Modeling of the substances dynamics within a self-consistent model of a systemic circle of blood circulation

M. V. Abakumov ; S. I. Mukhin ; K. M. Mysova ; A. Yu. Pokladyuk ; A. B. Khrulenko

Matematičeskoe modelirovanie, Tome 35 (2023) no. 11, pp. 3-20.

Résumé

The considered model is based on quasi-one-dimensional equations of hemodynamics and a model of convection-diffusion substance transfer along a closed highly detailed graph of elastic vessels corresponding to a systemic circulation. The complexity of the considered graph led to the further development of the used CVSS software package and to the new detailing of the constructed models, that describe the functions of absorption and secretion of substances by organs and tissues. The obtained models enable the conduction of a series of numerical experiments that reproduce the process of insulin and glucose transfer in the systemic circulatory circle.

Détail

Keywords: mathematical modeling, substance transfer, hemodynamics, insulin, glucose.

Bibliographie

[1] A. Quarteroni, A. Veneziani, P. Zunino, “Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls”, Siam J. Numer. Anal., 39:5 (2001), 1488–1511 | DOI | MR | Zbl

[2] G. Pontrelli, A. Tatone, “A mathematical model of solute dynamics in a curved artery”, WIT Transactions on Biomedicine and Health, 8 (2005), 319–329 | DOI

[3] M. Prosi, K. Perktold, Z. Ding, M. H. Friedman, “Influence of curvature dynamics on pulsatile coronary artery flow in a realistic bifurcation model”, Journal of Biomechanics, 37:11 (2004), 1767–1775 | DOI

[4] N. Sun, N. B. Wood, X. Y. Xu., “Computational modelling of mass transport in large arteries”, Modelling and Simulation, 2008, 555–580

[5] S. S. Simakov, A. S. Kholodov, “Computational study of oxygen concentration in human blood under low frequency disturbances”, Mathematical Models and Computer Simulations, 1:2 (2009), 283–295 | DOI | MR

[6] T. Köppl, R. Helmig, B. Wohlmuth, “A multi-scale model for mass transport in arteries and tissue”, Recent Trends in Computational Engineering, 2015, 197–213 | DOI

[7] M. V. Abakumov, I. V. Ashmetkov i dr., “Metodika matematicheskogo modelirovania serdechno-sosudistoi sistemy”, Matematicheskoe modelirovanie, 12:2 (2000), 106–117 | Zbl

[8] V. B. Koshelev, S. I. Mukhin, N. V. Sosnin, A. P. Favorskii, Matematicheskie modeli kvazi-odnomernoi gemodinamiki, Metodicheskoe posobie, MAKS Press, M., 2010

[9] I. V. Ashmetkov, A. Ia. Bunicheva i dr., “Matematicheskoe modelirovanie krovoobrashcheniia na osnove programmnogo kompleksa CVSS”, Kompiuternye modeli i progress meditsiny, Nauka, M., 2001, 194–218

[10] A. G. Borzov, S. I. Mukhin, N. V. Sosnin, “Conservative Schemes of Matter Transport in a System of Vessels Closed by the Heart”, Differential Equat., 48:7 (2012), 919–928 | DOI | MR | Zbl

[11] A. G. Borzov, A. V. Dreval, S. I. Mukhin, “Modelirovanie dinamiki gliukozy krovi c uchetom topologii bolshogo kruga krovoobrashcheniia”, Matem. modelirovanie, 27:2 (2015), 3–24 | Zbl

[12] T. R. Zhaleev, V. A. Kubyshkin, S. I. Mukhin, A. F. Rubina, A. B. Khurelnko, “Mathematical modeling of the blood flow in hepatic vessels”, Comput. Mathematics and Modeling, 30:4 (2019), 364–377 | DOI | MR

[13] M. V. Abakumov, K. V. Gavriliuk i dr., “Matematicheskaia model gemodinamiki serdechno-sosudistoi sistemy”, Differentsialnye uravneniia, 33:7 (1997), 892–898 | MR | Zbl

[14] V. B. Koshelev, S. I. Mukhin, T. V. Sokolova, N. V. Sosnin, A. P. Favorskii, “Matematicheskoe modelirovanie gemodinamiki serdechno-sosudistoi sistemy s uchetom vliianiia neiroreguliatsii”, Matematicheskoe modelirovanie, 19:3 (2007), 15–28 | Zbl

[15] E. V. Shikin, A. V. Boreskov, Kompiuternaia grafika. Dinamika, realisticheskie izobrazheniia, Dialog MIFI, M., 1996

[16] D.F. Rogers, J.A. Adams, Mathematical elements for computer graphics, McGraw-Hill Publ, N.Y., 1990

[17] A. V. Pogorelov, Differentsialnaia geometriia, Nauka, M., 1974

[18] O. V. Bartenev, Sovremennyi Fortran, Dialog MIFI, M., 2005

[19] X. Pacheco, S. Teixeira, Delphi 5 Developer's Guide, SANS, IN., 2000

[20] M. Woo, T. Davis, D. Neider, D. Shreiner, OpenGL(R) Programming Guide: The Official Guide to Learning OpenGL, Version 1.4, 4th Ed., Addison-Wesley, MA., 2003

[21] A. I. Galushkin, Neironnye seti: osnovy teorii, Goriachaia liniia Telekom, M., 2012

[22] A. Ia. Bunicheva, V. B. Koshelev, S. I. Mukhin i dr., Matematicheskoe modelirovanie filtratsionnoi funktsii pochki, MAKS-Press, M., 2001, 1–18

[23] A. V. Dreval, Lechenie sakharnogo diabeta i soputstvuiushchikh zabolevanii, Eksmo, M., 2010

[24] V. M. Undritsov, I. M. Undritsov, L. D. Serova, “Sarkopeniia — novaia meditsinskaia nozologiia”, Fizkultura v profilaktike, lechenii i reabilitatsii, 31:4 (2009), 7–16

[25] J. Radziuk, S. Pye, “Hepatic glucose uptake, gluconeogenesis and the regulation of glycogen synthesis”, Diabetes/metabolism research and reviews, 17:4 (2001), 250–272 | DOI

[26] W. Waldhausl, P. Bratusch-Marrain, S. Gasic, A. Korn, P. Nowotny, “Insulin production rate following glucose ingestion estimated by splanchnic C-peptide output in normal man”, Diabetologia, 17:4 (1979), 221–227 | DOI

[27] A. Tura, U. Morbiducci, S. Sbrignadello, Y. Winhofer, G. Pacini, A. Kautzky-Willer, Shape of glucose, insulin, C-peptide curves during a 3-h oral glucose tolerance test: any relationship with the degree of glucose tolerance?, American J. of Physiology Regulatory, Integrative and Comparative Physiology, 300 (2011), 941–948 | DOI

[28] W. Zhou, Y. Gu, H. Li, M. Luo, “Assessing 1-h plasma glucose and shape of the glucose curve during oral glucose tolerance test”, European J. of Endocrinology, 155 (2006), 191–197 | DOI

[29] R.F. Schmidt, G. Thews, Human physiology, 2 ed., Springer-Verlag, Berlin, 1989, 827 pp.

[30] M. A. Abdul-Ghani, C. P. Jenkinson, D. K. Richardson, D. Tripathy, R. A. DeFronzo, “Insulin secretion and action in subjects with impaired fasting glucose and impaired glucose toler-ance: results from the Veterans Administration Genetic Epidemiology Study”, Diabetes, 55 (2006), 1430–1435 | DOI