Computational Study of the Cerebral Circulation Accounting for the Patient-specific Anatomical Features

  • Sergey Simakov
  • Timur Gamilov
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 133)


In this work, we present a dynamical one-dimensional network model of haemodynamics. We numerically study total convergence for the two types of implicit discretization of the compatibility conditions at the boundary nodes. We also present a practical biomedical application of the developed numerical model to the problem of blood flow variability in cerebral circulation due to anatomical features of the vascular network in the presence of the atherosclerosis disease.


Mathematical modeling Hemodynamics Circle of willis Atherosclerosis 



This work was supported by the Russian Science Foundation, grant No. 14-31-00024. The authors acknowledge the help of the staff of Sechenov University, especially N. Gagarina, E. Fominikh, and A. Dzunzya for collecting the data. We also acknowledge the help of R. Pryamonosov for processing CTA data.


  1. 1.
    Severov, D.S., Kholodov, A.S., Kholodov, J.A.: Comparison of packet-level and fluid models of IP networks. Math. Models Comput. Simulations 4(4), 385–393 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kholodov, Y.A., Vasiliev, M.O., Kholodov, A.S., Tzibulin, I.V.: A mathematical model of impurity propagation in ventilation networks. Math. Models Comput. Simul. 9(2), 142–154 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kholodov, J.A., Kholodov, A.S. Bordonos, A.K., Morozov, I.I.: Modeling of global energy networks. Math. Model. 21(6), 3–16 (2009). (In Russian)Google Scholar
  4. 4.
    Kholodov, A.S.: Some dynamic models of external breathing and blood circulation with consideration for their coherence and mass transfer. In: Belotserkovskii OM, Kholodov AS. (eds.) Computer Models and Medicine Progress, pp. 127–163. Nauka, Moscow (2001). (in Russian)Google Scholar
  5. 5.
    Kholodov, Y.A., Kholodov, A.S., Kovshov, N.V., Simakov, S.S., Severov, D.S., Bordonos, A.K., Bapayev, A.: Computational models on graphs for nonlinear hyperbolic and parabolic system of equations. In: Motasoares CA, Martins JAC, Rodrigues HC, Ambrósio JAC, Pina CAB, Motasoares, CM, Pereira EBR, Folgado J. (eds) III European Conference Computational Mechanics, p. 43. Springer, Dordrecht (2006)Google Scholar
  6. 6.
    Kholodov, A.S.: Construction of difference schemes with positive approximation for hyperbolic equations. USSR Comput. Math. Math. Phys. 18(6), 116–132 (1978)CrossRefGoogle Scholar
  7. 7.
    Magomedov, K.M., Kholodov A.S.: Grid-characteristics numerical methods. 2nd edn. Urait, Moscow (2018)Google Scholar
  8. 8.
    Alastruey, J., Moore, S.M., Parker, K.H., David, T., Peiro, J., Sherwin, S.J.: Reduced modelling of blood flow in the cerebral circulation: coupling 1-D, 0-D and cerebral auto-regulation models. Int. J. Numer. Methods Fluids 56(8), 1061–1067 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Spronck, B., Martens, E., Gommer, E., van de Vosse, F.: A lumped parameter model of cerebral blood flow control combining cerebral autoregulation and neurovascular coupling. Am. J. Physiol Heart Circulatory Physiol. 303, H1143–H1153 (2012)CrossRefGoogle Scholar
  10. 10.
    Liang, F., Fukasaku, K., Liu, H., Takagi, S.: A computational model study of the influence of the anatomy of the circle of willis on cerebral hyperperfusion following carotid artery surgery. BioMedical Eng. OnLine 10, 84 (2011). Scholar
  11. 11.
    Simakov, S.S., Kholodov, A.S.: Computational study of oxygen concentration in human blood under low frequency disturbances. Math. Models Comput. Simul. 1(2), 283–295 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Abakumov, M.V., Gavrilyuk, K.V., Esikova, N.B., Koshelev, V.B., Lukshin, A.V., Mukhin, S.I., Sosnin, NV., Tishkin, V.F., Favorskii, A.P.: Mathematical model for hemodynamics of cardiovascular system. Differ. Equations 33(7), 892–898 (1997)Google Scholar
  13. 13.
    Abakumov, M.V., Ashmetkov, I.V., Esikova, N.B., Koshelev, V.B., Mukhin, S.I., Sosnin, N.V., Tishkin, V.F., Favorskii, A.P., Khrulenko, A.B.: Strategy of mathematical cardiovascular system modeling. Matematicheskoe Modelirovanie 12(2), 106–117 (2000)zbMATHGoogle Scholar
  14. 14.
    Bunicheva, A.Y., Mukhin, S.I., Sosnin, N.V., Favorskii, A.P.: Numerical experiment in hemodynamics. Differ. Equations 40(7), 984–999 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Formaggia, L., Quarteroni, A., Veneziani, A.: Cardiovascular mathematics, vol. 1. Springer Heidelberg (2009)Google Scholar
  16. 16.
    Van de Vosse, F.N., Stergiopulos, N.: Pulse wave propagation in the arterial tree. Annu. Rev. Fluid Mech. 43, 467–499 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bessonov, N., Sequeira, A., Simakov, S., Vassilevski, Y., Volpert, V.: Methods of Blood Flow Modelling. Math. Model. Natural Phenomena 11(1), 1–25 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pedley, T.J., Luo, X.Y.: Modelling flow and oscillations in collapsible tubes. Theoret. Comput. Fluid Dyn. 10(1–4), 277–294 (1998)CrossRefGoogle Scholar
  19. 19.
    Vassilevski, Y.V., Salamatova, V.Y., Simakov, S.S.: On the elasticity of blood vessels in one-dimensional problems of hemodynamics. Comput. Math. Math. Phys. 55(9), 1567–1578 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ganong, W.F.: Review of Medical Physiology. Appleton and Lange, Stamford (1999)Google Scholar
  21. 21.
    Gamilov, T., Kopylov, Ph, Simakov, S.: Computational simulations of fractional flow reserve variability. Numerical Math. Adv. Appl. ENUMATH 2015, 499–507 (2015)zbMATHGoogle Scholar
  22. 22.
    Vassilevskii, Yu., Simakov, S., Salamatova, V., Ivanov, Yu., Dobroserdova, T.: Blood flow simulation in atherosclerotic vascular network using fiber-spring representation of diseased wall. Math. Model. Natural Phenomena 6(5), 333–349 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Vassilevskii, Yu., Simakov, S., Salamatova, V., Ivanov, Yu., Dobroserdova, T.: Vessel wall models for simulation of atherosclerotic vascular networks. Math. Model. Natural Phenomena 6(7), 82–99 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Simakov, S.S., Kholodov, A.S., Kholodov, Y.S., Nadolskiy, A.A., Shushlebin, A.N.: Computational study of the vibrating disturbances to the lung function. In: Proceedings III European Conference on Computational Mechanics, ID-1467 p. 205 (2006)Google Scholar
  25. 25.
    Vassilevski, Y, Simakov, S, Dobroserdova, T, Salamatova, V.: Numerical issues of modelling blood flow in networks of vessels with pathologies. Russian J. Numerical Anal. Math. Model. 26(6), 605–622 (2011)Google Scholar
  26. 26.
    Vassilevski, Y., Danilov, A., Ivanov, Yu., Simakov, S., Gamilov, T.: Personalized anatomical meshing of human body with applications. In: Quarteroni, A. (ed.) Modeling the Heart and the Circulatory System, pp. 221–236. Springer, London (2015)Google Scholar
  27. 27.
    Vassilevski, YuV, Danilov, A.A., Simakov, S.S., Gamilov, T.M., Ivanov, Y.A., Pryamonosov, R.A.: Patient-specific anatomical models in human physiology. Russian J Numerical Anal. Math. Model. 30(3), 185–201 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Danilov, A, Ivanov, Y, Pryamonosov, R, Vassilevski, Y.: Methods of graph network reconstruction in personalized medicine. Int. J. Numerical Methods Biomedical Eng. 32(8) (2016). Scholar
  29. 29.
    Prasad, N., Chhetri, P., Poudel, A.: Normal variants of the circle of willis in patients undergoing CT angiography. J. College Medical Sci.-Nepal 13(1), 102–190 (2017)Google Scholar
  30. 30.
    Gamilov, T., Ivanov, Y., Kopylov, P., Simakov, S., Vassilevski, Yu.: Patient specific haemodynamic modeling after occlusion treatment in leg. Math. Model Natural Phenomena. 9(6), 85–97 (2014)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Moscow Institute of Physics and Technology (MIPT)DolgoprudnyRussian Federation
  2. 2.Institute of Numerical Mathematics of the RASMoscowRussian Federation
  3. 3.Sechenov UniversityMoscowRussian Federation

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