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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020457
References:
[1]

A. AubertR. CostalatP. 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.  Google Scholar

[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.  Google Scholar

[3]

R. GuillevinA. Miranville and A. Perrillat-Mercerot, On a reaction-diffusion system associated with brain lactate kinetics, Electronic J. Diff. Eqn., 23 (2017), 1-16.   Google Scholar

[4]

C. GuillevinR. GuillevinA. 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.  Google Scholar

[5]

J. B. McGillenC. J. KellyA. Martínez-GonzálezN. K. MartinE. A. GaffneyP. 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.  Google Scholar

[6]

B. Mendoza-JuezA. Martínez-GonzálezG. 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.  Google Scholar

[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.   Google Scholar

[8]

A. Perrillat-MercerotN. BourmeysterC. GuillevinA. 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.  Google Scholar

show all references

References:
[1]

A. AubertR. CostalatP. 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.  Google Scholar

[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.  Google Scholar

[3]

R. GuillevinA. Miranville and A. Perrillat-Mercerot, On a reaction-diffusion system associated with brain lactate kinetics, Electronic J. Diff. Eqn., 23 (2017), 1-16.   Google Scholar

[4]

C. GuillevinR. GuillevinA. 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.  Google Scholar

[5]

J. B. McGillenC. J. KellyA. Martínez-GonzálezN. K. MartinE. A. GaffneyP. 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.  Google Scholar

[6]

B. Mendoza-JuezA. Martínez-GonzálezG. 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.  Google Scholar

[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.   Google Scholar

[8]

A. Perrillat-MercerotN. BourmeysterC. GuillevinA. 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.  Google Scholar

Figure 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.)
Figure 5.  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.)
Figure 2.  Tumor (left) and sum of the lactate concentrations $ \varphi+ \psi $ (right) at time t = 10
Figure 3.  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
Figure 4.  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)
Figure 6.  Tumor (left) and sum of the lactate concentrations $ \varphi+ \psi $ (right) at time t = 10
Figure 7.  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
Figure 8.  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)
[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[3]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[4]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[5]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[6]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[7]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[8]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[9]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[10]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[11]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[12]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[13]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

2019 Impact Factor: 1.233

Article outline

Figures and Tables

[Back to Top]