Complex multicellular systems and immune competition: new paradigms looking for a mathematical theory

This chapter deals with the modeling and simulation of large systems of interacting entities whose microscopic state includes not only geometrical and mechanical variables (typically position and velocity), but also biological functions or specific activities. The main issue looks at the development of a biological mathematical theory for multicellular systems. The first part is devoted to the derivation of mathematical structures to be properly used to model a variety of biological phenomena with special focus on immune competition. Then, some specific applications are proposed referring to the competition between neoplastic and immune cells. Finally, the last part is devoted to research perspectives towards the objective of developing a mathematical-biological theory. A critique is presented of what has already been achieved towards the above target and what is still missing with special focus on multiscale systems.