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

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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

Abstract
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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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[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.
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[5]	R. A. Gatenby and R. J. Gillies, Why do cancers have high aerobic glycolysis?, Nat. Rev. Cancer, 4 (2004), 891-899.
[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.
[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.
[8]	J. D. Murray, Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, 3rd edn, Springer, New York, 2002.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[4]	R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753.
[5]	R. A. Gatenby and R. J. Gillies, Why do cancers have high aerobic glycolysis?, Nat. Rev. Cancer, 4 (2004), 891-899.
[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.
[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.
[8]	J. D. Murray, Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, 3rd edn, Springer, New York, 2002.
[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.
[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.
[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.
[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.

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