A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model

Niklas Kolbe, Kat’uchová, Jana, Nikolaos Sfakianakis, Nadja Hellmann and Lukáčová-Medvid’ová, Mária

Applied Mathematics and Computation, 2016, vol. 273, issue C, 353-376

Abstract: In the present work we investigate a model that describes the chemotactically and proteolytically driven tissue invasion by cancer cells. The model is a system of advection–reaction–diffusion equations that takes into account the role of the serine protease urokinase-type plasminogen activator. The analytical and numerical study of such a system constitutes a challenge due to the merging, emerging, and traveling 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 show 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∞ bounds for the solutions. We also studied 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.

Keywords: Cancer modeling; Chemotaxis; Merging and emerging concentrations; Finite volume method; IMEX; Adaptive mesh refinement (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:273:y:2016:i:c:p:353-376

DOI: 10.1016/j.amc.2015.08.023

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