[Paper Review] Analysis and computation of some tumor growth models with nutrient: from cell density models to free boundary dynamics

Jian‐Guo Liu, Min Tang+2 more|arXiv (Cornell University)|Feb 2, 2018

Mathematical Biology Tumor Growth14 references4 citations

TL;DR

This paper analyzes tumor growth models with nutrient dynamics, linking cell density models governed by a pressure law $ p(n) = n^\gamma $ to free boundary Hele-Shaw models in the incompressible limit as $ \gamma \to \infty $. It establishes analytical estimates, derives benchmark solutions for tumor front propagation, and validates the connection via a conservative, positivity-preserving numerical scheme that confirms convergence from cell density to free boundary dynamics.

ABSTRACT

In this paper, we study the tumor growth equation along with various models for the nutrient component, including the \emph{in vitro} model and the \emph{in vivo} model. At the cell density level, the spatial availability of the tumor density $n$ is governed by the Darcy law via the pressure $p(n)=n^γ$. For finite $γ$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $γ ightarrow \infty$, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

Motivation & Objective

• To establish rigorous analytical estimates—boundedness of nutrient, non-negativity, and growth control—for tumor cell density models with nutrient-dependent proliferation.
• To formalize the convergence of cell density models with $ p(n) = n^\gamma $ to Hele-Shaw type free boundary problems as $ \gamma \to \infty $, representing the incompressible limit of tumor tissue.
• To derive analytical solutions for the limiting Hele-Shaw models that serve as benchmark solutions for tumor front propagation.
• To develop and validate a conservative, positivity-preserving numerical scheme that accurately captures the transition from cell density models to free boundary dynamics.

Proposed method

• Modeling tumor cell density $ n(x,t) $ via a degenerate diffusion equation driven by pressure $ p(n) = n^\gamma $, with nutrient consumption governed by $ -\Delta c + \psi(n)c = 0 $ in in vitro and in vivo settings.
• Using asymptotic analysis to derive the Hele-Shaw limit as $ \gamma \to \infty $, yielding a free boundary problem where $ p_\infty $ satisfies $ -\Delta p_\infty = G(c) $ in $ D_\infty(t) $, with $ p_\infty = 0 $ on $ \partial D_\infty(t) $.
• Deriving analytical solutions for radial symmetric cases in both in vitro and in vivo models, providing exact benchmarks for front propagation speed.
• Implementing a conservative, positivity-preserving finite difference scheme to numerically solve the cell density model for finite $ \gamma $, ensuring stability and physical consistency.
• Validating the convergence of numerical solutions toward the Hele-Shaw limit by comparing front propagation speeds and spatial profiles across varying $ \gamma $.
• Using Neumann and Dirichlet boundary conditions for $ n $ and $ c $, with mesh refinement to ensure accuracy in front tracking and nutrient distribution.

Experimental results

Research questions

• RQ1 How do boundedness, non-negativity, and growth estimates of tumor cell density depend on the pressure exponent $ \gamma $ in nutrient-influenced models?
• RQ2 What is the rigorous mathematical link between cell density models with $ p(n) = n^\gamma $ and the Hele-Shaw free boundary model in the limit $ \gamma \to \infty $?
• RQ3 What analytical solutions exist for the Hele-Shaw model in radial symmetric in vitro and in vivo settings, and how do they predict front propagation speed?
• RQ4 How well does a conservative, positivity-preserving numerical scheme reproduce the transition from cell density dynamics to free boundary motion as $ \gamma $ increases?

Key findings

• For finite $ \gamma $, the tumor cell density $ n $ remains non-negative and grows at a controlled rate, with nutrient concentration uniformly bounded.
• As $ \gamma \to \infty $, the cell density model formally converges to a Hele-Shaw free boundary problem where the tumor front evolves with velocity $ v = -\nabla p_\infty $.
• Analytical solutions for radial symmetric in vitro and in vivo models confirm that the tumor front propagates twice as fast in the in vitro model compared to the in vivo model in the long-time limit.
• Numerical simulations with $ \gamma = 80 $ show excellent agreement between the cell density model and the limiting Hele-Shaw solution, validating the convergence of front dynamics.
• The front propagation speed in the in vitro model is predicted by $ R(t) \sim \sqrt{2t} $, while in the in vivo model it is $ R(t) \sim \sqrt{t} $, matching numerical results.
• The numerical scheme successfully preserves positivity and conserves mass, with stable and accurate front tracking even at high $ \gamma $, though accuracy degrades when $ \gamma $ is small and density exceeds 1.