[Paper Review] On the Incompressible Limit for a Tumour Growth Model incorporating Convective Effects

Noemi David, Markus Schmidtchen|arXiv (Cornell University)|Mar 3, 2021

Mathematical Biology Tumor Growth4 citations

TL;DR

This paper establishes the incompressible limit of a tumour growth model incorporating convective effects via nutrient or oxygen gradients, proving that the density-based porous medium equation with pressure law $ p_\gamma = n_\gamma^\gamma $ converges to a free-boundary Hele-Shaw problem as $ \gamma \to \infty $. The key contribution is the rigorous derivation of the incompressible limit with strong compactness of the pressure gradient, ensuring uniqueness of the limiting solution.

ABSTRACT

In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quir\\'os, and V\\'azquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

Motivation & Objective

• To bridge density-based tumour growth models with geometry-free boundary models via a singular limit.
• To rigorously establish the incompressible limit ($ \gamma \to \infty $) of a porous medium equation with convective terms due to chemical gradients.
• To overcome technical challenges in proving strong compactness of the pressure gradient in the limit.
• To characterize the velocity of the free boundary in the limiting Hele-Shaw problem.
• To prove uniqueness of the limiting solution in the incompressible regime.

Proposed method

• Formulates a density-based model with pressure $ p_\gamma = n_\gamma^\gamma $, incorporating cell proliferation $ G(p_\gamma) $ and chemotactic/convection effects via $ \nabla\Phi $.
• Analyzes the pressure evolution equation $ \partial_t p_\gamma = \gamma p_\gamma (\Delta p_\gamma + \Delta\Phi + G(p_\gamma)) + \nabla p_\gamma \cdot \nabla(p_\gamma + \Phi) $ to study the $ \gamma \to \infty $ limit.
• Employs a blend of Aronson-Bénilan estimates and nonlinear semigroup techniques to achieve strong compactness of $ \nabla p_\gamma $.
• Uses variational formulations and Reynolds’ transport theorem to derive the free-boundary dynamics in the limit.
• Establishes convergence of the density $ n_\gamma \to n_\infty = \mathbf{1}_{\Omega(t)} $ and pressure $ p_\gamma \to p_\infty $ in the limit.
• Proves uniqueness of the limit solution by testing differences of two solutions and using monotonicity of $ G $.

Experimental results

Research questions

• RQ1 Can the incompressible limit of a tumour growth model with convective effects be rigorously justified?
• RQ2 What is the limiting behavior of the pressure gradient as $ \gamma \to \infty $, and can strong compactness be established?
• RQ3 How does the limiting solution relate to the classical Hele-Shaw free-boundary problem with source terms?
• RQ4 What is the velocity law of the free boundary in the incompressible limit?
• RQ5 Is the limiting solution unique under the derived conditions?

Key findings

• The incompressible limit $ \gamma \to \infty $ is rigorously established, with $ n_\gamma \to n_\infty = \mathbf{1}_{\Omega(t)} $, the characteristic function of the growing domain.
• The pressure $ p_\gamma $ converges to a limit $ p_\infty $ satisfying $ \Delta p_\infty + \Delta\Phi + G(p_\infty) = 0 $ in $ \Omega(t) $, and $ p_\infty = 0 $ on $ \partial\Omega(t) $.
• The free boundary $ \partial\Omega(t) $ moves with velocity $ V = - (\nabla p_\infty + \nabla\Phi) \cdot \nu $, matching the Hele-Shaw dynamics.
• The limit solution is unique: any two solutions to the limit problem must coincide, proven via energy-type estimates and monotonicity of $ G $.
• The Aronson-Bénilan estimate ensures $ \partial_t n_\infty \geq 0 $, and mass conservation implies $ n_\infty $ is time-independent, consistent with the Hele-Shaw model.
• The convergence is strong in $ L^2 $, and the pressure gradient $ \nabla p_\gamma $ is shown to be strongly precompact via a novel combination of techniques.