From short-range repulsion to Hele-Shaw problem in a model of tumor growth
Authored by Sebastien Motsch, Diane Peurichard
Date Published: 2018
DOI: 10.1007/s00285-017-1143-4
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Abstract
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.
Tags
Agent-based models
Evolution
behavior
Tumor growth
System
Flow
Solid tumor
2-phase model
Porous media equation
Hele-shaw
problem
Navier-stokes equations
Traffic jams
Aggregation model
Crowd motion