In this manuscript we compare two theoretical models which account for trabecular alignment in current models of bone adaptation. Trabecular alignment results in continuum material properties which depend on direction (anisotropy). The general approach consists of adding an evolution (i.e., remodeling) rule for the full anisotropic stiffness tensor (all 21 independent terms) to the density evolution rule from an existing isotropic (non-directional) theory. Two criteria are explored for determining the form of the stiffness evolution; first, an assumption that bone is driven to satisfy the existing density remodeling criteria in the most efficient manner, second, an assumption that stiffness changes take place in the directions of principal stress such that high stress magnitudes lead to stiffening and low stress magnitudes lead to a loss of stiffness.
Both approaches have been implemented with the finite element method and applied to a two-dimensional model of the proximal femur. The scalar evolution rule for density was selected such that the resulting anisotropic remodeling formulation reduces to a previously studied isotropic model [cite{beaupr-90a}] in certain special cases. In this way, the adjustable parameters involved in the isotropic formulation are incorporated directly into the isotropic formulation without any additional parameters. The density distributions predicted with both approaches are very similar to those predicted with a purely isotropic theory. The assumption of a mechanically efficient response lead to transverse cortical stiffnesses that were much lower than observed experimentally, indicating that the transverse stiffness of cortical bone is not adaptive in nature, but rather induced by longitudinal adaptation. The principal stress-based formulation did not have this property, leading to reasonable predicted directional stiffnesses.