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2016, 13(1): 193-207. doi: 10.3934/mbe.2016.13.193

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, Spain

Received  October 2014 Revised  May 2015 Published  October 2015

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.
Keywords: numerical simulations., existence of solutions, steady states, Reaction diffusion equations, morphogenesis.
Mathematics Subject Classification: Primary: 35B35, 35B40, 92C50; Secondary: 35A01, 35K55, 35K5.
Citation: Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193
References:
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J. B. McGillen, E. A. Gaffney, N. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, J. Math. Biol., 68 (2014), 1199-1224. doi: 10.1007/s00285-013-0665-7.  Google Scholar

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show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Schmeisser, Triebel (Eds.), Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

J. J. Casciari, S. V. Sotirchos and R. M. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH, J. Cell Physiol., 151 (1992), 386-394. Google Scholar

[3]

A. Fasano, M. A. Herrero and M. R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Math. Biosci., 220 (2009), 45-56. doi: 10.1016/j.mbs.2009.04.001.  Google Scholar

[4]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. Google Scholar

[5]

R. A. Gatenby and R. J. Gillies, Why do cancers have high aerobic glycolysis?, Nat. Rev. Cancer, 4 (2004), 891-899. Google Scholar

[6]

R. J. Gillies, D. Verduzco and R. A. Gatenby, Evolutionary dynamics of cancer and why targeted therapy does not work, Nat. Rev. Cancer, 12 (2012), 487-493. Google Scholar

[7]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl. 23, Providence, RI, 1968.  Google Scholar

[8]

J. D. Murray, Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, 3rd edn, Springer, New York, 2002.  Google Scholar

[9]

J. B. McGillen, E. A. Gaffney, N. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, J. Math. Biol., 68 (2014), 1199-1224. doi: 10.1007/s00285-013-0665-7.  Google Scholar

[10]

M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discr. Cont. Dyn. Syst. B, 18 (2013), 2669-2688. doi: 10.3934/dcdsb.2013.18.2669.  Google Scholar

[11]

H. J. Park, J. C. Lyons, T. Ohtsubo and C. W. Song, Acidic environment causes apoptosis by increasing caspase activity, British J. Cancer, 80 (1999), 1892-1897. Google Scholar

[12]

Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Analysis: RWA, 12 (2011), 418-435. doi: 10.1016/j.nonrwa.2010.06.027.  Google Scholar

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