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

authored by: Vahid Mohammadi, Mehdi Dehghan, Amirreza Khodadadian, Nima Noii, Thomas Wick
Abstract: 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 non-convex domains using both uniform and quasi-uniform points.
Organisation(s): Institute of Applied Mathematics
Institute of Continuum Mechanics
External Organisation(s): Amirkabir University of Technology
École normale supérieure Paris-Saclay (ENS Paris-Saclay)
Type: Article
Journal: Applied mathematical modelling
Volume: 104
Pages: 826-849
No. of pages: 24
ISSN: 0307-904X
Publication date: 04.2022
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Modelling and Simulation, Applied Mathematics
Electronic version(s): https://doi.org/10.1016/j.apm.2021.12.011 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{b38ea35d064941c2ab6a71c2e579b719,
title = "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",
abstract = "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 non-convex domains using both uniform and quasi-uniform points.",
keywords = "Asymptotic analysis, Generalized minimal residual method, Generalized moving least squares approximation, Mathematical oncology, Prostate tumor growth model",
author = "Vahid Mohammadi and Mehdi Dehghan and Amirreza Khodadadian and Nima Noii and Thomas Wick",
note = "Funding Information: We wish to express our deep gratitude to anonymous reviewers for their helpful comments, which have significantly improved the quality of the paper. We also appreciate the Associate Editor for giving useful comments and suggestions to improve some parts of the paper related to the non-dimensional form of the mathematical model and the proposed analysis given in Section 3.",
year = "2022",
month = apr,
doi = "10.1016/j.apm.2021.12.011",
language = "English",
volume = "104",
pages = "826--849",
journal = "Applied mathematical modelling",
issn = "0307-904X",
publisher = "Elsevier Inc.",
}