Trabecular bone strength predictions using finite element analysis of micro-scale images at limited spatial resolution
Introduction
Finite element models have the potential to provide non-invasive in vivo estimates of bone biomechanical properties [1], [2], [3], which could aid in osteoporotic fracture risk prediction and assessment of therapeutic efficacy. Recent advances in micro-CT and micro-MRI scanning technology have made in vivo clinical imaging possible at voxel sizes of about 80 µm at peripheral sites [1], [4] and 150–300 µm at the vertebral body [5]. While such images may not be sufficient to provide accurate measures of apparent mechanical properties in the context of high-resolution micro-CT-based finite element analysis [6], [7], [8], they may still be able to provide information on bone strength that would be relevant in a clinical-type setting. Furthermore, variations in the strength–density relationships as predicted by these models may enable researchers to detect microarchitecture-related bone quality effects [9] associated with aging, disease, or treatment.
A promising avenue with finite element predictions of bone strength using images at limited spatial resolution is the added benefit of performing nonlinear analyses to obtain measures of failure stress over linear analyses that only supply stiffness or elastic modulus. While elastic modulus is typically well correlated with strength across a range of bone densities [10], [11], [12], estimates of yield stress (that can only be directly obtained from nonlinear finite element modeling) may improve predictions of strength since failure behavior by definition involves nonlinear phenomena such as material yielding and geometrically large deformations. To the best of our knowledge, only one study to date has implemented nonlinear modeling techniques in combination with HR-pQCT-based images and compared strength predictions from linear versus nonlinear finite element models [3], although nonlinear analysis was only performed for a small cohort of specimens. Results indicated that linear models correlated well with strength over a large data range, but that errors in absolute strength prediction were reduced appreciably when nonlinear models were used [3]. However, as imaging technologies improve and are applied to low-density or osteoporotic bone it is unclear whether similar trends will hold true. For example, the relative benefits of performing nonlinear versus linear finite element analysis for predicting bone strength may be influenced by factors such as image resolution (influencing both geometric discretization and finite element model convergence), volume fraction, and anatomic site, since nonlinear phenomena are present to differing degrees in low-density sites and low volume fraction bone [13], [14].
In this context, our overall goal was to assess the ability of both linear and nonlinear high-resolution finite element models to predict bone strength for human trabecular bone from multiple anatomic sites that display a wide range of volume fractions. Our specific objectives were to: 1) determine the predictive capability of coarsened finite element models relative to 20 µm models for apparent elastic and yield properties; 2) assess the added benefits of performing nonlinear versus linear finite element analysis for predicting trabecular bone strength; 3) determine the dependence of convergence behavior of linear and nonlinear models on bone volume fraction; and 4) quantify the differences in strength–density relationships as predicted by finite element models at varying resolutions. This study is, to our knowledge, the first to examine the convergence behavior of nonlinear micro-finite element models of trabecular bone, and to examine the differences in predictive abilities for linear versus nonlinear finite element models across a range of image resolutions, bone volume fractions, and anatomic sites.
Section snippets
Methods
Forty-six cylindrical specimens (∼ 8.1 mm diameter and 20 mm long) of trabecular bone were selected from the human femoral neck (n = 14), greater trochanter (n = 14), and vertebral body (n = 18). All specimens were machined such that the main trabecular orientation was aligned with the axis of the core [11]. None of the donors had a history of metabolic bone disease or cancer and all specimens showed no radiographic evidence of damage or bone pathologies.
Specimens were scanned using micro-computed
Results
The convergence characteristics for yield stress and elastic modulus were qualitatively similar, and showed comparable trends with respect to both bone volume fraction and anatomic site. Errors in predicted apparent yield stress for the 80 and 120 µm models were relatively small (< 15%) for the femoral neck, but could be appreciable (20–70%) for the more porous greater trochanter and vertebral trabecular bone (Fig. 2). Analysis of covariance (including BV/TV and anatomic site as independent
Discussion
The objectives of this study were to determine the convergence behavior of linear versus nonlinear high-resolution finite element models of trabecular bone, and to compare the ability of such coarsened models to predict trabecular bone strength as estimated by very high-resolution models. We found that the convergence behavior for predictions of yield stress and elastic modulus were qualitatively similar, and showed comparable trends with respect to both bone volume fraction and anatomic site.
Acknowledgments
Funding for this work was provided by Merck & Co., Inc., and the National Institutes of Health (AR43784). Cadaveric material was obtained from NDRI and UCSF Willed Body Program. Super-computing resources were obtained from the National Partnership for Advanced Computational Infrastructure (NPACI UCB266). Dr. Keaveny has a financial interest in O.N. Diagnostics and both he and the company may benefit from the results of this research.
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