Anisotropic bone remodelling model based on a continuum damage-repair theory.

The purpose of this paper is to present and discuss a new model for internal bone remodelling. We state that the evolution of the internal variables of the bone microstructure and its incidence on the modification of the elastic constitutive parameters may be formulated following the exact principles of Continuum Damage Mechanics (CDM), although no actual damage is considered. A remodelling tensor, analogous to the standard damage tensor, is proposed which completely characterises the state of the homogenised bone microstructure and, therefore, its stiffness. This tensor is defined in terms of the apparent density and the "fabric tensor" associated with porosity and directionality of the trabeculae, respectively. Contrary to standard damage mechanics, its variation may be negative to allow for material "repair". The different elements that compose the theory are then established, such as the mechanical stimulus which, as is common in CDM, was chosen as the tensor that is thermodynamically associated to the remodelling one. The resorption and apposition criteria (similar to the damage criterion) were also expressed in terms of this mechanical stimulus and, finally, the evolution law, to define the change rate of the remodelling tensor. An associated flow rule is proposed that fulfils a principle of minimum mechanical dissipation for convex remodelling criteria (the case in question here). Some other important experimental features are deduced naturally from the model, such as the coincidence of the principal directions of the fabric tensor with those of the elasticity tensor, or the fact that the principal directions of the fabric tensor tend to align with the principal directions of stress. This model is applied to the study of the remodelling evolution of the proximal extremity of the femur, obtaining results very similar to experimental data.