Analysis of a finite volume method for a bone growth system in vivo

This paper is devoted to the convergence analysis of a finite volume method and numerical simulations of a reaction–cross diffusion system arising from a bone growth model. This model describes the evolution of mesenchymal stem cells, osteoblasts, bone matrix and osteogenic growth factor. We propose a numerical scheme based on an implicit finite volume method constructed on an orthogonal mesh. Lack of the regularity of the approximate system is overcome by stability results which allow to obtain estimates on the translates, apply the Kolmogorov theorem in order to get compactness and show the convergence of the proposed scheme. The efficiency and robustness of the scheme are shown in simulating a situation in bone growth: the healing of a skull fracture in rats.