On a mathematical model for tissue regeneration

We propose a PDE-ODE model for tissue regeneration, obtained by parabolic upscaling from kinetic transport equations written for the mesoscopic densities of mesenchymal stem cells and chondrocytes which evolve in an artificial scaffold impregnated with hyaluron. Due to the simple chosen turning kernels, the effective equations obtained on the macroscopic level are of the usual reaction-diffusion-taxis type. We prove global existence of solutions to the coupled macroscopic system and perform a stability and bifurcation analysis, which shows that the observed patterns are driven by taxis. Numerical simulations illustrate the model behavior for various tactic sensitivities and initial conditions