# Near-Wall Flow in Cerebral Aneurysms

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Patient Anatomies

#### 2.2. Simulation Setup

#### 2.3. Newtonian and Generalised Newtonian Model for Blood Rheology

#### 2.4. Methods for Analysing the Flow Field in the Near-Wall Region

#### 2.5. Numerical Computation of the Near-Wall Flow

## 3. Results and Discussion

- WSS < 1 Pa suggests slow tangential flow, (→)
- WSS > 1 Pa suggests faster tangential flow, (⇉)
- WSSdiv < -1000 Pa m${}^{-1}$ suggests fast perpendicular flow to the wall, (⇊)
- WSSdiv > 1000 Pa m${}^{-1}$ suggests fast perpendicular flow from the wall, (⇈)
- |WSSdiv| < 1000 Pa m${}^{-1}$ suggests slow perpendicular flow, (⇵)

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

WSS | Wall shear stress—indication of near-wall flow in the plane of the wall |

WSSdiv | Divergence of wall shear stress—indication of near-wall flow normal to the wall |

## Appendix A. Tabulated Results for Different Flow Rate Profiles

Time (s) | ${\mathbf{\lambda}}_{1}$ | ${\mathbf{\lambda}}_{2}$ | ${\mathbf{\lambda}}_{1}/{\mathbf{\lambda}}_{2}$ | WSSdiv | Num. Critical Points | |
---|---|---|---|---|---|---|

(Strongest Persistent Focus Critical Point) | (Node/Saddle)-(Focus) | |||||

0.051 | - | - | - | - | 9-0 | |

Case 2 | 0.261 | −2200 | 3170 | −0.70 | 4260 | 10-1 |

Healthy | 0.794 | −7910 | 5880 | −1.34 | 17,000 | 3-1 |

Carreau | 0.051 | −3810 | 2430 | −1.57 | 6900 | 7-2 |

Case 2 | 0.261 | −6990 | 3160 | −2.21 | 14,800 | 8-4 |

Healthy | 0.794 | −9630 | 7420 | −1.30 | 19,300 | 7-3 |

0.246 | −11,200 | 7550 | −1.49 | 20,700 | 3-2 | |

Case 2 | 0.330 | −12,500 | 8610 | −1.46 | 25,000 | 4-1 |

Patient1 | 0.800 | −8770 | 7450 | −1.18 | 20,200 | 3-1 |

0.086 | −7750 | 5440 | −1.42 | 14,100 | 3-2 | |

Case 2 | 0.298 | −22,100 | 9300 | −2.38 | 49,500 | 4-1 |

Patient3 | 0.800 | −9340 | 4490 | −2.08 | 17,100 | 3-1 |

0.051 | −650,000 | 304,000 | −2.14 | 1,220,000 | 92-23 | |

Case 3 | 0.261 | −90,800 | 49,300 | −1.84 | 144,000 | 44-12 |

Healthy | 0.794 | −4200 | 2990 | −1.40 | 9040 | 35-7 |

0.246 | −119,000 | 70,600 | −1.69 | 243,000 | 82-19 | |

Case 3 | 0.330 | −139,000 | 56,800 | −2.44 | 232,000 | 65-20 |

Patient1 | 0.800 | −28,800 | 1440 | −2.00 | 60,500 | 44-11 |

0.086 | −81,200 | 37,500 | −2.17 | 145,000 | 67-16 | |

Case 3 | 0.298 | −45,700 | 10,900 | −4.18 | 88,300 | 44-13 |

Patient3 | 0.800 | −36,800 | 8570 | −4.3 | 95,900 | 37-10 |

0.051 | −689,000 | 768,000 | −0.90 | 998,000 | 37-10 | |

Case 6 | 0.261 | - | - | - | - | 27-3 |

Healthy | 0.794 | −104,000 | 59,600 | −1.75 | 195,000 | 19-4 |

0.246 | −317,000 | 117,000 | −2.71 | 608,000 | 42-5 | |

Case 6 | 0.330 | −111,000 | 61,100 | −1.82 | 228,000 | 31-5 |

Patient1 | 0.800 | −45,800 | 48,600 | −0.94 | 101,000 | 20-5 |

0.086 | −105,000 | 54,600 | −1.93 | 185,000 | 16-1 | |

Case 6 | 0.298 | −46,100 | 27,600 | −1.67 | 96,800 | 27-3 |

Patient3 | 0.800 | −104,000 | 111,000 | −0.93 | 184,000 | 13-4 |

Time (s) | WSS < 1 | WSS > 1 | ||||||
---|---|---|---|---|---|---|---|---|

WSSdiv < −1000 | |WSSdiv| < 1000 | WSSdiv > 1000 | WSSdiv < −1000 | |WSSdiv| < 1000 | WSSdiv > 1000 | |||

(→ ⇊) | (→ ⇵) | (→ ⇈) | (⇉ ⇊) | (⇉ ⇵) | (⇉ ⇈) | |||

0.051 | 0.37 | 21.43 | 1.34 | 14.05 | 41.90 | 9.48 | ||

Case 2 | 0.261 | 0.16 | 0.80 | 0.80 | 32.20 | 11.57 | 38.97 | |

Healthy | 0.794 | 0.40 | 1.83 | 0.64 | 21.43 | 31.38 | 27.99 | |

Carreau | 0.051 | 0.41 | 4.70 | 0.83 | 26.59 | 25.14 | 28.63 | |

Case 2 | 0.261 | 0.18 | 0.26 | 0.77 | 32.45 | 12.14 | 39.98 | |

Healthy | 0.794 | 0.42 | 4.99 | 1.73 | 24.81 | 29.48 | 25.93 | |

0.246 | 0.00 | 0.00 | 0.09 | 29.35 | 7.61 | 46.80 | ||

Case 2 | 0.330 | 0.00 | 0.03 | 0.63 | 33.10 | 7.66 | 41.98 | |

Patient1 | 0.800 | 0.82 | 4.83 | 1.75 | 21.25 | 29.79 | 25.20 | |

0.086 | 0.90 | 3.24 | 0.56 | 20.70 | 36.07 | 20.98 | ||

Case 2 | 0.298 | 0.30 | 0.21 | 0.72 | 31.75 | 8.91 | 40.98 | |

Patient3 | 0.800 | 0.99 | 5.93 | 1.33 | 21.05 | 27.28 | 27.18 | |

0.051 | 0.05 | 0.00 | 0.01 | 39.77 | 0.10 | 47.32 | ||

Case 3 | 0.261 | 0.93 | 0.35 | 0.52 | 38.88 | 1.82 | 43.26 | |

Healthy | 0.794 | 0.15 | 3.65 | 2.31 | 26.20 | 11.36 | 39.83 | |

0.246 | 0.38 | 0.14 | 0.07 | 38.81 | 1.08 | 46.50 | ||

Case 3 | 0.330 | 0.20 | 0.31 | 0.49 | 36.37 | 1.22 | 46.93 | |

Patient1 | 0.800 | 0.61 | 2.00 | 1.60 | 33.11 | 3.14 | 44.36 | |

0.086 | 0.10 | 0.06 | 0.49 | 38.73 | 1.19 | 46.95 | ||

Case 3 | 0.298 | 0.14 | 0.03 | 0.49 | 34.71 | 2.17 | 46.69 | |

Patient3 | 0.800 | 0.16 | 4.14 | 0.66 | 32.89 | 4.39 | 42.27 | |

0.051 | 0.00 | 0.00 | 0.00 | 40.65 | 0.00 | 51.80 | ||

Case 6 | 0.261 | 0.00 | 0.00 | 0.05 | 41.37 | 0.36 | 49.57 | |

Healthy | 0.794 | 0.01 | 0.00 | 0.37 | 36.57 | 1.22 | 51.68 | |

0.246 | 0.00 | 0.00 | 0.00 | 43.40 | 0.07 | 48.16 | ||

Case 6 | 0.330 | 0.03 | 0.00 | 0.13 | 42.12 | 0.06 | 49.15 | |

Patient1 | 0.800 | 0.00 | 0.00 | 0.06 | 39.83 | 0.64 | 50.40 | |

0.086 | 0.00 | 0.00 | 0.00 | 40.35 | 0.11 | 51.18 | ||

Case 6 | 0.298 | 0.01 | 0.00 | 0.09 | 38.43 | 0.32 | 52.19 | |

Patient3 | 0.800 | 0.02 | 0.01 | 0.03 | 37.71 | 0.60 | 51.20 |

Time (s) | Average WSSdiv if WSS <1 | Average WSSdiv if WSS >1 | |||
---|---|---|---|---|---|

if WSSdiv < −1000 | if WSSdiv > 1000 | if WSSdiv < −1000 | if WSSdiv > 1000 | ||

0.051 | −5250 | 1710 | −8610 | 3450 | |

Case 2 | 0.261 | 1660 | 3650 | −5080 | 4240 |

Healthy | 0.794 | −2150 | 3440 | −3580 | 2650 |

Carreau | 0.051 | −1460 | 2080 | −7220 | 4330 |

Case 2 | 0.261 | −3160 | 3180 | −5530 | 4460 |

Healthy | 0.794 | −1660 | 1930 | −3090 | 2880 |

0.246 | - | 7020 | $-8800$ | 5390 | |

Case 2 | 0.330 | −1520 | 3550 | −7000 | 5420 |

Patient1 | 0.800 | −1910 | 1740 | −3670 | 2780 |

0.086 | −1780 | 2380 | −3750 | 2510 | |

Case 2 | 0.298 | −3120 | 3470 | −6530 | 5170 |

Patient3 | 0.800 | −1880 | 1850 | −3750 | 2700 |

0.051 | −8210 | 16,100 | −75,700 | 73,400 | |

Case 3 | 0.261 | −13,500 | 25,000 | −18,600 | 16,400 |

Healthy | 0.794 | −4130 | 2920 | −9850 | 7270 |

0.246 | −33,900 | 28,000 | −34,400 | 27,600 | |

Case 3 | 0.330 | −9630 | 6000 | −23,200 | 18,000 |

Patient1 | 0.800 | −8600 | 6840 | −14,000 | 10,800 |

0.086 | −30,800 | 9300 | −26,700 | 23,800 | |

Case 3 | 0.298 | −7580 | 8190 | −21,000 | 16,700 |

Patient3 | 0.800 | −5520 | 4430 | −12,900 | 10,200 |

0.051 | - | - | −41,500 | 57,200 | |

Case 6 | 0.261 | - | - | −62,000 | 68,700 |

Healthy | 0.794 | −1140 | 5290 | −21,700 | 31,100 |

0.246 | −45,800 | 33,900 | −78,200 | 61,200 | |

Case 6 | 0.330 | −20,200 | 13,600 | −60,100 | 44,800 |

Patient1 | 0.800 | −7980 | 8290 | −63,000 | 54,000 |

0.086 | −13,800 | 53,800 | −65,400 | 53,400 | |

Case 6 | 0.298 | −34,900 | 14,500 | −51,900 | 37,500 |

Patient3 | 0.800 | −14,200 | 15,700 | −67,700 | 53,600 |

## References

- Cebral, J.R.; Mut, F.; Gade, P.; Cheng, F.; Tobe, Y.; Frosen, J.; Robertson, A.M. Combining data from multiple sources to study mechanisms of aneurysm disease: Tools and techniques. Int. J. Numer. Methods Biomed. Eng.
**2018**, 34, e3133. [Google Scholar] [CrossRef] [PubMed] - Nixon, A.M.; Gunel, M.; Sumpio, B.E. The critical role of hemodynamics in the development of cerebral vascular disease: A review. J. Neurosurg.
**2010**, 112, 1240–1253. [Google Scholar] [CrossRef] [PubMed] - Sforza, D.M.; Putman, C.M.; Cebral, J.R. Hemodynamics of cerebral aneurysms. Ann. Rev. Fluid Mech.
**2009**, 41, 91–107. [Google Scholar] [CrossRef] - Xiang, J.; Tutino, V.M.; Snyder, K.V.; Meng, H. CFD: Computational fluid dynamics or confounding factor dissemination? The role of hemodynamics in intracranial aneurysm rupture risk assessment. Am. J. Neuroradiol.
**2014**, 35, 1849–1857. [Google Scholar] [CrossRef] - Shojima, M.; Oshima, M.; Takagi, K.; Torii, R.; Hayakawa, M.; Katada, K.; Morita, A.; Kirino, T. Magnitude and role of wall shear stress on cerebral aneurysm: Computational fluid dynamic study of 20 middle cerebral artery aneurysms. Stroke
**2004**, 35, 2500–2505. [Google Scholar] [CrossRef] - Shimogonya, Y.; Ishikawa, T.; Imai, Y.; Mori, D.; Matsuki, N.; Yamaguchi, T. Formation of saccular cerebral aneurysms may require proliferation of the arterial wall: Computational investigation. J. Biomech. Sci. Eng.
**2008**, 3, 431–442. [Google Scholar] [CrossRef] - Miura, Y.; Ishida, F.; Umeda, Y.; Tanemura, H.; Suzuki, H.; Matsushima, S.; Shimosaka, S.; Taki, W. Low wall shear stress is independently associated with the rupture status of middle cerebral artery aneurysms. Stroke
**2013**, 44, 519–521. [Google Scholar] [CrossRef] [PubMed] - Jamous, M.A.; Nagahiro, S.; Kitazato, K.T.; Satoh, K.; Satomi, J. Vascular corrosion casts mirroring early morphological changes that lead to the formation of saccular cerebral aneurysm: An experimental study in rats. J. Neurosurg.
**2005**, 102, 532–535. [Google Scholar] [CrossRef] [PubMed] - Meng, H.; Wang, Z.; Hoi, Y.; Gao, L.; Metaxa, E.; Swartz, D.D.; Kolega, J. Complex hemodynamics at the apex of an arterial bifurcation induces vascular remodeling resembling cerebral aneurysm initiation. Stroke
**2007**, 38, 1924–1931. [Google Scholar] [CrossRef] - Kadirvel, R.; Ding, Y.H.; Dai, D.; Zakaria, H.; Robertson, A.M.; Danielson, M.A.; Lewis, D.A.; Cloft, H.J.; Kallmes, D.F. The influence of hemodynamic forces on biomarkers in the walls of elastase-induced aneurysms in rabbits. Neuroradiology
**2007**, 49, 1041–1053. [Google Scholar] [CrossRef] - Cebral, J.R.; Mut, F.; Weir, J.; Putman, C. Quantitative characterization of the hemodynamic environment in ruptured and unruptured brain aneurysms. Am. J. Neuroradiol.
**2011**, 32, 145–151. [Google Scholar] [CrossRef] [PubMed] - Ojha, M. Wall shear stress temporal gradient and anastomotic intimal hyperplasia. Circ. Res.
**1994**, 74, 1227–1231. [Google Scholar] [CrossRef] [PubMed] - Kleinstreuer, C.; Lei, M.; Archie, J.P. Flow input waveform effects on the temporal and spatial wall shear stress gradients in a femoral graft-artery connector. J. Biomech. Eng.
**1996**, 118, 506–510. [Google Scholar] [CrossRef] [PubMed] - Wootton, D.M.; Ku, D.N. Fluid mechanics of vascular systems, diseases, and thrombosis. Annu. Rev. Biomed. Eng.
**1999**, 1, 299–329. [Google Scholar] [CrossRef] [PubMed] - Gambaruto, A.M.; Doorly, D.J.; Yamaguchi, T. Wall Shear Stress and Near-Wall Convective Transport: Comparisons with Vascular Remodelling in a Peripheral Graft Anastomosis. J. Comput. Phys.
**2010**, 229, 5339–5356. [Google Scholar] [CrossRef] - Gambaruto, A.M.; Joao, A. Flow structures in cerebral aneurysms. Comput. Fluids
**2012**, 65, 56–65. [Google Scholar] [CrossRef][Green Version] - Goubergrits, L.; Thamsen, B.; Berthe, A.; Poethke, J.; Kertzscher, U.; Affeld, K.; Petz, C.; Hege, H.C.; Hoch, H.; Spuler, A. In vitro study of near-wall flow in a cerebral aneurysm model with and without coils. Am. J. Neuroradiol.
**2010**, 31, 1521–1528. [Google Scholar] [CrossRef] [PubMed] - Mantha, A.; Karmonik, C.; Benndorf, G.; Strother, C.; Metcalfe, R. Hemodynamics in a cerebral artery before and after the formation of an aneurysm. Am. J. Neuroradiol.
**2006**, 27, 1113–1118. [Google Scholar] - Ku, D.N.; Giddens, D.P.; Zarins, C.K.; Glagov, S. Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. Arteriosclerosis
**1985**, 5, 293–302. [Google Scholar] [CrossRef] - Shimogonya, Y.; Ishikawa, T.; Imai, Y.; Matsuki, N.; Yamaguchi, T. Can temporal fluctuation in spatial wall shear stress gradient initiate a cerebral aneurysm? A proposed novel hemodynamic index, the gradient oscillatory number (GON). J. Biomech.
**2009**, 42, 550–554. [Google Scholar] [CrossRef] - 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. J. Biomech.
**2013**, 46, 2651–2658. [Google Scholar] [CrossRef] - Qian, Y.; Takao, H.; Umezu, M.; Murayama, Y. Risk analysis of unruptured aneurysms using computational fluid dynamics technology: Preliminary results. Am. J. Neuroradiol.
**2011**, 32, 1948–1955. [Google Scholar] [CrossRef] - Takao, H.; Murayama, Y.; Otsuka, S.; Qian, Y.; Mohamed, A.; Masuda, S.; Yamamoto, M.; Abe, T. Hemodynamic differences between unruptured and ruptured intracranial aneurysms during observation. Stroke
**2012**, 43, 1436–1439. [Google Scholar] [CrossRef] [PubMed] - Papaharilaou, Y.; Ekaterinaris, J.A.; Manousaki, E.; Katsamouris, A.N. A decoupled fluid structure approach for estimating wall stress in abdominal aortic aneurysms. J. Biomech.
**2007**, 40, 367–377. [Google Scholar] [CrossRef] [PubMed] - Oeltze-Jafra, S.; Cebral, J.R.; Janiga, G.; Preim, B. Cluster analysis of vortical flow in simulations of cerebral aneurysm hemodynamics. IEEE Trans. Vis. Comput. Graph.
**2016**, 22, 757–766. [Google Scholar] [CrossRef] [PubMed] - Morbiducci, U.; Gallo, D.; Cristofanelli, S.; Ponzini, R.; Deriu, M.A.; Rizzo, G.; Steinman, D.A. A rational approach to defining principal axes of multidirectional wall shear stress in realistic vascular geometries, with application to the study of the influence of helical flow on wall shear stress directionality in aorta. J. Biomech.
**2015**, 48, 899–906. [Google Scholar] [CrossRef] [PubMed] - Lee, S.W.; Antiga, L.; Steinman, D.A. Correlations among indicators of disturbed flow at the normal carotid bifurcation. J. Biomech. Eng.
**2009**, 131, 061013. [Google Scholar] [CrossRef] [PubMed] - Baek, H.; Jayaraman, M.V.; Richardson, P.D.; Karniadakis, G.E. Flow instability and wall shear stress variation in intracranial aneurysms. J. R. Soc. Interface
**2009**, 7, 967–988. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cebral, J.R.; Castro, M.A.; Burgess, J.E.; Pergolizzi, R.S.; Sheridan, M.J.; Putman, C.M. Characterization of cerebral aneurysms for assessing risk of rupture by using patient-specific computational hemodynamics models. Am. J. Neuroradiol.
**2005**, 26, 2550–2559. [Google Scholar] - Sangalli, L.M.; Secchi, P.; Vantini, S.; Veneziani, A. A case study in exploratory functional data analysis: Geometrical features of the internal carotid artery. J. Am. Stat. Assoc.
**2009**, 104, 37–48. [Google Scholar] [CrossRef] - Cebral, J.R.; Mut, F.; Weir, J.; Putman, C.M. Association of hemodynamic characteristics and cerebral aneurysm rupture. Am. J. Neuroradiol.
**2011**, 32, 264–270. [Google Scholar] [CrossRef] - Sasaki, T.; Kakizawa, Y.; Yoshino, M.; Fujii, Y.; Yoroi, I.; Ichikawa, Y.; Horiuchi, T.; Hongo, K. Numerical Analysis of Bifurcation Angles and Branch Patterns in Intracranial Aneurysm Formation. Neurosurgery
**2018**. [Google Scholar] [CrossRef] - Millan, R.D.; Dempere-Marco, L.; Pozo, J.M.; Cebral, J.R.; Frangi, A.F. Morphological characterization of intracranial aneurysms using 3-D moment invariants. IEEE Trans. Med Imaging
**2007**, 26, 1270–1282. [Google Scholar] [CrossRef] - Tiago, J.; Gambaruto, A.; Sequeira, A. Patient-specific blood flow simulations: Setting Dirichlet boundary conditions for minimal error with respect to measured data. Math. Model. Nat. Phenom.
**2014**, 9, 98–116. [Google Scholar] [CrossRef] - Sequeira, A.; Tiago, J.; Guerra, T. Boundary Control Problems in Hemodynamics. In Trends in Biomathematics: Modeling, Optimization and Computational Problems; Springer: Cham, Switzerland, 2018; pp. 27–48. [Google Scholar]
- Tricerri, P.; Dedè, L.; Gambaruto, A.; Quarteroni, A.; Sequeira, A. A numerical study of isotropic and anisotropic constitutive models with relevance to healthy and unhealthy cerebral arterial tissues. Int. J. Eng. Sci.
**2016**, 101, 126–155. [Google Scholar] [CrossRef][Green Version] - Torii, R.; Oshima, M.; Kobayashi, T.; Takagi, K.; Tezduyar, T.E. Fluid–structure interaction modeling of blood flow and cerebral aneurysm: Significance of artery and aneurysm shapes. Comput. Methods Appl. Mech. Eng.
**2009**, 198, 3613–3621. [Google Scholar] [CrossRef] - Lei, M.; Giddens, D.P.; Jones, S.A.; Loth, F.; Bassinouny, H. Pulsatile flow in an end-to-side vascular graft model: Comparison of computations with experimental data. ASME J. Biomech. Eng.
**2001**, 123, 80–87. [Google Scholar] [CrossRef] - Meng, H.; Tutino, V.M.; Xiang, J.; Siddiqui, A. High WSS or low WSS? Complex interactions of hemodynamics with intracranial aneurysm initiation, growth, and rupture: Toward a unifying hypothesis. Am. J. Neuroradiol.
**2014**, 35, 1254–1262. [Google Scholar] [CrossRef] [PubMed] - Hansen, K.B.; Arzani, A.; Shadden, S.C. Mechanical platelet activation potential in abdominal aortic aneurysms. J. Biomech. Eng.
**2015**, 137, 041005. [Google Scholar] [CrossRef] [PubMed] - Hansen, K.B.; Shadden, S.C. A reduced-dimensional model for near-wall transport in cardiovascular flows. Biomech. Model. Mechanobiol.
**2016**, 15, 713–722. [Google Scholar] [CrossRef] [PubMed] - Cebral, J.; Ollikainen, E.; Chung, B.J.; Mut, F.; Sippola, V.; Jahromi, B.R.; Tulamo, R.; Hernesniemi, J.; Niemelä, M.; Robertson, A.; et al. Flow conditions in the intracranial aneurysm lumen are associated with inflammation and degenerative changes of the aneurysm wall. Am. J. Neuroradiol.
**2017**, 38, 119–126. [Google Scholar] [CrossRef] [PubMed] - Chalouhi, N.; Ali, M.S.; Jabbour, P.M.; Tjoumakaris, S.I.; Gonzalez, L.F.; Rosenwasser, R.H.; Koch, W.J.; Dumont, A.S. Biology of intracranial aneurysms: Role of inflammation. J. Cereb. Blood Flow Metab.
**2012**, 32, 1659–1676. [Google Scholar] [CrossRef] [PubMed] - Orr, A.W.; Helmke, B.P.; Blackman, B.R.; Schwartz, M.A. Mechanisms of mechanotransduction. Dev. Cell
**2006**, 10, 11–20. [Google Scholar] [CrossRef] - Arzani, A.; Gambaruto, A.M.; Chen, G.; Shadden, S.C. Lagrangian wall shear stress structures and near-wall transport in high-Schmidt-number aneurysmal flows. J. Fluid Mech.
**2016**, 790, 158–172. [Google Scholar] [CrossRef][Green Version] - Arzani, A.; Gambaruto, A.M.; Chen, G.; Shadden, S.C. Wall shear stress exposure time: A Lagrangian measure of near-wall stagnation and concentration in cardiovascular flows. Biomech. Model. Mechanobiol.
**2017**, 16, 787–803. [Google Scholar] [CrossRef] - Perry, A.E.; Chong, M.S. A series-expansion study of the Navier-Stokes equations with applications to three-dimensional separation patterns. J. Fluid Mech.
**1986**, 173, 207–223. [Google Scholar] [CrossRef] - Haller, G. Lagrangian coherent structures. Annu. Rev. Fluid Mech.
**2015**, 47, 137–162. [Google Scholar] [CrossRef] - Arzani, A.; Shadden, S.C. Wall shear stress fixed points in cardiovascular fluid mechanics. J. Biomech.
**2018**, 73, 145–152. [Google Scholar] [CrossRef] [PubMed] - Gambaruto, A.M. Processing the image gradient field using a topographic primal sketch approach. Int. J. Numer. Methods Biomed. Eng.
**2015**, 31, e02706. [Google Scholar] [CrossRef] - João, A.; Gambaruto, A.M.; Pereira, R.; Sequeira, A. Robust and effective automatic parameter choice for medical image filtering. Comput. Methods Biomech. Biomed. Eng. Imaging Vis.
**2009**. under review. [Google Scholar] - Gambaruto, A.M.; Taylor, D.J.; Doorly, D.J. Modelling nasal airflow using a Fourier descriptor representation of geometry. Int. J. Numer. Methods Fluids
**2009**, 59, 1259–1283. [Google Scholar] [CrossRef][Green Version] - Ramalho, S.; Moura, A.; Gambaruto, A.M.; Sequeira, A. Sensitivity to outflow boundary conditions and level of geometry description for a cerebral aneurysm. Int. J. Numer. Methods Biomed. Eng.
**2012**, 28, 697–713. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ramalho, S.; Moura, A.B.; Gambaruto, A.M.; Sequeira, A. Influence of blood rheology and outflow boundary conditions in numerical simulations of cerebral aneurysms. In Mathematical Methods and Models in Biomedicine; Springer: New York, NY, USA, 2013; pp. 149–175. [Google Scholar]
- Pedley, T.J. Mathematical modelling of arterial fluid dynamics. J. Eng. Math.
**2003**, 47, 419–444. [Google Scholar] [CrossRef] - Moyle, K.R.; Antiga, L.; Steinman, D.A. Inlet conditions for image-based CFD models of the carotid bifurcation: Is it reasonable to assume fully developed flow? J. Biomech. Eng.
**2006**, 128, 371–379. [Google Scholar] [CrossRef] - Stamatopoulos, C.; Papaharilaou, Y.; Mathioulakis, D.S.; Katsamouris, A. Steady and unsteady flow within an axisymmetric tube dilatation. Exp. Therm. Fluid Sci.
**2010**, 34, 915–927. [Google Scholar] [CrossRef] - He, X.; Ku, D.N. Unsteady entrance flow development in a straight tube. J. Biomech. Eng.
**1994**, 116, 355–360. [Google Scholar] [CrossRef] - Marzo, A.; Singh, P.; Larrabide, I.; Radaelli, A.; Coley, S.; Gwilliam, M.; Wilkinson, I.D.; Lawford, P.; Reymond, P.; Patel, U.; et al. Computational hemodynamics in cerebral aneurysms: The effects of modeled versus measured boundary conditions. Ann. Biomed. Eng.
**2011**, 39, 884–896. [Google Scholar] [CrossRef] [PubMed] - Gambaruto, A.M.; Janela, J.; Moura, A.; Sequeira, A. Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Math. Biosci. Eng. MBE
**2011**, 8, 409–423. [Google Scholar][Green Version] - Gambaruto, A.M.; Janela, J.; Moura, A.; Sequeira, A. Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms. Math. Biosci. Eng. MBE
**2013**, 10, 649–665. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bandaru, V.; Kolchinskaya, A.; Padberg-Gehle, K.; Schumacher, J. Role of critical points of the skin friction field in formation of plumes in thermal convection. Phys. Rev.
**2015**, 92, 043006. [Google Scholar] [CrossRef] [PubMed] - Surana, A.; Grunberg, O.; Haller, G. Exact theory of three-dimensional flow separation. Part 1. Steady separation. J. Fluid Mech.
**2006**, 564, 57–103. [Google Scholar] [CrossRef] - Chong, M.S.; Perry, A.E.; Cantwell, B.J. A general classification of three-dimensional flow fields. Phys. Fluids Fluid Dyn.
**1990**, 2, 765–777. [Google Scholar] [CrossRef] - Chong, M.S.; Monty, J.P.; Chin, C.; Marusic, I. The topology of skin friction and surface vorticity fields in wall-bounded flows. J. Turbul.
**2012**, 13, N6. [Google Scholar] [CrossRef] - Hornung, H.; Perry, A.E. Some aspects of three-dimensional separation. i-streamsurface bifurcations. Z. Flugwiss. Weltraumforsch.
**1984**, 8, 77–87. [Google Scholar] - Tobak, M.; Peake, D.J. Topology of three-dimensional separated flows. Annu. Rev. Fluid Mech.
**1982**, 14, 61–85. [Google Scholar] [CrossRef] - Lighthill, M.J. Attachment and separation in three-dimensional flow. In Laminar Boundary Layers; Rosenhead, L., Ed.; Oxford University Press: Oxford, UK, 1963; pp. 72–82. [Google Scholar]
- Nielson, M.N.; Jung, I.H. Tools for computing tangent curves for linearly varying vector fields over tetrahedral domains. IEEE Trans. Vis. Comput. Graph.
**1999**, 5, 360–372. [Google Scholar] [CrossRef][Green Version] - Kipfer, P.; Reck, F.; Greiner, G. Local exact particle tracing on unstructured grids. Comput. Graph Forum
**2003**, 22, 133–142. [Google Scholar] [CrossRef] - Scheuermann, G.; Bobach, T.; Hagen, H.; Mahrous, K.; Hamann, B.; Joy, K.I.; Kollmann, W. A tetrahedra-based stream surface algorithm. In Proceedings of the IEEE Visualization, San Diego, CA, USA, 21–26 October 2001. [Google Scholar]
- Chakraborty, P.; Balachandar, S.; Adrian, R.J. On the relationships between local vortex identification schemes. J. Fluid Mech.
**2005**, 535, 189–214. [Google Scholar] [CrossRef] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom.
**1985**, 16, 285–317. [Google Scholar] [CrossRef][Green Version] - Ottino, J.M. Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech.
**1990**, 22, 207–254. [Google Scholar] [CrossRef] - Dallmann, U. Topological Structures of Three-Dimensional Flow Separation; DFVLR Report IB 221-82-A07; Institut fuer Theoretische Stroemungsmechanik: Göttingen, Germany, 1983. [Google Scholar]
- Oswatitsch, K. Die Ablösungsbedingung von Grenzschichten. In Grenzschichtforschung/Boundary Layer Research; Springer: Berlin/Heidelberg, Germany, 1958; pp. 357–367. [Google Scholar]

**Figure 1.**Cross-section through aneurysm of Case 1, showing details of the prism layers and polyhedral mesh.

**Figure 2.**Inflow profiles obtained from [59], based on four subject datasets. The profiles from [59] were scaled to have mean Reynolds number of 450, as shown in the left plot. The right plot shows the flow rate profile for Case 6. Please note that the flow rate profiles for each Case will differ, due to geometric differences, however the Reynolds number profiles are the same for all Cases. The cardiac time period is approximately 0.8 s for each profile. The flow rate profile is marked with time instances, which approximately relate to peak systole ($t=0.181$ s), end systole (dicrotic notch) ($t=0.389$ s) and end diastole ($t=0.768$ s). Times in brackets are related to the waveform for Patient 5. Results of the numerical simulations are presented for these times.

**Figure 3.**Detail of surface shear lines, instantaneous streamlines and wall shear stress critical points. Wall shear stress critical points are marked by coloured dots, such that

**green**indicates a focus, hence complex conjugate pair solution,

**blue**indicates a saddle or node, hence real solution, and

**red**are locations a small distance along the eigenvectors (hence principal directions of dividing surface shear lines), see Equation (6). Left plot: solution for Case 1 at time = 0.181 s; velocity streamlines are coloured blue if originating near a focus WSS critical point, and red if originating near a saddle/node WSS critical point. Middle plot: detail of stable focus, showing how the flow moves in a spiral manner in the plane of the wall, gradually moving away from the wall. The red streamline indicates the spiralling core. We observe an interaction with the free-stream flow soon after, the vortex compresses before stretching again and continuing further into the aneurysm. Right plot: detail of an unstable node, showing how flow impinges on the wall and spreads out. The red streamline ends at the stagnation point. Red dots indicate direction of principal axes (the eigenvectors) on one side only of the critical point.

**Figure 4.**Solution for Case 1. Plots of surface shear lines and WSS critical points (colouring scheme as in Figure 3).

**Figure 5.**Solution for Case 2. Plots of surface shear lines and WSS critical points (colouring scheme as in Figure 3).

**Figure 6.**Solution for Case 3. Plots of surface shear lines and WSS critical points (colouring scheme as in Figure 3).

**Figure 7.**Solution for Case 4. Plots of surface shear lines and WSS critical points (colouring scheme as in Figure 3).

**Figure 8.**Solution for Case 5. Plots of surface shear lines and WSS critical points (colouring scheme as in Figure 3).

**Figure 9.**Solution for Case 6. Plots of surface shear lines and WSS critical points (colouring scheme as in Figure 3).

**Figure 10.**Cross-section plots of shear rate magnitude $\dot{\gamma}$ (s${}^{-1}$) of Case 3, for Newtonian and Carreau rheological models. Location of the cross-section is shown.

**Figure 11.**Cross-section plots of shear rate magnitude $\dot{\gamma}$ (s${}^{-1}$) of Case 6, for Newtonian and Carreau rheological models. Location of the cross-section is shown.

**Figure 12.**Detail of aneurysm dome for Case 3 at $t=0.181$ s. Cross-section location and selection of unstable node critical points are shown in left column. Detail of cross-sections present a snapshot of in-plane velocity streamlines, with plots of velocity magnitude and shear rate magnitude, for Newtonian and Carreau rheological models.

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
---|---|---|---|---|---|---|

Gender | Female | Female | Female | Female | Female | Female |

Age | 69 | 63 | 95 | 48 | 53 | 41 |

Presentation | Rupture | Incidental | Rupture | Incidental | Incidental/Thrombosis | Rupture |

Localisation | PIC | IC | AC | MC | Basilar | AC |

Side | Left | Right | N/A | Left | N/A | N/A |

Unique/Multiple | Unique | Multiple | Unique | Unique | Unique | Unique |

Other Vasc. Anom. | No | MCA aneurysm | - | No | No | hypoplasia A1 right |

Risk Factor | No | HAT | HAT | No | HAT/D/DLD | No |

Neck Area (mm${}^{2}$) | 12.6 | 33.6 | 41.8 | 66.1 | 93.5 | 18.0 |

Neck Diameter (mm) | 2.6 | 5.5 | 5.0 | 4.7 | 10.0 | 3.7 |

Max Height (mm) | 5.4 | 7.1 | 12.1 | 5.8 | 22.1 | 6.6 |

Perpend Height (mm) | 5.3 | 7.1 | 7.4 | 4.8 | 18.9 | 6.2 |

Aspect Ratio | 2.0 | 1.3 | 1.5 | 1.0 | 1.9 | 1.7 |

Adjac. art. diam. (mm) | 1.1/1 | 2.23/2.32-1.28 | 1.41/1.49-1.34 | 0.23/0.2-0.21 | 2.37/2.72 | 1.2/0.75-1.28 |

(inlet/outlet(s)) | ||||||

Adjac. art. angle (${}^{\circ}$) | 42/136 | 54/130-48 | 68/95-99 | 15/94-94 | 24/152 | 13/52-60 |

(inlet/outlet(s)) | ||||||

Volume (cm${}^{3}$) | 0.063 | 0.340 | 0.300 | 0.113 | 1.78 | 0.094 |

Medical Image Dataset | $512\times 512\times 367$ | $512\times 512\times 189$ | $512\times 512\times 380$ | $512\times 384\times 90$ | $512\times 512\times 383$ | $512\times 512\times 176$ |

Medical Image (mm) | $0.5\times 0.5\times 0.4$ | $0.39\times 0.39\times 0.5$ | $0.36\times 0.36\times 0.4$ | $0.39\times 0.39\times 0.8$ | $0.24\times 0.24\times 0.63$ | $0.24\times 0.24\times 0.6$ |

**Table 2.**Results for the strongest (based on maximum |WSSdiv|) persistent focus critical point located within the aneurysm. Note: for Case 4 no persistent focus critical point was present and hence different points are reported; for Case 6 the persistent focus critical point only appeared from end systole to end diastole. The total number of critical points within the aneurysm are also reported at each time instance (right column). Units: WSSdiv (Pa m${}^{-1}$), WSS (Pa). Results are presented to three significant figures.

Time (s) | ${\mathbf{\lambda}}_{1}$ | ${\mathbf{\lambda}}_{2}$ | ${\mathbf{\lambda}}_{1}/{\mathbf{\lambda}}_{2}$ | WSSdiv | Num. Critical Points | |
---|---|---|---|---|---|---|

(Strongest Persistent Focus Critical Point) | (Node/Saddle)-(Focus) | |||||

0.181 | −227,000 | 107,000 | −2.13 | 420,000 | 7-5 | |

Case 1 | 0.389 | −483,000 | 121,000 | −4.00 | 1,010,000 | 12-6 |

0.768 | −12,000 | 13,000 | −0.93 | 18,200 | 8-4 | |

0.181 | −6420 | 3270 | −1.96 | 11,800 | 8-2 | |

Case 2 | 0.389 | −4980 | 9350 | −0.53 | 5890 | 7-1 |

0.768 | −3920 | 1780 | −2.20 | 7490 | 6-1 | |

Carreau | 0.181 | −9500 | 7220 | −1.32 | 20,900 | 4-1 |

Case 2 | 0.389 | −6990 | 7360 | −0.95 | 13,900 | 8-1 |

0.768 | −3750 | 2350 | −1.60 | 7492 | 3-2 | |

0.181 | −379,000 | 242,000 | −1.57 | 1,100,000 | 56-18 | |

Case 3 | 0.389 | −23,300 | 17,100 | −1.37 | 54,200 | 40-14 |

0.768 | −24,500 | 18,800 | −1.30 | 60,400 | 35-8 | |

Carreau | 0.181 | −409,000 | 237,000 | −1.7. | 713,000 | 77-20 |

Case 3 | 0.389 | −47,500 | 28,200 | −1.68 | 99,600 | 71-19 |

0.768 | −24,400 | 13,200 | −1.85 | 39,500 | 50-10 | |

0.181 | - | - | - | - | 20-7 | |

Case 4 | 0.389 | −28,100 | 23,600 | −1.19 | 49,600 | 24-9 |

0.768 | 2870 | 695 | 4.12 | −7320 | 18-5 | |

0.181 | −2090 | 1090 | −1.91 | 4580 | 11-5 | |

Case 5 | 0.389 | −3430 | 1810 | −1.9 | 6550 | 20-10 |

0.768 | −694 | 507 | −1.37 | 1620 | 8-6 | |

0.181 | - | - | - | - | 19-1 | |

Case 6 | 0.389 | −130,000 | 39,000 | −3.34 | 282,000 | 22-9 |

0.768 | −8230 | 8450 | −0.97 | 17300 | 17-5 | |

Carreau | 0.181 | −350,000 | 361,000 | −0.97 | 743,000 | 25-6 |

Case 6 | 0.389 | −322,000 | 160,000 | −2.01 | 678,000 | 29-8 |

0.768 | −10,100 | 9510 | −1.06 | 28,300 | 21-4 |

**Table 3.**Percentage area of aneurysm which satisfies conditions on WSS and WSSdiv. WSS < 1 suggests slow tangential flow (→), WSS > 1 suggests faster tangential flow (⇉), WSSdiv < −1000 suggests fast perpendicular flow to the wall (⇊), WSSdiv > 1000 suggests fast perpendicular flow from the wall (⇈), |WSSdiv| < 1000 suggests slow flow perpendicular to the wall (⇵). The partitioning of the area is based on the underlying discrete surface mesh, the vertices of which hold the function values. If any vertex of a mesh element did not satisfy the partitioning criteria, it was not included. Consequently the sum of the areas in the rows does not add to 100%, as we have in effect also excluded partial perimeters to the partitioned regions. This does not affect the analysis. Units: WSSdiv (Pa m${}^{-1}$), WSS (Pa). Results are presented to two decimal places.

Time (s) | WSS < 1 | WSS > 1 | |||||
---|---|---|---|---|---|---|---|

WSSdiv < −1000 | |WSSdiv| < 1000 | WSSdiv > 1000 | WSSdiv < −1000 | |WSSdiv| < 1000 | WSSdiv > 1000 | ||

(→ ⇊) | (→ ⇵) | (→ ⇈) | (⇉ ⇊) | (⇉ ⇵) | (⇉ ⇈) | ||

0.181 | 0.02 | 0.01 | 0.04 | 50.67 | 0.22 | 36.85 | |

Case 1 | 0.389 | 0.01 | 0.02 | 0.06 | 40.87 | 0.30 | 47.22 |

0.768 | 0.35 | 1.02 | 1.47 | 39.71 | 2.19 | 40.73 | |

0.181 | 0.94 | 4.43 | 0.76 | 25.75 | 21.71 | 30.00 | |

Case 2 | 0.389 | 0.00 | 0.00 | 0.20 | 29.68 | 8.19 | 46.66 |

0.768 | 0.84 | 3.47 | 1.82 | 24.14 | 25.68 | 28.61 | |

Carreau | 0.181 | 0.51 | 1.32 | 0.28 | 29.29 | 14.01 | 40.64 |

Case 2 | 0.389 | 0.00 | 0.02 | 0.21 | 33.01 | 9.89 | 42.66 |

0.768 | 0.68 | 5.06 | 0.72 | 23.96 | 26.29 | 28.92 | |

0.181 | 0.14 | 0.00 | 0.15 | 39.13 | 0.33 | 45.49 | |

Case 3 | 0.389 | 0.87 | 1.17 | 1.04 | 34.88 | 3.79 | 41.57 |

0.768 | 0.83 | 2.16 | 1.06 | 31.99 | 5.40 | 41.24 | |

Carreau | 0.181 | 0.02 | 0.00 | 0.06 | 39.84 | 0.50 | 45.93 |

Case 3 | 0.389 | 0.86 | 0.50 | 0.70 | 36.20 | 3.32 | 42.38 |

0.768 | 1.18 | 5.66 | 1.57 | 32.05 | 6.36 | 37.27 | |

0.181 | 0.56 | 0.58 | 0.53 | 44.23 | 2.48 | 36.41 | |

Case 4 | 0.389 | 0.09 | 0.01 | 0.13 | 38.00 | 0.69 | 46.83 |

0.768 | 0.88 | 0.47 | 0.99 | 37.74 | 5.67 | 37.55 | |

0.181 | 0.54 | 19.71 | 0.67 | 15.94 | 42.51 | 9.83 | |

Case 5 | 0.389 | 0.94 | 9.20 | 1.42 | 17.70 | 34.73 | 19.96 |

0.768 | 0.39 | 32.37 | 1.18 | 9.57 | 39.14 | 6.49 | |

0.181 | 0.00 | 0.00 | 0.03 | 47.90 | 0.13 | 42.73 | |

Case 6 | 0.389 | 0.00 | 0.00 | 0.01 | 40.73 | 0.03 | 50.10 |

0.768 | 0.10 | 0.00 | 0.30 | 39.10 | 1.90 | 46.13 | |

Carreau | 0.181 | 0.01 | 0.00 | 0.01 | 43.84 | 0.07 | 47.68 |

Case 6 | 0.389 | 0.01 | 0.00 | 0.04 | 42.61 | 0.26 | 47.46 |

0.768 | 0.03 | 0.09 | 0.22 | 41.07 | 1.13 | 47.52 |

**Table 4.**Average values of WSSdiv as partitioned by WSS = 1 threshold. Only the aneurysm surface is considered. Units: WSSdiv (Pa m${}^{-1}$), WSS (Pa). Results are presented to three significant figures.

Time (s) | Average WSSdiv if WSS <1 | Average WSSdiv if WSS >1 | |||
---|---|---|---|---|---|

if WSSdiv < −1000 | if WSSdiv > 1000 | if WSSdiv < −1000 | if WSSdiv > 1000 | ||

0.181 | −20,400 | 10,900 | −38,400 | 34,600 | |

Case 1 | 0.389 | −14,700 | 14,500 | −48,200 | 37,000 |

0.768 | −5470 | 3240 | −15,700 | 11,100 | |

0.181 | −1720 | 2130 | −5520 | 3540 | |

Case 2 | 0.389 | - | 7340 | −7870 | 5030 |

0.768 | −1970 | 1950 | −3320 | 2650 | |

Carreau | 0.181 | −2330 | 2090 | −6040 | 3690 |

Case 2 | 0.389 | - | 5740 | −6410 | 4980 |

0.768 | −2120 | 2010 | −3390 | 2770 | |

0.181 | −6790 | 10,600 | −59,200 | 46,800 | |

Case 3 | 0.389 | −3500 | 4640 | −27,200 | 21,000 |

0.768 | −3160 | 4420 | −18100 | 12800 | |

Carreau | 0.181 | −6670 | 9950 | −46,100 | 43,500 |

Case 3 | 0.389 | −3620 | 5170 | −21,700 | 18,200 |

0.768 | −4050 | 3190 | −12,700 | 10,900 | |

0.181 | −5050 | 5880 | −22,700 | 16,000 | |

Case 4 | 0.389 | −14,100 | 9350 | −30,300 | 21,700 |

0.768 | −4900 | 3520 | −11,000 | 7900 | |

0.181 | −1860 | 1810 | −3040 | 2310 | |

Case 5 | 0.389 | −2100 | 1730 | −4180 | 2990 |

0.768 | −1700 | 1360 | −2120 | 1790 | |

0.181 | −66,600 | 21,800 | −74,600 | 60,900 | |

Case 6 | 0.389 | −36,700 | 28,080 | −77,900 | 60,600 |

0.768 | −10,000 | 8680 | −37,200 | 23,400 | |

Carreau | 0.181 | −15,300 | 32,800 | −57,600 | 33,400 |

Case 6 | 0.389 | −9420 | 9360 | −28,100 | 17,900 |

0.768 | −5270 | 11,500 | −19,300 | 11,300 |

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Goodarzi Ardakani, V.; Tu, X.; Gambaruto, A.M.; Velho, I.; Tiago, J.; Sequeira, A.; Pereira, R.
Near-Wall Flow in Cerebral Aneurysms. *Fluids* **2019**, *4*, 89.
https://doi.org/10.3390/fluids4020089

**AMA Style**

Goodarzi Ardakani V, Tu X, Gambaruto AM, Velho I, Tiago J, Sequeira A, Pereira R.
Near-Wall Flow in Cerebral Aneurysms. *Fluids*. 2019; 4(2):89.
https://doi.org/10.3390/fluids4020089

**Chicago/Turabian Style**

Goodarzi Ardakani, Vahid, Xin Tu, Alberto M. Gambaruto, Iolanda Velho, Jorge Tiago, Adélia Sequeira, and Ricardo Pereira.
2019. "Near-Wall Flow in Cerebral Aneurysms" *Fluids* 4, no. 2: 89.
https://doi.org/10.3390/fluids4020089