Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Alina Chertock; Pierre Degond; Sophie Hecht; Jean-Paul Vincent; Alina Chertock; Pierre Degond; Sophie Hecht; Jean-Paul Vincent

doi:10.3934/mbe.2019290

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5804-5835. doi: 10.3934/mbe.2019290

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Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

1.
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
2.
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
3.
Francis Crick Institute, 1 Midland Rd, London NW1 1AT, UK

Received: 27 February 2019 Accepted: 17 June 2019 Published: 22 June 2019

This paper proposes a model for the growth of two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Contrasting with earlier works which assume that the two populations are initially segregated, our model can deal with initially mixed populations as it explicitly incorporates a repul-sion force that enforces segregation. To balance segregation instabilities potentially triggered by the repulsion force, our model also incorporates a fourth order diffusion. In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion. Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.
- tissue growth,
- two cell populations,
- incompressible limit,
- free boundary problem
Citation: Alina Chertock, Pierre Degond, Sophie Hecht, Jean-Paul Vincent. Incompressible limit of a continuum model of tissue growth with segregation for two cell populations[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5804-5835. doi: 10.3934/mbe.2019290

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Abstract

This paper proposes a model for the growth of two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Contrasting with earlier works which assume that the two populations are initially segregated, our model can deal with initially mixed populations as it explicitly incorporates a repul-sion force that enforces segregation. To balance segregation instabilities potentially triggered by the repulsion force, our model also incorporates a fourth order diffusion. In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion. Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.

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