The Hele-Shaw Asymptotics for Mechanical Models of Tumor Growth

Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. Our first goal here is to 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 the stiff limit in the pressure law of state. In contrast with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. A complete description requires the equation on the cell number density. We then go on to consider a more complex model including the supply of nutrients through vasculature, and we study the stiff limit for the involved coupled system.