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

Youshan Tao; J. Ignacio  Tello

doi:10.3934/mbe.2016.13.193

Article Contents

2016, Volume 13, Issue 1: 193-207. Doi: 10.3934/mbe.2016.13.193

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

Youshan Tao^1, and
J. Ignacio Tello^2,

1.
Department of Applied Mathematics, Dong Hua University, Shanghai 200051
2.
Departamento de Matemática Aplicada, E.T.S.I. Sistemas Informáticos, Universidad Politécnica de Madrid, 28031 Madrid

Received: September 30, 2014

Revised: April 30, 2015

Published: September 2015

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Abstract

This work studies a general reaction-diffusion model for acid-mediated tumor invasion, where tumor cells produce excess acid that primarily kills healthy cells, and thereby invade the microenvironment. The acid diffuses and could be cleared by vasculature, and the healthy and tumor cells are viewed as two species following logistic growth with mutual competition. A key feature of this model is the density-limited diffusion for tumor cells, reflecting that a healthy tissue will spatially constrain a tumor unless shrunk. Under appropriate assumptions on model parameters and on initial data, it is shown that the unique heterogeneous state is nonlinearly stable, which implies a long-term coexistence of the healthy and tumor cells in certain parameter space. Our theoretical result suggests that acidity may play a significant role in heterogeneous tumor progression.

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Mathematics Subject Classification: Primary: 35B35, 35B40, 92C50; Secondary: 35A01, 35K55, 35K57.

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