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One of the main difficulties in the modeling of complex systems such as fish schooling, crowd motion or tumor growth is the lack of fundamental laws. It is unknown how two agents (e.g. cells, pedestrians, birds) interact, we only have access to the result of the interactions. But there is always one rule that agents have to satisfy: two agents cannot overlap, i.e. they cannot occupy the same position in space. Despite the simplicity of this rule, non-overlapping constraints have several intriguing effects and raise several challenges both analytically and numerically, as they induce non-convex problems.
Project:
Several methods have been proposed to encompass non-overlapping constraints. At the microscopic level (i.e. agent-based models), a common method is to introduce (short-range) repulsion dynamics: two cells move away from each other when they are too close. At the macroscopic level (i.e. Partial Differential Equations (PDE)), non-overlapping can be expressed as a density constraint, i.e. the density has to stay below a given threshold. The goal of this work is to link the two descriptions starting from a simple model of tumor growth [1]. The microscopic model combines short-range repulsion and cell division. The associated macroscopic dynamics obtained thourgh usual derivation leads here to a porous media type equation which 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 cell-cell 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 microand 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
Collaborations:
S. Motsch (Arizona State University)
Related publications:
S. Motsch, D. Peurichard, Large scale dynamics of short-range repulsion and cell division, to appear in Journal of Mathematical Biology (2017), lien arXiv
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