Femoral bone mesoscale structural architecture prediction using...

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International Biomechanics

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Femoral bone mesoscale structural architectureprediction using musculoskeletal and finiteelement modelling

Andrew T.M. Phillips, Claire C. Villette & Luca Modenese

To cite this article: Andrew T.M. Phillips, Claire C. Villette & Luca Modenese (2015) Femoralbone mesoscale structural architecture prediction using musculoskeletal and finite elementmodelling, International Biomechanics, 2:1, 43-61, DOI: 10.1080/23335432.2015.1017609

To link to this article: http://dx.doi.org/10.1080/23335432.2015.1017609

© 2015 The Author(s). Published by Taylor &Francis

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Femoral bone mesoscale structural architecture prediction using musculoskeletal and finiteelement modelling

Andrew T.M. Phillipsa,b* , Claire C. Villettea,b and Luca Modenesea,c

aStructural Biomechanics, Department of Civil and Environmental Engineering, Imperial College London, Skempton Building, SouthKensington Campus, London SW7 2AZ, UK; bThe Royal British Legion Centre for Blast Injury Studies at Imperial College London,Imperial College London, London, UK; cSchool of Allied Health Sciences, Menzies Health Institute Queensland, Griffith University,

Gold Coast, Australia

(Received 19 October 2014; accepted 11 December 2014)

Through much of the anatomical and clinical literature bone is studied with a focus on its structural architecture, while itis rare for bone to be modelled using a structural mechanics as opposed to a continuum mechanics approach in the engi-neering literature. A novel mesoscale structural model of the femur is presented in which truss and shell elements areused to represent trabecular and cortical bone, respectively. Structural optimisation using a strain-based bone adaptationalgorithm is incorporated within a musculoskeletal and finite element modelling framework to predict the structure of thefemur subjected to two loading scenarios; a single load case corresponding to the frame of maximum hip joint contactforce during walking and a full loading regime consisting of multiple load cases from five activities of daily living. Theuse of the full loading regime compared to the single load case has a profound influence on the predicted trabecular andcortical structure throughout the femur, with dramatic volume increases in the femoral shaft and the distal femur, andregional increases at the femoral neck and greater trochanter in the proximal femur. The mesoscale structural model sub-jected to the full loading regime shows agreement with the observed structural architecture of the femur while the struc-tural approach has potential application in bone fracture prediction, prevention and treatment. The mesoscale structuralapproach achieves the synergistic goals of computational efficiency similar to a macroscale continuum approach and aresolution nearing that of a microscale continuum approach.

Keywords: bone adaptation; structural optimisation; structure; architecture; femur; finite element; musculoskeletal

1. Introduction

Bone structure and mechanics have been studied exten-sively, from as early as the seventeenth century, whenGalilei (1638) proposed the dimensional scaling laws.The primary function of the skeletal system is thestructural support of the body, while bone may adaptits geometry and structure to fulfil this function andresist the loads placed upon it (Toridis 1969). Knowl-edge of skeletal structure is fundamental for assessmentof the mechanical environment within the musculoskele-tal system (Viceconti 2011), which in turn may informprediction, prevention and treatment of orthopaedicdisorders as well as design of protective devices andprosthetics. Historically, anatomists and engineers haveobserved the structure of trabecular bone in the proxi-mal femur, hypothesising that it follows trajectories ofcompressive and tensile stress (Culmann 1866; vonMeyer 1867; Wolff 1869; Koch 1917). Comparisonshave been made between the internal structure of afrontally sectioned proximal femur and the sketchedstress trajectories of a curved (Fairbairn) crane (Skedros& Baucom 2007). It is generally accepted that boneadapts to its mechanical environment (Wolff 1869,

1986; Frost 2003), leading to a structure optimised towithstand the forces acting on it (including muscleforces, joint contact forces (JCFs) and inertial loading)using a minimum volume of material. This study pre-sents a predictive mesoscale structural model of thefemur in which trabecular and cortical bone structure isoptimised based on the strain environment present dueto daily living activities.

1.1. Continuum modelling approaches

Finite element (FE) modelling using geometries andmaterial properties extracted from medical imaging (typi-cally computed tomography (CT) data) is a preferred toolfor investigating the behaviour of bone at both macro-scale (Taddei et al. 2006) and microscale (Hambli 2013).It is common at both the macroscopic and microscopicscales to model bone using solid continuum elements. Acontinuum model is considered to be either macroscaleor microscale when the solid element size is larger orsmaller, respectively, than the size of an individual struc-tural component of bone such as a trabeculae (Nägeleet al. 2004; Phillips 2012).

*Corresponding author. Email: [email protected]

© 2015 The Author(s). Published by Taylor & Francis.This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (http://creativecommons.org/licenses/by-nc/4.0/),which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

International Biomechanics, 2015Vol. 2, No. 1, 43–61, http://dx.doi.org/10.1080/23335432.2015.1017609

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http://orcid.org/0000-0001-6618-0145http://orcid.org/0000-0001-6618-0145http://orcid.org/0000-0001-6618-0145http://orcid.org/0000-0002-1638-258Xhttp://orcid.org/0000-0002-1638-258Xhttp://orcid.org/0000-0002-1638-258Xhttp://orcid.org/0000-0003-1402-5359http://orcid.org/0000-0003-1402-5359http://orcid.org/0000-0003-1402-5359mailto:[email protected]://creativecommons.org/licenses/by-nc/4.0/http://dx.doi.org/10.1080/23335432.2015.1017609

1.1.1. Macroscale continuum FE modelling

At the macroscale bone is considered as a continuumwithout voids, with material properties assigned acrosselements based on empirical relationships between CTattenuation values, density and Young’s modulus (Carter& Hayes 1977; Helgason et al. 2008). Macroscale con-tinuum models can run in a matter of minutes on a stan-dard workstation but present a limited resolution andtypically overlook anisotropic material properties.

The macroscale continuum approach has been usedin a number of studies investigating modelling (Beaupréet al. 1990a, 1990b) and remodelling (Huiskes et al.1987; Bitsakos et al. 2005; García-Aznar et al. 2005) ofbone, using a variety of stress and strain stimuli to guidethe bone apposition and resorption algorithm. Predictivestudies have successfully extended the material constitu-tive relationship to include orthotropy and anisotropy intwo-dimensional planar (Doblaré & Garćia 2001; Milleret al. 2002) and three-dimensional spacial models of thefemur (Geraldes & Phillips 2014).

1.1.2. Microscale continuum FE modelling

At the microscale bone is generally treated as a binarysystem with bone either being present or not. Homoge-neous material properties are generally used although dif-ferent values of Young’s modulus may be adopted forcortical and trabecular bone (Verhulp et al. 2006). Thegeometry of the model is typically derived from lCT orlMRI scans through thresholding on the attenuation val-ues (Ulrich et al. 1998). Although microscale modelsallow for fine resolution of bone structure, they are extre-mely computationally demanding, requiring multiple pro-cessors and run times of several days. In addition, thesignificant radiation dose associated with current lCTacquisition technologies limits its application in vivo(Pankaj 2013).

The microscale continuum approach has been used ina small number of studies investigating modelling of theproximal femur (Jang & Kim 2008; Tsubota et al. 2009;Boyle & Kim 2011), which found good agreementbetween predicted and observed trabecular bone trajecto-ries. These studies generally used a limited number ofsimplified load cases to represent the varying mechanicalenvironment present in the proximal femur due to a widerange of activities. As with microscale continuum modelsderived from lCT imaging, the predictive modelsprovide a higher degree of resolution than macroscalecontinuum models at the cost of being extremely com-putationally demanding.

In addition to macroscale and microscale FE mod-elling approaches, a small number of studies haveinvestigated multiscale modelling approaches, where dis-placement distributions at the macroscale are used to

drive modelling algorithms at the microscale (Coelhoet al. 2009; Kowalczyk 2010). This approach has theadvantage of increasing computational efficiency,although it does not result in a complete microscalemodel of the bone being investigated.

1.2. Structural modelling approaches

An alternative to both macroscale and microscale contin-uum FE modelling of bone is to adopt a structural FEmodelling approach where a combination of idealisedelements such as trusses, beams and shells are used torepresent structural components of bone. At the micro-scale van Lenthe et al. (2006) skeletonised a voxel-basedlCT to produce corresponding structural and continuummodels, the structural model being composed of individ-ual or small groups of beams representing trabecularbone. The structural model had a reduced CPU time byover a thousandfold compared to the continuum model,while results from both models were in excellent correla-tion (R2 ¼ 0:97).

Representing bone as a structure allows FE mod-elling to take place at the mesoscale, where individualstructural elements may be larger than those foundin vivo, while being capable of capturing the overallstructural behaviour of bone. The aim of this study wasto develop a mesoscale structural model of the femurbased on a physiological loading regime. With theexception of a small number of previous studies (Dudaet al. 1998; Polgar et al. 2003; Speirs et al. 2007;Phillips 2009) FE models of the femur have utilised sim-plified boundary conditions and loading, resulting innon-physiological strain and stress distributions. In addi-tion, the majority of studies have utilised a single loadcase or a combined load case (Beaupré et al. 1990a,1990b; Miller et al. 2002) to drive the bone adaptationalgorithm. This approach fails to address the role of boneas a structure, required to resist the variety of load casesplaced on it during daily living activities.

Hence, two principal development stages areinvolved in the presented novel approach to predictingbone structural architecture in the femur. Firstly, anequilibrated set of loads (including muscle forces, JCFsand inertial loading) sampling five daily living activitieswas derived from an updated version of a validatedmusculoskeletal model (Modenese et al. 2011). It isbelieved that these simulations captured a fair representa-tion of the physiological daily loading conditions experi-enced by the femur. Secondly, a strain-driven boneadaptation algorithm was used to optimise the bonestructure subject to the derived loading regime. Theresulting model was expected to be biofidelic, presentinga computational efficiency similar to macroscale contin-uum FE models and a spacial refinement approachingthat of microscale continuum FE models.

44 A.T.M. Phillips et al.

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2. Methods

The mesoscale structural model of the femur is obtainedthrough iterative adaptation of a base FE model subjectto a loading regime derived from musculoskeletal sim-ulations of the following daily activities: walking, stairascent and descent, and sit-to-stand and stand-to-sit. Themodelling framework is illustrated in Figure 1.

2.1. Musculoskeletal modelling

The load cases applied to the structural FE model werederived from musculoskeletal simulations of five daily liv-ing activities. Experimental data were collected on a volun-teer (Male, age: 26 years, weight: 74 kg, height: 175 cm)for the purpose of this study. The chosen activities: walk-ing, stair ascent, stair descent, sit-to-stand and stand-to-sitare consistent with the most frequent daily living activitiesidentified by Morlock et al. (2001) through the use of aportable monitoring system. Aspects of the musculoskele-tal model that should be highlighted are the use of identi-cal femoral geometry in both the musculoskeletal modeland in the FE model in order to ensure that the load casesderived using the musculoskeletal model could be appliedin the FE analysis as described in Section 2.2.1, and theuse of an OpenSim (National Center for Simulation inRehabilitation Research, Stanford, CA, USA) plugin devel-oped by the authors to extract muscle forces derived usingthe musculoskeletal model as vectors to be applied in theFE model (van Arkel et al. 2013) (available to downloadat https://simtk.org/home/force_direction).

The musculoskeletal model of the lower limb is basedon the anatomical dataset published by Klein Horsmanet al. (2007) and implemented in OpenSim (Delp et al.

2007). The ipsilateral model includes six segments (pelvis,femur, patella, tibia, hindfoot and midfoot plus phalanxes)connected by five joints (pelvis-ground connection, acetab-ulofemoral (hip) joint, tibiofemoral (knee) joint, patellofe-moral joint and ankle joint). The pelvis is connected toground with a free joint (6 degrees of freedom (DOF)), thehip is represented as a ball and socket joint (3 rotationalDOF), the knee and ankle joints are modelled as hinges(1 DOF each) while the patella rotates around a patellofe-moral axis as a function of the knee flexion angle. Thepatellar ligament was included in the model to allow forcetransmission between the patella and the tibia. Thirty-eightmuscles of the lower extremity are represented through onehundred sixty-three actuators, whose path is enhanced byfrictionless via points and wrapping surfaces. The localreference systems of the body segments were definedaccording to the recommendations of the InternationalSociety of Biomechanics (Wu et al. 2002). The muscleattachment coordinates were the same as in Modenese et al.(2011) for all segments except the femur, for which theywere mapped directly onto a femoral mesh identical to theone used for the FE simulations. This operation was per-formed using NMSBuilder (Martelli et al. 2011) and thevisual guidance of anatomical atlases (Gray 1918; Platzer2008) and the muscle standardised femur (Viceconti et al.2003). Additional wrapping surfaces were included torepresent the hip joint capsule as in Brand et al. (1994)to prevent the quadriceps from penetrating the femur and toimprove the gluteal muscle paths (Modenese et al. 2013) asreported in van Arkel et al. (2013). The musculoskeletalmodel is shown during sit-to-stand in Figure 2.

Full body gait data were collected for a healthyvolunteer with no history of joint pain or articular diseases,

Figure 1. Musculoskeletal and finite element modelling framework.

International Biomechanics 45

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https://simtk.org/home/force_direction

performing five daily living activities. The trajectories of59 reflective markers positioned on bony landmarks andtechnical clusters were tracked using a Vicon system(Oxford Metrics, Oxford, UK) equipped with 10 infraredcameras. External forces (ground reaction forces (GRFs))were measured using three Kistler force plates (Type9286BA, sampling rate 1000 Hz) (Kistler Instruments Ltd,Hook, UK). An instrumented walkway was used forrecording GRFs during walking (speed: 1.22 m/s, stridelength: 1.29 m, cadence: 113.4 steps/min). An instru-mented staircase consisting of 3 steps (step height 15 cmand step depth 25 cm, resulting in an inclination of 36.8degrees) was used for recording GRFs going upstairs anddownstairs, with the three force plates placed on subse-quent steps. A stool with a seat height of 50.7 cm from thefloor was used for recording GRFs during sit-to-stand andstand-to-sit, with force plates placed at both feet and at theseat. All gait data were collected in the Biodynamics Labin the Imperial College Research Labs at Charing CrossHospital and processed using Vicon Nexus (Version 1.7.1)and the Biomechanical ToolKit (Barre & Armand 2014).

The body segments of the musculoskeletal modelwere scaled to the anatomical dimensions of the volun-teer by calculating ratios from lengths between sets ofvirtual and experimental markers; the inertial propertiesof the body segments were updated according to theregression equations of Dumas et al. (2007). Joint anglesdescribing the motion for each of the investigated dailyliving activities were calculated from the experimentalmarkers using an inverse kinematics approach (Lu &O’Connor 1999). Muscle forces were estimated by

minimising the sum of muscle activations squared foreach frame of the kinematics under the constraints ofjoint moment equilibrium and physiological limits forthe muscle forces (Modenese et al. 2011; Modenese &Phillips 2012). Finally, JCFs were calculated at the hip,knee and patellofemoral joint. All musculoskeletal sim-ulations were performed in OpenSim (Version 3.0.1)(Delp et al. 2007).

For each of the investigated activities, all loads act-ing on the femur were determined with respect to thesegment reference system in order to be applied to theFE model. The inertial load and the gravitational forcewere calculated at the thigh centre of mass based on thesegment kinetics, the joint contact forces were calculatedat the joint centres using the JointReaction analysis toolavailable in OpenSim (Steele et al. 2012), while thefemoral attachment point coordinates of each muscleactuator, together with the direction and magnitude ofthe muscle force, were extracted using the plugin devel-oped by the authors (van Arkel et al. 2013).

2.2. Finite element base model

The base structural model of the femur was created usinga similar methodology to Phillips (2012). A CT scan ofa Sawbones (Pacific Research Laboratories, Inc., VashonIsland, WA, USA) fourth generation medium compositefemur (#3403) was processed in Mimics (Materialise,Leuven, Belgium) to create a volumetric mesh composedof 113103 four-noded tetrahedral elements with an aver-age edge length of 3.9 mm. The mesh was uniformly

Figure 2. The developed musculoskeletal model, (a) during sit-to-stand, (b) close up of the femoral mesh identical to that used inthe FE simulations. Forces from those muscles highlighted in red are applied in the FE simulations. Ground reaction forces beneatheach foot are shown. Wrapping surfaces are omitted for clarity. Background objects available from https://simtk.org/home/simgym.


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scaled to the femoral segment length required for thevolunteer specific musculoskeletal model. The subse-quent volumetric mesh was adapted using MATLAB(MathWorks, Natick, MA, USA) to create an initialstructural mesh. The nodes and element faces of theexternal surface of the volumetric mesh were used todefine three-noded linear triangular shell elements, takento be representative of cortical bone, with the externalsurface of the shell elements corresponding to the exter-nal geometry of the femur. These were arbitrarilyassigned an initial thickness of 0.1 mm. Each of theinternal nodes was considered in turn and used to definetwo-noded truss elements connecting between the nodeunder consideration and the nearest sixteen neighbouringnodes, with the resulting network taken to be representa-tive of trabecular bone. These were arbitrarily assigned acircular cross-section with an initial radius of 0.1 mm.With a minimum connectivity of 16 at each node, it isbelieved that a sufficient range of element directionalitieswere available to allow region specific trabecular direc-tionalities to develop during the bone adaptation process.It should be noted that while the minimum connectivitywas 16, the maximum was 42; mean 21.30 (SD 5.51).Figure 3 shows a 2.5 mm thick slice of the proximal

femur for the base model, composed of 10410 corticalshell elements and 218703 trabecular truss elements. Lin-ear isotropic material properties were assigned for all ele-ments, E ¼ 18000 MPa, m ¼ 0:3 based on reportedvalues for bone at the tissue level (Turner et al. 1999).

2.2.1. Loading

The muscle tensions estimated by the musculoskeletalmodel were applied as point loads at the nodes corre-sponding to the muscle insertion points in the FE model.JCFs and the inertial load, calculated at the joint centresand body segment centre of mass, were applied throughspecific constructs (‘load applicators’ and the ‘inertiaapplicator’) designed to spread the loads over the jointcontact surfaces and the whole bone surface, respec-tively. The use of load applicators provides a significantreduction in CPU time in comparison with the inclusionof contact at the joint surfaces. The load applicators areshown in Figure 4.

The load applicators were composed of constructsmade of four layers of six-noded linear continuumwedge elements superposed to the external surface of theappropriate regions of the base model. The load applica-tors, in combination with the surface elements of thebase model, were taken to represent the bone-cartilage-cartilage-bone interfaces at the joints. They were gener-ated through the projection of the nodes of the regionsof interest along the direction defined by the considerednodes and the centre of the joint, directed outwards. Thethickness of each of the layers was 1 mm. The bottomtwo layers, taken to represent cartilage, were assignedE ¼ 10 MPa, m ¼ 0:49. The top two layers wereassigned stiffer material properties; bone for the acetab-ulofemoral (hip) and tibiofemoral (knee) joints and aone order of magnitude softer material (E ¼ 1800 MPa,m ¼ 0:3) for the patellofemoral joint (the patella as asesamoid bone embedded in ligament is considered to beless stiff than the acetabular and tibial joint surfaces).

The hip joint presents three rotational DOF, hence itwill transfer forces but not moments. The acetab-ulofemoral load applicator was hence completed by con-necting each of the external nodes of the applicator tothe centre of the joint (as defined in the musculoskeletalmodel) using truss elements. The JCFs derived from themusculoskeletal simulations were applied at the centre ofthe joint. The knee and patellofemoral joints each presenta single rotational DOF, hence moments may be trans-ferred at both joints about the directions perpendicular totheir rotation axes. In order to facilitate the transfer ofmoments at the knee and patellofemoral joints withoutintroducing local moment transfer between the loadapplicators and the underlying bone, moments wereapplied via force couples on two points located on thejoint axes either side of the respective joint centres (as

Figure 3. 2.5 mm slice of the proximal femur for the basemodel. Triangular shell elements representing cortical bone areshown in grey; truss elements representing trabecular bone areshown in red.


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defined in the musculoskeletal model). The definition ofthe hip and knee joint load applicators corresponds tothe respective joint contact surfaces, over the range ofmotion for all activities. To allow for patella movementacross the surface of the femur during knee flexion, thepatellofemoral load applicator was defined as a bandpassing between the two condyles prolongated over thedistal portion of the frontal shaft. The tibiofemoral andpatellofemoral load applicators were completed in a simi-lar manner to the acetabulofemoral load applicator, byconnecting each of the external nodes of the applicatorto each of the points on the respective joint axes. Trusselements for all of the load applicators were given acircular cross section with a radius of 2.5 mm (similar tothe edge length of the surface elements). The hip andknee joint trusses were assigned the material propertiesof bone (E ¼ 18000 MPa, m ¼ 0:3). For consistency withthe top two layers of the load applicator, the patellofe-moral trusses were assigned a one order of magnitudesofter material (E ¼ 1800 MPa, m ¼ 0:3).

An ‘inertia applicator’ was designed based on thesame concept as the load applicators. It is composed ofsoft truss elements (radius: 2.5 mm, E ¼ 5 MPa, m ¼ 0:3)linking every node of the femoral surface with the centreof mass of the leg, where the inertial load is applied.

Young’s modulus was set to a low value to ensure thatstiffening of the model was negligible. The use of ahigher value could result in reduced deformation alongthe length of the femur. Spreading the inertial load overthe whole volume rather than the surface was considered,but ruled out at this stage due to the severe increase inCPU time (up to a five times higher) involved.

Loading conditions from a subset of frames, derivedfrom the musculoskeletal model, representative of eachactivity were selected to increase the computational effi-ciency of the FE model. Frame selection was done usingan ‘integration limit error’ approach based on the hipJCF. The evolution of the hip JCF was integrated usingthe trapezoidal method on the full set of frames. Frameswere then successively removed from the sample and thecorresponding integration between remaining framescompared to that obtained from the full frame set. Theprocess was repeated until no further frames could beremoved without generating a difference in integrationbetween two adjacent sampled frames of more than 1%of the integration of the full frame set. Figure 5 showsthe selected frames as well as the hip JCF derived fromthe musculoskeletal model, alongside the average hipJCF as reported by Bergmann et al. (2001) for the sameactivities, for comparison. The magnitudes of the

Figure 4. Load applicators at the, (a) hip, (b) knee and (c) patellofemoral joints. Shell elements representing cortical bone and thewedge elements of the applicators are shown as semi-transparent to highlight the truss elements linking the applicator constructs tothe hip joint centre, the knee and patellofemoral joint axes, respectively. Dashed lines show the joint axes for the knee and patellofe-moral joints.


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predicted hip JCFs for all activities were found to bewithin the ranges recorded by Bergmann et al. (2001),with the exception of the second peak during walking,which was higher for the musculoskeletal model. Adirect comparison is difficult as the hip JCFs derivedfrom the musculoskeletal model are for a young healthysubject (26 years), while those recorded by Bergmannet al. (2001) are for four older patients (51–76 years)who had undergone hip replacement surgery.

The load cases (including muscles forces, JCFs andinertia forces) corresponding to the selected time framesof the different activities were applied in consecutiveanalysis steps of the FE simulation.

2.2.2. Boundary conditions

Specific ‘fixator’ constructs were designed at the acetab-ulofemoral and the tibiofemoral joints to allow boundaryconditions compatible with the DOF present in themusculoskeletal model to be applied, based on the sameconcept as load applicators. The acetabulofemoral fixatorconsists of truss elements linking the nodes of the exter-nal surface of the acetabulofemoral load applicator backto a point superposed with the centre of the hip joint.The tibiofemoral fixator consists of truss elements link-ing the nodes of the external surface of the tibiofemoralload applicator back to two points superposed with theforce couple points on the knee joint axis describedpreviously. From consideration of the musculoskeletalmodel, it is clear that no moment can develop either atthe centre of the hip joint or about the knee joint axis.Hence the centre of the acetabulofemoral joint wasrestrained against displacement along any of the threefemoral axes (Wu et al. 2002). At the tibiofemoral joint,the medial of the two points on the joint axis wasrestrained against displacement in the plane perpendicu-lar to the joint axis, while the lateral of the two pointswas restrained against displacement in the direction cor-responding to the cross product of the vectors definingthe joint axis and the femoral X-axis (anterior-posterior)(Wu et al. 2002). Thus the FE model was restrainedagainst translation in the minimum number of DOF (six)required to define a stable structure. It should be notedthat although the points of load application and points ofrestraint application were coincident in space for theundeformed model, they were defined as separate points,which displaced with respect to each other whenthe model was subjected to load. Truss elements forboth of the fixators were given a circular cross sectionwith a radius of 2.5 mm and material properties,E ¼ 1000 MPa, m ¼ 0:3. The modulus of the fixatortrusses was set one order of magnitude lower than themodulus of the load applicator trusses in order to preventartificial stiffening of the model close to the joint sur-faces.

2.3. Bone adaptation algorithm

Adopting the Mechanostat hypothesis (Frost 2003), succes-sive iterations of the base model were subjected to theloading regime derived from the musculoskeletal model,with the cross-sectional area of each truss element and thethickness of each shell element adjusted with each iterationaccording to the resulting strain environment. The iterativeprocess was controlled using a combination of MATLABand Python (Python Software Foundation, Beaverton, OR,USA) scripts, while successive FE models were run usingthe Abaqus/Standard solver (Dassault Systèmes Simulia,Johnston, RI, USA), until convergence was achieved.

For the ith iteration the maximum absolute strain for thejth truss and the jth shell element over k ¼ 1; . . .; n loadcases was defined using Equations 1 and 2 respectively:

j�i;jjmax ¼ max j�11;j;kj� �

(1)

where �11;j;k is the axial strain in the jth truss element forthe load case k,

j�i;jjmax ¼ max j�tmax;j;kj; j�tmin;j;kj; j�bmax;j;kj; j�bmin;j;kj� �

(2)

where �tmax;j;k, �tmin;j;k and �

bmax;j;k, �

bmin;j;k are the maxi-

mum and minimum principal strains in the top and bot-tom surfaces respectively of the jth shell element for theload case k.

The adopted strain ranges associated with the deadzone, bone resorption, the lazy zone and bone apposition(Frost 2003; Phillips 2012) are given in Equation 3.

/i;j ¼

1; for 0� j�i;jjmax � 250l� ðDead zoneÞ1; for 250\j�i;jjmax\1000l� ðBone resorptionÞ0; for 1000� j�i;jjmax � 1500l� ðLazy zoneÞ1; for j�i;jjmax [ 1500l� ðBone appositionÞ

8>>><>>>:

(3)

For the ðiþ 1Þth iteration, the cross-sectional area, Aof the jth truss element and the thickness, T of the jthshell element were adjusted according to Equations 4and 5 respectively, adopting a target strain, �t of1250 l�, at the centre of the lazy zone. The target strainand range of the lazy zone were considered reasonablebased on in vivo surface strain measurements on thehuman femur, taken below the greater trochanter byAamodt et al. (1997) for two subjects during single legstance, walking and stair climbing, finding peak valuesin the range of 1000–1500 l� across all activities.

if /i;j ¼ 1; Aiþ1;j ¼ Ai;jj�i;jjmax

�telse Aiþ1;j ¼ Ai;j

(4)

if /i;j ¼ 1; Tiþ1;j ¼Ti; j2

1þ j�i;jjmax�t

� �

else Tiþ1;j ¼ Ti;j(5)


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Equation (5) compared to Equation (4) preferencesadaptation of trabecular bone compared to cortical boneover each individual iteration. This was done to avoidoscillation of the predicted thickness values of the shellelements representing cortical bone during the initialiterations of the FE simulation.

With the aim of reducing the complexity of themodel, hence increasing its computational efficiency, thetrabecular cross-sectional area and shell cortical thicknessdomains were linearly discretised into 255 and 256 cate-gories, respectively. The trabecular cross-sectional areawas discretised between lower and upper limits corre-sponding to circular cross sections of radii 0.1 and 2 mm(cross sectional areas of pð0:1Þ2 mm2 and pð2Þ2 mm2).This range was considered to correlate on the mesoscalewith bone volume fraction measurements (the ratio ofbone volume to total volume (BV/TV)) recorded for tra-becular bone samples using lCT (Nägele et al. 2004).The cortical thickness was discretised between lower andupper limits of 0.1 and 8 mm (Stephenson & Seedhom1999; Treece et al. 2010). Based on Aiþ1;j or Tiþ1;j eachelement was assigned the cross-sectional area or thick-ness value corresponding to the closest discrete value ofthe respective truss and shell domains.

For the trabecular truss elements a 256th discretecircular cross section was added with a radius of 1lm,allowing for effective removal of elements from themodel, making their stiffness contribution to the modelnegligible while maintaining numerical stability, subjectto Equation (6).

if Ai;j ¼ pð0:1Þ2&j�i;jjmax � 250l�;Aiþ1;j ¼ pð0:001Þ2(6)

These elements were allowed to regenerate subject toEquation (7).

if Ai;j ¼ pð0:001Þ2&j�i;jjmax � 2500000l�;Aiþ1;j ¼ pð0:1Þ2(7)

where the value of 2500000l� was decided based on theratio of cross-sectional areas for radii of 0.1 mm and 1lm.

3. Results

Figures 6 and 7 show a selection of 5 mm thick slicesthrough the converged mesoscale femoral structuralarchitecture for the model subjected to a single load casetaken at the maximum hip JCF during walking and themodel subjected to the full loading regime described inSection 2.2.1, respectively. It can be seen that the struc-ture is more substantial in the full loading regime model,compared to the single load case model, in particular inthe distal region of the femur.

The resulting bone architectures for the single loadcase and the full loading regime models were compared

to literature and lCT imaging available to the authors.Figure 6 shows that in the proximal femur a substantialproportion of the clinically observed architecture can bepredicted based on a single load case. Figure 8 highlightsthe five normal groups of trabeculae identified by Singhet al. (1970) for the frontal proximal slice shown inFigure 6(a). Ward’s triangle (Singh et al. 1970; Kimet al. 2009) can also be seen. The cortex at the hip jointand at the greater trochanter is thin, thickening in theshaft and the inferior femoral neck as expected fromclinical observations. The arched arrangement of trabecu-lae in the proximal metaphysis is clear, consistent withGarden (1961). Truss elements with a radius of 0.1 mmare clustered at the hip joint surface allowing force trans-fer perpendicular to the cortex. In the femoral shaft it isobserved that the single load case (Figure 6(c)–(g)) pro-vides a reasonable prediction of cortical thickness in themedial and lateral aspects, but a poor prediction in theanterior and posterior aspects compared to clinicalobservations (Stephenson & Seedhom 1999; Treece et al.2010). A number of large trabecular elements runningparallel to the femoral shaft are observed within thethickness of the cortex on the medial and lateral aspects,while a number of smaller trabecular elements areobserved running perpendicular to the femoral shaft inthe anterior and posterior aspects. These results areconsistent with the femur bending about the anterior-posterior axis during walking. In the distal femur for themodel subjected to the single load case (Figure 6(h)–(i)),the trabecular structure is sparse in comparison to clini-cal observations (Takechi 1977). However, the structuralarchitecture that is observed in the transverse plane inparticular (Figure 6(i)) is consistent with the principaltrabeculae group reported by Takechi (1977) with tra-beculae originating from the posterior condyle andpatella articular surfaces, arranged close to parallel to themedial and lateral perimeter surfaces of the condyles.

Comparing the structural architecture of the proximalfemur obtained with a single load case (Figure 6(a) and(b)) to that obtained with the full loading regime(Figure 7(a) and (b)) it is observed that the full loadingregime results in increased trabecular architecture in thefemoral neck and greater trochanter in particular. Figure 9shows for the same selection of slices which of the dailyloading activities is most influential over the structuralarchitecture in different regions of the model subjectedto the complete loading regime.

The activity mapping (Figure 9(a) and (b)) indicatesthat walking and stair ascent are primary responsible forthe thickness of the cortex in the femoral neck, whilestair ascent and stand-to-sit are responsible for theincrease in the trabecular structure in the femoral neckcompared to the frame of maximum hip JCF duringwalking alone. The additional structure in the greatertrochanter region is influenced by stair descent and


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Figure6.

Selected5mm

slices

fortheconv

ergedmesoscale

structural

mod

elsubjectedto

asing

leload

case

takenat

maxim

umhipJC

Fdu

ring

walking

.Shellelem

ents

repre-

sentingcortical

bone

areshow

nin

grey,trusselem

ents

representin

gtrabecular

bone

with

aradius

r>

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areshow

nin

redandtrusselem

ents

with

aradius

r¼

0:1mm

are

show

nin

thebackgrou

ndin

blue.Truss

elem

entswith

aradius

r¼

1lm

areom

itted

forclarity.


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Figure7.

Selected5mm

slices

fortheconv

ergedmesoscale

structural

mod

elsubjectedto

thefullloadingregime.

Shellelem

ents

representin

gcortical

bone

areshow

nin

grey,

trusselem

ents

representin

gtrabecular

bone

with

aradius

r>

0:1mm

areshow

nin

redandtrusselem

ents

with

aradius

r¼

0:1mm

areshow

nin

thebackgrou

ndin

blue.Truss

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entswith

aradius

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areom

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forclarity.


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stand-to-sit activities. Of particular note is the increasedcortical thickness in the anterior aspect of the greater tro-chanter region due to stand-to-sit and to a lesser extentsit-to-stand. Comparing the predicted structural architec-ture in the femoral shaft for the single load case(Figure 6(c)–(g)) and the full loading regime(Figure 7(c)–(g)) it is observed that the inclusion of addi-tional load cases causes a thickening of the cortex aswell as the development of an increased number of largetrabecular elements running perpendicular to the femoralshaft in the anterior and posterior aspects. The activitymapping (Figure 9(c)–(g)) indicates that walking influ-ences the thickness of the medial cortex throughout themajority of the femoral shaft, stair ascent influences thecortex thickness in the anterior, posterior and lateralaspects through various regions of the femoral shaft,while stair descent and sit-to-stand have increasing influ-ence in the distal region of the femoral shaft. The resultsare consistent with the addition of activities which placethe knee in flexion causing bending about the medial-lateral axis. In the distal femur, the full loading regime

(Figure 7(h) and (i)) is seen to produce a considerableincrease in the trabecular architecture in comparison tothe single load case (Figure 6(h) and (i)). The activitymapping (Figure 9(h) and (i)) indicates that sit-to-standand stand-to-sit have a significant influence over the tra-becular architecture of the distal femur, with sit-to-standcausing the development of trabeculae in the lateral con-dyle in particular. It is observed that many of the tra-beculae associated with stand-to-sit run perpendicular tothe main trabecular structure providing additional stiff-ness to the structure as a whole. For the full loadingregime in particular, a large number of trabecular ele-ments with r ¼ 0:1 mm are observed in the femoral shaft(Figure 7(c)–(g)). It is thought that this is due to thedead zone limit being set at 250l�. Although not shownhere there was a significant reduction in the occurrenceof these elements when the limit was raised to reducethe range between the dead zone and the lazy zone.

4. Discussion

There are a preponderance of studies, several of whichare referenced in this work, which focus on adaptationof the proximal femur under a single or combined loadcase. As discussed by Skedros and Baucom (2007), itmay be suggested that there has been ‘an unfortunatehistorical emphasis on the human proximal femur’ withthe role of multiple load cases in influencing the struc-tural architecture of the femur obfuscated. The results ofthis work indicate that the inclusion of a range of dailyliving activities has a profound influence on the pre-dicted architectural structure not only of the distal femurbut also of the femoral shaft and regions of the proximalfemur.

It is observed in Figures 7 and 9 that in certainregions of the converged structural model trabecular trusselements are enclosed within the volume of cortical shellelements. In order to compare the converged full loadingregime model with lCT images, the visual thickness ofthe cortex in these regions was altered in incorporate thevolume of material contained in the enclosed trabecularelements. Figure 10 shows proximal and distal slices ofthe altered cortical thickness model alongside equivalentlCT slices for an adult male.

Examining the coronal slices of the proximal(Figure 10(a) and (b)) and distal femur (Figure 10(c) and(d)) it can be seen that the predicted structure comparesfavourably to the observed structure in the proximalregion, while the comparison is not as favourable for thedistal femur. There is a sparse trabecular structure beneaththe trochlear grove in the lCT slice, while the sameregion in the predicted model has quite a dense trabeculararchitecture. This may be due to the specific implementa-tion of the patellofemoral load applicator. In future work,the design of the patellofemoral load applicator will be

Figure 8. 5 mm slice for the converged mesoscale structuralmodel subjected to a single load case taken at maximum hipJCF during walking (as shown in Figure 6(a)), highlighting thefive normal groups of trabeculae identified by Singh et al.(1970) and Ward’s triangle.


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Figure9.

Selected5mm

slices

fortheconv

ergedmesoscale

structural

mod

elsubjectedto

thefullloadingregime.

Shellandtrusselem

ents

arecolour-m

appedaccordingto

the

activ

itymostinfluentialin

determ

iningtheirgeom

etry.Truss

elem

entswith

aradius

r�0:1mm

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Figure10

.Selected5m

mslices

forthealteredthickn

essconv

ergedmesoscale

structural

mod

elsubjectedto

thefullloadingregime(a,c,

e,g),show

nalon

gsidecorrespo

nding

lCTslices

(b,d,

f,h).Shellandtrusselem

ents

with

aradius

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0:1mm

arecoloured

light

grey.Truss

elem

ents

with

aradius

r¼

0:1mm

arecoloured

dark

grey.Truss

ele-

mentswith

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areom

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forclarity.Allslices

areshow

nas

semi-transparentto

high

light

thestructurethroug

hthedepthof

theslice.


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altered to better represent separate areas of patella contacton the two sides of the articular surface. The absence ofknee ligaments in the musculoskeletal and FE models isalso highlighted, potentially resulting in the scant trabecu-lar structure at the medial and lateral perimeters of thecondyles seen in the predicted model compared to thelCT slice. The superior part of the femoral head has adenser structure in the lCT slice than the predictedmodel. This may be due to the large area for force trans-fer provided by the hip load applicator which surroundsthe femoral head in the FE model, while the contact areabetween the femoral head and the acetabulum duringeach activity will be smaller in practice. In the slices run-ning parallel to the femoral neck (Figure 10(e) and (f)),there is remarkable agreement in the cortical thicknessdistribution between the predicted model and the lCTobservations, while the trabecular architectural arrange-ment is similar between the two slices. In the distal trans-verse plane slices (Figure 10(g) and (h)), the trabeculararrangement shows similarities between the predictedmodel and the lCT slice, although the trajectories aremore pronounced in running parallel to the perimeter ofthe condyles in the lCT slice. This may also be related tothe design of the patellofemoral load applicator with thetrabeculae focusing towards the trochlear groove inthe predicted model. Quantitative comparison betweenthe predicted model and the lCT observations isimpractical due to the difference in geometries betweenthe two femurs and the difficult in selecting equivalentcorresponding slices. However, with the exceptionsdescribed, it can be seen that there is reasonable agree-ment between the predicted and observed trabecular andcortical structural architecture.

The converged mesoscale structural model was foundto have a low computational cost (229113 elements,77229 design variables (nodal DOF), with a run time of52 s on a workstation PC with two Intel Xeon E5-26031.80 GHz processors and 16 GB of RAM). The adapta-tion run times for the model subjected to a single loadcase and the full loading regime were around 1 and10 h, respectively. Although run times are not reported,Tsubota et al. (2009) developed microscale models ofthe proximal femur with around 12 million elements at a175 lm resolution, and around 93 million elements at a87:5 lm resolution, reporting converged structures visu-ally similar to those found using the mesoscale structuralmodel. Boyle and Kim (2011) developed a similarmicroscale model of the proximal femur, utilising around23.3 million elements at a 175 lm resolution, equivalentto around 15.7 million design variables. Subjecting themodel to a single combined load case, they reported anadaptation run time of around 343 h on a computingcluster. Although the presented structural model has notbeen implemented at the microscale it seems reasonableto conclude that it is efficient, with a low computational

cost in comparison to microscale continuum models,while providing an improved structural representation incomparison to a macroscale continuum model with asimilar number of design variables. In future work poten-tial efficiency gains may be realised by generating an ini-tial structural model, with fewer elements, based onstress and strain tensors found using a macroscale contin-uum model, aligning structural elements with principalstress directions and basing initial sizing on principalstrain values (Geraldes & Phillips 2014).

A number of limitations must be acknowledged inthe study. While some of these are associated with theuse of the structural modelling approach many are gen-eric to the utilisation of musculoskeletal and finite ele-ment modelling methodologies in the combinedmodelling approach (Wagner et al. 2010; Cronskär et al.2015). While the approach is considered to provide amore physiological mechanical environment, comparedto models in which simplified boundary conditions andloading are utilised, deficiencies are exposed in bothmodelling methodologies through the process ofdeveloping corresponding models. The development ofload applicators, fixators and application of correspond-ing boundary conditions in the finite element modelhighlight the assumptions made in the development ofthe musculoskeletal model, treating the tibiofemoral andpatellofemoral joints as hinges, with the position of thepatella depending on the knee flexion angle, omitting thepossibility of displacements in other degrees of freedom.

It has been demonstrated in previous studies of thefemur that inclusion of physiological loading (Bitsakoset al. 2005; Speirs et al. 2007) and boundary conditions

Figure 11. Anatomical and effective lines of action, force vec-tors FA and FE, and insertions A and E respectively, for thegastrocnemius medialis muscle.


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(Phillips 2009) is crucial for bone adaptation simulationsas they have a significant influence on the resultingmechanical fields. Deriving the load cases for the FE sim-ulation from a musculoskeletal model with an identicalfemoral geometry is therefore seen as essential and appro-priate in the context that the estimated JCFs (Figure 5)are of comparable magnitude to those measured usinginstrumented hip prostheses (Bergmann et al. 2001) whilethe activation profiles found using the original muscu-loskeletal model (Modenese et al. 2011) are similar tomeasured electromyographic profiles. However, a limita-tion of the combined modelling approach is the use of anequilibrated load set, derived from a rigid multibody sys-tem, applied to a deformable FE model, with displace-ment compromising the equilibrium condition.

While wrapping surfaces and via points in the muscu-loskeletal model allow for a more physiological repre-sentation of muscles paths, compared to a straight lineapproach, they are not replicated as constructs in thefinite element model, resulting in a further compromise ofthe equilibrium condition. When a muscle force is appliedin the FE simulations, a choice must be made betweenusing the ‘anatomical’ line of action (originating from themuscle attachment on the bone surface) or the ‘effective’line of action (originating off the bone surface, whichdetermines its mechanical effect on the joints and its con-tribution to the equilibrium equations (Yamaguchi 2005)).This choice of muscle lines of action is illustrated for thegastrocnemius medialis muscle in Figure 11. In this work,the anatomical lines of action were used. In future work,the authors plan to incorporate wrapping surface con-structs within the finite element model in order to facili-tate the transfer of compressive and traction muscleloading to the bone (Grosse et al. 2007; Favre et al.2010). It is hypothesised that this will provide animproved strain environment with which to drive thebone adaptation algorithm and allow the use of the use ofthe effective line of action avoiding violation of the equi-librium condition. Although other studies, conducted fora range of anatomical constructs, have used similarmethodologies to that described here (Speirs et al. 2007;Halloran et al. 2010; Wagner et al. 2010; Kunze et al.2012), this limitation was either inapplicable due to theabsence of wrapping surfaces or not explicitly discussed.

The principal limitation of the structural modellingapproach as applied in this study is the use of truss ele-ments to represent trabecular bone, in preference to beamelements, or a combination of beam and shell elements.The decision to use truss elements was considered reason-able as under loading an optimised structure can beexpected to maximise axial forces while minimising bend-ing moments and shear forces, as these are less efficientlyresisted through the cross section of a structural element,while truss elements are computationally efficient incomparison to beam elements. In order to assess the effect

of using truss rather than beam elements, the convergedmodel was adapted by replacing the truss elements in turnwith two-noded hermite-cubic Euler–Bernoulli beam ele-ments and three-noded quadratic Timoshenko elements,with the third node placed at the midpoint of the element.The original and adapted versions of the converged modelwere then subjected to a simplified load case, with a dis-tributed vertical load applied at the femoral head, andfixed boundary conditions applied at the knee joint. Thedisplacement in both the beam models was found to be1.4% greater than the displacement in the truss model,while all three models deformed in a similar manner. TheTimoshenko and Euler–Bernoulli beam models had runtimes of 214 and 189 s, respectively, on the workstationPC. A limitation of the structural model, albeit one that isinherent to the majority of phenomenological boneadaptation studies, is the adoption of particular values forthe target strain, the lazy zone and the dead zone. It ispossible that these values should be varied for differentregions of the skeletal system, while they may also beinfluenced by a multitude of factors including age, sex,ethnicity and disease conditions such as osteoarthritis andosteoporosis. An additional limitation of the structuralmodel is the adoption of particular ranges for the trabecu-lar cross-sectional area and the cortical thickness. Whilethe range of cortical thickness may be justified by compar-ison to clinical observations (Stephenson & Seedhom1999; Treece et al. 2010), the range of trabecular cross-sectional area was considered reasonable given the mesos-cale nature of the model. Future work will assess theapplication of the approach at the microscale. Thedevelopment of a microscale structural model withphysiological length and thickness ranges (Hildebrandet al. 1999; Nägele et al. 2004) for individual trabeculaewill allow for direct comparison with lCT data.

A robust structural approach to bone adaptation hasbeen presented as part of a combined musculoskeletaland finite element modelling framework. Future workwill extend the approach to the other skeletal structuresof the lower limb including the pelvis (Phillips et al.2007). The work has highlighted the importance ofincluding multiple load cases in bone adaptation studies,with a range of daily loading activities influencing thestructural architecture of different regions of the femur. Itis believed that the work has relevance to the study andpotential treatment of diseases of the musculoskeletalsystem including osteoporosis and osteoarthritis. As anexample, the risk of femoral neck fracture in osteoporo-sis may be reduced by introducing additional activities,other than walking, promoting bone structure formationin the femoral neck, into a protective exercise regime(Martelli et al. 2014). Preliminary work by the authorshas also indicated that the structural approach hasapplication in the computationally efficient modelling offracture initiation and progression due to traumatic


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loading such as that experienced during falls or jumpsfrom height, vehicular collision and blast injury.

The development of a mesoscale structural model,rather than a continuum model, allows for additivemanufacturing of the resulting structure. With suitablemanipulation of the bone adaptation algorithm, three-dimensional printing in materials including a wide rangeof polymers and metals, permits the manufacture offrangible bone simulants for use in experimental testing,as well as the potential design and manufacture of biore-sorbable scaffolds and orthopaedic implants, sympatheticto the remaining skeletal structural architecture.

AcknowledgementsThe authors thank the Human Performance and Musculoskele-tal Biomechanics groups at Imperial College London for assis-tance with the gait analysis, and Imperial Blast for providingthe lCT data. The authors also thank the volunteer. Theauthors acknowledge and thank Alfred Thibon for the workcarried out during his MSc Dissertation in the Department ofCivil and Environmental Engineering at Imperial CollegeLondon, which assisted in developing the work presented here.

Disclosure statementNo potential conflict of interest was reported by the authors.

FundingThis work was supported by the Royal British Legion Centrefor Blast Injury Studies at Imperial College London, and theEngineering and Physical Sciences Research Council through aDoctoral Training Award and a Project Award [grant numberEP/F062761/1].

ORCID

Andrew T.M. Phillips http://orcid.org/0000-0001-6618-0145Claire C. Villette http://orcid.org/0000-0002-1638-258XLuca Modenese http://orcid.org/0000-0003-1402-5359

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10.1243/095441199153488910.1243/095441199153488910.1109/TBME.2006.87947310.1109/TBME.2006.87947310.3109/1745367770899481610.1016/j.media.2010.01.00310.1016/j.media.2010.01.00310.1016/j.jbiomech.2009.02.03010.1016/S0021-9290(98)00177--810.1016/S0021-9290(98)00177--810.1016/S0021-9290(98)00118-310.1002/jor.2236410.1002/jor.2236410.1016/j.bone.2006.06.03310.1016/j.jbiomech.2005.10.02710.1016/j.jbiomech.2005.10.02710.1243/0954411036057931210.1243/0954411036057931210.1007/s11999-011-2042-410.1108/1573610101109511810.1007/s11999-011-2041-510.1016/S0021-9290(01)00222--6

Abstract1. Introduction1.1. Continuum modelling approaches1.1.1. Macroscale continuum FE modelling1.1.2. Microscale continuum FE modelling

1.2. Structural modelling approaches

2. Methods2.1. Musculoskeletal modelling2.2. Finite element base model2.2.1. Loading2.2.2. Boundary conditions

2.3. Bone adaptation algorithm

3. Results4. DiscussionAcknowledgements Disclosure statementFundingORCIDReferences

Femoral bone mesoscale structural architecture prediction using...

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