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Models of Cell Motion and Tissue Growth

  • Benoît PerthameEmail author
Chapter
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Part of the Lecture Notes in Mathematics book series (LNM, volume 2260)

Abstract

The mathematical description of cell movement, from the individual scale to the collective motion, is a rich and complex domain of biomathematics which leads to several types of questions and partial differential equations. For instance, bacteria move by run-and-tumble movement, which is well described, at the cell scale, by a kinetic equation in the phase coordinates. At the population scale, chemotactic effects lead to the famous parabolic Keller–Segel system, and the many improvements of it that have been addressed recently.

When considering living tissues, concepts issued from mechanics arise. Notions of pressure, phases, incompressibility are used in systems which carry the typical parabolic and hyperbolic characters of fluid mechanics. Their complexity is directly related to the details in the biological description and opens numerous mathematical questions which are poorly understood.

The various process involved in cell movements can be considered at the cell scale, at the population scale and, for tissues, at the organ scale. This leads to study singular perturbation problems of various types. For tumor growth, the tumor boundary can appear as a free boundary or as an internal layer.

Notes

Acknowledgements

The author has received partial funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740623) and partial funding from the ANR blanche project Kibord ANR-13-BS01-0004 funded by the French Ministry of Research.

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Authors and Affiliations

  1. 1.Sorbonne Université, CNRS, Université de Paris, InriaLaboratoire Jacques-Louis LionsParisFrance

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