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 journalArticle

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

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Tumor Growth
Growth Model
Free Boundary
Nutrients
Tumors
Cell
Tumor
Hele-Shaw Flow
Model
Incompressible Limit
Front Propagation
Darcy's Law
Nonnegativity
Dynamical Model
A Priori Estimates
Positivity
Numerical Scheme
Boundedness
Analytical Solution
Availability

Keywords

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

Cite this

Analysis and computation of some tumor growth models with nutrient : From cell density models to free boundary dynamics. / Liu, Jian Guo; Tang, Min; Wang, Li; Zhou, Zhennan.

In: Discrete and Continuous Dynamical Systems - Series B, Vol. 24, No. 7, 07.2019, p. 3011-3035.

Research output: Contribution to journalArticle

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