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

Niklas Kolbe, Jana Kat'Uchová, Nikolaos Sfakianakis*, Nadja Hellmann, Mária Lukáčová-Medvid'Ová

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)353-376
Number of pages24
JournalApplied Mathematics and Computation
Volume273
DOIs
Publication statusPublished - 15 Jan 2016

Keywords

  • Adaptive mesh refinement
  • Cancer modeling
  • Chemotaxis
  • Finite volume method
  • IMEX
  • Merging and emerging concentrations

Fingerprint

Dive into the research topics of 'A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model'. Together they form a unique fingerprint.

Cite this