meyer e 2011 civil msc thesis

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IM

Departm

Macro-scale p

the

Submitted in fulfilment of th

ERIAL COLLEGE LONDON

Faculty of Engineering

nt of Civil and Environmental Engineering

edictive structural mo

usculo-skeletal system

Meyer, Edouard

September 2011

e requirements for the MSc and the Diploma of Imperial

elling of

College London


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Declaration:

This submission is my own work. Any quotation from, or description of, the work of others isacknowledged herein by reference to the sources, whether published or unpublished.

Signature : ___________________________________


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Acknowledgements

I would like to thank my project supervisor, Dr Andrew Phillips, for his precious help,advices and his availability throughout this project.

He managed to share his interest on this vast subject with me and for that, I am grateful.

My thanks also to Luca Modenese for his help with the muscle data.

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AbstractBone Modelling has been the subject of a lot of scientific studies and has attracted the

interest because of its numerous applications in the medical field.

These studies are mainly based on finite element models which give accurate results.Nevertheless, they have the disadvantages to be computational expensive.

In this dissertation, we introduce a 3D model based on the beam theory and the stiffnessmethod to represent the varying cross-sectional geometry of the human femur, a femur thatwe represent as a series of interconnected beam elements.

As many models, it is based on an iterative algorithm that takes into consideration thebone adaptation to its surrounding mechanical environment. A combination of 68 loadcases representing three main motions: walking, stair climbing and sitting-to-standing, areconsidered to create an anatomically correct structure.

This model aims to be computational efficient comparatively to the models based onfinite elements, while providing satisfactory results despite the several assumptions consid-ered to elaborate the algorithm.

Sensitivity studies are conducted to assess the validity of the model and to study thedependence of the algorithm on the inputs of the problem. Suggested further research andmodel developments are proposed at the end of this report.

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Contents

I Introduction 10

II Objectives and Methods 13

III Literature Review 14

1 Bone Remodelling 141.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Wolffs Law and Bone Remodelling . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Mechanostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Bone Properties 182.1 Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 The femur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.3 The patella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.4 Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Bone Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Muscle loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Current models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

IV Method 31

3 Theory 313.1 Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 The 3D stiffness method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 The local stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Transformation matrix from the local coordinate system to the global

one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Computing the global nodal displacements . . . . . . . . . . . . . . 373.2.4 Calculation of the xx strain value . . . . . . . . . . . . . . . . . . . 38

4 Model 394.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Mechanical and Geometrical properties . . . . . . . . . . . . . . . . . . . . 414.3 Muscle and ligament loading . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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4.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 The iterative algorithm 485.1 Presentation and objective of the algorithm . . . . . . . . . . . . . . . . . 485.2 Optimisation of the angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Optimisation of the three geometrical parameters R1, R2 and t . . . . . . . 52

5.3.1 Optimisation of the parameter t . . . . . . . . . . . . . . . . . . . . 525.3.2 Optimisation of the parameter R1 . . . . . . . . . . . . . . . . . . . 545.3.3 Optimisation of the parameter R2 . . . . . . . . . . . . . . . . . . . 55

V Results 56

6 Sensitivity studies 566.1 Dependence on the number of elements . . . . . . . . . . . . . . . . . . . . 566.2 Dependence on the parameters optimisation order . . . . . . . . . . . . . . 586.3 Dependence on the number of load cases considered . . . . . . . . . . . . . 606.4 Dependence on the load cases considered to fix the parameter . . . . . . 626.5 Dependence on the constants minR and maxR . . . . . . . . . . . . . . . . 66

7 Discussion and Comparison to the geometry of a real femur 68

VI Development potential 71

8 Muscle loads 71

9 Geometry, Material Properties and Trabecular Bone 71

10 The algorithm 72

VII Conclusion 74

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List of Figures

1 Gross structure of a long bone (human femur) . . . . . . . . . . . . . . . . 112 Culmann and von Mayer works . . . . . . . . . . . . . . . . . . . . . . . . 153 Bone Atrophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Threshold values of strain associated with the dead zone, bone resorption,

the lazy zone, and bone apposition, as well as the target strain [Phillips, 2011] 175 Hip Joint A: Articular surfaces; B: Movement of the neck of the femur during

medial and lateral rotation [Drake et al., 2004] . . . . . . . . . . . . . . . . 186 Knee Joint [Drake et al., 2004] . . . . . . . . . . . . . . . . . . . . . . . . . 197 The proximal end of the femur [Drake et al., 2004] . . . . . . . . . . . . . . 208 The distal of the femur [Drake et al., 2004] . . . . . . . . . . . . . . . . . . 209 The patella [Drake et al., 2004] . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Transverse section through the femur [Drake et al., 2004] . . . . . . . . . . 2111 Muscles of the anterior compartment of the femur [Drake et al., 2004] . . . 2412 Muscles of the medial compartment of the femur [Drake et al., 2004] . . . . 2413 Muscles of the posterior compartment of the femur [Drake et al., 2004] . . 2414 Diagrammatic representation of the five normal groups of the trabecular

bone in the upper end of the femur [Singh et al., 1970] . . . . . . . . . . . 2615 Comparison of the complex (left) and most simplified model of the hip mus-

culature [Heller et al., 2005] . . . . . . . . . . . . . . . . . . . . . . . . . . 2716 Boundary conditions applied to the diaphysis (left), the condyles (center)

and the joint nodes (right) . . . . . . . . . . . . . . . . . . . . . . . . . . 2817 The meso scale approach [Phillips, 2011] . . . . . . . . . . . . . . . . . . . 29

18 The Euler-Bernoulli beam theory and its assumptions . . . . . . . . . . . 3119 The local coordinate system (x,y,z) of the beam element . . . . . . . . . 3220 The direction cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3521 A line, in red, is plotted passing through the center of each part of the femur 3922 The geometry of the femur considered during this dissertation, plotted in

the global coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 4023 An hollow ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4224 The complete model: in red, the femur; in blue, the rigid elements linking

the muscle attachment points to the femur . . . . . . . . . . . . . . . . . . 4325 The modelling of the hip joint (left) and the patella joint (right) . . . . . 43

26 The representation of the frame number 22 of the walking case: in purple,the muscle loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4427 The boundary condition at the knee joint considered in the dissertation . . 4728 Four variables are chosen to parameter totally the cross-section . . . . . . 4829 Optimisation of the angle. (y,z) is the local coordinate system of the section

and (y1,z1) is the bending principal axes of the section. At the end, themaximum strain value is located at the minor principal bending axis. . . . 49

30 Difference between the localization of the maximum principal axis and theZero strain neutral axis [Demes, 2007] . . . . . . . . . . . . . . . . . . . . 50

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31 Hollow ellipse after rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 5532 Dependence of the algorithm on the number of elements used to represent

the shaft of the femur: in blue, 20 elements; in red, 40 elements . . . . . . 5733 Dependence of the algorithm on the optimisation order of the parameters . 5934 Dependence of the algorithm on the number of load cases considered: in

blue, 68 load cases; in red, 3 load cases . . . . . . . . . . . . . . . . . . . . 6135 Dependence of the algorithm on the number of load cases considered to

calculate : in blue, 28 walking load cases; in red, 68 load cases . . . . . . 6336 Dependence of the algorithm on the constants maxR and minR: in blue,

minR=12mm and maxR=30mm; in red, minR=8mm and maxR=15mm . . 6737 The maximum strain value . . . . . . . . . . . . . . . . . . . . . . . . . . 6838 GSA Model with maxR=30mm . . . . . . . . . . . . . . . . . . . . . . . . 6939 Alfred Thibons Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

40 GSA Model with maxR=15mm . . . . . . . . . . . . . . . . . . . . . . . . 7341 Alfred Thibons Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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List of Tables

1 Muscles of the anterior compartment of the femur [Drake et al., 2004] . . . 222 Muscles of the medial compartment of the femur [Drake et al., 2004] . . . . 233 Muscles of the posterior compartment of the femur [Drake et al., 2004] . . 234 Muscle forces for the frame number 22 of the walking case, corresponding

to t=0.047s Part 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Muscle forces for the frame number 22 of the walking case, corresponding

to t=0.047s (Part 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 The optimisation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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Part I

Introduction

With the progress made in biomechanics and computing science, research studies aboutthe human body are more and more numerous all over the world. Unconsciously or not,people attach a great value to everything related to the human body because understand-ing what happens inside us has always been an interesting challenge.

The human body is a complex structure with no less than ten trillion of living cells.Basis of this sophisticated architecture, bones are rigid organs that constitute part of ourskeleton. In a mechanical point of view, they can serve, for instance, to provide a frameto keep the body supported and enable it to move in three dimensional space. They formwith skeletal muscles, tendons, ligaments and joints, a complex and powerful system tosupport the human body.

Our study is relatively focused on the femur. This choice is an arbitrary one for severalreasons. First, the femur can be viewed as a long and important bone, allowing , amongother things, the weight of the body to be transferred from the hip joint to the knee joint.For instance, the femur must carry around 200-250% of the body weight. It also acts as

a stiff structure about which muscles act to facilitate movement at both the hip and kneejoints. [Phillips, 2009]

Moreover, the femur has been subjected to numerous experimental works in the lasttwo decades in order to study the material properties of human long bones [M.Martenset al., 1980]. Thus, a lot of data are available in the literature in order to check the resultswe find during this study.

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Figure 1: Gross structure of a longbone (human femur)

As we will discuss about the femur geometry through-out the report , it will be preferable to describe briefly,in this introduction, the characteristics of this bone.As a long bone, it is characterized by a shaft, the dia-physis, made up of compact bone surrounding a cen-tral cavity containing cancellous bone together withmarrow and fat. As we can see in figure I, it can bedivided into two main parts: the epiphysis and thediaphysis. The epiphysis is the rounded end of thebone and the diaphisys is the cylindrical shaft of thebone [Nather et al., 2005].

One of the main characteristics of the femur is that

the cross-section changes along the length of the bone.Thus, the cortical bone is thickest in the mid-portionof the shaft, the cancellous bone being relatively di-minished in this portion of the bone. At the end por-tion of the bone, it is the opposite. [Nather et al.,2005]

This particular structure of the femur can be explained by the simple fact that the bonetries to resist economically to the stresses and the loads produced by the surrounding en-vironment. As J.Koch described in his article Laws of Bone Architecture, the form of boneis generally adapted to its function and, thus, the external form and inner architecture of

normal bone represent the adaptation to normal function. [Koch, 1917]

To go further, bones are actually adapting to the mechanical requirements appliedthrough a life-long process. To better understand this idea, it is common to considerthe femur as a long strut with complex boundary conditions at the hip and knee joints.During the human life, various forces are applied to its surface and the bone is subjectedto numerous stress solicitations. Since the mechanical needs of locomotion, weight-bearingare different from one individual to another, it is obvious that these loads acting on thefemur differ in individuals.

Following the same idea, we deduce that the organisation and the mass of bone must

differ in individuals practising different activities. However, since these informations areunknown during the intrauterine time when the prescribed shape of the femur is only dueto genetic information and even during the childhood, a dynamic regulatory system in bonemust exist. [Skerry, 2006]

This idea is the basis of the bone remodelling concept first described by Wolffs Law.The bone shape is the result of an iterative process in order to withstand the appliedforces using a minimal amount of material by relocating the bone tissue in an optimalway. [Phillips, 2011].

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To control this iterative process, Frost developed Wolffs concepts and thought about amechanostat (like a thermostat) as a model describing bone growth and bone loss. Thus,bone changes in structure is stimulated by the local mechanical elastic deformation ofbone, i.e the local strain values (like the temperature for a thermostat). This adaptationis considered to be a life-long process. [Frost, 1998]

It is based on the idea that bones contain cells that make them be true living organs,capable of growing and healing sustained fractures. These cells are very sensitive to me-chanical stimuli and thats why bone morphology and structure change and adapt to themechanical environment. [Huiskes, 2006]

This iterative concept is the basis of a lot of studies about modelling the structureof bones. With the progress made in computing science, the majority of them use finiteelements and continuum models to output qualitative and precise descriptions of the ex-

ternal form and the inner architecture of bones. Nevertheless, as these models use a lotof elements to model precisely the structure of bones, the computational time can be verylong and may become an obstacle to create bigger models.

In this study, we want to model the structure of bones and compute the iterative processby using beam theory and the siffness matrix method. By using civil engineering theories,we simplify the problem and thus, we reduce considerably the computational time allowingus to work on bigger models.

The aim of this dissertation is to validate this new approach of bone structure modelling.To do that, we will first focus on the femur, and, by using the 3D stiffness matrix method,we will represent optimal cross-sections of this bone. The results we will find, will becompared to in-vivo values selected in other articles.

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Part II

Objectives and Methods

This study aims to conceive an algorithm in order to predict the geometry of the femur.This algorithm must be computational efficient comparatively to other bone modellingmodels based on finite element theory. It must also provide a powerful tool to obtainpreliminary results associated with various load cases acting on the bone (during walking,stair climbing or even, during a fall or vehicle accident for instance).

As being one of the first studies about the macro-scale approach, this dissertation mustlay the foundations of a powerful algorithm and must target precisely the need for futureimprovements.

To achieve these objectives, we will use the Euler-Bernoulli beam theory associated withthe 3D stiffness method. The femur will be modelled using interconnected beam elements.

Using the Mechanostat Concept developed by H.M. Frost based on the idea that thebones structure adapts to their surrounding mechanical environment, the algorithm willbe an iterative process based on the strain values of the elements. So, an element will beoptimised if its absolute maximum strain value lies in a particular interval, called the lazyzone, between [1000; 1500]. To do that, the algorithm forces the absolute maximumstrain value of each element to converge to a target strain value equal to 1250.

The femoral sections will be modelled with hollow elliptical sections defined by fourparameters and these parameters will be optimised by the algorithm in order to obey theMechnostat Concept.

In order to create an anatomically correct structure, various load cases will be consid-ered.

The algorithm will be tested through some sensitivity studies and suggested furtherresearch and model developments will be proposed for future studies.

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Part III

Literature Review

To find the optimal femur cross-sections, we will use the Bone Remodelling Concept onwhich our iterative algorithm is based. The aim of this part is to describe this general ideafirst described by J. Wolff.

Then, once the algorithm concept described, we need some specific values related tobones (Young Modulus, loads, boundary conditions, spatial geometry...) to implement ourprogram. These data are used to initiate the model and are found in the literature that

we will describe in the following.

1 Bone Remodelling

To find the cross-sections of bones, an iterative algorithm is used based on the conceptsof Bone Remodelling and Mechanostat. The aim of this part is to introduce and explainthese ideas which form the basis of our study.

1.1 General Observations

In this section, we just want to give some examples that illustrate the fact that bonesadapt the surrounding environment by changing their structure in an optimal way to with-stand the forces acting on them, through a life-long regulatory process.

Bones are living organs capable of growing and healing sustained fractures due to com-bined actions of living cells called osteoclasts and osteoblats. [Huiskes, 2006]

To do that, bones are able to grow longitudinally but also to enlarge transversally bymeans of bone tissues redistributions that change the bone structure depending on its needs(the mechanical needs of locomotion, weight-bearing...). Despite of all these redistributions,the bones structure remains an arbitrary and optimal one . To illustrate this idea, we canfocus on long bones structure. As we can see in the figure I, the diaphysis with its weaklyvarying load cases of bending, torsion and longitudinal forces, is made of with a solidcylindrical shape, surrounding the medullary canal, formed by the strong cortical bonetissue .

On the contrary, loads of varying amplitudes and orientations at the bone ends arecarried by spongy trabecular bone tissues allowing for more flexibility in the three spatialdirections . [Ruberg, 2003]

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Figure 2: Culmann and von Mayer works

To give another example of optimal bonetissues redistribution, the works of Culmannand von Mayer in 1866 can be mentioned(figure 2). The two scientists highlightedthe correspondence between cancellous bonestructure and the mechanical loads to whichit is exposed. They noticed that the stresspatterns of a curved column structure lookedsimilar to the stress patterns of the inter-nal trabecular bone tissues of a long bone.These patterns are the result of bone tis-sues distribution in bones in order to with-stand the mechanical loads in an optimal

way.

Furthermore, at birth, the bones designis only due to our genetic information and,so, the bones structure differences between people are minor at this time. However,throughout life, people practise different activities and bones from one individual to an-other are not solicited in the same way. Thus, it is obvious that the structure of bones cannot remain similar as what it was at the beginning of our life and must differ in individuals.In conclusion, a dynamic mechanism of regulation and redistribution must exist in bonesin order to deal with the numerous mechanical solicitations in an optimal way. [Skerry, 2006]

1.2 Wolffs Law and Bone Remodelling

All these above observations were formulated in Wolffs works in which the GermanAnatomist/Surgeon stated that bone adapted to the loads acted on it in accordance withmathematical laws. Thus, when bones are subjected to heavy loads, they have to remodelitself overtime to become stronger in order to withstand that type of loadings. For example,the tennis players have one arm (the racquet holding arm) more developed and strongerthan the other one, since this arm is placed under high stress levels.

The opposite is also true: if bones are placed under low stress levels, bones become

weaker since there is no presence of enough mechanical stimuli to maintain the initial bonemass. This situation may possibly lead to their atrophy (figure 3).Despite the fact that J.Wolff stated about the relationship between bones structure and

mathematical laws, the German scientists didnt give any theoretical formula to predictbones shape. Instead of that, he only gave a collection of concepts. By carrying out someexperiments on the legs of animals, he noticed that any change in static conditions of thelegs resulted in modifications of both forms and architecture. [Wolff, 1987]

More precisely, there is a first change in the inner architecture of the bone, followingby a change in the outer form. [Setzer, 2004]

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Figure 3: Bone Atrophy

In his book Das Gesetz der Transformation der Knochen, J.Wolff mentioned Cullmannand von Mayers works. He criticized their works by noticing that the stress patterns inthe trabeculae must meet at right angles when bones are subjected to an unique load case,a fact which was not imply in Culmanns studies. This remark implies that these patternsfollow the lines of principal stresses. Nevertheless, if bones are subjected to various loadcases, non-orthogonal trabecular intersections offering increased shear resistance are moresuitable. [Phillips, 2011]

In conclusion, Wolffs works show that bones are subjected to an ongoing change in their

inner architecture and their outer form, a change which depends on mechanical solicitations.Generally, this phenomenon is called either modelling or remodelling. These two notions areoften confused for obvious reasons. Nevertheless, modelling is used to describe alterationsin bone shape and remodelling involves a temporal change without any increase or decreasein bone mass.

1.3 Mechanostat

Thanks to Wolffs works, we know that bones structure change over time, trying toadapt to their surrounding environment. But, now, the problem is to know how this pro-

cess is possible and what activates and controls it.

These explanations are due to H.M. Frost who created the concept of a mechanostatto explain how the iterative process is controlled. Although this Mechnostat idea can beunclear and obscure, this concept is similar to a more familiar one, the thermostat.

As everyone knows, a thermostat is a component of a control system whose the aim isto maintain the temperature of a room at a desired value. The thermostat achieves its goalby switching heating and cooling devices on or off to maintain the correct temperature.

The Mechanostat concept is based exactly on the same principle. Instead of working

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on temperature, the mechanostat is sensitive to mechanical stimuli, particularly strains.This idea is similar to what J. Wolff described in his book but Frost went further andnoticed the existence of strain thresholds for which bone is created or resorbed. It is obvi-ous that these thresholds data provide a good definition for convergence for our iterativealgorithm. [Frost, 1998]

The figure 4 will help us to better illustrate this idea.

Figure 4: Threshold values of strain associated with the dead zone, bone resorption, the

lazy zone, and bone apposition, as well as the target strain [Phillips, 2011]

As we can see in the above figure, we notice the existence of a lower threshold fromwhich bones start to resorb. In the same idea, bone remodelling starts from a higher strainthreshold. Fracture and micro-damage also occurs at the highest strain values. Betweenthese two extremes is the lazy zone which describe a range of strain values for which thereis no change in bone structure.[Phillips, 2011]

In this project, we consider our target strain as 1250, which lies at the midpoint of thelazy zone. Our iterative algorithm converge until the maximum bone strain values reachthis target value. Once this value reached, we know that the structure of the femur does

not change and so the optimal cross-sections are found (see part Objectives and Methods).Other scientists have appropriate the Mechanostat concept to study bones and create

other models. For instance, Skerry noticed that it is not possible that there is a singlemechanostat controlling the skeleton of each of us. He considered that our skeleton containsa lot of bone units, each of them having its own mechanostat system. [Skerry, 2006]

Moreover, Frost kept working on this concept by doing several updates. [Frost, 2003]

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2 Bone Properties

In the previous part, we describe the concepts which are the basis of our iterationalgorithm. Now, we need a model to perform the algorithm. As we underline in theIntroduction, the femur is the bone we study during this dissertation.

The aim of this part is to provide a description of this particular bone. Some data fromthis part will be used to implement our algorithm.

A lot of information are issued from the book Grays Anatomy for Students. [Drakeet al., 2004]

2.1 Anatomy

The boundary conditions and the forces acting on the femur are due to its surrounding

environment, knowing muscles, joints and ligaments. Thats why it is important to mentionthem in this part.

2.1.1 Joints

The Hip Joint

The hip joint is a synovial articulation between the head of the femur and the acetabu-lum of the pelvus bone (figure 5). The joint is similar to a multiaxial ball enabling to dealwith weight bearing and stability issues. Movements at the joint include flexion, extension,abduction, adduction, medial and lateral rotation, and circumduction.[Drake et al., 2004]

Figure 5: Hip Joint A: Articular surfaces; B: Movement of the neck of the femur duringmedial and lateral rotation [Drake et al., 2004]

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The Knee Joint

The Knee Joint consists of the articulation of the femur and the tibia and of the ar-ticulation of the femur and the patella (figure 6). This joint can be considered as a hingepoint allowing for extension and flexion movements. It is consolidated by strong ligamentsin order to maintain it in position and stabilize the hinge-like motion of the knee.[Drakeet al., 2004]

Figure 6: Knee Joint [Drake et al., 2004]

2.1.2 The femur

The femur is the longest bone in the human body. It spans between the hip joint and

the knee joint, allowing the weight of the upper body to be transferred. It can be dividedinto two main regions: the proximal femur and the distal femur.

The proximal end is mainly characterized by a neck and a head. The head shape isroughly spherical and articulates with the acetabulum of the pelvic bone. The neck is acylindrical strut that connects the shaft and the head of the femur (figure 7). The radiusof the neck is approximatively equal to 24mm.

The femur shaft is bowed from the neck to the distal end of the bone. The middle partof the shaft is characterized by a smooth medial, lateral and anterior surfaces and features

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between a triangular and cylindrical cross section (figure 8). In the proximal and distalregions of the bone, the linea aspera widens to create an additional posterior surface.[Drakeet al., 2004] The average outer diameter of the shaft is approximatively equal to 27mm.

At the distal region of the femur, there are two large condyles, which form an articula-tion with the proximal head of the tibia (figure 8). Finally, epicondyles are bony elevationson the surfaces of the condyles. [Drake et al., 2004]

Figure 7: The proximal end of the femur [Drake et al., 2004]

Figure 8: The distal of the femur [Drake et al., 2004]

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2.1.3 The patella

The patella is not an ordinary bone by the fact that it lies within the tendon of a muscle

knowing the quadriceps femoris muscle (figure 9). It is characterized by a triangular crosssection.[Drake et al., 2004]

Figure 9: The patella [Drake et al., 2004]

2.1.4 Muscles

Muscles attached to the femur are arranged in three compartments, knowing the ante-rior, medial and posterior compartments (figure 10).[Drake et al., 2004]

x

Figure 10: Transverse section through the femur [Drake et al., 2004]

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The following tables issued from the book Grays Anatomy for Students describe themain muscles attached to the femur:

Anterior compartment(table 1)

Muscle Origin Insertion Function

Psoas MajorPosterior abdominalwall

Lesser trochanter of femurFlexes the thigh at the hip

joint

IlliacusPosterior abdominalwall

Lesser trochanter of femurFlexes the thigh at the hip

joint

Vastus Medialis

Femur medial part ofintertrochanteric line,pectineal line, mediallip of the linea aspera,medial supracondylarline

Quadriceps femoris tendon andmedial border of patella

Extends the leg at the kneejoint

Vastus IntermediusFemur-upper-two-thirds of anterior andlateral surfaces

Quadriceps femoris tendon andlateral margin of patella

Extends the leg at the kneejoint

Vastus Lateralis

Femur-lateral partof intertrochantericline, margin of greatertrochanter, lateralmargin of glutealtuberosity, lateral lipof the linea aspera

Quadriceps femoris tendonExtends the leg at the knee

joint

Rectus femoris

Straight head origi-nates from the anteriorinferior iliac spine;reflected head origi-nates from the ilium

just superior to the

acetabulum

Quadriceps femoris tendonFlexes the thigh at the hip

joint and extends the knee joint

SartoriusAnterior superior iliacspine

Anterior surface of tibia justinferomedial to tibial tuberos-ity

Flexes the thigh at the hipjoint and extends the knee joint

Table 1: Muscles of the anterior compartment of the femur [Drake et al., 2004]

Medial compartment(table 2)


Gracilis

A line on the externalsurfaces of the body ofthe pubis, the inferior

pubic ramus, and theramus of the ischium

Medial surface of proximalshaft of the tibia

Adducts thigh at hip joint andflexes leg at knee joint

PectineusPectineal line and adja-cent bone of pelvis

Oblique line extending frombase of lesser trochanter tolinea aspera on posterior sur-face of proximal femur

Adducts and flexes thigh at hipjoint

Adductor longus

External surface ofbody of pubis (tri-angular depressioninferior to pubic crestand lateral to pubicsymphysis)

Linea aspera on middle one-third of shaft of femur

Adducts and medially rotatesthigh at hip joint

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Adductor brevis

External surface ofb ody of pubis andinferior pubic ramus

Posterior surface of proximalfemur and upper one-third oflinea aspera

Adducts thigh at hip joint

Adductor magnus

Adductor part-

ischiopubic ramus

Posterior surface of proximal

femur, linea aspera, medialsupracondylar line

Adducts and medially rotates

thigh at hip joint

Obturator externus

External surface of ob-turator membrane andadjacent bone

Trochanteric fossa

Table 2: Muscles of the medial compartment of the femur [Drake et al., 2004]

Posterior compartment(table 3)


Biceps femoris

Long head-inferomedial partof the upper area ofthe ischial tuberosity;short head-lateral lipof linea aspera

Head of fibula

Flexes leg at knee joint; ex-tends and laterally rotatesthigh at hip joint and laterallyrotates leg at knee joint

Semitendinosus

Inferomedial part ofthe upper area of theischial tuberosity

Medial surface of proximaltibia

Flexes leg at knee joint and ex-tends thigh at hip joint; medi-ally rotates the thigh at the hip

joint and leg at the knee joint

Semimembranosus

Superolateral impres-sion on the ischialtuberosity

Groove and adjacent bone onmedial and posterior surface ofmedial tibial condyle

Flexes leg at knee joint and ex-tends thigh at hip joint; me-dially rotates thigh at the hip

joint and leg at the knee joint

Table 3: Muscles of the posterior compartment of the femur [Drake et al., 2004]

Other muscles in the gluteal region must be taken into consideration. So, the piriformis,

the quadratus femoris and the gluteus can be quoted as important muscles that must beput in the model.

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Figure 11: Muscles of the anterior compartment of the femur [Drake et al., 2004]

Figure 12: Muscles of the medial com-partment of the femur [Drake et al., 2004]

Figure 13: Muscles of the posterior com-partment of the femur [Drake et al., 2004]

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2.2 Bone Mechanical properties

After having described the femur and its surrounding environment, we now focus on

the bone mechanical properties knowing Youngs modulus, Poissons ratio...This aspect of bone properties is important for our analysis to implement the mechan-

ical parameters of the algorithm.

Long bones are generally composed of an intern cellular component and an extracellularstiff matrix, respectively the trabecular and the cortical bones. These two materials areliving organs and so their mechanical behaviours are difficult to predict and to describesince they depend on some parameters.

As we saw in the first part of this Literature Review, the bone structure is highly depen-dent on the forces acting on it [Wolff, 1987]. Following this idea, the mechanical propertiesof trabecular bone must be dependent on anatomic location and function. Studies demon-strate this idea by cataloguing the distribution of trabecular bone material properties frommultiple locations in the human body[Goldstein, 1987]. Moreover, statistical analyses ofthese mechanical properties were first carried out by Dr Evans et al., who mentioned a cor-relation between the physical properties of trabecular bones and their anatomic location[Evans and King, 1961].

The mechanical properties depend also on age. Over life and with the repetition ofloading cycles, change in mechanical behaviour of bone tissue with increasing age has beennoticed in several studies. Indeed, Melick et al. mentioned, for instance, that the tensile

strength of an adult femoral cortex decreases by 4 per cent per decade [Melick and Miller,1966].

Although this issue has been studied in a lot of articles, it seems that the observationsare unanimous. Likewise, Burstein et al. noticed consistent decreases for the main bonemechanical properties with age and underlined the fact that the tibia and the femur donthave the same response to the ageing process. This observation is due to the fact thatthese two bones are not subjected to the same forces over time [Burstein et al., 1976].

If the femur is the studied bone, as mentioned earlier, the trabecular bone at theproximal end has a specific architecture as a result of an anisotropic and heterogeneousdistribution of bone tissues in order to withstand the forces in an optimal way. Singh etal. noticed the presence of five normal groups of the trabecular bone in the upper end ofthe femur whose the spatial arrangement ensures maximum strength with the availablematerial [Singh et al., 1970].

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Figure 14: Diagrammatic representation of the five normal groups of the trabecular bonein the upper end of the femur [Singh et al., 1970]

Moreover, unlike a lot of bone modelling algorithm assume, it seems that bone is notisotropic. Miller et al. noticed that the trabecular bone is effectively orthotropic that ex-plained the internal architecture of the trabecular bone we described earlier [Miller et al.,2001]. Likewise, Huiskes et al. mentioned the fact that the cortical bone behaves as atransversely isotropic material [Huiskes et al., 1981].

In the same way, the bone mechanical coefficients knowing Youngs Modulus and Pois-

sons ratio depend also on some parameters. For instance, Rho et al. noticed that thereare a correlation between these coefficients and the density, porosity and architecture ofboth cortical and cancellous bone [Rho et al., 1995].

Turner et al. determined approached values of these mechanical coefficients using acous-tic and nanoindentation. It was found that the Youngs Modulus of femoral trabecular boneis between 17.5 and 18.14 GPa and between 17.73 and 20.02 GPa for femoral cortical bone[Turner et al., 1998].

In our study, as we only deal with the cortical bone, we will use the cortical bonemechanical coefficients mentioned and used by Phillips knowing E = 18GPa and = 0.3

[Phillips, 2009]

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2.3 Muscle loads

In this part, we are interested to describe the main forces that we have to apply on the

femur.As is the case for any bone modelling algorithms, the muscles loads and the joint con-

tacts at the hip, the patella and the knee represent the boundary conditions, essential forsolving the problem.

As we mentioned earlier, the thigh bone is surrounded by a lot of muscles, each ofthem applying more or less loads depending on the course of regular activity. Obviously,the magnitude and the directions of the forces applying on the femur are expected to bedifferent during staircase ascent and during a walking pace. For instance, Bergmann et al.mentioned the fact that the hip joint of an average patient is loaded with 238% of the bodyweight when walking at about 4 km/h and with 251% of the body weight when climbingupstairs [Bergmann et al., 2001].

Figure 15: Comparison of thecomplex (left) and most simpli-fied model of the hip musculature[Heller et al., 2005]

The number of muscles attached on the femur isthe issue here and it is obviously difficult to predictthe magnitude and the direction of the forces appliedby them during the loading cycles (walking pace, stair-case ascent...). To deal with this problem, Heller etal. suggested a simplification by grouping function-ally similar hip muscles, leading to a reduction in thenumber of the muscles that we consider in the analysis

(figure 15) [Heller et al., 2005] .Likewise, in an attempt to simplify the muscle loads

modelling, we notice that some studies only considera set of three main forces to describe what happens atthe proximal end of the femur: the abductor, iliotibialtract and joint contact force [Phillips, 2011].

In our case, we will use the muscle and contactforces obtained by an adaptive model of the lower legstudied by Luca Modenese, a PhD student at the Im-perial College of London. This model provides theforces throughout different activities (gait, stair ascentand sit to stand) [Modenese et al., 2011].

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2.4 Boundary conditions

The introduction of boundary conditions in any modelling algorithms is essential to

solve the problem by the way that it allows the reduction of the problem degrees of freedom.Nevertheless, for two different boundary conditions, the result can be significantly dif-

ferent and thats why it represents a critical issue in the analysis. For instance, Speirs etal. noticed that the boundary conditions applied vary widely in studies of bone loading,leading generally to excessive femoral deformations [Speirs et al., 2007].

The objective is to use boundary conditions which represent the in-vivo conditions.Speirs et al. studied five boundary condition cases applied to a Finite Element model ofthe femur and compared the deflections found for each case to the physiological one. Theyconcluded on the fact that the use of physiological boundary conditions, consisting offixing the centre of the condyles in the three spatial directions and forcing the femoralhead to move along an axis towards the knee center and fixing a node on the distal lateralepicondyle, presents the best results [Speirs et al., 2007].

Figure 16: Boundary conditions applied to the diaphysis (left), the condyles (center) andthe joint nodes (right)

On the other hand, Phillips proposed a model in which the fixed constraints from thedistal femur are replaced by an array of muscles to control displacements . The author used

a set of 26 muscles and 7 ligamentous structures via spring elements to create a completemusculoskeletal system between the hip and knee joints. The found results matched closerto in-vivo data [Phillips, 2009].

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2.5 Current models

Bones structure is studied by means of computational and experimental approaches.

Experimental studies often consist of a series of experiments carried out on real bonessubjected to various load cases. Generally, these studies are favoured to deal with issuesthat computational approaches cannot solve [Szivek et al., 2000].

Figure 17: The meso scaleapproach [Phillips, 2011]

Most bone modelling algorithms use finite elements anda continuous modelling approach to predict the structure ofbone. The results provided by these methods are satisfac-tory but depend a lot on the assumptions done (boundaryconditions, contact and muscle loads considered...).

We notice two types of finite element approaches. Thefirst one is the macro-scale approach and is characterised bythe use of solid elements to represent bone, which are largerthan the individual structural elements in the bone. The sec-ond one is the micro-scale approach where the elements usedare smaller than the individual structural elements [Phillips,2011].

These two methods present some advantages. The macro-scale approach provides results in an efficient computationaltime and the micro-scale one provides accurate results.

However, they also have some disadvantages. Macro scale

analyses are based on several simplified hypotheses and as-sume, for instance, that the cortical structure is an isotropicmaterial. Micro scale ones are computationally expensive[Phillips, 2011].

In order to benefit from the advantages of these two ap-proaches, Phillips suggested a meso scale approach, wherebystructural elements are used to represent trabecular and cor-tical elements. Thanks to that, even if the elements used arelarger than the individual structural ones, the overall struc-tural behaviour of the femur can be captured [Phillips, 2011].

In the other hand, beam theories have been used in order to study the architecture ofbone. This supposes the specification of geometrical and mechanical properties of femoralsections. For a given human femur, Huiskes et al. compared experimental strain-gaugeand theoretical stress analysis methods based on the beam theory. It was concluded thatthe modelling of the cortical bone as a beam is a good approximation [Huiskes et al., 1981].

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This dissertation revisits the macro-scale approach to speed up the computational pro-cess while providing correct correlation to the actual femur.

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Part IV

Method

To study and predict the structure of bones, we use, in this dissertation, an algorithmbased on the beam theory and the 3D stiffness method, and the Mechanostat conceptdescribed in the Literature Review.

The 3D stiffness method allows us to calculate the displacements, the principal strainsacross the elements representative of the cross sections of the femur. The Mechanostatserves us as a basis of the iteration process in order to find the most suitable femoral

sections that correspond to the problem.This method has been a very little used since no article was found explaining or relating

this approach of Bone Modelling.

The aim of this part is to explain how the algorithm works and to describe the modelconsidered to run it. The algorithm is written in MATLAB.

3 Theory

3.1 Beam Theory

With this approach, the femur is divided into several elements. Each element is con-sidered as acting as a beam. Throughout this study, we assume that each element has anisotropic behaviour and we neglect the shear deformation, so that we only consider theEuler-Bernoulli beam bending theory.

The Euler-Bernoulli beam theory covers the case for small displacements and, so, isbased on several assumptions.

First, we assume that, for any loads applied on the element, the plane sections remainplane during deformation. Secondly, the deformed plane remains perpendicular to theneutral axis.

Figure 18: The Euler-Bernoulli beam theory and its assumptions

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In three dimensions, each node of an element is characterised by six degrees of freedomknowing three displacements (ux, uy and uz) and three rotations (x, y and z) where(x,y,z) is the local coordinate system of the element.

By convention, the z-axis is the minor bending axis and the y axis, the major one.

In the local coordinate system of the beam (figure 19), the displacements of a point,whose the coordinates are (x, y, z), belonged to the beam section are equal to:

Ux(x, y, z) = ux(x

) y z(x) + z y(x

)

Uy(x, y, z) = uy(x

) z y(x)

Uz(x, y, z) = uz(x

) + y z(x)

So, we deduce the strain components:

xx(x, y, z) =

ux

x(x) y

z

x(x) + z

y

x(x) =

axialstrain ux

x(x)

bendingstrain y z + z

y

(1)

yy(x, y, z) =

ux

x(x) z

y

x(x) =

ux

x(x) z y (2)

zz (x, y, z) =

ux

x(x) + y

z

x(x) =

ux

x(x) + y z (3)

y (resp. z) is the curvature around the y-axis (resp. the z-axis).

In this study, we are only interested in the xx-component of the strain tensor.

Figure 19: The local coordinate system (x,y,z) of the beam element

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3.2 The 3D stiffness method

To solve the problem, we use the 3D stiffness method. This method is a very used

method to compute the nodal displacements of any mechanical models.To do that, the model must be divided into several simple beam elements interconnected

at the nodes.Knowing the nodal forces and the geometrical and mechanical properties of the ele-

ments, the nodal displacements can be found.

3.2.1 The local stiffness matrix

We first consider the local coordinate system of a beam element as we described in theprevious section.

The two nodes have six degrees of freedom and the element is characterised by its geo-metrical parameters:

the length of the element, Le the area of the element section, Ae

It is also characterised by its mechanical parameters:

the Youngs Modulus, E the Shear Modulus, G

the second moment of inertia about the y-axis, Iy the second moment of inertia about the z-axis, Iz the torsion constant of the element, J

All these parameters are computed into a single matrix (12 12) called the stiffnessmatrix. In fact, this matrix is the characterisation of the geometrical and mechanicalproperties of an element. The element stiffness matrix is defined in the local coordinatesystem of the beam element.

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This matrix for one element Klocale is the following:

AeE

Le 0 0 0 0 0

AeE

Le 0 0 0 0 0

012EIz

L3e0 0 0

6EIz

L2e0

12EIz

L3e0 0 0

6EIz

L2e

0 012EIy

L3e0

6EIy

L2e0 0 0

12EIy

L3e0

6EIy

L2e0

0 0 0JG

Le0 0 0 0 0

JG

Le0 0

0 0 6EIy

L2e0

4EIy

Le0 0 0

6EIy

L2e0

2EIy

Le0

06EIz

L2e0 0 0

4EIz

Le0

6EIz

L2e0 0 0

2EIz

Le

AeE

Le0 0 0 0 0

AeE

Le0 0 0 0 0

0 12EIz

L3e0 0 0

6EIz

L2e0

12EIz

L3e0 0 0

6EIz

L2e

0 0

12EIy

L3e 0

6EIy

L2e 0 0 0

12EIy

L3e 0

6EIy

L2e 0

0 0 0 JG

Le0 0 0 0 0

JG

Le0 0

0 0 6EIy

L2e0

2EIy

Le0 0 0

6EIy

L2e0

4EIy

Le0

06EIz

L2e0 0 0

2EIz

Le0

6EIz

L2e0 0 0

4EIz

Le

Because the assumed behaviour of the mechanical system is linear, the equations al-lowing to link the nodal forces and displacements must be linear. Furthermore, if all thedisplacements vanish, so do the forces (assuming that there is no prestress or no initialstrain).

So, the equilibrium equation of an element is Fe = KeUe(4), where Fe is the nodal

forces matrix and Ue is the nodal displacements matrix.To better describe these two last matrix, we consider an element e with its two nodes

1 and 2. The nodal displacements matrix and the nodal forces matrices are

UTe =x1; y

1; z

1; x 1; y 1; z 1; x

2; y

2; z

2; x 2; y 2; z 2

FTe =Fx 1; Fy 1; Fz 1; Mx 1; My 1; Mz 1; Fx 2; Fy 2; Fz 2; Mx 2; My 2; Mz 2

where (x, y, z) is the local coordinate system of the considered element.

3.2.2 Transformation matrix from the local coordinate system to the global

one

For each element, the nodal forces are generally known in the global coordinate systemand the stiffness matrix is written in the local one.

The equation (4) is only correct when the nodal forces and the nodal displacementsmatrices are written in the same coordinate system. So, following this idea, in the localcoordinate system of the element,

Flocale = Klocale U

locale

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To calculate the nodal displacements in the global coordinate system, a system wherethe nodal forces are known, a 3D-transformation matrix Re from the global system to thelocal system is needed for each element.The rotation matrix Re is defined so that:

Flocale = Re Fglobal

e and Ulocale = Re U

globale

With that, the stiffness matrix in global coordinates is obtained as:

Kglobale = RTe K

locale Re

The transformation matrix is a square matrix (12 12) and is characterized as orthog-onal matrices with determinant 1. Considering the element stiffness matrix, the transfor-mation one can be written into the following form:

r 0 0 0

0 r 0 00 0 r 00 0 0 r

where r is the direction cosines matrix associated with the x axis (the axis of the beamelement):

Cxx Cyx CzxCxy Cyy CzyCxz Cyz Czz

and

CXx = cosxx

where angles xx, yx and zx are measured from global axes x,y and z with respect to thelocal axis x, respectively (figure 20) .

Figure 20: The direction cosines

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The next step is to determine the direction cosines from the global coordinates of thenodes of the model.

We consider a beam element with its two nodes called node1 and node2. The globalcoordinates of node1 are (x1, y1, z1) and those of node2 are (x2, y2, z2).

First, consider the two following particular cases where x1 = x2 and y1 = y2.If z2 > z1, the direction cosines matrix is

0 0 10 1 01 0 0

If z1 > z2, the direction cosines matrix is

0 0 10 1 01 0 0

In a more general case, we first consider the rotation with respect to the local axis x of theelement. In this case, the direction cosines are:

Cxx = cosxx =x2 x1

Lewhere Le is the length of the element

Cyx = cosyx =y2 y1

Le

Czx = coszx =z2 z1

Le

The rest of the direction cosines is deduced as follows:

Cxy = cosxy = Cyx

Dwhere D =

Cxx Cxx + Cyx Cyx

Cyy = cosyy =Cxx

D

Czy = coszy = 0 by convention

Cxz = cosxz = Cxx Czx

D

Cyz = cosyz = Cyx Czx

D

Czz = coszz = D

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3.2.3 Computing the global nodal displacements

At this stage of the analysis, a global stiffness matrix for each element is created by

using a transformation matrix.To find the nodal displacements, the overall mechanical system need to be considered

and so, the general matrices of the system must be calculated. To do that, the total num-ber of nodal degrees of freedom N and the degree of kinetic indeterminacy n, must bedetermined. This is done by considering the system boundary conditions. The number ofsupported or restrained degrees of freedom is N-n.

The key operation of the assembly process is the placement of the contribution of eachmember to the master equation representative of the equilibrium of the overall system.To deal with this issue, the nodes, the elements and the degrees of freedom of the overallsystem are numbered and classified in order to get a partition into three subsets:

1. Unknown displacements (L)

2. Displacements that are known and that are not equal to zero (P)

3. Displacements that are known and that are equal to zero (S)

The first subset has n terms and the two other ones have together N-n terms .

This classification depends on the boundary conditions of the overall system. The par-tition of the degrees of freedom of the system induces a partition of the general matrices:nodal forces matrix, stiffness matrix and nodal displacements matrix:

F =

[FL][FP]

[FS]

and U =

[UL][UP]

[US]

and K =

[KLL] [KLP] [KLS][KP L] [KP P] [KP S]

[KSL] [KSP] [KSS]

The general matrices of the system will be written in bolt in all the report.

As all the equations used are linear, the rows and the columns associated with zerodisplacements are never taken into account into the general matrices. So, the equilibriumequations are the following:

[FL][FP] = [KLL] [KLP][KP L] [KP P] [UL][UP] The unknown displacements are the solutions of the following equation systems:

[KLL] [UL] = [FL] [KLP] [FP]

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3.2.4 Calculation of the xx strain value

Knowing the nodal displacements in the global coordinate system, the transformation

matrix defined previously is used to determine these displacements in the local coordinatesystem of each element.

For each element e with its two nodes 1 and 2, the nodal displacements in the globalsystem are considered in the following form:

UTe g lobal =x1; y1; z1; x 1; y 1; z 1; x2; y2; z2; x 2; y 2; z 2

where (x,y,z) is the global coordinate system.

The nodal displacements in the local system of the element e are given by:

Ue local = Re Ue global

where: UTe local =x1; y

1; z

1; x 1; y 1; z 1; x

2; y

2; z

2; x 2; y 2; z 2

and (x,y,z) is the local coordinate system of the beam element e.

To determine the xx strain value for each element e of the system, the equation 1 isused.

xx(x, y, z) =

ux

x(x) y

z

x(x) + z

y

x(x)

To calculate the derivative functions from the nodal local displacements, a finite differ-ence method is also used. Hence, the terms of the above equation are calculated for eachelement as follows:

ux

x(x) =

x2 x

1

Lewhere Le is the length of the element e

y

z

x (x

) = y

z 2 z 1Le z

y

x(x) = z

y 2 y 1

Le

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4 Model

In this part, we will describe the model used during the dissertation. The geometry,the mechanical and geometrical parameters, the boundary conditions and the load caseswill be mentioned.

4.1 Geometry

To determine the geometry of the femur considered in this study, a 3D Abaqus modelof an actual femur was used in order to get the exact dimensions of the bone. It is based ona CT scan of a Sawbones fourth generation composite femur and provided by Dr Phillips[Phillips, 2011].

Based on that, a line passing through the center of each part of the femur is plotted as

illustrated in the following pictures (figure 21).

Figure 21: A line, in red, is plotted passing through the center of each part of the femur

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The equation of each part of this line is computed in order to define the geometry ofthe femur. This is illustrated in the following graphs in three dimensions (figure 22):

Figure 22: The geometry of the femur considered during this dissertation, plotted in theglobal coordinate system

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Every portion will be divided into several elements to match with beam theory and the3D stiffness method described previously.

4.2 Mechanical and Geometrical properties

In this study, it is important to recall the fact that we only consider the cortical bone,and thus we neglect the influence of the trabecular bone on the mechanical behaviour ofthe femur. It is obvious that this assumption will have to be taken into consideration whenthe results are discussed.

For the cortical bone, as mentioned in the Literature Review, the Youngs Modulus istaken as equal to E = 18 GPa and the Poissons ratio equal to = 0.3. The Shear Modulus

is then calculated with the following formula: G =E

2 (1 + )

.

Concerning the geometry of the bone sections, hollow elliptical sections are considered.Even if we mentioned in the Literature Review that the femoral sections feature between atriangular and cylindrical cross section, an hollow elliptical shape is a good approximation.

The mechanical and geometrical parameters of the hollow ellipse drawn in the figure23 are:

Ix =

4 b +t

2a +t

23

4 bt

2 at

23

Iy =

4

b +

t

2

3

a +

t

2

4

b

t

2

3

a

t

2

Area =

a +

t

2

b +

t

2

a

t

2

b

t

2

Perimeter =

2

a +t

2

2+

b +

t

2

2

(a b)2

2.2

2

at

2

2+

b

t

2

2

(a b)2

2.2

J=2 (2tab)2

t Perimeter

For non symmetric sections as this is the case for the sections considered during this

study, local and principal axes dont always coincide. So, they may be subjected to asym-metric bending since the term Ixy doesnt vanish.

However, all the above formula and the mechanical equations (1) to (3) are only validwhen the hollow ellipse is in canonical position (center at origin, major axis along they-axis...) as it is illustrated in the figure 23. In this case, the local and the principal axescoincide and the term Ixy vanishes.

So, when we consider all the above formula, we have to make sure that the local andthe principal axes of the hollow ellipse coincide.

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Figure 23: An hollow ellipse

Moreover to calculate the strain at any point of the section by using the equation 1, itcan be useful to determine the coordinates of a point M(x,y) located at the outer surface ofthe hollow ellipse (figure 23). Taking the angle as a general parameter, the coordinatesof M are equal to:

x =

b +

t

2

cos() (5)

y= a +

t

2 sin() (6)This parametrisation is also only valid when the local and the principal axes of the

hollow ellipse coincide as it is the case for the one represented in the figure 23.

4.3 Muscle and ligament loading

In this study, 30 muscles are considered to act on the femur. These muscles are rep-resented by a series of 115 point loads which correspond to the attachment points of themuscles.

The aim of this representation is to include in the analysis as muscles as possible, so as

to create a more physiologically correct set of loadings, as we mentioned in the LiteratureReview (part 2.3).

The muscle loads are obtained by an adaptive model of the lower leg studied by LucaModenese, based on the research of Klein Horsman et al. (2007) [Modenese et al., 2011].A total of 30 muscles is considered and, as some muscles act over large areas of the bonessurface, 115 attachment points are also considered to deal with this problem.

Using the same model, the knee, hip and patella joints are also calculated as well asthe inertia and weight forces.

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In order to integrate the muscle loads and the attachments sites to our model, we usebeam elements to join the attachment points to the femur as illustrated in the figure 24.

Figure 24: The complete model: in red, the femur; in blue, the rigid elements linking themuscle attachment points to the femur

In order to transfer the loads to the femur and not to interfere in the mechanical be-haviour of the femur, the beam elements linking the attachment points to the femur mustbe extremely rigid. So, we decide that the Youngs Modulus of these elements is equal to18000 GPa, which is one thousand times the Youngs Modulus of the femoral cortical bone.

We must mention that a too high value of the Youngs Modulus of these elements

compared to the elements that form the femur, can lead to a singular stiffness matrix.

Figure 25: The modelling of the hip joint (left) and the patella joint (right)

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4.4 Load cases

During this study, we analyse three frequent human activities: walking, stair climbing

and sitting-to-standing movements. For each of these activities, the femur is subjected todifferent forces.

Considering all these cases allows us to optimise the modelling of the femur. Indeed,as the structure of the femur depends strongly on the loads acting on it, it is appropriateto consider as many load cases as possible to produce an optimised model of the femur.

Lucas model divides each of these three motions into two hundred frames, each framerepresenting a load case (30 muscles loads, 115 attachment points, 3 joint loads for eachframe).

During this dissertation, it was time consuming to consider, in the model, all the framesfor each of the three motions. It is decided to consider a total number of frames equal to68 with 28 frames for the walking case, 20 frames for the stair climbing case and 20 framesfor the sit-to-stand one.

A discussion about the dependence of the model on the number of frames consideredwill be done in the part 6.3.

A example of one load case with all the muscle loads considered is given in the tables4 and 5 and illustrated in the figure 26.

Figure 26: The representation of the frame number 22 of the walking case: in purple, themuscle loads

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Table 4: Muscle forces for the frame number 22 of the walking case, corresponding tot=0.047s Part 1)

Muscles Fx(N) Fy(N) Fz(N)ADD BREV PROS 1 0.133 0.340 0.696ADD BREV PROS 2 0.121 0.372 0.665

ADD BREV MID 1 0.056 0.408 0.573ADD BREV MID 2 0.059 0.436 0.544

ADD BREV DIST 1 -0.002 0.442 0.464ADD BREV DIST 2 -0.007 0.462 0.437

ADD LONG 1 0.083 0.831 0.486ADD LONG 2 0.070 0.836 0.473ADD LONG 3 0.051 0.847 0.449ADD LONG 4 0.037 0.856 0.426ADD LONG 5 0.022 0.866 0.403ADD LONG 6 0.007 0.872 0.385

ADD MAG DIST 1 -0.644 3.234 0.099

ADD MAG DIST 2 -0.605 3.235 0.125ADD MAG DIST 3 -0.496 3.243 0.203ADD MAG MID 1 -0.398 1.325 0.320ADD MAG MID 2 -0.381 1.338 0.307ADD MAG MID 3 -0.336 1.333 0.302ADD MAG MID 4 -0.323 1.342 0.291ADD MAG MID 5 -0.286 1.336 0.291ADD MAG MID 6 -0.277 1.342 0.284

ADD MAG PROX 1 -0.121 0.346 0.363ADD MAG PROX 2 -0.149 0.416 0.265ADD MAG PROX 3 -0.113 0.426 0.256ADD MAG PROX 4 -0.087 0.436 0.238

BIC FEM CB 1 -0.770 -12.245 -0.811BIC FEM CB 2 -0.997 -11.301 -1.330BIC FEM CB 3 -1.279 -9.505 -1.736GASTR LAT 1 -7.581 -4.618 -0.585

GASTR MED 1 -136.696 -50.090 -23.351GEM INF 1 -3.010 2.506 3.513GEM SUP 1 -1.204 2.565 2.033

GLUT MAX SUP 1 2.018 8.074 3.547GLUT MAX SUP 2 0.818 12.425 5.171GLUT MAX SUP 3 -2.115 16.532 6.699GLUT MAX SUP 4 2.318 7.412 3.244GLUT MAX SUP 5 1.632 11.703 4.850GLUT MAX SUP 6 -0.706 16.207 6.535GLUT MAX INF 1 -0.495 1.950 1.374GLUT MAX INF 2 -0.414 1.657 1.030GLUT MAX INF 3 -0.402 1.449 0.785GLUT MAX INF 4 -0.387 1.954 1.335GLUT MAX INF 5 -0.331 1.683 1.009GLUT MAX INF 6 -0.352 1.478 0.775

GLUT MED ANT 1 2.043 0.903 0.804GLUT MED ANT 2 2.077 0.758 0.844GLUT MED ANT 3 2.102 0.576 0.911GLUT MED ANT 4 2.068 0.893 0.732GLUT MED ANT 5 2.079 0.794 0.798GLUT MED ANT 6 2.076 0.667 0.905

GLUT MED POST 1 2.822 2.384 1.210GLUT MED POST 2 2.534 3.270 2.072GLUT MED POST 3 2.947 5.796 4.433GLUT MED POST 4 2.754 2.515 1.217GLUT MED POST 5 2.489 3.485 2.241GLUT MED POST 6 3.015 5.846 4.271

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Table 5: Muscle forces for the frame number 22 of the walking case, corresponding tot=0.047s (Part 2)

Muscles Fx(N) Fy(N) Fz(N)GLUT MIN ANT 1 2.592 2.100 1.723GLUT MIN MID 2 1.668 2.272 1.756

GLUT MIN POST 3 1.241 3.025 2.431ILIACUS LAT 1 0.711 0.594 0.437ILIACUS LAT 2 0.711 0.594 0.437ILIACUS LAT 3 0.711 0.594 0.437ILIACUS MID 1 1.382 1.154 0.849ILIACUS MID 2 1.382 1.154 0.849ILIACUS MID 3 1.382 1.154 0.849

ILIACUS MED 1 0.817 0.682 0.502ILIACUS MED 2 0.817 0.682 0.502ILIACUS MED 3 0.817 0.682 0.502

OBT EXT SUP 1 0.661 -0.654 3.181OBT EXT SUP 2 0.673 -0.666 3.236OBT EXT SUP 3 0.693 -0.686 3.334

OBT INT 1 -6.979 7.784 11.718OBT INT 2 -6.979 7.784 11.718OBT INT 3 -6.979 7.784 11.718

PECTINEUS 1 0.264 0.415 0.547PECTINEUS 2 0.231 0.411 0.548PECTINEUS 3 0.194 0.430 0.528PECTINEUS 4 0.233 0.307 0.610PIRIFORME 1 -2.057 10.337 11.083PLANTARIS 1 -0.690 -0.606 -0.025POPLITEUS 1 -1.005 -1.473 0.899POPLITEUS 2 -1.005 -1.473 0.899PSOAS MAJ 1 2.042 1.705 1.255PSOAS MAJ 2 2.042 1.705 1.255PSOAS MAJ 3 2.042 1.705 1.255QUAD FEM 1 -0.786 0.366 2.469QUAD FEM 2 -0.780 0.743 2.381

QUAD FEM 3 -0.835 1.044 2.070QUAD FEM 4 -0.867 1.271 1.633

VASTUS INTERM 1 0.697 -2.235 0.219VASTUS INTERM 2 0.536 -2.276 0.241VASTUS INTERM 3 0.438 -2.293 0.277VASTUS INTERM 4 0.612 -2.252 0.282VASTUS INTERM 5 0.468 -2.288 0.270VASTUS INTERM 6 0.401 -2.300 0.274VASTUS INTERM 7 0.831 -2.199 -0.032VASTUS INTERM 8 0.875 -2.166 0.266VASTUS LAT INF 1 0.400 -0.523 0.062VASTUS LAT INF 2 0.290 -0.591 0.062VASTUS LAT INF 3 0.144 -0.638 -0.101VASTUS LAT INF 4 0.166 -0.639 -0.037VASTUS LAT INF 5 0.168 -0.638 0.051VASTUS LAT INF 6 0.146 -0.639 0.089

VASTUS LAT SUP 1 1.643 -10.651 1.740VASTUS LAT SUP 2 1.201 -10.791 1.135VASTUS MED INF 1 0.966 -1.227 0.924VASTUS MED INF 2 1.152 -1.319 0.476

VASTUS MED MID 1 1.639 -3.631 1.602VASTUS MED MID 2 1.843 -3.873 0.183VASTUS MED SUP 1 0.547 -1.517 0.395VASTUS MED SUP 2 0.467 -1.590 0.103VASTUS MED SUP 3 0.432 -1.554 0.392VASTUS MED SUP 4 0.364 -1.613 0.145VASTUS MED SUP 5 0.299 -1.625 0.161VASTUS MED SUP 6 0.299 -1.625 0.161

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4.5 Boundary conditions

As we mentioned in the Literature Review (part 2.4), the choice of the boundary con-

ditions applied to the system has an important influence on the mechanical results. Wenoticed that fixing the knee joint (localised at the centre of the condyles) in three spatialdirections and forcing the femoral head to move along an axis towards the knee center andfixing a node on the lateral epicondyle present the best results [Speirs et al., 2007].

In this study, we decide to fix only the knee joint in its six degrees of freedom. Adoptingthis boundary condition, the mechanical problem is entirely determined.

Even if this boundary condition is not the most physiological one, it has the advan-tage to be easy to implement in the algorithm. A further study about the influence of theboundary conditions on the model can be interested in the future.

The boundary condition considered in the model is illustrated in the figure 27.

Figure 27: The boundary condition at the knee joint considered in the dissertation

The reactions forces and the moments at the knee joint will be computed in order to becompared to those given by Lucas model. This comparison will allow us to explain someresults.

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5 The iterative algorithm

5.1 Presentation and objective of the algorithmThe iterative algorithm is used to calculate the optimal femoral sections and it is based

on the Mechanostat Concept described in the Literature Review (part 1.3).According to the Mechanostat concept, we notice the existence of strain thresholds for

which bone is created or resorbed (figure 4). In the lazy zone between 1000 and 1500,there is no change in bone structure and we can consider that the structure of the bonewhose the strain values lie in this zone, is stable and wont change. Bones are only sensitiveto the highest mechanical solicitations and stimuli, and that why we only work with themaximum strain values.

The objective of the iterative algorithm is to optimise the femoral cross sections in

order that the absolute maximum strain values of all the cross-sections due to various loadcombinations lie in the lazy zone. In all this dissertation, we work with absolute strainvalues.

To optimise the hollow elliptical cross-sections, we consider four variables to parametertotally the problem. Each cross-section is characterised by two radius R1 and R2, by aconstant thickness t and by an angle (figure 28).

Figure 28: Four variables are chosen to parameter totally the cross-section

The algorithm will change the values of these four variables for each cross-section ofthe model in order that its absolute maximum strain value lies in the lazy zone at the endof the analysis.

Nevertheless, if we analyse the problem carefully, there are four variables to optimiseas the only constraint of the problem is the strain value for each element. Because of that,

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there is not an unique solution but several couples (R1, R2, , t) can satisfy the problem, aproblem that becomes relatively sensitive to the initial conditions considered.

Hence, this problem seems to be undetermined and some assumptions need to be takenin order to solve it. Each variable will be optimised independently step-by-step in order toconverge to a solution. In order to make sure that the solution is the optimal one for ourspecific problem, we limit the values taken by the variables based on the actual geometryof the femur. For instance, the thickness value of a cross-section must lie in an interval[0.1, maxt] where maxt is defined by us and reflects the maximum thickness value of anactual femur. Likewise, the radius parameters must lie in an interval defined by two con-stants minR and maxR so that: minR R1, R2 maxR.

The algorithm has changed several times during the dissertation and we only describeits general idea in the following. All the mechanical calculations (nodal displacements,

strain values...) are performed by the 3D stiffness method we described previously.

5.2 Optimisation of the angle

We begin the iterative algorithm by optimizing the angle of the hollow ellipse. Tooptimise this parameter, we make a common assumption described in some articles. Weassume that the orientation of maximum bending rigidity reflects the orientation of peakor habitual bending forces the bone experiences [Lieberman et al., 2004]. In other words,the principal bending axes of the section are orientated according to the maximum bendingstrain value reached by the section due to the surrounding forces. By making sure thatthe maximum strain value of the section is located at the minor principal bending axis, we

assume that the bone section is oriented in such a way that it can withstand the forces inan optimal way.

Figure 29: Optimisation of the angle. (y,z) is the local coordinate system of the sectionand (y1,z1) is the bending principal axes of the section. At the end, the maximum strainvalue is located at the minor principal bending axis.

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In the literature, some experimental studies show that this assumption is not reallycorrect for long bones as illustrated in the figure 30. Demes carried out experimentalstudies on the tibia. She calculated the localization of both the Zero Strain neutral axisand the maximum principal axis and she determined the angle between the two axes.[Demes, 2007]. The assumption we make to optimise states that these two axes coincidewhich is not always the case.

Nevertheless, the differences between the experimental results and those found by con-sidering the assumption are not really important (between 10 and 20 depending on theload cases that are considered). Moreover, it represents the only way to fix the angularparameter.

Figure 30: Difference between the localization of the maximum principal axis and the Zero

strain neutral axis [Demes, 2007]

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The iterative algorithm consists in making sure that the maximum strain value calcu-lated for a section is located at the minor principal bending axis.

We initialise the algorithm with R1

= 13mm, R2

= 15mm and t = 5mm, making they-axis, the major principal bending axis.

The following diagram shows the working of the algorithm to optimise the angle of thehollow ellipse:

For each element, this process is repeated for all the load cases we want to consider.So, at the end, for each element, there are N maximum bending strain values for N framesconsidered. We choose the maximum of these values and we rotate the element to theangle associated.

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5.3 Optimisation of the three geometrical parameters R1, R2 andt

Once the angle for each element is determined, the sections are rotated once and forall (see figure 31), so that all the equations described in the parts 4.2 and 3.2.4 can beapplied without any problem. The local and the principal axes coincide.

The objective is now to determine the three variables so that, when all the load casesare considered, the maximum strain value reached by each element lies in the lazy zone.As we notice previously, this problem is undetermined because we have to deal with threeparameters and we have only one constraint.

To find a solution, we optimise step-by-step these three variables independently. Thegeneral idea of the algorithm is described in the table 6.

The stage 5 of the algorithm (see table 6) is an important one. To explain it, we considera step in which the variable t is optimised. At the end of the stage 4, for each element, wedetermine N optimised values of t corresponding to the N load cases considered.

At the stage 5, we determine the maximum of these values for each element and westate that the element thickness is equal to this maximum. So, by doing that, we are surethat the structure withstands all the possible combination of loads applied on it.

The optimisation is finished once the three parameters converged.

5.3.1 Optimisation of the parameter t

In this part, we describe the stage 4 of the algorithm (see table 6). We consider a stepin which the variable t is optimised and a particular load case, i.

We define a target strain value t equal to 1250 which is the center of the lazy zone.The aim of that is to force the absolute maximum total strain value of each element toconverge towards this target value.

As said previously, the maximum total strain value is decomposed into two contribu-tions : the axial strain a and the maximum bending strain b.

In this step, we want to optimise the thickness of all the elements for the load case, i,

considered. To do that, we define a target axial strain value at = t |a

a + b|.

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Table 6: The optimisation algorithm

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The cross sectional area for each beam element bears direct proportionality to its axialstiffness according to the stiffness matrix and so, the section area is adjusted by using thetarget axial strain value:

Ai+1 = Ai |a

at|

where Ai is the cross sectional area of the previous step.

Knowing the new area value of each element, we can determine the thickness value byusing the equation linking the area of the hollow ellipse to its thickness.

To do that, the Newton Method is used and the thickness value is controlled in orderto lie in the following interval: [0.1; maxt] as mentioned in the part 5.1.

This process is repeated for all the N load cases we want to consider. At the end of

the stage 4, there are N optimized thickness values for each element associated with the Nload cases.

5.3.2 Optimisation of the parameter R1

We now consider a step in which the parameter R1 is optimised. Following the sameprinciple mentioned previously, we defined a target bending strain value bt equal to

t |b

a + b|

The maximum bending strain value of an element can be also decomposed into twocontributions:

y z + z y = bz + by

Based on that, we can also define a new target strain value based on these two contri-butions:

byt = bt |

by

by + bz|

bzt = bt |bz

by + bz|

According to the stiffness matrix, bending stiffness has a direct proportionality with

the second moment of area of the section. Changing the second moment of area regulatesthe bending strains. As the second moment of area about the z-axis is more sensitive tochanges in R1 as defined in the figure 31, we consider Iz to optimise R1. So:

Ii+1z = Iiz |

bz

bzt|

Knowing the new second moment of area value Iz of each element, we can determinethe new radius R1 value by using the equation linking the second moment of area of thehollow ellipse to this radius.

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The solution R1 is also found by using the Newton Method and must lie in the interval[minR, maxR] as mentioned in the part 5.1.

5.3.3 Optimisation of the parameter R2

We now consider a step in which the parameter R2 is optimised. As the second momentof area about the y-axis is more sensitive to changes in R2 as defined in the figure 30, weconsider Iy to optimise R2. So:

Ii+1y = Iiy |

by

byt|

Knowing the new second moment of area value Iy of each element, we can determinethe radius R2 value by using the equation linking the second moment of area of the hollowellipse to this radius by using the Newton Method. We control that R2 lies in the interval

[minR, maxR] as mentioned in the part 5.1.

Figure 31: Hollow ellipse after rotation

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Part V

Results

6 Sensitivity studies

In this part, we want to study the dependence of our algorithm on some inputs of themodel: the number of load cases considered, the number of elements used to represent thefemur....

As a modelling algorithm, there are a multitude of options available to run it and it is

interested as a first approach to study their influence.

All values are plotted against the vertical coordinate as measured from the distal end ofthe femur. The 3D stiffness method implemented in the algorithm was tested independentlywith GSA.

6.1 Dependence on the number of elements

In this part, we want to study the dependence of the algorithm on the number of ele-ments used. The aim of the algorithm is to reduce the computational time comparativelyto the finite element models. This requires that the number of elements used to model thefemur must be also minimize.

To analyse this influence, we consider two cases. The only difference between these twomodels is the number of elements used to represent the shaft of the bone.

For the first case, we use 20 elements and for the other one, we use 40. The inputs ofthe algorithm which dont change, are summarised in the following table:

Rinitial1 = 13mm Rinitial2 = 15mm

tinitial = 5mm number of load cases considered = 68mint = 0.1mm maxt = 8mm

minR = 12mm maxR = 30mmNumber of load cases used to fix = 28 walking load cases .

The results are illustrated in the figure 32.

It is interesting to notice that there is little difference between the results found forthe two cases. Despite some small differences, the two cases give similar variations of theparameters across the elements representative of the shaft of the femur.

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Figure 32: Dependence of the algorithm on the number of elements used to represent theshaft of the femur: in blue, 20 elements; in red, 40 elements

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This result is relatively important to mention because it means that 20 elements for theshaft is sufficient to have significant results. Nevertheless, as the attachment points of themuscles are directly linked to the nodes of the shaft, we cant reduce too much the numberof elements in order not to sacrifice precision regarding to the musc

meyer e 2011 civil msc thesis

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