Numerical Analysis of Bifurcation Angles and Branch Patterns in Intracranial Aneurysm Formation

Abstract

Background:
Hemodynamic factors, especially wall shear stress (WSS), are generally thought to play an important role in intracranial aneurysm (IA) formation. IAs frequently occur at bifurcation apices, where the vessels are exposed to the impact of WSS.

Objective:
To elucidate the relationship between bifurcation geometry and WSS for IA formation.

Methods:
Twenty-one bifurcation models varying in branch angles and branch diameters were made with 3-dimensional computer-aided design software. In all models, the value of maximum WSS (WSSMAX), the area of high WSS (AREA), and the magnitude of wall shear force over AREA ($| {{{\vec{F}}_w}} |$) were investigated by the steady-flow simulation of computational fluid dynamics.

Results:
On the basis of statistical analysis, WSSMAX tended to be high when the bifurcation angle and/or branch diameter was small. AREA and $| {{{\vec{F}}_w}} |$ significantly increase as the bifurcation and/or the branch angle became larger.

Conclusion:
The magnitude of WSS strongly correlated with bifurcation geometry. In addition to high WSS, AREA and $| {{{\vec{F}}_w}} |$ were thought to affect IA formation. Observed bifurcation geometry may predict IA formation. Large branch angles and small branch may increase the risk of IA formation.

Full text

RESEARCH—HUMAN—CLINICAL STUDIES
Numerical Analysis of Bifurcation Angles and
Branch Patterns in Intracranial Aneurysm Formation
Tet su o Sasaki, MD∗
Yukinari Kakizawa, MD, PhD∗
Masato Yoshino, PhD‡§
Yasuhiro Fujii, ME¶
Ikumi Yoroi, BE¶
Yozo I chi kaw a, M D ∗
Tetsuyoshi Horiuchi, MD,
PhD∗
Kazuhiro Hongo, MD, PhD∗
∗Department of Neurosurgery, Shin-
shu University School of Medicine, Matsu-
moto, Japan; ‡Institute of Engineering,
Academic Assembly, Shinshu University,
Nagano, Japan; §Institute of Carbon
Science and Technology,I nterdisciplinary
Cluster for Cutting Edge Research,
Shinshu University, Nagano, Japan;
¶Department of Mechanical Systems
Engineering, Shinshu University, Nagano,
Japan
Correspondence:
Tet suo S asa ki, M D,
Department of Neurosurgery,
Shinshu University School of Medicine,
3-1-1 Asahi,
Matsumoto 390–8621, Japan.
E-mail: sasakit@shinshu-u.ac.jp
Received, November 9, 2017.
Accepted, July 24, 2018.
Copyright C
2018 by the
Congress of Neurological Surgeons
BACKGROUND: Hemodynamic factors, especially wall shear stress (WSS), are generally
thought to play an important role in intracranial aneurysm (IA) formation. IAs frequently
occur at bifurcation apices, where the vessels are exposed to the impact of WSS.
OBJECTIVE: To elucidate the relationship between bifurcation geometry and WSS for IA
formation.
METHODS: Twenty-one bifurcation models varying in branch angles and branch
diameters were made with 3-dimensional computer-aided design software. In all models,
the value of maximum WSS (WSSMAX ), the area of high WSS (AREA), and the magnitude of
wall shear force over AREA (|

Fw|)wereinvestigatedbythesteady-owsimulationofcompu-
tational uid dynamics.
RESULTS: On the basis of statistical analysis, WSSMAX tended to be high when the bifur-
cation angle and/or branch diameter was small. AREA and |

Fw|signicantly increase as the
bifurcation and/or the branch angle became larger.
CONCLUSION: The magnitude of WSS strongly correlated with bifurcation geometry. In
addition to high WSS, AREA and |

Fw|were thought to aect IA formation. Observed bifur-
cation geometry may predict IA formation. Large branch angles and small branch may
increase the risk of IA formation.
KEY WORDS: Bifurcation, Computational uid dynamics, Geometry, Intracranial aneurysm, Wall shear stress
Neurosurgery 0:1–9, 2018 DOI:10.1093/neuros/nyy387 www.neurosurgery-online.com
Hemodynamic factors play important
roles in intracranial aneurysm (IA)
formation.1-6IAs frequently occur at
bifurcation apices, where the vessels are exposed
to the impact of wall shear stress (WSS).5-12
Recent studies show that high WSS regulates
vessel endothelium function and causes inflam-
matory reactions in the vessel wall underlying
aneurysm formation and growth.5,6,9,13-16
Some studies have shown that bifurcation
angles or branch diameters affect IA devel-
opment.17-19 Alnæs et al18 used computational
fluid dynamics (CFD) to investigate the impact
of vessel radius and bifurcation angle varia-
tions on pressure and WSS in the complete
ABBREVIATIONS: 3D CAD, 3-dimensional
computer-aided design; AREA, theareaofhigh
WSS; CFD, computational uid dynamics; |

Fw|,
the magnitude of wall shear force over AREA;
IA, intracranial aneurysm; WSS, wall shear stress;
WSSMAX,the value of maximum WSS; WSSG, WSS
gradient
circle of Willis. They found that deviations from
normal anatomy resulted in redistribution of
wall pressures and increased WSS. Although
WSS magnitude likely depends on bifurcation
geometry and may be a leading factor of IA
formation, there are no detailed analyses of the
relationship between bifurcation geometry and
WSS. Therefore, we constructed basic bifur-
cation models with many variations and eluci-
dated how bifurcation geometry influences IA
formation by examining the WSS increase and
distribution using CFD simulations.
METHODS
Geometric Modeling
Many variations of 3-dimensional computer-aided
design (3D CAD) models were made using the
3D CAD engineering software (SolidWorks2009;
Dessault Systèms SolidWorks Corp, Waltham,
Massachusetts) (Figure 1, Left). All models had a
parent vessel (D0) 4 mm in diameter to approximate
major intracranial arteries, where IAs frequently occur.
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SASAKI ET AL
FIGURE 1. Left: An example of the bifurcation model with 3D CAD. The diameter of the parent vessel (D0) is fixed at 4
mm, and the parent vessel divided into small branch (D1) and large branch (D2). Branch angles are represented as φLand
φR, respectively. The bifurcation angle (φL+R) is denoted by the sum of φLand φR. Right: All 21 models with variations of
the bifurcation geometry. Bifurcation angles (φL+R)aresetin5patternsat60
◦,90
◦, 120◦, 150◦, and 180◦. Branch angles
(φLor φR)arevariedby30
◦from 0◦to 90◦.A, The 8 models have equal branches in diameter as the basic variations. Both
branch diameters are 3.175 mm. B, The 13 models have different branches in diameter. The diameter of small branch (D1)
is 1.600 mm, and that of large branch (D2) is 3.913 mm. Each branch diameter is decided by Murray’s law.
Bifurcation angles (φL+R)weresetin5patternsat60
◦,90
◦, 120◦, 150◦,
and 180◦. Branch angles (φLor φR) were varied by 30◦from 0◦to 90◦.
Eight models (type A) had equal-diameter (3.175 mm) branches as the
basic variations (Figure 1, Right-A). Additionally, 13 models (type B)
had different-diameter branches (Figure 1, Right-B). The small-branch
diameter (D1) was 1.600 mm, and the large-branch diameter (D2)was
3.913 mm. Branch diameters were determined according to Murray’s
law,20 which is derived based on the basis of the mass conservation in
the bifurcation. That is, r03=r13+r23,wherer0is the radius of the
parent artery, and r1and r2are the radii of the branching arteries.
Numerical Simulation
The whole domain was divided into tetrahedral elements, and
body-fitting meshes were used near the wall boundaries to perform
accurate WSS calculation. The number of elements used in this study
ranged from 900 000 to 1 150 000. Blood was assumed as an incom-
pressible Newtonian fluid with a density of 1060 kg/m3and viscosity
of 4.24 ×10−3Pa ·s. The vessel wall was considered rigid with a no-
slip condition. A recent study showed that steady-state CFD solution
virtually agrees (<3% WSS difference) with the average pulsatile CFD
solution in animal models.21 Indeed, although pulsatile-flow simula-
tions should be done, our preliminary computations also indicated that
the WSS magnitude trends were captured in steady-flow simulations.
Therefore, steady-flow simulations were conducted for simplicity. At
the inlet boundary, the uniform velocity was set to 0.425 m/s as the
average peak systole and end diastole in the internal carotid artery.22
At the outlet boundary, the flow-rate ratio of each branch was specified
in proportion to the cross-sectional branch area ratio. The calculated
Reynolds number was 425, defined by the uniform inflow velocity and
the parent vessel’s diameter; hence the flow was assumed laminar.23 The
continuity and Navier-Stokes equations for incompressible fluids with
boundary conditions were solved by the commercial software ANSYS
FLUENT 12.1 (ANSYS Inc, Canonsburg, Pennsylvania). The numerical
method was based on the SIMPLE algorithm24 and the second-order
upwind scheme for the convection terms. No turbulent models were
used in computation. Steady-flow computations were repeated until a
convergence criterion that the relative errors of the velocity compo-
nents became <10−5for all grid points. In simulations, WSS magni-
tudes on each geometric model’s boundary were calculated. Additionally,
maximum value of WSS (WSSMAX), area of high WSS (AREA), and
magnitude of wall shear force over AREA (|

Fw|) were investigated. Note
that AREA was defined as the area where WSS magnitude was ≥15 Pa,
using a previously described threshold.25 When AREA was continuous
over both branches, it bisected the bifurcation angle to calculate the
AREAofeachbranch(Figure2, left).

Fwmagnitude was given as follows:




Fw

=
n

i=1



τwi
Ai
where |
τwi|is the magnitude of WSS vector on the boundary surface of
the i-th element, and Aiis the area of the element (i=1,2,..., N).
Briefly, WSS (Pa) and WSSMAX (Pa) are the forces per unit area, which
are applied to one point on the vessel wall, AREA (mm3)istheareaof
the vessel wall under high WSS(≥15 Pa), and |

Fw|(10−6N) is the sum
of WSS over the AREA.
Statistical Analysis
WSSMAX,AREA,and|

Fw|were collected for all models. Each
WSS parameter was compared against φL+Ror either branch angle of
interest (φLor φR) using univariate linear regression analysis. Dependent
variables (WSSMAX, AREA, and |

Fw|) were treated as continuous variables
each to BLand BR, respectively. Independent variables (φL+R,φLand
φR) were treated as continuous variables. Since dependent variables
were treated as continuous variables, univariate linear regression analysis
was used. The Pvalues of the Wald test were described as the test of
univariate analysis. P<0.05 was considered statistically significant in
each test using commercial software JMP 9 (SAS Institute Inc, Cary,
North Carolina). Furthermore, multivariate linear regression analyses
were added as independent variables of φLand φR.φL+Rwas not added as
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BIFURCATION GEOMETRY ASSOCIATED ANEURYSM FORMATION
FIGURE 2. Left: An example of the area of high WSS with ≥15 Pa (AREA). When AREA is continuous over both branches
as this sample, it is divided in bisector of a bifurcation angle to calculate AREA of each branch. Right: The distribution of
wall shear stress (WSS) visualized with color-coded magnitudes in the 3D geometric models. A, The basic models having
equal diameter branches, and B, another models having different diameter branches. Peak WSS is found near the terminus of
bifurcations in each model. Maximum value of WSS (WSSMAX) is shown as each arrow except for symmetrical models (A-1,
A-5, A-8).
an independent variable in multivariate linear regression analyses because
of the sum of φLand φR.
RESULTS
Figure 2(Right) shows WSS visualized with color-coded
magnitudes in the 3D geometric models. Peak WSS was found
near the terminus of bifurcations in each model.
Table 1shows bifurcation geometries and each WSS parameter
for type A models. In the symmetrical models (A-1, A-5, A-8),
WSSMAX was highest in the model with the smallest φL+R(A-
1), while AREA and |

Fw|increased as φL+Rincreased. The site
of WSSMAX shifted distally from the apex as φL+Rincreased.
In asymmetrical models with different branch angles (A-2,
A-3, A-4, A-6, A-7), WSSMAX, AREA, and |

Fw|were higher
with large-branch than with small-branch angles. WSSMAX was
high when φL+Rwas small. There was a negative correlation
between WSSMAX of the interest branch and φL+Rstatistical
significance with univariate linear regression analysis (Table 2).
From multivariate linear regression analysis, association between
WSSMAX of the BLand φL+Rdepended on φR,largerbranch
angle (Table 3). Association between WSSMAX of the BRand
φL+Rtended to depend on φR(Table 3). A positive correlation
was shown between AREA of the interest branch and φL+Ror
the branch angle of the interest branch with univariate linear
regression analysis (Table 2). From multivariate linear regression
analysis, association between AREA of the BLand φL+Rdepended
on φL(Table 3). Association between AREA of the BRand φL+R
depended on both φLand φR(Table 3). There was also a positive
correlation between |

Fw|of the interest branch and φL+Ror
the branch angle of the interest branch with univariate linear
regression analysis (Table 2). From multivariate linear regression
analysis, association between |

Fw|of the BLand φL+Rdepended
on φL(Table 3). Association between |

Fw|of the BRand φL+R
depended on both φLand φR(Table 3). For type A, WSSMAX
was significantly higher when φL+Rwas small or branch angle
was large. AREA and |

Fw|were significantly higher when φL+R
or the branch angle of the interest branch was larger.
For type B, irrespective of branch angles, WSSMAX was high
on small branches when φL+Rwas ≤120◦(except for B-9) and
on large branches when φL+Rwas ≥150◦.AREAand|

Fw|
were greater for large branches in models having equal branch
angles (B-1, B-8, B-13) and when φL+Rwas ≥150◦(B-11, B-12);
these indices were greater for large branch angles in other models
(Table 4). There was a negative correlation between WSSMAX of
the small branch (BL)andφL+Rwith univariate linear regression
analysis (Table 5). From multivariate linear regression analysis,
association between WSSMAX of the BLand φL+Rdepended on
both φLand φR(Table 6). Association between WSSMAX of the
BRand φL+Rdepended on φL(Table 6). Irrespective of branch
diameter, there was a positive correlation between AREA and
φL+Rwith univariate linear regression analysis (Table 5). The
relationship between AREA and φLwas not observed, while
there was a positive correlation between AREA and φRwith
univariate linear regression analysis (Table 5). From multivariate
linear regression analysis, association between AREA of the BL
and φL+Rdepended on both φLand φR(Table 6). Association
between AREA of the BRand φL+Rdepended on φR(Table 6).
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SASAKI ET AL
TAB LE 1. The Data of Bifurcation Geometries and Hemodynamic Parameters in Type A Models
φL+R(◦)φL(◦)φR(◦)WSS
MAX (Pa) AREA (mm2)|

Fw|(10−6N)
BLBRBLBRBLBR
A-1 60 30 30 42.1 43.9 3.52 3.53 85.8 85.7
A-2 60 0 60 36.5 43.1 2.32 4.78 55.1 114
A-3 90 30 60 26.5 27.6 5.76 7.54 114 151
A-4 90 0 90 23.8 27.4 3.74 10.6 70.8 215
A-5 120 60 60 21.5 21.5 10.2 10.2 183 183
A-6 120 30 90 20.8 22.5 6.30 13.5 109 254
A-7 150 60 90 18.0 20.9 9.17 15.0 150 264
A-8 180 90 90 24.3 24.3 15.6 15.6 285 285
φL+R: bifurcation angle, φL: angle of the left branch, φR: angle of the right branch, BL: the left branch, BR: the right branch, WSS: wall shear stress, WSSMAX:thevalueofmaximum
WSS, AREA: the area of high WSS (≥15 Pa), |

Fw|: wall shear force of over the area of high WSS (≥15 Pa).
TAB LE 2. Statistical Analysis for Testing Correlation Between Bifurcation Geometries and Hemodynamic Parameters in Type A Models;
Regression Analysis by Univariate Linear Regression Model
φL+RφLφR
Coecient (95% CI) P-value Coecient (95% CI) P-value Coecient (95% CI) P-value
WSSMAX BL–0.146 (–0.252, 30.344) 3.518E-02∗–0.114 (–0.308, 21.737) 2.951E-01 –0.303 (–0.477, 35.415) 1.404E-02∗
BR–0.175 (–0.283, 35.369) 1.956E-02∗–0.166 (–0.367, 25.582) 1.571E-01 –0.305 (–0.535, 33.638) 4.007E-02∗
AREA BL0.097 (0.067, –6.978) 7.443E-04∗0.135 (0.100, 0.373) 2.637E-04∗0.087 (–0.055, –9.613) 2.765E-01
BR0.100 (0.069, –4.384) 7.129E-04∗0.093 (0.002, 2.332) 9.159E-02 0.180 (0.104, –8.355) 3.572E-03∗
|

Fw|BL1.557 (0.889, –115.032) 3.812E-03∗2.284 (1.687, 17.593) 2.920E-04∗1.155 (–1.356, –137.079) 4.020E-01
BR1.563 (0.971, –44.429) 2.059E-03∗1.334 (–0.216, 70.447) 1.426E-01 3.014 (1.997, –96.313) 1.142E-03∗
φL+R: bifurcation angle, φL: angle of the left branch, φR: angle of the right branch, BL: the left branch, BR: the right branch, CI: condence interval, WSS: wall shear stress, WSSMAX:the
value of maximum WSS, AREA: the area of high WSS (≥15 Pa), |

Fw|: wall shear force of over the area of high WSS (≥15 Pa).
∗Statistically signicant.
A similar tendency was shown in the relationship between |

Fw|
and φL+Ror branch angles. For type B, WSSMAX of the small
branch was significantly higher when φL+Rwas small. AREA and
|

Fw|significantly correlated with φL+Rand the angle of the large
branch.
Our results suggest that (1) WSSMAX tended to be high when
bifurcation angle and/or branch diameter was small; and (2)
AREA and |

Fw|were significantly increased as bifurcation and/or
branch angle increased.
DISCUSSION
Common risk factors for IA formation such as hypertension,
smoking, familial predisposition, and hemodynamic stress have
been identified.5Hemodynamic factors are generally recognized
to play an important role on IA formation.1-6IAs frequently occur
in the circle of Willis, and in particular at apices of arterial bifur-
cations or at the branching points of a parent artery, where the
vessels are exposed to the impact of WSS.5-12 Hashimoto et al1
demonstrated that increased flow and systemic hypertension are
required to create experimental IAs in rats. Observations from
animal models showed that elevations of WSS caused alterations
in endothelial phenotype, endothelial damage, and fragmentation
of the internal elastic lamina.2-4,8,10,11 ,26 Meng et al8reported
histopathological and hemodynamic analysis using IA models
in dogs, in which aneurysmal initiation was observed at the
site of high WSS and high WSS gradient (WSSG). Kulcsár et
al27 analyzed CFD for 3 human-specific models in which IAs
occurred, and demonstrated that both WSS and WSSG increased
at the regions where IAs developed. Moreover, Alfano et al12
indicated that high WSS and high WSSG were found at bifur-
cations where IAs frequently occur. Accordingly, many studies
support that high WSS is associated with the first stage in IA
formation.5-12,26 ,27
The present study also showed that WSSMAX tended to be
high when a branch diameter was small as the previous reports.18
However, the observation suggested that WSSMAX was high when
a bifurcation angle was small in the present study although the
previous studies have shown that large branch angle was a risk
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BIFURCATION GEOMETRY ASSOCIATED ANEURYSM FORMATION
TAB LE 3. Statistical Analysis for Testing Correlation Between Bifurcation Geometries and Hemodynamic Parameters in Type A Models;
Regression Analysis by Multivariate Linear Regression Model
Dependent variables Independent variables
φLφR
Coecient (95% CI) P-value Coecient (95% CI) P-value
WSSMAX BL–0.067 (–0.194, –0.282) 3.531E-01 –0.282 (–0.460, –0.282) 2.661E-02∗
BR–0.121 (–0.273, –0.266) 1.783E-01 –0.266 (–0.478, –0.266) 5.672E-02
AREA BL0.127 (0.101, 0.045) 2.270E-04∗0.045 (0.009, 0.045) 6.020E-02
BR0.066 (0.046, 0.158) 1.463E-03∗0.158 (0.130, 0.158) 1.148E-04∗
|

Fw|BL2.211 (1.598, 0.442) 8.742E-04∗0.442 (–0.411, 0.442) 3.563E-01
BR0.879 (0.588, 2.731) 1.964E-03∗2.731 (2.326, 2.731) 4.422E-05∗
φL: angle of the left branch, φR: angle of the right branch, BL: the left branch, BR: the right branch, WSS: wall shear stress, WSSMAX: the value of maximum WSS, AREA: the area of high
WSS (≥15 Pa), |

Fw|: wallshear force of over the area of high WSS (≥15 Pa).
∗Statistically signicant.
TAB LE 4. The Data of Bifurcation Geometries and Hemodynamic Parameters in Type B Models
φL+R(◦)φL(◦)φR(◦) Branch diameter (mm) WSSMAX (Pa) AREA (mm2)|

Fw|(10−6N)
BL(D1)B
R(D2)B
LBRBLBRBLBR
B-1 60 30 30 1.600 3.913 55.1 27.0 2.21 2.96 58.3 59.5
B-2 60 0 60 1.600 3.913 58.2 32.7 2.11 5.61 55.8 113
B-3 60 60 0 1.600 3.913 55.7 21.7 2.15 1.08 54.5 19.6
B-4 90 30 60 1.600 3.913 32.1 24.5 4.63 10.3 97.3 190
B-5 90 60 30 1.600 3.913 29.2 18.7 4.05 3.46 82.6 57.5
B-6 90 0 90 1.600 3.913 29.2 24.1 3.91 21.3 79.3 392
B-7 90 90 01.600 3.913 29.3 13.5 3.52 070.5 0
B-8 120 60 60 1.600 3.913 22.7 17.5 5.56 8.75 100 142
B-9 120 30 90 1.600 3.913 21.9 23.7 6.53 23.9 115 436
B-10 120 90 30 1.600 3.913 22.1 15.1 5.31 0.023 97 0.348
B-11 150 60 90 1.600 3.913 20.0 24.2 7.99 25.6 132 459
B-12 150 90 60 1.600 3.913 20.1 22.3 10.0 15.7 172 274
B-13 180 90 90 1.600 3.913 21.5 26.8 11.7 25.5 199 473
φL+R: bifurcation angle, φL: angle of the left branch, φR: angle of the right branch, BL: the left branch, BR: the right branch, D1: small branch diameter, D2: large branch diameter, WSS:
wall shear stress, WSSMAX: the value of maximum WSS, AREA: the area of high WSS (≥15 Pa), |

Fw|: wall shear force of over the area of high WSS (≥15 Pa).
factor of IA formation.2,17,19 This paradoxical result may be
explained by the following hypotheses: (1) actual cerebral arteries,
particularly in the circle of Willis, hardly have sharp bifurca-
tions17; (2) other hemodynamic parameters except for WSSMAX
may also affect IA formation. Mean arterial WSS in the straight
segments of large arteries is recognized to be within the range
of 1.5 to 2.0 Pa.5,13,27 Although peak of WSS was observed
near the terminus of bifurcations, the range of WSSMAX by
changes of bifurcation angles was not so large in the present study.
In contrast, AREA and |

Fw|were greater as bifurcation and/or
branch angle became larger with strong correlation. Conse-
quently, speculation would suggest that AREA and |

Fw|affect IA
formation as well as high WSS because a risk of IA formation
seems to be higher by exposure of high WSS consistently and
widely. AREA and |

Fw|can be 2 of the factors to support the
clinical observation that large bifurcation angle is a risk of IA
formation. On the other hand, in type B models having different
branches in diameter, WSSMAX tended to be higher on small
branch by a correlation analysis, whereas there was no correlation
between a branch diameter and AREA or |

Fw|. These observa-
tions might be brought by the difference of the area of high
velocity gradient near the vessel wall between different branches
in diameter. That is, in a part of type B models, AREA of a large
branch would be greater than that of a small branch because the
area of high velocity gradient near the vessel wall in a large branch
was greater than that in a small branch. We thought that further
studies to investigate the relationships between WSS and a branch
diameter would be needed, using additional models having varia-
tions of branch diameters. The present study suggested that a
small branch would be a risk factor of IA formation because
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SASAKI ET AL
TAB LE 5. Statistical Analysis for Testing Correlation Between Bifurcation Geometries and Hemodynamic Parameters in Type B Models;
Regression Analysis by Univariate Linear Regression Model
φL+RφLφR
Coecient (95% CI) P-value Coecient (95% CI) P-value Coecient (95% CI) P-value
WSSMAX BL(D1) –0.316 (–0.440, 51.699) 4.055E-04∗–0.221 (–0.445, 29.925) 8.064E-02 –0.204 (–0.434, 28.715) 1.101E-01
BR(D2) –0.015 (–0.096, 14.973) 7.185E-01 –0.102 (–0.175, 23.362) 1.912E-02∗0.081 (0.000, 13.104) 7.576E-02
AREA BL(D1)0.077 (0.066, –4.024) 2.049E-08∗0.049 (0.004, –0.085) 5.800E-02 0.054 (0.011, –0.223) 3.338E-02∗
BR(D2) 0.181 (0.067, –20.952) 9.949E-03∗–0.038 (–0.218, 2.021) 6.839E-01 0.282 (0.210, –8.310) 9.694E-06∗
|

Fw|BL(D1) 1.111 (0.900, –40.718) 5.526E-07∗0.701 (0.016, 21.590) 7.004E-02 0.793 (0.145, 18.915) 3.536E-02∗
BR(D2) 3.229 (1.109, –379.451) 1.240E-02∗–0.803 (–4.075, 41.901) 6.402E-01 5.142 (3.839, –152.123) 9.010E-06∗
φL+R: bifurcation angle, φL: angle of the left branch, φR: angle of the right branch, BL: the left branch, BR: the right branch, D1: small branch diameter, D2: large branch diameter, WSS:
wall shear stress, WSSMAX: the value of maximum WSS, AREA: the area of high WSS (≥15 Pa), |

Fw|: wall shear force of over the area of high WSS (≥15 Pa).
∗Statistically signicant.
TAB LE 6. Statistical Analysis for Testing Correlation Between Bifurcation Geometries and Hemodynamic Parameters in Type B Models;
Regression Analysis by Multivariate Linear Regression Model
Dependent variables Independent variables
φLφR
Coecient (95% CI) P-value Coecient (95% CI) P-value
WSSMAX BL–0.322 (–0.481, –0.310) 2.682E-03∗–0.310 (–0.469, –0.310) 3.442E-03∗
BR–0.084 (–0.158, 0.054) 4.878E-02∗0.054 (–0.020, 0.054) 1.834E-01
AREA BL0.075 (0.062, 0.079) 7.361E-07∗0.079 (0.065, 0.079) 4.717E-07∗
BR0.061 (–0.010, 0.302) 1.236E-01 0.302 (0.231, 0.302) 7.836E-06∗
|

Fw|BL1.077 (0.807, 1.146) 1.435E-05∗1.146 (0.876, 1.146) 8.276E-06∗
BR0.990 (–0.320, 5.467) 1.693E-01 5.467 (4.157, 5.467) 9.713E-06∗
φL: angle of the left branch, φR: angle of the right branch, BL: the left branch, BR: the right branch, WSS: wall shear stress, WSSMAX: the value of maximum WSS, AREA: the area of high
WSS (≥15 Pa), |

Fw|: wall shear force of over the area of high WSS (≥15 Pa).
∗Statistically signicant.
FIGURE 3. Computational tomography angiogram in the patient of 71-yr-
old man showing the right middle cerebral artery unruptured aneurysm. The
aneurysmal neck distributes from the bifurcation apex to the small branch
having larger branch angle.
statistical significance was shown between elevation of WSSMAX
and small branch. Actually, aneurysmal necks often ride the side of
small branch at bifurcation, such as the middle cerebral artery and
the posterior communicating artery (Figures 3and 4). Therefore,
care should be taken of bifurcation geometry to avoid recur-
rence in aneurysmal clipping, such as obliteration of aneurysmal
neck, especially of the side of small branch and addition of
wrapping distally to bifurcation apices. Tight packing for the area
of high WSS which occurs to aneurysmal orifice after aneurysmal
obliteration is recommended in endovascular coiling for cerebral
aneurysms.
Recently, although other hemodynamic parameters
contributing to IA formation have been proposed including
WSSG,8,9,12,27 oscillatory shear index,25 ,28 aneurysm formation
indicator,29 and gradient oscillatory number,30 these indices are
short of evidences compared with WSS. However, WSSG has
been considered to be one of leading factors in IA formation,
and the research on relationship between WSSG and bifurcation
geometry should be our future subject.
Although a number of CFD studies were analyzed using the
realistic vessel models created by angiography of patients or
healthy volunteers, we considered the following problems of
CFD simulations to investigate the relationship between bifur-
cation geometry and WSS using the patient-specific models: (1)
it is complicated to produce the models varied bifurcation angle
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BIFURCATION GEOMETRY ASSOCIATED ANEURYSM FORMATION
FIGURE 4. A patient-based model from computational tomography
angiogram in the case (58-yr-old woman) of the left internal carotid artery.
A, Unruptured aneurysm. B, An aneurysm removal model. C, A steady-
flow simulation model for WSS. Numerical analysis was conducted under
the same conditions as the present study. High WSS was observed from the
apex of the bifurcation to the posterior communicating artery (arrow).
or branch diameter; (2) measurement errors between imaging
modalities in modeling can occur.31 In contrast, exact adjustment
of angles and diameters is possible in simple models as the present
study, and production of many models is also easy. Moreover,
with simple models used, comparison of hemodynamic indices
between each model should be advantageous, and numerical
reproducibility can be high.
Recent studies have disclosed that high WSS regulates the
functions of the vessel endothelium and it causes inflammatory
reactions in the vessel wall underlying aneurysm formation
and growth.5,6,9,13-16 Furthermore, the medicine with an anti-
inflammatory effect is thought to have a possibility of cure for
IAs. Aoki et al32,33 demonstrated that statins could inhibit the
progression of IAs in animal models. The present study suggests
that high and regional WSS was shown when a bifurcation angle
was small and when a branch diameter was small. In contrast, the
area exposed to high WSS was greater as a branch angle became
larger. By getting to know characteristic vessel geometries which
have a potential risk of IA formation, intervention of preventive
medication as well as close follow-up may be recommended for
such cases. Furthermore, in cases where the vessels have risky
bifurcation geometries, careful follow-up should be done and it
may be considered to perform wrapping if there is an oppor-
tunity of direct observation in craniotomy. Although the simula-
tions using the patient-specific models seem to be a better way
when investigating a risk of IA formation, it generally requires
complicated processes to calculate the hemodynamic parameters.
Actually, we think that it is simple and practical to make bifur-
cation geometry into an indicator of IA formation.
Limitations
There are several limitations in the present study. The first
limitation is the difference of bifurcation geometry between
simple models and human vessels. We herein designed the bifur-
cation geometry only including branches in a 2D plane because
of very complicated analyses in a 3D model. Actual bifurcations
of the cerebral arteries have complex structures such as tortuous
vessels, irregular vessel diameters, and others. Additionally, to
include the vessel elasticity into CFD simulations is techni-
cally difficult. Furthermore, boundary conditions are changed
by a range of vessel length or flow rate,34,35 so it is difficult
to measure the values of hemodynamic parameters correctly.
Therefore, the results of the present study can not necessarily
suit human vessels. In future work, further investigations using
more complicated geometry models and many patient-specific
models are required to verify the relationship between these
models and clinical IA formation. The second limitation is that
similar hemodynamic change is not necessarily observed in side-
wall aneurysms occurring at the nonbranching site. In side-wall
aneurysms, it is unclear whether WSS in the site of IA formation
is high or low.27,30 IA formation has a multifactorial etiology,
so other factors may affect side-wall aneurysms although WSS is
a strong candidate of IA formation.7Hemodynamic analysis of
nonbranching vessels remains as a future subject. Many limita-
tions still remain in CFD studies, whereas by the development of
computer technology and biorheology, it is expected that those
problems will be solved in the near future.
CONCLUSION
The magnitude of WSS strongly correlated with bifurcation
geometry. The present study suggested that high and regional
WSS was shown when a bifurcation angle was smaller and when
a branch diameter was small. In contrast, the area exposed to
high WSS was greater as bifurcation and/or branch angle became
larger. In addition to high WSS, the area of high WSS and
the magnitude of wall shear force over the area were thought
to affect IA formation. Observed bifurcation geometry would
be a predictor for IA formation. Large branch angles and small
branches can be a potential risk factor of IA formation.
Disclosures
The corresponding author was supported by Grants-in-Aid for Young Scien-
tists (B); grant number 20791003 from the Ministry of Education, Culture,
Sports, Science and Technology of Japan. The authors have no personal, financial,
or institutional interest in any of the drugs, materials, or devices described in this
article.
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COMMENTS
In the present computational fluid dynamic analysis of morphologi-
cally varying arterial bifurcations, the authors hypothesize that vessel
wall shear stress (WSS) is correlated with different branch point geome-
tries. In a well-designed and executed assessment, they conclude that both
smaller bifurcation angles and smaller branch diameters correlate with
arterial vessel WSS.
As with any computational model, the authors make several simplistic
assumptions, which must be fully considered prior to extrapolating the
results to human physiology. First, the study assesses vessel bifurcation in
only 2 dimensions (x and y), and second, the authors implement a steady-
flow simulation, rather than a physiological pulsatile-flow simulation.
While necessary to simplify computational modeling, these assumptions
do not completely represent normal physiology. As mentioned in the
manuscript, the authors acknowledge these assumptions, but it should
be recognized that these results might not hold true with more physio-
logically accurate modeling.
Regardless, with these limitations considered, this CFD study allows
us to hypothesize further about the relationship between artery branch
point geometry and aneurysm formation. Further advanced fluid
dynamic testing is certainly necessary, but based on these data, we can
better predict arterial anatomy at high risk of aneurysm formation, and
likely rupture.
Kurt Yaeger
J Mocco
New York, New York
The authors present a well-written manuscript describing how various
changes in bifurcation angles and branch patterns may affect
the hemodynamic forces that eventually lead to intracranial aneurysm
formation. To do so, the authors created 3D printed models with
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different branch angles and vessel diameters, and employed computa-
tional flow dynamic calculations in order to determine how changes
in bifurcation angles and branch vessel diameter affects the magnitude
of wall shear stress on blood vessels, a potential surrogate of aneurysm
formation. The authors conclude that maximal wall shear stress was
greatest when bifurcation angles were small and branch diameters were
smaller. Calculations of AREA (wall shear stress) and |

Fw|(magnitude
of wall shear stress force over area) were significantly increased as bifur-
cation and/or branch angle increased. As such, the authors conclude
that observed bifurcation geometry may be a predictor for aneurysm
formation.
Josh Osbun
Ralph Dacey
St. Louis, Missouri
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