Single-Trabecula Building Block for Large-Scale Finite Element Models Of
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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 À 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 cosÀ1 3 À a À À (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 Á t À 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 Á (1 þ a) 3 6 cos cosÀ1 3 À a À À The goal of the present paper was to present and characterise a L 2 2 Á t À 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 À b L sin g 3L 2 Methods V ¼ p Á À (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 cosÀ1 3 À a À t, L40 enough for the insertion of implants and for the conducting of 2 2t À 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 À18C 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).