American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?
  • MBE Home
  • This Issue
  • Next Article
    A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis
October  2018, 15(5): 1225-1242. doi: 10.3934/mbe.2018056

Analysis of a mathematical model for brain lactate kinetics

Carole Guillevin 1, , Rémy Guillevin 1, , Alain Miranville 2, and  Angélique Perrillat-Mercerot 2,,

1. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Equipe DACTIM-MIS, CHU de Poitiers, 2 Rue de la Milétrie, F-86021 Poitiers, France
2. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Equipe DACTIM-MIS, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

* Corresponding author: angelique.perrillat@math.univ-poitiers.fr

Received  October 20, 2017 Revised  January 27, 2018 Published  May 2018

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 $\varepsilon$ and compare them with experimental data.

Keywords: Brain lactate kinetics, fast-slow system, regularity, well-posedness, linear stability, limit system, simulations, in vivo data.
Mathematics Subject Classification: 34A34, 35B09, 35Q92.
Citation: Carole Guillevin, Rémy Guillevin, Alain Miranville, Angélique Perrillat-Mercerot. Analysis of a mathematical model for brain lactate kinetics. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1225-1242. doi: 10.3934/mbe.2018056
References:
[1]

A. Aubert and R. Costalat, Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism, Journal of Cerebral Blood Flow & Metabolism, 25 (2005), 1476-1490.  doi: 10.1038/sj.jcbfm.9600144.  Google Scholar

[2]

A. Aubert, R. Costalat, P. Magistretti, J. Pierre and L. Pellerin, Brain lactate kinetics: modeling evidence for neuronal lactate uptake upon activation, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 16448-16453.  doi: 10.1073/pnas.0505427102.  Google Scholar

[3]

M. Cloutier, F. B. Bolger, J. P. Lowry and P. Wellstead, An integrative dynamic model of brain energy metabolism using in vivo neurochemical measurements, Journal of Computational Neuroscience, 27 (2009), 391-414.  doi: 10.1007/s10827-009-0152-8.  Google Scholar

[4]

R. Costalat, J.-P. Françoise, C. Menuel, M. Lahutte, J.-N. Vallée, G. De Marco, J. Chiras and R. Guillevin, Mathematical modeling of metabolism and hemodynamics, Acta Biotheoretica, 60 (2012), 99-107.  doi: 10.1007/s10441-012-9157-1.  Google Scholar

[5]

C. E. Griguer, C. R. Oliva and G. Y. Gillespie, Glucose metabolism heterogeneity in human and mouse malignant glioma cell lines, Journal of Neuro-oncology, 74 (2005), 123-133.  doi: 10.1007/s11060-004-6404-6.  Google Scholar

[6]

R. Guillevin, C. Menuel, J.-N. Vallée, J.-P. Françoise, L. Capelle, C. Habas, G. De Marco, J. Chiras and R. Costalat, Mathematical modeling of energy metabolism and hemodynamics of WHO grade Ⅱ gliomas using in vivo MR data, Comptes rendus biologies, 334 (2011), 31-38.  doi: 10.1016/j.crvi.2010.11.002.  Google Scholar

[7]

M. Lahutte-Auboin, R. Costalat, J.-P. Françoise, R. Guillevin, Dip and Buffering in a fast-slow system associated to Brain Lactacte Kinetics, preprint, arXiv: 1308.0486. Google Scholar

[8]

M. Lahutte-Auboin, R. Guillevin, J.-P. Françoise, J.-N. Vallée and R. Costalat, On a minimal model for hemodynamics and metabolism of lactate : application to low grade glioma and therapeutic strategies, Acta Biotheoretica, 61 (2013), 79-89.  doi: 10.1007/s10441-013-9174-8.  Google Scholar

[9]

P. J. Magistretti and I. Allaman, A cellular perspective on brain energy metabolism and functional imaging, Neuron, 86 (2015), 883-901.  doi: 10.1016/j.neuron.2015.03.035.  Google Scholar

[10]

S. Mangia, G. Garreffa, M. Bianciardi, F. Giove, F. Di Salle and B. Maraviglia, The aerobic brain: Lactate decrease at the onset of neural activity, Neuroscience, 118 (2003), 7-10.  doi: 10.1016/S0306-4522(02)00792-3.  Google Scholar

[11]

J. R. Mangiardi and P. Yodice, Metabolism of the malignant astrocytoma, Neurosurgery, 26 (1990), 1-19.   Google Scholar

show all references

References:
[1]

A. Aubert and R. Costalat, Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism, Journal of Cerebral Blood Flow & Metabolism, 25 (2005), 1476-1490.  doi: 10.1038/sj.jcbfm.9600144.  Google Scholar

[2]

A. Aubert, R. Costalat, P. Magistretti, J. Pierre and L. Pellerin, Brain lactate kinetics: modeling evidence for neuronal lactate uptake upon activation, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 16448-16453.  doi: 10.1073/pnas.0505427102.  Google Scholar

[3]

M. Cloutier, F. B. Bolger, J. P. Lowry and P. Wellstead, An integrative dynamic model of brain energy metabolism using in vivo neurochemical measurements, Journal of Computational Neuroscience, 27 (2009), 391-414.  doi: 10.1007/s10827-009-0152-8.  Google Scholar

[4]

R. Costalat, J.-P. Françoise, C. Menuel, M. Lahutte, J.-N. Vallée, G. De Marco, J. Chiras and R. Guillevin, Mathematical modeling of metabolism and hemodynamics, Acta Biotheoretica, 60 (2012), 99-107.  doi: 10.1007/s10441-012-9157-1.  Google Scholar

[5]

C. E. Griguer, C. R. Oliva and G. Y. Gillespie, Glucose metabolism heterogeneity in human and mouse malignant glioma cell lines, Journal of Neuro-oncology, 74 (2005), 123-133.  doi: 10.1007/s11060-004-6404-6.  Google Scholar

[6]

R. Guillevin, C. Menuel, J.-N. Vallée, J.-P. Françoise, L. Capelle, C. Habas, G. De Marco, J. Chiras and R. Costalat, Mathematical modeling of energy metabolism and hemodynamics of WHO grade Ⅱ gliomas using in vivo MR data, Comptes rendus biologies, 334 (2011), 31-38.  doi: 10.1016/j.crvi.2010.11.002.  Google Scholar

[7]

M. Lahutte-Auboin, R. Costalat, J.-P. Françoise, R. Guillevin, Dip and Buffering in a fast-slow system associated to Brain Lactacte Kinetics, preprint, arXiv: 1308.0486. Google Scholar

[8]

M. Lahutte-Auboin, R. Guillevin, J.-P. Françoise, J.-N. Vallée and R. Costalat, On a minimal model for hemodynamics and metabolism of lactate : application to low grade glioma and therapeutic strategies, Acta Biotheoretica, 61 (2013), 79-89.  doi: 10.1007/s10441-013-9174-8.  Google Scholar

[9]

P. J. Magistretti and I. Allaman, A cellular perspective on brain energy metabolism and functional imaging, Neuron, 86 (2015), 883-901.  doi: 10.1016/j.neuron.2015.03.035.  Google Scholar

[10]

S. Mangia, G. Garreffa, M. Bianciardi, F. Giove, F. Di Salle and B. Maraviglia, The aerobic brain: Lactate decrease at the onset of neural activity, Neuroscience, 118 (2003), 7-10.  doi: 10.1016/S0306-4522(02)00792-3.  Google Scholar

[11]

J. R. Mangiardi and P. Yodice, Metabolism of the malignant astrocytoma, Neurosurgery, 26 (1990), 1-19.   Google Scholar

Figure 1.  Schematic representation of lactate exchanges in a local brain part. There is a cotransport through the brain-blood barrier, a blood flow, cell creation and consumption and interactions between a cell and its neighborhood. Interactions are described in the main text
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The functions $F$ and $J$; $F$ is a periodic function while $J$ is a monotone decreasing function of $u$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Intracellular and capillary lactate dynamics with nonconstant functions $J$ and $F$. On the left, the intracellular lactate trajectory is upper bounded. On the right, the capillary lactate trajectory is upper bounded too, but has an initial dip. At the bottom the orbit is typical of fast-slow systems
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Intracellular and capillary lactate dynamics with constant functions $J$ and $F$. The intracellular lactate trajectory (on the left) and the capillary lactate trajectory (on the right) are both upper bounded and reach the corresponding steady state. The trajectories for the original system (with $\varepsilon >0$) are also lower bounded. On the right top corner the capillary lactate of the original system has an initial dip, while it does not exist on the capillary lactate curve of the limit system (right bottom corner)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Dynamics for different values of $\varepsilon$. On the left intracellular; the lactate trajectories seem not to differ a lot. On the right, the value of $\varepsilon$ is related to the dip stiffness for the capillary lactate trajectories
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Dynamics for different values for $J$. On the right, the intracellular lactate trajectories are divided into two groups : for $J \in \{1, 0.1, 0.01 \}$, the concentration seems to explode, while for $J \in \{0.001, 0.0001\}$, it seems more stable. On the right the capillary lactate trajectories are devided into these two groups. For the first one, we can see a dip, while, for the second one, the steady state is not quickly reached
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Lactate concentration changes in a local brain part. Lactate concentration is given in mM (vertical axis) and time in days (horizontal axis). The red dots stand for medical data values, while the model simulations are displayed in continuous lines. While the four first patients exhibit Grompertz growth of their brain lactate concentration, patient 5 lactate concentration decreases in time. All the dynamics simulations tend to the steady state given in section 2
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Parameters for $F$ and $J$
ParameterValueUnit
$F_0$0.012s$^{-1}$
$\alpha_f$0.5 $1$
$t_i$50 $s$
$t_f$100 $s$
$C_J$5.7*10$^{-5}$ $mM^2.s^{-1}$
$\varepsilon_J$0.001 $mM$
$G_J$0.002 $mM.s^{-1}$
$L_J$0.001 $mM.s^{-1}$
Table Options

Download as excel

ParameterValueUnit
$F_0$0.012s$^{-1}$
$\alpha_f$0.5 $1$
$t_i$50 $s$
$t_f$100 $s$
$C_J$5.7*10$^{-5}$ $mM^2.s^{-1}$
$\varepsilon_J$0.001 $mM$
$G_J$0.002 $mM.s^{-1}$
$L_J$0.001 $mM.s^{-1}$
Table 2.  Parameters values
ParameterValueUnit
$T$0.01mM.s$^{-1}$
$k$3.5mM
$k'$3.5mM
$L$0.3mM
$\varepsilon$0.001s$^{-1}$
Table Options

Download as excel

ParameterValueUnit
$T$0.01mM.s$^{-1}$
$k$3.5mM
$k'$3.5mM
$L$0.3mM
$\varepsilon$0.001s$^{-1}$
Table 3.  Parameters values
ParameterValueUnit
$T$0.01mM.s$^{-1}$
$k$3.5mM
$k'$3.5mM
$L$0.3mM
$J$0.0057mM.s$^{-1}$
$F$0.0272s$^{-1}$
$\varepsilon$0.1s$^{-1}$
Table Options

Download as excel

ParameterValueUnit
$T$0.01mM.s$^{-1}$
$k$3.5mM
$k'$3.5mM
$L$0.3mM
$J$0.0057mM.s$^{-1}$
$F$0.0272s$^{-1}$
$\varepsilon$0.1s$^{-1}$
Table 4.  Parameters values
ParameterValueUnit
$T$0.1mM.d$^{-1}$
$k$3.5mM
$k'$3.5mM
$L$0.3mM
$F$0.0272d$^{-1}$
$\varepsilon$0.1d$^{-1}$
Table Options

Download as excel

ParameterValueUnit
$T$0.1mM.d$^{-1}$
$k$3.5mM
$k'$3.5mM
$L$0.3mM
$F$0.0272d$^{-1}$
$\varepsilon$0.1d$^{-1}$
Table 5.  Fitted values of $\bar{u}_0$, $\bar{v}_0$ and $J$
Patient $\bar{u}_0$ (mM) $\bar{v}_0$ (mM) $J$ (mM.d$^{-1}$)
$1$ $0.025$ $0.329$ $0.026$
$2$ $0.017$ $0.320$ $0.010$
$3$ $0.034$ $0.338$ $0.001$
$4$ $0.146$ $0.460$ $0.036$
$5$ $1.817$ $2.291$ $0.007$
Table Options

Download as excel

Patient $\bar{u}_0$ (mM) $\bar{v}_0$ (mM) $J$ (mM.d$^{-1}$)
$1$ $0.025$ $0.329$ $0.026$
$2$ $0.017$ $0.320$ $0.010$
$3$ $0.034$ $0.338$ $0.001$
$4$ $0.146$ $0.460$ $0.036$
$5$ $1.817$ $2.291$ $0.007$
[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]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377
[3]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[4]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361
[5]

Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021002
[6]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2020393
[7]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077
[8]

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
[9]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[10]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[11]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021006
[12]

Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021001
[13]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030
[14]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003
[15]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
[16]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[17]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[18]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[19]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145
[20]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (167)
  • HTML views (366)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences