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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
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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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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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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.
46 A.T.M. Phillips et al.
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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.
48 A.T.M. Phillips et al.
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Figure5.
Hip
JCFsderivedfrom
themusculoskeletal
mod
elforsing
lecycles
of(a)walking
,(b)stairascent,(c)stairdescent,(d)sit-to-stand
and(e)stand-to-sitareshow
nas
blacksolid
lines.Selectedfram
esfrom
each
activ
ity,used
intheFEsimulations,areindicatedusingsolid
circles.
Average
hipJC
Fsacross
allsubjects
foralltrials,as
recorded
andrepo
rted
byBergm
annet
al.( 200
1)forthesameactiv
ities,areshow
nas
reddash-dot
lines
(fulldetails
ofthecustom
ised
averagingprocessareavailableon
theHIP98
data-
setaccompany
ingBergm
annet
al.( 200
1)).Due
todifferencesin
theselectionof
thestartandfinish
points
ofsomeactiv
itycycles,theaveragehipJC
Fsrepo
rted
byBergm
ann
etal.( 200
1)fortheseactiv
ities
areshifted(stairdescent)or
plottedov
erapprox
imatelycorrespo
ndingperiod
s(sit-to-stand
andstand-to-sit)
foreasier
visual
comparison.
International Biomechanics 49
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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>
0:1mm
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
elem
entswith
aradius
r¼
1lm
areom
itted
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
areom
itted
forclarity.
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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
r>
0:1mm
arecoloured
light
grey.Truss
elem
ents
with
aradius
r¼
0:1mm
arecoloured
dark
grey.Truss
ele-
mentswith
aradius
r¼
1lm
areom
itted
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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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