A new asymptotic model of multilayer tumor growth
Rafael Granero-Belinchón, Martina Magliocca
TL;DR
This work develops a new weakly nonlinear asymptotic model for a multilayer tumor colony with two free boundaries, valid in the regime where the width-to-length ratio $\varepsilon$ is small. By combining a dimensionless ALE formulation with a fixed-domain reformulation, the authors derive a nonlinear, nonlocal high-order PDE system governed by operators $\Theta_1$ and $\Theta_2$ and coupled top/bottom boundary dynamics. A systematic $\varepsilon$-expansion yields explicit leading-order and first-order corrections, culminating in an asymptotic two-boundary model for $U=h^{(0)}+\varepsilon h^{(1)}$ and $V=b^{(0)}+\varepsilon b^{(1)}$ with nonlinear, nonlocal couplings and forcing terms $K^{(0)},K^{(1)},J^{(1)}$, as well as a fixed-domain Poisson problem that closes the system. An existence result for a simplified fixed-domain two-boundary problem in Wiener spaces demonstrates local well-posedness under small initial data, providing mathematical justification for the proposed framework and paving the way to study potential finite-time interface collisions in multilayer tumor dynamics. The methodology offers a rigorous, nonlocal PDE toolkit to analyze geometry-driven tumor growth in elongated domains and informs the validity of continuum descriptions in near-flat, multi-layer configurations.
Abstract
In this paper we study the growth of a tumor colony of multilayer type and focus on how the tumor grows from a near flat (when compared to the length of the tumor as, for instance, in the case of a bone tumor in a femur) initial colony. In particular we derive and study a new weakly nonlinear asymptotic model of multilayer tumor growth. The model takes the form of a nonlinear and nonlocal high order system of PDEs. Finally, motivated by the possibility of a finite time collision of the interfaces, we study the well-posedness of this system.