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Analysis of a mathematical model for brain lactate kinetics
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
J. R. Mangiardi and P. Yodice, Metabolism of the malignant astrocytoma, Neurosurgery, 26 (1990), 1-19. Google Scholar |







Parameter | Value | Unit |
| 0.012 | s |
| 0.5 | |
| 50 | |
| 100 | |
| 5.7*10 | |
| 0.001 | |
| 0.002 | |
| 0.001 | |
Parameter | Value | Unit |
| 0.012 | s |
| 0.5 | |
| 50 | |
| 100 | |
| 5.7*10 | |
| 0.001 | |
| 0.002 | |
| 0.001 | |
Parameter | Value | Unit |
| 0.01 | mM.s |
| 3.5 | mM |
| 3.5 | mM |
| 0.3 | mM |
| 0.001 | s |
Parameter | Value | Unit |
| 0.01 | mM.s |
| 3.5 | mM |
| 3.5 | mM |
| 0.3 | mM |
| 0.001 | s |
Parameter | Value | Unit |
| 0.01 | mM.s |
| 3.5 | mM |
| 3.5 | mM |
| 0.3 | mM |
| 0.0057 | mM.s |
| 0.0272 | s |
| 0.1 | s |
Parameter | Value | Unit |
| 0.01 | mM.s |
| 3.5 | mM |
| 3.5 | mM |
| 0.3 | mM |
| 0.0057 | mM.s |
| 0.0272 | s |
| 0.1 | s |
Parameter | Value | Unit |
| 0.1 | mM.d |
| 3.5 | mM |
| 3.5 | mM |
| 0.3 | mM |
| 0.0272 | d |
| 0.1 | d |
Parameter | Value | Unit |
| 0.1 | mM.d |
| 3.5 | mM |
| 3.5 | mM |
| 0.3 | mM |
| 0.0272 | d |
| 0.1 | d |
Patient | | | |
| | | |
| | | |
| | | |
| | | |
| | | |
Patient | | | |
| | | |
| | | |
| | | |
| | | |
| | | |
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