A Three-Dimensional Tumor Growth Model and Its Boundary Instability

doi:10.1007/s42967-024-00443-5

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (3): 1034-1073.doi: 10.1007/s42967-024-00443-5

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A Three-Dimensional Tumor Growth Model and Its Boundary Instability

Jian-Guo Liu¹, Thomas Witelski¹, Xiaoqian Xu², Jiaqi Zhang²

1 Mathematics Department, Duke University, Durham, NC, USA;
2 Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, Kunshan 215316, Jiangsu, China

Received:2023-12-27 Revised:2024-05-27 Accepted:2024-06-28 Online:2025-09-20 Published:2025-05-23
Supported by:
This project is partially supported by the National Key R&D Program of China, Project Number 2021YFA1001200. J. Zhang is partially supported by the Summer Research Scholar program at Duke Kunshan University. X. Xu is partially supported by the National Science Foundation of China Youth Program, Project Number 12101278, and Kunshan Shuangchuang Talent Program, Project Number kssc202102066. The authors would also like to thank the helpful discussion with Yu Feng, Dang Xing Chen, and Lin Jiu.

Abstract

Abstract: In this paper, we investigate the instability of growing tumors by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al. (Z Angew Math Phys 74:107, 2023). Building upon the insights derived from the analytical reconstruction of key results in the aforementioned work in one dimension and two dimensions, we extend our analysis to three dimensions. Specifically, we focus on the determination of boundary instability using perturbation and asymptotic analysis along with spherical harmonics. Additionally, we have validated our analytical results in a two-dimensional (2D) framework by implementing the Alternating Direction Implicit (ADI) method. Our primary focus has been on ensuring that the numerical simulation of the propagation speed aligns accurately with the analytical findings. Furthermore, we have matched the simulated boundary stability with the analytical predictions derived from the evolution function, which will be defined in subsequent sections of our paper. This alignment is essential for accurately determining the stability or instability of tumor boundaries.

Key words: Tumor growth, Boundary instability, Asymptotic analysis, Spherical harmonics, Bessel functions

CLC Number:

Jian-Guo Liu, Thomas Witelski, Xiaoqian Xu, Jiaqi Zhang. A Three-Dimensional Tumor Growth Model and Its Boundary Instability[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1034-1073.

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A Three-Dimensional Tumor Growth Model and Its Boundary Instability

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