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A nonlocal equation describing tumor growth

Rafael Granero-Belinchón, Martina Magliocca

TL;DR

The paper addresses modeling tumor growth by deriving a reduced, nonlocal PDE for the evolution of a one-phase tumor boundary in early avascular or near-vascular settings. Starting from a free boundary reaction-diffusion system with nutrient and inhibitor dynamics, the authors reformulate in ALE coordinates and perform an asymptotic expansion in a small parameter $\varepsilon$, culminating in a 1D nonlocal evolution equation for the boundary function $g(x_1,t)$ that involves the Hilbert transform $H$ and a nonlocal operator $\Lambda$. The main contributions are a new asymptotic, nonlinear, nonlocal PDE with explicit error control, a local well-posedness result for a simplified depth-dependent case in Wiener spaces, and a discussion of numerical tractability via spectral methods. These results offer a mathematically rigorous, computationally efficient framework for simulating multilayer tumor growth under biologically relevant nutrient/inhibitor exchange regimes. The work thus provides theoretical and practical insights into boundary-driven tumor dynamics and paves the way for more general, higher-dimensional analyses.

Abstract

Cancer is a very complex phenomenon that involves many different scales and situations. In this paper we consider a free boundary problem describing the evolution of a tumor colony and we derive a new asymptotic model for tumor growth. We focus on the case of a single phase tumor colony taking into account chemotactic effects in an early stage where there is no necrotic inner region. Thus, our model is valid for the case of multilayer avascular tumors with very little access to both nutrients and inhibitors or the case where the amount of nutrients and inhibitors is very similar to the amount consumed by the multilayer tumor cells. Our model takes the form of a single nonlocal and nonlinear partial differential equation. Besides deriving the model, we also prove a well-posedness result.

A nonlocal equation describing tumor growth

TL;DR

The paper addresses modeling tumor growth by deriving a reduced, nonlocal PDE for the evolution of a one-phase tumor boundary in early avascular or near-vascular settings. Starting from a free boundary reaction-diffusion system with nutrient and inhibitor dynamics, the authors reformulate in ALE coordinates and perform an asymptotic expansion in a small parameter , culminating in a 1D nonlocal evolution equation for the boundary function that involves the Hilbert transform and a nonlocal operator . The main contributions are a new asymptotic, nonlinear, nonlocal PDE with explicit error control, a local well-posedness result for a simplified depth-dependent case in Wiener spaces, and a discussion of numerical tractability via spectral methods. These results offer a mathematically rigorous, computationally efficient framework for simulating multilayer tumor growth under biologically relevant nutrient/inhibitor exchange regimes. The work thus provides theoretical and practical insights into boundary-driven tumor dynamics and paves the way for more general, higher-dimensional analyses.

Abstract

Cancer is a very complex phenomenon that involves many different scales and situations. In this paper we consider a free boundary problem describing the evolution of a tumor colony and we derive a new asymptotic model for tumor growth. We focus on the case of a single phase tumor colony taking into account chemotactic effects in an early stage where there is no necrotic inner region. Thus, our model is valid for the case of multilayer avascular tumors with very little access to both nutrients and inhibitors or the case where the amount of nutrients and inhibitors is very similar to the amount consumed by the multilayer tumor cells. Our model takes the form of a single nonlocal and nonlinear partial differential equation. Besides deriving the model, we also prove a well-posedness result.
Paper Structure (11 sections, 1 theorem, 114 equations, 2 figures)

This paper contains 11 sections, 1 theorem, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free boundary problems describing tumor growth
  3. The free boundary problem in Arbitrary Lagrangian-Eulerian coordinates
  4. Derivation of the asymptotic model
  5. Methodology and a toy problem
  6. Ansatz
  7. The case $n=0$
  8. The case $n=1$
  9. The asymptotic model
  10. An explicit particular case
  11. Discussion

Key Result

Theorem 5.1

Let $\mathbcal{g}\in A^1{\textcolor{black}{(\mathbb{T})}}$ be such that Then, there exists a time $T$ and a unique solution of eq:particular.

Figures (2)

  • Figure 1: The one-phase case with only the tumor colony and infinite depth
  • Figure 2: The reference domain.

Theorems & Definitions (2)

  • Theorem 5.1: Well-posedness
  • proof