Bone remodeling description based on micro mechanical/biological effects

This paper is devoted to formulate a model for bone remodeling capable to account for the microscopical phenomena which drive resorption and rebuilt of cancellous bone tissues, by merging micro-mechanics and optimization. The behaviour of the effective bone is described by means of a coarse model which catches the main topological, mechanical and biological characteristics of the structure via a micromechanical analysis of the stress state in a properly defined Reference Elementary Volume (RVE) and provides a description of remodeling evolution according to some optimality conditions for properly defined control functions. Adopting a macroscopic approach bone remodeling arises from admissible variations of the constitutive parameters and is associated to a non-trivial dissipation inequality; our goal here is to identify the basic set of control parameters, associated to cancellous bone architecture, which affect the macroscopic constitutive relations, and to describe their time evolution according to suitable control laws. The control parameters are chosen in such a way to be related to local mechanical and biological phenomena. From the point of view of micro-mechanics the idea is to describe the architecture of the bone by suitably oriented RVEs inside which ellipsoidal inclusions aligned along the same main axes can be detected; the centers of the ellipsoid are equally spaced along the common main directions. It is therefore the distribution of orientation of the RVEs which gives the grade of anisotropy of the problem together with the shape of the inclusions. This simplified description of bone architecture allows us to account for biology, since it seems almost well grounded that remodeling occurs on the surface of the inclusions; on the other hand no periodicity of the structure is considered, which means that no homogenization techniques are going do be developed.

Every RVE is uniformly strained (or stressed) through its boundary by the corresponding macroscopic fields; this means that the characterization of the microscopic state of strain in the RVE (micro problem) provides the macroscopic constitutive law when identifying the average strain energy of the RVE with the energy associated to the average strain. In the description of the micro problem we assume the material to be wherever spread inside the RVE with a non-homogeneous distribution of the constitutive parameters: the RVE is replaced by a kind of grey material whose constituive coefficients are not uniform over the domain but fit the effective spatially discontinuous distribution of the elastic moduli which characterized the effective RVE. The size of the inclusions, their spatial distribution inside the RVE and the orientation of the RVE itself are regarded as the control parameters of remodeling which affect the behaviour of the macroscopic constitutive relations. A properly defined mechanical constraint for the fourth order elasticity tensor guarantees via an optimization procedure to determine the missing evolution equations for the control functions.