Skip to main content

Evidence for non-Newtonian behavior of intracranial blood flow from Doppler ultrasonography measurements


Computational fluid dynamics (CFD) studies of intracranial hemodynamics often use Newtonian viscosity model to close the shear rate term in the Navier-Stokes equation. This is based on a commonly accepted hypothesis which state that non-Newtonian effects can be neglected in intracranial blood flow. This study aims to examine the validity of such hypothesis to guide future CFD studies of intracranial hemodynamics. Doppler ultrasonography (DUS) measurements of systolic and diastolic vessel diameter and blood velocity were conducted on 16 subjects (mean age 50.6). The measurements were conducted on the internal carotid (ICA), middle cerebral (MCA), and anterior communicating (AComA) arteries. Systolic and diastolic wall shear stress (WSS) values were calculated via the Hagen-Poiseuille exact solution using Newtonian and three different non-Newtonian models: namely Carreau, power-law and Herschel-Bulkley models. The Weissenberg-Rabinowitsch correction for blood shear-thinning viscosity was applied to the non-Newtonian models. The error percentage between the two sets of models was calculated and discussed. The Newtonian hypothesis was tested statistically and discussed using paired t tests. Significant differences (P < 0.0001) were found between the Newtonian and non-Newtonian WSS in ICA. In MCA and AComA, similar differences were found except in the systole and diastole for the Herschel-Bulkley and power-law models (P = 0.0669, P = 0.7298), respectively. The error between the Newtonian and non-Newtonian models ranged from − 27 to 30% (0.2 to 2.2 Pa). These values could affect the physical interpretation of IA CFD studies. Evidence suggests that the Newtonian assumption may be inappropriate to investigate intracranial hemodynamics.

The WSS estimation error resulting from using the Newtonian assumption compared to three non-Newtonian models for ICA, MCA, and AComA in systole and diastole conditions, based on TCCD measurements of 16 subjects. The error due to the Newtonian assumption ranged from 0.2 to 2.2 Pa (− 27 to 30%). These values could affect the physical interpretation of IA CFD studies.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4



anterior communicating artery


computational fluid dynamics


Doppler ultrasonography


internal carotid artery


middle cerebral artery


transcranial color-coded Doppler


wall shear stress


normalized wall shear stress


time-averaged wall shear stress


  1. 1.

    Almuhanna K, Zhao L, Kowalewski G, Beach KW, Lal BK, Sikdar S (2012) Investigation of cerebral hemodynamics and collateralization in asymptomatic carotid stenoses. In: Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS. pp 5618–5621. doi:

  2. 2.

    Avari H, Savory E, Rogers KA (2016) An in vitro hemodynamic flow system to study the effects of quantified shear stresses on endothelial cells. Cardiovasc Eng Technol 7:44–57.

    Article  PubMed  Google Scholar 

  3. 3.

    Boussel L, Rayz V, McCulloch C, Martin A, Acevedo-Bolton G, Lawton M, Higashida R, Smith WS, Young WL, Saloner D (2008) Aneurysm growth occurs at region of low wall shear stress. Stroke 39:2997–3002

    Article  PubMed  PubMed Central  Google Scholar 

  4. 4.

    Brooks DE, Goodwin JW, Seaman GV (1970) Interactions among erythrocytes under shear. J Appl Physiol 28:172–177

    Article  CAS  PubMed  Google Scholar 

  5. 5.

    Campo-Deano L, Oliveira MSN, Pinho FT (2015) A review of computational hemodynamics in middle cerebral aneurysms and rheological models for blood flow. Appl Mech Rev 67:030801.

    Article  Google Scholar 

  6. 6.

    Carreau PJ (1972) Rheological equations from molecular network theories. Trans Soc Rheol 16:99–127.

    Article  CAS  Google Scholar 

  7. 7.

    Cebral JR, Castro MA, Burgess JE, Pergolizzi RS, Sheridan MJ, Putman CM (2005) Characterization of cerebral aneurysms for assessing risk of rupture by using patient-specific computational hemodynamics models. Am J Neuroradiol 26:2550–2559

    PubMed  Google Scholar 

  8. 8.

    Cebral JR, Mut F, Raschi M, Scrivano E, Ceratto R, Lylyk P, Putman CM (2011) Aneurysm rupture following treatment with flow-diverting stents: computational hemodynamics analysis of treatment. Am J Neuroradiol 32:27–33

    Article  CAS  PubMed  Google Scholar 

  9. 9.

    Cebral JR, Mut F, Weir J, Putman C (2011) Quantitative characterization of the hemodynamic environment in ruptured and unruptured brain aneurysms. Am J Neuroradiol 32:145–151.

    Article  CAS  PubMed  Google Scholar 

  10. 10.

    Chalouhi N, Ali MS, Jabbour PM, Tjoumakaris SI, Gonzalez LF, Rosenwasser RH, Koch WJ, Dumont AS (2012) Biology of intracranial aneurysms: role of inflammation. J Cereb Blood Flow Metab 32:1659–1676.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  11. 11.

    Chiu J-J, Chien S (2011) Effects of disturbed flow on vascular endothelium: pathophysiological basis and clinical perspectives. Physiol Rev 91:327–387.

    Article  PubMed  Google Scholar 

  12. 12.

    De Verdier MC, Wikström J (2016) Normal ranges and test-retest reproducibility of flow and velocity parameters in intracranial arteries measured with phase-contrast magnetic resonance imaging. Neuroradiology 58:521–531

    Article  Google Scholar 

  13. 13.

    DePaola N, Davies PF, Pritchard WF, Florez L, Harbeck N, Polacek DC (1999) Spatial and temporal regulation of gap junction connexin43 in vascular endothelial cells exposed to controlled disturbed flows in vitro. Proc Natl Acad Sci U S A 96:3154–3159

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  14. 14.

    Dewey JCF, Bussolari SR, Gimbrone JMA, Davies PF (1981) The dynamic response of vascular endothelial cells to fluid shear stress. J Biomech Eng 103:177–185.

    Article  PubMed  Google Scholar 

  15. 15.

    Dimmeler S, Hermann C, Galle J, Zeiher AM (1999) Upregulation of superoxide dismutase and nitric oxide synthase mediates the apoptosis-suppressive effects of shear stress on endothelial cells. Arterioscler Thromb Vasc Biol 19:656–664

    Article  CAS  PubMed  Google Scholar 

  16. 16.

    Dolan JM, Meng H, Singh S, Paluch R, Kolega J (2011) High fluid shear stress and spatial shear stress gradients affect endothelial proliferation, survival, and alignment. Ann Biomed Eng 39:1620–1631.

    Article  PubMed  PubMed Central  Google Scholar 

  17. 17.

    Dolan JM, Sim FJ, Meng H, Kolega J (2012) Endothelial cells express a unique transcriptional profile under very high wall shear stress known to induce expansive arterial remodeling. Am J Physiol Cell Physiol 302:C1109–C1118.

    Article  CAS  PubMed  Google Scholar 

  18. 18.

    Dolan JM, Meng H, Sim FJ, Kolega J (2013) Differential gene expression by endothelial cells under positive and negative streamwise gradients of high wall shear stress. Am J Physiol Cell Physiol 305:C854–C866.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. 19.

    Drazin F, Riley N (2006) The Navier-Stokes equations: a classification of flows and exact solutions. Cambridge University Press, Cambridge

  20. 20.

    Enzmann DR, Ross MR, Marks MP, Pelc NJ (1994) Blood flow in major cerebral arteries measured by phase-contrast cine MR. Am J Neuroradiol 15:123–129

    CAS  PubMed  Google Scholar 

  21. 21.

    Frolov SV, Sindeev SV, Liepsch D, Balasso A (2016) Experimental and CFD flow studies in an intracranial aneurysm model with Newtonian and non-Newtonian fluids. Technol Health Care 24:317–333.

    Article  CAS  PubMed  Google Scholar 

  22. 22.

    Frösen J, Tulamo R, Paetau A, Laaksamo E, Korja M, Laakso A, Niemelä M, Hernesniemi J (2012) Saccular intracranial aneurysm: pathology and mechanisms. Acta Neuropathol 123:773–786.

    Article  PubMed  Google Scholar 

  23. 23.

    Galdi GP, Rannacher R, Robertson AM, Turek S (2008) Hemodynamical flows: modeling, analysis and simulation. Birkhäuser, Basel

  24. 24.

    Gassner M, Killu K, Bauman Z, Coba V, Rosso K, Blyden D (2015) Feasibility of common carotid artery point of care ultrasound in cardiac output measurements compared to invasive methods. J Ultrasound 18:127–133.

    Article  PubMed  Google Scholar 

  25. 25.

    Haematology ICfSi (1984) Recommendation for a selected method for the measurement of plasma viscosity. International Committee for Standardization in Haematology. J Clin Pathol 37:1147

    Article  Google Scholar 

  26. 26.

    Harloff A, Albrecht F, Spreer J, Stalder AF, Bock J, Frydrychowicz A, Schöllhorn J, Hetzel A, Schumacher M, Hennig J, Markl M (2009) 3D blood flow characteristics in the carotid artery bifurcation assessed by flow-sensitive 4D MRI at 3T. Magn Reson Med 61:65–74.

    Article  CAS  PubMed  Google Scholar 

  27. 27.

    Herschel WH, Bulkley R (1926) Konsistenzmessungen von Gummi-Benzollösungen. Kolloid-Zeitschrift 39:291–300.

    Article  Google Scholar 

  28. 28.

    Hippelheuser JE, Lauric A, Cohen AD, Malek AM (2014) Realistic non-Newtonian viscosity modelling highlights hemodynamic differences between intracranial aneurysms with and without surface blebs. J Biomech 47:3695–3703.

    Article  PubMed  Google Scholar 

  29. 29.

    Hoi Y, Meng H, Woodward SH, Bendok BR, Hanel RA, Guterman LR, Hopkins LN (2004) Effects of arterial geometry on aneurysm growth: three-dimensional computational fluid dynamics study. J Neurosurg 101:676–681.

    Article  PubMed  Google Scholar 

  30. 30.

    Hoskins PR, Lawford PV, Doyle BJ (2017) Cardiovascular biomechanics. Springer International Publishing.

  31. 31.

    Husain I, Labropulu F, Langdon C, Schwark J (2013) A comparison of Newtonian and non-Newtonian models for pulsatile blood flow simulations. J Mech Behav Mater 21. doi:

  32. 32.

    Ieuan Owen JG, Escudier M, Poole R (2009) The importance of the non-Newtonian characteristics of blood in flow modelling studies. Journal of Applied Fluid Mechanics Vol 2

  33. 33.

    Isaksen JG, Bazilevs Y, Kvamsdal T, Zhang Y, Kaspersen JH, Waterloo K, Romner B, Ingebrigtsen T (2008) Determination of wall tension in cerebral artery aneurysms by numerical simulation. Stroke 39:3172–3178

    Article  PubMed  Google Scholar 

  34. 34.

    Jou LD, Lee DH, Morsi H, Mawad ME (2008) Wall shear stress on ruptured and unruptured intracranial aneurysms at the internal carotid artery. Am J Neuroradiol 29:1761–1767.

    Article  PubMed  Google Scholar 

  35. 35.

    Kataoka H (2015) Molecular mechanisms of the formation and progression of intracranial aneurysms. Neurol Med Chir 55:214–229.

    Article  Google Scholar 

  36. 36.

    Kellawan JM, Harrell JW, Schrauben EM, Hoffman CA, Roldan-Alzate A, Schrage WG, Wieben O (2016) Quantitative cerebrovascular 4D flow MRI at rest and during hypercapnia challenge. Magn Reson Imaging 34:422–428

    Article  Google Scholar 

  37. 37.

    Li Y-SJ, Haga JH, Chien S (2005) Molecular basis of the effects of shear stress on vascular endothelial cells. J Biomech 38:1949–1971.

    Article  PubMed  Google Scholar 

  38. 38.

    Macosko CW (1994) Rheology: principles, measurements, and applications. Wiley-VCH, New York

  39. 39.

    Meng H, Wang Z, Hoi Y, Gao L, Metaxa E, Swartz DD, Kolega J (2007) Complex hemodynamics at the apex of an arterial bifurcation induces vascular remodeling resembling cerebral aneurysm initiation. Stroke 38:1924–1931.

    Article  PubMed  PubMed Central  Google Scholar 

  40. 40.

    Molla MM, Paul MC (2012) LES of non-Newtonian physiological blood flow in a model of arterial stenosis. Med Eng Phys 34:1079–1087.

    Article  CAS  PubMed  Google Scholar 

  41. 41.

    Morales HG, Larrabide I, Geers AJ, Aguilar ML, Frangi AF (2013) Newtonian and non-Newtonian blood flow in coiled cerebral aneurysms. J Biomech 46:2158–2164.

    Article  PubMed  Google Scholar 

  42. 42.

    Otani T, Ii S, Hirata M, Wada S (2017) Computational study of the non-Newtonian effect of blood on flow stagnation in a coiled cerebral aneurysm. Nihon Reoroji Gakkaishi 45:243–249.

    Article  CAS  Google Scholar 

  43. 43.

    Partridge J, Carlsen H, Enesa K, Chaudhury H, Zakkar M, Luong L, Kinderlerer A, Johns M, Blomhoff R, Mason JC, Haskard DO, Evans PC (2007) Laminar shear stress acts as a switch to regulate divergent functions of NF-κB in endothelial cells. FASEB J 21:3553–3561.

    Article  CAS  PubMed  Google Scholar 

  44. 44.

    Rashad S, Sugiyama S-I, Niizuma K, Sato K, Endo H, Omodaka S, Matsumoto Y, Fujimura M, Tominaga T (2018) Impact of bifurcation angle and inflow coefficient on the rupture risk of bifurcation type basilar artery tip aneurysms. J Neurosurg 128:723–730.

    Article  PubMed  Google Scholar 

  45. 45.

    Schirmer CM, Malek AM (2010) Critical influence of framing coil orientation on intra-aneurysmal and neck region hemodynamics in a sidewall aneurysm model. Neurosurgery 67:1692–1702.

    Article  PubMed  Google Scholar 

  46. 46.

    Sforza DM, Putman CM, Cebral JR (2009) Hemodynamics of cerebral aneurysms. Annu Rev Fluid Mech 41:91–107.

    Article  PubMed  PubMed Central  Google Scholar 

  47. 47.

    Shojima M, Oshima M, Takagi K, Torii R, Hayakawa M, Katada K, Morita A, Kirino T (2004) Magnitude and role of wall shear stress on cerebral aneurysm. Stroke 35:2500–2505

    Article  PubMed  Google Scholar 

  48. 48.

    Shojima M, Oshima M, Takagi K, Torii R, Hayakawa M, Katada K, Morita A, Kirino T (2004) Magnitude and role of wall shear stress on cerebral aneurysm. Computational Fluid Dynamic Study of 20 Middle Cerebral Artery Aneurysms. Stroke 35:2500–2505.

    Article  PubMed  Google Scholar 

  49. 49.

    Suzuki T, Takao H, Suzuki T, Suzuki T, Masuda S, Dahmani C, Watanabe M, Mamori H, Ishibashi T, Yamamoto H, Yamamoto M, Murayama Y (2017) Variability of hemodynamic parameters using the common viscosity assumption in a computational fluid dynamics analysis of intracranial aneurysms. Technol Health Care 25:37–47.

    Article  PubMed  Google Scholar 

  50. 50.

    Taba Y, Sasaguri T, Miyagi M, Abumiya T, Miwa Y, Ikeda T, Mitsumata M (2000) Fluid shear stress induces lipocalin-type prostaglandin D2 synthase expression in vascular endothelial cells. Circ Res 86:967–973

    Article  CAS  PubMed  Google Scholar 

  51. 51.

    Takeda Y (1999) Ultrasonic Doppler method for velocity profile measurement in fluid dynamics and fluid engineering. Exp Fluids 26:177–178

    Article  Google Scholar 

  52. 52.

    Tressel SL, Huang R-P, Tomsen N, Jo H (2007) Laminar shear inhibits tubule formation and migration of endothelial cells by an angiopoietin-2-dependent mechanism. Arterioscler Thromb Vasc Biol 27:2150–2156.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  53. 53.

    Turjman AS, Turjman F, Edelman ER (2014) Role of fluid dynamics and inflammation in intracranial aneurysm formation. Circulation 129:373–382.

    Article  PubMed  PubMed Central  Google Scholar 

  54. 54.

    Ujiie H, Tachibana H, Hiramatsu O, Hazel AL, Matsumoto T, Ogasawara Y, Nakajima H, Hori T, Takakura K, Kajiya F (1999) Effects of size and shape (aspect ratio) on the hemodynamics of saccular aneurysms: a possible index for surgical treatment of intracranial aneurysms. Neurosurgery 45:119–130.

    CAS  Article  PubMed  Google Scholar 

  55. 55.

    Valencia A, Morales H, Rivera R, Bravo E, Galvez M (2008) Blood flow dynamics in patient-specific cerebral aneurysm models: the relationship between wall shear stress and aneurysm area index. Med Eng Phys 30:329–340.

    Article  PubMed  Google Scholar 

  56. 56.

    Walburn FJ, Schneck DJ (1976) A constitutive equation for whole human blood. Biorheology 13:201–210.

    Article  CAS  PubMed  Google Scholar 

  57. 57.

    Wang C, Tian Z, Liu J, Jing L, Paliwal N, Wang S, Zhang Y, Xiang J, Siddiqui AH, Meng H, Yang X (2016) Hemodynamic alterations after stent implantation in 15 cases of intracranial aneurysm. Acta Neurochir 158:811–819.

    Article  PubMed  Google Scholar 

  58. 58.

    Westerhof N, Stergiopulos N, Noble MIM (2010) Snapshots of hemodynamics: an aid for clinical research and graduate education. Springer.

  59. 59.

    Xiang J, Natarajan SK, Tremmel M, Ma D, Mocco J, Hopkins LN, Siddiqui AH, Levy EI, Meng H (2010) Hemodynamic–morphologic discriminants for intracranial aneurysm rupture. Stroke 42:144

    Article  PubMed  PubMed Central  Google Scholar 

  60. 60.

    Xiang J, Natarajan SK, Tremmel M, Ma D, Mocco J, Hopkins LN, Siddiqui AH, Levy EI, Meng H (2011) Hemodynamic-morphologic discriminants for intracranial aneurysm rupture. Stroke 42:144–152.

    Article  PubMed  Google Scholar 

  61. 61.

    Xiang J, Tremmel M, Kolega J, Levy EI, Natarajan SK, Meng H (2012) Newtonian viscosity model could overestimate wall shear stress in intracranial aneurysm domes and underestimate rupture risk. J Neurointerv Surg 4:351–357.

    Article  PubMed  Google Scholar 

  62. 62.

    Zhao X, Zhao M, Amin-Hanjani S, Du X, Ruland S, Charbel FT (2015) Wall shear stress in major cerebral arteries as a function of age and gender-a study of 301 healthy volunteers. J Neuroimaging 25:403–407.

    Article  PubMed  Google Scholar 

  63. 63.

    Zhou G, Zhu Y, Yin Y, Su M, Li M (2017) Association of wall shear stress with intracranial aneurysm rupture: systematic review and meta-analysis. Sci Rep 7:5331.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

Download references


Ohta, Saqr and Tupin acknowledge the support from IFS, Tohoku University, JAPAN. Mansour acknowledges the support from Alexandria University hospital for conducting the ultrasonography measurements.


Saqr and Hassan acknowledge the support of the Science and Technology Development Fund (STDF), Egypt, under project (Project ID: 5219)

Author information



Corresponding author

Correspondence to Khalid M. Saqr.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Saqr, K.M., Mansour, O., Tupin, S. et al. Evidence for non-Newtonian behavior of intracranial blood flow from Doppler ultrasonography measurements. Med Biol Eng Comput 57, 1029–1036 (2019).

Download citation


  • Doppler ultrasound
  • Non-Newtonian flow
  • Blood viscosity
  • Intracranial hemodynamics
  • Blood rheology