From short-range repulsion to Hele-Shaw problem in a model of tumor growth
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We investigate the large time behavior of an agent based model describing tumor growth. The microscopic model combines short-range repulsion and cell division. As the number of cells increases exponentially in time, the microscopic model is challenging in terms of computational time. To overcome this problem, we aim at deriving the associated macroscopic dynamics leading here to a porous media type equation. As we are interested in the long time behavior of the dynamics, the macroscopic equation obtained through usual derivation method fails at providing the correct qualitative behavior (e.g. stationary states differ from the microscopic dynamics). We propose a modified version of the macroscopic equation introducing a density threshold for the repulsion. We numerically validate the new formulation by comparing the solutions of the micro- and macro- dynamics. Moreover, we study the asymptotic behavior of the dynamics as the repulsion between cells becomes singular (leading to non-overlapping constraints in the microscopic model). We manage to show formally that such asymptotic limit leads to a Hele-Shaw type problem for the macroscopic dynamics. We compare the micro- and macro- dynamics in this asymptotic limit using explicit solutions of the Hele-Shaw problem (e.g. radially symmetric configuration). The numerical simulations reveal an excellent agreement between the two descriptions, validating the formal derivation of the macroscopic model. The macroscopic model derived in this paper therefore enables to overcome the problem of large computational time raised by the microscopic model, but stays closely linked to the microscopic dynamics.