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There is currently a great deal of cancer research that has been devoted to the modeling and numerical simulation of the cancer cell invasion of the extracellular matrix (ECM) or biological tissue [1–8]. According to these studies, the invasion of the cancer N-octanoyl-L-Homoserine lactone into the ECM is driven by the movement of cancer cells as a result of their interactions with the proteins in the ECM, their diffusion, the gradients of the chemicals (chemotaxis) within the ECM, and the gradient in the structure of the ECM . Therefore, the mathematical models describing the process of the invasion of the cancer cells into the ECM are typically given as nonlinear systems of reaction–diffusion-taxis partial differential equations (PDEs), for example see [2,4,5,9–14].
Developing an efficient numerical method to accurately compute the solutions of the cancer cell invasion models is particularly challenging. This is due to the fact that the solutions of the models are unstable, exhibit very rapid variations at the boundary of the healthy and cancer cells, and can blow-up within a finite period of time [5,15,16,16,17]. Moreover, the solution of these models can become negative over time when only standard schemes are used for the spatial discretization. This raises several computational and theoretical challenges. Over the last decade, several numerical r> ∗ Corresponding author. E-mail addresses: [email protected] (M. Sulman), [email protected] (T. Nguyen).
Fig. 1. Diagram illustrating the uPA system interactions. Solid lines show direct effect of the interactions between two species on their concentrations.
Dashed lines show indirect effect of two species interactions on other concentrations.
methods have been developed for solving the cancer cell invasion models, for example see [11,18–22]. For a fixed uniform grid method, a very fine grid resolution is required in order to accurately resolve the spatial gradients of the solution in the regions of large solution variations. As a result, the computation can become prohibitively expensive and inefficient. Therefore, it is important to use an adaptive grid method in order to compute the solutions of the cancer cell invasion models more accurately.
In this paper, we propose a positivity preserving adaptive moving mesh finite difference method for the numerical solution of the cancer cell invasion models. More precisely, in this effort we consider employing the adaptive mesh for solving the recently developed cancer cell invasion model of Andasari et al. [4,5]. The model is a nonlinear system of five reaction–diffusion-taxis partial differential equations describing the time evolution of the cancer cells density and four densities of proteins within the ECM. The large errors of the solution are expected to occur in the regions of large solution gradients. The adaptive mesh method uses a fixed number of grid points that are continuously redistributed in time so that the mesh points are concentrated in regions of large solution variations in the physical domain. Thus, the spatial derivatives of the solution are approximated more accurately which improves the overall accuracy of the solution of the model while reducing the computational cost significantly. The adaptive mesh is computed as the solution of the optimal mass transfer problem, also known as Monge–Kantorovich problem (MKP). The optimal solution of the MKP is computed using the parabolic Monge–Ampére (PMA) method described in . The use of a positivity preserving scheme for the spatial discretization is critical in order to ensure that the solution of the model will remain positive at all time levels.