A Positivity Preserving Adaptive Moving Mesh Method for Cancer Cell Invasion Models.

In this paper, we present an efficient adaptive moving mesh finite difference method for the modeling of early stage cancer cell invasion of tissue or extracellular matrix (ECM). The cancer cell invasion model is a nonlinear system of reaction–diffusion-taxis partial differential equations (PDEs) describing the time evolution of the cancer cells density and concentrations of proteins of the ECM. The solution of the model exhibits very rapid variations at the boundary of the healthy and cancer cells. Thus, using a uniform grid method to solve the cancer cell invasion model requires a very large number of grid points in order to resolve the steep gradients of the solution precisely. As a result, the computation can become prohibitively expensive and inefficient. In this effort, we propose a positivity preserving scheme for the spatial discretization of the model on an adaptive mesh to accurately capture the sharp structures of the solution. The adaptive mesh is generated by a coordinate transformation obtained as the solution of the optimal mass transfer problem. Several numerical experiments are presented to demonstrate the performance of the adaptive mesh method for solving the cancer cell invasion model. The numerical results show that the proposed adaptive mesh method is effective in capturing and predicting the correct behavior of the cancer cells movement within the ECM.