A new biological bone remodeling in silico model combined with advanced discretization methods

Spatial extension of a bone remodeling dynamics model and its finite element analysis

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.

Sprouting Angiogenesis: A Numerical Approach with Experimental Validation

A functional vascular network is essential to the correct wound healing. In sprouting angiogenesis, vascular endothelial growth factor (VEGF) regulates the formation of new capillaries from pre-existing vessels. This is a very complex process and mathematical formulation permits to study angiogenesis using less time-consuming, reproducible and cheaper methodologies. This study aimed to mimic the chemoattractant effect of VEGF in stimulating sprouting angiogenesis. We developed a numerical model in which endothelial cells migrate according to a diffusion-reaction equation for VEGF. A chick chorioallantoic membrane (CAM) bioassay was used to obtain some important parameters to implement in the model and also to validate the numerical results. We verified that endothelial cells migrate following the highest VEGF concentration. We compared the parameters-total branching number, total vessel length and branching angle-that were obtained in the in silico and the in vivo methodologies and similar results were achieved (p-value smaller than 0.5; n = 6). For the difference between the total capillary volume fractions assessed using both methodologies values smaller than 15% were obtained. In this study we simulated, for the first time, the capillary network obtained during the CAM assay with a realistic morphology and structure.

Analysis of a bone remodeling model with myeloma disease arising in cellular dynamics

In this work we study a bone remodeling model for the evolution of the myeloma disease. The biological problem is written as a coupled nonlinear system consisting of parabolic partial differential equations. They are written in terms of the concentrations of osteoblasts and osteoclasts, the density of the relative bone and the concentration of the tumor cells. Then, we deal with the numerical analysis of this variational problem, introducing a numerical approximation by using the finite element method and a hybrid combination of both implicit and explicit Euler schemes. We perform some a priori error estimates and show a few numerical simulations to demonstrate the accuracy of the approximation. Finally, we present the comparison with previous works and the behavior of the solution in two-dimensional examples.