On a Minimal Model for Hemodynamics and Metabolism of Lactate: Application to Low Grade Glioma and Therapeutic Strategies

Abstract

WHO II low grade glioma evolves inevitably to anaplastic transformation. Magnetic resonance imaging is a good non-invasive way to watch it, by hemodynamic and metabolic modifications, thanks to multinuclear spectroscopy (1)H/(31)P. In this work we study a multi-scale minimal model of hemodynamics and metabolism applied to the study of gliomas. This mathematical analysis leads us to a fast-slow system. The control of the position of the stationary point brings to the concept of domain of viability. Starting from this system, the equations bring to light the parameters that push glioma cells out of their domain of viability. Four fundamental factors are highlighted. The first two are cerebral blood flow and the rate of lactate transport through monocarboxylate transporters, which must be reduced in order to push glioma out of its domain of viability. Another factor is the intra arterial lactate, which must be increased. The last factor is pH, indeed a decrease of intra cellular pH could interfere with glioma growth. These reflections suggest that these four parameters could lead to new therapeutic strategies for the management of low grade gliomas.

Analysis of a mathematical model for brain lactate kinetics

“…Bounds on the solutions are important as they are related with the viability domain of the cell. Furthermore, as mentioned in [8], a therapeutic perspective is to have the steady state outside the viability domain where cell necrosis occurs. Additionnally, we present numerical simulations with different values of the small parameter ε and compare them with experimental data.…”

A singular reaction‐diffusion equation associated with brain lactate kinetics

“…The following system of ODE's: where ε is a small parameter, was proposed and studied as a model for brain lactate kinetics ( and ). In this context, x and y correspond to the lactate concentrations in an interstitial (i.e., extra‐cellular) domain and in a capillary domain, respectively.…”

“…Actually, in , the authors considered fast‐slow dynamics, meaning that – are written in the form where ε ′ is a second small parameter. The corresponding reaction‐diffusion system then reads The mathematical analysis of – (and, in particular, the well‐posedness) appears to be challenging for negative initial data, because of the coupling terms (note that the well‐posedness for the system of ODE's – is also challenging; in and , questions related to equilibria are addressed); see however Remark in the succeeding discussions for some comments on the full system –.…”

“…D FL, , F, L > 0, (1.2) where is a small parameter, was proposed and studied as a model for brain lactate kinetics ( [1][2][3] and [4]). In this context, x and y correspond to the lactate concentrations in an interstitial (i.e., extra-cellular) domain and in a capillary domain, respectively.…”

“…The mathematical analysis of (1.3)-(1.4) (and, in particular, the well-posedness) appears to be challenging for negative initial data, because of the coupling terms (note that the well-posedness for the system of ODE's (1.1)-(1.2) is also challenging; in [1][2][3] and [4], questions related to equilibria are addressed); see however Remark 3.8 in the succeeding discussions for some comments on the full system (1.3)-(1.4). Now, note that u and v are still concentrations and are thus expected to be nonnegative.…”