Modeling the pressure distribution in a spatially averaged cerebral capillary network

Andrey Kovtanyuk; Alexander Chebotarev; Nikolai Botkin; Varvara Turova; Irina Sidorenko; Renée Lampe

doi:10.3934/mcrf.2021016

Article Contents

2021, Volume 11, Issue 3: 643-652. Doi: 10.3934/mcrf.2021016

This issue Previous Article External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms Next Article Second order directional shape derivatives of integrals on submanifolds

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

^* Corresponding author: Andrey Kovtanyuk
^* Corresponding author: Andrey Kovtanyuk

Received: March 2019

Revised: March 2020

Early access: March 2021

Published: September 2021

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

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Abstract

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:

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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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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).
[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.