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Incompressible limit for an age-structured tumor model

Maeve Wildes

TL;DR

The main contribution of this paper is establishing the convergence of solutions of the age-structured model to a limit satisfying a Hele-Shaw free boundary problem that describes the geometric motion of the tumor as it grows according to a nonlinear Darcy's law.

Abstract

In this paper, we consider an age-structured mechanical model for tumor growth. This model takes into account the life-cycle of tumor cells by including an age variable. The underlying process for tumor growth is the same as classical tumor models, where growth is driven by pressure-limited cell proliferation, and cell movement away from regions of high pressure. The main contribution of this paper is establishing the convergence of solutions of the age-structured model to a limit satisfying a Hele-Shaw free boundary problem. This limiting problem describes the geometric motion of the tumor as it grows according to a nonlinear Darcy's law.

Incompressible limit for an age-structured tumor model

TL;DR

The main contribution of this paper is establishing the convergence of solutions of the age-structured model to a limit satisfying a Hele-Shaw free boundary problem that describes the geometric motion of the tumor as it grows according to a nonlinear Darcy's law.

Abstract

In this paper, we consider an age-structured mechanical model for tumor growth. This model takes into account the life-cycle of tumor cells by including an age variable. The underlying process for tumor growth is the same as classical tumor models, where growth is driven by pressure-limited cell proliferation, and cell movement away from regions of high pressure. The main contribution of this paper is establishing the convergence of solutions of the age-structured model to a limit satisfying a Hele-Shaw free boundary problem. This limiting problem describes the geometric motion of the tumor as it grows according to a nonlinear Darcy's law.
Paper Structure (8 sections, 15 theorems, 122 equations)

This paper contains 8 sections, 15 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Age-structured mechanical model
  3. Motivations and Related Work
  4. Definitions and notation
  5. Outline of the paper
  6. Assumptions and Main Results
  7. Existence of solutions to the model
  8. Incompressible Limit

Key Result

Theorem 2.1

Let $(n_m,p_m,\rho_m)$ be a weak solution of nm, under the assumptions a.beta, a.r, a.nu, and v.assum. Assume that the initial data satisfies assumpinit-initconv. Further, assume that $n_{0,m}$ is compactly supported in $x$ and $\theta$, uniformly in $m$. Then, up to a subsequence, as $m \rightarrow

Theorems & Definitions (32)

  • Theorem 2.1: Weak Free-Boundary Limit
  • Remark 2.2
  • Theorem 2.3: Complementarity Formula
  • Theorem 3.1: llmw, Theorem 2.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 22 more