A novel approach to estimate trabecular bone anisotropy from stress tensors

Abstract

Continuum finite element (FE) models of bones and bone-implant configurations are usually based on clinical CT scans. In virtually all of these models, material properties assigned to the bone elements are chosen as isotropic. It has been shown, however, that cancellous bone can be highly anisotropic and that its elastic behavior is best described as orthotropic. Material models have been proposed to derive the orthotropic elastic constants from measurements of density and a fabric tensor. The use of such relationships in FE models derived from CT scans, however, is hampered by the fact that the measurement of such a fabric tensor is not possible from clinical CT images since the resolution of such images is not good enough to resolve the trabecular micro-architecture. In this study, we explore an alternative approach that is based on the paradigm that bone adapts its micro-architecture to the loading conditions, hence that fabric and stress tensors should be aligned and correlated. With this approach, the eigenvectors and eigenvalues of the element continuum-level stress tensor are used as an estimate of the element fabric tensor, from which the orthotropic material properties then are derived. Using an iterative procedure, element orthotropic material properties and fabric tensors are updated until a converged situation is reached. The goals of this study were to investigate the feasibility and accuracy of such an iterative approach to derive orthotropic material properties for a human proximal femur and to investigate whether models derived in this way can provide more accurate results than isotropic models. Results were compared to those obtained from models of the same femurs for which the fabric was measured from micro-CT scans. It was found that the iterative approach could well estimate the orientation of the fabric principal directions. When comparing the stress/damage values in the models with material properties based on estimated and measured fabric tensors, the differences were not significant, suggesting that the material properties based on the estimated fabric tensor well reflected those based on the measured fabric tensor. Errors were less than those obtained when using isotropic models. It is concluded that this novel approach can provide a reasonable estimate of anisotropic material properties of cancellous bone. We expect that this approach can lead to more accurate results in particular for models used to study implants, which are usually anchored in highly anisotropic cancellous bone regions.

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Acknowledgments

Funding from the European Union for the osteoporotic virtual physiological human project (VPHOP FP7-ICT2008-223865) is gratefully acknowledged.

Conflict of interest

Bert van Rietbergen is a consultant for Scanco Medical AG.

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Correspondence to Bert van Rietbergen.

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Hazrati Marangalou, J., Ito, K. & van Rietbergen, B. A novel approach to estimate trabecular bone anisotropy from stress tensors. Biomech Model Mechanobiol 14, 39–48 (2015). https://doi.org/10.1007/s10237-014-0584-6

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Keywords

  • Anisotropy
  • Trabecular bone
  • Fabric tensor
  • Finite element analysis