Due the complexity and the importance of this phenomenon, tumor growth has raised the attention of many mathematicians. Indeed, cancer has become a major cause of death in western countries. Partial differential equations (PDE) may be an interesting tool to model and describe the expansion of tumoral cells. The aim of this course is to review some classes of PDE models used to describe tumor growth. In particular, we will distinguish two main classes of models: cell mechanical model and free boundary model. In the former, the tumor is described through the dynamics of cell density of tumoral cells. In the latter, we focus on the dynamics of the domain of the tumor subjected to internal pressure. Then we will explain a link between both classes of model.
The course will be divided into 4 sessions. No prerequisite are necessary.
Thursday 13th October:
– Course 1: Modelling of spatial tumor growth
In this first session, we will first recall some useful mathematical tools. Then we will review spatial models of tumor growth.
– Course 2 : Derivation of Stefan free boundary problems
This second session is devoted to a known example of derivation of free boundary model. We will explain the different steps to establish rigorously this derivation.
Friday 14th October:
– Course 3 : Derivation of Hele-Shaw models for tumor growth
The derivation of a free boundary model of Hele-Shaw type used to describe tumor growth will be presented. This derivation is obtained thanks to an incompressible limit from a cell mechanical model.
– Course 4 : Some extension
Finally, in the last session we will present some extension to a more physical model with viscosity, for which we need a different approach.