Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion

  • Received: 01 October 2014 Accepted: 29 June 2018 Published: 01 October 2015
  • MSC : Primary: 35B35, 35B40, 92C50; Secondary: 35A01, 35K55, 35K57.

  • This work studies a general reaction-diffusion model foracid-mediated tumor invasion, where tumor cells produce excess acidthat primarily kills healthy cells, and thereby invade the microenvironment. The acid diffuses and could be cleared byvasculature, and the healthy and tumor cells are viewed as twospecies following logistic growth with mutual competition. A keyfeature of this model is the density-limited diffusion for tumorcells, reflecting that a healthy tissue will spatially constrain atumor unless shrunk. Under appropriate assumptions on modelparameters and on initial data, it is shown that the uniqueheterogeneous state is nonlinearly stable, which implies a long-term coexistence of the healthy and tumor cells in certainparameter space. Our theoretical result suggests that acidity mayplay a significant role in heterogeneous tumor progression.

    Citation: Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion[J]. Mathematical Biosciences and Engineering, 2016, 13(1): 193-207. doi: 10.3934/mbe.2016.13.193

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  • This work studies a general reaction-diffusion model foracid-mediated tumor invasion, where tumor cells produce excess acidthat primarily kills healthy cells, and thereby invade the microenvironment. The acid diffuses and could be cleared byvasculature, and the healthy and tumor cells are viewed as twospecies following logistic growth with mutual competition. A keyfeature of this model is the density-limited diffusion for tumorcells, reflecting that a healthy tissue will spatially constrain atumor unless shrunk. Under appropriate assumptions on modelparameters and on initial data, it is shown that the uniqueheterogeneous state is nonlinearly stable, which implies a long-term coexistence of the healthy and tumor cells in certainparameter space. Our theoretical result suggests that acidity mayplay a significant role in heterogeneous tumor progression.


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