Over one hundred years ago, Wolff hypothesized that trabecular bone aligned itself with directions of principal stress according to an undetermined set of mathematical laws. The bulk of the current mathematical models of bone remodeling simulate the redistribution of bone mass (apparent density), as regulated by mechanical loading, by predicting the resulting inhomogeneity of the bony structure. However, they fail to account for the directional nature of the material behavior. This study presents a general theory for extending an isotropic model of adaptation to an anisotropic model. This theory takes the form of an equation for the rate-of-change of the general stiffness matrix in terms of the principal stress magnitudes which supplements the existing equation for the rate-of-change of apparent density.
The method does not rely on morphological measures of trabecular orientation (e.g., mean intercept length ellipsoid, directed volume, fabric). Thus, no additional adjustable parameters are required which might otherwise be difficult to measure or estimate. The approach of supplementing an existing isotropic formulation that describes inhomogeneity allows all of the response parameters involved in the isotropic formulation to be directly incorporated into the new anisotropic formulation. This allows one to exploit all of the experimental verification studies conducted with the isotropic formulation. Ad hoc assumptions of material symmetry are not required (e.g., the common assumption of orthotropy), and any observed regions of orthotropy, transverse isotropy, or isotropy are the result entirely of the functional adaptation of the bone and not the consequence of an a priori assumption of the model. This approach has been implemented with the finite element method and applied to a two-dimensional model of the proximal femur with encouraging results.