renner_2007a_determination of muscle forces acting on the femur and stress analysis

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.


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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.


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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

)(M

if : i’th muscle force

Subjected to: dCf = (6.5)

,0)( ≥M

if )(,...,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)


- 33 -

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 i

M

i Nf ≤≤ )(0 (6.8)

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

M

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


- 34 -

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 joint total

Hip 2 3 6 Knee 2 5 10

Ankle 2 4 8 ˬ 24


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6.2.2 Marker driven model

As mentioned in the previous section the gait model needs 18 drivers. The Gait3D model is driven by markers and therefore it needs 18 marker coordinates. The centres of the joints are needed for describing the movements of the patients. Due to fact that markers cannot be put directly at the joints location inside the body of the patient, external markers are used and their positions are measured throughout each investigated activity. The next figure 6.9 illustrates an exemplary system of infrared reflecting markers which are used on the trunk and the single segments of the legs [24]:

Figure 6.9: Infrared reflecting markers on a human body

A Vicon system with six cameras and a sample rate of 50 Hz was used to measure the position of body markers which were smoothed by 5th order splines. The coordinates of these markers were recorded in a fixed “laboratory coordinate system” shown in the above figure. The marker positions relative to the palpable bony landmarks were measured on the patient. The location of joint centres and additional reference points to these landmarks, used for calculation of rotations, were determined using individual CT data [24]. Thus, the calculation of the coordinates of joint centres and reference points relative to the laboratory coordinate system from the measured marker positions are possible.

Figure 6.10: Visualized markers

However, the motion data for the Gait3D model are grabbed from the book by Vaughan et al: Dynamics of Human Gait [9], which are available through the public domain [25]. In the Gait3D model only as many driver coordinates as necessary are selected to avoid kinematical over-determinacy. Segments for each of the markers are created and driven according to the data from the motion capture experiment. Therefore, text files with the coordinate data from the motion capture system for every marker (grey spheres) for each time step is read in (see figure 6.10). Corresponding markers are defined on the bones (blue spheres). The model is driven by requiring coincidence between the free floating markers (grey) and the markers on the bones (blue) for selected DOFs.


- 38 -

6.2.3 Force plates: Ground reaction forces

The most common force-measuring instrument used in gait analysis is the force plate. Force plates are available in many configurations, sizes, and with various performance characteristics. For the “Gait3D” model in ANYBODY, the ground reaction forces (GRF), acting from the floor to the supporting leg(s), were measured by two force plates. To get a better understanding how these force plates work the figure 6.11 is presented:

Figure 6.11: Typical force plate

The force plate has four output sensors as seen in the figure 6.11 above. The typical transducers used in piezoelectric-based force plates are four identical three component force transducers; one placed at each corner of the plate. Walking direction is the positive y-direction. The four output sensors produce the following eight outputs: Channel 1: force in x-direction measured by sensor 1 and sensor 2 → Fx1 + Fx2

Channel 2: force in x-direction measured by sensor 3 and sensor 4 → Fx3 + Fx4

Channel 3: force in y-direction measured by sensor 1 and sensor 4 → Fy1 + Fy4

Channel 4: force in y-direction measured by sensor 2 and sensor 3 → Fy2 + Fy3 Channel 5-8: force in z-direction measured by sensor 1-4 → Fz1, Fz2, Fz3, Fz4

These eight channels represent the four individual vertical forces measured, two shear forces in the x-direction and two shear forces in the y-direction. To receive six ground reactions and moments, the data are further reduced as follows: ( ) ( )4321 xxxxx FFFFF +++= (6.10)

( ) ( )3241 yyyyy FFFFF +++= (6.11)

4321 zzzzz FFFFF +++= (6.12)

( ) ( )[ ] bFFFFM zzzzx ⋅−−++= 4321 (6.13)

( ) ( )[ ] aFFFFM zzzzy ⋅−−++= 4132 (6.14)

( ) ( )[ ] ( ) ( )[ ] aFFFFbFFFFM yyyyxxxxz ⋅−−+++⋅−−++= 32412143 (6.15)

The loads transmitted between the force plate and a body in contact with it can be determined by a resultant force and a resultant moment.


- 39 -

When only compressive forces in the z-direction act on the force plate, the couple that can be transmitted to the force plate would be in the x-y plane. This couple is referred as the “free moment”. To perform a 3D gait analysis also the point of application of the resultant force and the free moment has to be known [26] and can be calculated from the measured force and moment components expressed in the Cartesian coordinate system:

z

xy

F

azFMax

0⋅+−= (6.16)

z

yx

F

azFMay

0⋅+= (6.17)

ayFaxFMT xyzz ⋅+⋅−= (6.18)

The point of the resultant force application is also known as the centre of pressure (COG). The ground reaction data used in the Gait3D model are also grabbed from the book by Vaughan et al: Dynamics of Human Gait [9].

6.2.4 Boundary condition

In general the question arises, especially for the latter calculation of the centre of mass, why does the model work without the upper body. The following boundary conditions are applied to the model:

• The feet are loaded with the measured forces from force plate experiments [9]. • The pelvis is supported by six reaction forces (they do not supply any motion!). • The reaction force provides the missing reactions between the lower extremity and the

upper body. Theoretically it is possible to perform gait analysis without force plate data. Therefore, the upper body has to be modelled which requires upper body movements and a high accuracy.

6.3 Expected displacement of the body centre of mass during gait

The body centre of mass (COM) is the point in the human body about which all the body segments balance each other:

• All linear forces acting on the human body are balanced, i.e. Σ F = 0 • All rotary forces acting on the human body are balanced, i.e. Σ M = 0


- 40 -

During the gait cycle the body centre of mass translates along the walking direction but also moves in a sinusoidal pattern in the lateral and vertical direction [10] as illustrated as follows:

Figure 6.12: Scheme of vertical / lateral displacement during one gait cycle

Considering the sagittal plane, the body centre of mass moves vertically through two full oscillations during each gait cycle. Therefore, the vertical displacement curve has two peaks and two troughs. The two minimum heights of the COM occur at loading response (5%) and at preswing (55%) and the two maximum heights of the COM occur at midstance (30%) and again at midswing (80%). The total excursion of the vertical displacement curve at average walking speed is approximately 5cm. Considering the transverse plane, the body centre of mass also oscillates laterally in a sinusoidal pattern with a total excursion at average walking speed of approximately 4cm []. Only one full lateral oscillation of the body centre of mass occurs during one gait cycle and the minimum right position of the COM is at the end of midstance (30%). The maximum left position of the COM is at midswing (80%). The following figure 6.13 illustrates the displacements in the saggital and in the transverse (horizontal) plane during one gait cycle:

Figure 6.13: Displacements of COM: saggital and horizontal view


- 41 -

It is also important to mention at this point, that the effect of walking speed plays a role on the body centre of mass displacement. Investigations on the COM displacement in vertical and lateral directions have been made and it was found out that the vertical displacement increases as walking speed increases [27]. In opposite to the decreasing lateral displacement, the walking speed increases. Thus, slow walking speeds require significant balance response due to the large lateral displacement of the COM. It was also shown that vertical displacement of the COM is highly correlated with energy consumption and therefore energetically and metabolically costly. The locomotor system has several methods to try to reduce the amplitude of the COM. These methods are the six so-called determinants of gait, but they will not be discussed here.

6.3.1 Calculation of the body centre of mass

As mentioned in one of the previous sections, in the “Gait3D” model the measured reaction forces provided the missing reactions between the lower extremity and the upper body. Therefore, it was possible to just model the lower extremity. In ANYBODY there exists a class called “AnyKinCom”, which is able to calculate the centre of mass for an arbitrary number of desired segments. However, this class cannot be used due to the fact that just the lower segments (pelvis, thigh and shank) are available in the model. Another way has to be found. 6.3.1.1 Force platform method A number of methodologies for calculating the body centre of mass exist. Some of these methods uses kinematic data obtained from markers that are placed on the body and others uses data obtained from force platforms. Due to the fact that the “Gait3D” model is driven by force plate data the force platform method will be used. A step is defined as the interval from initial contact of one foot to the initial contact of the other foot [28]. The vertical acceleration )(ta z is computed from the summed vertical ground reaction

forces )(tFz , less the body weight gm ⋅ , and divided by the body mass m :

m

gmtFta z

z

⋅−=

)()( (6.19)

The vertical velocity of the body centre of mass is calculated by integrating the acceleration over a single step:

τττ dgmtFm

vdavtv

t

z

t

zz ))((1

)()(0

0

0

0 ⋅−+=+= ∫∫ (6.20)

where 0v is the integration constant at the beginning of the step cycle. The integration

constant was determined by requiring the average vertical body centre of mass velocity to be zero.


- 42 -

The vertical position of the body centre of mass )(tzB is found by integrating the vertical

velocity over a single step:

ττ dvztz

t

zB )()(0

0 ∫+= (6.21)

where 0z is the integration constant at the beginning of the step cycle, which is set to zero

since only the vertical displacement of the body mass centre is of interest. In the “Gait3D” model two force platforms are used and both vertical forces given by the textfiles and the total vertical force of both of them are presented:

Vertical forces given by the two force plate textfiles

-100

0

100

200

300

400

500

600

700

800

900

1000

0 0,5 1 1,5 2

time [s]

forc

e [

N]

vertical force: force plate 1

vertical force: force plate 2

total vertical force

Figure 6.14: Vertical ground reaction forces given by textfiles

In the following figure 6.15 the start position and the end position during walking of the human model in ANYBODY of one gait cycle is shown:

Figure 6.15: Start and end position of the gait cycle in AnyBody


- 43 -

Comparing the definition of the step interval from initial contact of one foot to the initial contact of the other foot, on which the vertical displacement calculations are based on, with the start and end position of the AnyBody model, it can be seen that the AnyBody model starts with the initial contact of the right foot but does not end with the initial contact of the other foot (again the right foot). According to the definition of the step interval there should be force data available of three double support intervals and two single support intervals illustrated as follows:

Figure 6.16: Definition of one step interval of the force plate method

As it can be observed from figure 6.14 there exists just one double support phase. Therefore, the calculation of the vertical displacement of the COM with the force platform method is not possible because the force under the other leg during two of the three periods of double support is not recorded. The force platform method is also error prone, if the integration factors are not chosen accurately.

6.3.1.2 Full body model Due to the fact that the “Gait3D” model just consists of the lower extremity as mentioned earlier the “AnyKinCom” class of AnyBody to calculate the COM cannot be used. Therefore, a full body model with no muscles from the BRep was loaded into the “Gait3D” model as the next figure 6.17 illustrates:

Figure 6.17: Full body model (no muscles)


- 44 -

In the “mannequin.any” file, variables were set to control the rotation of the thorax with respect to the pelvis. These values are overtaken and modified from the “StandingModel” mannequin file from the ARep. The human model contains 29 body segments, which are put in the class “AnyKinCom” to compute the COM. The “AnyKinCom” class calculates the position from full body kinematics for every time step. Therefore, the total body centre of mass for one time step is the weighted sum of the centre of mass of every segment of the body:

∑

∑ ⋅

=

j

j

jij

j

m

pm

COM

,

(6.22)

mj: mass of segment j

pi,j: ith component (i = x, y, z) of the position vector of its centre of mass

Then a kinematical analysis was performed to get the position values for every time step. The received result values can just be considered as approximated position values because of the missing movement information of the arms. Therefore, the arm movement is neglected in the analysis, which is assumed to play a secondary role in the determination of the COM. Different arm postures were tried out, giving nearly all the same lateral/vertical displacements as presented:

Vertical / Lateral displacement of the COM

Brep: Full body model no muscles

0,00

0,20

0,40

0,60

0,80

1,00

1,20

0% 20% 40% 60% 80% 100% 120%

gait cycle [%]

dis

pla

cem

ent [m

]

vertical displacement

lateral displacement

Figure 6.18: Vertical/Lateral displacement of the COM (AnyBody)

As mentioned earlier the two peaks of the sinusoidal vertical displacement curve should be at midstance (13 – 31%) and midswing (76 – 86%). The two troughs should appear at loading response (0 – 12%) and at preswing (51 – 62%).The above presented sinusoidal pattern of the vertical displacement of figure 6.18 shows good agreement according to the peaks and troughs. The amplitude of the vertical displacement curve received from AnyBody using a male human model is 3.4cm.


- 45 -

Only one full lateral oscillation of the body centre of mass should occur during one gait cycle and the minimum right position of the COM should be at the end of midstance (30%). The maximum left position of the COM should be at midswing (80%). Comparing with the above figure 6.18 the result can be assumed not to be completely correct. Thus, also marker coordinates for the upper body have to be used from motion capture systems to drive the upper body to give correct results. It can be stated that the vertical displacement of the COM is not as strong influenced by the arm movement as the lateral displacement of the COM. 6.3.1.3 Visualization of COM position during gait cycle The position coordinates (x, y and z) of the COM for every time step of the gait cycle were copied from the output data received from the “AnyKinCom” class, including 29 body segments, into a textfile. The first column of the textfile is the time followed by the columns with the x, y and z coordinate values. First, a segment with mass = 0 and principle moments of inertia = 0 is created with the “AnySeg” class including the visualization of a node for the COM: AnySeg Segment = r0 =.ComData(Main.Study.tStart); //initial position Mass = 0; Jii = 0.0,0.0,0.0; //principle axis of inertia AnyRefNode node = sRel = 0,0,0; //relative position vector AnyDrawNode drw = //visualization ScaleXYZ = 0.03,.03,.03;RGB = 1,0,0;; ;;

In the next step two objects, “AnyKinLinear” and “AnyKinRotational”, are created to measure the translation, the position of one reference frame with respect to the global reference frame, and the rotations. AnyKinLinear LinCom = AnyRefFrame &ref1 =..GlobalRef; // the reference frames that define the vector AnyRefFrame &ref2 = .Segment; ; AnyKinRotational RotCom = AnyRefFrame &ref1 =..GlobalRef; // measures 3D orientation/rotation AnyRefFrame &ref2=.Segment; Type=RotVector; ;


- 46 -

Then a “kinematic measure” is used to for driving the motion of the COM by adding a kinematic driver to the “AnyKinLinear” object. AnyKinEqInterPolDriver LinDrvCom = // interpolates between data points spaced in time AnyKinLinear &ref1 = .LinCom; // measure that should be used Type=Bspline; BsplineOrder = 8; FileName =.datafile; //reference to the name of the data ; file to use

The “AnyKinEqInterPolDriver” interpolates between the given data points; it is a very useful driver for the import of sampled kinematic data. In addition, an “AnyEqSimpleDriver” is used, which drives the “AnyKinRotational” object. This kind of driver is often used to set constant values or constant velocities. Therefore, the model is again kinematical determinate.

6.4 Musculoskeletal Modelling in ANYBODY

A muscle model is a description of how a muscle behaves under different operation conditions. A traditional muscle model takes an activation signal and a present muscle state as input and produces a force as output. AnyBody, instead of taking an activation signal as input produces the muscle active state as output by using inverse dynamics. Thus, it is necessary to know the muscle properties involved. AnyBody has three different muscle models included, based on the classical work by A.V.Hill in 1938, which are ranging from simple to more complicated physiological behaviour:

1. AnyMuscleModel: assuming constant strength of the muscle regardless of its working condition

2. AnyMuscleModel3E: a three element model taking serial and parallel elastic

elements into account along with fibre length and contraction velocity

3. AnyMuscleModel2ELin: a bilinear model taking length and contraction velocity

into account The “AnyGait3D” model is using the muscle model “AnyMuscleModel3E”. Therefore, just this muscle model will be considered more in detail. To represent a muscle contraction in a mechanical formulation the so-called three component Hill model is often used in simulations. This model consists of the following three components:

• contractile element (CE) : representing the active force of the muscle fibres • two non-linear elastic elements:

o parallel elastic element (PE): representing passive properties of surrounding tissue

o series elastic element (T): representing the tendon and other elastic tissues


- 47 -

The “AnyMuscleModel3E” is a muscle-tendon actuator model based on the three component Hill-model proposed by Zajac in 1998, including a pennation angle γ . The next figure 6.19 illustrates the scheme of the “AnyMuscle3E” model:

Figure 6.19: The generic three component Hill-type muscle-tendon model described by Zajac

Received from experimental studies muscle architectural properties of the muscle force-fibre length, the muscle force-fibre velocity and tendon force-tendon length relationships, together with peak isometric force, optimal muscle fibre length and pennation angle at optimal fibre length, tendon slack length and maximum shortening velocity are used as input data. In the AnyBody Modeling system, muscles mechanically consist of two separate computational models, the strength model such as the three previous mentioned models and the kinematical model. Three different kinematical models are existing in AnyBody, which are defining the muscles’ path from origin to insertion depending on the posture of the body:

1. AnyViaPointMuscle 2. AnyShortestPathMuscle 3. AnyGeneralMuscle

In this project just the kinematical model “AnyViaPointMuscle” will be used and explained in the following two sections.

6.4.1 Direction of “AnyViaPointMuscle” pull

Muscles can be just defined by two points (a straight line), from origin to insertion. In these cases the direction of the pull is from the origin point to the insertion point as illustrated:

Figure 6.20: Muscle from origin to insertion


- 48 -

There are also existing muscles that are modelled not only with the insertion and origin point. These muscles or its tendon is bent out of a straight line by a bony process or ligament so that it runs over a pulley-like arrangement, the so called Via-points in AnyBody. The direction of muscle pull is then naturally bent out of line as illustrated in the next figure 6.21:

Figure 6.21: Via-point muscles

6.4.2 “ViaPointMuscles” of the “Stemmkörpermodel”

Anatomically, via point muscles are mostly found in the lower extremities and the spine. Therefore, all the muscles of the lower extremity which will contribute to the verification of the “Stemmkörpermodel” are modelled as ViaPointMuscles. ViaPointMuscles are passing through at least a set of two points on their way from origin to insertion. As many Via points as needed can be used, they just have to be attached to a segment or the global reference frame of the model. The approximation between these constraining points is a piecewise straight line. The first and the last point of the set defining the muscle path are fixing the muscle rigidly to them and transfer forces in its longitudinal direction to them. Conversely, the muscle passes through the interior via points transfer only forces to interior via points along a line that bisects the angle formed by the muscle on the two sides of the via point. The muscles of the “Gait3D”are modelled according to the data by Scott Delp [25]. The only two muscles with just origin and insertion point are “Gluteus minimus” and “Gluteus medius”. Both muscles are divided into three parts with the same insertion point around the greater trochanter as figure 6.22 shows:

Figure 6.22: gluteus minimus (green) and gluteus medius (blue)

Insertion points

Origin points


- 49 -

All the other muscles which contribute to the “Stemmkörpermodel” are modelled with at least one Via-point:

• gluteus maximimus: o part 1: two via-points o part 2: three via-points o part 3: one via-point

• tensor fascia latae: one via-point • vastus lateralis: two via-points • vastus medialis: two via-points • vastus intermedius: two via-points • rectus femoris: two via-points

The next figure 6.23 presents the second part of the gluteus maximus muscle (three via-points):

Figure 6.23: gluteus maximus (second part): insertion / origin / three via-points

For the determination of the muscle forces of the “Stemmkörpermodel” different things were tried out. For example, it was tried out to shift the origin point of the muscle vastus lateralis up to the greater trochanter, where it anatomically originates [13]. Due to the fact that the vastus lateralis is a pennation muscle, which means if one elongates it, more muscle fibres are added, which in turn created a much larger muscle force than the original model. Therefore, it was decided to use the original muscle model according to the data from Scott Delp. However, when all the segments of the different muscles were considered, the calculated muscles resultant, for example Msc‘, which should pull in downward direction to the knee, was bended in an outward direction of the femur, which is physiologically not possible. Due to this fact some segments of the ViaPoint muscles were not considered. For the muscles vastus lateralis, vastus intermedius and vastus medialis an assumption was done, which considers that the vastus lateralis, vastus intermedius and vastus medialis are one muscle, which originates at the greater trochanter region.

insertion

via-points

origin


- 50 -

6.5 Determination of the hip joint force

According to Heimkes “Stemmkörpermodel” the hip joint force is the resultant of the summation of the abductor muscles M and the partial body weight wh. It is also possible to get the hip joint force directly from AnyBody. In AnyBody the 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.

6.5.1 Comparison of the hip joint force: Heimkes/AnyBody

First of all the computed hip joint force Rh according to Heimkes “Stemmkörpermodel” is compared with the output force vector of the hip joint getting from ANYBODY. The following figure shows the comparison of the magnitudes of Rh:

Comparison: calculated hip joint force R (Stemmkörpermodel)

and output hip joint force AnyBody

0

500

1000

1500

2000

2500

3000

3500

4000

0% 20% 40% 60% 80% 100%

gait cycle [%]

hip

co

nta

ct

forc

e [

N]

R: ANYBODY

R: Stemmkörpermodel

Figure 6.24: Comparison of the hip joint force Heimkes/AnyBody

The two above presented curves of figure 6.24 are showing in shape a quite good agreement. However, the magnitude of the computed hip joint force according to the “Stemmkörpermodel” is much lower until approximately 70% of the gait cycle, especially between 13% and 20% of the gait cycle. Thus, a muscle or several muscles are assumed to be missing in the calculation of the hip joint force of the “Stemmkörpermodel”.

6.5.2 Correction of the hip joint force

To correct the hip joint force of the “Stemmkörpermodel” the missing muscle or muscles have to be found. In a dissertation [29] found, the muscles that are involved in the calculation of the hip joint force in the one-legged stance were investigated and also the percentage of each single muscle force involved was given according to the sum of all muscle forces.


- 51 -

Therefore, the muscle activity of the muscles ilipsoas and rectus femoris are checked in AnyBody:

Activity patter of rectus femoris and ilipsoas

-2,00E-01

0,00E+00

2,00E-01

4,00E-01

6,00E-01

8,00E-01

1,00E+00

0% 20% 40% 60% 80% 100%

gait [%]

acti

vit

y

ilipsoas

rectus femoris

Figure 6.25: Activity patter of ilipsoas and rectus femoris

The activation patter of the muscles ilipsaos and rectus femoris is looking like it could be the missing part of the difference in the calculation of the hip joint force. Thus, the muscles ilipsoas and rectus femoris are also included into the calculation of the hip joint force and the result is presented as follows:

Comparison: calculated hip joint force R

(Stemmkörpermodel modified)

and output hip joint force AnyBody

0

500

1000

1500

2000

2500

3000

3500

4000

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

gait cycle [%]

hip

co

nta

ct

forc

e [

N]

R: ANYBODY

R: Stemmkörpermodel

R: Stemmkörpermodel +ilipsoas + rectus femoris

Figure 6.26: Comparison of the corrected hip joint force Heimkes/AnyBody


- 52 -

As it can be observed from the above figure 6.26, the output hip joint force from AnyBody and the calculated hip joint force of the “Stemmkörpermodel” plus the muscles ilipsoas and rectus femoris are now fitting quite well. This in turn means that a small correction of the “Stemmkörpermodel” has to be done and also a new nomenclature of it will be introduced in the next section.

6.5.2.1 New nomenclature: “Stemmkörpermodel”

The following new nomenclature is introduced for the “Stemmkörpermodel” and will be used in the proceeding of this project:

Figure 6.27: New nomenclature “Stemmkörpermodel”

As it can be seen from the above described new nomenclature, the first subscript index of the muscle forces F is “m” standing for muscles. The second subscript index indicates either which muscles are acting or the location they are acting.

Fmh

Rh

wh

Fmt

RT

Fmk

Fmh: all hip equilibrates • gluteus minimus • gluteus medius • gluteus maximus • tensor fascia latae • rectus femoris • ilipsoas

wh: partial body weight Rh: hip joint force → sum of Fmh and wh Fmt: all hip equilibrates acting on the trochanter

• gluteus minimus • gluteus medius • gluteus maximus • tensor fascia latae

Fmk: knee muscles + additional muscles

• gluteus maximus • tensor fascia latae • vastus lateralis • vastus medialis • vastus intermedius

RT: trochanter resultant force → sum of Fmt and Fmk


- 53 -

6.6 Verification of the “Stemmkörpermodel”

Resulting from the done investigations and computations described in chapter 3 a table of resultant force vectors acting on the femur in the one-legged stance (presenting the loading response phase) with the new nomenclature is presented:

Figure 6.28: Computed force vectors dependent on the body weight

The above presented table of the resultant force vector values should be verified by the computed muscle force vector values received from the AnyBody modelling system. The following table shows the expected resultant force vector values according to the human body model with a body weight of 64.9 kg, which is used in the “Gait3D” model of AnyBody:

Figure 6.29: Expected force values according to the body weight of 64.9 kg

Due to the correction of the “Stemmkörpermodel” the expected force magnitude value of Rh is expected to be higher and in turn also the force magnitude value of RD. The angles of these muscle forces may also change. All other muscle force magnitudes should be approximately the same as figure 6.29 shows.


- 54 -

6.6.1 Verification of the force vectors

The following figure presents a summarized table of the muscle force vectors Fmt, Rh, Fmk, RT and RD in kg:

Fmt = 150.57 [kg]

Rh = 201.19 [kg]

Fmk = 209.63 [kg]

RT = 111.63 [kg]

RD = 262.85 [kg]

1% 36,79 90,3 70,43 44,64 121,42

2% 57,92 125,03 97,02 57,77 161,57

3% 85,15 155,17 128,79 74,32 197,3

4% 123,13 198,96 169,39 96,15 244,64

5% 155,32 238,97 194,5 110,42 278,1

6% 147,61 241,23 210,27 121,83 303,3

7% 134,94 233,94 217,33 132,04 315,57

8% 118,48 225,83 216,89 138,58 322,97

9% 134,03 247,87 215,61 133,25 330,41

10% 128,42 247,06 208,85 128,50 328,07

11% 118,58 239,19 201,4 124,96 322,28

12% 105,3 228,82 194,09 122,97 317,56

Figure 6.30: Loading response: Computed muscle force vector values in kg

The calculated resultant muscle forces of the “Stemmkörpermodel” showed good agreement with the expected values. However, the values of Rh and RD are higher than expected due to the additional muscles ilipsoas and rectus femoris.

6.6.2 Verification of the angles

The next figure 6.31 illustrates the global AnyBody coordinate system and the hip joint resultant force vector Rh :

Figure 6.31: Global coordinate system of AnyBody and resultant force vector Rh

x

Inclination to the perpendicular: α

d

z

y

R


- 55 -

To verify the inclination to the perpendicular the resultant force vectors have to be converted into an angle representation. The calculation will be illustrated for the resultant force vector R at 9% of the gait cycle (loading response):

=

)(

)(

)(

RZ

RY

RX

R ⇒ R = (223.64, -499, -2369.28)T

22 )()( RYRXd += ⇒ 83.546)499(64.223 22 =−+=d (6.23)

−=

)(arctan

RZ

dα ⇒ °=

−−= 13

)28.2369(

83.546arctanα (6.24)

The inclination to the perpendicular, α = 13°, of the resultant force vector R calculated from the computed values from AnyBody is a little bit lower than the expected value of 17°. Thus, it can be said that the inclination to the perpendicular of the resultant force vector R at 9% of the gait cycle is verified, but with a small deviation. The next figure 6.32 shows the summarized table of all calculated inclination angles of Fmt, Rh, Fmk, RT and RD during the loading response phase, which means from 0%-12% of the gait cycle:

Fmt = 23° Rh = 17° Fmk = 8° RT = 52° RD = 6,4°

1% 12,76 5,15 25,81 51,93 14,67

2% 18,12 8,19 28,56 58,52 17,69

3% 22,81 12,22 28,53 62,2 19,21

4% 25,08 15,14 28,67 67,3 20,49

5% 25,78 16,29 28,93 74,03 20,93

6% 24,47 14,36 28,66 65,69 20,64

7% 24,14 13,16 28,25 58,85 20,23

8% 23,24 11,23 27,81 53,22 19,48

9% 26,37 13,0 27,15 57,42 18,53

10% 25,4 11,9 26,35 56,39 17,49

11% 24,56 10,68 25,85 54,03 16,85

12% 23,69 9,17 24,83 49,59 15,76

Figure 6.32: Loading response: Angles to the perpendicular of the computed force vectors

The angles of the computed force vectors are in general too high, especially the angles to the perpendicular of Fmk and RD.


- 56 -

Due to the fact that the “Stemmkörpermodel” is actually a 2D model, the planar angle to the perpendicular in the frontal plane is calculated and summarized again in a table:

Fmt = 23° Rh = 17° Fmk = 8° RT = 52° RD = 6,4°

1% 12,06 4,87 3,8 23,36 2,06

2% 17,76 7,67 5,43 40,45 3,37

3% 21,8 11,19 5,48 50,63 3,76

4% 23,69 13,74 5,67 59,78 4,18

5% 24,32 14,79 6,0 69,47 4,52

6% 23,31 13,09 6,14 57,18 4,72

7% 23,29 12,14 6,26 47,3 4,82

8% 22,67 10,43 6,67 39,29 5,09

9% 25,35 11,89 7,15 48,44 5,48

10% 24,47 10,96 7,57 47,51 5,63

11% 23,77 9,89 7,62 44,36 5,7

12% 23,02 8,56 7,78 39,21 5,8

Figure 6.33: Loading response: planar angle to the perpendicular in the frontal plane

As it can be seen from the above figure 6.33 the planar angle of the frontal plane are now in well agreement with the inclination angles to the perpendicular of the 2D “Stemmkörpermodel” and therefore also verified like the magnitudes of the force vectors.

6.6.3 Visualization of the force parallelogram at the greater trochanter

During the loading response of the gait cycle, the greater trochanter apophysis should be subject to pressure stress from a craniolateral direction. The visualization of the trochanter force resultant acting on the greater trochanter apophysis ensured the craniolateral force direction during the whole loading response (0%-12%) of the gait cycle. The next figure 6.34 shows the visualized force parallelogram acting on the greater trochanter apophysis in the frontal view and the lateral view at 9% of the gait cycle to present the craniolateral force direction:

Figure 6.34: Trochanter resultant in the frontal and lateral view at 9%


- 57 -

Considering the complete gait cycle the trochanter resultant is highest in the loading response like expected.

6.7 Coordinate Transformation

The calculated muscle force vectors are all given in the global coordinate system of AnyBody. This global coordinate system is fixed to the ground where the “human body model” is walking on. For the force application in ANSYS the force vectors have to be transformed from the global coordinate system to the local coordinate system of the femur. The local coordinate system of the femur is located at the centre of gravity of the bone:

Figure 6.35: Global and local (technical coordinate system) coordinate system of the femur

For the transformation of the force vector from the global coordinate system of AnyBody to the local coordinate system of the femur a rotation matrix is necessary. AnyBody has for every segment an output option “Axes” which is the rotation matrix. The rotation matrix of the femur describes the orientation of the local coordinate system “L” with respect to the global coordinate system “G”, it can also be calculated when the location of the unit vectors of system “L” are expressed in system “G”. The three unit vectors are the columns of the rotation matrix:

( )

==

zzyzxz

zyyyxy

zxyxxx

zyx

eee

eee

eee

eeeAxes

Therefore, a muscle force vector F defined in system “G” can be rotated into system “L” according to the following formula:

( ) GL FAxesF ⋅=−1 (6.25)

By using this formula all muscle force vectors are now available also in the local coordinate system of the femur.

z

y x

7 Generation of the FE model of the femur

- 58 -


7.1 Simpleware software

Simpleware provides three software options for the conversion of 3D images to CAD, Rapid Prototyped and Finite Element models. A core image processing platform called “ScanIP” forms the basis of the software including modules for CAD integration and mesh generation. The next figure 7.1 illustrates the relationship between the different simpleware products:

Figure 7.1: Simpleware software products

In this project “ScanIP” and “ScanFE” are used to construct the FE model of the femur from the CT-data of the femur.

7.1.1 ScanIP: image processing software

A left femur bone was taken and scanned by a CT scanner. In a CT scanning process the amount of X-rays absorbed by the femur bone is measured and at each sampling point within the volume these values are converted into greyscale pixel values. The unit of these pixel values is called “Hounsfield unit” (HU).The femur bone is scanned by 661 slices and the images are saved in a format called “DICOM”(Digital Imaging and Communications in Medicine) which is the standardized format for exchanging digital pictures in medicine. The 661 DICOM images are imported in “ScanIP” and the different slices can be visualized by their greyscale pixel values in different views. The following figure 7.2 shows one part of the slice of the femur in the x-z plane:

Figure 7.2: Slice of the femur in the x-z plane


- 59 -

First a segmentation has to be done to distinguish between the compact bone region and the spongiosa bone region of the femur. The segmentation is performed by an operation called “FloodFill” which creates a so-called mask1 consisting of the pixels of greyscale values chosen. The range for the greyscale value was chosen from 1 (to preserve the marrow) to the highest greyscale value of the femur. Therefore, the femur without the marrow is defined as one mask. The problem of the “FloodFill” operation is that there were still remaining small holes like presented in the following figure 7.3:

Figure 7.3: cavities in mask1

To get rid of these small holes of the mask1 while keeping the marrow as a cavity of the femur again the “Floodfill” operation as cavity fill will be applied. Due to the fact that all pixels which are not belonging to the exterior of the object will be filled by this operation the marrow is connected with the outside of the bone by a “padding” operation. Then another mask2 for the interior of the bone (marrow) is created by “unpadding” the marrow and another “FloodFill” operation on mask1. In order to connect the marrow structures in Mask 2, a morphological close is applied. The result is presented in the next figure 7.4 containing mask1 and mask2:

Figure 7.4: The marrow of the femur of one slice

To separate the compact bone region and the spongiosa bone region of the femur another mask for the spongiosa region has to be defined. This mask will then be subtracted from mask1 and the result will be a new mask containing the compact bone region. A “FloodFill” operation with a different range of greyscale pixel values is done to receive one mask with the spongiosa region of the femur. The operation was not performing well and manual corrections had to be done in every slice. As a result the outer shell of the femur (compact bone) was thicker than the original outer femur shell not to get problems in meshing the compact bone region.


- 60 -

The received three different masks are shown in the following figure 7.5:

Figure 7.5: Three defined masks of the femur

Having a closer look to the surface of the rendered femur model it can be observed that the surface has still some artefacts. To smooth the data a Recursive Gaussian Filter is applied and the comparison before and after applying the filter is presented:

Figure 7.6: Comparison of the femur before and after applying the Recursive Gaussian Filter

The femur with the manually corrected masks was exported as STL file to ANSYS ICEM and meshed with tetra-and hexahedral elements. The original femur shape containing one mask including the marrow was exported to “ScanFE”as SFH files.


- 61 -

7.1.2 ScanFE: mesh generation / material assignment module

“ScanFE” offers a robust approach to the conversion of segmented 3D image data into multi-part volumetric and/or surface meshes. The generated meshes can be imported into a range of commercial FE programs such as ANSYS. The model was resampled in “ScanIP” not to get too many elements. The exported material model is visualized as voxel model in “ScanFE” as shown:

Figure 7.7: Voxel model of the proximal femur

With the help of the CT images an inhomogeneous material distribution of the femur can be considered. According to the Beer-Lambert law the attenuation of the x-ray intensity by penetrating the human bone is proportional to the density of the bone. In turn, the density of the bone can be set in relation to the values of absorption (in Hounsfield units [HU]) of the CT images and again the E-modulus of the bone is a function of the density of the bone. Therefore, for every element or group of elements an own value of the density and E-modulus can be assigned. The program “ScanFE” also has the possibility to assign material properties throughout the femur based on the Hounsfield number. It can be set of how many different material types the femur model shall consist (upper limit 256 different material types). For the femur model 100 different material types were chosen. The mass density, the young’s modulus and the poisson’s ration were calculated according to the following formula:

GS

bandaGSbadensitymass

⋅+=⇒

==→⋅+=

385.01000

385.01000:

ρ

ρ

GS: greyscale value

34.0

03,4.0,0:mod'

ρ

ρρ

⋅=⇒

=====→⋅+⋅+=

E

edandcbadbaEulussyoungec

4.0

00,0,4.0:'

=⇒

=====→⋅+⋅+=

µ

ρρµ edandcbadbaratiospoissonec

The coefficient values of a, b, c, d and e were taken from the ScanFE documentation [30].


- 62 -

7.1.2.1 Correction of the material properties

After the first FEM analysis with 100 different material types (loading according Heimkes) at 9% of the gait cycle was performed, a look at the displacements of the femur revealed that there is something wrong with the femur model (maximum displacement: 0.675411 m) as illustrated:

Figure 7.8: Displacement of the femur at 9% of the gait cycle (loading according Heimkes)

Due to the large maximum displacement, the material properties (density and E-modulus) have been regarded more in detail and the E-modulus values [Mpa] are plotted versus the density values [kg/m3] as follows:

E-Modulus versus density

0

20

40

60

80

100

120

140

160

0 500 1000 1500 2000

density [kg/cm3]

E-M

od

ulu

s [

MP

a]

ten different materials

hundred differentmaterials

Figure7.9: assigned material properties from “ScanFE”; E-modulus versus density

The calculated E-modulus values from “ScanFE” seemed to be very low. Therefore, they had to be compared with E-modulus values of the femur recorded in the literature.

DMX = 0.675411 m


- 63 -

However, in the literature there exists no common conversion from the original CT data to the values of E-modulus utilized for the FEM model as the next figure 7.10 illustrates:

Figure 7.10: Comparison of the conversion formula [31]

Curve A: valid for the complete bone

Curve B: spongiosa part of the bone

Curve C: cortical part of the bone

Curve D: two parts, spongiosa and cortical bone

The comparison of the E-modulus values [Mpa] computed from “ScanFE” and the E-modulus values from the literature confirmed that the E-modulus values received from “ScanFE” are too low. Therefore, a correction has to be done. A European project about “Virtual Animation of the Kinematics of the Human for Industrial, Educational and Research Purposes” (VAKHUM) was found [32], which provides also datasets of the femur.


- 64 -

The E-modulus [Mpa] versus density [kg/m3] of the VAKHUM femur model is plotted:

E-modulus versus density

y = 0,00000377545225x3 -

0,00001140543178x2 +

0,01228931065013x - 4,28249233909230

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0 500 1000 1500 2000 2500

density[kg/m3]

E-m

od

ulu

s [M

pa]]

VAKHUM: femur

Polynomisch(VAKHUM: femur)

Figure 7.11: E-modulus versus density (VAKHUM: femur)

To correct the E-modulus values a program was written in C++. The lowest and the highest density values of the femur of the VAKHUM data were taken and also the highest and lowest grey scale values (GS) from the available output data from “ScanFE”. Then, these values were set in the following linear equations to recalculate the coefficient values “a” and “b”:

GSba ⋅+=ρ (7.1)

⇒ )1292(2174

)692(566

⋅+=

−⋅+=

ba

ba

⇒ a = 1126.85 and b = 0.81

According to the recalculated coefficients “a” and “b” the density values of the femur were recalculated. In the next step, a polynomial E-modulus versus density curve fitting (see figure 7.11) was done and with the received coefficient values the new E-modulus was calculated:

2825.450130122893106.031780000114054.052250000037754.0][ 23 −⋅+⋅−⋅= ρρρMPaE (7.2)

The assignment of the material properties of the femur according to this described procedure is expected to present now the separation between the cortical and spongiosa bone quite well.


- 65 -

To ensure that the separation of the compact and spongiosa bone of the femur is correct a transversal cut in approximately the middle of the femur was done and the elements with the lowest E-modulus (representing spongiosa bone) and highest E-modul (representing cortical bone) were plotted as follows:

Figure 7.12: blue elements (cortical bone); red elements (spongiosa bone)

The maximum displacement of the FEM femur model with 100 different material types (loading according Heimkes) at 9% of the gait cycle was also checked again:

Figure 7.13: Displacement of the femur at 9% of the gait cycle (loading according Heimkes)

The FEM model shows realistic results and therefore the stress analysis in Ansys can be performed now.

DMX = 0.018 m

8 Stress analysis using ANSYS

- 66 -


8.1 Applied loads

In the stress analysis performed with Ansys, the Pauwels’ model is compared with the “Stemmkörpermodel” at 9% of the gait cycle. This step of the loading response is chosen due to the fact that the trochanter resultant has its highest magnitude there. Therefore, the loads, which are applied for the “Stemmkörpermodel”are the following:

• abductor muscles Fmt • hip joint force Rh • trochanter resultant RT

The applied loads of the Pauwels’ model differ in the absence of the vastus lateralis/intermedius/medialis muscle forces:

• abductor muscles Fmt • hip joint force Rh • tractus forces (gluteus maximus and tensor fascia latae)

8.1.1 Hip joint force

Comparing the output of the hip joint force given by AnyBody with the calculated hip joint force according to Heimkes, a difference in the direction can be observed. Considering, for example, the planar angle to the perpendicular in the frontal plane, the planar angle of the output hip joint force is approximately 30° and the calculated planar angle according to Heimkes is just 13°. Thus, theses values have to be compared with data from the literature. Bergmann [24] investigated nine different activities (slow/normal/fast walking, up/down stairs, standing up, sitting down, standing on 2-1-2 legs and knee bend) which are assumed to cause high hip joint loads. The angle Ay to the perpendicular in the frontal plane of the hip contact force was also investigated and illustrated:

Figure 8.1: Hip joint force in the frontal plane [24]

For all the investigated activities the hip joint force directions in the frontal plane are very similar and their variation is remarkable small as it can be seen from the above figure 8.1. Small forces act more from medial than large ones.

x

z


- 67 -

The found peak force is in a small range of 12°-16° for all activities except standing on one leg, where the peak force has an angle of 7° to the perpendicular. The other question which is arising is, if the very high force values of the output hip joint force especially between 15% and 19% of the gait cycle are realistic. In another study [33] the hip reaction forces calculated in AnyBody using the gait analysis data from Vaughan were compared with data from literature. It was concluded that the calculated hip joint values from AnyBody, except the large lateral force at HS, are possible due to individual variation between patients. Due to the fact that at 9% of the gait cycle the left leg already quitted the ground (toe-off) the force values from AnyBody can be taken. Thus, the calculated force values according to Heimkes are applied.

8.1.1.1 Determination of the hip joint load application point

The joint reaction force R at the hip should hit exactly the femur head centre. Therefore, the load application point at a surface node, which is fulfilling this condition, has to be found. Due to the fact that the femur head is not exactly spherical, an approximation has to be done for every time step. From the middle plane of the femur head approximately the centre node was chosen and the opposite direction of the load vector is applied to have an outer node of the femur. Then, from the centre node and the outer node of the femur a link element was created as shown in the next figure:

Figure 8.2: Link elements for determination of load application point

Considering the above presented figure 8.2, the load application point can now be determined easily. Therefore, the hip contact force R is applied to the node, which is nearest to the point, where the link element is penetrating the bone surface.

8.2 Constraints

In the FE model the femur is locked distally, which is the preferable standard in the literature.

8.3 Postprocessing in Ansys

In order to calculate the normal and shear stresses on defined cutting planes some postprocessing steps in Ansys have to be performed and are described in the next sections.


- 68 -

8.3.1 Stress tensor

Stress is a second-order tensor with nine components, but can be fully described with six components due to symmetry for elastic isotropic materials. From ANSYS the 2nd order stress tensor can be obtained:

[ ]

=

zzzyzx

yzyyyx

xzxyxx

σσσ

σσσ

σσσ

σ

8.3.2 Stress vector

In order to calculate the effective stresses a stress vector t has to be generated. The stress vector belongs to the stress tensor and a plane of arbitrary orientation (normal vector of the plane) as desired for example for a cut through the femoral head. The following figure 8.3 illustrates an arbitrary cutting plane in 2D and the stress vector:

Figure 8.3: illustration of the stress vector defined on a cutting plane

Unit vector normal to the plane:

i

i gnn = ji

ij nngn = (8.1)

Stress vector t:

nt ⋅= σ (8.2) j

jij

ijj

jij

ijj

jji

ij

i

igngggngngggt δσσσ =⋅=⋅=

j

ijint σ=

ti are the coordinates of the stress tensor t in the covariant basis. Since the base vectors in ANSYS (Cartesian coordinate system) form a set of orthogonal and normal vectors the coefficients of the stress tensor related to them are physical quantities. Therefore, no further calculations are required. However, the components of a tensor depend on the chosen coordinate frame, but tensors themselves do not depend on a chosen coordinate frame.


- 69 -

8.3.3 Calculation of normal and shear stresses of a defined cutting plane

For the investigation of the femoral head and neck, normal and shear stresses are of significant interest. For this aim the stress vector t has to be divided into vectors of the corresponding directions as the next figure 8.4 shows:

Figure 8.4: shear/normal part of the stress vector

Value of the normal stress:

j

ij

i

j

j

i

ij

ij

i

j

jji

iji

i nnggggnngngggn

nnnt

σσσ

σσ

=⋅⋅=⋅⋅=

==⋅= (8.3)

Value of the shear stress:

j

ij

i

j

j

i

ij

ij

i

j

jji

iji

i npggggnpgngggp

np

σσσ

στ

=⋅⋅=⋅⋅=

== (8.4)

with: p unit vector in direction of the investigated shear stress

or: 22 στ −= t (8.5)

The value of normal and shear stresses depends on the investigated cutting plane. Thus, for the different cuts through the femur calculations for the normal and shear stresses in ANSYS have to be done.

8.4 Analytical solution

The analytical solution is developed by the help of GiD personal postprocessing library in C++. This part of the project is achieved independent of Ansys pre- and postprocessing data.

8.4.1 Bending stress of a cutting section

The analytical solution of the bending stress is calculated for an arbitrary cutting section through the femur. Before applying the analytical bending stress formula the second moment of inertia and the total area of the cross sections are calculated numerically. The components of the external forces being perpendicular and parallel to the cutting sections are assumed to be uniformly distributed. Thus, the procedure is not completely analytical.

t

normalt

sheart


- 70 -

The classic formula for determining the bending stress in a member is [34]:

m

bI

cM ⋅=σ (8.6)

It is simplified for a beam of any cross section to:

bσ : bending stress

M: moment at the neutral axis c: perpendicular distance to the neutral axis

mI : second moment of inertia with respect to the applied moment axis

The second moment of inertia can be calculated numerically as:

∑∫ ∆⋅≈⋅=nodes

i

i

A

m AyxcdAyxcI#

22 ),(),( (8.7)

In order to find the discretisized areas ˝A, the node cloud of the transversal cut at z=0.2m through the body of the femur is taken. This point cloud is triangulated with the “delaunay triangulation algorithm”. The elemental areas after triangulation are taken as the discrete areas, and used to calculate the second moment of inertia and the total area of the complex cross section. Delaunay triangulation (DT) for a set P of points in the plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid "sliver" triangles [wikipedia]. The illustration of the “delaunay triangulation” in a plane is as follows:

Figure 8.5: scheme of delaunay triangulation in a plane

As it can be observed from the above figure 8.5,none of the points in the group enters the circumcircles of the candidate elements.


- 71 -

Summarized, a point cloud of a transversal cut through the body of the femur, the triangulation of this point cloud and the calculated total area is presented:

Figure 8.6: point cloud (left); triangulated point cloud (middle); total area (right)

The bending moment is calculated by considering the moment effect of the forces applied in the unconstrained region of the femoral head. The procedure is accomplished for both biomechanical models: the “Stemmkörpermodel” and the Pauwels’ model. The resultant moment of the forces in the unconstrained part of femur to the centre of the cross section determines the bending direction. Due to the fact that the bending moment is zero along this axis, the neutral axis is chosen as the moment direction. Therefore, the “c” values and second moment of inertia are calculated with respect to this line of action. Beside of that, only the planar components of the applied moment are decisive on this direction. The component of the moment being parallel to the cross section normal drives the torsion effect causing the shear stress but not the bending stress. The next figure 8.7 illustrates the above described procedure:

Figure 8.7: illustration of the procedure of the calculation of the bending moment


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The same procedure is also performed with a hollow circle cross section with an approximated outer and inner diameter as follows:

Figure 8.8: approximated inner and outer diameter of the cross section

To proof that the results visualized in GID are trustable, the numerically calculated second moment of inertia is compared with the analytical moment of inertia for the hollow circle: analytical:

444 827491.3)(64

mmedDI −=−Π

= (8.8)

numerical:

4#

22 815195.3),(),( mmeAyxcdAyxcInodes

i

i

A

m −=∆⋅≈⋅= ∑∫ (8.9)

From the analytical and numerical calculated second moment of inertia it can be said that the results are quite confidential.

8.4.2 Shear stress of a cutting section

As mentioned also in the previous section the solution is apart from the calculation of the polar moment of inertia and the calculation of the cutting section area is analytical. The calculated values will be compared with the postprocessed FEM results.

J

rT

A

Fs

torsionsheartotal

⋅+=+= τττ e

n

e

e ArdArJ ∆≈= ∑∫Ω

22 (8.10)

shearτ : shear stress due to the pure shear force

torsionτ : shear stress due to the torsion effect

T: torsion with respect to the centre of gravity of the cross section r: distance to the centre of gravity point

sF : shear force parallel to the cross section

J: polar moment of inertia w.r.t. the centre of gravity of the cross section


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8.4.3 Visualization of the results in GiD

8.4.3.1 Total normal stress The total normal stress of the transversal section at z = 0.2m is calculated as the sum of the normal stress and the bending stress:

m

bendingnormaltotalI

cM

A

F

W

M

A

F ⋅+=+=+= σσσ (8.11)

For the “Stemmkörpermodel” and the Pauwels’ model with the hollow circle cross section the following results of the total normal stress of the transversal section through the body of the femur are expected:

Figure 8.9: total normal stress [Pa] in the hollow circle cross section: Pauwels(left);

Heimkes(right)

The total normal stress is also shown for the real complex cross section of the femur:

Figure 8.10: total normal stress in the real complex cross section: Pauwels(left);

Heimkes(right)

total normal stress [Pa] total normal stress [Pa]

total normal stress [Pa] total normal stress [Pa]


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The next figure 8.11 illustrates a comparison of the total stress values [MPa] over the full cross-section at z = 0.2m versus the number of defined colours of the Pauwels model and the “Stemmkörpermodel”:

Compression/tension of the full cross-section

at z=0.2m

-80,00

-60,00

-40,00

-20,00

0,00

20,00

40,00

60,00

80,00

1 2 3 4 5 6 7 8 9 10

number of colours of the cross-section

tota

l str

es

s [

MP

a]

Pauwels

"Stemmkörpermodel"

Figure 8.11: comparison compression/tension values over the cross-section

As it can be obviously seen from the above figure the Pauwels model has higher tension values, but lower compression values than the “Stemmkörpermodel”.

8.4.3.2 Total shear stress

In the following the expected total shear stress and the two components of it of the transversal cutting section at z = 0.2m of the homogeneous “Stemmkörpermodel” is visualized according to the analytical solution:

Figure 8.12: Heimkes: total shear stress (left); tau_yz (middle); tau_xz (right); unit [Pa]

:

= +


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For the same transversal cut the expected total shear stress and the two components of it are presented for the homogeneous Pauwels’ model:

Figure 8.13: Pauwels: total shear stress (left); tau_yz (middle); tau_xz (right); unit [Pa]

8.5 Equilibrium check

The static equilibrium is simply checked by regarding the reaction forces given by Ansys. For the “Stemmkörpermodel” and the Pauwels model the following force component values are expected to be verified by the reaction forces: Heimkes: Fx = 52.69 N Fy = 413.3 N Fz = -1118.14 N Reaction forces (Heimkes): TOTAL VALUES

VALUE -52.69 -413-3 1118.1

Pauwels: Fx = 31.7 N Fy = 383.0 N Fz = -1061.19 N Reaction Forces (Pauwels): TOTAL VALUES

VALUE -31.700 -383.00 1061.2

ջ static equilibrium of the femur model

8.5.1 Internal equilibrium

The internal equilibrium of the femur model is also checked at two different cross-sections, at a transversal cross section through the body of the femur and at a cross section through the femoral head.

= +


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First, the internal equilibrium of a transversal cross section was checked in Ansys with the following surface commands: sucr,cut1,cplane,3 // create cutting plane at the location of the working plane sumap,sz,s,z // mapping of z-direction stress to the surface and name it SZ sueval,f1,sz,intg //get axial force by integrating axial stress and name it f1 The calculated axial force values for different transversal cross sections, for example of the “Stemmkörpermodel” are: z = 0.2 m: Fz = -3239.27 N z = 0.25 m: Fz = -3229.74 N z = 0.35 m: Fz = -3227.94 N This calculated internal axial force value of the transversal cross sections should be equal to the above applied external force components in the axial direction. The axial summed external force components have a value of -3225.14 N. Thus, the internal equilibrium is fulfilled for the axial force with a small error.. It can be stated that the physical stresses normal and parallel to some critical planes found by Ansys are fitting together with the analytical solutions. To verify statical equilibrium of the external forces with the internal stresses, some additional operations are performed in C++. In this way, the accuracy of the solutions in terms of the stresses was ensured quantitatively. The verification is done by comparing the analytical bending moment exerted by the external muscle forces with the internal stress traction vector moments. For this accomplishment some physical area values were necessary. For this purpose, the nodal normal stress values obtained from the previous postprocessing works are interpolated to obtain the elemental stress values:

Figure 8.14: illustration of the interpolation normal stress values

ii A∆⋅σ 1σ

2σ

3σ

2d

3d

1d

iσ

3

3

2

2

1

1

321

111111

ddddddi ⋅+⋅+⋅=

++⋅ σσσσ (8.12)


- 77 -

Where, the 3,2,1σ values are standing for the nodal total normal stress resultants. The force

equilibrium of the internal-external balance is verified similarly. According to the following formulas the errors can be calculated:

∑

∑

∆⋅−≡

∆⋅⋅−×≡

i

iienormalplan

extern

i

iiionplane

extern

AFerrorForce

AcFrerrorBending

σ

σ

r

rr

_

_

(8.13)

This procedure is applied to different cross sections with different models studied for the project. The algorithm is implemented in femur_cuts.dsw. The following figure 8.15 presents the verified internal equilibrium for a cut through the femoral head and a transversal cut through the body of the femur visualized in GiD:

Figure 8.15: internal equilibrium; (top) cut through the femoral head; (bottom) cut through

the body of the femur

bending-analytical/ansys:3.82068/3.99402[Nm]

normalforce-analytical/ansys:-1910.42/-1908.76[N]

bending-analytical/ansys: 144.036/141.739 [Nm]

normalforce-analytical/ansys: -3225.14 /-3244.82 [N]


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8.6 Results: Comparison Pauwels model / “Stemmkörpermodel”

The results presented in this chapter are all corresponding to the loading response at 9% of the gait cycle.

8.6.1. Transversal cut: FEM and analytical solution

For the purpose of comparison, the transversal cut is done like in the analytical solution through the body of the femur at z=0.2m. First, the FEM results of the stress in z-direction of the Pauwels model and the “Stemmkörpermodel” are presented in the view from above:

Figure 8.16: stress in z-direction [MPa] of the Pauwels model

Figure 8.17: stress in z-direction [MPa] of the “Stemmkörpermodel”

Comparing the results of the analytical solution with the FEM results, it can be also observed that the compression values of the “Stemmkörpermodel” are higher and the tension values are lower compared with the Pauwels model. However, the FEM stress values of both models are much higher than those of the analytical solution. The explanation of these higher values could be that the outer layer elements of both FEM models are not carrying the highest values like expected. It seems like the scanned femur has an outer porous layer. To ensure that the higher stress values of the FEM femur models are caused by the inhomogeneous material distribution, a new analysis is performed with homogenous material distribution.


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Next, the FEM results of the homogenous models according to Pauwels and Heimkes are compared again with the analytical solution:

Figure 8.18: Pauwels model: view from above

left: FEM results with corresponding colour legend at the bottom

right: analytical solution with corresponding colour legend

Figure 8.19: “Stemmkörpermodel”: view from above

left: FEM results with corresponding colour legend at the bottom

right: analytical solution with corresponding colour legend

Both FEM models show nearly perfect agreement with the analytical solution of the transversal cut through the body of the femur at z=0.2m. Thus, the higher stress values of the FEM models including inhomogeneous material distribution is therefore explained.


- 80 -

Next, the “pure moment” results by subtracting the axial stress are obtained from Ansys for the homogenous Pauwels’ model and the “Stemmkörpermodel” from the view from above:

Figure 8.20: comparison pure moment result; (left) “Stemmkörpermodel”; (right) Pauwels’

As it can be obviously seen from the above figure 8.20 the pure moment results of the “Stemmkörpermodel” are slightly higher. Inspite the total shear stress is not of special interest for this transversal cut, a comparison between the FEM solution of the homogenous Pauwels’ model and the “Stemmkörpermodel” is done:

Figure 8.21: Ansys total shear force; (left) Pauwels; (right) Heimkes

The maximum of the total shear force in the “Stemmkörpermodel” is slightly higher than in the Pauwels’ model, but considering the minimum value of the total shear force it is vice versa.


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8.6.2. Cut through the femoral head/neck

In this section two cuts will be done through the proximal femur. One through the head and the other one through the neck as illustrated:

Figure 8.22: defined cutting planes of the proximal femur

Due to the fact that the stresses existing in the femoral head are only influenced by the hip joint force, just the FEM results of the homogenous “Stemmkörpermodel” are presented:

Figure 8.23: Heimkes: normal and shear stresses approximately in the middle of the head

[Pa]

The distribution of the normal stresses look like expected, just compression. However, the normal stresses of a cut approximately through the femoral head can not be compared with the analytical solution because it is based on the assumption that the hip joint force is distributed uniquely. The same is valid for the analytical solution of the shear stress, but the FEM result of the shear force is like it is expected from the literature, with the largest shear value in the middle decreasing with the increasing radius of the spherical head. Here, it should be mentioned that the cut is not exactly through the hip joint centre.

cutting plane through the femoral neck

cutting plane through the femoral head


- 82 -

Next, a cut through the femoral neck for the inhomogeneous Pauwels’ model and the “Stemmkörpermodel” is done and the normal stress is investigated. The following figure 8.24 presents the normal stress in comparison:

Figure 8.24: Normal stresses [Pa] of the cut through the femoral neck;

(right) Pauwels; (left) Heimkes

It can be concluded from the above figure 8.24 that the normal stresses of a cutting section through the femoral neck of the Pauwels’ and the “Stemmkörpermodel” are approximately the same.

9 Summary and discussion

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9 Summary and discussion

The determined magnitudes of the muscle force vectors and their inclination to the perpendicular corresponding to the “Stemmkörpermodel” using AnyBody showed good agreement to the expected values based on anatomical, radiological and computational results. However, the “Stemmkörpermodel” was slightly modified by the addition of the two muscles ilipsoas and rectus femoris. Actually, if these additional two muscles have to be included in the “Stemmkörpermodel” is dependent on the antetorsion of the femur and therefore dependent on the individual patient. However, it should also be mentioned here, that the calculated hip joint force according to the vectorial force parallogram of the modified “Stemmkörpermodel” was differing in angle with the output hip joint reaction force given by AnyBody. In turn, the output hip joint reaction force received from AnyBody was also a little bit differing compared to the hip joint contact force values and angles given in the literature. The computation of the hip joint force in AnyBody due to inverse dynamics is based on the ground reaction forces input data. The accuracy of this data plays a crucial role for the computations of the joint forces and moments. However, the hip joint force is not influencing the muscle force parallelogram at the apophysis of the greater trochanter. The length of the generated FE model based on CT images of the femur was with a small deviation the same as the length of the femur used in AnyBody. However, the shape of the femur for every individual patient is different. The application point of the muscle forces acting on the apophysis of the greater trochanter according to the “Stemmkörpermodel” is assumed at one point, which is of course anatomically not correct. Therefore, in the FE model the forces are applied at a single node at the greater trochanter, which does not play a role for the investigation of a transversal cut through the body of the femur due to the St. Venant principle. In the stress analysis performed in Ansys the change in the bending in a transversal cut through the body of the femur was shown comparing the Pauwels’ model with the “Stemmkörpermodel”. Considering the applied muscle forces of both of these models, the difference of them is the absence of the vastus lateralis, vastus intermedius and vastus medialis muscle in the Pauwels’ model. It was found, that the bending of the “Stemmkörpermodel” of a transversal cut through the body of the femur can be said to be nearly the same (slightly higher through the higher compressive normal forces) as the Pauwels’ model. Thus, it can not be concluded that the bending of the body of the femur, which was the main investigation point of the stress analysis is reduced in the “Stemmkörpermodel” compared to the Pauwels’ model.

10 References

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10 References

1. Wikimedia Commons Category: Anatomy,

http://www.commons.wikimedia.org/wiki/Anatomy (accessed august 2007) 2. Jean-Pierre Kassi (2004) Musculoskeletal Loading and Pre-clinical Analysis of Primary

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