|Abstract:|| Tissue growth, as in solid tumors, can be described at a number of different scales from the individual cell to the organ. For a large number of cells, the 'fluid mechanical' approach has been advocated recently by many authors in mathematics, physics and biomecanics. Several levels of mathematical descriptions are commonly used, including elastic or visco-elastic effects, nutrients, active movement, surrounding tissue, vasculature remodeling and several other features.
We will focus on the links between two types of mathematical models. The `compressible' description is at he cell population density level and a more macroscopic, description is based on a free boundary problem close to the classical Hele-Shaw equation. Asymptotic analysis is a tool to derive these Hele-Shaw free boundary problems from cell density systems in the stiff pressure limit. This modeling also opens other questions as circumstances in which instabilities develop and questions on the regularity of the interface between heathy and cancerous tissues.
This presentation follows collaborations with F. Quiros and J.-L. Vazquez (Universidad Autonoma Madrid), M. Tang (SJTU), N. Vauchelet (LJLL), A. Mellet (College Park), A. Lorz (LJLL) and T. Lorenzi (St Andrews).