An asymptotic analysis and numerical simulation of a prostate tumor growth model via the generalized moving least squares approximation combined with semi-implicit time integration

This paper focuses on presenting an asymptotic analysis and employing simulations of a model describing the growth of local prostate tumor (known as a moving interface problem) in two-dimensional spaces. We first demonstrate that the proposed mathematical model produces the diffuse interfaces with the hyperbolic tangent profile using an asymptotic analysis. Next, we apply a meshless method, namely the generalized moving least squares approximation for discretizing the model in the space variables. The main benefits of the developed meshless method are: First, it does not need a background mesh (or triangulation) for constructing the approximation. Second, the obtained numerical results suggest that the proposed meshless method does not need to be combined with any adaptive technique for capturing the evolving interface between the tumor and the neighboring healthy tissue with proper accuracy. A semi-implicit backward differentiation formula of order 1 is used to deal with the time variable. The resulting fully discrete scheme obtained here is a linear system of algebraic equations per time step solved by an iterative scheme based on a Krylov subspace, namely the generalized minimal residual method with zero-fill incomplete lower-upper preconditioner. Finally, some simulation results based on estimated and experimental data taken from the literature are presented to show the process of prostate tumor growth in two-dimensional tissues, i.e., a square, a circle and nonconvex domains using both uniform and quasi-uniform points. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study focuses on the growth of local prostate tumors in two-dimensional spaces, using asymptotic analysis and simulations to demonstrate the formation of diffuse interfaces. The developed meshless method has advantages such as not requiring a background mesh for approximation and not needing to be combined with adaptive techniques. Results show that the proposed method effectively captures the evolving interface between the tumor and neighboring healthy tissue.