2017/01/03 by Sébastien Motsch, Motsch, Sebastien, Diane Peurichard +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Biology Tumor Growth #Micro and Nano Robotics #Soft Condensed Matter (cond-mat.soft) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1701.00671
openalex publication_date 2017/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the large time behavior of an agent based model describing\ntumor growth. The microscopic model combines short-range repulsion and cell\ndivision. As the number of cells increases exponentially in time, the\nmicroscopic model is challenging in terms of computational time. To overcome\nthis problem, we aim at deriving the associated macroscopic dynamics leading\nhere to a porous media type equation. As we are interested in the long time\nbehavior of the dynamics, the macroscopic equation obtained through usual\nderivation method fails at providing the correct qualitative behavior (e.g.\nstationary states differ from the microscopic dynamics). We propose a modified\nversion of the macroscopic equation introducing a density threshold for the\nrepulsion. We numerically validate the new formulation by comparing the\nsolutions of the micro- and macro- dynamics. Moreover, we study the asymptotic\nbehavior of the dynamics as the repulsion between cells becomes singular\n(leading to non-overlapping constraints in the microscopic model). We manage to\nshow formally that such an asymptotic limit leads to a Hele-Shaw type problem\nfor the macroscopic dynamics.\n The macroscopic model derived in this paper therefore enables to overcome the\nproblem of large computational time raised by the microscopic model and stays\nclosely linked to the microscopic dynamics.\n