A Novel Approach to Estimate Trabecular Bone Anisotropy using Fabric Tensors Derived from Stress Tensors

Javad Hazrati Marangalou, Bert van Rietbergen, and Keita Ito


anisotropy, trabecular bone, fabric tensor, stress tensor


Continuum finite element models of bones and bone-implant configurations are usually based on clinical CT scans. In virtual all of these models, material properties assigned to the bone elements were 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 laws have been proposed to derive the orthotropic elastic constants from measurements of density and fabric. The use of such laws 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 in which we assumed that bone fabric reflects the anisotropy of the continuum-level stress tensor. 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 its anisotropic material properties are derived. Using an iterative procedure, element anisotropic material properties and fabric tensors are updated until a converged situation is reached. The goal of this study was to investigate the feasibility and accuracy of such an iterative approach to derive anisotropic material properties for a human proximal femur. In order to investigate the accuracy, the estimated fabric tensors were compared with those measured from micro-CT scans of the same femur. It was found that the iterative approach could well estimate the orientation of the fabric principal directions (average error 0.19 radians), but the correlation between estimated and measured degree of anisotropy was poor (r=0.41). Nevertheless, when comparing the principal stresses in a model with material properties based on estimated and measured fabric tensors, a high correlation was found (r=0.90), suggesting that the material properties based on the estimated fabric tensor well reflects those based on the measured fabric tensor. 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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