Analysis of a mathematical model for brain lactate kinetics

Abstract

The aim of this article is to study the well-posedness and properties of a fast-slow system which is related with brain lactate kinetics. In particular, we prove the existence and uniqueness of nonnegative solutions and obtain linear stability results. We also give numerical simulations with different values of the small parameter ε and compare them with experimental data.2010 Mathematics Subject Classification. 34A34, 35B09, 35Q92.

What mathematical models can or cannot do in glioma description and understanding

“…When a glioma forms, it first received nutrient from the existing vessels and capillaries changing the first distribution plan. To understand nutritive changes in the brain when a glioma grows, Aubert, Costalat, Jolivet, Guillevin and coworkers proposed systems based on nonlinear ODEs focused on glucose, lactate and oxygen dynamics [13,68,56,104]. Their models also take into account energy creation, mitochondrial respiration, calcium or potassium and the venous balloon.…”

“…Their models also take into account energy creation, mitochondrial respiration, calcium or potassium and the venous balloon. Acidity plays a key role in tumor development and is intimately linked with lactate exchanges [117,56]. It could be modelled using a diffusion process [83].…”

“…In particular nuclear magnetic resonance spectroscopy (MRS) may help to know the glioma mixture evolution. Therefore, coupled with modeling, it can describe or predict glioma behavior evolution [56,104].…”

“…In the first case a program is realized to simulate the dynamic of a given situation. Because it helps to comprehend a population dynamics in time, mathematical modeling can underlie in silico models [92,56]. Simulations and in silico models are able to take into account kinetic properties but also spatial-temporal considerations to understand spatial interactions and build more accurate models.…”

Global Dynamics of a Piecewise Smooth System for Brain Lactate Metabolism

“…Furthermore the input F stands for the Capillary Blood Flow through capillaries from arterial to venous, and L represents arterial Lactate. In these previous articles, different time scales were considered on the evolution of the two variables and the asymptotics of fast-slow dynamical systems was used (see also a more recent reference [6]). Here, our results are independent of this scaling.…”