Effects of size and elasticity on the relation between flow velocity and wall shear stress in side-wall aneurysms: A lattice Boltzmann-based computer simulation study

Autoři: Haifeng Wang aff001;  Timm Krüger aff002;  Fathollah Varnik aff001
Působiště autorů: Interdisciplinary Center for Advanced Materials Simulation (ICAMS), Ruhr-Universität Bochum, Bochum, Germany aff001;  School of Engineering, Institute for Multiscale Thermofluids, University of Edinburgh, Edinburgh, Scotland, United Kingdom aff002
Vyšlo v časopise: PLoS ONE 15(1)
Kategorie: Research Article
doi: 10.1371/journal.pone.0227770


Blood flow in an artery is a fluid-structure interaction problem. It is widely accepted that aneurysm formation, enlargement and failure are associated with wall shear stress (WSS) which is exerted by flowing blood on the aneurysmal wall. To date, the combined effect of aneurysm size and wall elasticity on intra-aneurysm (IA) flow characteristics, particularly in the case of side-wall aneurysms, is poorly understood. Here we propose a model of three-dimensional viscous flow in a compliant artery containing an aneurysm by employing the immersed boundary-lattice Boltzmann-finite element method. This model allows to adequately account for the elastic deformation of both the blood vessel and aneurysm walls. Using this model, we perform a detailed investigation of the flow through aneurysm under different conditions with a focus on the parameters which may influence the wall shear stress. Most importantly, it is shown in this work that the use of flow velocity as a proxy for wall shear stress is well justified only in those sections of the vessel which are close to the ideal cylindrical geometry. Within the aneurysm domain, however, the correlation between wall shear stress and flow velocity is largely lost due to the complexity of the geometry and the resulting flow pattern. Moreover, the correlations weaken further with the phase shift between flow velocity and transmural pressure. These findings have important implications for medical applications since wall shear stress is believed to play a crucial role in aneurysm rupture.

Klíčová slova:

Aneurysms – Arteries – Blood pressure – Deformation – Fluid flow – Fluids – Mechanical energy – Shear stresses


1. Bederson JB, Awad IA, Wiebers DO, Piepgras D, Haley EC Jr, Brott T, et al. Recommendations for the Management of Patients With Unruptured Intracranial Aneurysms: A Statement for Healthcare Professionals From the Stroke Council of the American Heart Association. Stroke. 2000;31:2742–2750. doi: 10.1161/01.str.31.11.2742 11062304

2. Grotberg JB, Jensen OE. Biofluid mechanics in flexible tubes. Annu Rev Fluid Mech. 2004;36:121–147. doi: 10.1146/annurev.fluid.36.050802.121918

3. Castro MA, Putman CM, Cebral JR. Patient-specific computational fluid dynamics modeling of anterior communicating artery aneurysms: a study of the sensitivity of intra-aneurysmal flow patterns to flow conditions in the carotid arteries. AJNR Am J Neuroradiol. 2006;27(10):2061–8. 17110667

4. Castro MA. Understanding the role of hemodynamics in the initiation, progression, rupture, and treatment outcome of cerebral aneurysm from medical image-based computational studies. ISRN Radiology. 2013;013:602707.

5. Furukawa K, Ishida F, Tsuji M, Miura Y, Kishimoto T, Shiba M, et al. Hemodynamic characteristics of hyperplastic remodeling lesions in cerebral aneurysms. PLoS ONE. 2018;13(1):e0191287. doi: 10.1371/journal.pone.0191287 29338059

6. Xu L, Liang F, Zhao B, Wan J, Liu H. Influence of aging-induced flow waveform variation on hemodynamics in aneurysms present at the internal carotid artery: A computational model-based study. Comput Biol Med. 2018;101:51–60. doi: 10.1016/j.compbiomed.2018.08.004 30099239

7. Diagbouga MR, Morel S, Bijlenga P, Kwak BR. Role of hemodynamics in initiation/growth of intracranial aneurysms. Eur J Clin Invest. 2018;48(9):e12992. doi: 10.1111/eci.12992 29962043

8. Humphrey JD, Taylor CA. Intracranial and abdominal aortic aneurysms: similarities, differences, and need for a new class of computational models. Annu Rev Biomed Eng. 2008;10:221–46. doi: 10.1146/annurev.bioeng.10.061807.160439 18647115

9. Kolega J, Gao L, Mandelbaum M, Mocco J, Siddiqui AH, Natarajan SK, et al. Cellular and molecular responses of the basilar terminus to hemodynamics during intracranial aneurysm initiation in a rabbit model. J Vasc Res. 2011;48(5):429–42. doi: 10.1159/000324840 21625176

10. Meng H, Tutino VM, Xiang J, Siddiqui A. High WSS or low WSS? Complex interactions of hemodynamics with intracranial aneurysm initiation, growth, and rupture: toward a unifying hypothesis. AJNR Am J Neuroradiol. 2014;35(7):1254–62. doi: 10.3174/ajnr.A3558 23598838

11. Oshinski J, Ku D, Mukundan S Jr, Loth F, Pettigrew R. Determination of wall shear stress in the aorta with the use of MR phase velocity mapping. J Magn Reson Imaging. 1995;5:640–647. doi: 10.1002/jmri.1880050605 8748480

12. Carvalho J, Nielsen J, Nayak K. Feasibility of in vivo measurement of carotid wall shear rate using spiral fourier velocity encoded MRI. Magn Reson Med. 2010;63:1537–1547. doi: 10.1002/mrm.22325 20512857

13. Szajer J, Ho-Shon K. A comparison of 4D flow MRI-derived wall shear stress with computational fluid dynamics methods for intracranial aneurysms and carotid bifurcations—A review. Magn Reson Imaging. 2018;48:62–69. doi: 10.1016/j.mri.2017.12.005 29223732

14. Novitzke J. The basics of brain aneurysms: A guide for patients. J Vasc Interv Neurol. 2008;1(3):89–90. 22518230

15. Hassan T, Timofeev EV, Saito T, Shimizu H, Ezura M, Matsumoto Y, et al. A proposed parent vessel geometry-based categorization of saccular intracranial aneurysms: Computational flow dynamics analysis of the risk factors for lesion rupture. J Neurosurg. 2005;103:662–680. doi: 10.3171/jns.2005.103.4.0662 16266049

16. Ujiie H, Tachibana H, Hiramatsu O, Hazel AL, Matsumoto T, Ogasawara Y, et al. Effects of size and shape (aspect ratio) on the hemodynamics of saccular aneurysms: a possible index for surgical treatment of intracranial aneurysms. Neurosurgery. 1999;45(1):119–29. doi: 10.1097/00006123-199907000-00028 10414574

17. Baharoglu MI, Schirmer CM, Hoit DA, Gao BL, Malek AM. Aneurysm inflow-angle as a discriminant for rupture in sidewall cerebral aneurysms: morphometric and computational fluid dynamic analysis. Stroke. 2010;41(7):1423–30. doi: 10.1161/STROKEAHA.109.570770 20508183

18. Xu J, Wu Z, Yu Y, Lv N, Wang S, Karmonik C, et al. Combined Effects of Flow Diverting Strategies and Parent Artery Curvature on Aneurysmal Hemodynamics: A CFD Study. PLoS ONE. 2015;10(9):e0138648. doi: 10.1371/journal.pone.0138648 26398847

19. Duan Z, Li Y, Guan S, Ma C, Han Y, Ren X, et al. Morphological parameters and anatomical locations associated with rupture status of small intracranial aneurysms. Scientific Reports. 2018;8(1):6440. doi: 10.1038/s41598-018-24732-1 29691446

20. Tateshima S, Chien A, Sayre J, Cebral J, Viñuela F. The effect of aneurysm geometry on the intra-aneurysmal flow condition. Neuroradiology. 2010;52(12):1135–41. doi: 10.1007/s00234-010-0687-4 20373097

21. Long Y, Yu H, Zhuo Z, Zhang Y, Wang Y, Yang X, et al. A geometric scaling model for assessing the impact of aneurysm size ratio on hemodynamic characteristics. Biomed Eng Online. 2014;17:13–17.

22. Valencia Alvaro, Solis Francisco. Blood flow dynamics and arterial wall interaction in a saccular aneurysm model of the basilar artery. Computers & Structures. 2006;84(21):1326–1337. doi: 10.1016/j.compstruc.2006.03.008

23. Yamaguchi R, Tanaka G, Liu H. Effect of Elasticity on Flow Characteristics Inside Intracranial Aneurysms. Int J Neurol Neurother. 2016;3:049. doi: 10.23937/2378-3001/3/3/1049

24. Xu L, Sugawara M, Tanaka G, Ohta M, Liu H, Yamaguchi R. Effect of elasticity on wall shear stress inside cerebral aneurysm at anterior cerebral artery. Technol Health Care. 2016;24(3):349–57. doi: 10.3233/THC-161135 26835728

25. Chadwick P. The existence and uniqueness of solutions of two problems in the Mooney-Rivlin theory for rubber. Journal of Elasticity. 1972;2(2):123–128. doi: 10.1007/BF00046061

26. Skalak R, Tozeren A, Zarda RP, Chien S. Strain Energy Function of Red Blood Cell Membranes. Biophys J. 1973;13(3):245–264. doi: 10.1016/S0006-3495(73)85983-1 4697236

27. Fung YC. Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag. 1981.

28. Volokh KY, Vorp DA. A model of growth and rupture of abdominal aortic aneurysm. J Biomech. 2008;41(5):1015–1021. doi: 10.1016/j.jbiomech.2007.12.014 18255074

29. Rodríguez J, Merodio J. A new derivation of the bifurcation conditions of inflated cylindrical membranes of elastic material under axial loading: Application to aneurysm formation. Mechanics Research Communications. 2011;38(3):203–210. doi: 10.1016/j.mechrescom.2011.02.004

30. Pierce DM, Maier F, Weisbecker H, Viertler C, Verbrugghe P, Famaey N, et al. Human thoracic and abdominal aortic aneurysmal tissues: Damage experiments, statistical analysis and constitutive modeling. J Mech Behav Biomed Mater. 2015;41:92–107. doi: 10.1016/j.jmbbm.2014.10.003 25460406

31. Charalambous HP, Roussis PC, Giannakopoulos AE. The Effect of Strain Hardening on the Dynamic Response of Human Artery Segments. Open Biomed Eng J. 2017;26(11):85–110. doi: 10.2174/1874120701711010085

32. Krüger T, Varnik F, Raabe D. Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method. Comput Math with Appl. 2011;61:3485–3505. doi: 10.1016/j.camwa.2010.03.057

33. Kamm R, Grodzinsky A. 20.310J Molecular, Cellular, and Tissue Biomechanics. Spring 2015. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu.

34. Krafczyk M, Cerrolaza M, Schulz M, Rank E. Analysis of 3D transient blood flow passing through an artificial aortic valve by Lattice-Boltzmann methods. J Biomech. 1998;31(5):453–462. doi: 10.1016/s0021-9290(98)00036-0 9727343

35. Cebral JR, Castro MA, Appanaboyina S, Putman CM, Millan D, Frangi AF. Efficient pipeline for image-based patient-specific analysis of cerebral aneurysm hemodynamics: Technique and sensitivity. IEEE Trans Med Imaging. 2005;(24):457–467. doi: 10.1109/tmi.2005.844159 15822804

36. Shishir SS, Miah MAK, Islam AKMS, Hasan ABMT. Flow Dynamics in Cerebral Aneurysm—A CFD Simulation. Procedia Engineering. 2015;105:919–927.

37. Ariane M, Allouche MH, Bussone M, Giacosa F, Bernard F, Barigou M, Alexiadis A. Discrete multi-physics: A mesh-free model of blood flow in flexible biological valve including solid aggregate formation. PLoS ONE. 2017;12(4):e0174795. doi: 10.1371/journal.pone.0174795 28384341

38. Han HC, Fung YC. Longitudinal strain of canine and porcine aortas. J Biomech. 1995;28(5):637–41. doi: 10.1016/0021-9290(94)00091-h 7775500

39. Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A, Larsen J. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng. 2000;28(11):1281–99. doi: 10.1114/1.1326031 11212947

40. Krüger T, Gross M, Raabe D, Varnik F. Crossover from tumbling to tank-treading-like motion in dense simulated suspensions of red blood cells. Soft Matter. 2013;9(37):9008–9015. doi: 10.1039/c3sm51645h 25353617

41. Gross M, Krüger T, Varnik F. Fluctuations and diffusion in sheared athermal suspensions of deformable particles. EPL (Europhysics Letters). 2014;108(6):68006. doi: 10.1209/0295-5075/108/68006

42. Gross M, Krüger T, Varnik F. Rheology of dense suspensions of elastic capsules: normal stresses, yield stress, jamming and confinement effects. Soft Matter. 2014;10(24):4360–4372. doi: 10.1039/c4sm00081a 24796957

43. Gompper G, Kroll D. Random Surface Discretizations and the Renormalization of the Bending Rigidity. J Phys I France. 1996;6(10):1305–1320. doi: 10.1051/jp1:1996246

44. Taylor CL, Yuan Z, Selman WR, Ratcheson RA, Rimm AA. Cerebral arterial neurysm formation and rupture in 20,767 elderly patients: hypertension and other risk factors. J Neurosurg. 1995;83(5):812–819. doi: 10.3171/jns.1995.83.5.0812 7472548

45. Juvela S. Prehemorrhage risk factors for fatal intracranial aneurysm rupture. Stroke. 2003;34(8):1852–7. doi: 10.1161/01.STR.0000080380.56799.DD 12829865

46. Kim MO, Adji A, O’Rourke MF, Avolio AP, Smielewski P, Pickard JD, et al. Principles of cerebral hemodynamics when intracranial pressure is raised: lessons from the peripheral circulation. J Hypertens. 2015;33(6):1233–41. doi: 10.1097/HJH.0000000000000539 25764046

47. Lallemand P, Luo LS. Theory of the Lattice Boltzmann Method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys Rev E. 2000;61(6):6546–62. doi: 10.1103/PhysRevE.61.6546

48. d’Humières D, Ginzburg I. Viscosity independent numerical errors for Lattice Boltzmann models: From recurrence equations to “magic” collision numbers. Comput Math Appl. 2009;58:823–840. doi: 10.1016/j.camwa.2009.02.008

49. Asinari P, Karlin I. Quasiequilibrium lattice Boltzmann models with tunable bulk viscosity for enhancing stability. Phys Rev E. 2010;81:016702. doi: 10.1103/PhysRevE.81.016702

50. Guo Z, Zheng C, Shi B. Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys Rev E. 2002;65:046308. doi: 10.1103/PhysRevE.65.046308

51. Krüger T, Varnik F, Raabe D. Shear stress in lattice Boltzmann simulations. Phys Rev E. 2009;79:046704. doi: 10.1103/PhysRevE.79.046704

52. Peskin CS. Flow patterns around heart valves: A digital computer method for solving the equations of motion. Ph.D. thesis, Albert Einstein College of Medicine. 1972.

53. Womersley JR. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J Physiol. 1955;127(3):553–563. doi: 10.1113/jphysiol.1955.sp005276 14368548

54. Gilmanov A, Sotiropoulos F, Balaras E. A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids. J Comput Phys. 2003;191:660–669. doi: 10.1016/S0021-9991(03)00321-8

55. Arthurs KM, Moore LC, Peskin CS, Pitman EB, Layton HE. Modeling arteriolar flow and mass transport using the immersed boundary method. J Comput Phys. 1998;147(2):402–440. doi: 10.1006/jcph.1998.6097

56. Peskin CS. The immersed boundary method. Acta Numerica. 2002;11:479–517. doi: 10.1017/S0962492902000077

57. Smith KM, Moore LC, Layton HE. Advective transport of nitric oxide in a mathematical model of the afferent arteriole. Am J Physiol Renal Physiol. 2003;284(5):F1080–96. doi: 10.1152/ajprenal.00141.2002 12712988

58. Updegrove A, Wilson NM, Merkow J, Lan H, Marsden AL, Shadden SC. SimVascular—An open source pipeline for cardiovascular simulation. Ann Biomed Eng. 2017;45(3):525–541. doi: 10.1007/s10439-016-1762-8 27933407

59. Shapiro AH. Steady flow in collapsed tubes. Trans. ASME K: J Biomech Engng. 1977;99:126–147.

60. Jensen OE, Pedley TJ. The existence of steady flow in a collapsed tube. J Fluid Mech. 1989;206:339–374. doi: 10.1017/S0022112089002326

61. Gibo H, Carver CC, Rhoton AL Jr, Lenkey C, Mitchell RJ. Microsurgical anatomy of the middle cerebral artery. J Neurosurg. 1981;54(2):151–69. doi: 10.3171/jns.1981.54.2.0151 7452329

62. Mouches P, Forkert ND. A statistical atlas of cerebral arteries generated using multi-center MRA datasets from healthy subjects. Scientific Data. 2019;6:29. doi: 10.1038/s41597-019-0034-5 30975990

63. Isaksen JG, Bazilevs Y, Kvamsdal T, Zhang Y, Kaspersen JH, Waterloo K, et al. Determination of wall tension in cerebral artery aneurysms by numerical simulation. Stroke. 2008;39(12):3172–8. doi: 10.1161/STROKEAHA.107.503698 18818402

64. Aaslid R, Markwalder TM, Nornes H. Noninvasive transcranial Doppler ultrasound recording of flow velocity in basal cerebral arteries. J Neurosurg. 1982;57(6):769–74. doi: 10.3171/jns.1982.57.6.0769 7143059

65. Krejza J, Szydlik P, Liebeskind DS, Kochanowicz J, Bronov O, Mariak Z, et al. Age and sex variability and normal reference values for the V(MCA)/V(ICA) index. AJNR Am J Neuroradiol. 2005;26(4):730–5. 15814913

66. Ogoh S, Fadel PJ, Zhang R, Selmer C, Jans Ø, Secher NH, et al. Middle cerebral artery flow velocity and pulse pressure during dynamic exercise in humans. Am J Physiol Heart Circ Physiol. 2005;288(4):H1526–31. doi: 10.1152/ajpheart.00979.2004 15591094

67. Bouvy WH, Geurts LJ, Kuijf HJ, Luijten PR, Kappelle LJ, Biessels GJ, et al. Assessment of blood flow velocity and pulsatility in cerebral perforating arteries with 7-T quantitative flow MRI. NMR Biomed. 2016;29(9):1295–304. doi: 10.1002/nbm.3306 25916399

68. Warriner RK, Johnston KW, Cobbold RS. A viscoelastic model of arterial wall motion in pulsatile flow: implications for Doppler ultrasound clutter assessment. Physiol Meas. 2008;29(2):157–79. doi: 10.1088/0967-3334/29/2/001 18256449

Článek vyšel v časopise


2020 Číslo 1