Spatial extension of a bone remodeling dynamics model and its finite element analysis. (10th January 2021) - Baldonedo, Jacobo G.; Fernández, José R.; Segade, Abraham
Abstract: There are many works dealing with the dynamics of bone remodeling, proposing increasingly complex and complete models. In the recent years, the efforts started to focus on developing models that not only reproduce the temporal evolution, but also include the spatial aspects of this phenomenon. In this work, we propose the spatial extension of an existing model that includes the dynamics of osteocytes. The spatial dependence is modeled in terms of a linear diffusion, as proposed in previous works dealing with related problems. The resulting model is then written in its variational form, and fully discretized using the well‐known finite element method and a combination of the implicit and explicit Euler schemes. The numerical algorithm is then analyzed, proving some a priori error estimates and its linear convergence. Finally, we extend the examples already published for the temporal model to one and two dimensions, showing the dynamics of the solution in the spatial domain. Abstract : In this paper, we studied, from the numerical point of view, a new spatio‐temporal bone remodeling model. It led to a nonlinear system written in terms of the concentrations of osteocytes, osteoblasts, preosteoblasts and osteoclasts, and the bone density. A fully discrete approximation was introduced by using the finite element method and the implicit Euler scheme, and an a priori error analysis was provided. The numerical simulations have shown the accuracy of the approximation and a comparison with other works.