erratum tissue stresses and strain in trabeculae of a canine

*Corresponding author. Tel.: 0031 24 361 4476; fax: 0031 24 4540555; e-mail: [email protected].

1Original PII: S0021-9290(98)00150-X.2Present address: Institute for Biomedical Engineering, University of

Zurich and Swiss Federal Institute of Technology (ETH), Zurich, Swit-zerland.

3Present address: Orthopedic Biomechanics Laboratory, HarvardMedical School (BIDMC), Boston, USA.

Journal of Biomechanics 32 (1999) 443—451

Erratum1

An error occured on p. 169 of the above article, whereby Fig. 4 was reproduced incorrectly. The complete paper with the correct version of Fig. 4 isreprinted below.

The publisher apologises most sincerely to the authors and the readers for any inconvenience caused by this error.

Tissue stresses and strain in trabeculae of a canine proximalfemur can be quantified from computer reconstructions

B. Van Rietbergen!,2, R. Muller",3, D. Ulrich", P. Ruegsegger", R. Huiskes!,*

! Orthopaedic Research Lab, Institute of Orthopaedics, University of Nijmegen, P.O. Box 9101, 6500 HB Nijmegen, The Netherlands"Institute for Biomedical Engineering, University of Zu( rich and Swiss Federal Institute of Technology (ETH), Zu( rich, Switzerland

Received 29 December 1997; accepted 21 September 1998

Abstract

A quantitative assessment of bone tissue stresses and strains is essential for the understanding of failure mechanisms associated withosteoporosis, osteoarthritis, loosening of implants and cell- mediated adaptive bone-remodeling processes. According to Wolff’strajectorial hypothesis, the trabecular architecture is such that minimal tissue stresses are paired with minimal weight. This paradigmat least suggests that, normally, stresses and strains should be distributed rather evenly over the trabecular architecture. Althoughbone stresses at the apparent level were determined with finite element analysis (FEA), by assuming it to be continuous, there is nodata available on trabecular tissue stresses or strains of bones in situ under physiological loading conditions. The objectives of thisproject were to supply reasonable estimates of these quantities for the canine femur, to compare trabecular-tissue to apparent stresses,and to test Wolff ’s hypothesis in a quantitative sense. For that purpose, the newly developed method of large-scale micro-FEA wasapplied in conjunction with micro-CT structural measurements.

A three-dimensional high-resolution computer reconstruction of a proximal canine femur was made using a micro-CT scanner.This was converted to a large-scale FE-model with 7.6 million elements, adequately refined to represent individual trabeculae. Usinga special-purpose FE-solver, analyses were conducted for three different orthogonal hip-joint loading cases, one of which representedthe stance-phase of walking. By superimposing the results, the tissue stress and strain distributions could also be calculated for otherforce directions. Further analyses of results were concentrated on a trabecular volume of interest (VOI) located in the center of the head.

For the stance phase of walking an average tissue principal strain in the VOI of 279 strain was found, with a standard deviation of212 lstrain. The standard deviation depended not only on the hip-force magnitude, but also on its direction. In more than 95% of thetissue volume the principal stresses and strains were in a range from zero to three times the averages, for all hip-force directions. Thisindicates that no single load creates even stress or strain distributions in the trabecular architecture. Nevertheless, excessive valuesoccurred at few locations only, and the maximum tissue stress was approximately half the value reported for the tissue fatiguestrength. These results thus indicate that trabecular bone tissue has a safety factor of approximately two for hip-joint loads that occurduring normal activities. ( 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Trabecular bone; Bone mechanical properties; Computed tomography; Finite element analyses; Bone architecture

0021-9290/99/$ — see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 0 2 4 - X

1. Introduction

The main function of trabecular bone is to distributemechanical loads from articular surfaces to the diaphysesof the long bones. The load-transfer pathway is largelydetermined by the internal architecture of the bone, sincethe individual trabeculae constitute the actual load-car-rying structure. According to Wolff ’s trajectorial hypoth-esis, the trabecular architecture is such that minimaltissue stresses are paired with minimal weight. This para-digm at least suggests that, normally, stresses and strainsshould be distributed rather evenly over the trabeculararchitecture. So far, however, there have been no possi-bilities for a quantitative evaluation of this paradigm. Itis not known, for example, if, and to what extent, an evendistribution is possible for the actual tissue stresses andstrains, or, if this is only possible for the average tissuesstresses and strains during a loading cycle. Nor is itknown to what extent this expected even distribution oftissue stresses and strains is affected if the bone architec-ture or the external loads are changed and what the‘safety factor’ of the bone is for changes in its loading.Such changes in architecture or loading can be due to, forexample, osteoporosis or the placement of an implant.Quantitative knowledge about the tissue stress and straindistribution thus could be a key factor to quantify boneintegrity according to Wolff ’s hypothesis, but is alsoessential for the understanding of failure mechanismsassociated with osteoporosis, osteoarthritis, loosening ofimplants and cell-mediated load adaptive bone remodel-ing processes.

So far, however, no methods have been developed thatcan be used to measure tissue stresses or strains, not evenin vitro. As an alternative, methods based on the finiteelement analysis (FEA) have been used to calculaterather than measure tissue stress and strain conditions.In these studies, however, the bone tissue was consideredas a continuum. Such continuum models can only beused to calculate average tissue stresses and strains, andnot those in the individual trabeculae.

Several authors have attempted to obtain informationabout trabecular stresses and strains, but only with re-spect to small bone samples. In an early study, trabeculararchitecture was represented as a repetitive structure ofunit cells (Beaupre and Hayes, 1985). More recent studieshave used new techniques that enable the FE-analysis ofrealistic trabecular architectures in detail (Fyhrie et al.,1992; Hollister et al., 1993; Van Rietbergen et al., 1995).These techniques were based on high-resolution imagingtechniques, such as serial sectioning (Odgaard et al.,1994) or micro-CT scanning (Feldkamp et al., 1989;Ruegsegger et al., 1996), in combination with newly de-veloped, iterative FE-solvers (Hollister and Kikuchi,1993; Van Rietbergen et al., 1995, 1996). With thesetechniques, the detailed three-dimensional architecture ofbone samples can be digitized and converted to large-

scale FE-models from which tissue stresses can be cal-culated. Applying these methods to cubic bone samples,it was found that the actual tissue stresses can be muchhigher than those calculated from continuum models(Hollister and Kikuchi, 1993; Van Rietbergen et al.,1995). Although these studies have produced valuableinformation about load transfer in trabecular architec-tures, this cannot be transferred to the situation of bonein situ, as the boundary forces for an excised specimen arenot representative for the intact situation. In other stud-ies, a homogenization sampling procedure, incorporatinglarge-scale FE-models representing small samples of tra-becular bone in detail, was applied to obtain the trabecu-lar tissue stresses throughout larger pieces of bones orwhole joints (Hollister et al., 1993; Hollister and Gold-stein, 1993). With this procedure, however, assumptionsabout boundary conditions can seriously influence theresults as well. To obtain dependable information abouttrabecular stresses and strains, in short, one cannot es-cape the necessity of representing the structure and itsloading conditions realistically.

The objective of this study was to determine physiolo-gical in situ stresses and strains in trabecular bone tissueof a proximal femur. For this purpose, a microstructuralfinite element model was used, that represents a wholeproximal femur in detail. The question was asked of howthe load is transferred through the trabecular architec-ture for a variety of hip-joint forces, if an external hip-joint force exists that provides a uniform distribution oftissue stresses and strains as predicted by Wolff’s law;how much variety in trabecular stresses and strains oc-curs for other force directions; whether the extent of thisrange can be estimated from average tissue values asdetermined from apparent-level FE models; and what the‘safety factor’ of the bone architecture is for hip-jointloads that occur during normal activities. The modelused in the present study represents a canine proximalfemur. The choice for this model was based on sizelimitations of present computer-reconstruction methodsand FE-solvers, and on the fact that detailed in vivo jointforce data are available for the dog.

2. Methods

The right femur of a small dog was selected froma stock of six. On micro-radiographs, this femurlooked average in shape and trabecular architecture andshowed no remains of a growth plate. A body weight of13 kg was estimated for the dog, based on the size of thefemur.

A computer reconstruction of the proximal 27 mm ofthe femur was made using a micro-CT scanner (ScancoMedical, Bassersdorf, Switzerland). Since the dimensionsof the proximal femur in the medial-lateral directionexceeded the bore size of the micro-CT scanner (16 mm),

444 B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451

Fig. 1. Comparison of a micro-radiograph of the proximal part of the dog femur (left), with a simulated radiograph of the 3-D reconstruction (right).The 3-D reconstruction is exactly the same as the FE-model but without the cup. The simulated radiograph was made by summing element densities inthe direction normal to the projection. The gray levels in the plot are linearly scaled with the calculated projected density, such that white correspondsto the highest value and black to the lowest. Note that typical structures seen on the real radiograph can be recognized in the simulated radiograph,although some loss of detail is apparent due to the limited resolution of the computer reconstruction.

a part of the trochanter was cut from the bone beforescanning, using a 0.3 mm wire-saw, and was scannedseparately. The nominal resolution of the scanner is14 lm (Ruegsegger et al., 1996), but to reduce the data setand scan time a voxel size of 35 lm in all directions waschosen. With this voxel size, the total scan time was closeto 6 h. Before segmentation the voxel size was increasedto 70 lm to further reduce the number of voxels. Boththree-dimensional reconstructions were merged intoa new voxel grid and positioned such that the cut faceswere at a 0.3 mm distance. The cut trabeculae and corti-cal regions on either side of the gap could be identified onthe sequential images that form the voxel grid, andthe gap was repaired by manually editing each image.The resulting computer reconstruction measured 29.3]19.3]27.0 mm and was represented by 418]275]385voxels of which 7.3 million represented bone tissue.

The quality of the reconstruction was judged by com-paring a simulated micro-radiograph made from thecomputer reconstruction to a real one made earlier(Fig. 1). The simulated radiograph was produced bysumming voxel densities in the anterior-posterior direc-tion, assigning a density of 1 for voxels representing bonetissue and zero for the voxels representing marrow. Thesummed values were linearly scaled to represent a graylevel in a digital image such that white represents thehighest value and black represents zero. By comparison ofthe simulated and real micro-radiograph, it was concludedthat the computer reconstruction adequately representedtypical features as shown on the real micro-radiograph,although some loss of detail is apparent due to the limitedresolution (70 lm) of the computer reconstruction.

An artificially created cup was merged with the recon-struction of the femur in order to apply realistic loading

conditions later. The cup was modeled separately asa 7 mm thick half-hemisphere built of voxels that had thesame sizes and orientations as those used for the bonereconstruction. The inner radius of the cup was chosenapproximately one voxel smaller than the radius of thefemoral head such that cup and femoral head wouldoverlap when their centers are aligned. The cup waspositioned in the bone reconstruction grid such that itcovered the anterior—medial—superior quadrant of thefemoral head, which is the region in which the resultanthip-joint force acts (Bergmann et al., 1984; Page et al.,1993). After merging both reconstructions the bone—cupinterface disappeared at most locations since the recon-struction of the cup and that of the bone slightly overlap.In this way an artificial cup was created that was fixed tothe femoral head at most of its inner surface. In someregions, however, (notably the region were the femoralfovea and the round ligament are located) the overlapbetween the cup and the femoral head reconstruction wastoo small to bridge the gap, and in these regions a gapbetween the cup and the femoral head remained. Forceswere applied to the cup in the medial, anterior andsuperior directions. Near the force application points, thethickness of the cup was gradually increased to forma plateau for a smooth distribution of the loads to thefemoral head.

The three-dimensional computer reconstruction wasconverted to a large-scale FE-model by simply convert-ing all voxels that represent bone tissue or cup to eight-node brick elements in the FE-model. This resulted ina FE-model with a total of 7.6 million elements and 9.1million nodes (Fig. 2). The same linear elastic and iso-tropic material properties, with a Young’s modulus of15 GPa and a Poisson’s ratio of 0.3 were assigned to

B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451 445

Fig. 3(a). Simulated radiographs of the FE-model with cup in the AP and ML direction with the forces applied for the three load cases. Also indicatedis the VOI in the femoral head. Fig. 3(b). The coordinate system used to describe the direction of the resultant joint force. Two coordinate angles / andh and three orthonormal forces F

1, F

2and F

3are used to define the direction and magnitude of the resultant force. The center of the sphere

corresponds to the center of the femoral head. Only resultant forces acting in the highlighted quadrant are investigated.

Fig. 2. The FE-model of the dog’s femur with artificial cup. The modelis built of 7.6 million cubic brick elements of 70 lm, in size and hasa total of 27.3 million degrees of freedom. In the plot, some of theposterior cross sections are removed to show how the trabecular bone ismodeled with the brick elements.

elements representing trabecular bone, cortical bone andthe cup. A special-purpose iterative FE-solver, imple-menting a ‘Row-By-Row’ matrix—vector multiplicationalgorithm was used to solve this large FE-problem (VanRietbergen et al., 1996). Approximately 4 GByte memoryand 30 h of CPU-time were needed on the Cray C90supercomputer which was used for the calculations.

The FE-problem was solved three times to representthree load cases out of a daily loading cycle (Bergmannet al., 1984; Page et al., 1993). In the first load case

(Fig. 3a), a 40 N force directed laterally was applied tothe medial side of the cup, in the second case a 50 N forcedirected posteriorly was applied to the anterior side, andin the third case a 200 N force directed distally to thesuperior side of the cup. The last case represents the hipjoint force during the stance phase of walking. For allcases, the displacements of nodes at the distal end of thefemur were fully constrained in all three spatial direc-tions. Since this is a linear FE-model, any resultant forceacting towards the center of the head in the anterior—medial—superior quadrant can be approximated by scal-ing and superimposing the results of these three analyses.In this way, results were calculated for another 88 load casesrepresenting a resultant force in the anterior—medial—superior quadrant at 10° intervals (Fig. 3). For each ele-ment, the superimposed strain vector was calculated from

e"e1

FR

F1

cos / cos h#e2

FR

F2

sin/ cos h#e3

FR

F3

sin h ,

(1)

where eiis the strain vector for load case i, F

Rthe magnitude

of the resultant force and /, h, F1, F

2and F

3as defined in

Fig. 3. The magnitude of the resultant force FR

waslinearly interpolated from the three load cases using

FR"f

1F

1#f

2F

2#f

3F3

(2)

with interpolation functions:

f1"A1!

/

90BA1!h90B , f

2"

/

90A1!h90B , f

3"

h90

.

The notion that results for the tissue level stressesand strains can be obtained by superimposing those


Fig. 4. Contour plot of the strain energy density distribution for a 200 N load acting on the superior side of the cup, representing the stance phase ofloading. Red areas indicate high values whereas in white areas the SED is close to zero. Some of the posterior cross sections are removed to show howthe load is transferred through the trabecular bone. Also indicated in this plot is the location of the VOI for which histograms of the tissue stresses andstrains are calculated.

calculated from three orthogonal hip-joint forces wasbased on recent findings that the contact areas in thehuman hip-joint are located near the periphery of the cupwith no contact in the region near the center of theacetabular cup where the femoral fovea and the roundligament inhibit contact (Eisenhart-Rothe et al., 1997;Bay et al., 1997), and on the observation that the anat-omy of the pelvis favors load transfer in orthogonaldirections rather than in other directions.

In the analyses of the data we concentrated on a 7 mmcubic volume of interest (VOI) in the center of the head(Fig. 3a). The number of elements in this volume was608,463 and the number of nodes was 815,967. Fromthese numbers a finite element fractal dimension of 2.58was calculated (Van Rietbergen et al., 1996). Since thispart has a plate-like architecture, the fractal dimensionindicates that, on average, three elements were present ina cross section of the trabeculae. For this VOI, histo-grams for the absolute value of the maximal principalstress and strain, for the Von Mises equivalent stress andfor the Strain Energy Density (SED) distribution werecalculated. For each of these distributions, the average

values and standard deviations were calculated to quantifya range for the tissue stresses and strains. In the followingwe will simply write ‘principal stress’ where we mean ‘thelargest component (in an absolute sense) of the principalstress’, and similarly for ‘principal strain’. To compare theactual trabecular values determined here with those thatcan be obtained from continuous, apparent-level FE mod-els, the average values over the VOI were considered.

For the 200 N force representing the stance phase ofwalking, principal strains were calculated as well ina 1.4 mm cubic volume located within the cortical boneclose to the periosteal surface, on the medial side of thefemoral neck. This volume was chosen since it enablescomparison of calculated strains with literature valuesmeasured from in vivo strain-gauge measurements indogs (Page et al., 1993).

3. Results

A contour plot of the SED distribution for the loadcase representing the stance phase of walking (case 3)


Fig. 5. Histograms representing the tissue principal stress, principal strain, Von Mises stress and SED distribution in the VOI for the 200 N loadrepresenting the stance-phase of walking. The average value, standard deviation and maximum value found for any element in the VOI are indicated aswell.

demonstrates how the load is transferred from the cup,through the trabecular network to the cortex (Fig. 4). Thevalues and distribution of the SED in the cortical regionare very similar to those determined in earlier studies,using continuum models of a canine femur (Weinans etal., 1993). For the VOI, however, it was found that theactual tissue principal stress, principal strain, Von Misesstress and SED value could deviate considerably fromthe average tissue value (Fig. 5). In the histograms ofFig. 5 the average values, the standard deviations and themaximum values calculated for any element in the VOIare indicated as well. For the principal strain, forexample, the average value in the VOI waseAV

"279 lstrain, the standard deviation eSD"212

lstrain, and the highest value for any element in the VOIwas 3731 lstrain. Similar to normal distributions, how-ever, it was found that for 96% of all tissue material theprincipal strains were in the range defined byeAV

$2]eSD

, and for 69.2% of the tissue, the values didnot even exceed the range e

AV$e

SD. The principal strain

in the medial cortex of the femoral neck was larger thanthat in the trabecular bone tissue and averaged958 lstrain. The shape of the tissue stress distribution inthe VOI resembles that of the strain distribution. Theaverage tissue stress was 3.88 MPa, the standard devi-ation 3.04 MPa and the largest value found was60.2 MPa.

The average values of the principal strain in the VOIfor all 91 load cases, plotted in the contour plot of Fig. 6a,are largely determined by the magnitude of the resultantforce vector. One can think of this plot as a projection ofthe average principal strain onto a sphere, as a functionof the two coordinate angles / and h. The values rangefrom 57 lstrain for a 41 N resultant force in the /"10,h"0 direction, to 279 lstrain for the 200 N force di-rected distally at h"90. A plot of the standard devi-ations as a function of the coordinate angles would looksimilar, since the magnitude of the standard deviations islinearly related to that of the average value. If, however,the standard deviation is normalized by the averagevalue, the effect of the force magnitude is eliminated anda plot results in which high values correspond to result-ant force directions for which a relatively large standarddeviation for the trabecular tissue strain distribution inthe VOI is found, and low values correspond with forcedirections for which a more narrow distribution is found.The normalized standard deviations for the principalstrain distribution curves ranged from 54.4% of the aver-age tissue strain for a resultant force in the /"30°,h"50° direction to 99.0% of the average for a resultantforce in the /"80°, h"10° direction (Fig. 6b). The highvalues for forces acting on the anterior side (the yellowarea in Fig. 6b) thus indicate that the trabecular architec-ture is not well adapted to forces acting in the posterior


Fig. 6. (a) Contour plot of the average tissue principal strain in the VOI as a function of the resultant force direction, projected on the quadrant of thesphere shown in Fig. 3a. The values found are largely determined by the magnitude of the resultant force, which was maximal (200 N) for a resultantforce working in the distal direction (at h"90°). 6(b). Projected contour plot of the standard deviation for the tissue principal strain in the VOInormalized by the average principal strain. The standard deviations in the yellow areas are relatively high, those in the red areas low. Since a low valueindicates a more uniform strain distribution in the VOI, these are the preferred loading directions.

direction. The lower values found for forces acting nearthe medial—superior side (indicated by the dark-red spotin Fig. 6b) indicate that loads acting in the lateral—inferior direction produce a more uniform strain distri-bution in the VOI. However, no single force directionproduces uniform tissue strains.

4. Discussion

The objective of this study was to estimate physiolo-gical in situ stresses and strains in trabecular bone tissueof a proximal canine femur. With the new techniquesapplied here it is now, for the first time, possible to obtainrealistic estimates for in situ trabecular tissue stresses andstrains. This accomplishment allowed us to quantitat-ively test the trajectorial hypothesis put forward byWolff. It should be noted, however, that a number ofassumptions and simplifications were made in this studythat need some discussion.

First, no muscle forces were applied to the model.Although muscle forces will add to the tissue stresses andstrains in the trochanteric region and in the femoral shaft,the geometry of the femur is such that muscle forces alone(that is in the absence of joint forces) will not induce

significant stresses or strains in the femoral head. Itshould be noted that the net effect of the muscle forces onthe hip joint force is accounted for in the model, since thehip joint forces were obtained from in vivo measure-ments. Nevertheless, it is possible that muscle forcesattaching to the proximal femur can slightly affect thelocal stress/strain calculation in the femoral head. Sec-ond, a unique Young’s modulus for all tissues was used inthe model, whereas several investigators have found thatthe trabecular tissue properties are less than those ofcortical bone (Rho et al., 1993; Choi et al., 1990; Kuhn etal., 1989). Hence, it is possible that the trabeculae aresomewhat too stiff. Third, the modeling of trabecularbone with unique cubic elements produces ‘jagged’ surfa-ces. It has been shown that this can lead to errors in thestress-strain calculations, resulting in oscillating values,in particular at the bone surfaces (Jacobs et al., 1993;Guldberg and Hollister, 1994, Camacho et al., 1997).However, in an earlier study we found that the height ofany of the bars in stress and strain histograms calculatedfrom FE-models with an element size of 80 lm, differedby less than 7% from those calculated from FE-modelsrepresenting the same structure with 20 lm elements(Van Rietbergen et al., 1995). Other researchershave found errors in the same range (1%—7%) for the


calculated average Von Mises stress when comparingresults from digital based models with those from con-ventional smooth FE-models (Guldberg and Hollister,1994). Consequently, although local oscillations and er-rors in the calculated stresses and strains might exist,their effect on the histograms and the calculated averageand standard deviations will be small. We thus concludedthat the model was sufficiently converged and that onlya minor part of the wide variation in tissue stresses andstrains found in this study can be due to numericalartifacts related to these jagged surfaces. Fourth, no carti-lage was modeled and the cup, added for more realisticload transfer, was fixed to the femoral head. This impliesthat, in theory, shear stresses can be transferred as well.Since, however, the orthonormal forces were directedtowards the center of the head, the interface shear stresseswere small and the net shear stresses were zero. Fifth, thecalculation of tissue stresses and strains by superimpos-ing results does not account for the changes in contactarea during flexion/extension of the femur. It is thuspossible that the superimposed results are less accuratethan those for the orthogonal load directions. Finally, noaccurate information about the magnitude of the hip-joint force was available for this particular dog. Thetissue strains calculated in the cortical region(958 lstrain) are larger than those measured from in vivostrain-gauge measurements during normal walking forthe same region (325—502 lstrain; Page et al., 1993).These differences could not be explained by the fact thatthe strains in the model were not calculated exactly at thebone surface, since it would be expected that strains onthe surface would be even slightly higher due to bendingof the femur. It is possible, however, that the 200 N forcechosen to represent the stance phase of loading is chosensomewhat too high, and better represents the forces dur-ing running or other more strenuous activities.

Nevertheless, within the context of these limitations, itis possible to address the questions posed in the introduc-tion. One of the questions was how large the range for thetissue stresses and strains is, and if this range can beestimated from average tissue values as determined fromapparent-level FE models. It was found that the range forthe tissue principal strains, as quantified by the standarddeviations, depends on the force direction. The largeststandard deviation for the principal strain distribution forany of the resultant force directions investigated was al-most equal to the average tissue value. This indicates thatfor some 95% of the tissue in the VOI, the tissue stressesand strains are in a range defined by zero to three timestheir average tissue value. The smallest standard deviationwas found for a force acting on the medial-superior quad-rant in the region indicated by the red spot in Fig. 6b. Forthis force direction, the standard deviations were onlyhalf of the average value, indicating that with forces inthis direction 95% of the tissue principal strains are ina range given by zero to twice their average tissue value.

One of the most interesting results is that no singleforce creates uniform stress or strain distributions in thetrabeculae. Even in the most optimal direction, the rangeof tissue stresses and strains is still rather large. This mayseem to contradict the hypothesis put forward by Wolff.However, the hip joint force in the dog during a walkingcycle varies in a large number of directions and evenupward directed forces are possible during the swingphase (Bergmann et al., 1984; Bergmann, 1997). Hence, itis likely that the bone is adapted to withstand this totalrange of forces rather than one single force. The sameconclusion was drawn in other studies based on thefinding that non-orthogonal trabecular architectures arefrequently found near many joints (Hert, 1992; Pidapartiand Turner, 1997). Since Wolff’s law, which was basedupon assumptions drawn from unidirectional loading,can only explain the existence of orthogonal architec-tures, it was suggested in these studies that an optimalcancellous structure may appear differently under multi-directional joint loads than the ‘trajectorial’ organizationproposed by Wolff (Pidaparti and Turner, 1997). Theassumption that bone adapts to multiple load directionsis also supported by the results of earlier load adaptivebone remodeling simulation studies, where it was foundthat multiple femoral loads are required to reproduce thenatural bone density in a computer model of the femur(Carter et al., 1989; Weinans et al., 1992). It is thuspossible that a more uniform distribution is found fortissue stresses and strains integrated over a whole walk-ing cycle. Since the superimposing of loads is only pos-sible within the quadrant indicated in Fig. 3b, this wasnot investigated in the present study.

Although the range of tissue stresses and strains for theforce representing the stance-phase is rather large, excess-ive values are found at few locations, and the maximumtissue stress found (60.2 MPa) is still less than valuesreported for the tissue fatigue strength (100—140 MPa)(Choi and Goldstein, 1992). These results indicatethat trabecular bone has a safety factor of approximatelytwo for forces that occur during normal activities.This value may seem rather low. As indicated before,however, the loads applied in the present analysis mightbe more representative for strenuous activities. Further-more, it is possible that the micro damage generatedin the bone is repaired by remodeling of the bone beforefractures will occur. Finally, it should be noted thatthe calculated maximum tissue stress might be somewhataffected by the inaccuracies due to the ‘jagged’surface. Jacobs et al. (1993) found that errors inthe stresses calculated near the surface could be as largeas 66%, although it was found in an earlier study(Guldberg and Hollister, 1994) that large errors are onlyfound in regions of low stress. While recognizing theselimitations, we think that the value found in the presentstudy can be a reasonable first estimate for the safetyfactor of bone.


Although the determination of tissue stresses andstrains from large-scale FE-models has its limitations, itis shown in this study that this FE-approach can providenew information about the tissue loading conditions, thatcannot be obtained by other methods. This new informa-tion can be of great importance for a better understand-ing of mechanically induced processes in bone.

Acknowledgements

This study was supported by NCF (Dutch NationalComputer Facilities) and a Cray Research University grant.

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erratum tissue stresses and strain in trabeculae of a canine

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