Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Lyubimova A. A., Kuchumov A. G. The influence of the rheological models of blood on the hemodynamic characteristics of the flow in the Circle of Willis. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2026, vol. 26, iss. 2, pp. 236-250. DOI: 10.18500/1816-9791-2026-26-2-236-250, EDN: PPRMVU

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Heading: 
Mechanics
Article type: 
Article
UDC: 
517.98
DOI: 
10.18500/1816-9791-2026-26-2-236-250
EDN: 
PPRMVU

The influence of the rheological models of blood on the hemodynamic characteristics of the flow in the Circle of Willis

Autors: 
Lyubimova Alexandra Alekseevna, Perm National Research Polytechnic University
Kuchumov Alexei Gennad'evich, Perm National Research Polytechnic University
Abstract: 

Hemodynamic disturbances and the geometric features of blood vessels play an important role in the onset and progression of various vascular pathologies, such as aneurysms. Cerebral aneurysms are particularly dangerous due to their specific location. Experimental and clinical methods often fail to adequately assess a patient's current hemodynamic status or predict disease progression in vivo. Numerical modeling of hemodynamics can become a key tool for assessing the risk of growth and rupture of cerebral aneurysms. The accuracy of such models depends on many factors, including the choice of a rheological blood model. Despite the widespread use of the Newtonian model, its adequacy for cerebral arteries requires verification in comparison with more complex non-Newtonian models that account for shear-dependent viscosity. The aim of this study was to conduct a comparative analysis of the influence of three rheological blood models (Newtonian, Carreau, and Casson) on the hemodynamic characteristics in eight anatomical variants of the Circle of Willis with aneurysms. In this study, the geometry of the cerebral vasculature was obtained by segmenting CT images. Computational fluid dynamics (CFD) methods were used to simulate blood flow. Velocity profiles based on intracranial Doppler ultrasound data were set at the inlets, and a three-element Windkessel model was applied at the outlets. The distributions of velocity, pressure, wall shear stress (WSS), and its oscillatory shear index (OSI) were investigated. The vessel walls were considered rigid. It was found that in the large arteries, the differences between the rheological models in the calculations of velocity, pressure, WSS, and OSI are insignificant. All models showed a similar systematic deviation from clinical Doppler data. Inside the aneurysm domes, blood flow velocities are low, and the profiles for all three rheological models are practically identical in shape, with the Newtonian model tending to overestimate the values. The results indicate that for modeling hemodynamics in the large vessels of the Circle of Willis, the use of a Newtonian model is a permissible simplification.

Key words: 
aneurysm
circle of Willis
CFD
hemodynamics
simulation
rheological models
Acknowledgments: 
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (project No. FSNM-2025-0001).
References: 
  1. Krylov V. V. Khirurgiya anevrizm golovnogo mozga [Surgery of brain aneurysms. Vol. 1]. Moscow, IP “T. A. Alekseeva”, 2011. 432 p. (in Russian).
  2. Bjorkman J., Frosen J., Tahtinen O., Backes D., Huttunen T., Harju J., Huttunen J., Kurki M. I., von und zu Fraunberg M., Koivisto T., Manninen H., Jaaskelainen J. E., Lindgren A. E. Irregular shape identifies ruptured intracranial aneurysm in subarachnoid hemorrhage patients with multiple aneurysms. Stroke, 2017, vol. 48, iss. 7, pp. 1986–1989. DOI: https://doi.org/10.1161/STROKEAHA.117.017147
  3. Cebral J., Ollikainen E., Chung B. J., Mut F., Sippola V., Jahromi B. R., Tulamo R., Hernesniemi J., Niemela M., Robertson A., Frosen J. Flow conditions in the intracranial aneurysm lumen are associated with inflammation and degenerative changes of the aneurysm wall. American Journal of Neuroradiology, 2017, vol. 38, iss. 1, pp. 119–126. DOI: https://doi.org/10.3174/ajnr.a4951
  4. Li H., Pan R., Wang H., Rong X., Yin Z., Milgrom D. P., Shi X., Tang Y., Peng Y. Clipping versus coiling for ruptured intracranial aneurysms: A systematic review and meta-analysis. Stroke, 2013, vol. 44, iss. 1, pp. 29–37. DOI: https://doi.org/10.1161/STROKEAHA.112.663559
  5. Brinjikji W., Pereira V. M., Khumtong R., Kostensky A., Tymianski M., Krings T., Radovanovich I. PHASES and ELAPSS scores are associated with aneurysm growth: A study of 431 unruptured intracranial aneurysms. World Neurosurgery, 2018, vol. 114, pp. 425–432. DOI: https://doi.org/10.1016/j.wneu.2018.03.003
  6. Dol A. V. Biomechanics of neck and head arteries: The development of aneurysms and the separation of atherosclerotic plaques with combined pathologies. Russian Journal of Biomechanics, 2024, vol. 28, iss 3, pp. 19–30. DOI: https://doi.org/10.15593/rjbiomech/2024.3.02, EDN: BWBEXL
  7. Kuchumov A. G., Nyashin Y. I., Samartsev V. A. Modelling of peristaltic bile flow in the papilla ampoule with stone and in the papillary stenosis case: Application to reflux investigation. In: Goh J., Lim C. (eds.) 7th WACBE World Congress on Bioengineering 2015. IFMBE Proceedings, vol. 52. Springer, Cham, 2015, pp. 158–161. DOI: https://doi.org/10.1007/978-3-319-19452-3_42
  8. Kuchumov A. G., Gilyov V. G., Popov V. A., Samartsev V. A., Gavrilov V. A. Experimental study of the rheology of pathological bile. Russian Journal of Biomechanics, 2011, vol. 15, iss. 3, pp. 52—60 (in Russian). EDN: OJFBYL
  9. Brambila-Solorzano A., Mendez-Lavielle F., Naude J. L., Martinez-Sanchez G. J., Garcia-Rebolledo A., Hernandez B., Escobar-Del Pozo C. Influence of blood rheology and turbulence models in the numerical simulation of aneurysms. Bioengineering (Basel), 2023, vol. 10, iss. 10, art. 1170. DOI: https://doi.org/10.3390/bioengineering10101170
  10. Zylka M., Gorski G., Zylka W., Gala-Bladzinsk A. Numerical analysis of blood flow in the abdominal aorta under simulated weightlessness and earth conditions. Scientific Reports, 2024, vol. 14, art. 15978. DOI: https://doi.org/10.1038/s41598-024-66961-7
  11. Prokop V., Kozel K. Numerical simulation of Newtonian and non-Newtonian flows in bypass. Mathematics and Computers in Simulation, 2010, vol. 80, iss. 8, pp. 1725–1733. DOI: https://doi.org/10.1016/j.matcom.2009.06.001
  12. Zhu Z., Ji S., Liang L., Wang H., Xia H., Tang P. Hemodynamic study of blood flow in the aorta during the interventional robot treatment using fluid-structure interaction. Biomechanics and Modeling in Mechanobiology, 2023, vol. 22, iss. 6, pp. 1857–1872. DOI: https://doi.org/10.1007/s10237-023-01737-y
  13. Razavi S. E., Sahebjam R. Numerical Simulation of the blood flow behavior in the circle of Willis. Bioimpacts, 2014, vol. 4, iss. 2, pp. 89–94. DOI: https://doi.org/10.5681/bi.2014.008
  14. Backes D., Vergouwen M. D., Tiel Groenestege A. T., Bor A. S., Velthuis B.K., Greving J. P., Algra A., Wermer M. J., van Walderveen M. A., terBrugge K. G., Agid R., Rinkel G. J., PHASES score for prediction of intracranial aneurysm growth. Stroke, 2015, vol. 46, iss. 5, pp. 1221–1226. DOI: https://doi.org/10.1161/STROKEAHA.114.008198
  15. Valen-Sendstad K., Steinman D. A. Mind the gap: Impact of computational fluid dynamics solution strategy on prediction of intracranial aneurysm hemodynamics and rupture status indicators. American Journal of Neuroradiology, 2014, vol. 35, iss. 3, pp. 536–543. DOI: https://doi.org/10.3174/ajnr.a3793
  16. Landau L. D., Lifshitz E. M. Teoreticheskaya fizika. T. 6. Gidrodinamika [Theoretical Physics. Vol. 6. Hydrodynamics]. Moscow, Nauka, 1986. 736 p. (in Russian).
  17. Wilcox C. D. Turbulence modeling for CFD. San Diego, Birmingham Press, Inc., 2006. 536 p.
  18. Launder B. E., Sharma B. I. Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Letters in Heat and Mass Transfer, 1974, vol. 1, iss. 2, pp. 131–138. DOI: https://doi.org/10.1016/0094-4548(74)90150-7
  19. Westerhof N., Lankhaar J. W., Westerhof B. E. The arterial Windkessel. Medical and Biological Engineering and Computing, 2009, vol. 47, pp. 131–141. DOI: https://doi.org/10.1007/s11517-008-0359-2
  20. Egnor M., Yang L., Mani R. M., Fiore S. M., Djuriс P. M. A quantitative model of the cerebral windkessel and its relevance to disorders of intracranial dynamics. Journal of Neurosurgery: Pediatrics, 2023, vol. 32, iss. 3, pp. 302–311. DOI: https://doi.org/10.3171/2023.1.PEDS22372
  21. Chien S. Rheology in the microcirculation in normal and low flow states. Advances in Shock Research, 1987, vol. 8, pp. 71–80.
  22. Gijsen F. J. H., van de Vosse F. N., Janssen J. D. The influence of the non-Newtonian properties of blood on the flow in large arteries: Steady flow in a carotid bifurcation model. Journal of Biomechanics, 1999, vol. 32, iss. 6, pp. 601–608. DOI: https://doi.org/10.1016/s0021-9290(99)00015-9
  23. Casson N. A flow equation for pigment oil-suspensions of the printing ink type. In: Mill C. C. (ed.) Rheology of Disperse Systems. London, Pergamon Press, 1959, pp. 84–104.
  24. Wright J. Calculate wall shear gradient from velocity gradient. Available at: https://www.jameswright.xyz/post/20200813/calculate_wall_shear_from_velocity_gradient/ (accessed August 23, 2025).
  25. Peiffer V., Sherwin S. J., Weinberg P. D. Computation in the rabbit aorta of a new metric — the transverse wall shear stress — to quantify the multidirectional character of disturbed blood flow. Journal of Biomechanics, 2013, vol. 46, iss. 15, pp. 2651–2658. DOI: https://doi.org/10.1016/j.jbiomech.2013.08.003
  26. Zhao Y., Wang H., Chen W., Sun W., Yu X., Sun C., Hua G. Time-resolved simulation of blood flow through left anterior descending coronary artery: Effect of varying extent of stenosis on hemodynamics. BMC Cardiovascular Disorders, 2023, vol. 23, iss. 1, art. 156. DOI: https://doi.org/10.1186/s12872-023-03190-2
  27. Carvalho V., Rodrigues N., Ribeiro R., Costa P. F., Teixeira J. C. F., Lima R. A., Teixeira S. F. C. F. Hemodynamic study in 3D printed stenotic coronary artery models: Experimental validation and transient simulation. Computer Methods in Biomechanics and Biomedical Engineering, 2021, vol. 24, iss. 6, pp. 623–636. DOI: https://doi.org/10.1080/10255842.2020.1842377
  28. Babikian V. L., Wechsler L. R. Transcranial doppler ultrasonography. St. Louis, Mosby-Year Book, 1993. 323 p.
  29. Zheng R., Zhang S., Zhu C., Zhang C., Hong W. Impact of anatomical variations of the circle of Willis on the blood flow within unruptured intracranial aneurysm. Quantitative Imaging in Medicine and Surgery, 2025, vol. 15, iss. 8, pp. 6667–6681. DOI: https://doi.org/10.21037/qims-2025-55
Received: 
23.08.2025
Accepted: 
12.01.2026
Published: 
01.06.2026
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