# The Effect of Blood Rheology and Inlet Boundary Conditions on Realistic Abdominal Aortic Aneurysms under Pulsatile Flow Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Image Segmentation, Surface Reconstruction, and Meshing

#### 2.2. Simulation Setup, Boundary Conditions, and Rheology Models

**U**,P, respectively, depended on blood density $\mathsf{\rho}$ and stress tensor $\mathsf{\tau}.$ For the needs of this study, the stress tensor was expressed in terms of the rate-of-deformation tensor

**D**and the shear rate $\dot{\mathsf{\gamma}}$ as follows

^{−4}. The SIMPLEC algorithm was chosen for pressure–velocity coupling and a fixed time step of 0.005 s was adopted for a cardiac cycle of 1 s. In order to ensure that all transient effects were washed-out, four cycles were simulated before results were collected. All simulations assumed a rigid wall and the no-slip condition was prescribed at the wall boundary. Furthermore, a transient inlet velocity and outlet pressure were prescribed in all simulations; both profiles (Figure 2) closely follow Olufsen et al. [30].

^{−1}, all models converge to the Newtonian case. Specifications were selected following a recent classification of 16 rheology models [38], which revealed a partition in three main homogeneous groups (clusters) and six in total, with the Newtonian model appearing as an outlier. The Carreau–Yasuda and the Casson models were members of the largest cluster, whereas the Power law was the best representative of the second largest cluster and Herschel–Bulkley a satisfactory representative of the third cluster.

#### 2.3. Hemodynamic and Flow Parameters

**WWS**) over the cardiac cycle, T. These include the time average wall shear stress (TAWSS, Pa), oscillatory shear index (OSI), and relative residence time (RRT, Pa

^{−1}), defined by the following expressions

**WSS**. As a result, uniaxial flows yield $\mathrm{OSI}=0$ while flows with no preferred direction correspond to OSI = 0.5. Finally, Himburg et al. [49] presented RRT to quantify the time that blood resides close to the wall while accounting for the effect of both TAWSS and OSI. The abovementioned variables provide valuable information for various diseased states, such as thrombogenic stimulating environments for TAWSS < 0.4 Pa [50], OSI > 0.3 [49], and RRT > 10 Pa

^{−1}[51].

_{A}, and flow dispersion, f

_{D}; both quantities were calculated on random planes (to ensure that results were not affected in a systematic way) of the aneurysm sac and quantify the eccentricity and broadness of flows, respectively. Following [18,52], the centroid coordinates $\left({\mathrm{x}}_{0},{\mathrm{y}}_{0},{\mathrm{z}}_{0}\right)$ of the top 15% peak systolic velocity $\left({\mathrm{V}}_{\mathrm{max}}^{15\%}\right)$ were compared with respect to the centroid $\left({\mathrm{x}}_{\mathrm{c}},{\mathrm{y}}_{\mathrm{c}},{\mathrm{z}}_{\mathrm{c}}\right)$ of the plane under consideration. Flow asymmetry is formulated as

#### 2.4. Statistical Analysis

_{m}, respectively, produced a balanced design, as the sample sizes within each level of the independent factors are equal. Fourteen response variables that characterize vascular flows were examined: flow asymmetry (FA

_{pct}), flow dispersion (FD

_{pct}), percentage areas with thrombus-prone conditions, designated by the levels of TAWSS (TAWSS

_{pct}: % area with TAWSS < 0.4 Pa), OSI (OSI

_{pct}: % area with OSI > 0.3), and RRT (RRT

_{pct}: % area with RRT > 10 Pa

^{−1}), and the observed average levels, minima and maxima of hemodynamic variables, namely TAWSS (TAWSS

_{ave}, TAWSS

_{min}, TAWSS

_{max}), OSI (OSI

_{ave}, OSI

_{min}, OSI

_{max}), and RRT (RRT

_{ave}, RRT

_{min}, RRT

_{max}).

_{m}. If an ANOVA test signifies substantial evidence against the null hypothesis of equal means across factor levels, the analysis proceeds in the second stage, which comprises multiple pairwise comparisons between the group means: the latter determine if specific group pairs (for R

_{m}, say P versus N, or for IVD, Plug versus Parabolic) are significantly different. On the other hand, if the p-values from an ANOVA test do not provide evidence against the null hypothesis of equal means across groups, there is no need to conduct a post hoc test to determine which groups are different from each other.

_{m}choices. In what follows, to evaluate evidence that favors the alternative hypothesis, we use p-values following the recent recommendation in [56]: values between 0.005 and 0.05 offer weak or “suggestive” evidence, whereas values lower than 0.005 provide strong evidence against the null hypothesis. It should be stressed that p-values are often misinterpreted in ways that lead to overstating the evidence against the null hypothesis when conventional thresholds (e.g., p-value = 0.05) that signify “statistical significance” are utilized [57]. As shown in [56], conventional levels of significance do not actually provide strong evidence against the null hypothesis. When the prior probabilities of the null and the alternative hypotheses are equal, the upper bound on the posterior probability of the alternative hypothesis equals 0.89 for a p-value of 0.01, which is often considered “highly significant”; hence, there remains at least an 11% chance that the null hypothesis is true. For a p-value of 0.005, the upper bound on the posterior probability of the alternative hypothesis equals 0.933, which reduces the chance that the null is true by about 50% relative to when the p-value equals 0.01 [56].

## 3. Results

#### 3.1. Mesh Convergence

#### 3.2. ANOVA Models

_{m}groups. Figure 5a supports this finding, with substantial overlapping in the group-specific boxplots.

_{pct}. Specifically, the difference in FD

_{pct}trimmed-means equals −6.982 for Parabolic vs. Plug (95% CI [−12.674, −1.289]); the corresponding p-value equals 0.012, which provides weak evidence against the null hypothesis of equal average levels. A noteworthy, although also weak, dissimilarity is also observed between Parabolic and Womersley specifications: the difference in FD

_{pct}trimmed-means equals −8.314 (95% CI [−15.626, −1.003]) and the corresponding p-value = 0.015.

_{pct}equals −7.920, CI [−10.313, −10.313], p-value < 0.001) and Plug vs. Parabolic (difference in OSI

_{pct}equals 10.109, CI [6.914, 13.304], p-value < 0.001). This is clearly observed in Figure 6b: the distributions of OSI

_{pct}computed when the Plug specification is adopted lie clearly below the ones from Womersley and Parabolic. On the other hand, there is no evidence of different outcomes for Parabolic vs. Womersley distribution profiles (difference in OSI

_{pct}equals 2.189, CI [−1.017, 5.396], p-value = 0.096). RRT-based assessments differ mainly in the IVD factor pair Parabolic vs. Plug (difference in RRT

_{pct}equals 6.997, CI [3.092, 10.902], p-value < 0.001) with weak evidence against the null hypothesis for N vs. P rheologies (difference in RRT

_{pct}equals 6.083, CI [0.133, 12.032], p-value = 0.046).

_{ave}equals 0.026, CI [0.012, 0.041], p-value < 0.001) and Plug vs. Womersley (difference in OSI

_{ave}equals −0.019, CI [−0.032, −0.005], p-value = 0.003). Observed OSI maxima differ for the pairs Parabolic vs. Plug (difference in OSI

_{max}equals 0.004, CI [0.002, 0.005], p-value < 0.001) and Parabolic vs. Womersley (difference in OSI

_{max}equals 0.004, CI [0.002, 0.005], p-value < 0.001). Interestingly, the observed differences in average levels of OSI

_{min}, RRT

_{ave}, RRT

_{min}, and RRT

_{max}do not provide evidence against the null hypothesis, for all R

_{m}and IVD specifications examined. The pairwise tests provided very weak evidence against the null hypothesis of equal RRT

_{ave}means across groups for the Parabolic vs. Plug IVD pair (difference in RTT

_{ave}equals 1.241, CI [0.005, 2.478], p-value = 0.05).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

U | fluid velocity |

P | fluid pressure |

$\mathsf{\rho}$ | fluid density |

$\mathsf{\tau}$ | stress tensor |

$\mathbf{D}$ | rate-of-deformation tensor |

$\dot{\mathsf{\gamma}}$ | shear rate |

μ | dynamic viscosity |

IVD | Inlet Velocity Distribution |

CY | Carreau–Yasuda |

Cs | Casson |

HB | Herschel–Bulkley |

N | Newtonian |

P | Power law |

WSS | wall shear stress |

TAWSS | time average wall shear stress |

OSI | oscillatory shear index |

RRT | relative residence time |

f_{A} | flow asymmetry |

f_{D} | flow dispersion |

${\mathrm{V}}_{\mathrm{max}}^{15\%}$ | top 15% peak systolic velocity |

ANOVA | Analysis of variance |

CI | Confidence Interval |

p | order of convergence |

r | grid refinement ratio |

GCI | Grid Convergence Index |

${\mathrm{F}}_{\mathrm{s}}$ | safety factor |

## References

- Mendis, S.; Puska, P.; Norrving, B.; World Health Organization; World Heart Federation; World Stroke Organization. Global Atlas on Cardiovascular Disease Prevention and Control; World Health Organization: Geneva, Switzerland, 2011; p. 155. [Google Scholar]
- Minino, A.M.; Heron, M.P.; Murphy, S.L.; Kochanek, K.D.; Centers for Disease Control and Prevention, National Center for Health Statistics, National Vital Statistics System. Deaths: Final data for 2004. Natl. Vital. Stat. Rep.
**2007**, 55, 1–119. [Google Scholar] [PubMed] - Bown, M.; Sutton, A.J.; Bell, P.R.F.; Sayers, R.D. A meta-analysis of 50 years of ruptured abdominal aortic aneurysm repair. Br. J. Surg.
**2002**, 89, 714–730. [Google Scholar] [CrossRef] [PubMed] - Nordon, I.M.; Hinchliffe, R.J.; Loftus, I.M.; Thompson, M.M. Pathophysiology and epidemiology of abdominal aortic aneurysms. Nat. Rev. Cardiol.
**2011**, 8, 92–102. [Google Scholar] [CrossRef] [PubMed] - Soerensen, D.D.; Pekkan, K.; de Zelicourt, D.; Sharma, S.; Kanter, K.; Fogel, M.; Yoganathan, A.P. Introduction of a New Optimized Total Cavopulmonary Connection. Ann. Thorac. Surg.
**2007**, 83, 2182–2190. [Google Scholar] [CrossRef] [PubMed] - Taylor, C.A.; Draney, M.T.; Ku, J.P.; Parker, D.; Steele, B.N.; Wang, K.; Zarins, C.K. Predictive medicine: Computational techniques in therapeutic decision-making. Comput. Aided Surg.
**1999**, 4, 231–247. [Google Scholar] [CrossRef] [PubMed] - Sauceda, A. A contemporary review of non-invasive methods in diagnosing abdominal aortic aneurysms. J. Ultrason.
**2021**, 21, 332–339. [Google Scholar] [CrossRef] - Adams, L.C.; Brangsch, J.; Reimann, C.; Kaufmann, J.O.; Nowak, K.; Buchholz, R.; Karst, U.; Botnar, R.M.; Hamm, B.; Makowski, M.R. Noninvasive imaging of vascular permeability to predict the risk of rupture in abdominal aortic aneurysms using an albumin-binding probe. Sci. Rep.
**2020**, 10, 3231. [Google Scholar] [CrossRef] [Green Version] - Elhanafy, A.; Guaily, A.; Elsaid, A. Numerical simulation of blood flow in abdominal aortic aneurysms: Effects of blood shear-thinning and viscoelastic properties. Math. Comput. Simul.
**2019**, 160, 55–71. [Google Scholar] [CrossRef] - Tzirakis, K.; Kamarianakis, Y.; Metaxa, E.; Kontopodis, N.; Ioannou, C.V.; Papaharilaou, Y. A robust approach for exploring hemodynamics and thrombus growth associations in abdominal aortic aneurysms. Med. Biol. Eng. Comput.
**2017**, 55, 1493–1506. [Google Scholar] [CrossRef] - Xenos, M.; Labropoulos, N.; Rambhia, S.; Alemu, Y.; Einav, S.; Tassiopoulos, A.; Sakalihasan, N.; Bluestein, D. Progression of Abdominal Aortic Aneurysm Towards Rupture: Refining Clinical Risk Assessment Using a Fully Coupled Fluid–Structure Interaction Method. Ann. Biomed. Eng.
**2015**, 43, 139–153. [Google Scholar] [CrossRef] [Green Version] - Philip, N.T.; Patnaik, B.S.V.; Sudhir, B.J. Hemodynamic simulation of abdominal aortic aneurysm on idealised models: Investigation of stress parameters during disease progression. Comput. Methods Programs Biomed.
**2022**, 213, 106508. [Google Scholar] [CrossRef] [PubMed] - Neofytou, P.; Tsangaris, S. Flow effects of blood constitutive equations in 3D models of vascular anomalies. Int. J. Numer. Methods Fluids
**2006**, 51, 489–510. [Google Scholar] [CrossRef] - Arzani, A. Accounting for residence-time in blood rheology models: Do we really need non-Newtonian blood flow modelling in large arteries? J. R. Soc. Interface
**2018**, 15, 20180486. [Google Scholar] [CrossRef] [Green Version] - Bilgi, C.; Atalık, K. Numerical investigation of the effects of blood rheology and wall elasticity in abdominal aortic aneurysm under pulsatile flow conditions. Biorheology
**2019**, 56, 51–71. [Google Scholar] [CrossRef] [PubMed] - Skiadopoulos, A.; Neofytou, P.; Housiadas, C. Comparison of blood rheological models in patient specific cardiovascular system simulations. J. Hydrodyn. Ser. B
**2017**, 29, 293–304. [Google Scholar] [CrossRef] - Morbiducci, U.; Ponzini, R.; Gallo, D.; Bignardi, C.; Rizzo, G. Inflow boundary conditions for image-based computational hemodynamics: Impact of idealized versus measured velocity profiles in the human aorta. J. Biomech.
**2013**, 46, 102–109. [Google Scholar] [CrossRef] [PubMed] - Youssefi, P.; Gomez, A.; Arthurs, C.; Sharma, R.; Jahangiri, M.; Figueroa, C.A. Impact of Patient-Specific Inflow Velocity Profile on Hemodynamics of the Thoracic Aorta. J. Biomech. Eng.
**2018**, 140, 011002. [Google Scholar] [CrossRef] [Green Version] - Madhavan, S.; Kemmerling, E.M.C. The effect of inlet and outlet boundary conditions in image-based CFD modeling of aortic flow. Biomed. Eng. Online
**2018**, 17, 66. [Google Scholar] [CrossRef] [Green Version] - Fuchs, A.; Berg, N.; Wittberg, L.P. Pulsatile Aortic Blood Flow—A Critical Assessment of Boundary Conditions. ASME J. Med. Diagn.
**2020**, 4, 011002. [Google Scholar] [CrossRef] - Jiang, Y.; Zhang, J.; Zhao, W. Effects of the inlet conditions and blood models on accurate prediction of hemodynamics in the stented coronary arteries. AIP Adv.
**2015**, 5, 057109. [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] [PubMed] - Marzo, A.; Singh, P.; Reymond, P.; Stergiopulos, N.; Patel, U.; Hose, R. Influence of inlet boundary conditions on the local haemodynamics of intracranial aneurysms. Comput. Methods Biomech. Biomed. Eng.
**2009**, 12, 431–444. [Google Scholar] [CrossRef] [PubMed] - Hardman, D.; Semple, S.I.; Richards, J.M.; Hoskins, P.R. Comparison of patient-specific inlet boundary conditions in the numerical modelling of blood flow in abdominal aortic aneurysm disease. Int. J. Numer. Methods Biomed. Eng.
**2013**, 29, 165–178. [Google Scholar] [CrossRef] [PubMed] - Metaxa, E.; Kontopodis, N.; Vavourakis, V.; Tzirakis, K.; Ioannou, C.V.; Papaharilaou, Y. The influence of intraluminal thrombus on noninvasive abdominal aortic aneurysm wall distensibility measurement. Med. Biol. Eng. Comput.
**2015**, 53, 299–308. [Google Scholar] [CrossRef] [PubMed] - Yushkevich, P.A.; Piven, J.; Hazlett, H.C.; Smith, R.G.; Ho, S.; Gee, J.C.; Gerig, G. User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability. Neuroimage
**2006**, 31, 1116–1128. [Google Scholar] [CrossRef] [Green Version] - Antiga, L.; Piccinelli, M.; Botti, L.; Ene-Iordache, B.; Remuzzi, A.; Steinman, D.A. An image-based modeling framework for patient-specific computational hemodynamics. Med. Biol. Eng. Comput.
**2008**, 46, 1097–1112. [Google Scholar] [CrossRef] [Green Version] - Taubin, G. Curve and surface smoothing without shrinkage. In Proceedings of the Fifth International Conference on Computer Vision, IEEE Computer Society, Cambridge, MA, USA, 20–23 June 1995; pp. 852–857. [Google Scholar]
- De Santis, G.; Mortier, P.; De Beule, M.; Segers, P.; Verdonck, P.; Verhegghe, B. Patient-specific computational fluid dynamics: Structured mesh generation from coronary angiography. Med. Biol. Eng. Comput.
**2010**, 48, 371–380. [Google Scholar] [CrossRef] - Olufsen, M.S.; Peskin, C.S.; Kim, W.Y.; Pedersen, E.M.; Nadim, A.; Larsen, J. Numerical Simulation and Experimental Validation of Blood Flow in Arteries with Structured-Tree Outflow Conditions. Ann. Biomed. Eng.
**2000**, 28, 1281–1299. [Google Scholar] [CrossRef] - Rana, M.S.; Rubby, M.F.; Hasan, A.T. Study of Physiological Flow Through an Abdominal Aortic Aneurysm (AAA). Procedia Eng.
**2015**, 105, 885–892. [Google Scholar] [CrossRef] [Green Version] - Kaewchoothong, N.; Algabri, Y.A.; Assawalertsakul, T.; Nuntadusit, C.; Chatpun, S. Computational Study of Abdominal Aortic Aneurysms with Severely Angulated Neck Based on Transient Hemodynamics Using an Idealized Model. Appl. Sci.
**2022**, 12, 2113. [Google Scholar] [CrossRef] - Leung, J.H.; Wright, A.R.; Cheshire, N.; Crane, J.; Thom, S.A.; Hughes, A.D.; Xu, Y. Fluid structure interaction of patient specific abdominal aortic aneurysms: A comparison with solid stress models. Biomed. Eng. Online
**2006**, 5, 33. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Finol, E.; Amon, C.H. Flow dynamics in anatomical models of abdominal aortic aneurysms: Computational analysis of pulsatile flow. Acta Cient. Venez.
**2003**, 54, 43–49. [Google Scholar] [PubMed] - Boyd, A.J.; Kuhn, D.C.; Lozowy, R.J.; Kulbisky, G.P. Low wall shear stress predominates at sites of abdominal aortic aneurysm rupture. J. Vasc. Surg.
**2016**, 63, 1613–1619. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Egelhoff, C.; Budwig, R.; Elger, D.; Khraishi, T.; Johansen, K. Model studies of the flow in abdominal aortic aneurysms during resting and exercise conditions. J. Biomech.
**1999**, 32, 1319–1329. [Google Scholar] [CrossRef] [PubMed] - Khanafer, K.M.; Bull, J.L.; Upchurch, G.R., Jr.; Berguer, R. Turbulence significantly increases pressure and fluid shear stress in an aortic aneurysm model under resting and exercise flow conditions. Ann. Vasc. Surg.
**2007**, 21, 67–74. [Google Scholar] [CrossRef] [PubMed] - Tzirakis, K.; Kamarianakis, Y.; Kontopodis, N.; Ioannou, C.V. Classification of blood rheological models through an idealized bifurcation. Symmetry, 2023; submitted. [Google Scholar]
- Cho, Y.I.; Kensey, K.R. Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: Steady flows. Biorheology
**1991**, 28, 241–262. [Google Scholar] [CrossRef] - Weddell, J.C.; Kwack, J.; Imoukhuede, P.I.; Masud, A. Hemodynamic Analysis in an Idealized Artery Tree: Differences in Wall Shear Stress between Newtonian and Non-Newtonian Blood Models. PLoS ONE
**2015**, 10, e0124575. [Google Scholar] [CrossRef] [Green Version] - Fung, Y.C. Mechanical properties and active remodeling of blood vessels. In Biomechanics: Mechanical Properties of Living Tissues; Springer: Berlin/Heidelberg, Germany, 1993. [Google Scholar]
- Valant, A.Z.; Ziberna, L.; Papaharilaou, Y.; Anayiotos, A.; Georgiou, G.C. The infuence of temperature on rheological properties of blood mixtures with different volume expanders-implications in numerical arterial hemodynamics simulations. Rheol. Acta
**2011**, 50, 389–402. [Google Scholar] [CrossRef] - Soulis, J.V.; Giannoglou, G.D.; Chatzizisis, Y.S.; Seralidou, K.V.; Parcharidis, G.E.; Louridas, G.E. Non-Newtonian models for molecular viscosity and wall shear stress in a 3D reconstructed human left coronary artery. Med. Eng. Phys.
**2008**, 30, 9–19. [Google Scholar] [CrossRef] - Souza, M.S.; Souza, A.; Carvalho, V.; Teixeira, S.; Fernandes, C.S.; Lima, R.; Ribeiro, J. Fluid Flow and Structural Numerical Analysis of a Cerebral Aneurysm Model. Fluids
**2022**, 7, 100. [Google Scholar] [CrossRef] - Molla, M.; Paul, M. LES of non-Newtonian physiological blood flow in a model of arterial stenosis. Med. Eng. Phys.
**2012**, 34, 1079–1087. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Luo, X.; Kuang, Z. A study on the constitutive equation of blood. J. Biomech.
**1992**, 25, 929–934. [Google Scholar] [CrossRef] [PubMed] - Husain, I.; Labropulu, F.; Langdon, C.; Schwark, J. A comparison of Newtonian and non-Newtonian models for pulsatile blood flow simulations. J. Mech. Behav. Biomed Mater.
**2013**, 21, 147–153. [Google Scholar] [CrossRef] - He, X.; Ku, D.N. Pulsatile Flow in the Human Left Coronary Artery Bifurcation: Average Conditions. J. Biomech. Eng.
**1996**, 118, 74–82. [Google Scholar] [CrossRef] [PubMed] - Himburg, H.A.; Grzybowski, D.; Hazel, A.; LaMack, J.; Li, X.-M.; Friedman, M.H. Spatial comparison between wall shear stress measures and porcine arterial endothelial permeability. Am. J. Physiol. Heart Circ. Physiol.
**2004**, 286, H1916–H1922. [Google Scholar] [CrossRef] [Green Version] - Malek, A.M.; Alper, S.L.; Izumo, S. Hemodynamic Shear Stress and Its Role in Atherosclerosis. JAMA
**1999**, 282, 2035–2042. [Google Scholar] [CrossRef] - Morbiducci, U.; Gallo, D.; Ponzini, R.; Massai, D.; Antiga, L.; Montevecchi, F.M.; Redaelli, A. Quantitative Analysis of Bulk Flow in Image-Based Hemodynamic Models of the Carotid Bifurcation: The Influence of Outflow Conditions as Test Case. Ann. Biomed. Eng.
**2010**, 38, 3688–3705. [Google Scholar] [CrossRef] - Mahadevia, R.; Barker, A.; Schnell, S.; Entezari, P.; Kansal, P.; Fedak, P.; Malaisrie, S.C.; McCarthy, P.; Collins, J.; Carr, J.; et al. Bicuspid Aortic Cusp Fusion Morphology Alters Aortic Three-Dimensional Outflow Patterns, Wall Shear Stress, and Expression of Aortopathy. Circulation
**2014**, 129, 673–682. [Google Scholar] [CrossRef] [Green Version] - Lawson, J. Design and Analysis of Experiments with R, 1st ed.; Chapman and Hall/CRC: New York, NY, USA, 2014; pp. 65–150. [Google Scholar]
- Wilcox, R. Introduction to Robust Estimation and Hypothesis Testing, 4th ed.; Academic Press: Cambridge, MA, USA, 2012. [Google Scholar] [CrossRef]
- Mair, P.; Wilcox, R. Robust statistical methods in R using the WRS2 package. Behav. Res. Methods
**2020**, 52, 464–488. [Google Scholar] [CrossRef] - Benjamin, D.J.; Berger, J.O. Three Recommendations for Improving the Use of p-Values. Am. Stat.
**2019**, 73, 186–191. [Google Scholar] [CrossRef] [Green Version] - Wasserstein, R.; Lazar, N. The ASA’s statement on p-values: Context, Process, and Purpose. Am. Stat.
**2017**, 70, 129–133. [Google Scholar] [CrossRef] [Green Version] - Roache, P.J. Verification and Validation in Computational Science and Engineering; Hermosa Pub: Albuquerque, Mexico, 1998; p. 464. [Google Scholar]
- ASME. Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer; ASME: New York, NY, USA, 2008; p. 100. [Google Scholar]

**Figure 1.**(

**A**): Inlet mesh and O-Grid for the construction of boundary layer. (

**B**): Surface mesh of the aneurysm and bifurcation area, where it can be seen that the mesh density increases while moving close to the bifurcation, in order to capture non-trivial flow dynamics.

**Figure 2.**Mean inlet velocity (

**a**) and outlet pressure (

**b**) for all simulations considered in this study.

**Figure 3.**Parabolic (

**a**), Plug (

**b**), and Womersley (

**c**) inlet velocity distribution profiles at the beginning of the cardiac cycle.

**Figure 4.**Patient-specific geometries and corresponding planes where flow asymmetry and dispersion were calculated.

**Figure 5.**Flow asymmetry and dispersion boxplots for alternative rheological models and inlet velocity distributions.

**Figure 6.**Boxplots of thrombogenic region percentages for alternative rheological models and inlet velocity distributions.

**Figure 7.**Boxplots of averages of hemodynamic variables for alternative rheological models and inlet velocity distributions.

**Figure 8.**Boxplots of observed maxima of hemodynamic variables for alternative rheological models and inlet velocity distributions.

**Figure 9.**Boxplots of observed minima of hemodynamic variables for alternative rheological models and inlet velocity distributions.

Case | Inlet Radius (m) | Mean Re | Max Re | α |
---|---|---|---|---|

2B | 0.01148 | 452.0 | 2026.5 | 15.87 |

7A | 0.01167 | 459.4 | 2060.0 | 16.13 |

14B | 0.01168 | 459.8 | 2061.8 | 16.15 |

16A | 0.01157 | 455.5 | 2042.4 | 16.00 |

31A | 0.01157 | 455.5 | 2042.4 | 16.00 |

41B | 0.01152 | 453.5 | 2033.5 | 15.93 |

63A | 0.01145 | 450.8 | 2021.2 | 15.83 |

Name (Abbreviation) | Expression | Parameter Values | References |
---|---|---|---|

Carreau–Yasuda (CY) | $\mathsf{\mu}\left(\dot{\mathsf{\gamma}}\right)={\mathsf{\mu}}_{\infty}+\frac{\left({\mathsf{\mu}}_{0}-{\mathsf{\mu}}_{\infty}\right)}{\left[1+{\left(\mathsf{\lambda}\dot{\mathsf{\gamma}}\right)}^{\mathsf{\alpha}}\right]{}^{\left(1-\mathrm{n}\right)/\mathsf{\alpha}}}$ | $\begin{array}{l}{\mathsf{\mu}}_{\infty}=0.00345,{\mathsf{\mu}}_{0}=0.056\\ \mathsf{\lambda}=1.902,\mathrm{n}=0.22,\mathsf{\alpha}=1.25\end{array}$ | [14,39,40] |

Casson (Cs) | $\mathsf{\mu}\left(\dot{\mathsf{\gamma}}\right)={\left(\sqrt{\mathrm{k}}+\frac{\sqrt{{\mathsf{\tau}}_{0}}}{\sqrt{\dot{\mathsf{\gamma}}}}\right)}^{2}$ | $\mathrm{k}=0.00345,{\mathsf{\tau}}_{0}=0.005$ | [39,41] |

Herschel–Bulkley (HB) | $\mathsf{\mu}\left(\dot{\mathsf{\gamma}}\right)=\mathrm{k}{\dot{\mathsf{\gamma}}}^{\mathrm{n}-1}+\frac{{\mathsf{\tau}}_{0}}{\dot{\mathsf{\gamma}}}$ | $\begin{array}{l}{\mathsf{\tau}}_{0}=0.00345,\mathrm{k}=0.008\\ \mathrm{n}=0.8375\end{array}$ | [10,42] |

Newtonian (N) | $\mathsf{\mu}\left(\dot{\mathsf{\gamma}}\right)={\mathsf{\mu}}_{\infty}$ | ${\mathsf{\mu}}_{\infty}=0.00345$ | [43,44] |

Power law (P) | $\mathsf{\mu}\left(\dot{\mathsf{\gamma}}\right)=\mathrm{k}{\dot{\mathsf{\gamma}}}^{\mathrm{n}-1}$ | $\mathrm{k}=0.01467,\mathrm{n}=0.7755$ | [45,46,47] |

**Table 3.**Convergence Grid Index for the 31A, Cs, Parabolic case, and convergence rates for TAWSS, OSI, and RRT.

# Elements | 194,392 | 388,080 | 776,000 | $\mathit{p}$ | $\mathit{C}\mathit{G}{\mathit{I}}_{12}$ | $\mathit{C}\mathit{G}{\mathit{I}}_{23}$ | $\mathit{C}\mathit{G}{\mathit{I}}_{23}/\left({\mathit{r}}^{\mathit{p}}\mathit{C}\mathit{G}{\mathit{I}}_{12}\right)$ |
---|---|---|---|---|---|---|---|

TAWSS | 0.6507 | 0.6523 | 0.6530 | 1.25533 | 0.093225 | 0.222779 | 1.0010 |

OSI | 0.2209 | 0.2227 | 0.2231 | 2.11008 | 0.069600 | 0.301029 | 1.0019 |

RRT | 5.0077 | 5.1355 | 5.1875 | 1.29873 | 0.857794 | 2.131632 | 1.0101 |

**Table 4.**p-values derived from heteroscedasticity- and outlier-robust, trimmed-mean-based ANOVA analyses for the effects of alternative inlet velocity distributions and rheological models at the hemodynamic behavior of real aneurysm geometries. Response variables related to hemodynamic behavior are shown in rows with the two main factors and their interaction in columns. p-values very close to zero, shown in bold, provide strong evidence against the null hypothesis of equal means across factor groups.

R_{m} | IVD | R_{m}:IVD | |
---|---|---|---|

FA% | 0.999 | 0.719 | 0.999 |

FD% | 0.415 | 0.002 | 0.999 |

TAWSS% | 0.009 | 0.392 | 0.999 |

OSI% | 0.237 | 0.001 | 0.999 |

RRT% | 0.008 | 0.001 | 0.987 |

TAWSSave | 0.030 | 0.916 | 0.999 |

OSIave | 0.065 | 0.001 | 0.999 |

RRTave | 0.045 | 0.017 | 0.999 |

TAWSSmax | 0.072 | 0.999 | 0.999 |

OSImax | 0.671 | 0.001 | 0.691 |

RRTmax | 0.166 | 0.201 | 0.659 |

TAWSSmin | 0.001 | 0.016 | 0.996 |

OSImin | 0.104 | 0.710 | 0.890 |

RRTmin | 0.108 | 0.995 | 0.999 |

Peak Systole | Flow Asymmetry | Flow Dispersion | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Cs | CY | HB | N | P | Cs | CY | HB | N | P | ||

2B | Parabolic | 38.45 | 32.87 | 42.09 | 39.87 | 32.08 | 5.11 | 5.22 | 4.79 | 5.94 | 6.28 |

Plug | 79.61 | 83.54 | 74.70 | 78.37 | 83.02 | 7.05 | 5.99 | 8.90 | 7.75 | 6.03 | |

Womersley | 46.05 | 53.12 | 50.48 | 30.63 | 60.54 | 16.78 | 15.85 | 15.43 | 10.29 | 13.34 | |

7A | Parabolic | 57.87 | 60.21 | 61.78 | 57.14 | 65.57 | 5.87 | 6.31 | 6.06 | 6.97 | 6.94 |

Plug | 69.90 | 71.08 | 69.35 | 68.82 | 71.07 | 17.96 | 16.94 | 19.00 | 19.73 | 16.48 | |

Womersley | 73.84 | 73.97 | 74.44 | 74.70 | 73.72 | 12.91 | 13.79 | 12.95 | 10.87 | 12.79 | |

14B | Parabolic | 85.62 | 85.24 | 85.76 | 86.72 | 84.88 | 5.79 | 6.03 | 5.87 | 5.42 | 6.10 |

Plug | 83.75 | 83.85 | 83.28 | 85.32 | 83.36 | 6.53 | 6.60 | 7.19 | 6.21 | 6.89 | |

Womersley | 82.90 | 83.14 | 82.60 | 82.39 | 82.65 | 6.75 | 6.83 | 7.31 | 6.97 | 6.99 | |

16A | Parabolic | 88.11 | 87.08 | 88.77 | 90.74 | 84.92 | 6.63 | 6.97 | 6.29 | 5.64 | 7.61 |

Plug | 79.80 | 79.12 | 80.12 | 80.64 | 78.19 | 11.24 | 12.07 | 11.24 | 8.50 | 12.20 | |

Womersley | 80.87 | 81.48 | 80.37 | 81.59 | 80.46 | 10.82 | 9.90 | 9.84 | 9.53 | 10.89 | |

31A | Parabolic | 34.44 | 35.62 | 34.59 | 35.67 | 37.35 | 18.06 | 19.03 | 17.31 | 15.91 | 19.13 |

Plug | 35.06 | 36.10 | 36.25 | 34.27 | 36.81 | 42.48 | 42.30 | 39.95 | 40.81 | 40.41 | |

Womersley | 35.38 | 37.43 | 40.41 | 39.89 | 37.44 | 38.37 | 39.65 | 35.53 | 32.68 | 36.35 | |

41B | Parabolic | 28.96 | 30.63 | 30.40 | 54.89 | 39.39 | 17.29 | 17.83 | 14.86 | 7.44 | 17.85 |

Plug | 30.10 | 24.99 | 29.96 | 27.29 | 16.99 | 22.64 | 23.49 | 26.74 | 9.87 | 36.70 | |

Womersley | 29.15 | 22.94 | 19.83 | 20.77 | 22.09 | 32.74 | 35.03 | 38.73 | 14.63 | 43.93 | |

63A | Parabolic | 23.69 | 23.50 | 22.99 | 10.07 | 25.16 | 9.07 | 13.15 | 8.56 | 8.96 | 15.48 |

Plug | 7.69 | 9.67 | 6.59 | 4.91 | 4.87 | 21.77 | 22.26 | 21.56 | 13.53 | 19.72 | |

Womersley | 7.61 | 8.75 | 8.15 | 7.26 | 13.94 | 20.78 | 19.83 | 22.89 | 22.97 | 18.90 |

% Area with TAWSS < 0.4 Pa | % Area with OSI > 0.3 | % Area with RRT > 10 Pa^{−1} | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cs | CY | HB | N | P | Cs | CY | HB | N | P | Cs | CY | HB | N | P | ||

2B | Parabolic | 16.74 | 15.78 | 24.27 | 25.29 | 12.06 | 17.10 | 16.99 | 17.20 | 19.82 | 17.34 | 12.34 | 11.76 | 15.21 | 16.78 | 11.74 |

Plug | 23.97 | 23.52 | 27.47 | 27.98 | 19.00 | 6.686 | 6.257 | 7.248 | 9.636 | 6.135 | 3.778 | 3.438 | 4.792 | 4.810 | 2.801 | |

Womersley | 24.34 | 24.75 | 27.86 | 27.58 | 20.75 | 18.78 | 18.52 | 18.92 | 18.59 | 18.12 | 9.646 | 7.513 | 15.09 | 16.20 | 4.266 | |

7A | Parabolic | 17.14 | 16.50 | 72.52 | 74.26 | 14.04 | 20.25 | 19.98 | 21.52 | 51.21 | 19.72 | 13.68 | 13.07 | 17.54 | 17.85 | 12.42 |

Plug | 20.73 | 20.23 | 74.24 | 75.87 | 17.03 | 10.08 | 9.522 | 10.99 | 40.00 | 9.337 | 6.328 | 5.990 | 7.590 | 8.158 | 5.203 | |

Womersley | 21.93 | 21.87 | 75.82 | 75.99 | 19.21 | 17.88 | 17.47 | 18.80 | 47.79 | 17.56 | 9.891 | 8.384 | 13.70 | 14.16 | 5.767 | |

14B | Parabolic | 6.703 | 6.420 | 7.536 | 8.149 | 5.482 | 15.96 | 15.74 | 16.74 | 17.56 | 15.35 | 6.121 | 5.616 | 8.872 | 9.348 | 5.575 |

Plug | 4.920 | 4.483 | 6.405 | 6.670 | 3.220 | 9.223 | 9.037 | 9.428 | 10.99 | 8.367 | 1.400 | 1.392 | 1.382 | 1.759 | 1.369 | |

Womersley | 7.594 | 7.446 | 8.310 | 8.411 | 6.815 | 16.02 | 15.94 | 16.75 | 17.31 | 15.07 | 5.050 | 4.969 | 8.009 | 7.514 | 3.603 | |

16A | Parabolic | 30.67 | 30.54 | 33.91 | 34.32 | 27.94 | 17.42 | 16.75 | 18.81 | 20.31 | 16.58 | 14.14 | 13.45 | 16.99 | 18.42 | 11.50 |

Plug | 31.82 | 31.32 | 36.25 | 36.74 | 28.69 | 10.02 | 9.818 | 10.15 | 10.55 | 9.889 | 12.61 | 12.10 | 13.86 | 15.73 | 10.62 | |

Womersley | 34.05 | 33.44 | 38.53 | 38.78 | 30.06 | 14.37 | 13.70 | 15.23 | 15.29 | 13.30 | 11.97 | 12.00 | 14.16 | 15.61 | 10.58 | |

31A | Parabolic | 11.66 | 11.44 | 16.58 | 15.71 | 9.094 | 19.45 | 19.16 | 19.80 | 19.72 | 19.11 | 10.83 | 10.14 | 14.20 | 14.27 | 9.897 |

Plug | 14.89 | 13.71 | 21.33 | 21.05 | 10.63 | 7.300 | 6.747 | 7.570 | 8.490 | 6.607 | 3.100 | 2.941 | 3.857 | 4.802 | 2.950 | |

Womersley | 18.14 | 17.83 | 22.86 | 22.84 | 13.99 | 17.58 | 17.26 | 18.38 | 18.57 | 17.03 | 8.299 | 7.894 | 13.28 | 13.21 | 6.488 | |

41B | Parabolic | 31.07 | 30.68 | 41.50 | 41.10 | 27.22 | 26.66 | 26.00 | 27.96 | 28.10 | 25.21 | 18.42 | 17.17 | 22.60 | 24.36 | 14.79 |

Plug | 30.75 | 29.74 | 40.77 | 41.60 | 26.20 | 13.63 | 12.84 | 13.77 | 15.79 | 12.40 | 11.74 | 10.75 | 13.07 | 15.90 | 10.09 | |

Womersley | 33.64 | 31.79 | 45.37 | 45.39 | 26.13 | 20.90 | 18.34 | 22.49 | 25.27 | 16.78 | 13.57 | 12.37 | 15.90 | 18.55 | 10.97 | |

63A | Parabolic | 34.81 | 34.00 | 41.01 | 40.22 | 31.08 | 26.73 | 26.29 | 27.41 | 26.26 | 27.45 | 23.36 | 22.88 | 26.78 | 26.35 | 22.33 |

Plug | 40.31 | 39.84 | 43.62 | 43.44 | 36.81 | 16.85 | 16.66 | 16.64 | 19.41 | 17.07 | 19.10 | 18.46 | 20.09 | 20.77 | 17.37 | |

Womersley | 42.09 | 41.95 | 44.94 | 44.85 | 40.02 | 26.04 | 26.40 | 25.68 | 26.25 | 27.32 | 22.12 | 19.81 | 27.81 | 28.78 | 15.87 |

Average Values | TAWSS | OSI | RRT | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cs | CY | HB | N | P | Cs | CY | HB | N | P | Cs | CY | HB | N | P | ||

2B | Parabolic | 0.558 | 0.561 | 0.498 | 0.502 | 0.586 | 0.221 | 0.219 | 0.228 | 0.236 | 0.213 | 5.363 | 5.204 | 6.236 | 6.562 | 5.012 |

Plug | 0.548 | 0.550 | 0.486 | 0.494 | 0.574 | 0.196 | 0.193 | 0.202 | 0.208 | 0.187 | 4.075 | 3.893 | 4.678 | 4.742 | 3.629 | |

Womersley | 0.546 | 0.546 | 0.487 | 0.496 | 0.571 | 0.218 | 0.215 | 0.224 | 0.229 | 0.208 | 4.801 | 4.673 | 5.599 | 5.689 | 4.241 | |

7A | Parabolic | 0.443 | 0.447 | 0.340 | 0.397 | 0.467 | 0.266 | 0.259 | 0.273 | 0.285 | 0.253 | 7.685 | 7.341 | 9.214 | 9.063 | 7.119 |

Plug | 0.438 | 0.443 | 0.392 | 0.392 | 0.466 | 0.241 | 0.233 | 0.246 | 0.259 | 0.227 | 5.702 | 5.423 | 6.419 | 6.986 | 5.030 | |

Womersley | 0.433 | 0.438 | 0.389 | 0.390 | 0.460 | 0.257 | 0.250 | 0.264 | 0.273 | 0.243 | 6.297 | 5.938 | 7.199 | 7.603 | 5.487 | |

14B | Parabolic | 0.765 | 0.761 | 0.674 | 0.708 | 0.773 | 0.210 | 0.208 | 0.216 | 0.218 | 0.203 | 4.584 | 4.464 | 5.071 | 4.882 | 4.457 |

Plug | 0.777 | 0.773 | 0.684 | 0.718 | 0.784 | 0.186 | 0.185 | 0.190 | 0.194 | 0.178 | 2.783 | 2.761 | 3.097 | 3.144 | 2.607 | |

Womersley | 0.767 | 0.763 | 0.674 | 0.710 | 0.775 | 0.202 | 0.201 | 0.209 | 0.211 | 0.195 | 3.388 | 3.327 | 4.065 | 4.417 | 3.079 | |

16A | Parabolic | 0.530 | 0.532 | 0.471 | 0.480 | 0.553 | 0.222 | 0.220 | 0.229 | 0.233 | 0.213 | 5.781 | 5.703 | 6.627 | 7.135 | 5.298 |

Plug | 0.531 | 0.533 | 0.470 | 0.480 | 0.554 | 0.202 | 0.200 | 0.206 | 0.212 | 0.193 | 5.987 | 5.408 | 6.398 | 6.375 | 5.054 | |

Womersley | 0.524 | 0.526 | 0.464 | 0.473 | 0.547 | 0.213 | 0.212 | 0.220 | 0.223 | 0.206 | 5.364 | 5.390 | 6.123 | 6.230 | 5.065 | |

31A | Parabolic | 0.653 | 0.652 | 0.575 | 0.597 | 0.670 | 0.223 | 0.222 | 0.230 | 0.232 | 0.217 | 5.188 | 5.106 | 5.731 | 5.532 | 4.977 |

Plug | 0.654 | 0.655 | 0.573 | 0.593 | 0.675 | 0.192 | 0.190 | 0.199 | 0.203 | 0.185 | 3.457 | 3.364 | 3.897 | 4.022 | 3.243 | |

Womersley | 0.644 | 0.644 | 0.566 | 0.585 | 0.663 | 0.215 | 0.214 | 0.224 | 0.227 | 0.207 | 4.380 | 4.265 | 5.237 | 5.407 | 3.922 | |

41B | Parabolic | 0.510 | 0.512 | 0.454 | 0.463 | 0.532 | 0.247 | 0.244 | 0.254 | 0.257 | 0.239 | 7.573 | 7.145 | 8.525 | 8.727 | 6.453 |

Plug | 0.515 | 0.518 | 0.457 | 0.465 | 0.540 | 0.216 | 0.212 | 0.220 | 0.226 | 0.208 | 6.203 | 5.817 | 6.791 | 7.595 | 5.536 | |

Womersley | 0.505 | 0.508 | 0.448 | 0.455 | 0.529 | 0.235 | 0.230 | 0.241 | 0.249 | 0.225 | 6.829 | 6.324 | 7.452 | 9.815 | 5.893 | |

63A | Parabolic | 0.499 | 0.501 | 0.445 | 0.458 | 0.519 | 0.248 | 0.245 | 0.254 | 0.253 | 0.242 | 8.257 | 7.884 | 9.734 | 8.800 | 7.831 |

Plug | 0.486 | 0.489 | 0.431 | 0.438 | 0.509 | 0.226 | 0.223 | 0.230 | 0.235 | 0.219 | 7.532 | 7.224 | 8.326 | 9.404 | 6.685 | |

Womersley | 0.484 | 0.486 | 0.431 | 0.439 | 0.506 | 0.244 | 0.242 | 0.249 | 0.253 | 0.237 | 7.446 | 7.035 | 8.267 | 8.589 | 6.653 |

Maximum Values | TAWSS | OSI | RRT | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cs | CY | HB | N | P | Cs | CY | HB | N | P | Cs | CY | HB | N | P | ||

2B | Parabolic | 2.389 | 2.337 | 2.049 | 2.207 | 2.299 | 0.490 | 0.491 | 0.491 | 0.493 | 0.492 | 152.1 | 227.5 | 348.6 | 223.3 | 271.9 |

Plug | 2.438 | 2.332 | 1.971 | 2.356 | 2.235 | 0.487 | 0.488 | 0.491 | 0.493 | 0.489 | 260.5 | 178.0 | 338.5 | 247.6 | 162.1 | |

Womersley | 2.511 | 2.398 | 2.080 | 2.426 | 2.244 | 0.485 | 0.491 | 0.489 | 0.492 | 0.482 | 185.3 | 387.2 | 836.6 | 290.3 | 268.0 | |

7A | Parabolic | 3.467 | 3.348 | 2.790 | 3.249 | 3.149 | 0.495 | 0.493 | 0.497 | 0.494 | 0.496 | 249.6 | 200.0 | 426.6 | 202.7 | 476.4 |

Plug | 3.440 | 3.363 | 2.814 | 3.193 | 3.151 | 0.489 | 0.488 | 0.488 | 0.488 | 0.490 | 203.0 | 203.2 | 203.5 | 190.4 | 262.3 | |

Womersley | 3.393 | 3.328 | 2.818 | 3.095 | 3.099 | 0.491 | 0.487 | 0.491 | 0.490 | 0.489 | 186.4 | 180.8 | 217.6 | 412.7 | 166.6 | |

14B | Parabolic | 4.098 | 4.010 | 3.227 | 3.888 | 3.600 | 0.497 | 0.495 | 0.492 | 0.491 | 0.495 | 419.9 | 326.4 | 523.0 | 392.8 | 596.1 |

Plug | 3.998 | 3.945 | 3.097 | 3.783 | 3.498 | 0.488 | 0.490 | 0.485 | 0.488 | 0.490 | 118.7 | 223.9 | 85.62 | 138.5 | 128.6 | |

Womersley | 4.010 | 3.952 | 3.145 | 3.860 | 3.496 | 0.488 | 0.490 | 0.491 | 0.496 | 0.490 | 124.8 | 141.7 | 389.7 | 454.1 | 199.3 | |

16A | Parabolic | 3.369 | 3.352 | 2.726 | 3.140 | 3.045 | 0.486 | 0.486 | 0.487 | 0.487 | 0.489 | 295.7 | 242.1 | 258.0 | 274.6 | 191.3 |

Plug | 3.409 | 3.362 | 2.766 | 3.180 | 3.076 | 0.494 | 0.488 | 0.491 | 0.492 | 0.484 | 701.2 | 253.6 | 516.4 | 579.2 | 218.0 | |

Womersley | 3.447 | 3.350 | 2.813 | 3.305 | 3.021 | 0.484 | 0.487 | 0.488 | 0.487 | 0.485 | 231.4 | 244.3 | 279.8 | 120.9 | 477.8 | |

31A | Parabolic | 2.598 | 2.560 | 2.029 | 2.380 | 2.439 | 0.494 | 0.495 | 0.494 | 0.494 | 0.496 | 301.9 | 359.8 | 327.8 | 238.3 | 521.6 |

Plug | 2.627 | 2.568 | 2.141 | 2.439 | 2.397 | 0.491 | 0.490 | 0.489 | 0.492 | 0.491 | 150.4 | 146.5 | 165.6 | 270.7 | 181.5 | |

Womersley | 2.585 | 2.545 | 2.109 | 2.425 | 2.384 | 0.492 | 0.492 | 0.490 | 0.491 | 0.492 | 174.8 | 200.6 | 204.0 | 263.4 | 177.5 | |

41B | Parabolic | 2.420 | 2.388 | 2.056 | 2.207 | 2.291 | 0.495 | 0.496 | 0.494 | 0.494 | 0.494 | 284.9 | 257.4 | 273.5 | 441.5 | 200.2 |

Plug | 2.363 | 2.379 | 2.002 | 2.194 | 2.302 | 0.489 | 0.485 | 0.490 | 0.493 | 0.485 | 428.3 | 297.5 | 373.5 | 473.0 | 227.5 | |

Womersley | 2.354 | 2.376 | 1.977 | 2.214 | 2.293 | 0.489 | 0.484 | 0.492 | 0.495 | 0.488 | 247.3 | 203.5 | 319.8 | 703.1 | 276.9 | |

63A | Parabolic | 3.317 | 3.237 | 2.805 | 3.198 | 2.938 | 0.493 | 0.494 | 0.494 | 0.492 | 0.494 | 478.0 | 493.5 | 671.8 | 646.5 | 724.2 |

Plug | 3.232 | 3.155 | 2.698 | 3.115 | 2.919 | 0.493 | 0.491 | 0.491 | 0.491 | 0.492 | 481.5 | 352.8 | 394.8 | 420.5 | 362.5 | |

Womersley | 3.245 | 3.165 | 2.724 | 3.139 | 2.921 | 0.492 | 0.491 | 0.492 | 0.489 | 0.492 | 370.2 | 312.8 | 690.5 | 379.0 | 328.5 |

Minimum Values | TAWSS | OSI | RRT | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cs | CY | HB | N | P | Cs | CY | HB | N | P | Cs | CY | HB | N | P | ||

2B | Parabolic | 0.131 | 0.147 | 0.118 | 0.089 | 0.165 | 0.005 | 0.003 | 0.005 | 0.005 | 0.004 | 0.425 | 0.433 | 0.496 | 0.462 | 0.439 |

Plug | 0.119 | 0.130 | 0.106 | 0.099 | 0.149 | 0.002 | 0.001 | 0.002 | 0.001 | 0.002 | 0.412 | 0.431 | 0.510 | 0.427 | 0.449 | |

Womersley | 0.142 | 0.138 | 0.141 | 0.109 | 0.155 | 0.003 | 0.002 | 0.003 | 0.005 | 0.002 | 0.401 | 0.419 | 0.485 | 0.418 | 0.449 | |

7A | Parabolic | 0.120 | 0.129 | 0.110 | 0.095 | 0.127 | 0.003 | 0.002 | 0.003 | 0.004 | 0.002 | 0.297 | 0.309 | 0.371 | 0.317 | 0.331 |

Plug | 0.084 | 0.101 | 0.083 | 0.065 | 0.098 | 0.002 | 0.001 | 0.002 | 0.005 | 0.001 | 0.300 | 0.308 | 0.367 | 0.323 | 0.330 | |

Womersley | 0.098 | 0.108 | 0.092 | 0.083 | 0.110 | 0.003 | 0.001 | 0.002 | 0.003 | 0.002 | 0.303 | 0.310 | 0.366 | 0.334 | 0.335 | |

14B | Parabolic | 0.113 | 0.129 | 0.102 | 0.078 | 0.143 | 0.003 | 0.003 | 0.004 | 0.005 | 0.001 | 0.256 | 0.261 | 0.344 | 0.286 | 0.293 |

Plug | 0.298 | 0.308 | 0.271 | 0.259 | 0.321 | 0.004 | 0.004 | 0.007 | 0.007 | 0.002 | 0.262 | 0.266 | 0.349 | 0.289 | 0.301 | |

Womersley | 0.313 | 0.319 | 0.274 | 0.256 | 0.336 | 0.005 | 0.005 | 0.005 | 0.007 | 0.002 | 0.261 | 0.266 | 0.347 | 0.288 | 0.302 | |

16A | Parabolic | 0.088 | 0.099 | 0.082 | 0.067 | 0.105 | 0.001 | 0.001 | 0.001 | 0.002 | 0.001 | 0.299 | 0.300 | 0.370 | 0.322 | 0.330 |

Plug | 0.105 | 0.117 | 0.098 | 0.091 | 0.116 | 0.001 | 0.001 | 0.004 | 0.005 | 0.001 | 0.296 | 0.301 | 0.366 | 0.318 | 0.328 | |

Womersley | 0.115 | 0.126 | 0.105 | 0.089 | 0.131 | 0.007 | 0.001 | 0.001 | 0.001 | 0.001 | 0.292 | 0.301 | 0.359 | 0.305 | 0.333 | |

31A | Parabolic | 0.123 | 0.132 | 0.112 | 0.086 | 0.139 | 0.000 | 0.000 | 0.001 | 0.001 | 0.000 | 0.386 | 0.392 | 0.494 | 0.423 | 0.410 |

Plug | 0.140 | 0.149 | 0.132 | 0.110 | 0.162 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.381 | 0.391 | 0.469 | 0.412 | 0.418 | |

Womersley | 0.146 | 0.161 | 0.138 | 0.108 | 0.164 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.390 | 0.395 | 0.478 | 0.418 | 0.421 | |

41B | Parabolic | 0.121 | 0.128 | 0.118 | 0.093 | 0.140 | 0.002 | 0.002 | 0.002 | 0.004 | 0.002 | 0.420 | 0.425 | 0.497 | 0.460 | 0.443 |

Plug | 0.130 | 0.133 | 0.122 | 0.106 | 0.139 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.431 | 0.428 | 0.511 | 0.464 | 0.443 | |

Womersley | 0.135 | 0.138 | 0.125 | 0.106 | 0.144 | 0.003 | 0.003 | 0.003 | 0.004 | 0.002 | 0.432 | 0.428 | 0.515 | 0.459 | 0.444 | |

63A | Parabolic | 0.132 | 0.147 | 0.120 | 0.096 | 0.161 | 0.001 | 0.005 | 0.001 | 0.001 | 0.000 | 0.308 | 0.316 | 0.366 | 0.318 | 0.349 |

Plug | 0.088 | 0.104 | 0.080 | 0.061 | 0.112 | 0.001 | 0.000 | 0.001 | 0.001 | 0.000 | 0.315 | 0.323 | 0.378 | 0.326 | 0.351 | |

Womersley | 0.132 | 0.141 | 0.120 | 0.114 | 0.150 | 0.001 | 0.000 | 0.001 | 0.001 | 0.000 | 0.314 | 0.322 | 0.375 | 0.324 | 0.350 |

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## Share and Cite

**MDPI and ACS Style**

Tzirakis, K.; Kamarianakis, Y.; Kontopodis, N.; Ioannou, C.V.
The Effect of Blood Rheology and Inlet Boundary Conditions on Realistic Abdominal Aortic Aneurysms under Pulsatile Flow Conditions. *Bioengineering* **2023**, *10*, 272.
https://doi.org/10.3390/bioengineering10020272

**AMA Style**

Tzirakis K, Kamarianakis Y, Kontopodis N, Ioannou CV.
The Effect of Blood Rheology and Inlet Boundary Conditions on Realistic Abdominal Aortic Aneurysms under Pulsatile Flow Conditions. *Bioengineering*. 2023; 10(2):272.
https://doi.org/10.3390/bioengineering10020272

**Chicago/Turabian Style**

Tzirakis, Konstantinos, Yiannis Kamarianakis, Nikolaos Kontopodis, and Christos V. Ioannou.
2023. "The Effect of Blood Rheology and Inlet Boundary Conditions on Realistic Abdominal Aortic Aneurysms under Pulsatile Flow Conditions" *Bioengineering* 10, no. 2: 272.
https://doi.org/10.3390/bioengineering10020272