An accurate front capturing scheme for tumor growth models with a free boundary limit

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

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when the pressure-density relationship becomes highly nonlinear. In particular, the constitutive law connecting pressure p and density ρ is p(ρ)=[Formula presented]ρm−1, and when m≫1, the cell density ρ may evolve its support according to a pressure-driven geometric motion with sharp interface along its boundary. The nonlinearity and degeneracy in the diffusion bring great challenges in numerical simulations. Prior to the present paper, there is lack of standard mechanism to numerically capture the front propagation speed as m≫1. In this paper, we develop a numerical scheme based on a novel prediction-correction reformulation that can accurately approximate the front propagation even when the nonlinearity is extremely strong. We show that the semi-discrete scheme naturally connects to the free boundary limit equation as m→∞. With proper spatial discretization, the fully discrete scheme has improved stability, preserves positivity, and can be implemented without nonlinear solvers. Finally, extensive numerical examples in both one and two dimensions are provided to verify the claimed properties in various applications.

Original languageEnglish (US)
Pages (from-to)73-94
Number of pages22
JournalJournal of Computational Physics
Volume364
DOIs
StatePublished - Jul 1 2018
Externally publishedYes

Bibliographical note

Funding Information:
J. Liu is partially supported by KI-Net NSF RNMS grant No. 11-07444 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. M. Tang and L. Wang would like to thank Prof. Jose Carrillo for fruitful discussions.

Funding Information:
J. Liu is partially supported by KI-Net NSF RNMS grant No. 11-07444 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 . M. Tang and L. Wang would like to thank Prof. Jose Carrillo for fruitful discussions.

Publisher Copyright:
© 2018 Elsevier Inc.

Keywords

  • Free boundary problem
  • Front capturing scheme
  • Hele–Shaw equation
  • Prediction-correction method
  • Tumor growth model

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