Details
| Original language | English |
|---|---|
| Pages (from-to) | 826-849 |
| Number of pages | 24 |
| Journal | Applied mathematical modelling |
| Volume | 104 |
| Early online date | 13 Dec 2021 |
| Publication status | Published - Apr 2022 |
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
ASJC Scopus subject areas
- Mathematics(all)
- Modelling and Simulation
- Mathematics(all)
- Applied Mathematics
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In: Applied mathematical modelling, Vol. 104, 04.2022, p. 826-849.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - 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
AU - Mohammadi, Vahid
AU - Dehghan, Mehdi
AU - Khodadadian, Amirreza
AU - Noii, Nima
AU - Wick, Thomas
N1 - 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.
PY - 2022/4
Y1 - 2022/4
N2 - 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.
AB - 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.
KW - Asymptotic analysis
KW - Generalized minimal residual method
KW - Generalized moving least squares approximation
KW - Mathematical oncology
KW - Prostate tumor growth model
UR - http://www.scopus.com/inward/record.url?scp=85122280584&partnerID=8YFLogxK
U2 - 10.1016/j.apm.2021.12.011
DO - 10.1016/j.apm.2021.12.011
M3 - Article
AN - SCOPUS:85122280584
VL - 104
SP - 826
EP - 849
JO - Applied mathematical modelling
JF - Applied mathematical modelling
SN - 0307-904X
ER -