Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics

Jian Guo Liu, Min Tang, Li Wang, Zhennan Zhou

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = n γ . For finite γ, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As γ → ∞, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

Original languageEnglish (US)
Pages (from-to)3011-3035
Number of pages25
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume24
Issue number7
DOIs
StatePublished - Jul 2019

Bibliographical note

Funding Information:
J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135.

Funding Information:
2010 Mathematics Subject Classification. 35K55, 35B25, 76D27, 92C50. Key words and phrases. Tumor growth model, free boundary limit, Hele-Shaw flow model. J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135. ∗ Corresponding author: Zhennan Zhou.

Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.

Keywords

  • Free boundary limit
  • Hele-shaw flow model
  • Tumor growth model

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