# FRB

## Seminar

### Plenary Lecture 14: Free boundary problems for mechanical models of tumor growth

Seminar Room 1, Newton Institute

#### Abstract

Mathematical models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem. We first formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain singular limit which leads to a Hele-Shaw type problem. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the limit problem. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. Using this theory as a basis, we go on to consider a more complex model including nutrients. Here, technical difficulties appear, that reduce the generality and detail of the description. We prove uniqueness for the system, a main mathematical difficulty. Joint work with Benoit Perthame, Paris, and Fernando Quiros, Madrid#### Video

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