Contribution of muscular weakness to osteoporosis: …msbm/resources/Clin_Biomech... · 2011. 7. 8. · the different parts of the proximal femur can be provided. To model the mechanical

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Clinical Biomechanics 20 (2005) 984–997

Contribution of muscular weakness to osteoporosis:Computational and animal models

M. Be�ery-Lipperman, A. Gefen *

Department of Biomedical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Received 19 August 2004; accepted 27 May 2005

Abstract

Background. Chronic weakness of the femoral musculature with old age may result in prolonged exposure of bone to criticalunderstressing and, thus, cause osteoporotic changes. This study aims at quantifying long-term changes in thickness and mechanicalproperties of trabecular bone at the proximal femur due to muscular weakness.Methods. We utilized computational models of typical planar trabecular lattices at the proximal femur for simulating long-term

changes in morphological and mechanical properties of trabecular bone. Incorporating cellular communication network with osteo-cytes as mechanosensors, the models were able to mimic mechanotransduction and consequent thickening and/or thinning of indi-vidual trabeculae in response to altered gluteus muscle and hip joint loads. We also studied a rat model (n = 14) in which wesurgically detached the gluteus muscle, to �50% or completely.Findings. The computational simulations showed that when the force of the gluteus decreased (with or without simultaneous

decrease in hip joint load), the most dramatic degradation in bone density, strength and stiffness occurred at the greater trochanter.Animal studies also demonstrated significant thinning of femoral trabeculae after 19 weeks of adaptation. Specifically, Tukey–Kra-mer analysis showed that rats subjected to partial surgical detachment of the gluteus had femoral trabeculae that were 22% thinnerthan controls (P < 0.05).Interpretation. The present study showed that in both the computer and animal models, manipulation of muscle loading pro-

duced a significant stimulus for bone to adapt, i.e., a stimulus that is beyond its irresponsive �lazy zone�. Accordingly, the resultsobtained herein indicate that muscular weakness may be an important factor contributing to osteoporosis.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Trabecular bone remodeling; Tissue adaptation; Biomechanical model; Femoral neck fracture; Finite element analysis

1. Introduction

Muscular weakness, defined as the inability of mus-cles to maintain a reasonably expected force output, in-creases the risk for hip fractures because of its effect onthe risk of falls. It has been suggested that muscularweakness and inactivity may also be responsible forsome of the bone loss and osteoporotic-like changes intrabecular bone that were previously associated with

0268-0033/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.clinbiomech.2005.05.018

* Corresponding author.E-mail address: [email protected] (A. Gefen).

aging per se (Sinaki, 1998). Specifically, muscular weak-ness may alter the mechanical stress flow in bones, there-by causing understressing or overstressing of specificskeletal sites and thus interrupting the homeostatic con-ditions for normal bone mass maintenance.

Deprivation of mechanical stress was shown to in-duce loss of cortical, and mainly, of trabecular bone atthe understressed sites (Riggs et al., 1982; Weiss et al.,1991; Seeman, 2003). Ongoing resorptive response willoccur only if exposure to stress deprivation is drasticand prolonged enough so that bone is required to func-tion beyond its ‘‘lazy zone’’ of mass regulation (where

M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997 985

bone is said to be insensitive to small perturbations inthe level of mechanical stimuli (Huiskes et al., 1987;van Rietbergen et al., 1993)). We hypothesize thatchronic weakness of the musculature of the femur,e.g., being the consequence of muscular atrophy withold age, may cause prolonged exposure to criticalunderstressing beyond the ‘‘lazy zone’’ of bone mass reg-ulation and thus, trigger trabecular bone resorption.

To test this hypothesis, we first utilized biomechani-cal computer models of typical planar trabecular latticesat the proximal femur. The models simulated long-termchanges in morphological and mechanical properties oftrabecular bone, due to gluteal muscle weakness. Incor-porating a cellular communication network with osteo-cytes as mechanosensors, the models were able tomimic mechanotransduction and consequent thickeningand/or thinning of individual trabeculae in response toaltered gluteal and hip joint forces. Second, we con-ducted animal studies using a rat model in which we sur-gically manipulated the gluteal attachment. Thesereductions in musculoskeletal loads in the computa-tional and animal models are considered to representchronic underloading of the proximal femur.

2. Methods

To investigate the adaptation response of trabecularbone to muscular weakness, a hybrid methodologyinvolving a combination of computer and animal studieswas adopted. The computer studies were aimed to pre-dict the association between the external (gluteus mus-cle, hip joint) mechanical loading applied to the grossfemur and internal mechanical stresses in bone tissueat the continuum and tissue levels, which constitutethe pattern of bone adaptation in response to the exter-nal loads. The animal studies were aimed to mimic somesimulation conditions of gluteal weakness experimen-tally, so that the computer model predictions forhumans could be supported.

2.1. Computer studies

The computer studies of trabecular adaptation undermuscular weakness must incorporate all the bone andmuscle structures that are involved in maintaining bonemass, including geometry of the whole bone (corticaland trabecular), the loads applied to it, the process ofmechanotransduction in bone (Huiskes et al., 2000)and the response to abnormal loads at the cellular andtissue levels. Accordingly, the effects of musculoskeletalloads are studied in different (but complementary) modelsystems with increasing characteristic scales, and incor-porating different functional tissue units: from singlebone cells, to an individual trabecula, to a lattice of tra-beculae, to a continuum of trabecular bone. Thus, the

computer models span from a scale of lm (single cells)to mm scale (cell-matrix constructs in a single trabeculaand in small-size trabecular lattices) and to cm scale(the proximal part of the femur). This hierarchial systemof models is used to study an effect of altered loading onmechanical stimuli applied to individual bone cells, theprocesses occurring in the matrix surrounding these cellsas a result of the stimuli and the overall changes causedin the bone architecture. Through coupling of these phe-nomena, which take place in different scales, we couldextrapolate information to the level of bulk tissue andhence, give clinically relevant predictions on tissue degra-dation and osteoporotic-like changes in the femur undergluteus muscle weakness conditions.

The basic element in the hierarchial system of compu-tational models is the single trabecula (Dagan et al.,2004). For this trabecula, we developed an adaptationalgorithm which mimics the physiological process ofdeposition or resorption of bone mass in response toaltered forces that are applied through the ends of thetrabecula. The process of bone mass gain or loss is reg-ulated at the cellular level of the trabecula. This wasaccounted for by including ‘‘bone cell’’ elements in thetrabecula, which are sites where strain and strain energyper unit bone mass (SEPUBM) generated due to theloading from the ends are monitored. It is assumed thatthe bone cells communicate the strain informationthrough the canalicular system of the trabecula, andso, an overall SEPUBM stimulus is calculated as a driv-ing force for adaptation sensed accumulatively by allcells (Huiskes et al., 2000). This driving force determinesthe response in terms of extent of mass gain or loss. Atrabecula thickens if forces applied at its ends cause anoverall driving force that is significantly greater thanthe homeostatic conditions. Likewise, a trabecula be-comes thinner if these forces cause overall driving forcethat is significantly less than that of homeostasis. Aftersuccessful development of the single trabecula model,this model was expanded for describing small trabecularlattices in specific regions at the proximal femur. Pathsof trabeculae in these lattices were identified on an ante-rior–posterior high-resolution X-ray and trabeculaewere positioned to align with these paths. Models atthe lattice level of hierarchy were loaded at their bound-aries with abnormally altered loads, which triggeredadaptation of trabeculae in the lattice. To determinethese loads, we developed a model of the whole proximalfemur, which is set at the highest level of hierarchy. Thewhole femur model is loaded by abnormal gluteus mus-cle and hip joint loads. These interventions cause under-loading or overloading at specific sites of the femur,which can be quantified using the finite element (FE)method. The extent of overloading or underloading, interms of a new system of local loading, is fed to themicro-scale trabecular lattices, inducing adaptation ofindividual trabeculae which is analyzed using a Matlab

Fig. 1. Modeling individual trabeculae: (a) Potential profile shapes oftrabeculae in proximal femora of humans (Dagan et al., 2004). Theaverage thickness and length were set as 283 and 1000 lm, respectively.(b) Distribution of 3610 osteocytes (OCY) in a model of an individualtrabecula. Average distance between neighboring OCY along theradial direction was 2.5 lm. Average distance between neighboringOCY along the longitudinal direction was 91 lm.

986 M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997

computer code (Mathworks Co., MA, USA). When newhomeostatic conditions are obtained in the lattices, theapparent lattice properties (density, stiffness, strength)are specified, so that evaluations of bone quality overthe different parts of the proximal femur can beprovided.

To model the mechanical behavior and adaptation re-sponse of individual trabeculae, several assumptionswere taken. First, we assumed that at the tissue level,trabecular bone is a homogeneous and isotropic mate-rial, with density of 1.7 g/cm3 and elastic modulus of20 GPa (Turner et al., 1999). Second, we assumed thatbone mass gain or loss occurs only on external surfacesof a trabecula (which are accessible to osteoblasts orosteoclasts) but mechanosensing occurs within the tra-becula, in osteocytes (OCY) (Huiskes et al., 2000).Third, we assumed that bone gain or loss is regulatedby the magnitude of SEPUBM communicated byOCY to the surface of a trabecula (Huiskes et al.,2000). Bone gain or loss occurs only when significantSEPUBM changes, i.e., exceeding ±25% the homeo-static SEPUBM, occur (Weinans et al., 1993). Theobjective function of the process of mass/geometry reg-ulation is hence to return to a state where all SEPUBMsignals in all trabeculae are again at the homeostaticlevel. Fourth, we idealized individual trabeculae to besymmetric with respect to their central and longitudinalaxes. The geometrical model of individual trabeculae inour simulations is given elsewhere (Dagan et al., 2004).Briefly, we developed a parametric model describingthe profile of a trabecula as a co-sinusoidal surfacebased on the average thickness and length of the trabec-ula. This geometrical model is able to describe the com-plete spectrum of shapes of trabeculae in humanproximal femora (Dagan et al., 2004; Fig. 1(a)).

In order to simulate the geometrical and mechanicaladaptation of a single trabecula to an altered compoundloading system, where compression forces F and/orshear forces V and bending moments M1 and M2 aremodified (Fig. 1(a)), we formulated a software code(Matlab 6.1) that runs an algorithm as follows. Withinthe volume of revolution representing a single trabecula(0.05 mm3 for the neutral condition before adaptation;Puzas, 1996; Dagan et al., 2004), we distributed 3610mechanosensors representing OCY (Fig. 1(b)). Thisnumber of OCY per trabecula generally conforms re-ported density of OCY in mammalian trabeculae (inrats: 93,200 per mm3 and in bovine 31,900 per mm3;Mullender et al., 1996a), however, it is mildly higherthan that reported by Mullender et al. (1996b) forhumans (�1000 OCY per 0.05 mm3), to compensatefor errors in histology-based cell counting which tendto underestimate cell numbers due to finite histologicalslice thicknesses.

The general formulation of the strain energy densityUi at the location of OCY i is 2Ui = rijeij, however, the

symmetry assumed for trabeculae along their z and raxes (as defined in Fig. 1) simplifies the above generaltensor formulation so that only the normal stressesalong the longitudinal direction z of the trabecula(rzz) and the shear stresses in the cross-section of thetrabecula (szr) contribute to the local strain energydensity Ui

Ui ¼ rzzezz þ szrczr ¼1

E� FA�Mr

I

� �2

þ 1

GVA

� �2

ð1Þ

where the material properties of bone tissue are E theelastic modulus (20 GPa; Turner et al., 1999), and G

the shear modulus (7.7 GPa). The term in the firstparentheses is the strain energy density from axialstrains along the z-axis of a trabecula rzz, due to axialstress F/A and due to bending stress Mr/I, where A(z)and I(z) are the local cross-sectional area and localmoment of inertia along the z-axis, respectively. Theterm in the second parentheses is the strain energydensity due to shear stress V/A in the cross-section.Axial forces F in Eq. (1) can be either tensile or com-pressive. The stress stimulus Si at the location of OCYi is calculated on each iteration by Si = Ui/qtr (Weinanset al., 1992) where qtr is the last tissue density valueobtained for the trabecula. Each OCY signals to thelining cells (LnC) located at the surface of the trabeculain proportion to the SEPUBM at its location, andsignals over the external layer are accumulatedby a stimulus value (Mullender and Huiskes, 1997;Mullender et al., 1998):

M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997 987

UðtÞavg ¼

Xsurface OCY

j

Xinner OCY

i

fij � ðSi � ð1� kÞSref iÞ( ),

surfaceOCY ð2aÞ

if Si > ð1þ kÞSref ) dqdt

> 0;

i:e:; thickness of trabeculae increases ð2bÞ

if ð1� kÞSref 6 Si 6 ð1þ kÞSref ) dqdt

¼ 0;

i:e:; thickness of trabeculae is not affected ð2cÞ

if Si < ð1� kÞSref ) dqdt

< 0;

i:e:; thickness of trabeculae decreases; ð2dÞ

where k = 0.25 is the half-width of a �lazy zone� in whichno adaptive response occurs (Huiskes et al., 1987; vanRietbergen et al., 1993), Sref is the reference SEPUBMcalculated for a state of stress/strain where the femuris loaded with 100% gluteal muscle and 100% hip jointloads (i.e., normal musculoskeletal load level) and fij isa spatial influence function, given by (Mullender, 1997;Mullender and Huiskes, 1997; Mullender et al., 1998):

fij ¼ e�dij=D ð3ÞEq. (3) represents the attenuation of signals from distantOCY as these are transmitted via the network of canalic-uli in the trabecula. Hence, dij is the distance of mecha-nosensor i from the LnC j located at the surface, and D

is an attenuation coefficient taken as 0.1 mm. The coef-ficient D represents the distance from OCY at which theeffect of mechanosensing of that OCY on activation ofsurface bone cells (osteoclasts/osteoblasts) is reducedto e�1 (�37%). It is also required that the influence func-tion (Eq. (3)) will be assigned a value of unity when thestimulus signal is monitored directly at the OCY andthat the function will decay asymptotically to zero forlarge distance from the osteocyte. Considering that tra-beculae in our lattice models are all �1 mm long, themaximum possible value of dij can be specified, andwas found to be 0.824 mm. We therefore need to selecta value D that is less than 0.824 mm to obtain an effec-tive influence function. A value D = 0.1 mm providedrealistic adaptation responses in preliminary simula-tions. Moreover, the final thickness in preliminary simu-lations with a single trabecula, aimed at testing thesensitivity to model parameters, did not differ when D

was modified in the range of 0.001–0.5 mm. All trabec-ular lattice simulations were therefore performed withD = 0.1 mm.

In response to a modified compound loading, an al-tered stimulus will be calculated. Outside of the lazyzone, this stimulus, transmitted by the OCY network,will trigger osteoblastic/osteoclastic activation (Huiskeset al., 1987; van Rietbergen et al., 1993). The effective re-sult of many resorption bays (generated by osteoclasts)

or deposition sites (by osteoblasts) is manifested as achange in the peripheral density qs (of the external layer)of the trabecula over a time interval Dt, according to therelation (Mullender, 1997)

qsðt þ DtÞ ¼ qsðtÞ þ s � UavgðtÞ ð4Þwhere s is a constant converting stress stimulus to den-sity change, which is taken as 105 Pa m�2. Interdepen-dency exists between the number of OCY, theattenuation coefficient D and the conversion constants. The values given here are selected to produce realisticand refined simulations of adaptation of trabeculae inthe proximal femur, which converge to a homeostaticcondition. Specifically, we found that s values greaterby two orders of magnitude induced too large differ-ences adaptation steps and hence sometimes, causedthe simulation to miss the point of convergence (ornew homeostasis). Likewise, s values which were smallerby two orders of magnitude required too much compu-tation time, with no beneficial additional information onthe adaptation process and outcome. With the selectedadaptation rate, s = 105 Pa m�2, all simulations ade-quately converged with a reasonable computation time(�4 min of adaptation per trabecula in a lattice, whenboundary loads are reduced by 50%, on a Pentium IIIPC with 512 kb RAM).

As a consequence of the change in density of the tra-becula (Eq. (4)), the elastic modulus of its peripheral tis-sue layer will be modified according to the relation(Currey, 1988; Rice et al., 1988)

Esðt þ DtÞ ¼ a � qsðt þ DtÞ½ �b ð5Þwith constants a = 11.4 and b = 1.06. The effective elas-tic modulus of the whole trabecula is calculated using acomposite material model, with different properties forthe surface layer (that is subjected to osteoblastic/osteo-clastic activity) and internal tissue

Etr ¼V tr

s Etrs þ V tr

intEtrint

V trs þ V tr

int

ð6Þ

where Etrs is the elastic modulus of the surface layer of

the trabecula from Eq. (5), V trs is the volume of the sur-

face layer (Fig. 1(a)), Etrint is the modulus of tissue under

the surface layer and V trint is the volume of internal tissue

(i.e., volume of the trabecula excluding the surface layer,Fig. 1(a)). Bone tissue density qs of 1.7 g/cm3 andsurface elastic modulus Etr

s of 20 GPa were set at eachtrabecula as initial (homeostatic) conditions. The com-posite material model (Eq. (6)) was then employed toupdate the effective elastic modulus of trabeculae Etr

in computational iterations during the adaptation simu-lations. The elastic modulus of internal bone tissue in atrabecula, Eint

tr , was kept constant, 20 GPa, correspond-ing to bone tissue density of 1.7 g/cm3 (Eq. (5)).

The process of calculating the OCY mechanostimuliand fitting the peripheral density and elasticity with

Fig. 2. The gross (finite element) model of the human proximal femur:(a) The loading system, constraints and meshing. Five regions ofinterest (ROIs) are marked with rectangles, and the trabecular latticein ROI 2 is shown. For ROI 2, the circle and triangle mark constraintsfor vertical displacement and all translational displacements, respec-tively. Force components at each trabecular junction on the perimeter(marked with arrows) are calculated from the gross femur model (inNewtons). (b) Homeostatic apparent density of trabecular bone ateach ROI, calculated as the ratio of tissue weight to the volumeoccupied by the respective trabecular lattice. (c) von Mises stressdistribution under the musculoskeletal loading shown in (a).

988 M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997

respect to the overall sensed stimulus value (Eq. (2a))continues until the local tissue density at the surfacelayer of the trabecula exceeds 1.7 g/cm3 or drops below0.1 g/cm3. In the first case, trabecular thickening willtake place, and the next thicker trabecular profile (asin Fig. 1(a)) will be selected for the continuing iterations,in order to simulate the resulted mass deposition. In thesecond case, trabecular thinning will take place, and thenext thinner profile will be selected (Fig. 1(a)) to accountfor mass resorption. The simulation terminates whenone of the following conditions is met: (i) a steady stateis reached, i.e., the mechanical stimulus for adaptation(Uavg) is within the lazy zone; (ii) the trabecula is totallyresorbed; (iii) the trabecula breaks, i.e., maximal stressesexceed the ultimate stress (which was taken as that ofcortical bone tissue): 200 MPa in compression and130 MPa in tension (Reilly and Burstein, 1975).

The above-described adaptation algorithm for a sin-gle trabecula was expanded to allow simulation of adap-tation of planar trabecular lattices containing 38trabeculae (20 trabeculae in the longitudinal directionand 18 trabeculae in the transverse direction of the lat-tice). Five regions of interest (ROIs) in the trabecularbone of the proximal femur were selected on a longitu-dinal cross-section of the femur (Fig. 2(a); Table 1), atthe lateral aspect of the proximal femur (ROI 1), lowerfemoral neck (ROI 2), higher femoral neck (ROI 3),greater trochanter (ROI 4) and ball of the joint (ROI5). At the scale of the whole bone, trabecular bone canbe treated as a continuum and predictions of local inter-nal stress are meaningful as long as the ROI is not smal-ler than 5 · 3 mm (Harrigan et al., 1988). All 5 latticeswere approximately 5 mm long (length of 5 trabeculae)and 3 mm wide (length of 3 trabeculae) and so, FE anal-ysis of the gross proximal femur is a valid method forproviding the loading conditions at the boundaries ofthese lattices, both in the homeostatic loading conditionand during muscular weakness.

We used a high-resolution anterior–posterior X-rayimage of the human right proximal femur (Weiss,1988) for developing the computational model. Thehigh-resolution X-ray could distinguish between thecortical and trabecular components, and the trabecularorientation at all ROIs was also demonstrated. The con-tours of the bone cortex and trabecular content were de-

Table 1Location of the regions of interest (ROIs) associated with the trabecular ori

Location

ROI 1 Lateral proximalROI 2 Lower neckROI 3 Higher neckROI 4 TrochanterROI 5 Ball of joint

Trabecular path classification is adopted from Singh et al. (1970) and Lin et

tected and segmented into solid sections (SolidWorks,2003) and the contour data were transferred to a FEsolver for plane stress analysis (NASTRAN 2001,MSC software Co., CA, USA) (Fig. 2(a) and (c)). Theproximal femur model was constrained for vertical andhorizontal displacements and for planar rotation at thedistal part of the shaft (Fig. 2(a)). Utilizing St. Venant�s

entation

Dominant trabecular paths

Secondary tensile group; secondary compressive groupSecondary compression; principal tensile groupPrincipal compressive group; secondary compression groupGreater trochanter groupPrinciple compressive group

al. (1999). Volume of bone only per volume of tissue.

M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997 989

principle, the five ROIs are selected to be far enoughfrom the base of the shaft (i.e., at a distance exceedingthe shaft thickness; Beer et al., 2002). Hence, artificial ef-fects on simulated bone adaptation response as relatedto selection of boundary conditions are negligible (Beeret al., 2002).

The model was loaded with a combination of muscleand joint forces (Fig. 2(a)) which included an equivalentgluteal muscles force of 703 N whose resultant wasdirected 28� medially, and a joint reaction force of2317 N whose resultant was directed 24� laterally (Tsu-bota et al., 2002). This simplified joint-muscle force sys-tem was selected because it involves just one variable todefine muscular weakness, % decrease in force outputof the gluteus, which is beneficial in a first model thataims at determining the overall effect of muscle weak-ness on bone response, considering that there are nodata in the literature that quantitatively characterizeweakness of individual femur muscles, or muscle com-pensation patterns in patients suffering muscular weak-ness. The joint force was sinusoidally distributed overthe joint surface and the muscle force was sinusoidallydistributed over the greater trochanter to avoid non-physiological stress concentrations (Tsubota et al.,2002).

At this whole bone scale, bone material was assumedto be homogenous, isotropic and linear-elastic withYoung�s moduli of 20 and 1 GPa for the cortical andtrabecular components, respectively (Turner et al.,1999). Poisson�s ratio was taken as m = 0.35 for bothcomponents. The proximal femur model was meshedto a total of 76,392 three-node triangular linear elements(Fig. 2(a)). The cortex was meshed into 6606 elementsand the trabecular bulk was meshed into 69,786 ele-ments (Fig. 2(a)). We verified that finer meshes did notaffect the stress predictions. The mesh of each ROIwas further refined by setting the distance betweennodes to 0.001 mm, which is 0.1-fold the distance be-tween nodes outside the ROIs. The interface betweencortical and trabecular bone was set as �no slip�.

The gross femur model was first solved for the nor-mal magnitudes and orientations of hip joint and glutealmuscle loads (Fig. 2(c)). The results from the gross fe-mur model are the forces acting on the perimeters ofthe five selected ROIs. The loads on perimeters of theROIs under normal muscle/joint femoral loads were as-signed as the homeostatic loads on the lattices, i.e., theSEPUBM stimuli produced by these loads (Eq. (2a))were set as the ones for which no adaptation responseoccurs. Subsequently, we conducted simulations inwhich the gluteus muscle and hip joint loads weremanipulated, separately or together, as described later.Manipulating the muscle and joint loads in the grossfemur model, in order to simulate various degrees ofseverity of muscle/joint dysfunction resulted in changesin continuum stress in the trabecular bulk and therefore,

in many cases, modified the forces acting on the perim-eters of the ROIs. By analyzing the adaptation of eachtrabecular lattice to the boundary conditions of modi-fied local forces we were able to provide site-specific pre-dictions of bone quality and mechanical properties invarious scenarios of muscular weakness. The methodof applying the forces acting on the perimeters asboundary conditions to the lattices corresponding tothe 5 ROIs is described next.

We constructed 5 planar trabecular lattices corre-sponding to the selected ROIs in the 2D gross femurmodel by arranging individual trabecula elements(Dagan et al., 2004) so that they aligned with the dom-inant trabecular paths at each ROI site (e.g., see trabec-ular lattice of ROI 2 in Fig. 2(a)). For this purpose, weidentified the trabecular paths on the same high-resolu-tion X-ray that was used to construct the gross FE fe-mur model (Weiss, 1988). Next, polynoms of the second-order were fitted to the trabecular paths crossing eachROI using the method described in Gefen and Seliktar(2004). By arranging 38 trabeculae along the polynomi-als derived from the radiograph at a site (Fig. 2(a)) wecompleted each lattice. These trabeculae slightly varyin length (around 1 mm) because they had to be posi-tioned along curved polynomial functions that describethe trabecular paths; as with real adjacent trabecularpaths, these polynomials do not maintain a constant dis-tance between them.

At the homeostatic (initial) condition all trabeculaewere assign the same nominal thickness (283 lm).Therefore, differences between the 5 lattices at the initial(homeostatic) condition were 2-fold: (i) the 5 latticesdiffered in orientation and length of trabeculae, (ii)the 5 lattices differed in boundary conditions, whichcorresponded to the site-specific force distribution inthe femur, during homeostasis as well as if homeostasiswas interrupted. Lattices at the different ROIs wereloaded with external forces acting at the trabecularjunctions on the perimeter of each ROI (Fig. 2(a)),which resulted from the FE analysis of the gross femurmodel. We assumed that all forces are transferredthrough the trabecular architecture and that the mar-row contained in the inter-trabecular spaces carries noloads. To allow solution of internal forces in the inter-nal junctions of a lattice, we used structural analysis.Through this method, it is first necessary to calculatethe forces which act at each junction of the lattice. Be-cause the boundary forces acting on the perimeter of thetrabecular lattice are provided for each femoral loadingcondition by the FE gross femur model, internal forcescan also be obtained.

A stiffness matrix was first calculated for each indi-vidual trabecula. For this purpose, trabeculae were con-sidered as uniform rods subjected to axial and flexuralstresses, and their displacements were described usingthird-order polynomial shape functions. This process,

990 M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997

which led to construction of a global stiffness matrix [K]for each planar trabecular lattice, is detailed in Chapter7 of McGuire et al. (2000). The global stiffness matrix ofthe lattice [K] is the assembly of all stiffness matrices [Ki]of the individual trabeculae. Next, we calculated a ma-trix of displacements [u] of the junctions in the latticeunder external loading [P], as:

fug ¼ ½K��1fPg ð7ÞFinally, the vector of loads {Fi} acting on the edges eachtrabecula i was calculated from the structural displace-ments at the edges of the trabecula {ui} and its individ-ual stiffness matrix [Ki]:

fF ig ¼ ½Ki�fuig ð8Þ

Thus, global loads taken by the whole proximal femurwere converted to microstructural forces in the trabecu-lae of the femur in each of the five ROIs. This allowed tofurther calculate strain energy densities and SEPUBMdistributions at each trabecula (Eq. (1)) in order to sim-ulate the adaptation of entire lattices. The algorithm ofadaptation of individual trabeculae described earlier washence extended for predicting structural changes in atrabecular lattice in response to a disruption in the exter-nal loading matrix [P]. Expending the adaptation algo-rithm for adaptation of a lattice required additionalcondition for terminating the adaptation simulations,which was considered together with those already men-tioned for the single trabecula case: A trabecula in thelattice cannot be thicker than the given inter-trabecularspace. This implies that when all other (surrounding)trabeculae are in homeostatic condition, a trabeculawhich undergoes adaptation cannot be thicker than500 lm at its base, or cannot be of average thicknessgreater than 417 lm. Iteratively, the algorithm calcu-lated internal forces in the junctions between trabeculaeand the resulted thickness and properties of all trabecu-lae for each ROI until reaching a homeostatic conditionin all the ROIs.

To simulate the effect of muscular weakness on tra-becular bone in the proximal femur, we manipulatedthe gluteal loading and the hip joint force in the grossFE model of the femur so that boundary conditionson perimeters of each lattice at the ROI were alteredfrom the neutral, homeostatic condition. Specifically,we reduced the force output of the gluteus muscle (inseparate simulations) to 80%, 50% and 0% its normalforce output (703 N). Since muscular weakness is poten-tially accompanied by reduced hip joint reaction forces,we repeated the above simulations for joint reactions of100%, 90% and 80% of the normal joint reaction magni-tude (2317 N).

At the end of adaptation simulations, apparentdensity of trabecular bone was calculated for eachROI as the product of mean bone tissue density andbone volume fraction. Apparent mechanical properties

were then approximated for each ROI utilizing pub-lished experimental regression relations of apparent den-sity to apparent elastic modulus and strength, fortrabecular bone in the human femur (Lotz et al., 1990).

2.2. Animal studies

Considering that human studies cannot isolate thecontribution of muscular weakness to osteoporosis,and while appreciating the many anatomical and func-tional differences between the human and rat femora,animal studies were conducted to answer the same basicquestion investigated using the numerical model, i.e.,can muscular weakness per se be an important contrib-utor to trabecular bone loss in osteoporosis. The proto-col of animal experiments was designed so that it wouldbe possible to compare the conclusions from the abovecomputer-aided simulation studies with the conclusionsfrom the animal studies in light of the above researchquestion, but not so that animal models validate thecomputer model.

Pre-clinical research established that skeletal reac-tions and artificially induced osteopenia in rodentsare similar to those in humans (Mosekilde, 1995).Accordingly, rodents, and rats specifically, were usedas models of bone adaptation in numerous studies.Typically, the hind-limbs are underloaded using thetechnique of tail suspension (Hutton et al., 2002).However, no rodent model was developed so that re-duced muscle loads will act on the femur in a mannerthat allows to analyze consequent bone adaptation. Anew rodent model was therefore developed. To con-form the computer simulations, we reduced the forcesapplied on the proximal femur of the rat by the hipmusculature using surgical intervention. By surgicallydisconnecting a controlled cross-sectional area of mus-culature from the hip of the rat, it was therefore possi-ble to decrease the hip muscular loading, as in ourcomputer simulations. This not only allowed compari-son of bone response with predictions of the computermodels, but also yielded new experimental data onbone adaptation to muscular dysfunction that werepreviously not available.

The following protocol was approved by the Institu-tional Animal Care and Use Committee (IACUC) of TelAviv University and was carried out in compliance withinstitutional guidelines for care and use of animalmodels (IACUC approval #M-03-59). A group of 15Sprague Dawley adult male rats (age: 3–4 months,weight: 350–400 g) was assigned for this study. Animalswere divided into 3 sub-groups: (i) controls (n = 5), (ii)animals in which �50% of the cross-section of the glu-teal muscle was detached from the right trochanter(n = 6), and (iii) animals in which all the musculatureconnected to the right trochanter was detached (n = 4).Animals from groups (ii) and (iii) were anesthetized

M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997 991

prior to surgery using Ketamine (90 mg/kg) and Xyla-zine (10 mg/kg) that were injected intraperitonially.The skin above the right hind-limb was carefully shavedand reflected, and the muscles gluteus superficial andgluteus medius were cut using a scalpel at the transversedirection above the right trochanter (to �50% the cross-section for group (ii) and across their full thickness ingroup (iii)). In rats assigned for partial reduction ofthe gluteal cross-section, muscle thickness was gentlymeasured using a caliper before and after transverse cut-ting. During the following 24 h, all animals whichunderwent surgical interventions were treated with anal-gesic drugs and with antibiotics to prevent infections.After recovery, animals were housed in cages (3 percage) for 19 weeks to allow adaptation of trabecularbone to the reduced muscular load of the right hind-limb. This period was reported to be sufficient for man-ifesting bone adaptation in the rat (Mosekilde et al.,2000). After 19 weeks in which animals were monitoredand were all seen active and ambulant, animals wereeuthanized with an overdose of KCl (intracardial) andthe right femurs were harvested for microscopic evalua-tion of the microarchitecture of trabecular bone. Duringthe recovery time (after 4 weeks), one animal from thefully detached gluteus group died and its femur wasnot considered in the analysis.

The harvested femora were cleaned from soft connec-tive tissues, and a slice (�2 mm thick) was cut from eachfemur in the direction of the femoral neck (using a dia-mond coated disk saw, Dremel Co.) for evaluation ofthe trabecular microarchitecture under optical digitalmicroscopy (Axiolab A, Zeiss Co.). Each slice wasscanned for trabeculae and each identified trabeculawas characterized by its length (L), base thickness (tmax)and minimal thickness (tmin) (Dagan et al., 2004). Thesmall number of trabeculae in the proximal femur ofthe rat does not allow meaningful classification oftrabeculae to sites analogous to the ROIs defined forhumans (Fig. 2(a)) and, therefore, comparisons weremade across the whole proximal femur. We first com-pared measurements of L, tmax and tmin between contra-lateral femora (where muscles were left intact) in thegroup subjected to complete muscle detachment, andcontrol animals. Trabeculae in the contralateral femoraof operated rats were statistically indistinguishable indimensions from those in control animals. Thus, we con-clude that muscle detachment was the cause for loss oftrabecular thickness in the operated side. Accordingly,we compared trabecular dimensions between controlsand the 2 operated groups, using one-way analysis ofvariance (ANOVA) for the factor of gluteal cross-sectional area (100% (controls), 50% or 0%). Post hocTukey–Kramer analysis was utilized for multiple pair-wise comparisons of dimensions of trabeculae amongthe 3 rat groups. A P value less than 0.05 was consideredsignificant.

3. Results

3.1. Computer studies

We analyzed computer simulation cases of (i) mildgluteal dysfunction (80% the normal load), (ii) moderategluteal dysfunction (50% the normal load) and (iii) se-vere gluteal dysfunction (0%). For each gluteal dysfunc-tion case, we conducted simulations with a normal hipjoint force (Fig. 3), as well as with hip joint forces thatwere reduced to 90% and 80% (Fig. S1, supplementaryonline material) the normal magnitude, because inter-dependencies between muscle force output and jointload transfer are likely. Weakness of the gluteus causedsubstantial degradation of trabecular bone architectureat the proximal femur, and the effect of bone degrada-tion was more profound when hip joint forces were alsoreduced. Specifically, degradation in trabecular bonedensity in simulated cases of gluteal weakness was mostdramatic at the site adjacent to the gluteal attachmentsurface at the greater trochanter (ROI 4: maximal de-crease in apparent density = 65%), substantial at nearbysites at the lower femoral neck (ROI 2: maximal de-crease = 36%) and lateral proximal femur (ROI 1: max-imal decrease = 32%), and minimal at the upper neck(ROI 3: maximal decrease = 10%). Density of the ROIat the ball of the joint (ROI 5) was not affected whenthe gluteal force output was reduced from 100% to0%, but was substantially affected (27% decrease) whenthe force transfer through the hip joint was also reducedto 80% its normal value. We conclude that trabecularbone density at the trochanter, lower neck and lateralproximal femur is mostly influenced by the force outputof the gluteal musculature, but density at the ball of thehip joint and upper neck is dominated by the forcestransferred through the hip joint. Interestingly, whenthe gluteus force was reduced, trabeculae that supportcompression stresses in the upper neck, ROI 3 (i.e., thatbelong to the principal compressive group, Singh et al.,1970; Lin et al., 1999; Table 3) resorbed, but trabeculaethat bear tension in ROI 3 thickened, likely because themodified bending moment on the femoral neck pro-duced more bending-related tension stresses and lessbending-related compression.

Because the trochanter region (ROI 4) demonstratedthe most dramatic decrease in bone density, we com-pared the apparent properties at that region across allsimulation cases (Figs. 4 and 5). It was suggested thatosteoporotic-like trabecular bone at the proximal femuris characterized by mean thickness of trabeculae that islower than 218 lm, apparent density that is lower than0.35 g/cm3 and apparent strength of trabecular bonethat is less than 2 MPa (Li and Aspden, 1997). Our anal-ysis showed that dysfunction of the gluteus resultedadapted thickness of trabeculae that is lower than thecritical thickness for an extent of gluteal dysfunction

Fig. 3. Simulated architecture of trabecular bone at the five regions of interest (ROIs) in the proximal femur following adaptation to weakness of thegluteus muscle. Gluteus force output was set as (a) 80%, (b) 50% and (c) 0%. Hip joint forces were kept at the normal level. For each ROI,the simulation provided the distribution of thickness of trabeculae and a scheme of the lattice with trends of resorption and apposition in response tothe modified musculoskeletal loading.

992 M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997

Fig. 5. Apparent properties of trabecular bone at the center of thetrochanter region (ROI 4) following adaptation to gluteal muscularweakness: (a) apparent density, (b) apparent ultimate stress (strength),and (c) apparent elastic modulus. Shaded areas indicate normal rangeof properties, as reported in Li and Aspden (1997) for trabecular bonefrom human femora of 58–93-year-old subjects (median of 79 years).Apparent strength values for our lattices were calculated using theempirical relation of Lotz et al. (1990): r = 25 Æ (Apparent Density)1.8,which was obtained based on data from 49 specimens harvested fromfemora of 4 human cadavers. Age of subjects in Lotz� study was 25–82years.

Fig. 4. Thickness of trabeculae in the lattice located at the center of thegreater trochanter region (ROI 4) as function of the force output of thegluteus muscle, for a normal hip joint force (solid line) and for hip jointforces that are 90% (dashed line) and 80% (dotted line) the normal one.Bars (depicted with corresponding line types) indicate SDs from themean thickness of trabeculae in the ROI 4 lattice on each simulationcase.

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below 25%, when transfer of forces through the hip jointis 90% the normal one or above (Fig. 4). When the forcetransfer through the hip joint was 80%, gluteal weaknessto an extent of 40% the normal gluteal force was suffi-cient to induce osteoporotic-like trabecular thickness(Fig. 4). Considering also the criteria of apparent den-sity (<0.35 g/cm3) and apparent strength (<2 MPa) forosteoporotic-like trabecular bone, it is demonstrated inFig. 5 that complete dysfunction of the gluteus inducesosteoporotic-like trabecular bone in terms of both struc-tural and mechanical property aspects. However, be-cause the ‘‘normal’’ range of properties shown inFig. 5 was obtained from studies of cadavers that weremostly of elderly (ages 58–93, and median of 79-year-old, Li and Aspden, 1997), we cannot rule out formationof osteoporotic-like bone if gluteal function is abovezero. Hence, considering that muscle force output andjoint loading closely relate, we conclude that trabecularbone in the femur may start manifesting osteoporotic-like characteristics when the gluteal force output chron-ically drops below 40%.

3.2. Animal studies

In a rat model, we partially detached gluteus muscleto �50% the original cross-section (n = 6) or, in anothergroup, completely detached the gluteus (n = 3). Wecompared dimensions of trabeculae in the proximal fe-murs connected to the manipulated gluteus in these ani-mals (after allowing bone adaptation for 19 weeks) tothose of controls (n = 5). One-way ANOVA for the fac-tor of muscle cross-sectional reduction, run separately

for each dimension type (tmax (averaged for left andright edges of trabeculae), tmin, and average (nominal)thickness of trabeculae calculated from 1

2ðtmax þ tminÞ) re-

vealed a significant effect of this factor on trabecularthickness. Accordingly, we conducted post hoc Tukey–Kramer multiple comparisons of trabecular dimensionsacross groups, which yielded that (i) thickness of trabec-ulae from the groups which underwent partial and com-plete detachment of the gluteus were statisticallyindistinguishable, and (ii) in both experimental groupsaverage thickness and minimal thickness of trabeculaewere significantly lower than those of controls(P < 0.05), by 22% and 20%, respectively (Fig. 6). Basethickness (tmax) of trabeculae from the experimentalgroups was also lower than that of controls, by 25%,

Fig. 6. Trabecular thickness in the proximal femora of rats, 19 weeksafter partial (�50%, n = 6) and complete (0%, n = 3) detachment ofthe gluteus muscle, with respect to thickness of trabeculae in controls(n = 5). Post hoc Tukey–Kramer analysis revealed that the decrease inaverage thickness (22%) and minimal thickness (20%) was significant(P < 0.05) in both experimental groups with respect to controls.Results from the group in which partial detachment of the gluteus wasperformed were indistinguishable from those of the group whichunderwent complete detachment of the gluteus.

994 M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997

but unfortunately, without statistical significance(Fig. 6). We conclude that surgical reduction of thecross-sectional area of the gluteus muscle by 50% orover caused chronic loss of trabecular bone mass in rats.

4. Discussion

The present study showed that in both the computerand animal models, manipulation of muscle loadingproduced a significant stimulus for bone to adapt, i.e.,a stimulus that is beyond its irresponsive �lazy zone�(Huiskes et al., 1987; van Rietbergen et al., 1993).Accordingly, the results obtained herein indicate thatmuscular weakness may be an important factor contrib-uting to osteoporosis and that this direction is worthfurther investigation.

Osteoporosis is a complex disease, which was attrib-uted to age, postmenopausal estrogen deficiency, clinicaldisorders, and extended bed rest (Marcus and Majum-der, 2001). In the recent years, it had been suggested thatchronic muscular weakness, specifically, is contributingto osteoporosis in the elders. Unfortunately, in humanstudies, it is very difficult to isolate the effect of muscularweakness from effects of other age-related factors, suchas metabolic, hormonal, inactivity, and illness (Marcusand Majumder, 2001). Computer modeling which cou-ples stress/strain analysis with a bone remodeling the-ory, however, can test the particular effect of muscle

weakness on bone morphology in order to determine ifmuscular weakness per se can be an important contrib-uting factor to trabecular bone loss in osteoporosis.Hence, the important unique features of the presentcomputer modeling studies were: (i) In the aspect ofmodel methodology—bone is modeled as a hierarchialstructure so that gross muscle loading from the gluteusis transferred to local mechanical stimulus at the levelsof a trabecula and of individual osteocytes. (ii) Fromthe standpoint of application—adaptation of trabecularmorphology at different sites in the proximal femur isstudied in response to changes in gluteus force outputand to potentially related changes in hip joint loads.

The distribution of apparent densities in the humanproximal femur was measured using micro-CT (Morganet al., 2003), and density values at the greater trochanterand femoral neck were found to be 0.22 and 0.56 g/cm3,respectively. For the trochanter, 23 samples were ex-tracted from 21 donors who were 49–101-year-old atthe time of death (mean age 70 years, SD 13 years).For the femoral neck, 27 samples were obtained from23 donors who were 57–101-year-old at the time ofdeath (mean age 74 years, SD 13 years). The apparentdensity values in Morgan�s studies (2003) are lower thanthose obtained with our trabecula elements (neutral con-ditions: ROI 4, trochanter: 0.81 g/cm3; ROI 2, lowerfemoral neck: 0.85 g/cm3). However, the values in Mor-gan et al. (2003) approximately match the apparent den-sities at the trochanter and lower neck which werepredicted in a simulation of the response of normal bone(with structural characteristics of a young adult) to zerogluteal force and 80% of the normal hip joint force(Fig. S1(c), supplementary online material). Consideringthat the mean age of subjects in Morgan�s studies (2003)was 70-year-old or over, the discrepancy between ourmodel results and his experimental studies reflect thedegradation of bone density in elderly. This also stressesthe contribution of musculoskeletal loads to bone qual-ity. Chronic weakness of the gluteus muscle and reducedhip joint load in the less active elderly are expected tocause substantial bone loss (demonstrated in Figs. 3and S1), which is reflected in the experimental findingsof Morgan et al. (2003). Specifically, when the gluteusmuscle load and the hip joint load were reduced in thesimulations, maximal bone loss occurred at the trochan-ter (ROI 4) and lower femoral neck (ROI 2), and indeed,these regions are the ones that are most susceptible tofracture among elderly who typically underload theirfemora (Gnudi et al., 1999).

The present computational and animal studiesrequired several assumptions/limitations which shouldbe recognized. First, in the calculation of apparentmechanical properties (Fig. 5), we used the same elas-ticity–density relation across all ROIs, but a very re-cent study (Morgan et al., 2003) indicated thatelasticity–density relations in the femur are site-specific.

M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997 995

Unfortunately, experimental data of elasticity–densityrelations are not yet available for all the ROIs consid-ered herein. With the accumulation of new experimen-tal data the modeling system can be improved toaccount for the site-specific elasticity–density relations.Second, a 2D gross femur model was utilized to con-form to the other model components in our hierarchialapproach, and particularly, to set planar force bound-ary conditions on the perimeters of our ROIs for thebone adaptation simulations at the trabecular latticelevel (since we considered planar trabecular lattices).However, the architecture of trabecular bone is 3D innature, and out-of-plane forces are transferred to tra-becular junctions; this must play some role in themechanical stimulus for trabecular bone adaptation.Third, trabecular geometry at the lattice scale is againsimplified with respect to the real-world situation.Although it was possible to study real bone micro-structures, e.g., reconstructed from a micro-CT, speci-men-specific geometry of trabecular bone may biasthe conclusions regarding large populations, becauseof anatomical variations or local damage which maybe included in the specific specimen. In other words,the specimen-specific geometries are only valid for aparticular bone and subject and they do not easily lendthemselves to be extrapolated or otherwise generalized.Hence, generic bone geometry is appropriate as aninitial condition for adaptation simulations where itis desired to study the mechanics of a ‘‘typical’’, ratherthan specific bone.

Taking this and other limitations of the planar anal-ysis of loads in the modeling system into account, weconsider this study as a first step towards a more com-plex, full 3D modeling system of the adaptation of thetrabecular microarchitecture in response to muscularweakness.

In order to support the computational predictions ofa thinning response of femoral trabeculae under muscu-lar weakness (Fig. 4) we conducted animal studies. Forthat purpose, we detached the gluteal muscle of rats,partially (to an extent of �50%) or fully. After 19 weeks,we sacrificed the animals and measured dimensions oftrabeculae in their femora. When partial detachmentof the gluteus is carried out, it is difficult to accuratelydetach the same proportion of cross-sectional area ofgluteal musculature across animals. Nevertheless, oursurgical intervention induced bone adaptation responsethat could be associated with reduced muscle forces, asopposed to the commonly used tail suspension approach(Hutton et al., 2002), and indeed, Tukey–Kramer analy-sis of dimensions of trabeculae established that (partialor total) detachment of the gluteus did cause a signifi-cant reduction in trabecular thickness. A second limita-tion of the animal model is the effect of post-surgicalrecovery on the process of bone adaptation. It is reason-able to assume that animals did not return to normal

activity level after surgery, which implies that the totalload on the limbs was reduced, and it is also likely thatthe right hind-limb (which was subjected to interven-tion) was the least loaded one. Unfortunately, the reduc-tion of skeletal (hip joint) forces could not be quantifiedwith our experimental design. However, it was possibleto compare the overall bone loss in the animals (22%loss of trabecular thickness for experimental groupswhich underwent partial or complete gluteus detach-ment) to that predicted by our computational sim-ulations of human femora subjected to reducedmusculoskeletal loads. While appreciating the inherentstructural and physiological differences between humanand rat femora, our computational simulations pre-dicted similar extent of bone loss in humans (21% reduc-tion in trabecular thickness) for a case where the humangluteal force was 50% the normal one and the hip jointforce was 80% the normal one. As mentioned above,considering that surgery of the limb in the rat very likelyinduced underloading of the hip joint as well, bone losspattern in the animals show excellent agreement with thecomputational predictions. Finally, we could not obtainsite-specific bone loss in proximal femora of rats becausetheir physical size was too small and trabecular densityin the rat is low (Mullender, 1997). Therefore, the sizesof our experimental groups required that we analyzedimensions of trabeculae pooled from the whole proxi-mal femur of the rat in order to obtain sufficient statis-tical power. Considering that our experiments were thefirst to quantify adaptive changes in dimensions of tra-beculae in response to surgically reduced musculatureforce, the advantage of our experimental design over-weighs the above limitations.

In closure, the present computer and animal studiesprovide new mechanistic insight into the shape andproperty adaptation behavior of trabecular bone underthe effect of muscular weakness. Specifically, the abilityof the computational model to consider coupled effectsof cellular signaling and gross mechanical underloadingmakes it a powerful test-bench not only for biomechan-ical analysis of the consequences of muscular weakness,chronic inactivity and immobilization, but also for eval-uating approaches for compensating the expected boneloss. For example, an important clinical question in thisregard is whether the degenerative response of trabecu-lar bone can be recovered if the weak muscles arestrengthen and trained. In a preliminary animal study(Be�ery and Gefen, 2003) comparing thickness of trabec-ulae at the proximal femur in 1-year-old (i.e., old-age)control (n = 3) and physically trained rats (which under-went 2 weeks of treadmill training, n = 3), we found thatmean trabecular thickness was statistically indistinguish-able between the two groups (P = 0.69). This is in agree-ment with computational simulations performed withthe model described herein, which predicted a very smallrise (of between 0.3% and 1.7%) in mean trabecular

996 M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997

thickness with respect to normal trabecular thickness, asa result of an increase, by 1.1–1.3-fold, of the force out-put of the gluteus muscle (Be�ery and Gefen, 2003).Additional computer and animal studies are now under-way in our Musculoskeletal Biomechanics Laboratoryto evaluate whether degraded trabecular morphologysubsequent to muscular weakness can be recovered withphysical exercise.

Acknowledgement

Dr. David Kastel, Dr. Mickey Scheinowitz and Shar-on Barzelai from the Neufeld Cardiac Research Instituteof Sheba Medical Center, and Dr. Naam Karib from theFaculty of Medicine of Tel Aviv University, Israel, arethanked for their help in surgery and handling of ani-mals. This study was supported by the Ela Kodesz Insti-tute for Medical Engineering and Physical Sciences andby the Internal Fund of Tel Aviv University, Israel.

Appendix A. Supplementary data

Supplementary data associated with this articlecan be found, in the online version, at doi:10.1016/j.clinbiomech.2005.05.018.

References

Beer, F.P., Johnston, E.R., DeWolf, J.T., 2002. Mechanics ofMaterials. McGraw-Hill, Boston.

Be�ery, M., Gefen, A., 2003. The role of muscular weakness inpathogenesis of osteoporosis in the elderly. In: 25th AnnualInternational Conference of the IEEE Engineering in Medicine andBiology Society, Cancun, Mexico, September 17–21.

Currey, J.D., 1988. The effect of porosity and mineral content on theYoung�s modulus of elasticity of compact bone. J. Biomech. 21,131–139.

Dagan, D., Be�ery, M., Gefen, A., 2004. A single-trabecula building-block for large-scale finite element models of cancellous bone. Med.Biol. Eng. Comput. 42, 549–556.

Gefen, A., Seliktar, R., 2004. Comparison of the trabecular architec-ture and the isostatic stress flow in the human calcaneus. Med. Eng.Phys. 26, 119–129.

Gnudi, S., Ripamonti, C., Gualtieri, G., Malavolta, N., 1999.Geometry of proximal femur in the prediction of hip fracture inosteoporotic women. Br. J. Radiol. 72, 729–733.

Harrigan, T.P., Jasty, M., Mann, R.W., Harris, W.H., 1988. Limita-tions of the continuum assumption in cancellous bone. J. Biomech.21, 269–275.

Huiskes, R., Ruimerman, R., van Lenthe, G.H., Janssen, J.D., 2000.Effects of mechanical forces on maintenance and adaptation ofform in trabecular bone. Nature 405, 704–706.

Hutton, W.C., Yoon, S.T., Elmer, W.A., Li, J., Murakami, H.,Minamide, A., Akamaru, T., 2002. Effect of tail suspension (orsimulated weightlessness) on the lumbar intervertebral disc: studyof proteoglycans and collagen. Spine 27, 1286–1290.

Huiskes, R., Weinans, H., Grootenboer, H.J., Dalstra, M., Fudala, B.,Slooff, T.J., 1987. Adaptive bone-remodeling theory applied toprosthetic-design analysis. J. Biomech. 20, 1135–1149.

Li, B., Aspden, R.M., 1997. Material properties of bone from thefemoral neck and calcar femorale of patients with osteoporosis orosteoarthritis. Osteoporosis Int. 7, 450–456.

Lin, J.C., Grampp, S., Link, T., Kothari, M., Newitt, D.C., Felsen-berg, D., Majumdar, S., 1999. Fractal analysis of proximal femurradiographs: correlation with biomechanical properties and bonemineral density. Osteoporosis Int. 9, 516–524.

Lotz, J.C., Gerhart, T.N., Hayes, W.C., 1990. Mechanical propertiesof trabecular bone from the proximal femur: a quantitative CTstudy. J. Comput. Assist. Tomogr. 14, 107–114.

Marcus, R., Majumder, S., 2001. The nature of osteoporosis. In: R.Marcus, D. Feldman, J. Kelsey (Eds.), Osteoporosis, vol. 1, seconded. Academic Press, San Diego.

McGuire, W., Gallagher, R.H., Ziemian, R.D., 2000. Matrix Struc-tural Analysis. Wiley, New York.

Morgan, E.F., Bayraktar, H.H., Keaveny, T.M., 2003. Trabecularbone modulus-density relationships depend on anatomic site.J. Biomech. 36, 897–904.

Mosekilde, Li., 1995. Assessing bone quality: animal models inpreclinical osteoporosis research. Bone 17, 343S–352S.

Mosekilde, Li., Thomsen, J.S., Mackey, M.S., Phipps, R.J., 2000.Treatment with risedronate or alendronate prevents hind-limbimmobilization induced loss of bone density and strength in adultfemale rats. Bone 27, 639–645.

Mullender, M., 1997. The process of mechanical adaptation intrabecular bone. Ph.D. Thesis, University of Nijmegen, TheNetherlands.

Mullender, M., Huiskes, R., 1997. Osteocytes and bone lining cells:which are the best candidates for mechano-sensors in cancellousbone? Bone 20, 527–532.

Mullender, M.G., Huiskes, R., Versleyen, H., Buma, P., 1996a.Osteocyte density and histomorphometric parameters in cancellousbone of the proximal femur in five mammalian species. J. Orthop.Res. 14, 972–979.

Mullender, M.G., van der Meer, D.D., Huiskes, R., Lips, P., 1996b.Osteocyte density changes in aging and osteoporosis. Bone 18, 109–113.

Mullender, M., van Rietbergen, B., Ruegsegger, P., Huiskes, R., 1998.Effect of mechanical set point of bone cells on mechanical controlof trabecular bone architecture. Bone 25, 125–131.

Puzas, J.E., 1996. Osteoblast cell biology—lineage and functions. In:Favus, M.J. (Ed.), Primer on the Metabolic Bone Diseases andDisorders of Mineral Metabolism, third ed. Lippincot Raven, pp.11–16.

Reilly, D.T., Burstein, A.H., 1975. The elastic and ultimate propertiesof compact bone tissue. J. Biomech. 8, 393–405.

Rice, J.C., Cowin, S.C., Bowman, J.A., 1988. On the dependence of theelasticity and strength of cancellous bone on apparent density.J. Biomech. 21, 155–168.

Riggs, B.L., Wahner, H.W., Seeman, E., Offord, K.P., Dunn, W.L.,Mazess, R.B., Johnson, K.A., Melton 3rd, L.J., 1982. Changes inbone mineral density of the proximal femur and spine with aging.Differences between the postmenopausal and senile osteoporosissyndromes. J. Clin. Invest. 70, 716–723.

Singh, M., Nagrath, A.R., Maini, P.S., 1970. Changes in trabecularpattern of the upper end of the femur as an index of osteoporosis.J. Bone Joint Surg. Am. 52, 457–467.

Sinaki, M., 1998. Musculoskeletal challenges of osteoporosis. Aging10, 249–262.

Seeman, E., 2003. Pathogenesis of osteoporosis. J. Appl. Physiol. 95,2142–2151.

SolidWorks Corporation, 2003. SolidWorks 2003 Training guides(Essentials, Advanced Part modeling, Advanced Assembly model-ing, and API Fundamentals), Concord MA, USA.

Tsubota, K., Adachi, T., Tomita, Y., 2002. Functional adaptation ofcancellous bone in human proximal femur predicted by trabecularsurface remodeling simulation toward uniform stress state. J. Bio-mech. 35, 1541–1551.

M. Be’ery-Lipperman, A. Gefen / Clinical Biomechanics 20 (2005) 984–997 997

Turner, C.H., Rho, J., Takano, Y., Tsui, T.Y., Pharr, G.M., 1999. Theelastic properties of trabecular and cortical bone tissues are similar:results from two microscopic measurement techniques. J. Biomech.32, 437–441.

van Rietbergen, B., Huiskes, R., Weinans, H., Summer, D.R., Turner,T.M., Galante, J.O., 1993. The mechanism of bone remodeling andresorption around press-fitted THA stems. J. Biomech. 26, 369–382.

Weinans, H., Huiskes, R., Grootenboer, H.J., 1992. The behavior ofadaptive bone-remodeling simulation models. J. Biomech. 25,1425–1441.

Weinans, H., Huiskes, R., van Rietbergen, B., Sumner, D.R., Turner,T.M., Galante, J.O., 1993. Adaptive bone remodeling aroundbonded noncemented total hip arthroplasty: a comparison betweenanimal experiments and computer simulation. J. Orthop. Res. II,500–513.

Weiss, A., Arbell, I., Steinhagen-Thiessen, E., Silbermann, M., 1991.Structural changes in aging bone: osteopenia in the proximalfemurs of female mice. Bone 12, 165–172.

Weiss, L., 1988. Cell and Tissue Biology. A Textbook of Histology.Urban and Schwarzenberg, Baltimore.