Post-graduate ThesisPost-graduate Thesis: RECONSTRUCTION OF : RECONSTRUCTION OF THE 3D STRUCTURE OF THE UPPER FEMUR THE 3D STRUCTURE OF THE UPPER FEMUR AND ANALYSIS OF ITS MECHANICAL AND ANALYSIS OF ITS MECHANICAL PROPERTIES WITH FINITE ELEMENT MESHESPROPERTIES WITH FINITE ELEMENT MESHES

• AIKATERINI AIKATERINI STAMOUSTAMOU

• Supervisor Supervisor

• DrDr Dimitrios Dimitrios EftaxiopoulosEftaxiopoulos

• School: National School: National Technical University Technical University of Athensof Athens

• Date: September Date: September 20092009

Objectives of the workObjectives of the work

• Reconstruction of the human upper femur from Reconstruction of the human upper femur from computational tomography imagescomputational tomography images

• Modeling its mechanics with Modeling its mechanics with finite elementsfinite elements

1.1.Introduction to Biomechanics Introduction to Biomechanics

2.2.Anatomy of the upper femurAnatomy of the upper femur

3.3.Segmentation of the computational tomography Segmentation of the computational tomography images with algorithms of the software bookcase images with algorithms of the software bookcase Insight Toolkit (ITK)Insight Toolkit (ITK)

4.4.Modeling and discretization of the upper femur Modeling and discretization of the upper femur Creation of the composite boneCreation of the composite bone

5.5.Solving differential equations in the discretized femur Solving differential equations in the discretized femur bone Specially, solving Navier-Stokes equations in bone Specially, solving Navier-Stokes equations in the bone which accepts static loadingthe bone which accepts static loading

6.6.ConclusionsConclusions

ContentsContents

1. 1. Introduction to BiomechanicsIntroduction to Biomechanics

Galileo (1638)Galileo (1638) Studied the mechanics of long bones and Studied the mechanics of long bones and analyzed the gross anatomical structure of the thighanalyzed the gross anatomical structure of the thigh

Wolf Wolf (1870) The structures were found not to be irregular in (1870) The structures were found not to be irregular in orientation, but aligned to form a sophisticated patternorientation, but aligned to form a sophisticated pattern

(1892) The bone structures were not only predetermined by (1892) The bone structures were not only predetermined by genetic factors but also by adaptation to mechanical load genetic factors but also by adaptation to mechanical load situationsituation

Koch (1917)Koch (1917) First tried to quantify the mechanical situation First tried to quantify the mechanical situation within the femur by calculating the stresses and strains within the femur by calculating the stresses and strains

Torodls (1969)Torodls (1969) The effect of the muscle loads on the The effect of the muscle loads on the

mechanical behavior of the femur was not as important as mechanical behavior of the femur was not as important as that of the body weight that of the body weight

Ghista (1976)Ghista (1976) A complete mathematical description of the A complete mathematical description of the

internal stresses in a leg during gaitinternal stresses in a leg during gait

2. Anatomy of the upper femur2. Anatomy of the upper femur The head: The head: The head which is The head which is

globular and forms rather more globular and forms rather more than a hemisphere Its surface is than a hemisphere Its surface is smooth, coated with cartilage in smooth, coated with cartilage in the fresh state, except over an the fresh state, except over an ovoid depression, the fovea capitis ovoid depression, the fovea capitis femoris, which is situated a little femoris, which is situated a little below and behind the center of the below and behind the center of the head head

The neck: The neck: In addition to In addition to projecting upward and medialward projecting upward and medialward from the body of the femur, the from the body of the femur, the neck projects forward; the amount neck projects forward; the amount of thof thee forward projection is forward projection is extremely variable, but on an extremely variable, but on an average is from 12° to 14°. average is from 12° to 14°.

The greater trochanter: The greater trochanter: A A large, large, irregular, quadrilateral eminence, irregular, quadrilateral eminence, situated at the junction of the neck situated at the junction of the neck with the upper part of the body with the upper part of the body

The lesser trochanter: The lesser trochanter: AA conical eminence, which projects conical eminence, which projects from the lower and back part of the from the lower and back part of the base of the neck. base of the neck.

3. 3. Insight Toolkit (ITK)Insight Toolkit (ITK) applicationsapplications

• Segmentation: the process of identifying and Segmentation: the process of identifying and classifying data found in a digitally sampling classifying data found in a digitally sampling representationrepresentation

• The basic approach of a region growing algorithm is The basic approach of a region growing algorithm is to start from a seed region (typically one or more to start from a seed region (typically one or more pixels) that are considered to be inside the object to pixels) that are considered to be inside the object to be segmented. The pixels neighboringbe segmented. The pixels neighboring

this region are evaluated to determine if they should this region are evaluated to determine if they should also be considered part of the object. If so, they are also be considered part of the object. If so, they are added to the region and the process continues as long added to the region and the process continues as long as new pixels are added toas new pixels are added to

the region. the region.

Region growing algorithms vary dependent on the Region growing algorithms vary dependent on the criteria: criteria:

• A) a pixel should be included in the region A) a pixel should be included in the region • B) the type connectivity used to determine neighborsB) the type connectivity used to determine neighbors• C) the strategy used to visit neighboring pixelsC) the strategy used to visit neighboring pixels

Connected Threshold algorithmConnected Threshold algorithm

• A simple criterion for including A simple criterion for including pixels in a growing region is to pixels in a growing region is to evaluate intensity value inside evaluate intensity value inside a specific interval.a specific interval.

• Most of the algorithmic complexity of Most of the algorithmic complexity of a region growing method comes from a region growing method comes from visiting neighboring pixels. visiting neighboring pixels.

• The algorithm is left to establish a The algorithm is left to establish a criterion to decide whether a criterion to decide whether a particular pixel should be included in particular pixel should be included in the current region.the current region.

• The criterion used by the The criterion used by the ConnectedThresholdImageFilter is based ConnectedThresholdImageFilter is based on an interval of intensity values provided on an interval of intensity values provided by the user. Values of lower and upper by the user. Values of lower and upper threshold should be provided. The region threshold should be provided. The region growing algorithm includes those pixels growing algorithm includes those pixels whose intensities are inside the interval.whose intensities are inside the interval.

∈ l(x) [lower,upper]l(x) [lower,upper]

Confidence-Connected algorithmConfidence-Connected algorithm

• The criterion used is based on simple The criterion used is based on simple statistics of the current region.statistics of the current region.

• First, the algorithm computes the mean and First, the algorithm computes the mean and standard deviation of intensity values for all the standard deviation of intensity values for all the pixels currently included in the region. pixels currently included in the region.

• A user-provided factor is used to multiply the A user-provided factor is used to multiply the standard deviation and define a range around standard deviation and define a range around the mean. Neighbor pixels whose intensity the mean. Neighbor pixels whose intensity values fall inside the range are accepted and values fall inside the range are accepted and included in the region. When no more included in the region. When no more neighboring pixels are found that satisfy the neighboring pixels are found that satisfy the criterion, the algorithm is considered to have criterion, the algorithm is considered to have finished its first iteration. finished its first iteration.

• Then, the mean and standard deviation of the Then, the mean and standard deviation of the intensity levels are recomputed using all the intensity levels are recomputed using all the pixels currently included in the region. This pixels currently included in the region. This mean and standard deviation defines anew mean and standard deviation defines anew intensity range that is used to visit current intensity range that is used to visit current region neighbors and evaluate whether their region neighbors and evaluate whether their intensity falls inside the range. intensity falls inside the range.

( ) [ ]σσ fm,fmXl +−∈

4. 4. SoftwareSoftware SlicerSlicer and segmentation methodand segmentation method

• It is used for the imaging and processing of MRI for It is used for the imaging and processing of MRI for the discretisation of the bone. Image DICOM is the discretisation of the bone. Image DICOM is processed from a segmentation with the simple region processed from a segmentation with the simple region growing method. growing method.

• Simple region growing method is described from a Simple region growing method is described from a statistic segmentation algorithm. statistic segmentation algorithm.

• The algorithm uses seeds as data.The algorithm uses seeds as data.

• The algorithm computes the mean and standard The algorithm computes the mean and standard deviation of intensity values for all the pixels deviation of intensity values for all the pixels currently included in the seed region. Using the currently included in the seed region. Using the standard deviation as a radius, the region extends standard deviation as a radius, the region extends and the pixels with intensity which belongs and the pixels with intensity which belongs

in the interval (range) are located (found) there.in the interval (range) are located (found) there.

Imaging the tomography in SlicerImaging the tomography in Slicer

Views of tomography in SlicerViews of tomography in Slicer

Creation of the volume of the marrow cavity Creation of the volume of the marrow cavity with automated segmentationwith automated segmentation

Volume models of the marrow cavityVolume models of the marrow cavity

• Model without smoothingModel without smoothing

• Model with big smoothingModel with big smoothing

100smooth =

Creation of the cancellous bone using Creation of the cancellous bone using module Editor module Editor

• Label mapLabel map of of the marrow the marrow cavitycavity

• Result of EditorResult of Editor on the on the cancellous bonecancellous bone

((anterior-anterior-posteriorposterior))

Creation of the composite bone using Creation of the composite bone using

module Editor module Editor

Volume modelVolume model of cancellous boneof cancellous bone

(Number of Iterations: 50, Decimate: 0.05)

Volume model of cortical boneVolume model of cortical bone

(Number of Iterations:50, Decimate: 0.05)

Discretisation of the bone from the software GmshDiscretisation of the bone from the software Gmsh • Gmsh is an automatic three-dimensional finite Gmsh is an automatic three-dimensional finite

element mesh generator with built-in pre- and element mesh generator with built-in pre- and post-processing facilities. Its design goal is to post-processing facilities. Its design goal is to provide a simple meshing tool for academic provide a simple meshing tool for academic problems.problems.

Main characteristics ofMain characteristics of GmshGmsh

• QQuickly describe simple and/or “repetitive” geometries, uickly describe simple and/or “repetitive” geometries, thanks to user-defined functions,thanks to user-defined functions, loops, conditionalsloops, conditionals. .

PParametrize these geometries.arametrize these geometries.

• GGenerate 1D, 2D and 3D simplicialenerate 1D, 2D and 3D simplicial finite element meshesfinite element meshes

• CCreate simple extruded geometries and meshesreate simple extruded geometries and meshes

• GGenerate complex animationsenerate complex animations

• VVisualize computational results in a great variety of isualize computational results in a great variety of ways. Gmsh can display scalar,vector and tensor ways. Gmsh can display scalar,vector and tensor datasets, and can perform various operations on the datasets, and can perform various operations on the resulting postprocessingresulting postprocessing viewsviews..

Meshing of surfaces and volumes on GmshMeshing of surfaces and volumes on Gmsh

Cancellous Bone

Clipping of the discretised cancellous boneClipping of the discretised cancellous bone

ChoiceChoice StatisticsStatistics ofof GmshGmsh for the composite for the composite

bonebone

• The frame model volume The frame model volume of the discretized of the discretized composite bone composite bone incorporatesincorporates

106888 Triangles106888 Triangles

397552 Tetrahedra397552 Tetrahedra

SurfacesSurfaces of the discretised composite of the discretised composite bonebone

Clipping of the discretised composite boneClipping of the discretised composite bone

• Inside from the choiceInside from the choice ToolsTools→→Clipping PlanesClipping Planes with with the choicethe choice Clipping PlaneClipping Plane 4 4 we can take the we can take the upper imageupper image..

• Volumes of bone undertaking clipping are imaged Volumes of bone undertaking clipping are imaged upper.upper.

Clippingplane 4

Discretized composite bone with defined Discretized composite bone with defined surfaces of zero degrees of freedom (DOF)surfaces of zero degrees of freedom (DOF)

5. 5. The softwareThe software FreeFem++FreeFem++

The FreeFem++ The FreeFem++

• Problem description (real or complex valued) by their Problem description (real or complex valued) by their variational formulations, with access to the internal variational formulations, with access to the internal vectors and matrices if needed.vectors and matrices if needed.

• Multi-variables, multi-equations, bi-dimensional , static Multi-variables, multi-equations, bi-dimensional , static or time dependent, linear or nonlinear coupled or time dependent, linear or nonlinear coupled systems; however the user is required to describe the systems; however the user is required to describe the iterative procedures which reduce the problem to a set iterative procedures which reduce the problem to a set of linear problems.of linear problems.

• High level user friendly typed input language with an High level user friendly typed input language with an algebra of analytic and finite element functions.algebra of analytic and finite element functions.

• A large variety of triangular finite elements : linear A large variety of triangular finite elements : linear and quadratic Lagrangian elements, discontinuous P1 and quadratic Lagrangian elements, discontinuous P1 and Raviart-Thomas and Raviart-Thomas

elements, elements of a non-scalar type, mini-element elements, elements of a non-scalar type, mini-element (but no quadrangles).(but no quadrangles).

Creation of the composite boneCreation of the composite bone

• CodeCode getmeshgetmesh..edpedp

mesh3 Th1("floioskat.mesh");mesh3 Th1("floioskat.mesh");

mesh3 Th2("spogkat.mesh");mesh3 Th2("spogkat.mesh");

int[int] r1=[3,5], r2=[1,7];int[int] r1=[3,5], r2=[1,7];

mesh3mesh3 Th3=tetgreconstructionTh3=tetgreconstruction (Th2,(Th2, switchswitch ="rYYCVVV",refface=r2,="rYYCVVV",refface=r2,

reftet=r1);reftet=r1);

mesh3 Th=Th1+Th3;mesh3 Th=Th1+Th3;

Solving equations of Elasticity with the effect of Solving equations of Elasticity with the effect of

gravitygravity Variation formulationVariation formulation

Put Put and and zero zero displacementdisplacement

exterior to the composite exterior to the composite bonebone

solvesolve Elasticity([u1,u2,u3],Elasticity([u1,u2,u3],

[v1,v2,v3],solver=CG)=int3d(T[v1,v2,v3],solver=CG)=int3d(Th)h)

((Ep*nu/((1.+nu)*(1.-2.*nu)))((Ep*nu/((1.+nu)*(1.-2.*nu)))

*div(u1,u2,u3) *div(v1,v2,v3)*div(u1,u2,u3) *div(v1,v2,v3) +2.*+2.*

(Ep/(Ep/(2.*(1+nu)))*(epsilon(u1,u2,(2.*(1+nu)))*(epsilon(u1,u2,u3)'*u3)'*

epsilon(v1,v2,v3)))epsilon(v1,v2,v3)))-int3d(Th)(--int3d(Th)(-v3)v3)

+on(1,u1=0,u2=0,u3=0);+on(1,u1=0,u2=0,u3=0);

( ) ( )

( ) ( ) ( )( ) ( )( )

( ) ( ) 0vfdxdxv:u2v.u.

vuv:u

0vfdxdxv:u

j,iijij

=−+∇∇

=

=−

∫∫

∑

∫∫

ΩΩ

ΩΩ

εµελ

εσεσ

εσ

1f =

SolutionSolution u2u2

SolvingSolving NavierNavier equations on the human upper femur equations on the human upper femur during walkingduring walking

Ph reg=region, Ep, nu;Ph reg=region, Ep, nu; int floios=reg(-141.642, -11.5074, 6.68015);int floios=reg(-141.642, -11.5074, 6.68015); Ep=600.+17000.*(region==floios);Ep=600.+17000.*(region==floios); nu=0.3-0.02*(region==floios); nu=0.3-0.02*(region==floios);

Region functionRegion function

Young’s modulusYoung’s modulus

real volume55= int3d(Th,55)(1.);real volume55= int3d(Th,55)(1.);real area70= int2d(Th,70)(1.);real area70= int2d(Th,70)(1.);real area115= int2d(Th,115)(1.);real area115= int2d(Th,115)(1.);real area190= int2d(Th,190)(1.);real area190= int2d(Th,190)(1.);solve Elasticity([u1,u2,u3],[v1,v2,v3],solver=CG)= solve Elasticity([u1,u2,u3],[v1,v2,v3],solver=CG)= int3d(Th)((Ep*nu/((1.+nu)*(1.-2.*nu)))*div(u1,u2,u3)*div(v1,v2,v3) int3d(Th)((Ep*nu/((1.+nu)*(1.-2.*nu)))*div(u1,u2,u3)*div(v1,v2,v3) +2.*(Ep/(2.*(1.+nu)))*(epsilon(u1,u2,u3)'*epsilon(v1,v2,v3)))+2.*(Ep/(2.*(1.+nu)))*(epsilon(u1,u2,u3)'*epsilon(v1,v2,v3)))-int3d(Th,55)(((-0.54/volume55)*836)*v1) -int3d(Th,55)(((-0.54/volume55)*836)*v1) -int3d(Th,55)(((-0.328/volume55)*836)*v2) -int3d(Th,55)(((-0.328/volume55)*836)*v2) -int3d(Th,55)(((-2.292/volume55)*836)*v3) -int3d(Th,55)(((-2.292/volume55)*836)*v3)

-int3d(Th,55)(((-0.081/volume55)*836)*v1) -int3d(Th,55)(((-0.081/volume55)*836)*v1) -int3d(Th,55)(((-0.128/volume55)*836)*v2) -int3d(Th,55)(((-0.128/volume55)*836)*v2) -int3d(Th,55)(((-0.782/volume55)*836)*v3) -int3d(Th,55)(((-0.782/volume55)*836)*v3) -int2d(Th,70)(((0.58/area70)*836)*v1) -int2d(Th,70)(((0.58/area70)*836)*v1) -int2d(Th,70)(((0.043/area70)*836)*v2) -int2d(Th,70)(((0.043/area70)*836)*v2) -int2d(Th,70)(((0.865/area70)*836)*v3) -int2d(Th,70)(((0.865/area70)*836)*v3) -int2d(Th,70)(((0.072/area70)*836)*v1) -int2d(Th,70)(((0.072/area70)*836)*v1) -int2d(Th,70)(((0.116/area70)*836)*v2)-int2d(Th,70)(((0.116/area70)*836)*v2)-int2d(Th,70)(((0.132/area70)*836)*v3)-int2d(Th,70)(((0.132/area70)*836)*v3)-int2d(Th,70)(((-0.005/area70)*836)*v1)-int2d(Th,70)(((-0.005/area70)*836)*v1) -int2d(Th,70)(((-0.007/area70)*836)*v2) -int2d(Th,70)(((-0.007/area70)*836)*v2)

-int2d(Th,70)(((-0.19/area70)*836)*v3) -int2d(Th,70)(((-0.19/area70)*836)*v3) -int2d(Th,190)(((-0.009/area190)*836)*v1)-int2d(Th,190)(((-0.009/area190)*836)*v1)-int2d(Th,190)(((0.185/area190)*836)*v2) -int2d(Th,190)(((0.185/area190)*836)*v2) -int2d(Th,190)(((-0.929/area190)*836)*v3) -int2d(Th,190)(((-0.929/area190)*836)*v3) +on(19,u1=0,u2=0,u3=0) +on(16,u1=0,u2=0,u3=0);+on(19,u1=0,u2=0,u3=0) +on(16,u1=0,u2=0,u3=0);

Norm of the displacementNorm of the displacement

• The maximum value is appeared in the headThe maximum value is appeared in the head and the zero and the zero value in the surfaces with zero Degrees of Freedomvalue in the surfaces with zero Degrees of Freedom..

First principal strainFirst principal strain

Big strains in the neckThe neck is characterized as a bending bar with tension in the upper part.The upper part of the neck is a possible point of beginning the fracture because the bending cortical is very thin.

• Big strains between the Big strains between the loading surfaces. loading surfaces.

It is due to the big tension It is due to the big tension which loads create locally, which loads create locally, but also the bending of the but also the bending of the diaphysis, which has the diaphysis, which has the tensile region outside tensile region outside because of the press of because of the press of pelvis to the head.pelvis to the head.

• Big strains in the small Big strains in the small loading surfaces due to the loading surfaces due to the small concentration of small concentration of strains as a result of being strains as a result of being the small loading surfaces.the small loading surfaces.

MaximumMaximum shearshear strainstrain

• Maximum values in the upper neck and exterior to Maximum values in the upper neck and exterior to the diaphysis.the diaphysis.

• Possible failure due to shear is assumed on these Possible failure due to shear is assumed on these regions.regions.

MinimumMinimum principal strainprincipal strain

• The maximum compression is developed in the The maximum compression is developed in the lower neck and interior to the diaphysis. lower neck and interior to the diaphysis.

Maximum shear stressMaximum shear stress

• Big shear strains are developed exterior to the Big shear strains are developed exterior to the diaphysis, around the muscle loading regions and diaphysis, around the muscle loading regions and interior to the diaphysis.interior to the diaphysis.

Strain energyStrain energy

• Maximum strain energy is developed interior to Maximum strain energy is developed interior to the diaphysis and in the lower neck.the diaphysis and in the lower neck.

Solving Navier-Stokes equations in the Solving Navier-Stokes equations in the human upper femur human upper femur

during stair-climbing during stair-climbing

solve Elasticity([u1,u2,u3],[v1,v2,v3],solver=CG)=solve Elasticity([u1,u2,u3],[v1,v2,v3],solver=CG)=int3d(Th)int3d(Th)((Ep*nu/((1.+nu)((Ep*nu/((1.+nu)* *

(1.-2.*nu))*div(u1,u2,u3)*div(v1,v2,v3)+2.*(Ep/(1.-2.*nu))*div(u1,u2,u3)*div(v1,v2,v3)+2.*(Ep/(2.*(1.+nu)))*(epsilon(u1,u2,u3)(2.*(1.+nu)))*(epsilon(u1,u2,u3)

'' *epsilon(v1,v2,v3)))*epsilon(v1,v2,v3))) -int3d(Th,55)(((-0.593/volume55)*847)*v1) -int3d(Th,55)(((-0.593/volume55)*847)*v1)

..........................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

-int2d(Th,115)(((-0.088/area115)*847)*v1) -int2d(Th,115)(((-0.088/area115)*847)*v1)

-int2d(Th,115)(((0.396/area115)*847)*v2) -int2d(Th,115)(((0.396/area115)*847)*v2)

-int2d(Th,115)(((-2.671/area115)*847)*v3) -int2d(Th,115)(((-2.671/area115)*847)*v3) ;; +on(19,u1=0,u2=0,u3=0)+on(19,u1=0,u2=0,u3=0) +on(16,u1=0,u2=0,u3=0);+on(16,u1=0,u2=0,u3=0);

Displacement in the direction of the x-axisDisplacement in the direction of the x-axis

• Zero values in the surfaces with zero Degrees of Zero values in the surfaces with zero Degrees of Freedom.Freedom.

• Maximum x-displacement in the neck, where the Maximum x-displacement in the neck, where the maximum load is implemented. maximum load is implemented.

Maximum principal strainMaximum principal strain

• Big strains interior and exterior to the diaphysis, also in the Big strains interior and exterior to the diaphysis, also in the upper and lower neck and in the conjunction of the neck upper and lower neck and in the conjunction of the neck and the head.and the head.

Maximum shear strainMaximum shear strain

• Big strains are developed in the lower part of the neck Big strains are developed in the lower part of the neck and interior to the diaphysis.and interior to the diaphysis.

Minimum principal strainMinimum principal strain

• Maximum compression is developed in the lower part of the Maximum compression is developed in the lower part of the neck and interior to the diaphysis.neck and interior to the diaphysis.

Von Misses stressVon Misses stress

• Bigger stresses are developed interior to the diaphysisBigger stresses are developed interior to the diaphysis. .

Strain energyStrain energy

• Maximum strain energy is developed interior to the Maximum strain energy is developed interior to the diaphysis.diaphysis.

6. 6. ConclusionsConclusionsWalkingWalking

The bigger tension is developed in the cortical bone, exterior to The bigger tension is developed in the cortical bone, exterior to the diaphysis and in the upper part of the neck. The possibility the diaphysis and in the upper part of the neck. The possibility of beginning the failure is bigger there, because the cortical of beginning the failure is bigger there, because the cortical bone is brittle and it appears small tolerance in tension. bone is brittle and it appears small tolerance in tension. Maximum shear is developed there.Maximum shear is developed there.

The bigger compression is developed interior to the diaphysis The bigger compression is developed interior to the diaphysis and in the lower part of the neck, in the cortical bone. The and in the lower part of the neck, in the cortical bone. The cortical bone appears big tolerance in compression. There is the cortical bone appears big tolerance in compression. There is the danger of failure due to compression interior to the diaphysis danger of failure due to compression interior to the diaphysis and in the lower part of the neck because of pathological and in the lower part of the neck because of pathological situations. situations.

The Von Misses stress and strain energy density are maximum The Von Misses stress and strain energy density are maximum interior to the diaphysis and in the lower part of the neck, in the interior to the diaphysis and in the lower part of the neck, in the cortical bone due to big compression there. cortical bone due to big compression there.

Stair-climbingStair-climbing Similar notes with those of walkingSimilar notes with those of walking Only quantified differences between walking and stair-climbingOnly quantified differences between walking and stair-climbing No qualified differences between walking and stair-climbingNo qualified differences between walking and stair-climbing

ReferencesReferences

Gray’s anatomyGray’s anatomy Henry Gray (1821-1865)Henry Gray (1821-1865) Anatomy of the human body Anatomy of the human body Determination of muscle loading at the hip joint for use in pre-Determination of muscle loading at the hip joint for use in pre-

clinical testingclinical testing M.O. Heller, G. Bergmann, J.-P. Kassi, L. Claes, N.P. Haas, G.N. M.O. Heller, G. Bergmann, J.-P. Kassi, L. Claes, N.P. Haas, G.N.

DudaDuda Magazine: Journal of Biomechanics 38 (2005) Magazine: Journal of Biomechanics 38 (2005) 1155–11631155–1163 FreeFem++ (2009FreeFem++ (2009)) Gmsh (2009)Gmsh (2009) ITK (Insight Segmentation and Registration Toolkit, 2009)ITK (Insight Segmentation and Registration Toolkit, 2009) Medit (2009)Medit (2009) Slicer (2009)Slicer (2009) Tetgen (2009)Tetgen (2009) Visible Human Project (2009)Visible Human Project (2009)