Abstract
At the continuous level, we consider two types of tumor growth models: the cell density model, based on the fluid mechanical construction, is more favorable for scientific interpretation and numerical simulations, and the free boundary model, as the incompressible limit of the former, is more tractable when investigating the boundary propagation. In this work, we aim to investigate the boundary propagation speeds in those models based on asymptotic analysis of the free boundary model and efficient numerical simulations of the cell density model. We derive, for the first time, some analytical solutions for the free boundary model with pressure jumps across the tumor boundary in multidimensions with finite tumor sizes. We further show that in the large radius limit, the analytical solutions to the free boundary model in one and multiple spatial dimensions converge to traveling wave solutions. The convergence rate in the propagation speeds are algebraic in multidimensions as opposed to the exponential convergence in one dimension. We also propose an accurate front capturing numerical scheme for the cell density model, and extensive numerical tests are provided to illustrate the analytical findings.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1052-1076 |
| Number of pages | 25 |
| Journal | SIAM Journal on Applied Mathematics |
| Volume | 81 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2021 |
Bibliographical note
Funding Information:∗Received by the editors October 30, 2019; accepted for publication (in revised form) March 2, 2021; published electronically June 1, 2021. https://doi.org/10.1137/19M1296665 Funding: The work of the first author was partially supported by KI-Net NSF RNMS grant 11-07444 and NSF grant DMS 1812573. The work of the second author was supported by Science Challenge Project TZZT2017-A3-HT003-F and NSFC 11871340. The work of the third author was partially supported by NSF grant DMS-1903420 and NSF CAREER grant DMS-1846854. The work of the fourth author was supported by NSFC grant 11801016. †Department of Mathematics and Department of Physics, Duke University, Durham, NC 27708 USA (jliu@phy.duke.edu). ‡School of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China (tangmin@sjtu.edu.cn). §School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (wang8818@ umn.edu). ¶Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China (zhennan@bicmr.pku.edu.cn).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.
Keywords
- Brinkman model
- Free boundary model
- Front capturing scheme
- Tumor growth models