Single-Trabecula Building Block for Large-Scale Finite Element Models Of

Single-trabecula building block for large-scale finite element models of cancellous bone

D. Dagan M. Be’ery A. Gefen Department of Biomedical Engineering, Faculty of Engineering, Tel Aviv University, Israel

Abstract—Recent development of high-resolution imaging of cancellous bone allows finite element (FE) analysis of bone tissue stresses and strains in individual trabe- culae. However, specimen-specific stress=strain analyses can include effects of anatomical variations and local damage that can bias the interpretation of the results from individual specimens with respect to large populations. This study developed a standard (generic) ‘building-block’ of a trabecula for large-scale FE models. Being parametric and based on statistics of dimensions of ovine trabeculae, this building block can be scaled for trabecular thickness and length and be used in commercial or custom-made FE codes to construct generic, large-scale FE models of bone, using less computer power than that currently required to reproduce the accurate micro-architecture of trabecular bone. Orthogonal lattices constructed with this building block, after it was scaled to trabeculae of the human proximal femur, provided apparent elastic moduli of 150 MPa, in good agreement with experimental data for the stiffness of cancellous bone from this site. Likewise, lattices with thinner, osteoporotic-like trabeculae could predict a reduction of 30% in the apparent elastic modulus, as reported in experimental studies of osteoporotic femora. Based on these comparisons, it is concluded that the single-trabecula element developed in the present study is well-suited for representing cancellous bone in large-scale generic FE simulations.

Keywords—Spongy bone, Trabecular tissue stiffness, Apparent elastic modulus, Constitutive properties, Osteoporosis

Med. Biol. Eng. Comput., 2004, 42, 549––556

1 Introduction models have represented trabecular bone as a continuum, and so only average tissue stresses and strains could be predicted. TRABECULAR BONE consists of delicate plates and struts of bone The recent development of high-resolution imaging of bone tissue, trabeculae, that branch and intersect to form a sponge-like (serial sectioning, micro-CT and micro-MRI scanning) opened a lattice. Individual trabeculae are the load-bearing elements of new field of study in bone mechanics: FE analysis of realistic cancellous (spongy) bone. The overall architecture of trabecular trabecular architectures. With these techniques, the detailed lattices aligns with the principal load-transfer pathways in bone three-dimensional (3D) architecture of trabecular bone samples under physiological loading. This makes the epiphysis in long can be digitised and converted to large-scale FE models from bones, which consist mainly of trabecular bone, an efficient which tissue stresses, displacements and strains in individual structure in distributing concentrated loads from the joint trabeculae can be obtained (VAN RIETBERGEN et al., 1995; 1999; surfaces to the diaphysis. 2003). Little is known about the distribution of mechanical stresses Such models can be validated experimentally using the and strains in the individual trabeculae of bone under physiolo- texture correlation technique, which extracts displacement gical loading. As direct measurements of stresses and strains in patterns from digitised contact radiographs of the samples individual trabeculae are not feasible, finite element (FE) models under load (BAY et al., 1999). However, if the specific bone of the micro-architecture of bone have been used for such specimens subjected to the stress=strain analysis contain some studies. Traditionally, to simplify the calculations, these anatomical variations or local damage, conclusions regarding large populations can be biased. This calls for the development of a complementary generic trabecular bone model that can be used where it is desired to study the mechanics of a ‘typical’, Correspondence should be addressed to Dr Amit Gefen; rather than specific, bone. email: [email protected] Moreover, epiphyses of human long bones contain thousands Paper received 17 November 2003 and in final form 13 April 2004 of trabeculae, each with irregular and unique geometry. MBEC online number: 20043907 Accordingly, meshing of these complex lattices in the FE # IFMBE: 2004 method produces vast databases that currently require a Medical & Biological Engineering & Computing 2004, Vol. 42 549 supercomputer or a cluster of computers for analysis This uniquely defines the base thickness tmax values for each (VAN RIETBERGEN et al., 1999). Although, with the constantly trabecula (Fig. 1b). The value of tmin was measured at the centre growing power of computers, it is expected that this would be of the trabecula, halfway between the two locations of its tmax less of a problem in the future, in many cases, the complexity of boundaries. We found significant linear correlation (R2 ¼ 0.71, the modelling and, mainly, of the FE meshing would be p50.05) between the base and minimum thickness dimensions substantially reduced if the jagged and irregular geometries of of individual trabeculae (tmax ¼ atmin þ b, where a ¼ 1.3736 and individual trabeculae were approximated to smoother and b ¼ 40.9 mm; see Fig. 1c). simpler standard elements. It has also been suggested that By means of cross-correlation of digitised trabecular profiles, smoother surfaces of trabeculae in FE models reduce solution we also found high degrees of symmetry of the curvature of artifacts (GULDBERG et al., 1998). individual trabeculae (Fig. 1a) around their longitudinal (z) and Several generic models of repeated cellular solid structures radial (r) axes (R2 ¼ 0.97 0.99), which made it possible to fit were developed to study the mechanical behaviour of trabecular cosinusoidal curves to the upper and lower trabecular profiles bone using different geometrical descriptions for the unit cell  r 2z 1 b 3 (GIBSON, 1985; WERNER et al., 1996; ANDERSON and CARMAN, ¼ cos cos1 3 a (1) 2000; KIM and AL-HASSANI, 2002; KOWALCZYK, 2003). The tmin L 2 tmin 2 more recent contributions accounted for the curved geometry of individual trabeculae rather than treating them as beams with Because tmin and tmax are linearly related, (1) can also be uniform (circular or rectangular) cross-sections. Specifically, formulated in terms of tmax. It is also possible to write both KIM and AL-HASSANI (2002) considered differences between tmin and tmax as functions of the average thickness of a trabecula t the base and central thicknesses of trabeculae and used linear (where t ¼ (tmin þ tmax)=2) and of the constants a, b. This made it regression equations for relating thickness and separation of possible to simplify the representation of trabecular profiles in trabeculae to the age of bone. Most recently, KOWALCZYK (1) so that only two parameters, the characteristic length L and (2003) described the shape of unit cells using Be´zier curves, average thickness t, are incorporated which made it possible to demonstrate a wide variety of 2 t b microstructural patterns. However, none of the published r ¼ generic models employed real architectural statistical data to 1 þa  define the unit cell geometry. 2z 1 b (1 þ a) 3 6 cos cos1 3 a The goal of the present paper was to present and characterise a L 2 2 t b 2 standard ‘building block’ of a trabecula for large-scale FE models. Being parametric and based on statistics of dimensions (2) of mammalian trabeculae, this building block can be scaled for Utilising our microscopy measurements, which yielded that the trabecular thickness and length and used in commercial or range of thickness of trabeculae was between 0.3 times and 2.86 custom-made FE codes to construct generic large-scale FE times the mean thickness (215 mm, see Fig. 2a), (2) allows models of bone that can serve as a ‘gold standard’ in basic representation of the complete spectrum of potential trabecular studies of bone mechanics, as well as during the design and profiles in sheep. performance evaluation of orthopaedic implants. This paper also The volume bounded within the surface of revolution derived provides basic statistical information on the geometrical varia- from (2) (i.e. the surface generated by rotating the positive curve tions between individual trabeculae. of (2) 360 about the z-axis) provides an estimate for the volume V of a trabecula with given nominal thickness t and length L  2t b L sin g 3L 2 Methods V ¼ p (3) 1 þ a g 2 2.1 Geometric model of a single trabecula where Sheep became a common orthopaedic model because, in  addition to being relatively inexpensive, their bones are large 1 b(1 þ a) g ¼ cos1 3 a t, L40 enough for the insertion of implants and for the conducting of 2 2t b mechanical property studies (AN and FRIEDMAN, 1999). Accordingly, we developed a single-trabecula geometric Equation (3) thus approximates the distribution of volumes of model based on statistical analysis of the dimensions of 200 trabeculae in sheep (Fig. 2c). rod-like trabeculae from the epiphyseal parts of six ovine Bone morphology studies across species suggest that the femora. Six specimens were transversely cut from the upper- shape of individual trabeculae is common to all mammals, third of the epiphysis of each femur, with an electrically powered although dimensions of trabeculae and their structural arrange- saw, after the bones had been dried at 85C for 4.5 h. The dried ment do differ between species (FAJARDO and MULLER, 2001). samples were kept at 18C and defrosted to room temperature Accordingly, to represent a single, normal rod-type trabecula of before measurement of the trabecular dimensions. the human proximal femur with mean dimensions, we scaled the Under digital optical microscopy* (magnification x30), we parameters of (2) and (3), so that L ¼ 1 mm and t ¼ 283 mm measured the length L, base thickness tmax (at the junctions of the (PUZAS, 1996). To represent further the spectrum of potential lattice) and minimum thickness tmin (at the centre of the strut) of trabecular profiles in a normal human femoral epiphysis, we each trabecula (Figs. 1 and 2) from a transverse view, using assumed that the extent of biological variation in trabecular designated software for microscopic measurements.{ The base thickness found in normal sheep (Fig. 2a) applies to normal thickness tmax at each edge of a trabecula was determined by humans, i.e. that the ratios of maximum-to-mean and minimum- containing the respective trabecular junction within a circle and to-mean thicknesses are similar in normal sheep and normal measuring the distance between points common to the profiles of human cancellous bone. This assumption allowed us to plot the the trabecula and the junction circle, as demonstrated in Fig. 1b. spectrum of potential geometric profiles of human trabeculae (Fig. 3). The 3D reconstruction ((2) and (3)) of a rod-type human *Axiolab A, ZEISS Co., trabecula of the femoral neck with mean thickness and length {MGI photosuite 3.0SE (t ¼ 283 mm; L ¼ 1) is shown in Fig. 4a. 550 Medical & Biological Engineering & Computing 2004, Vol. 42 a b

800 ttmax= a+b min abm= 1.3736, = 40.932 m 700 R2 = 0.71

600

500

m ,m 400

max

t 300

200

100

0 0 100 200 300 400 500 tmin,mm c

Fig. 1 (a) Basic geometric dimensions of rod-like trabecula: base thickness (tmax) and minimum thickness at the centre (tmin); (b) method of determination of base of trabeculae shown for junction between 3 trabeculae marked i, j and k, with corresponding base thickness values i j k tmax,tmax and tmax; (c) base and minimum thickness of individual trabeculae were found to be linearly correlated (p50.05). 2 tmax ¼ atmin þ b; a ¼ 1.3736; B ¼ 40.9 mm; R ¼ 0.71 (Micrograph magnifications: 630)

2.2 Characterisation of the elastic properties of orthogonal of the thickness range; see Fig. 3), 100, 130, 150, 186 and trabecular lattices 283 mm (mean thickness of trabeculae at the normal human proximal femur). Cancellous bone specimens containing trabe- Equations (2) and (3) describe a parametric geometric culae with mean thickness lower than 120 mm are considered building block of a trabecula for constructing large-scale FE osteoporotic (WERNER et al., 1996), and therefore the lattices models. To test whether FE models constructed with these comprising trabeculae with thicknesses of 86 and 100 mm elements present apparent elastic properties that are typical of represent osteoporotic bone quality. trabecular bone structures, we constructed computer models of To connect adjacent trabeculae by a junction to construct 3D six orthogonal lattices{ and performed computational FE experi- lattices, we developed a junction element (Fig. 4b). The six ments of uni-axial compression of these cubic lattices. Each connection ports on a junction element (Fig. 4b) have the same lattice comprised 144 trabecular building blocks (Fig. 4a) of the diameter as the base diameter of the attached trabeculae (t ), same geometry, connected at 64 junctions. The mean thickness t max and the fillet radii between connection ports were set as t =3. values used for trabeculae in the six lattices were 86 (lower limit max Fig. 4c shows a lattice constructed from trabecular building blocks with thickness of 283 mm. As the compressive loads {using SolidWorks 2001 applied to a face of the orthogonal lattice cube align with the Medical & Biological Engineering & Computing 2004, Vol. 42 551 Fig. 2 Histograms showing distributions of trabecular (a) mean thickness, (b) length and (c) volume (estimated from (3)) in femoral epiphyses of sheep. N ¼ 200 orientation of trabeculae, the orthogonal architecture provides obtained. For a model containing trabeculae with thickness of maximum structural resistance to uni-axial loading, and the 283 mm, this was achieved when each trabecula (with tissue resulting structural properties should be considered ideal. volume of 0.05 mm3; see (3)) was meshed into 276 elements. It The geometries of the six lattices were transferred to an FE was assumed that the tissue contained in each trabecula building software package** for analysis of structural (apparent) proper- block was a homogenous, isotropic and linear elastic material, ties. The lattice geometries (e.g. Fig. 4c) were meshed with four- with elastic modulus of 10 GPa and Poisson’s ratio of 0.3 node tetrahedral elements, and the density of the meshes was (TOWNSEND et al., 1975; RHO et al., 1993; WERNER et al., 1996). optimised by decreasing the size of elements until stable Gradually increasing compressive loads were applied quasi- solutions of the stress distribution under compression were statically at the junctions on one face of the cube, and the junctions on the opposite face were constrained for displace- **NASTRAN 2001 ments in the direction of loading (unconfined compression). 552 Medical & Biological Engineering & Computing 2004, Vol. 42 r The compressive loads ranged from zero to a maximum of 1.3 N and were uniformly distributed over the joints at the loaded face, averaged mm thickness generating a maximum pressure of 500 KPa. These pressures 810 caused small axial strains, of up to 0.3%. Within this stress– 650 z 510 strain range, all lattices behaved linearly, with a constant 370 proportion between the (applied) compressive load and (calcu- 280 230 lated) structural strain. This constant of proportion, the apparent 85 elastic modulus of the trabecular lattice, was calculated for each 0 thickness case. To estimate the degradation of the apparent elastic proper- Fig. 3 Potential surface profiles of rod-like trabeculae in proximal ties of lattices when the structural arrangement is not ideal (e.g. femora of humans, predicted from (2) after setting of average as a result of fracture of a trabecula or resorption of a trabecula thickness t and length L as 283 mm and 1000 mm, respectively in osteoporotic bone), we repeated the above simulations after

ab

cd

e

Fig. 4 Characterisation of elastic properties of lattices built with trabecular building block (2) and (3): (a) Volume of revolution representing idealised geometry of trabecula with average thickness of 283 mm (mean thickness of rod-like trabeculae in human femoral neck). (b) Junction element designed for building ideally organised lattices from building block shown in (a). (c) Ideally organised orthogonal lattice of trabecular building blocks comprising 144 trabeculae (64 junctions). (d) Example for testing effect of disconnecting trabecula on apparent elastic modulus of lattice. (e) Stress concentrations due to bending of trabeculae under compression in vicinity of missing trabecula in (d). Peak stresses in circled region are 1.8 times stress value at same site, in corresponding intact lattice Medical & Biological Engineering & Computing 2004, Vol. 42 553 4 Discussion This study described the geometric and mechanical character- istics of an idealised trabecula element that was developed based on empirical observations of anatomical variations in individual ovine trabeculae. Based on experimental studies of the trabe- cular architecture in mammals (FAJARDO and MULLER, 2001), we assumed that trabeculae from human and ovine proximal femora share a similar shape. The distributions of thickness and length of ovine trabeculae from the femoral epiphysis (Figs 2a and b) are right-skewed (log-normal distributions), demonstrating a large range of intraspecimen variation in the shape and size of trabeculae. Importantly, this statistical characterisation allowed develop- Fig. 5 Apparent elastic moduli Ea of ideally organised orthogonal lattice (144 trabeculae, 64 junctions) against average thick- ment of an anatomically and physiologically based geometric 0.256 ness t of trabeculae in lattice. Ea ¼ 36.4 t (for t40); characterisation of the single trabecula building block ((2) and R2 ¼ 0.98 (3) and Fig. 3). Designed for large-scale FE studies of cancellous bone, this trabecula model ((2) and (3)) is parametric for length and thickness and can therefore represent trabeculae in different removing a centre trabecula aligned parallel with the anatomical sites and account for the distribution of anatomical direction of load (Figs 4d and e) or perpendicular to the variations in shape and size of trabeculae. The idealisations in direction of load. this model also allow large-scale generic FE analyses of trabecular microstructures. Constantly improving computer technology should be able to provide the computational power for routine, patient-specific, 3 Results large-scale FE models in the future. An important advantage of patient-specific, large-scale models is their utility in computing For the lattice containing trabeculae with thickness character- patient-specific structural properties of trabecular bone (e.g. istic of normal human proximal femora (283 mm), we found an stiffness and strength). This allows for many new, clinically apparent elastic modulus of 151.9 MPa. We also found that a important evaluations in the diagnosis and prognosis of patients, reduction in the thickness t of trabeculae had a substantially such as assessment of the risk of suffering an osteoporosis- deteriorating effect on the apparent modulus Ea (Fig. 5). A lattice related fracture, or monitoring of the effect of a drug on the constructed with trabeculae of the minimum thickness consid- mechanical performance of individual bones. However, ered in this study, 86 mm (corresponding to our geometric and specimen-specific properties of bone may bias the conclusions statistical analyses of trabecular profile shapes (Fig. 3)), regarding large populations, because of anatomical variations or produced an apparent modulus of 111 MPa (27% reduction local damage that may be included in the specific specimen. from the ‘normal’ condition). Overall, the reduction in moduli Thus, to complement the patient-specific, large-scale type of with reduction in trabecular thickness could be described with a 2 model, there is also a need for generic FE models of trabecular power law (R ¼ 0.98) bone. E ¼ mtn (t40) (4) The advantage of a generic bone model is its utility where it is a necessary to study the mechanics of a ‘typical’, rather than where the constants are m ¼ 36.4 MPa and n ¼ 0.256. specific, bone. For example, reconstruction of the trabecular When a central trabecula was removed from a lattice structure architecture of the femur using the generic trabecula element and the lattice was loaded in the direction of the missing described herein, by assembling building block trabeculae on the trabecula, bending-related stress concentrations appeared at the trabecular paths of the femur, can provide an enhanced ‘stan- ‘necks’ of neighbouring trabeculae (Fig. 4e). These focal dardised femur’ model that, unlike the existing one (VICECONTI stresses were typically 1.8 times greater than stresses at the et al., 2003), accounts for bone architecture at the micro-scale. same site in the intact lattice. This bending phenomenon Such standardised generic models are useful for preliminary consistently reduced the apparent elastic moduli, by 8–9% implant design, where the effect of the geometry and material (Table 1). However, when a trabecula perpendicular to the properties of the implant on the bone micro-architecture can be loading direction was removed from the centre of the lattice, simulated systematically to minimise subsequent animal studies the apparent moduli were nearly unaffected (decreased by 1% and clinical tests. Importantly, evaluation and improvement of or less). implant performances through the design process cannot be

Table 1 Apparent elastic moduli Ea of trabecular lattices (each containing 144 trabeculae) against average trabecular thickness t. Moduli are calculated for intact lattice structures and for lattices missing centre trabecula (Fig. 4d) parallel to direction of load or perpendicular to direction of load

Direction of missing trabecula Ea of lattice with missing t, mm Ea of intact lattice, MPa with respect to loading trabecula, MPa 100 118.1 parallel 107.5 perpendicular 118.1 150 132.4 parallel 122.4 perpendicular 132 186 138.5 parallel 125.4 perpendicular 137

554 Medical & Biological Engineering & Computing 2004, Vol. 42 carried out using individual, subject-specific, large-scale FE In conclusion, we have presented a new, standard, parametric models: individual anatomy or the presence of local defects in building block that is useful for large-scale FE studies of the specimen can lead to misinterpretation of the analysis of cancellous bone. Being idealised and smooth, but also being mechanical performances of the bone-implant system with based on anatomical and physiological statistical data, this regard to a population of patients. This illustrates the need for trabecula model is useful for reconstructions and FE analyses both generic and specimen-specific bone models for practical of large-scale bone models that represent ‘typical’, rather than purposes, even when computer resources for routine, large-scale subject-specific bones. Orthogonal lattices built using this FE modelling become available. building block showed structural stiffness behaviour that is A literature review of experimental data reported for the very similar to that of real cancellous bone tested with small apparent elastic moduli of adult human trabecular bone from strains. the proximal femur (BANSE et al., 1996; LI and ASPDEN, 1997; Studies are currently being performed by our group to develop AUGAT et al., 1998; BROWN et al., 2002; HOMMINGA et al., FE models of the trabecular structures of the calcaneus (for which 2002; KOHLES and ROBERTS, 2002; MORGAN et al., 2003) we characterised the trabecular micro-architecture in detail demonstrates that the reported properties vary by as much as (GEFEN and SELIKTAR,2004)using this new trabecula model an order of magnitude. Minimum reported values are around as a building block. The trabecula building block will be 100 MPa, and maximal ones are in the order of 4000 MPa. Even duplicated along polynoms that were previously fitted to the within individual specimens, large ranges and high standard dominant trabecular paths in the calcaneus (GEFEN and deviations were reported (KRISCHAK et al., 1999). This substan- SELIKTAR,2004), to form two- and three-dimensional lattices tial variability has been attributed mainly to the inhomogeneous of ‘typical’ trabecular bone volumes in the calcaneus. These trabecular bone structure (KRISCHAK et al., 1999), but discre- generic models of regions of interest in the calcaneus will be pancies may also relate to differences in testing protocols (e.g. employed in FE studies of stresses and strains in the trabecular specimen shape and size, specimen preparation, strain magni- micro-architecture of the calcaneus during physiological load- tudes, strain rate etc.). Nevertheless, several experimental bearing. studies of the proximal femur (LI and ASPDEN, 1997; AUGAT et al., 1998; BROWN et al., 2002) reported apparent elastic Acknowledgments—This study was supported by the Ela moduli in the range of 100–200 MPa, which overlap the apparent Kodesz Institute for Medical Engineering and Physical elastic modulus of our computational lattice (150 MPa) with Sciences and by the Internal Fund of Tel Aviv University, Israel. femoral trabeculae of normal, average thickness (283 mm). We further compared our computed apparent modulus (150 MPa) with values provided by other structural models. The volume fraction Vf in our model (bone tissue volume calculated using (3) and divided by a total lattice volume of 3 27 mm )isVf ¼ 0.267 (for trabeculae that are 283 mm thick), References which is comparable with the study of VAN RIETBERGEN et al. (1995), who conducted FE analyses of real trabecular micro- AN, Y. H., and FRIEDMAN, R. J. (1999): ‘Animal models in orthopae- dic research’ (CRC Press, Boca Raton, Florida, USA, 1999) ¼ architecture from the human tibial plateau (Vf 0.20–0.33). The ANDERSON, I. A., and CARMAN, J. B. (2000): ‘How do changes to apparent moduli calculated by VAN RIETBERGEN et al. (1995) plate thickness, length, and face-connectivity affect femoral cancel- varied between 80 and 102 MPa, which is in good agreement lous bone’s density and surface area? An investigation using regular with the present findings. cellular models’, J. Biomech., 33, pp. 327–335 In contrast, for comparable Vf, our model predictions disagree AUGAT, P., LINK, T., LANG,T.F.,LIN, J. C., MAJUMDAR, S., and with the apparent moduli (of over 500 MPa) predicted by GENANT, H. K. (1998): ‘Anisotropy of the elastic modulus of WERNER et al. (1996), who used two truncated pyramids trabecular bone specimens from different anatomical locations’, facing each other to represent a single trabecula. Apparently, Med. Eng. 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(2002): ‘Regional differences in mechanical and building block trabeculae are suitable for representing the material properties of femoral head cancellous bone in health and apparent elasticity of trabecular bone. osteoarthritis’, Calcif. Tissue Int., 71, pp. 227–234 The computational lattices containing thin, osteoporotic-like FAJARDO, R. J., and MULLER, R. (2001): ‘Three-dimensional analysis femoral trabeculae with thicknesses of 100 mm and 86 mm of nonhuman primate trabecular architecture using micro-computed provided apparent moduli that were 22% (118.1 MPa) and tomography’, Am. J. Phys. Anthropol., 115, pp. 327–336 GEFEN, A., and SELIKTAR, R. (2004): ‘Comparison of the trabecular 27% (111 MPa) lower, respectively. This prediction is in architecture and the isostatic stress flow in the human calcaneus’, excellent agreement with experimental data showing a 30% Med. Eng. Phy., 26, pp. 119–129 reduction in the apparent elastic modulus of osteoporotic GIBSON, L. J. (1985): ‘The mechanical behaviour of cancellous bone’, cancellous bone from the proximal femur with respect to J. Biomech., 18, pp. 317–328 normal controls (LI and ASPDEN, 1997). GULDBERG, R. E., HOLLISTER, S. J., and CHARRAS, G. T. (1998): ‘The Based on these comparisons, we conclude that the single- accuracy of digital image-based finite element models’, J. Biomech. trabecula generic element developed in the present study is well Eng., 120, pp. 289–295 suited for representing cancellous bone in FE simulations HOMMINGA, J., MCCREADIE, B. R., CIARELLI, T. E., WEINANS, H., of osteoporotic changes. The latter results also support GOLDSTEIN, S. A., and HUISKES, R. 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Eng., 217, the femoral neck and calcar femorale of patients with osteoporosis pp. 105–110 or osteoarthritis’, Osteoporosos Int., 7, pp. 450–456 WERNER, H. J., MARTIN, H., BEHREND, D., SCHMITZ, K. P., and MORGAN, E. F., BAYRAKTAR, H. H., and KEAVENY, T. M. (2003): SCHOBER, H. C. (1996): ‘The loss of stiffness as osteoporosis ‘Trabecular bone modulus-density relationships depend on ana- progresses’, Med. Eng. Phys., 18, pp. 601–606 tomic site’, J. Biomech., 36, pp. 897–904 PUZAS, J. E. (1996): ‘Osteoblast cell biology – lineage and functions’, in FAV US, M. J. (Ed.): ‘Primer on the metabolic bone diseases and disorders Author’s biography of mineral metabolism, 3rd edn’ (Lippincot Raven, 1996), pp. 11–16 RHO,J.Y.,ASHMAN, R. B., and TURNER, C. H. (1993): ‘Young’s AMIT GEFEN is a Lecturer in the Department of Biomedical Engineer- modulus of trabecular and cortical bone material: ultrasonic and ing at the Faculty of Engineering of Tel Aviv University, Israel. He microtensile measurements’, J. Biomech., 26, pp. 111–119 received his BSc in mechanical engineering in 1994, his MSc in 1997, TOWNSEND, P. R., ROSE, R. M., and RADIN, E. L. (1975): and PhD in biomedical engineering (2001) from Tel Aviv University. ‘Buckling studies of single human trabeculae’, J. Biomech., 8, During 2002–3, Dr Gefen was a post-doctoral fellow at the Injury pp. 199–201 Biomechanics Laboratory of the Bioengineering Department at the VAN RIETBERGEN, B., WEINANS, H., HUISKES, R., and ODGAARD,A. University of Pennsylvania. His research interests are in the study of (1995): ‘A new method to determine trabecular bone elastic proper- normal and pathological effects of mechanical factors on the structure ties and loading using micromechanical finite-element models’, J. and function of human tissues, with an emphasis on the musculoske- Biomech., 28, pp. 69–81 letal system.

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