Composite Waves for a Cell Population System Modeling Tumor Growth and Invasion



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 

Mathematics Subject Classification

35J60 35K57 74J30 92C10 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  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

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