Derivation of macroscopic equations for individual cell-based models. A formal approach

Abstract

In this paper we review the theory of cells (particles) that evolve according to a dynamics determined by friction and that interact between themselves by means of suitable potentials. We derive by means of elementary arguments several macroscopic equations that describe the evolution of cell density. Some new results are also obtained — a formal derivation of a limit equation in the case of attractive potential as well as in the case of repulsive potential with a hard-core part are presented Finally, we discuss the possible relevance of those results within the framework of individual cell-based models. Several classes of potentials, including hard-core, repulsive and potentials with attractive parts are discussed. The effect of noise terms in the equation is also considered.