Modeling and calculation of bone healing. Application to a bioreactor model.
This thesis analyzes a bone healing model which is then coupled with a model of fluid flow to model bone growth in bioreactors (systems that mimic the in vivo environment). The proposed model of population dynamics take into account the rates of change of four populations: the mesenchymal stem cells, the osteoblasts, the bone matrix and the osteogenic growth factor. With the Faedo-Galerkin approximations, we proved that this model has at least one weak solution physically admissible. In this proof, getting the energy estimates is difficult because some populations do not have spatial terms. Then, we proposed a numerical scheme for admissible meshes. The discretization is classical except for the haptotaxis term discretized by an upstream scheme with an additional monotony property. For this scheme, we have proved existence and convergence of discrete solutions physically admissible toward a weak solution physically admissible of the system. We have computed some numerical simulations to validate the model. Finally, to model the bone growth in bioreactors, we have coupled the previous model with a model of fluid flow in porous medium. This new model takes into account the effects of the shear stress on the osteoblastic differentiation and the population advection by the flow.