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

    Related Papers:

  • 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.
    加载中
    [1] Teubner Texte zur Mathematik, 133 (1993), 9-126.
    [2] J. Cell Physiol., 151 (1992), 386-394.
    [3] Math. Biosci., 220 (2009), 45-56.
    [4] Cancer Res., 56 (1996), 5745-5753.
    [5] Nat. Rev. Cancer, 4 (2004), 891-899.
    [6] Nat. Rev. Cancer, 12 (2012), 487-493.
    [7] Amer. Math. Soc. Transl. 23, Providence, RI, 1968.
    [8] Interdisciplinary Applied Mathematics, 3rd edn, Springer, New York, 2002.
    [9] J. Math. Biol., 68 (2014), 1199-1224.
    [10] Discr. Cont. Dyn. Syst. B, 18 (2013), 2669-2688.
    [11] British J. Cancer, 80 (1999), 1892-1897.
    [12] Nonlinear Analysis: RWA, 12 (2011), 418-435.

    © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(28) PDF downloads(498) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog