DRBEM-FDM solution of a chemotaxis–haptotaxis model for cancer invasion

Abstract

In this paper, the invasion model focusing on the effect of chemotaxis and haptotaxis is considered which involves the interactions between the cancer cells, the normal cells and the matrix degrading enzyme (MDE) that is secreted by the cancer cells to invade the surrounding tissue. The corresponding model consists of a system of nonlinear reaction–diffusion-transport equations and a combined application of dual reciprocity boundary element method (DRBEM) and finite difference method (FDM) is used to solve this system numerically. For the space discretization of the equations for cancer cells, MDE, plasminogen activator inhibitor and plasmin DRBEM is made use of. To this end, the fundamental solution of Laplace equation is used and the rest of the terms are considered as the nonhomogenity. Due to chemotaxis and haptotaxis terms, the nonhomogenity for the cancer cell equation contains the space derivatives of other unknowns and FDM is used in combination with DRBEM for their discretization. The time dependent ordinary differential equations (ODEs) of the DRBEM and DRBEM-FDM discretized equations for the cancer cell density and MDE concentration, respectively, and the ODE for the normal cell density are solved by using a combination of forward and backward Euler methods. The difficulties of the numerical solution in two-space dimensions, due to the nonlinear terms in the modeling system, are overcome using the boundary-only nature of DRBEM which gives the solution with a lower computational cost. Effects of haptotaxis, chemotaxis, proliferation of cells and plasminogen activation system on the invasion are analyzed and the numerical solutions are in good agreement with the expected behavior of the modeling phenomena.