Influence of diffusion on interactions between malignant gliomas and immune system

Urszula Foryś¹

¹ Institute of Applied Mathematics and Mechanics Faculty of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Applicationes Mathematicae, Tome 37 (2010) no. 1, pp. 53-67.

Résumé

We analyse the influence of diffusion and space distribution of cells in a simple model of interactions between an activated immune system and malignant gliomas, among which the most aggressive one is GBM Glioblastoma Multiforme. It turns out that diffusion cannot affect stability of spatially homogeneous steady states. This suggests that there are two possible outcomes—the solution is either attracted by the positive steady state or by the semitrivial one. The semitrivial steady state describes the healthy state, while the positive one reflects the chronic disease and typically the level of tumour cells in this state is very high, exceeding the threshold of lethal outcome. Results of numerical simulation show that the initial tumour cells distribution has an essential impact on the dynamics of the system. If the positive steady state exists, then we observe bistability and the initial distribution decides to which steady state the solution tends.

Détail

DOI : 10.4064/am37-1-4

Keywords: analyse influence diffusion space distribution cells simple model interactions between activated immune system malignant gliomas among which aggressive gbm glioblastoma multiforme turns out diffusion cannot affect stability spatially homogeneous steady states suggests there possible outcomes solution either attracted positive steady state semitrivial semitrivial steady state describes healthy state while positive reflects chronic disease typically level tumour cells state high exceeding threshold lethal outcome results numerical simulation initial tumour cells distribution has essential impact dynamics system positive steady state exists observe bistability initial distribution decides which steady state solution tends