Mathematical modelling of blood flow in vascular system of the brain with pathological changes

A. Issakhov, A. Shubay


The paper presents numerical study of blood flow in vascular system without stenosis and with pathological changes. The Navier – Stokes system of equation is used for modeling of liquid flow in
blood vessels. The numerical algorithm was based on projection method. At first stage, it is assumed that the transfer of the movement of amount occurred is due to convection and diffusion. The intermediate regions of velocity are determined by using the 5-step Runge – Kutta method algorithm. At second stage the intermediate velocity field is used to define the pressure field. The Poisson equation for pressure field is solved by Jacobi method. The numerical results are checked by comparing analytical solutions for the Poiseuille flow with numerical results. The influence of the blood velocity flow on behavior of the liquid in vascular system of brain was investigated. One of the characteristic features of blood flow in blood vessels affected by stenosis is an appearance of vortex currents, which manifests as a pathology, as
opposite to normal blood circulation. The results of this work can be used in mathematical modeling
investigations of blood circulation in brain and solving problems in practical medicine.

Ключевые слова

Navier-Stokes equation, Poiseuille flow, atherosclerotic plaque, stenosis, pathological changes in the blood vessels

Полный текст:



Skоbtsov Yu. A., Rostov Yu. V., Overko B. S. Modelirovanie i vizualizatsia povedeniya potokov krovi pri patologicheskikh protsessakh – Donetsk: Izatel’ Zaslavskiy A. Yu. – 2008. – 212 p..

Zabielski L., Mestel A. Unsteady blood flow in a helically symmetric pipe // Journal of Fluid Mechanics. – 1998. – Vol. 370. – P. 321- 345.

Smirnov V. M. Fiziologiya cheloveka. izd – M.: Meditsina, 2002. – 608 p.

Tsvibel’ V., Pellerito Dzh. Ul’trazvukovoe issledovanie sosudov. izd. – M.Vidar, 2008. – 644 p.

Tazyukov F. Kh., Dzhafar M. Khassan, Khalaf Kh. A., SnigerevФ B. A., Safa Kh. Abdul Rakhman Techenie krovi v simmetrichnoi krovenosnoi arterii so stenozom // Rossiyskiy zhurnal biomekhaniki – 2012. – Vol. 16, No 1(55). – P. 46–56.

Schlichting G. Teotiya pogranichnego sloya. izd. – M.:Nauka, 1974. – 708 p.

Issakhov A., Mathematical modeling of the discharged heat water effect on the aquatic environment from thermal power plant // International Journal of Nonlinear Science and Numerical Simulation. – 2015. – Vol. 16, No 5, P. 229–238.

Issakhov A., Mathematical modeling of the discharged heat water effect on the aquatic environment from thermal power plant under various operational capacities // Applied Mathematical Modelling. – 2016. – Vol. 40, No 2. – P. 1082–1096

Issakhov A. Large eddy simulation of turbulent mixing by using 3D decomposition method // J. Phys.: Conf. Ser. – 2011. – Vol. 318, No 4. – P. 1282-1288.

Chorin A.J. Numerical solution of the Navier-Stokes equations // Math. Comp. – 1968. – Vol. 22. – P. 745-762.