[Paper Review] Numerical study of cancer cell invasion dynamics using adaptive mesh refinement: the urokinase model

Niklas Kolbe, Jana Kat+3 more|arXiv (Cornell University)|Aug 4, 2014

Mathematical Biology Tumor Growth53 references3 citations

TL;DR

This paper proposes an adaptive mesh refinement (AMR) finite volume method using h-refinement to simulate urokinase-driven cancer cell invasion with high accuracy and reduced computational cost. By employing gradient-based refinement and smooth refinement rules, the method achieves stable, second-order convergence while resolving complex dynamics like merging and emerging cell clusters in chemotactic and proteolytic environments.

ABSTRACT

In the present work we investigate the chemotactically and proteolytically driven tissue invasion by cancer cells. The model employed is a system of advection-reaction-diffusion equations that features the role of the serine protease urokinase-type plasminogen activator. The analytical and numerical study of this system constitutes a challenge due to the merging, emerging, and travelling concentrations that the solutions exhibit. Classical numerical methods applied to this system necessitate very fine discretization grids to resolve these dynamics in an accurate way. To reduce the computational cost without sacrificing the accuracy of the solution, we apply adaptive mesh refinement techniques, in particular h-refinement. Extended numerical experiments exhibit that this approach provides with a higher order, stable, and robust numerical method for this system. We elaborate on several mesh refinement criteria and compare the results with the ones in the literature. We prove, for a simpler version of this model, $L^\infty$ bounds for the solutions, we study the stability of its conditional steady states, and conclude that it can serve as a test case for further development of mesh refinement techniques for cancer invasion simulations.

Motivation & Objective

• Address the high computational cost of simulating cancer cell invasion with fine grids due to complex dynamics like merging and emerging concentrations.
• Develop a robust, high-order numerical method that maintains accuracy while reducing computational load for reaction-diffusion-advection systems in cancer invasion modeling.
• Justify a simplified chemotaxis-haptotaxis model with logistic growth as a test case for future mesh refinement techniques.
• Investigate the stability and qualitative behavior of solutions in both the full and reduced models to support methodological validation.
• Enable efficient, scalable simulations of 2D cancer invasion dynamics by integrating adaptive mesh refinement with IMEX time integration.

Proposed method

• Employ a higher-order finite volume method with IMEX3 time integration for solving the advection-reaction-diffusion system modeling cancer cell invasion.
• Implement h-refinement via cell bisection, dynamically adjusting grid resolution based on solution gradients to concentrate computational effort where needed.
• Apply smooth refinement constraints, limiting neighboring cells to at most one refinement level difference to ensure stability and efficiency.
• Use the gradient of cancer cell density as the primary error estimator for mesh refinement and coarsening decisions.
• Prove $L^{inity}$ bounds on solutions for both the full and reduced models to ensure numerical stability and convergence.
• Compare results from adaptive and uniform grids to validate accuracy and efficiency gains in 1D and 2D experiments.

Experimental results

Research questions

• RQ1 Can adaptive mesh refinement significantly reduce computational cost while preserving accuracy in simulating cancer cell invasion with complex concentration dynamics?
• RQ2 Which refinement indicator (e.g., cancer cell gradient) yields the most accurate and stable results in resolving merging and emerging concentration patterns?
• RQ3 Does the simplified chemotaxis-haptotaxis model with logistic growth exhibit similar qualitative dynamics to the full urokinase model, justifying its use as a test case?
• RQ4 What are the conditions under which the reduced model maintains $L^{inity}$ bounded solutions, ensuring numerical stability?
• RQ5 How does the choice of refinement strategy (e.g., smooth refinement) affect the convergence and robustness of the numerical solution?

Key findings

• The adaptive mesh refinement approach with gradient-based refinement and smooth refinement rules achieves stable, second-order convergence in numerical experiments.
• The method reduces computational cost significantly compared to uniform fine grids while accurately resolving complex dynamics such as merging and emerging cell clusters.
• The reduced chemotaxis-haptotaxis model with logistic growth exhibits similar transient behaviors—merging and emerging concentrations—as the full urokinase model, validating its use as a test case.
• Analytical $L^{inity}$ bounds were proven for both the full and reduced models, ensuring solution stability and enabling reliable numerical simulations.
• Numerical results show that the cancer cell front propagates into the extracellular matrix, degrading it and forming heterogeneous clusters by $t=200$, with no steady states observed.
• The best-performing refinement strategy combines cancer cell gradient as the estimator and enforces a maximum one-level difference between neighboring cells, ensuring robustness and accuracy.