Chinese Annals of Mathematics, Series B

, Volume 34, Issue 2, pp 295–318 | Cite as

Composite waves for a cell population system modeling tumor growth and invasion

  • Min Tang
  • Nicolas Vauchelet
  • Ibrahim Cheddadi
  • Irene Vignon-Clementel
  • Dirk Drasdo
  • Benoît Perthame


In the recent biomechanical theory of cancer growth, solid tumors are considered as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, with the latter depending on the local cell density (contact inhibition) or/and on the mechanical stress in the tumor.

For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling, the authors prove that there are always traveling waves above a minimal speed, and analyse their shapes. They appear to be complex with composite shapes and discontinuities. Several small parameters allow for analytical solutions, and in particular, the incompressible cells limit is very singular and related to the Hele-Shaw equation. These singular traveling waves are recovered numerically.


Traveling waves Reaction-diffusion Tumor growth Elastic material 

2000 MR Subject Classification

35J60 35K57 74J30 92C10 


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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Min Tang
    • 1
    • 2
  • Nicolas Vauchelet
    • 3
    • 4
  • Ibrahim Cheddadi
    • 3
    • 4
  • Irene Vignon-Clementel
    • 3
    • 4
  • Dirk Drasdo
    • 3
    • 4
  • Benoît Perthame
    • 3
    • 4
  1. 1.Department of mathematics, Institute of Natural Sciences and MOE-LSCShanghai Jiao Tong UniversityShanghaiChina
  2. 2.INRIA Paris RocquencourtParisFrance
  3. 3.Laboratoire Jacques-Louis LionsUPMC Univ Paris 06 and CNRS UMR 7598ParisFrance
  4. 4.INRIA Paris RocquencourtParisFrance

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