Modelling and numerical simulation of remodelling processes in cortical bone: An IGA approach to flexoelectricity-induced osteocyte apoptosis and subsequent bone cell diffusion

Remodelling is an important process in bones in order to maintain bone mass and to recover cracks which naturally develop within the bone material. There is agreement in the literature that the piezoelectric characteristic of bones is one possible initiator of bone remodelling. In recent experimental studies, however, it has been found that cortical bone also exhibits flexoelectric properties which can, in particular, lead to osteocyte apoptosis through the induction of electric fields and thereby initiate remodelling processes. This is especially the case in the vicinity of micro cracks where large strain gradients are present. In this contribution, a modelling approach for flexoelectricity-induced bone remodelling processes is presented. Due to the higher-order nature of the flexoelectric effect, isogeometric analysis is employed for a globally C¹-continuous approximation of the displacement field. The bone cells mainly involved in the remodelling process – osteocytes, osteoclasts and osteoblasts – are accounted for by the introduction of additional field variables so that the model includes chemo-electro-mechanical coupling. The migration of the latter two cell types is modelled by non-linear diffusion equations with generally anisotropic evolving diffusion tensors. It is shown that the proposed modelling approach can capture how flexoelectricity leads to osteocyte apoptosis in the vicinity of micro cracks and how bone cells subsequently move towards the remodelling site in response to particular signalling mechanisms. The simulation results indicate that, due to the size-dependency of the flexoelectric effect, its relevance with regard to bone remodelling increases on smaller scales and can potentially exceed piezoelectric influences.