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Modeling the pressure distribution in a spatially averaged cerebral capillary network
1. | Technische Universität München, 81675 Munich, Germany |
2. | Far Eastern Federal University, 690950 Vladivostok, Russia |
3. | Institute for Applied Mathematics, FEB RAS, 690041 Vladivostok, Russia |
A boundary value problem for the Poisson's equation with unknown intensities of sources is studied in context of mathematical modeling the pressure distribution in cerebral capillary networks. The problem is formulated as an inverse problem with finite-dimensional overdetermination. The unique solvability of the problem is proven. A numerical algorithm is proposed and implemented.
References:
[1] |
J. W. Baish, P. A. Netti and R. K. Jain,
Transmural coupling of fluid flow in microcirculatory network and interstitium in tumors, Microvasc. Res., 53 (1997), 128-141.
doi: 10.1006/mvre.1996.2005. |
[2] |
N. D. Botkin, A. E. Kovtanyuk, V. L. Turova, I. N. Sidorenko and R. Lampe,
Direct modeling of blood flow through the vascular network of the germinal matrix, Comp. Biol. Med., 92 (2018), 147-155.
doi: 10.1016/j.compbiomed.2017.11.010. |
[3] |
N. D. Botkin, A. E. Kovtanyuk, V. L. Turova, I. N. Sidorenko and R. Lampe, Accounting for tube hematocrit in modeling of blood flow in cerebral capillary networks, Comp. Math. Meth. Med., 2019 (2019), 4235937.
doi: 10.1155/2019/4235937. |
[4] |
F. Hecht,
New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
[5] |
A.-R. A. Khaled and K. Vafai,
The role of porous media in modeling flow and heat transfer in biological tissues, Int. J. Heat Mass Trans., 46 (2003), 4989-5003.
doi: 10.1016/S0017-9310(03)00301-6. |
[6] |
A. E. Kovtanyuk, A. Yu. Chebotarev, N. D. Botkin, V. L. Turova, I. N. Sidorenko and R. Lampe, Nonstationary model of oxygen transport in brain tissue, Comp. Math. Meth. Med., 2020 (2020), 4861654.
doi: 10.1155/2020/4861654. |
[7] |
A. E. Kovtanyuk, A. Yu. Chebotarev, N. D. Botkin, V. L. Turova, I. N. Sidorenko and R. Lampe,
Continuum model of oxygen transport in brain, J. Math. Anal. Appl., 474 (2019), 1352-1363.
doi: 10.1016/j.jmaa.2019.02.020. |
[8] |
A. E. Kovtanyuk, A. Yu. Chebotarev, A. A. Dekalchuk, N. D. Botkin and R. Lampe, Analysis of a mathematical model of oxygen transport in brain, Proc. Int. Conf. Days on Diffraction 2018, (2018), 187–191.
doi: 10.1109/DD.2018.8553338. |
[9] |
A. E. Kovtanyuk, A. Yu. Chebotarev, A. A. Dekalchuk, N. D. Botkin and R. Lampe, An iterative algorithm for solving an initial boundary value problem of oxygen transport in brain, Proc. Int. Conf. Days on Diffraction 2019, (2019), 99–104.
doi: 10.1109/DD46733.2019.9016443. |
[10] |
S. K. Piechnik, P. A. Chiarelli and P. Jezzard,
Modelling vascular reactivity to investigate the basis of the relationship between cerebral blood volume and flow under CO2 manipulation, NeuroImage, 39 (2008), 107-118.
doi: 10.1016/j.neuroimage.2007.08.022. |
[11] |
S.-W. Su and S. J. Payne, A two phase model of oxygen transport in cerebral tissue, Proc. 31st Ann. Int. Conf. IEEE Eng. Med. Bio. Soc. (EMBC 2009) (2009) 4921–4924.
doi: 10.1109/IEMBS.2009.5332469. |
[12] |
S.-W. Su, Modelling Blood Flow and Oxygen Transport in the Human Cerebral Cortex, Ph.D thesis, Oxford University, Depart. Eng. Sci. (2011). Google Scholar |
[13] |
W. J. Vankan, J. M. Huyghe, J. D. Janssen, A. Huson, W. J. G. Hacking and W. Schreiner,
Finite element analysis of blood flow through biological tissue, Int. J. Eng. Sci., 35 (1997), 375-385.
doi: 10.1016/S0020-7225(96)00108-5. |
[14] |
F. J. M. Walters, Intracranial pressure and cerebral blood flow, Update in Anaesthesia, 8 (1998), 18-23. Google Scholar |
show all references
References:
[1] |
J. W. Baish, P. A. Netti and R. K. Jain,
Transmural coupling of fluid flow in microcirculatory network and interstitium in tumors, Microvasc. Res., 53 (1997), 128-141.
doi: 10.1006/mvre.1996.2005. |
[2] |
N. D. Botkin, A. E. Kovtanyuk, V. L. Turova, I. N. Sidorenko and R. Lampe,
Direct modeling of blood flow through the vascular network of the germinal matrix, Comp. Biol. Med., 92 (2018), 147-155.
doi: 10.1016/j.compbiomed.2017.11.010. |
[3] |
N. D. Botkin, A. E. Kovtanyuk, V. L. Turova, I. N. Sidorenko and R. Lampe, Accounting for tube hematocrit in modeling of blood flow in cerebral capillary networks, Comp. Math. Meth. Med., 2019 (2019), 4235937.
doi: 10.1155/2019/4235937. |
[4] |
F. Hecht,
New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
[5] |
A.-R. A. Khaled and K. Vafai,
The role of porous media in modeling flow and heat transfer in biological tissues, Int. J. Heat Mass Trans., 46 (2003), 4989-5003.
doi: 10.1016/S0017-9310(03)00301-6. |
[6] |
A. E. Kovtanyuk, A. Yu. Chebotarev, N. D. Botkin, V. L. Turova, I. N. Sidorenko and R. Lampe, Nonstationary model of oxygen transport in brain tissue, Comp. Math. Meth. Med., 2020 (2020), 4861654.
doi: 10.1155/2020/4861654. |
[7] |
A. E. Kovtanyuk, A. Yu. Chebotarev, N. D. Botkin, V. L. Turova, I. N. Sidorenko and R. Lampe,
Continuum model of oxygen transport in brain, J. Math. Anal. Appl., 474 (2019), 1352-1363.
doi: 10.1016/j.jmaa.2019.02.020. |
[8] |
A. E. Kovtanyuk, A. Yu. Chebotarev, A. A. Dekalchuk, N. D. Botkin and R. Lampe, Analysis of a mathematical model of oxygen transport in brain, Proc. Int. Conf. Days on Diffraction 2018, (2018), 187–191.
doi: 10.1109/DD.2018.8553338. |
[9] |
A. E. Kovtanyuk, A. Yu. Chebotarev, A. A. Dekalchuk, N. D. Botkin and R. Lampe, An iterative algorithm for solving an initial boundary value problem of oxygen transport in brain, Proc. Int. Conf. Days on Diffraction 2019, (2019), 99–104.
doi: 10.1109/DD46733.2019.9016443. |
[10] |
S. K. Piechnik, P. A. Chiarelli and P. Jezzard,
Modelling vascular reactivity to investigate the basis of the relationship between cerebral blood volume and flow under CO2 manipulation, NeuroImage, 39 (2008), 107-118.
doi: 10.1016/j.neuroimage.2007.08.022. |
[11] |
S.-W. Su and S. J. Payne, A two phase model of oxygen transport in cerebral tissue, Proc. 31st Ann. Int. Conf. IEEE Eng. Med. Bio. Soc. (EMBC 2009) (2009) 4921–4924.
doi: 10.1109/IEMBS.2009.5332469. |
[12] |
S.-W. Su, Modelling Blood Flow and Oxygen Transport in the Human Cerebral Cortex, Ph.D thesis, Oxford University, Depart. Eng. Sci. (2011). Google Scholar |
[13] |
W. J. Vankan, J. M. Huyghe, J. D. Janssen, A. Huson, W. J. G. Hacking and W. Schreiner,
Finite element analysis of blood flow through biological tissue, Int. J. Eng. Sci., 35 (1997), 375-385.
doi: 10.1016/S0020-7225(96)00108-5. |
[14] |
F. J. M. Walters, Intracranial pressure and cerebral blood flow, Update in Anaesthesia, 8 (1998), 18-23. Google Scholar |



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