background information such as the terminology describing the musculoskeletal system, the architecture of the femur, the definition of the gait cycle etc. will be described. In chapter 3 the underlying theory of the mathematical Pauwels model, which determines the loading of the proximal femur will be presented. Based on the Pauwels model the “Stemmkörpermodel” from Prof. Dr. Heimkes will also be presented and the important facts which lead to its development will be explained.

1 Introduction

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Due to the fact that the femur is a subject of research since years, chapter 4 deals with the bone as biomaterial and the early steps of investigations done on the femur starting by Galileo. Additional, an overview of different loading conditions and applied constraints of performed femur analysis will be summarized. Chapter 5 illustrates in a flow chart the methodology of this project and the software’s which will be used to reach the aim of the project. The proceeding steps for the determination of the muscle forces (magnitudes and their inclination to the perpendicular) involved in the “Stemmkörpermodel” and the verification according to Prof. Dr. med. Heimkes computed/expected muscle forces will be done in chapter 6 by using the software ANYBODY. In addition, some background information to the software and the way of functioning of the gait model will be explained. After having determined the load case of the femur a finite element model of the femur considering inhomogeneous material distribution has to be built up from CT images. This procedure is described in chapter 7. In chapter 8 a stress analysis of the femur comparing the Pauwels’ model with the “Stemmkörpermodel” in the one-legged stance will be performed. In the last chapter 9 the found results will be discussed and also the realism of them will be considered. Furthermore, recommendation for future work will be done.

2 Background

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2 Background

2.1 Terminology describing the musculoskeletal system

For the description of the human body and its movement anatomical terminology is used. A three dimensional coordinate system consisting of three anatomical planes is defined as follows:

The three anatomical body planes are used to determine an anatomical position and the axes of motion. To describe the positions of structures relative to other structures or locations in the body directional and spatial terminology is necessary:

• superior or cranial: toward the head end of the body; upper • inferior or caudal: away from the head; lower • anterior or ventral: toward the front • posterior or dorsal: toward the back • medial: toward the midline of the body • lateral: away from the midline of the body • proximal: refers to structures nearer to the trunk

(e.g. the knee is proximal to the foot) • distal: refers to structures further from the trunk

(e.g. the foot is distal to the knee)

Coronal Plane (Frontal Plane):

• vertical plane running from side to side • divides the body in anterior and

posterior portions Sagittal Plane (Lateral Plane):

• vertical plane running from front to back

• divides the body into right and left side Axial Plane (Transversal Plane):

• horizontal plane • divides the body into upper and lower

parts Median Plane:

• sagittal plane through the midline of the body

• divides the body into right and left halves

Figure 2.1: Anatomical body planes [1]

2 Background

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In addition some important terms of movement will be explained next:

• Flexion / extension: increasing angle with frontal plane / decreasing angle with frontal plane

• Abduction / adduction: moving away from or toward the sagittal plane • Protraction / retraction: moving forward or backward along a surface • Elevation / depression: raising or lowering a structure • Medial rotation / lateral rotation: movement around an axis of a bone • Pronation / supination: placing palm backward or forward (in anatomical position) • Circumduction: combined movements of flexion, extension, abduction, adduction

medial and lateral rotation circumscribe a cone • Opposition: bringing tips of fingers and thumb together as in picking something up

2.2 Hip anatomy

In anatomy, the hip is the bony projection of the femur which is known as the greater trochanter, and the overlying muscle and fat. The following figure 2.2 is illustrating the hip anatomy:

Figure 2.2: hip anatomy [2]

The hip comprises the joint between the upper end of the femur and the pelvis or acetabulum. This joint is a so-called ball-and-socket joint, where the femoral head forms the ball portion and the round area of the lower pelvis known as acetabulum forms the socket portion. Thus, the femoral head fits inside the acetabulum and both bones are covered by a protective layer, which acts as a cushion, called cartilage. A diseased hip shows degeneration of the cartilage, the consequences are swelling, inflammation and pain. Sometimes even bone will begin to contact with bone. In turn, cartilage is covered by a synovial fluid, which has the function of a lubricant to reduce wear during joint motion. The primary function of the hip joint is to support the body weight in the static (e.g. standing) and dynamic (e.g. walking) posture.

2 Background

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2.3 The femur: a hollow bone

The femur or the so-called thigh bone is the longest, most voluminous and strongest bone in the human skeleton. The following figure 2.3 illustrates the ventral and dorsal view of the femur and presents the most important terms and definitions:

Figure 2.3: Femur in ventral view (left) and in dorsal view (right) [3]

The femur can be divided into three main sections:

1. the upper extremity (proximal extremity)consists of:

• a rounded head which articulates with the acetabulum of the hip bone to form the hip joint

• a relatively narrow neck • two protuberances for muscle attachment, the greater trochanter and

the lesser trochanter

2. the body or shaft (corpus femoris):

• it is almost cylindrical in form • it is a little bit broader above than in the centre • it is slightly arched: convex in front and concave behind

3. the lower extremity (distal extremity):

• it is larger than the upper extremity and consists of two oblong

eminences known as the condyles

head

neck

greater trochanter greater trochanter

lesser trochanter

corpus femoris

condyles

2 Background

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2.3.1 The inner architecture

The next figure 2.4 shows the two major kinds of bone structure of the femur, the trabecular (spongiosa/spongy) bone and cortical (compact) bone:

Figure 2.4: Trabecular and cortical bone of the femur [4]

The outside of the shaft of the femur consists of cortical (solid) bone and this type of bone forms the outer shell of all bones of the human body. Trabecular bone gives supporting strength to the ends of the weight-bearing bone and is found at the expanded heads of long bones. Considering the area near the hip joint of the femur, it is filled with a micro-framework of very fine small struts of bone (spongiosa). This fine framework near the joint has the function to distribute the load from the hard bone wall over a larger area and to act as a shock absorber or dash pot. During standing on one leg, the neck of the femur has to transmit about 2.5 to 6 times the body weight BW as axial loading due to lever relationships. To cope with this high load a well adapted design has to exist. The cortical bone has a higher density and stiffness than the spongiosa bone and is therefore better adapted to higher local stresses.

2.3.2 Classification according to the CCD angle

The angle formed between the neck and shaft of the femur is described as the CCD (Caput-Collum-Diaphysis) angle γ. According to this CCD angle a division between three cases can be made as shown [5]:

Figure 2.5: Coxa normala (left); coxa vara (middle); coxa valga (right)

Trabecular bone

Cortical bone

2 Background

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Coxa normale is defined at a CCD angle between 120° and 135°. At birth the CCD angle is normally between 135° and 140° and should gradually reduce with development to the adult norm of 125°. If the CCD angle reduces too much coax vara (CCD angle < 120°) is resulting and if the reduction is inadequate coxa valga (CCD angle > 135°) is resulting. Coxa vara and coxa valga are deformities of the hip which are of clinical importance, but out of the scope of this thesis.

2.3.2 Antetorsion and Retrotorsion

Looking in the direction of the mechanical axis of the femur (shaft axis) from proximal to distal, the projection of the neck of the femur is not covered by the axis of the condyles. Antetorsion is an increase in the angle of the head and neck of the femur relative to the femoral condyles and in healthy people the antetorsion is around 12° also called normal torsion. In turn, retrotorsion is a decrease in the angle of the head and neck of the femur relative to the femoral condyles. For purposes of clarity, the following figure 2.6 illustrates the superimposition of normal torsion, retrotorsion, and antetorsion [6]:

Figure 2.6: Antetorsion, normal torsion and retrotorsion

The resulting clinical consequences are out of the scope of the thesis and will not be discussed here.

2.4 Muscles

Muscles are the actuators of the human body. Muscles can be separated into three different types:

• Skeletal muscles • Cardiac muscles • Smooth muscles

For the movement of the body segments the skeletal muscles are responsible and their structure will be shortly explained in the next section.

2.4.1 Structure of skeletal muscle

Each person has around 600 skeletal muscles which have a wide range in size. A whole skeletal muscle is considered as an organ of the muscular system. It consists of skeletal muscle tissue, connective tissue, nerve tissue, and blood or vascular tissue.

normal torsion 12°

antetorion 30°

retrotorsion 8°

2 Background

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The following figure 2.7 shall just give an impression how the structure of an arbitrary skeletal muscle looks like:

Figure 2.7: Structure of an arbitrary skeletal muscle [7]

In some muscles the fibres are parallel to the long axis of the muscle, in some they converge to a narrow attachment, and in some they are oblique. Each muscle fibre is a single cylindrical cell and one individual muscle can contain hundreds, or even thousands of these fibres bundled together and wrapped in a connective tissue (Epimysium) covering [7]. Typically a muscle spans a joint and is attached to bones by tendons at both ends. One of the bones remains nearly fixed or stable while the other end moves as a result of muscle contraction. Muscle fibres are oriented either in the direction of the tendon or at an acute angle ˺ (pennation angle) to the tendon. The primary function of skeletal muscle is contraction. Before a skeletal muscle fibre can contract, it has to receive an impulse from a nerve cell. Therefore, skeletal muscles have a supply of blood vessels and nerves. Generally, an artery and at least one vein accompany each nerve that penetrates the epimysium of a skeletal muscle. Just this small insight in the structure of skeletal muscles reveals the complexity of this topic and will not be explained more in detail in this project.

2.4.2 Muscles of the lower extremity

This project deals just with the muscles of the lower extremity, especially the muscles acting on the femur. Therefore, a brief overview of the function of the muscles that move the thigh and the leg will be given: Thigh flexors: Iliacus and Psoas major Thigh adductors: Adductor magnus, Adductor longus and Gracilis Extensor of the thigh: Gluteus maximus and Tensor fascia latae Thigh abductors: Gluteus medius and Gluteus minimus Thigh extenders: Biceps femoris, Ssemimembranosus and Semitendinosus Knee extenders “Quadriceps femoris”: Rectus fermoris, Vastus lateralis, Vastus medialis and Vastus intermedius Further details about the anatomical location, origin and insertion points will be explained in the muscle modelling section in chapter 6.

2 Background

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a.) plane joint b.) hinge-joint c.) Radgelenk d.) condylar joint e.) elipsoid joint f.) saddle joint g.) ball and socket

joint

2.5 Different kinds of joints in the human body

The following figure 2.8 [5] gives an overview of the existing kinds of joints in the human body with their according degrees of freedom (DOF):

Figure 2.8: Different kind of joints in the human body

2.5.1 Joints of the lower extremity

In this project the lower extremity is of special interest. Therefore, just the according joints are considered more in detail:

1. hip joint: ball and socket joint (3 DOF)

The hip joint is the joint between the femur and the acetabulum of the pelvis and its primary function is to support the weight of the body in both static (e.g. standing) and dynamic (e.g. walking or running) postures. Seven different kinds of movement are possible in the hip joint:

• Abduction and adduction of the femur • Internal (medial) and external (lateral) rotation of the pelvis, thigh or

spine • Circumduction of the femur or pelvis • Flexion and extension on or from the spine (Wirbelsäule) or on or from

the thigh (Oberschenkel)

2. knee joint: condylar joint (2 DOF)

The condylar joint is a joint allowing primary movement in one plane flexion, extension) with small amounts of movement in another plane (rotation).

3. ankle joint: hinge joint (1DOF)

The hinge (ginglymus) joint allows movement in one plane (flexion, extension) and is termed uniaxial.

a) b) c) d)

e) f) g)

2 Background

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2.6 Gait cycle

The gait cycle is sometimes also called the walking cycle. The term “gait cycle” describes the whole activity, from the heel which is first put on the ground (heel strike HS) and the following heel contact with the same feet (HS) [8]. The next figure 2.9 represents one complete gait cycle:

Figure 2.9: One complete gait cycle

From the above presented figure 2.9 it is visible that the complete gait cycle can be divided into two main phases:

• stance phase • swing phase

The stance phase (ca.60%) takes longer than the swing phase (ca.40%). During the stance phase of one leg, the other leg is in the swing phase, which is the shorter phase. This results in the double limb support phase where both legs are on the ground. Double limb support occurs for two periods of 12% of the gait cycle in a normal walk. Double limb support occurs between heel strike of the limb and toe off of the contralateral limb. Single limb support where just one foot is on the ground occurs for two periods of 38% of the gait cycle in a normal walk In general, as walking speed increases double support time is reduced until it is eliminated and the gait changes to running.

2.6.1 Stance and Swing phase

The stance phase consists of five events based on the movement of the foot. The first event of the stance phase is the contact of the heel on the ground, heel strike (HS), and ends with the event where the toes of the same leg quitting the ground, toe-off (TO). In the heel strike the centre of mass is at lowest position. The three events between the heel strike and the toe-off are in order:

• Foot-flat (FF): plantar surface of the foot is on the ground • Midstance (MS): the swinging foot passes the stance foot, body centre of

gravity is at highest position • Heel-off (HO): the heel loses the contact with the ground

2 Background

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The swing phase starts in the moment when the stance phase is ending. It takes from the toes lift-off to the heel contact of the same leg and is described by three events:

• Acceleration: begins as soon as the foot leaves the ground • Midswing: same event as the event midstance of the stance phase, just

the feet are interchanged • Deceleration: action of the muscles slowing down the leg and stabilize

the foot for the next heel strike The next figure 2.10 summarizes all the events of the stance phase and the swing phase [9]:

Figure 2.10: All events of the stance / swing phase

2.6.2 Alternative Nomenclature

An alternative nomenclature describing the gait cycle is developed by the famous gait analyst Perry and her associates at Rancho Los Amigos Hospital in California (Cochran, 1982) shown in the next figure 2.11:

Figure 2.11: Alternative nomenclature for the gait cycle

In this project this alternative nomenclature [10] is more reasonable and will be used in the proceeding sections.

2 Background

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The different phases are given in percentage of one gait cycle:

1. Initial contact (0%) 2. Loading response (0 - 12%) 3. Midstance (13 - 31%) 4. Terminalstance (32 - 50%) 5. Pre-swing (51 - 62%) 6. Initial swing (63 - 75%) 7. Midswing (76 – 86%) 8. Terminal swing (87 – 100%)

3 Theory

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3 Theory

3.1 Mathematical model

In this section the biomechanical Pauwels model and the developed “Stemmkörpermodel” by Prof. Dr. med. Heimkes will be explained. These two models are the base of investigation in this project. Both models are two dimensional and illustrated in the frontal plane in the one-legged stance phase of the gait cycle (loading response).

3.1.1 “Pauwels’ model”

Braune and Fischer, two famous gait analysts, divided in their experiments one walking period into 31 phases. The next figure 3.1 shows the 16. Phase of one walking period, the so called “one legged stance”:

Figure 3.1: 16. Phase of walking after Fischer and Braune

The red point in the above figure represents the partial body weight, consisting of the trunk, the head, both arms and the swinging leg, which has to be balanced by the hip joint of the stance leg. According to Pauwels the hip joint force during standing or slow walking mainly depends on the torque around the hip joint centre H, caused by the partial body weight G5 (body weight minus the weight of the stance leg). Pauwels used gait data from Fischer to determine the center of gravity (d5) of the weight G5 during standing. He also located the point T where the abductor muscles M have their insertion point at the greater trochanter. The origin point A at the pelvic bone of these muscles was found in anatomical studies.

3 Theory

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Using all the available data made it possible to calculate the resultant hip joint force R from the equilibrium of forces and moments acting around the hip joint centre H [11]:

∑ Fx = 0 : Mx + Rx = 0 (3.1) ∑ Fz = 0: Mz + Rz + G5 = 0 (3.2) ∑MH = 0: M * m = G5 * d5 (3.3)

The following figure 3.2 represents Pauwels simplified static “one legged stance” model in the frontal plane (corresponding to the 16.phase from Braune and Fischer) [12]:

Figure 3.2: Pauwels mathematical model

All other muscle forces which may be active during the standing phase to stabilize the body concerning flexion and stretching of the hip joint are negligible. Pauwels proved that this model can be applied for the one legged stance and the period when the swinging leg passes the frontal plane during slow walking. Dynamic forces are not considered in this model. Pauwels also recognizes the problem of bending in the leg. He searched for a muscle which decreases the bending and found the muscle tensor fascia latae and gluteus maximus (tractus).

3 Theory

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3.1.2 Modification aspects of the “Pauwels” model

In the previous described biomechanical “Pauwels” model theoretical aspects of the growth-inducing forces acting on the capital epiphysis were studied, but no computations were done of the forces acting on the apophysis of the greater trochanter. Actually, until today it is still not completely clear whether the apophysis of the greater trochanter is a “traction apophysis” or a “pressure apophysis”. The aim of the investigations was to analyze the forces acting on the trochanter apophysis in the one-legged stance and integrate them into the biomechanical hip model of Pauwels. The development of a two-dimensional vectorial model of the load on the juvenile hip in the one-legged stance was based on two kinds of data:

1. Anatomical data: The anatomical data were received from 16 anatomic specimens from newborns to children aged 14 years. Additional 6 specimens from adults were taken to determine the normal cross-sectional areas of the muscles inserting on the greater trochanter.

2. Radiological data: The radiological data were received from 1350 hip joints of healthy children in a cross-sectional radiological study. In this study 11 biomechanical relevant angles and length were measured.

In the following points the anatomic and radiological results are listed:

1. Anatomic results:

• Insertion areas of the greater trochanter apophysis o lateral surface is trapezoidal o ventral surface is triangular

• Muscles insertion of the greater trochanter apophysis: o gluteus medius muscle has a ribbon-like insertion on the lateral surface o gluteus minimus muscle has the insertion on the triangular ventral surface o vastus lateralis muscle originates both from the lateral trapezoid and from the

ventral triangle The following figure 3.3 shows the femur end from the lateral view with the insertion areas of the greater trochanter apophysis [13]:

Figure 3.3: Femur end from the lateral view

ventral dorsal

Periost

Origin area of the muscle Vastus lateralis

Insertion area of the muscle Gluteus medius

Insertion area of the muscle Gluteus minimus

3 Theory

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• Normal cross-sections of muscles inserting on the apophysis of the greater trochanter (mean physiologic muscle cross-section of six anatomic specimens):

o gluteus medius muscle : 27.78 cm2 o gluteus minimus muscle : 10.05 cm2 o vastus lateralis muscle : 25.45 cm2 o vastus intermedius muscle : 10.69 cm2

• Inclination of the greater trochanter growth plate (the proximal end of the femur was

cut into 3 to 6 mm thick slices in the coronal plane): o the larger part of the apophyseal growth plate (90.4% of the total area on the

average) is oriented parallel to the femoral neck o the smaller, dorsal part of the apophyseal growth plate (9.6% of the total area

on the average) is almost oriented perpendicular to the shaft

2. Radiological results:

• most important results: o the epiphyseal angle EY increases steadily during growth → inclination of the

capital growth plate in the coronal plane o the apophyseal angle AY remains nearly constant during growth → inclination

of the greater trochanter growth plate in the coronal plane

The next figure 3.4 illustrates the position of the angles EY and AY in the femur and their change in angle during growth [14]:

Figure 3.4: Increase of the angle EY during growth (left); angle AY remains nearly constant

during growth (right)

years years

AY EY

3 Theory

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From the anatomic and radiological results the following conclusions can be drawn, on which subsequent computations are based:

1. Forces acting on the greater trochanter apophysis can be calculated from the muscle forces exerted by the gluteus maximus and the tensor fascia latae in tensing the iliotibial tract, the gluteus medius and minimus muscles, and parts of the knee extensors that are connected to the greater trochanter apophysis by the vastus lateralis muscle.

2. The results of the normal cross-sections of the muscles indicate that the muscles gluteus medius and gluteus minimus are connected to the counteracting vastus lateralis muscle by a common tendinous junction at the apophysis of the greater trochanter (so-called vastogluteal muscular sling).

3. Approximately 90% of the area of the greater trochanter growth plate is a plane oriented nearly parallel to the femoral neck → a force RT is acting on the greater trochanter apophysis (resultant of all muscular forces acting on the greater trochanter apophysis) which can be postulated to act perpendicular to the greater trochanter growth plate.

Considering all the facts of the anatomic and radiological investigations it can be stated that the greater trochanter apophysis is a “pressure” apophysis. After having computed the forces acting on the greater trochanter apophysis in the one-legged stance these forces can be integrated into the biomechanical hip model of Pauwels. The result is the “Stemmkörpermodel”, which will be presented in the next section.

3.1.3 Heimkes “Stemmkörpermodel”

The “Stemmkörpermodel” consists of two vectorial force parallelograms, which are reflecting the loading of the femur in the one-legged stance:

1. The Pauwels hip parallelogram G5/M/R with the hip joint resultant R described in section 3.1.1.

2. The trochanter parallelogram M/Mfsc/RT with the trochanter resultant RT, which reflects the loading of the greater trochanter.

Therefore, the femur is stressed by two resultant forces R and RT. The direction and magnitude of the partial body weight G5 is known and the direction of the hip resultant force R can be measured by the epiphyseal angle EY. The magnitude of R and the direction of M can be calculated with the magnitude of M received from measurements and computation of the ratio of the lever arms. The computation of the trochanter parallelogram forces M/Mfsc/RT acting on the trochanter apophysis included the following steps:

• All muscles pulling the greater trochanter apophysis in a cranial direction are summarized in the resultant force M. The muscles included in M are gluteus maximus, tensor fascia latae, gluteus medius and gluteus minimus. Due to the fact that the resultant force M is nearly equivalent to the muscle resultant force M of Pauwel’s parallelogram. Thus, M is known.

• The direction of the trochanter resultant RT is also known because it is perpendicular to the apophyseal angle AY.

3 Theory

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• The direction of all distal muscle forces acting on the greater trochanter apophysis, Mfsc, can be assumed to act parallel to the femoral axis. The resultant force Mfsc contains the muscles gluteus maximus, tensor fascia latae, vastus lateralis, vastus medialis and vastus intermedius.

Thus, if the angle and absolute value of M is known and also the angle of RT and Mfsc, then the absolute values of RT and Mfsc can be calculated by using the following formulas [14]:

)sin(

)sin(

fscT

fsc

TMR

MMMR

∠−∠

∠+∠= (3.4)

[ ]

)sin(

)(180sin

fscT

T

fscMR

MRMM

∠−∠

∠+∠−°= (3.5)

The next figure shows the scheme of the “Stemmkörpermodel” with the two vectorial force parallelograms:

gluteus medius gluteus minimus

gluteus maximus tensor fascia latae

Figure 3.5: Scheme of the two force parallelograms (Pauwels, Heimkes)

Msc T

1.

2.

3.

4.

M

Mpt

Msc

gluteus maximus tensor fascia latae vastus lateralis vastus intermedius

1.

2.

3.

4.

Mfsc

Msc’

Mfc

M

R

G5

Mpt M

Msc’

RT

Mfsc

Mfc

3 Theory

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3.1.3.1 Forces acting on the juvenile hip in the one-legged stance

Based on all previous described investigations and computations a table reflecting all the resultant forces of the “Stemmkörpermodel” depending on the body weight at the end of the growing period near the adult age in the one-legged stance can be presented [14]:

Figure 3.6: Computed force vectors dependent on the body weight

The magnitude of the forces and the inclination to the perpendicular of the force vectors of the above figure 3.6 should be verified in this project.

4 Subject of study in research: femur

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4.1 Biomaterial Bone

Bone tissue appears in a lot of microstructure forms, which have different material densities. Bone tissue with a density above 1.5 g/cm3 is referred as cortical (compact) bone and below a density of 1 g/cm3 as spongiosa bone [15]. The behaviour of bone tissue under compressive loading concerning the mechanical properties and the failure limit shows a significant dependency on the density of the material. Bone tissue is mechanically a two-phase material. First it behaves elastic and after a certain point plastic as the next figure 4.1 illustrates:

Figure 4.1: Force-deformation curve of bones with an elastic and plastic deformation region

The failure criterions can be determined by two points (see figure 4.1):

• transition between the elastic (reversible) and the inelastic behaviour (begin of the permanent damage): region between the points B and D → yield point

• stress at the ultimate failure point C → tensile strength

The mechanical material properties of bones are depending on the direction of loading. Thus, bones have an anisotropic structure. During body activities forces and moments are transmitted in different directions to the bony structure and produce tensile stress and strain, compressive stress and strain as well as shearing stress and strain. These forms of loading can appear in all combinations. The largest strength of the bone is presented by loading in the longitudinal direction (compression). A load case transversal to the central axis of the bone (shear stress) shows the lowest strength. Thus, the stiffness of the bone is largest in the longitudinal direction and lowest perpendicular to the central axis of the bone.


- 21 -

Spongiosa bone based on the elastic region has an E-modulus of approximately 1Gpa and the much stiffer cortical bone 20Gpa. For comparison, steel has an E-modulus value of 100Gpa. The failure limit of the bone material is different for tension and compression. Considering the cortical bone the failure limit for compression is circa 30-50% higher than for tension. As an evidence of the limit loading average values for the cortical and spongiosa bone were found in the literature [15] as the next figure 4.2 shows:

Cortical bone Spongiosa bone Compressive stress 200MPa 20MPa Tensile stress 135MPa 14MPa

Figure 4.2: Ultimate strength of the cortical and compact bone

Muscle forces can significantly influence the stresses and stress distributions in bones. Especially, shearing stresses can be reduced or completely eliminated. The consequence of additional muscle forces is an increase in the compressive stresses. Therefore, the tolerance limit of bones for compressive stresses is much larger than for shearing stresses. The direction of the force transmission to the bone is of great importance regarding the limit load exceeding. If the load is not transmitted in the direction of the longitudinal axis of the bone, the complete mechanical load is increasing on the bone surface. Joint forces, which are not acting along the longitudinal axis of the bones, are in general compensated by muscle forces. Thus, also the CCD angle of the femur plays an important role considering the hip joint force as illustrated [11]:

Figure 4.3: Direction/magnitude of the hip joint force in dependency of the CCD-angle

In this project, inhomogeneous material properties of the femur will be considered, which obey Hook’s law and the according procedure is described in chapter 7. For simplification, isotropic material behaviour will be assumed.


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4.2 The early steps

The load situation within the femur has been a subject of study in research for many years. In 1638, Galileo studied the mechanics of long bones and analyzed the cross anatomical structure of the femur [17]. His work based on the assumption that there exists a relationship between mechanical principles and the anatomical shape of the bones. Culmann, a German civil engineer, who has developed a technique called graphical statics for determining the direction of internal structures of complex systems, studied the drawing of the Swiss anatomist von Meyer, who set the beginning of serious research directed in uncovering the influence of the mechanical environment on trabecular structures. The drawings of von Meyer were based on the observations of cadaver specimens. The analyzed stress pattern of Culmanns curved column structure showed remarkable similarity to the trabeculae pattern of that von Meyers of the femur. Therefore, Culmann and von Meyer postulated that trabeculae are oriented along principle stresses. Around 1892, Wolff quoted the work of Culmann in which the advantages offered by the orientation of the trabecular structure could be interpreted by mechanical means. With his ideas about the relation of stimulus and anatomy Wolff brought outspread support to the idea of von Meyer. Wolff denoted his work as “Trajectorial Theory” of trabecular alignment and adapted von Meyers drawings according to the established fact in mechanics that internal stress directions must always intersect perpendicular for any load case. The next figure 4.4 compares the drawing of the internal structure of the proximal femur (trabeculae) adapted by Wolff with the stress trajectories in a crane design according to Culmann:

Figure 4.4: Stress trajectories in a crane design (left) compared

with trabeculae in a femur (right)

Although Wolff described the adaption of bone to mechanical stimulus (“Wolff’s law” or “Law of bone remodelling”), it was the American anatomist Koch (1917), who first performed an analysis of the femur by calculating the cross-sectional area and moment of inertia of seventy-five locations of the femur to quantify the stresses and strains.


- 23 -

A maximum shear force was present at the femoral head and decreased towards the lesser trochanter; the maximal bending moments occurred at the level of the lesser trochanter. Koch compared shear stress with density and trabeculae directions with principle stress directions. It can be seen from Koch’s lines of stress result that the upper femur is composed of two distinct systems of trabeculae arranged in curved paths as follows:

Figure 4.5: Lines of stress in the upper femur (Koch) [18]

• One with its origin in the medial (inner) side of the shaft and curving upward in a fan-like radiation to the opposite side of the bone

• The second, having origin in the lateral (outer) portion of the shaft and arching upward and medially to end in the upper surface of the greater trochanter, neck and head.

These two systems intersect each other at right angles. With this analysis Koch confirmed Wolff’s statements and he also recognized that bone density is highest in areas of highest shear stress. However, Koch’s analysis was only capable of representing femoral curvatures in the frontal plane and he neglected to include muscle activities. The exclusion of muscle activity led to an underestimation of the joint contact force and therefore also to an underestimation of the femoral loads. Nevertheless, Koch’s work is considered as the classical approach to femoral stress calculations which were the bases of numerous research studies on this subject. The first researcher, who included the effect of muscles on femoral loading in his analytical analysis, was Pauwels (1950) as described in the previous chapter. However, the three dimensional characteristic of the bone was ignored. Although the Pauwels model has been criticized [19], it is still accepted and can be found nearly in every biomechanical book until today. Summarized the important points are:

1. Trabeculae orientation according to principle stress directions 2. Highest density in areas of highest shear stress 3. Change in the loading of the bone causes adaptation of bone structure


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The next points are still the basis of many current research projects:

4. Bone cells may be regulated by local stresses 5. Results show no correlation of stress to cell activity 6. Relationship of bone structure to mechanics was derived without considering

physiological mechanisms

4.3 Loading Conditions simulated in the literature

In the literature a lot of different systems of forces can be found. One of the reasons for this variability is the fact that different load situations are simulated, for example, two-leg stance, single-legged stance, or different phases of gait. The most common set-up found in literature is the one that simulates a single force at the femoral head, neglecting everything else (e.g. Crowninshield, Huiskes et al., Koch, Rohlmann etc.). The second most common set-up in addition to the first most common set-up simulates the action of the abductors muscles in a single force (e.g. Crowninshield, Crick et al., Okumura et al., Rohlmann etc) [20]. However, a large variability considering the magnitude and the direction of the abducting force exists. The third most common set-up is the one which includes the hip joint force, the abducting force and the iliotibial tract (e.g. Finlay et al., Prati et al., Rohlmann et al. etc). In several works there exists incomplete information about the magnitude and direction of these forces. In very few studies, only numerical one’s, simulated a system of more than three forces (e.g. Ferre et al). In the following figure 4.6 the percentage of works in the literature that apply different set-up’s, as described earlier, is illustrated [20]:

Figure 4.6: Percent of the literature according to the applied load

In summary, only a few researchers included muscle forces as an important influence on the load conditions in the femur in their investigations. In all of the investigations, nearly no one tried to use more than three to four muscle groups in modelling and most of the works are restricted to two-dimensions. A complete description of the internal loads of the femur was not found, which incorporates the forces from all muscle attachments to the bone and also a documentation of a complete free body diagram of the femur was not found. However, methods to quantify the load condition in the femur are currently available, but none of the found publications considered all the forces acting on the femur.

Only hip joint force

Hip force + abductors

Three muscles or more

Hip force, abductors, iliotibial tract


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4.4 Constraints simulated in the literature

In a large number of cases existing in the literature it is not clear what sort of constraints is applied to the femur. The other large number of cases of constraints found in the literature is to lock the knee, thus assuming that this distal lock applies the constraints physiologically exerted by the knee muscles. Therefore, most finite element models are fully constrained distally, with the forces being applied proximally in the desired direction [20]. The corresponding experimental situation is when the femur is distally cemented and shear plates are used to avoid any undesirable horizontal force component. Thus, the distally constrained femur can be said to be the preferable standard due to the fact that it is easily reproducible experimentally and also in finite element models.

5 Methodology

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5 Methodology

To reach the aim of this project the following flow chart was constructed and will be followed:

Figure 5.1: Flow chart of the followed steps within the project

The first step will be to get familiar with the software “ANYBODY”, which will be used to calculate the muscle forces during walking. The loading response is of special interest due to the fact that the Pauwels model and the “Stemmkörpermodel” are referring to this phase of the gait cycle. The aim is the verification of the force magnitudes and the inclination to the perpendicular of the considered force vectors of the “Stemmkörpermodel” in the loading response with ANYBODY.

CT Scanning of the femur

Segmentation of the CT data

Reconstruction of the solid geometry

Smoothing the geometry

Material properties Gait analysis

Computation of the desired muscle forces

Load case determination

Musco-skeletal model

Verification of the “Stemmkörpermodel”

Importing geometry

Meshing

FEM anaylsis

5 Methodology

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However, it is also of interest how the trochanter resultant force RT of the “Stemmkörpermodel” is behaving in the other phases of the gait cycle. After the determination of the desired muscle forces for the “Stemmkörpermodel” the load case for the Finite Element Model of the femur is also defined. The second step will be to build up a finite element model of the femur. Therefore, an existing right leg femoral bone will be scanned by CT. From the CT images the finite element model will be reconstructed. The final step is the finite element analysis according to the determined load case by ANYBODY. Of special interest is the loading response phase of the gait cycle, where the two different load conditions of the Pauwels model and the “Stemmkörpermodel” will be analysed and compared.

6 Analysis using ANYBODY

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ANYBODY is a software system for modelling the mechanics of the human body developed by the Aalborg University in Denmark. It can handle models with hundreds of muscles on ordinary personal computers. As calculation output forces in individual muscles, joint forces, metabolism, elastic energy in tendons, antagonistic muscle actions, and many other useful properties of the working human body are accessible. Application fields of ANYBODY are automotive, medical/rehabilitation, aerospace, occupational health and sports.

6.1 Background AnyBody

6.1.1 Kinematical Analysis

An unconstrained segment in space has six degrees of freedom. Therefore, if the model has n segments, the model will have a total of 6n degrees of freedom unless some of them are constrained somehow. The purpose of a kinematical analysis is to determine the position of all segments at all times. Thus, kinematical analysis is about solving 6n equations with 6n unknowns. Adding joints, jntn (number of joints), to the model is a way to constrain the degrees of

freedom. However, to provide all 6n constraints by adding joints would make the system unmovable. Therefore, usually a body model will have enough joints to keep the segments together and few to let the model move. After the joints, jntn , have taken their part of degrees

of freedom, AnyBody utilizes so-called drivers, drvn (number of drivers), which are added to

the system to resolve the remaining unknowns in the system up to the required number of 6n. Joints can also be understood as kinematical measures equipped with drivers. For instance, a spherical joint is a distance between two points on two different segments that is driven to be zero. Solving the system of 6n equations, the system is said to be kinematically determinate and usually this is necessary to perform a kinematical analysis. However, there are also some cases where the system is solvable although the number of equations is different to 6n. In few cases the system cannot be solved even though there are 6n equations available. Both cases are connected with redundant constraints. According to the following formula: Determinacy d = drvjnt nnn −−6 (6.1)

Three cases according to the determinacy can be distinguished:

• d = 0: kinematically determinate (suitable for inverse dynamics) • d > 0: kinematically indeterminate (forward dynamics needed) • d < 0: kinematically over-determinate (statically indeterminate)

It is an important point to keep track of the number of constraints and the number of degrees of freedom is quite important.


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6.2.1 Inverse Dynamics

Essentially there are existing two types of problems in rigid body dynamics:

• Direct Dynamics Problem: Known forces are applied to a mechanical system and the objective is to determine the motion of the system.

• Inverse Dynamics Problem: The motion of a mechanical system is known and the objective is to determine the forces that cause the motion of the system.

Due to the fact that the direct measurement of the tension in muscles, the forces and moments transmitted by the joints of the human body, and the activation of the peripheral and central nervous system is difficult or sometimes impossible the “Inverse Dynamic” approach is used in the software ANYBODY. In the analysis using the inverse dynamic approach the human locomotor system is modelled as a kinematic chain, consisting of single bone segments which are connected by frictionless joints. The skeletal bones are represented by rigid body segments. Each segment has six degrees of freedom (three translational and three rotational), whereas through the connection of two segments according to the kind of joint a certain number of degrees of freedom is restricted. On each single segment there are acting muscle forces, forces as a result of gravity and inertia, joint contact forces and external forces (e.g. ground reaction force) as the next figure 6.1 represents [21]:

Figure 6.1:

Left: Mechanical model representing the musculosketal system (foot, tibia and femur),

the resultant volume forces of a segment act at the segment centre of mass

Middle: Free body diagram resulting from the sectional cut A-A and B-B, at the cutting

section muscle forces and joint contact forces

Right: Combination to a resultant joint force and resultant joint moment


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For every segment, according to the Newtonian laws, the following equations have to be fulfilled:

Equilibrium of forces: bbi rmF &&⋅=∑ (6.2)

Equilibrium of moments: bbii Fr ωθ &⋅=⋅∑ (6.3)

bm : Mass of the segment b

br&& : Acceleration of the segment b

iF : All forces acting on the segment b

bθ : Moment of inertia regarding the centre of gravity of the segment b

bω& : Angular acceleration of the segment b

br : Vector from the segment centre of gravity to the force point of application

The computational scheme proceeds distal to proximal and is based on three assumptions:

• a complete kinematics analysis has been carried out • the geometric and mechanical parameters of each segment are known • external forces are known (e.g. ground reaction forces)

Gait analysts are able to measure four components in the movement chain which are highlighted in the next figure 6.2:

Figure 6.2: Movement chain of gait analysis [9]

Electromyography

Anthropometry of Skeletal segments

Segment displacements

Ground Reaction forces

Tension in muscles

Joint forces and moments

Equation of motion

Velocities and accelerations

Segment masses and moments of inertia


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Electromyography (EMG) is not able to measure the muscles tension but can give an idea about the activation pattern of the muscles. Via skin-or needle electrodes relayed electrical signal shows the examiner an impression about sequence, force and intensity of the muscle activity during the gait cycle. The emitted EMG signals can be quantified manually, electronically or with a computer program and reflect a quite objective figure of the muscle activity. Thus, the electromyo-graphical investigations represent, which muscle in which moment during walking is active. As seen in the above figure 6.2, segment anthropometry may be used to generate the segment masses, whereas the displacements of the segments may be double differentiated to yield accelerations. Ground reaction forces are used with the segment masses and accelerations in the equations of motion which are solved in turn to give resultant joint forces and moments. Some gait analysts measure all of those four highlighted components. In inverse dynamics as mentioned above, the external loads on the body and the motion is assumed to be known (e.g. from motion capturing devices). The output of the computation is the “internal forces”. If the “internal forces” are just joint moments and joint reaction forces, this becomes then in most cases a straightforward procedure, involving the solution of a system of linear equilibrium equations. However, for the computation of individual muscle forces, inverse dynamics leads to a redundancy problem, which will be described in the next section.

6.1.2 Redundancy problem

Considering, for example, the elbow joint:

Figure 6.3: Simple arm model with three muscles

3 muscles for 1 DOF (rotation of lower arm around elbow joint) ջ redundant system A musculoskeletal system is usually a redundant system meaning that the number of muscles nM is greater than the number of degrees of freedom nDOF. Therefore, a desired motion can be achieved by an infinite number of activation patterns of muscles. In nature, the central nervous system (CNS) overtakes the task to determine how much force each individual muscle has to provide. Constructing an algorithm to determine the activation of each muscle therefore entails guessing the motives behind the CNS’s function. ANYBODY overcomes the redundancy problem with an optimization strategy. Mathematical optimization, applied to the distribution of muscle forces, is a well-known technique for simulation of the muscle recruitment.

biceps brachii

brachialis

brachioradialis

shoulder

upper arm

lower arm elbow joint


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6.1.4 Optimization in ANYBODY

Several optimization criteria can be found in the literature. For example, the so-called polynomial criterion, minimization of weighted sum of muscles loads raised to some power, has become popular [21]. This criterion leads to non-linear optimization problems which need to be solved by time-consuming and sophisticated numerical methods. Therefore, a criteria leading to linear optimization problem is generally more attractive in terms of more ease implementation and efficiency. Numerical efficiency in the development of the optimization algorithm has been a motivation factor with several aims. Efficient computations are convenient considering large models, for example full body models, analyst work with. Secondly, numerical efficiency is important for parameter optimization which can be used for several purposes such as model parameter estimation, design optimization of man-driven artefacts, and simulation of human motion patters. In the following the min/max muscle recruitment criterion used in ANYBODY is presented. Large body models contain of more than 100 muscles and are still solved in fractions of a second on a standard PC for one configuration of an inverse dynamics simulation.

6.1.4.1 Min/Max criterion

Based on the assumption that muscles are recruited according to an optimality criterion leads to the task choosing the right one. The min/max muscle recruitment [22] can be expressed mathematically as an optimization problem with an objective function called G. Minimize objective function:

)( )(MfG (6.4)

Mf : vector of all muscle forces

)(Mif : i’th muscle force

Subjected to: dCf = (6.5)

,0)( ≥Mif { })(,...,1 Mni ∈ (6.6) C is the coefficient matrix and d the right hand side of equation (6.5) consists of external forces, inertia forces, and passive elasticity in the tissues of the body. Due to the fact that muscles cannot push an additional equation (6.6) was formulated. In the AnyBody Modeling System, a min/max criterion is used for the objective function G:

��

��

�=

i

M

iM

N

ffG

)()( max)( (6.7)


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To change the non-smooth min/max problem into a linear problem an artificial variable, ˻, is introduced. This variable has the function as an upper bound on all muscle activities:

,)(

β≤i

M

i

N

f { })(,...,1 Mni ∈ with iMi Nf ≤≤ )(0 (6.8)

Muscle activity is defined as a relative measure of the muscle force, iM

i Nf /)( , where iN is the

momentary strength of the muscle i. Via the so-called “bound formulation” a linear programming problem can be obtained with muscle forces and joint reactions as free variables. Due to the fact that joint reactions are free in sign and without side constraints, they can be eliminated from the equation system. Thus, the result is a linear program with as many unknowns as there are muscles. Due to the fact that the min/max formulation only cares about the maximal activity of the muscles another problem arises. Therefore, only a subset of the muscles is actually represented in the objective which leads to indeterminacy. It is also observed that groups of sub-maximally activated muscles may not be determined uniquely. There exist three categories of such sub-maximal muscle groups:

• Counter-working muscles • Parallel muscles, i.e., muscles with the same function • Independent sub-systems (one limb in a multi-limb model)

The problem which arises from counter-working muscles can be solved by adding penalties to the objective G: (6.9) But these penalties don’t handle the remaining two other sub-maximal muscle groups. This problem called for an iterative solution scheme, where each iteration step eliminates muscles which are uniquely determined and removes their contribution to the support of the external load from the right hand side of the equation. The next iteration can then determine the sub-maximal muscles. This procedure continues until there are no muscles left in the system. Considering that an analysis involves many time steps and each time step involves the determination of thousand muscles it is obvious that this is a very demanding numerical task. To get a better understanding of how the procedure works a small example will be given. The following figure 6.4 shows a simple model of the arm with an applied load:

Figure 6.4: Simple arm model [22]

∑=

��

��

�+=

)(

1

)(Mn

i i

M

i

N

fG εβ


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Next a table [22] is presented (figure 6.5) which shows the relative muscles activities in percent for three cases for the angle ́ = 0:

1. raw min/max solution having counter-working muscles

2. elimination by linear penalty (as mentioned above)

3. iterative solution method Figure 6.5: Table showing the relative muscles activities

Case 1: simultaneous activity of the elbow extensors and flexors Case 2: solution is still not satisfactory, not all elbow flexors are active Case 3: all three mono-articular elbow flexors are activated equally, they are so-called parallel muscles Having a look at the next figure 6.6, a sub-maximal activated sub-system exists in one arm when a higher force is applied to one arm than to the other. The corresponding relative activities computed by the iterative method for one revolution of the external forces (F1 = 2F2) are shown as well:

Figure 6.6: model with applied forces (F1=2F2) (left); relative activities computed by the

iterative method (right)

It is visible from both activation profiles that they have the same form but different magnitudes by a factor of two (according to the applied forces). This proofs that the iterative min/max solution method does handle the two sub-systems uniquely, which would not have been the case we the raw min/max solution method.


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6.1.5 Programming language: AnyScript

AnyScript is the modelling language of ANYBODY for the development of multibody dynamics models. It is an object oriented modelling language with a fixed set of available data types (classes). This means it is not possible to define own data types and write, like in a programming language, algorithms. An AnyScript file is purely consisting of definitions (declarations) of objects and the internal data structure in ANYBODY maps these definitions completely. Syntactically it is not unlike Java or C++, but perhaps it resembles most of all JavaScript. An AnyScript model is roughly divided into two main sections:

• The body model containing the definition of the mechanical system • The study section containing lists of analyses and other operations which

can be done on the model (e.g. kinematical analyse)

6.1.6 ANYBODY Model Repository

Developing accurate human body models is a complex challenging task. Therefore a so-called library “ANYBODY Model Repository” [23] of models exist that scientists and other advanced users have developed and made available in the public domain. The ANYBODY Model Repository uses elements of the AnyScript language such as include files, parameters, and equipping parts with their own interface to make it possible:

• To change the model pieces to fit to a given task – preferably without tampering with the interior workings of the parts that are used.

• To combine existing body parts to larger models. • To find parts and attach them to construct bits of an own new model.

The Repository files are divided into two main groups:

Figure 6.7: Two parts of the Repository

ARep: “Application Repository”

The “Application Repository” contains various devices, environments, and working situations of different AnyScript models. For example, existing applications are models lifting a box or riding a bicycle. For analyzing these “application models” the main files can be loaded in the “ANYBODY Modelling System” and different studies can be performed.

Repository

ARep

BRep


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BRep: “Body Repository”

The “Body Repository” contains AnyScript models with no specification of movements, forces, supports and no attachments such as environmental devices. Due to the fact that the entire body model consists of hundreds of muscles which is heavy computational, the BRep directory is structured to link applications to subsets of the body model such as the lower extremities in this project.

6.2 “AnyGait” model

Movement analysis has now become an important clinical tool to diagnose gait disorders. For this purpose a universal gait model was developed [23]. However, an estimation of muscle forces during a gait cycle will give precise information for the treatment of gait pathologies, particularly, in surgical cases. The available “Gait3D” model in the “ARep” branch of the Repository consists just of the lower extremity with 7 rigid segments (thigh, shank and foot for both legs and the pelvis) and 35 muscles in each leg. The model contains three different kinds of joints which are connecting the segments:

1. HIP: spherical joint ջ it is a point constraint, also known as a ball-and-socket joint 2. KNEE: revolute joint ջ it is an ideal hinge joint that only allows rotation about one

axis 3. ANKLE: universal joint ջ it allows rotation about two perpendicular axis

6.2.1 Degrees of freedom

The next table lists the degrees of freedom per bone and the constraints per joint:

Figure 6.8: DOF’s per bone and constraints per joint

ջ 42 DOFs minus 24 joint constraints ջ 18 drivers are needed that the model is kinematically determinate (d = 0).

Segments # bones DOFs per bone total

Foot 2 6 12 Shank 2 6 12 Thigh 2 6 12 Pelvis 1 6 6

ˬ 42

Joint type # joints Constraints per

new determination of muscle forces acting on the femur and stress...

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