Nonequilibrium bone remodelling: Changes of mass density and of the axes of anisotropy

Abstract

This paper is concerned with some aspects of bone tissue remodelling brought about by a change of the daily loading history. The mathematical model of remodelling is based on the assumption that every deformation state corresponds with a single thermodynamically balanced state of bone tissue and that the rate of changes of all parameters characterizing the bone tissue is determined by its strain history. It is assumed that the rate of the change of the bone tissue density depends on the history of the difference between the equilibrium and the actual densities. If we know the experimentally ascertained relation between the equilibrium density and the formation deformation state, we can compute from the derived integral equation the density at any moment of time. The equilibrium orientation of the principal axes of anisotropy is coaxial with the principal axes of strain. The position of the equilibrium and actual systems of the axes of anisotropy with regard to the reference coordinate system can be expressed at any moment by the vectors of rotation of these systems. If we assume small rotations, the rate of rotation of the system of axes of anisotropy can be expressed by the history of the difference of these vectors. From the strain history and the derived vector integral equation it is possible to compute the vector determining the spatial orientation of the principal axes of anisotropy at any moment of time.