A comparison of enhanced continuum FE with micro FE models of human vertebral bodies☆
Introduction
Patient-specific continuum finite element (FE) models of vertebral bodies based on clinical quantitative computer tomography (QCT) images become increasingly attractive for evaluation of stiffness and strength in vivo (Faulkner et al., 1991, Crawford et al., 2003, Liebschner et al., 2003, Imai et al., 2006, Chevalier et al., 2008). Even damage susceptibility of specific human vertebral bodies for instance along the course of anti-resorbtive or anabolic treatments can be modeled (Keaveny et al., 2007, Graeff et al., 2007). Segmentation of the QCT images together with volume meshing provide the FE models. The stiffness of the individual finite elements in the spongious bone region. is related to calibrated mineral densities obtained from the QCT image (density-only). At these clinical resolutions, the cortical shell including the endplates cannot be properly segmented and the trabecular morphology beyond density remains invisible. Despite proper experimental calibration and the undeniable value of these continuum FE models, questions rise how their accuracy is affected by the lack of a subject-specific cortex and trabecular bone fabric.
In the laboratory, micro finite element () models based on high-resolution computer tomography ( or HR-pQCT) represent the current gold standard to investigate the stiffness of human vertebral bodies in vitro (Homminga et al., 2004, Eswaran et al., 2007). In such models, the CT voxel information is converted directly into 8-noded hexahedral elements, which result in large numbers of degrees of freedom. The bone tissue is assumed to be isotropic and the elastic parameters are typically taken from micro-indentations measurements. These models include the cortex and account naturally for bone morphology but require large computing resources.
In awareness of the limitations of current modeling approaches, we propose an enhanced continuum FE model based on high-resolution CT images which includes an anatomy-specific cortex and accounts for bone morphology by using fabric–elasticity relationships (Cowin, 1985, Zysset, 2003). This hybrid approach might be more robust than at intermediary CT image resolutions, requires less computing resources, is easily extendable to non-linear behavior and converges naturally to the density-only modeling for low clinical resolutions.
The main aim of this work is to investigate the accuracy of such an enhanced continuum FE model based on HR-pQCT. The elastic structural response of human vertebral body slices is predicted by taking based on the same HR-pQCT images as the gold standard. Taking as the gold standard avoids experimental errors and allows us to consider two sources of errors: (1) modeling of a smooth cortex, (2) the morphology–elasticity relationship necessary for assigning the trabecular bone properties. Additionally, the predictions of density-only models are compared to the enhanced model predictions.
We hypothesize that, unlike the models including a simple density-only morphology–elasticity relationship, the enhanced continuum FE model including both an anatomy-specific cortex modeling and a fabric–elasticity relationship for trabecular bone makes the closest structural predictions to those of models.
In this study, we use slices of vertebral bodies from HR-pQCT scans (similar to Eswaran et al., 2007) with a resolution of . In the first step, the error of extracting a smooth anatomical cortex is investigated by considering FE models of the isolated cortical shell alone. The errors due to the morphology–elasticity relationship were minimized a priori by calibrating them with virtual material tests of 60 cubical micro FE models cropped from the same vertebral bodies (“best case scenario”). Due to their significant impact on the homogenization results (Pahr and Zysset, 2008a), two types of boundary conditions are used: kinematic uniform boundary conditions (KUBC, Van Rietbergen et al., 1996) and periodicity compatible mixed uniform boundary conditions (PMUBC, Pahr and Zysset, 2008a). To quantify the errors due to the density-only and density-fabric-based morphology–elasticity relationship, we focused on the stiffness of the trabecular centrum alone. In the third step we compared the stiffness of the whole vertebral body slices for axial compression and antero–posterior shear loading.
Section snippets
CT-scanning
Twelve vertebral bodies were extracted from four human lumbar spines (spine/level/gender/age: A/L2-L4/M/47; B/L1-L5/M/70; C/L4-L5/M/66; D/L4-L5/M/83) and scanned individually in a water-filled container with a high-resolution pQCT system (XtremeCT, isotropic resolution, 59.4 kV, , Scanco Medical AG, Brüttisellen, Switzerland) to provide a three-dimensional map of bone mineral density across the vertebral bodies. Details of the scanning procedure can be found in Chevalier et al., 2006,
Results
Material model calibration results (Fig. 3 and Table 1) showed an improvement of the predictions in case of orthotropic models. Newly introduced PMUBCs based model results were less accurate than KUBCs based model results (higher scattering lower ). The orthotropic KUBC based model gave the highest coefficients of determination. In this case Young's modulus of the model (see Table 1) was very close to the tissue modulus of .
Results of the material property mapping were given
Discussion
The present work investigates the accuracy of an enhanced continuum FE model based on HR-pQCT images by comparing vertebral stiffnesses and taking results as gold standard. It attempts to answer the question whether a subject-specific cortex and an additional trabecular bone fabric information increase the accuracy of the predictions. A numerical comparison avoids errors from experiments and enables the focus on modeling errors. Two sources of numerical errors are considered: (1) errors
Conflict of interest statement
We declare that there are no issues that may be considered as potential conflicts of interest to this work.
Acknowledgments
Computer resources were available through Hochschuljubiläumsstiftung der Stadt Wien, Austria. Thanks to Yan Chevalier, TU-Vienna, for providing the HR-pQCT images and Cyril Flaig, ETH Zürich for the support with ParFE.
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All authors were fully involved in the study and preparation of the manuscript. This material has not been and will not be submitted for publication elsewhere.