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A TUMOR GROWTH MODEL WITH AUTOPHAGY: THE REACTION-(CROSS-)DIFFUSION SYSTEM AND ITS FREE BOUNDARY LIMIT

Publication ,  Journal Article
Dou, X; Liu, JG; Zhou, Z
Published in: Discrete and Continuous Dynamical Systems - Series B
March 1, 2023

In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a reaction-(cross-)diffusion system describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary system, describing the geometric motion of the tumor region. Analyzing the free boundary model in a special case, we show that the ratio of the two phases of cells exponentially converges to a “well-mixed” limit. Within this “well-mixed” limit, we obtain an analytical solution of the free boundary system which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. Numerical simulations are also provided to illustrate the analytical properties and to explore more scenarios.

Duke Scholars

Published In

Discrete and Continuous Dynamical Systems - Series B

DOI

ISSN

1531-3492

Publication Date

March 1, 2023

Volume

28

Issue

3

Start / End Page

1964 / 1992

Related Subject Headings

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

Citation

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Dou, X., Liu, J. G., & Zhou, Z. (2023). A TUMOR GROWTH MODEL WITH AUTOPHAGY: THE REACTION-(CROSS-)DIFFUSION SYSTEM AND ITS FREE BOUNDARY LIMIT. Discrete and Continuous Dynamical Systems - Series B, 28(3), 1964–1992. https://doi.org/10.3934/dcdsb.2022154
Dou, X., J. G. Liu, and Z. Zhou. “A TUMOR GROWTH MODEL WITH AUTOPHAGY: THE REACTION-(CROSS-)DIFFUSION SYSTEM AND ITS FREE BOUNDARY LIMIT.” Discrete and Continuous Dynamical Systems - Series B 28, no. 3 (March 1, 2023): 1964–92. https://doi.org/10.3934/dcdsb.2022154.
Dou X, Liu JG, Zhou Z. A TUMOR GROWTH MODEL WITH AUTOPHAGY: THE REACTION-(CROSS-)DIFFUSION SYSTEM AND ITS FREE BOUNDARY LIMIT. Discrete and Continuous Dynamical Systems - Series B. 2023 Mar 1;28(3):1964–92.
Dou, X., et al. “A TUMOR GROWTH MODEL WITH AUTOPHAGY: THE REACTION-(CROSS-)DIFFUSION SYSTEM AND ITS FREE BOUNDARY LIMIT.” Discrete and Continuous Dynamical Systems - Series B, vol. 28, no. 3, Mar. 2023, pp. 1964–92. Scopus, doi:10.3934/dcdsb.2022154.
Dou X, Liu JG, Zhou Z. A TUMOR GROWTH MODEL WITH AUTOPHAGY: THE REACTION-(CROSS-)DIFFUSION SYSTEM AND ITS FREE BOUNDARY LIMIT. Discrete and Continuous Dynamical Systems - Series B. 2023 Mar 1;28(3):1964–1992.

Published In

Discrete and Continuous Dynamical Systems - Series B

DOI

ISSN

1531-3492

Publication Date

March 1, 2023

Volume

28

Issue

3

Start / End Page

1964 / 1992

Related Subject Headings

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