Mathematical Modeling of Substrates Fluxes and Tumor Growth in the Brain

Abstract

The aim of this article is to show how a tumor can modify energy substrates fluxes in the brain to support its own growth. To address this question we use a modeling approach to explain brain nutrient kinetics. In particular we set up a system of 17 equations for oxygen, lactate, glucose concentrations and cells number in the brain. We prove the existence and uniqueness of nonnegative solutions and give bounds on the solutions. We also provide numerical simulations.

Appendix: Parameters Value for the Simulations

Initial values chosen for this simulation are given in Aubert et al. works. We refer the interested reader to Aubert and Costalat (2005) to get an explanation of these relevant values.

We give in following Tables 4, 5, 6 and 7 the main parameter values. The function F is a positive square wave function with a maximum value of $0.048\hbox { s}^{-1}$ (during $10\hbox { s}$) and a minimum value of $0.036\hbox { s}^{-1}$ (during $100\hbox { s}$). It reports the cerebral blood flow (Aubert et al. 2001; Aubert and Costalat 2005; Jolivet et al. 2015; Guillevin et al. 2018). It can be taken as follows:

$$\begin{aligned} F \left\{ \begin{array}{rl} {\mathbb {R}}^+&{}\longrightarrow {\mathbb {R}}^+\\ t&{}\longmapsto \left\{ \begin{array}{rl} F_0(1+\alpha _f) &{}\text { if }\exists N \in {\mathbb {N}}^* / (N-1)t_f+t_i<t<Nt_f, \\ F_0 &{}\text { if not.} \end{array} \right. \end{array} \right. \end{aligned}$$

with parameters from Table 4.

Table 4 Parameters for F

Table 5 Oxygen parameters

Oxygen parameter values are based on Valabrégue et al. (2003) and Aubert and Costalat (2005) works. The quadratic adjustement we have done is explained in the main text.

Table 6 Glucose parameters

Glucose parameter values are taken up from Jolivet et al. (2015) and Aubert and Costalat (2005) models. Chosen values are supported by the related literature (Gruetter et al. 1998).

Table 7 Lactate parameters

Lactate parameter values are also taken up from Jolivet et al. (2015) and Aubert and Costalat (2005) models. To the best of our knowledge there has been no measure of the lactate maximal transport rate between astrocyte and neuron yet. Therefore its value has been chosen arbitrarily here.

Moreover every tumor shows up a different mechanism concerning substrate use (Romero-Garcia et al. 2016). Gliomas are ad initio formed by degenarated astrocytic cells. Therefore we arbitrarily chose here to keep the same maximal transport rate for both astrocytes and tumor cells. We have:

$$\begin{aligned} \mu _{c,g}&=\mu _{c,a}=0.104 \text { s}^{-1},&\rho _{e,g}&=\rho _{e,a}=0.147 \text { mmoL}\times \text {L}^{-1}\times \text {s}^{-1},\\ \kappa _{g,n}&=\kappa _{a,n}= 0.5 \text { mmoL}\times \text {L}^{-1}\times \text {s}^{-1},&\kappa _{g,e}&=\kappa _{a,e}= 106.1 \text { mmoL} \times \text {L}^{-1}\times \text {s}^{-1},\\ \kappa _{g,c}&=\kappa _{a,c}= 0.00243 \text { mmoL}\times \text {L}^{-1}\times \text {s}^{-1}. \end{aligned}$$

Knowing that glioma cells are more efficient to take up substrates we arbitrarily chose for this simulation tumor Michaelis–Mentens constant times higher than astrocytic Michaelis–Mentens constant.

$$\begin{aligned} p_g = 3 \times p_a = 9 \text {mmoL}\times \text {L}^{-1},\quad k_g = 3 \times k_a = 7.5 \text {mmoL}\times \text {L}^{-1}. \end{aligned}$$

Finally, glucose, lactate and oxygen consumption rate of each compartment are arbitrarily chosen keeping in mind that astrocytes show higher abilities to take glucose and convert it into lactate than neurons. Contrariwise, neurons have better abilities to take lactate and convert it into energy.