# American Institute of Mathematical Sciences

August  2021, 14(8): 2729-2749. doi: 10.3934/dcdss.2020457

## Analysis of a model for tumor growth and lactate exchanges in a glioma

 1 La Rochelle Université, LaSIE UMR CNRS 7356, Avenue Michel Crépeau, F-17042 La Rochelle Cedex, France 2 Università di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Via Campi 213/B, I-41125 Modena, Italy 3 Laboratoire I3M et Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348 4 Equipe DACTIM-MIS, Site du Futuroscope - Téléport 2, 11 Boulevard Marie et Pierre Curie - Bâtiment H3 - TSA 61125, 86073 Poitiers Cedex 9, France 5 CHU de Poitiers, 2 rue de la Milétrie, 86000 Poitiers, France

* Corresponding author: Laurence Cherfils

Received  March 2020 Published  August 2021 Early access  November 2020

Our aim in this paper is to study a mathematical model for tumor growth and lactate exchanges in a glioma. We prove the existence of nonnegative (i.e. biologically relevant) solutions and, under proper assumptions, the uniqueness of the solution. We also state the permanence of the tumor when necrosis is not taken into account in the model and obtain linear stability results. We end the paper with numerical simulations.

Citation: Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2729-2749. doi: 10.3934/dcdss.2020457
##### References:
 [1] A. Aubert, R. Costalat, P. J. Magistretti and L. Pellerin, Brain lactate kinetics: Modeling evidence for neuronal lactate uptake upon activation, Proc. National Acad. Sci. USA, 102 (2005), 16448-16453.  doi: 10.1073/pnas.0505427102. [2] M. Conti, S. Gatti and A. Miranville, Mathematical analysis of a model for proliterative-to-invasive transition of hypoxic glioma cells,, Nonlinear Analysis, 189 (2019), Article 111572, 17 pages. doi: 10.1016/j.na.2019.111572. [3] R. Guillevin, A. Miranville and A. Perrillat-Mercerot, On a reaction-diffusion system associated with brain lactate kinetics, Electronic J. Diff. Eqn., 23 (2017), 1-16. [4] C. Guillevin, R. Guillevin, A. Miranville and A. Perrillat-Mercerot, Analysis of a Mathematical model for brain lactate kinetics, Mathematical Biosciences and Engineering, 15 (2018), 1225-1242.  doi: 10.3934/mbe.2018056. [5] J. B. McGillen, C. J. Kelly, A. Martínez-González, N. K. Martin, E. A. Gaffney, P. K. Maini and V. M. Pérez-García, Glucose-lactate metabolic cooperation in cancer: Insights from a spatial mathematical model and implications for targeted therapy, J. Theoret. Biol., 361 (2014), 190-203.  doi: 10.1016/j.jtbi.2014.09.018. [6] B. Mendoza-Juez, A. Martínez-González, G. F. Calvo and V. M. Peréz-García, A mathematical model for the glucose-lactate metabolism of in vitro cancer cells, Bull. Math. Biol., 74 (2012), 1125-1142.  doi: 10.1007/s11538-011-9711-z. [7] A. Miranville, Mathematical analysis of a parabolic-elliptic model for brain lactate kinetics in Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., 22 (2017), 379-403. [8] A. Perrillat-Mercerot, N. Bourmeyster, C. Guillevin, A. Miranville and R. Guillevin, Mathematical modeling of substrates fluxes and tumor growth in the brain, Acta Biotheoretica, 67 (2019), 149-175.  doi: 10.1007/s10441-019-09343-1.

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##### References:
 [1] A. Aubert, R. Costalat, P. J. Magistretti and L. Pellerin, Brain lactate kinetics: Modeling evidence for neuronal lactate uptake upon activation, Proc. National Acad. Sci. USA, 102 (2005), 16448-16453.  doi: 10.1073/pnas.0505427102. [2] M. Conti, S. Gatti and A. Miranville, Mathematical analysis of a model for proliterative-to-invasive transition of hypoxic glioma cells,, Nonlinear Analysis, 189 (2019), Article 111572, 17 pages. doi: 10.1016/j.na.2019.111572. [3] R. Guillevin, A. Miranville and A. Perrillat-Mercerot, On a reaction-diffusion system associated with brain lactate kinetics, Electronic J. Diff. Eqn., 23 (2017), 1-16. [4] C. Guillevin, R. Guillevin, A. Miranville and A. Perrillat-Mercerot, Analysis of a Mathematical model for brain lactate kinetics, Mathematical Biosciences and Engineering, 15 (2018), 1225-1242.  doi: 10.3934/mbe.2018056. [5] J. B. McGillen, C. J. Kelly, A. Martínez-González, N. K. Martin, E. A. Gaffney, P. K. Maini and V. M. Pérez-García, Glucose-lactate metabolic cooperation in cancer: Insights from a spatial mathematical model and implications for targeted therapy, J. Theoret. Biol., 361 (2014), 190-203.  doi: 10.1016/j.jtbi.2014.09.018. [6] B. Mendoza-Juez, A. Martínez-González, G. F. Calvo and V. M. Peréz-García, A mathematical model for the glucose-lactate metabolism of in vitro cancer cells, Bull. Math. Biol., 74 (2012), 1125-1142.  doi: 10.1007/s11538-011-9711-z. [7] A. Miranville, Mathematical analysis of a parabolic-elliptic model for brain lactate kinetics in Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., 22 (2017), 379-403. [8] A. Perrillat-Mercerot, N. Bourmeyster, C. Guillevin, A. Miranville and R. Guillevin, Mathematical modeling of substrates fluxes and tumor growth in the brain, Acta Biotheoretica, 67 (2019), 149-175.  doi: 10.1007/s10441-019-09343-1.
Tumor (left) and sum of the lactate concentrations $\varphi+ \psi$ (right) at time t = 0. (${ } u_0 (x, y) = 0.1e^{-10(x^2 +(y-2.5)^2)}$, $\varphi_0 = 0.025mM$ inside and $\varphi_0 = 3.854$ outside the tumor area, $\psi_0 = 0.329mM$ inside and $\psi_0 = 1.256mM$ outside the tumor area.)
Tumor (left) and sum of the lactate concentrations $\varphi+ \psi$ (right) at time t = 0. (${ } u_0 (x, y) = 0.1e^{-10(x^2 +(y-2.5)^2)}$, $\varphi_0 = 1.817mM$ inside and $\varphi_0 = 0.915mM$ outside the tumor area, $\psi_0 = 2.291mM$ inside and $\psi_0 = 0.557mM$ outside the tumor area.)
Tumor (left) and sum of the lactate concentrations $\varphi+ \psi$ (right) at time t = 10
Evolution of the tumor concentration (left) and the lactate concentrations (right) with respect to time at the center of the tumor (point $(0; 2.5)$). Evolution of the tumor diameter (below) with respect to time
Concerning the tumor (left), intracellular lactate (right) and capillary lactate (below), comparison between the concentrations at the center of the tumor (green) and at the point (0.8; 2.5), initially outside the tumor (blue)
Tumor (left) and sum of the lactate concentrations $\varphi+ \psi$ (right) at time t = 10
Evolution of the tumor concentration (left) and the lactate concentrations (right) with respect to time at the center of the tumor (point $(0; 2.5)$). Evolution of the tumor diameter (below) with respect to time
Concerning the tumor (left), intracellular lactate (right) and capillary lactate (below), comparison between the concentrations at the center of the tumor (green) and at the point (1; 2.5), initially outside the tumor (blue)
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