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

Publication ,  Journal Article
Liu, JG; Tang, M; Wang, L; Zhou, Z
Published in: Discrete and Continuous Dynamical Systems - Series B
July 1, 2019

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.

Duke Scholars

Published In

Discrete and Continuous Dynamical Systems - Series B

DOI

ISSN

1531-3492

Publication Date

July 1, 2019

Volume

24

Issue

7

Start / End Page

3011 / 3035

Related Subject Headings

  • Applied Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

Citation

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Liu, J. G., Tang, M., Wang, L., & Zhou, Z. (2019). Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete and Continuous Dynamical Systems - Series B, 24(7), 3011–3035. https://doi.org/10.3934/dcdsb.2018297
Liu, J. G., M. Tang, L. Wang, and Z. Zhou. “Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics.” Discrete and Continuous Dynamical Systems - Series B 24, no. 7 (July 1, 2019): 3011–35. https://doi.org/10.3934/dcdsb.2018297.
Liu JG, Tang M, Wang L, Zhou Z. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete and Continuous Dynamical Systems - Series B. 2019 Jul 1;24(7):3011–35.
Liu, J. G., et al. “Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics.” Discrete and Continuous Dynamical Systems - Series B, vol. 24, no. 7, July 2019, pp. 3011–35. Scopus, doi:10.3934/dcdsb.2018297.
Liu JG, Tang M, Wang L, Zhou Z. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete and Continuous Dynamical Systems - Series B. 2019 Jul 1;24(7):3011–3035.

Published In

Discrete and Continuous Dynamical Systems - Series B

DOI

ISSN

1531-3492

Publication Date

July 1, 2019

Volume

24

Issue

7

Start / End Page

3011 / 3035

Related Subject Headings

  • Applied Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics