On interfaces between cell populations with different mobilities

Tommaso Lorenzi; Alexander Lorz; Benoît Perthame

doi:10.3934/krm.2017012

Article Contents

2017, Volume 10, Issue 1: 299-311. Doi: 10.3934/krm.2017012

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On interfaces between cell populations with different mobilities

1.
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK
2.
Centre de Mathématiques et de Leurs Applications, ENS Cachan, CNRS, Cachan 94230 Cedex, France
3.
Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, UMR 7598, Laboratoire Jacques-Louis Lions, Équipe MAMBA, 4, place Jussieu 75005, Paris, France
4.
CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

* Corresponding author: Alexander Lorz
* Corresponding author: Alexander Lorz

Received: December 31, 2015

Revised: March 31, 2016

Published: September 2017

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Abstract

Partial differential equations describing the dynamics of cell population densities from a fluid mechanical perspective can model the growth of avascular tumours. In this framework, we consider a system of equations that describes the interaction between a population of dividing cells and a population of non-dividing cells. The two cell populations are characterised by different mobilities. We present the results of numerical simulations displaying two-dimensional spherical waves with sharp interfaces between dividing and non-dividing cells. Furthermore, we numerically observe how different ratios between the mobilities change the morphology of the interfaces, and lead to the emergence of finger-like patterns of invasion above a threshold. Motivated by these simulations, we study the existence of one-dimensional travelling wave solutions.

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Mathematics Subject Classification: 35C07, 35B35, 35B36, 92C10.

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Figure 1. Numerical observations in the case µ < ν

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Figure 2. Numerical observations in the case µ > ν

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Figure 4. Travelling waves of Theorem 3.1 for µ < ν

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Figure 5. Transient regime of Theorem 3.1 for µ > ν

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Figure 3. The profile of p for the travelling wave when n has a finite support that coincides with [0, r]

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Figure 6. Travelling waves of Theorem 4.1 for µ < ν

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Figure 7. Transient regime of Theorem 4.1 for µ > ν

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