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doi: 10.3934/mcrf.2021016

Modeling the pressure distribution in a spatially averaged cerebral capillary network

Andrey Kovtanyuk 1,2,, , Alexander Chebotarev 2,3, , Nikolai Botkin 1, , Varvara Turova 1, , Irina Sidorenko 1, and  Renée Lampe 1,

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

* Corresponding author: Andrey Kovtanyuk

Received  March 2019 Revised  March 2020 Published  March 2021

Fund Project: The work was supported by the Klaus Tschira Foundation, Buhl-Strohmaier Foundation, and Würth Foundation

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.

Keywords: Inverse problem for the Poisson equation, unique solvability, cerebral blood flow circulation.
Mathematics Subject Classification: Primary: 58J05; Secondary: 92B05.
Citation: Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021016
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

F. J. M. Walters, Intracranial pressure and cerebral blood flow, Update in Anaesthesia, 8 (1998), 18-23.   Google Scholar

Figure 1.  Stencil corresponding to the cubic topology
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Figure 2.  The pressure distribution
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Figure 3.  The absolute velocity distribution
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