Elastic Anisotropy of Trabecular Bone in the Elderly Human Vertebra

Abstract

Knowledge of the nature of the elastic symmetry of trabecular bone is fundamental to the study of bone adaptation and failure. Previous studies have classified human vertebral trabecular bone as orthotropic or transversely isotropic but have typically obtained samples from only selected regions of the centrum. In this study, the elastic symmetry of human vertebral trabecular bone was characterized using microfinite element (μFE) analyses performed on 1019 cubic regions of side length equal to 5 mm, obtained via thorough sampling of the centrums of 18 human L1 vertebrae (age = 81.17 ± 7.7 yr; eight males and ten females). An optimization procedure was used to find the closest orthotropic representation of the resulting stiffness tensor for each cube. The orthotropic elastic constants and orientation of the principal elastic axes were then recorded for each cube and were compared to the constants predicted from Cowin's fabric-based constitutive model (Cowin, 1985, "The Relationship Between the Elasticity Tensor and the Fabric Tensor," Mech. Mater., 4(2), pp. 137-147.) and the orientation of the principal axes of the fabric tensor, respectively. Deviations from orthotropy were quantified by the "orthotropic error" (van Rietbergen et al., 1996, "Direct Mechanics Assessment of Elastic Symmetries and Properties of Trabecular Bone Architecture," J. Biomech., 29(12), pp. 1653-1657), and deviations from transverse isotropy were determined by statistical comparison of the secondary and tertiary elastic moduli. The orthotropic error was greater than 50% for nearly half of the cubes, and the secondary and tertiary moduli differed from one another (p < 0.0001). Both the orthotropic error and the difference between secondary and tertiary moduli decreased with increasing bone volume fraction (BV/TV; p ≤ 0.007). Considering only the cubes with an orthotropic error less than 50%, only moderate correlations were observed between the fabric-based and the μFE-computed elastic moduli (R2 ≥ 0.337; p < 0.0001). These results indicate that when using a criterion of 5 mm for a representative volume element (RVE), transverse isotropy or orthotropy cannot be assumed for elderly human vertebral trabecular bone. Particularly at low values of BV/TV, this criterion does not ensure applicability of theories of continuous media. In light of the very sparse and inhomogeneous microstructure found in the specimens analyzed in this study, further work is needed to establish guidelines for selecting a RVE within the aged vertebral centrum.

Fig. 1

Schematic representation of the procedure adopted to find the best orthotropic representation of each cube of trabecular bone: All δ_ij terms in C^OPT (d) are set to zero to obtain C^ORT (e); no other changes are made when obtaining C^ORT from C^OPT. (a) μCT image of vertebra, (b) micro-FE analyses of the cube, (c) trabecular fabric-Eigenvectors of the cube, (d) nearly orthotropic stiffness matrix after optimization, and (e) orthotropic stiffness matrix after assigning nonorthotropic terms of zero.

Fig. 2

Histogram of orthotropic error (a) and correlations between orthotropic error and BV/TV (b) and between E₂/E₁ and BV/TV (c)

Fig. 3

Correlations of the Young's moduli ((a)–(c)), shear moduli ((d)–(f)), and Poisson's ratio ((g)–(i)) from the μFE model versus the morphology-based model: The correlation coefficients were nearly unchanged when the circled data point was excluded