Numerical simulation of a prostate tumor growth model by the RBF-FD scheme and a semi-implicit time discretization

Abstract

The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains.