Cite this article as:

Stiukhina E. S., Avtomonov Y. N., Postnov D. . Mathematical Model of Vascular Tone Autoregulation. Izvestiya of Saratov University. New series. Series Physics, 2018, vol. 18, iss. 3, pp. 202-214. DOI:


Mathematical Model of Vascular Tone Autoregulation


Background and Objectives: The conventional approach to study the blood circulat ion in the cardiovascular system of humans and animals is based on representation of the vascular system as a hierarchical structure of branching elastic tubes. While considerable progress has been achieved in the framework of this p aradigm, the other fails when one needs to analyze the dynamical patterns in networks of small arterial vessels. It mainly caused by the do minant contribution of cellular regulatory pathways that adjust a vascular tone in response to systemic si gnals and local metabolic demands. Since its complexity, these cellular mechanisms are typically studied (and modeled) separately from the blood flow modeling studies. We believe that the progress in the field essentially depends on the availability of simple enough, but still problem-relevant mathematical models that would provide the better understanding of the behavior of the vascular system as a complex network of nonlinear elements.

Results: In th is paper, we propose a minimized mathematical model of the process of autoregulation of the blood flow in the blood v essel segment. Being considerably simplifi ed our model still takes into account both the typical nonlinearities and the basic mechanisms of active regulation of a vascular tone. We verifiy our model in order to check whether the observed b ehavior is consistent with the known basic properties of rea l vessels. We show that the model successfully reproduces the effect of changes in the vessel radius and the corresponding stabilization of the flow with considerable (up to several times) pressure changes at the entrance to the segment. The oscillatory response of the radius of the segment on the pressure jump at the inlet has been revealed. This behavior possibly can underlie the complex types of reaction in small and medium microcirculatory networks. Next, we have studied the propagation of the pulse wave in the 100-segment model of the blood vessel. The nonlinear dependence of its pulse wave velocity on the pressure pulse amplitude applied to the first (input) segment of the model has been revealed.

Conclusion: We suggest that the si multaneous control of both the speed of the pulse wave and its pressure derivative is promising from the point of solving the practically important inverse problem being the pressure recovery from the measured pulse wave veloc ity.


1. Nasimi A. Hemo dynamics. In: The Cardiovascular System – Physiology, Diagnostics and Clinical Implications. Ed. D. C. Gaze. Rijeka, I nTech, 2012, pp. 95–111.

2. Morman D., Heller L. Cardiovascular physiology. Lange, 2013. 287 p.

3. Keener J., Sneyd J. Mathematical phys iology. New York, Springer- Verlag, 1998. 767 p.

4. Lishchuk V. A. Matematicheskaya teoriya krovoobrashcheniya [Mathematical theory of blood fl ow]. Moscow, Medicina Publ., 1991. 256 p. (in Russian).

5. Thiriet M., Parker K. H., Formaggia L., Perktold K., Quarteroni A., Fernandez M. A., Gerbeau J.-F., Antiga L., Peiro J., Steinman D. A., Doorly D., Sherwin S., Robertson A. M., Sequeira A., Owens R. G., Perktold K., Prosi M., Zunino P., Maday Y., Veneziani A., Arimon A., Balossino R., D’Angelo C., Dubini G., Giordana S., Migliavacca F., Pennati G., Vergara C., Vidrascu M. Cardiovascular mathematics. modeling and simulation of the circulatory system. Milano, Italia, Springer-Verlag, 2009. 522 p.

6. Berne R. M., Levy M. N. Physiologiya serdechnososudistoi sistemy. In: Fundamentalnaya i klinicheskaya phisiologiya [Phyndamental and clinical physiology]. Ed. Andrey G. Kamkin, Andrey A. Kamensky. Moscow, Akademiya Publ., 2004, pp. 513–703 (in Russian).

7. Abakumov M. V., Gavrilyuk K. V., Esikova N. B., Koshelev V. B., Lukshin A. B., Mukhin S. I., Sosnin N. V., Tishkin V. F., Favorsky A. P. Matematicheskaya model gemodinamiki serdechno-sosudistoi sistemy [Mathematical model of hemodynamics cardiovascular system]. Differencialnye Uravneniya, 1997, vol. 33, no. 7, pp. 892–898 (in Russian).

8. Gustafsson F., Nolstein-Rathlou N.-H. Conducted vasomotor responses in arterioles: characteristics, mechanisms and physiological significance. Acta Physiol. Scand., 1999, vol. 167, pp. 11–21. DOI:

9. Neganova A., Stiukhina E. S., Postnov D. E. Mathematical model of depolarization mechanism of conducted vasoreactivity. Proc. SPIE, 2015, vol. 9448, № 94481J. DOI:

10. Peng H., Matchkov V., Ivarsen A., Aalkjaer C., Nilsson H. Hypothesis for the initiation of vasomotion. Circ. Res., 2001, vol. 88, pp. 810–815.

11. Kapela A., Nagaraja S., Tsoukias N. M. A mathematical model of vasoreactivity in rat mesenteric arterioles. II. Conducted vasoreactivity. Amer. J. Physiol. Heart Circ. Physiol., 2010, vol. 298, no. 1, pp. H52–H65. DOI:

12. Neganova A. Dynamical characteristics of microvascular networks with a myogenic response gradient. Journal for Modeling in Ophthalmology, 2017, vol. 1, no. 4, pp. 43–61.

13. Postnov D. D., Marsh D. J., Postnov D. E., Braunstein T. H., Holstein-Rathlou N.-H., Martens E. A., Sosnovtseva O. Modeling of kidney hemodynamics: probability-based topology of an arterial network. PLoS Comput. Biol., 2016, vol. 12, no. 7, e1004922, pp. 1–28. DOI:

14. Gesche H., Grosskurth D., Küchler G., Patzak A. Continuous blood pressure measurement by using the pulse transit time: comparison to a cuff-based method. Eur. J. Appl. Physiol., 2012, vol. 112, pp. 309–315. DOI:

15. Hirata K., Kawakami M., O’Rourke M. Pulse wave analysis and pulse wave velocity. A review of blood pressure interpretation 100 years after Koro tkov. Circ. J., 2006, vol. 70, no. 10, pp. 1231–1239.

16. Nelson M. R., Stepanek J., Cevette M., Covalciuc M., Hurst R. T., Tajik A. J. Noninvasive measurement of central vascular pressures with arterial tonometry: Clinical Revival of the Pulse Pressure Wav eform? Mayo Clin. Proc., 2010, vol. 85, no. 5, pp. 460–472. DOI:

17. Wang R., Jia W., Mao Z. H, Sclabassi R. J., Sun M. Cufffree blood pr essure estimation using pulse transit time and heart rate. Intern. Conf. Signal Process Proc., 2014, pp. 115–118. DOI:

18. Hennig A., Patzak A. Continuous blood pressure measurement using pulse transit time. Somnologie, 2013, vol. 17, no. 2, pp. 104–110.

19. Churchland P. S., Koch C., Sejnowski T. J. What is computational neuroscience? In: Computational neuroscience. Cambridge, MIT Press, 1993, pp. 46–55.

20. Izikevich E. M. Dynamical systems in neuroscience: the geometry of excitability and bursting. Cambridge, MIT Press, 2007. 505 p.

21. Miller J., Bower J. M., Beeman D., Crook S. M., Davison A. P., Plesser H. E., Blackwell K., Calabrese R. L., Destexhe A., Bhalla U. S., Hasselmo M. E., Linster C., Cleland T. A., Ols hausen B. A., Montague P. R. 20 years of Computational neuroscience. New York, Springer-Verlag, 2013. 283 p. DOI:

22. Caro C. G., Pedley T. J., Schroter R. C., Seed W. A. The mechanics of the circulations. Cambridge, Cambridge University Press, 2012. 550 p.

23. Diep H. K., Vigmond E. J., Segal S. S., Welsh D. G. Defi ning electrical communication in skeletal muscle resistance arteries: a computational approach. J. Physiol., 2005, vol. 568, no. 1, pp. 267–281. DOI:

24. Liu C., Xin S., Liu C., Gu J., Yu M. Non-invasive measurement of arterial pressure-dependent compliance. In: 2007 Canadian Conference on Electrical and Computer Engineering. Vancouver, 2007, pp. 590–593. DOI:

25. Jeppesen P. Sanye-Hajari J. Increased blood pressure induces a diameter response of retinal arterioles that increases with decreasing arteriolar diameter. Invest. Ophthalmol. Vis. Sci., 2007, vol. 48, no. 1, pp. 328–331. DOI:

26. Yang J., Clark J. W. Jr., Bryan R. M., Robertson C. S. The myogenic response in isolated rat cerebrovascular arteries: vessel model. Med. Eng. Phys., 2003, vol. 25, no. 8, pp. 711–717.

Short text (in English): 
Full text (in Russian):