Differences in Trabecular Microarchitecture and Simplified Boundary Conditions Limit the Accuracy of Quantitative Computed Tomography-Based Finite Element Models of Vertebral Failure.

Vertebral fractures are common in the elderly, but efforts to reduce their incidence have been hampered by incomplete understanding of the failure processes that are involved. This study's goal was to elucidate failure processes in the lumbar vertebra and to assess the accuracy of quantitative computed tomography (QCT)-based finite element (FE) simulations of these processes. Following QCT scanning, spine segments (n = 27) consisting of L1 with adjacent intervertebral disks and neighboring endplates of T12 and L2 were compressed axially in a stepwise manner. A microcomputed tomography scan was performed at each loading step. The resulting time-lapse series of images was analyzed using digital volume correlation (DVC) to quantify deformations throughout the vertebral body. While some diversity among vertebrae was observed on how these deformations progressed, common features were large strains that developed progressively in the superior third and, concomitantly, in the midtransverse plane, in a manner that was associated with spatial variations in microstructural parameters such as connectivity density. Results of FE simulations corresponded qualitatively to the measured failure patterns when boundary conditions were derived from DVC displacements at the endplate. However, quantitative correspondence was often poor, particularly when boundary conditions were simplified to uniform compressive loading. These findings suggest that variations in trabecular microstructure are one cause of the differences in failure patterns among vertebrae and that both the lack of incorporation of these variations into QCT-based FE models and the oversimplification of boundary conditions limit the accuracy of these models in simulating vertebral failure.

Elastic Anisotropy of Trabecular Bone in the Elderly Human Vertebra.

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.

Effect of specimen-specific anisotropic material properties in quantitative computed tomography-based finite element analysis of the vertebra.

Intra- and inter-specimen variations in trabecular anisotropy are often ignored in quantitative computed tomography (QCT)-based finite element (FE) models of the vertebra. The material properties are typically estimated solely from local variations in bone mineral density (BMD), and a fixed representation of elastic anisotropy ("generic anisotropy") is assumed. This study evaluated the effect of incorporating specimen-specific, trabecular anisotropy on QCT-based FE predictions of vertebral stiffness and deformation patterns. Orthotropic material properties estimated from microcomputed tomography data ("specimen-specific anisotropy"), were assigned to a large, columnar region of the L1 centrum (n = 12), and generic-anisotropic material properties were assigned to the remainder of the vertebral body. Results were compared to FE analyses in which generic-anisotropic properties were used throughout. FE analyses were also performed on only the columnar regions. For the columnar regions, the axial stiffnesses obtained from the two categories of material properties were uncorrelated with each other (p = 0.604), and the distributions of minimum principal strain were distinctly different (p ≤ 0.022). In contrast, for the whole vertebral bodies in both axial and flexural loading, the stiffnesses obtained using the two categories of material properties were highly correlated (R2 > 0.82, p < 0.001) with, and were no different (p > 0.359) from, each other. Only moderate variations in strain distributions were observed between the two categories of material properties. The contrasting results for the columns versus vertebrae indicate a large contribution of the peripheral regions of the vertebral body to the mechanical behavior of this bone. In companion analyses on the effect of the degree of anisotropy (DA), the axial stiffnesses of the trabecular column (p < 0.001) and vertebra (p = 0.007) increased with increasing DA. These findings demonstrate the need for accurate modeling of the peripheral regions of the vertebral body in analyses of the mechanical behavior of the vertebra.

A new material mapping procedure for quantitative computed tomography-based, continuum finite element analyses of the vertebra.

Inaccuracies in the estimation of material properties and errors in the assignment of these properties into finite element models limit the reliability, accuracy, and precision of quantitative computed tomography (QCT)-based finite element analyses of the vertebra. In this work, a new mesh-independent, material mapping procedure was developed to improve the quality of predictions of vertebral mechanical behavior from QCT-based finite element models. In this procedure, an intermediate step, called the material block model, was introduced to determine the distribution of material properties based on bone mineral density, and these properties were then mapped onto the finite element mesh. A sensitivity study was first conducted on a calibration phantom to understand the influence of the size of the material blocks on the computed bone mineral density. It was observed that varying the material block size produced only marginal changes in the predictions of mineral density. Finite element (FE) analyses were then conducted on a square column-shaped region of the vertebra and also on the entire vertebra in order to study the effect of material block size on the FE-derived outcomes. The predicted values of stiffness for the column and the vertebra decreased with decreasing block size. When these results were compared to those of a mesh convergence analysis, it was found that the influence of element size on vertebral stiffness was less than that of the material block size. This mapping procedure allows the material properties in a finite element study to be determined based on the block size required for an accurate representation of the material field, while the size of the finite elements can be selected independently and based on the required numerical accuracy of the finite element solution. The mesh-independent, material mapping procedure developed in this study could be particularly helpful in improving the accuracy of finite element analyses of vertebroplasty and spine metastases, as these analyses typically require mesh refinement at the interfaces between distinct materials. Moreover, the mapping procedure is not specific to the vertebra and could thus be applied to many other anatomic sites.

Correlations between local strains and tissue phenotypes in an experimental model of skeletal healing.

Defining how mechanical cues regulate tissue differentiation during skeletal healing can benefit treatment of orthopaedic injuries and may also provide insight into the influence of the mechanical environment on skeletal development. Different global (i.e., organ-level) mechanical loads applied to bone fractures or osteotomies are known to result in different healing outcomes. However, the local stimuli that promote formation of different skeletal tissues have yet to be established. Finite element analyses can estimate local stresses and strains but require many assumptions regarding tissue material properties and boundary conditions. This study used an experimental approach to investigate relationships between the strains experienced by tissues in a mechanically stimulated osteotomy gap and the patterns of tissue differentiation that occur during healing. Strains induced by the applied, global mechanical loads were quantified on the mid-sagittal plane of the callus using digital image correlation. Strain fields were then compared to the distribution of tissue phenotypes, as quantified by histomorphometry, using logistic regression. Significant and consistent associations were found between the strains experienced by a region of the callus and the tissue type present in that region. Specifically, the probability of encountering cartilage increased, and that of encountering woven bone decreased, with increasing octahedral shear strain and, to a lesser extent, maximum principal strain. Volumetric strain was the least consistent predictor of tissue type, although towards the end of the four-week stimulation timecourse, cartilage was associated with increasingly negative volumetric strains. These results indicate that shear strain may be an important regulator of tissue fate during skeletal healing.