# Hemodynamic Effect of Unequal Anterior Cerebral Artery Flow

Authors: Thomas Rau; Xing He; Prem Venugopal; Fernando Viñuela; Gary Duckwiler; Daniel J. Valentino
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

Computational fluid dynamics techniques were used to investigate the hemodynamic effect
of unequal anterior cerebral artery flow rates on the anterior cerebral and anterior communicating
artery (ACA-ACOM) bifurcation. Hemodynamics have long been implicated as a major
factor in cerebrovascular disease. Using an idealized 2D symmetric model of the ACA-ACOM
geometry, the flow field and wall shear stress (WSS) at the bifurcation regions are assessed
for pulsatile inflows with left to right flow ratios of 1:1, 2:1, 3:1, and 4:1. Unequal flow rates
through the ACA parent arteries result in bifurcation of the higher flow parent stream and a
shifting of the impingement points along the A2-ACOM adjoining wall toward the contralateral ACA. Cross-flow through the ACOM is generally unstable and results in increased WSS
at the impingement region from the higher flow parent artery and a double amplitude peak
in the WSS at the contralateral bifurcation region from local recirculation effects. These results
suggest that asymmetry in ACA flow rates result in increased hemodynamic stresses at the ACA-ACOM bifurcation regions and suggest a possible factor for vessel weakening and
aneurysm formation.


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## Body

## 1. Introduction

Numerous
investigations on the hemodynamics in the Circle of Willis have been performed to elucidate the pathogenesis of aneurysm formation. It is known that over 85%
of aneurysms occur in the anterior circulation and form predominantly at the
apex of vessel bifurcations and around sharp bends. It is at these locations
where hemodynamic stresses are the greatest and the structural protein assembly
of the muscular layer of the vessel wall is most underdeveloped [1]. A generally accepted theory
is that an anomalous response of the vascular endothelium to oscillating wall
shear stresses leads to progressive and unstable degradation of the arterial
wall and consequent aneurysm formation [2]. The validity and reliability of computational methods
to model the hemodynamic processes has motivated numerous computational flow
studies to characterize the flow patterns and wall shear stresses associated
with pulsatile flows in these susceptible regions [3–9].The junction of the anterior cerebral (ACA) and
anterior communicating (ACOM)  arteries represents a unique and complex vascular
geometry with two closely apposed bifurcations (see Figure1). Studies in the past have shown
that a strong association exists between the development of aneurysms and their
neck sizes at the ACOM junction with the dominant A1 parent artery feeder
[10]. In this study, a
computational fluid dynamics (CFD) simulation was performed on an idealized 2D
symmetric model of the ACA and ACOM bifurcation to investigate the hemodynamic
effect of unequal flow rates through the A1 ACA segments. Simulations were run
with left to right inflow ratios fixed at 1 : 1, 2 : 1, 3 : 1, and 4 : 1. The overall
flow patterns as well as the wall shear stress (WSS) at various points along
the adjoining wall between the A2 and ACOM segments are analyzed spatially and
temporally near the impingement points where aneurysms are predicted to occur.
Unequal left-right A1 flow rates can occur whenever the total upstream flow
resistances are not equal bilaterally. Clinically, this may occur with
unilateral high grade stenotic lesions in upstream vessels to the Circle of
Willis. This might be seen after a thromboembolic event in stroke victims and
succeeding vasospasm in upstream arteries or the existence of a congenital
hypoplastic A1-ACA. Asymmetric flow rates between the A1 segments can also
occur transiently by iatrogenic means during a rapid intravenous infusion of
contrast media during diagnostic cerebral angiography. This study attempts to
characterize some hemodynamic stress pattern differences which may occur under
these circumstances.Figure 1
A 3D model depiction of the anterior cerebral
circulation upon which the simulation is based.


## 2. Methods

A 2D symmetric
model of the ACA and ACOM was constructed using the Solidworks [11] software seen in Figure 2.
The diameter was assigned a unit length of 1, and all other dimensions were
scaled accordingly and proportioned from other studies [12, 13]. The bifurcation angle from
the A1 to A2 segments of the anterior cerebral arteries was chosen to be 45
degrees, within the range of clinical models. The radii of curvature adjoining
the walls of the A1 to the A2 and ACOM segments were chosen such that a smooth
transition was made. The model was then imported into FLUENT's Gambit for
meshing. A finer-graded mesh was used at the boundary layer along all wall
surfaces to increase the resolution of the gradients near the wall. A total of
196 944 data points were obtained after meshing.(a) ACA-ACOM geometry with 45 degree
bifurcation angles. (b) Pulse waveform given by (3). The global maximum
occurring at 0.45 t/T will define the transition between the systolic and
diastolic phases of the flow waveform.
(a)(b)Substitution of the following nondimensional
expressions for velocityu and v,
length x and y,
time t,
pressure P,
and Reynolds number Re allows us to
undimensionalize the governing equations of continuity
and Navier-Stokes in the Cartesian coordinate system. We divide the flow
variables with the corresponding references variables denoted by subscript ref to produce the nondimensional variables
denoted by *. Let ρ be the density of blood,
μ the dynamic viscosity of blood, uref be the reference velocity, and Lref the reference length,
where
(1)u*=uuref,v*=vuref,x*=xLref,y*=yLref,t*=turefLref,P*=Pρuref2,Re=ρurefLrefμ.The equations of continuity and Navier-Stokes become
[14] (2)∂u*∂x*+∂v*∂y*=0,∂u*∂t*+u*∂u*∂x*+v*∂u*∂y*=−∂p*∂x*+1Re(∂2u*∂x*2+∂2u*∂y*2),∂v*∂t*+u*∂v*∂x*+v*∂v*∂y*=−∂p*∂y*+1Re(∂2v*∂x*2+∂2v*∂y*2).The usage of the governing flow equations in this form
eliminates dependence on absolute dimensions and makes the simulation dependent
only on the Reynold's number. The results obtained can, therefore, be more
generalized. Incompressible, Newtonian fluids were
assumed in the calculations of the nondimensional Navier-Stokes equations, and
the solutions were performed using OpenFOAM software [15]. A minimum Reynold's number
of 400 was set by allowing fluid density to be
equal 1, scaling the maximum of the velocity profile to 1, and setting fluid
viscosity to 0.0025. The Reynold's number was allowed to vary up to 800 by the
nature of the pulsatile velocity profile at the 2 inlets. Two-plug flow inlets
were assigned at the parent anterior cerebral arteries with the pulse arterial
waveform defined by (3). A no-slip and nonelastic condition was imposed on the
walls and the outflow boundary was set at a reference pressure of 0. The
simulations were allowed to run for three to four complete cycles at 60 beats
per minute to achieve convergence of solutions.Simulations were run with the ratio of the inflow
rates between the left and right A1 segments set at 1 : 1, 2 : 1, 3 : 1, and 4 : 1.
This was conducted by decreasing the time-dependent amplitude of the right A1
inflow segment by the appropriate fraction (0.5, 0.33, and 0.25).The velocity,U,
pulse profile used in these simulations was given by(3)U(t)=8.419*{0.08225+2(−0.013077cos(wt)−0.002868sin(wt))+2(0.001125cos(2wt)−0.001625sin(2wt))+2(−0.000172cos(3wt)+0.001631sin(3wt))+2(−0.001000cos(4wt))}, where w=2π/T, T=1 second.The transition point between the systolic and
diastolic phases of the velocity wave form will be defined at the global
maximum occurring at 0.45t/T.The wall shear stress (WSS) is obtained with the
following equation:(4)(WxWy)=μ(2∂u∂x∂u∂y+∂v∂x∂u∂y+∂v∂x2∂v∂y)(nxny),where nx and ny are the x and y components of the normal vector of vessel
wall.


## 3. Results

The relative inflow rates through the two A1 parent
artery segments greatly affect the overall flow pattern and the hemodynamic
stresses on the walls at the bifurcation region. When the left-right A1 inflow
rates are unequal, bifurcation of the higher velocity parent stream occurs and
the ACOM becomes a functional anastomosis. While flow through the A1 and A2 ACA
segments is generally laminar, unsteadiness is seen in the ACA-ACOM bifurcation
section especially during the deceleration phase of pulsation. Figures4, 5, and 6
show the bifurcation of the higher flow parent artery streams. The flow through
the junction appears more stabilized during the acceleration phase of the pulse
cycle.For equal left-right A1 flow rates, separation and
vortex shedding occur at the ACOM junction as the
streams bypass the anastomosis into the A2 segment. Thus, while the ACOM is
rendered a dead zone, a local unsteady region is created near the impingement
site resulting in irregular, low amplitude WSS patterns at the A2-ACOM
adjoining wall. Figure3 shows the bilateral oblique impingement zones of the
parent A1 segments onto the distal A2-ACOM adjoining wall. The WSS plots in
Figures 8 and 12 demonstrate smaller amplitude WSS and gradients compared to
the unequal ACA flow rate cases.Velocity field with equal left and right ACA
flow rates (QR=QL) during (a) systole and (b) early diastole. A
dead zone in the ACOM exists throughout the pulsation cycle. (b) Unsteadiness
exists bilaterally at the A2-ACOM impingement sites with flow separation and
vortices seen during the diastolic phase.
(a)(b)(a) Velocity field during systole and (b)
early diastole with ACA flow rates set atQR=0.5QL.
Left ACA impingement site moves rightward along the A2-ACOM adjoining wall as
compared to the case where QR=QL with bifurcation of flow into the A2-ACA and
ACOM segments. (b) A large vortex is generated on
the right bifurcation site during diastole.
(a)(b)(a) Velocity field during systole and (b)
early diastole with ACA flow rates set atQR=0.33QL.
Again bifurcation of the A1 parent flow artery into the ACOM and A2 segments is
seen. Left ACA impingement site moves further rightward along A2-ACOM adjoining
wall compared to that seen in the case where QR=0.5QL.
Flow separation is again seen on the right A2-ACOM
impingement site leading to local recirculation effects.
(a)(b)(a) Velocity field during systole and (b)
early diastole with ACA flow rates set atQR=0.25QL.
Bifurcation of the A1 parent flow artery into the ACOM and A2 segments is again
seen. Left ACA impingement site appears to move even further rightward along
A2-ACOM adjoining wall compared to that seen in the case where QR=0.33QL.
Flow separation is again seen on the right A2-ACOM
impingement site leading to local recirculation effects.
(a)(b)As the inflow rate from the right ACA is reduced
relatively to the left, the impingement point of
the left A1 stream moves toward the contralateral bifurcation along the A2-ACOM
adjoining wall segment and is divided. The
impingement point is also observed to oscillate toward the contralateral ACA
segment during the deceleration phase of the pulsation cycle. The majority of
the inflow continues into the A2 segment while the divided fraction enters the
ACOM segment and joins the flow through the right ACA. Flow through the ACOM
anastomosis is subject to separation as it negotiates the curvature of the ACOM
during the deceleration phase of the pulsation cycle. The flow separation results
in a large second peak in time in the WSS plots due to local recirculation
effects. The recirculation zone is most apparent in the case whereQR=0.5QL in Figure 4(b) but can be seen in the
other cases at high magnification views (not shown). The magnitude of the
second peak appears to increase as the ratio of the left to right flow rates is
increased and occurs earlier in the pulsation cycle. Thus, unequal A1 flow
rates impose additional stresses on the ACA-ACOM bifurcation region by the
degree of unbalanced left-right A1 inflow rates.The plots of WSS versus time and position are shown in
Figures8, 9, 10, and 11 for the various left to right flow ratios. The
positions along the left and right impingement
points are denoted by L1-L8 and R1-R8, respectively, and are shown in Figure 7.
The plots for unequal left to right flow rates in Figures 9–11 show the
double amplitude peak which occurs during the diastolic phase which is absent
when the left and right flow rates are equal (see Figure 8). Figure 12 plots
WSS versus time for a characteristic point at the left and right impingement
regions. Clearly shown in this figure is the double amplitude peak occurring
with unequal parent A1 flows and the time between successive peaks as the
imbalance between the A1 flows is greater.Figure 7
Labeled points on
the left and the right impingement points in the WSS plots.WSS versus position and time forQR=QL
on (a) the left
and (b) the right bifurcation regions.
(a)(b)WSS versus position and time forQR=0.5QL
on (a) the left
and (b) the right bifurcation regions.
(a)(b)WSS versus position and time forQR=0.33QL
on (a) the left
and (b) the right bifurcation regions.
(a)(b)WSS versus position and time forQR=0.25QL
on (a) the left
and (b) the right bifurcation regions.
(a)(b)WSS on (a) the
left and (b) the right impingement points.
(a)(b)


## 4. Discussion

The results
demonstrate that the flow behavior and hemodynamic stresses at the ACOM
bifurcation are highly dependent on the relative flow rate ratios between the
A1-ACA parent arteries. Asymmetrical flow conditions at the A1 segments in a
symmetrical geometry result in bifurcation of the higher flow parent artery
into the A2 and ACOM segments. Cross-flow through the ACOM from the higher flow
parent artery is generally unstable at the bifurcation regions especially
during diastole. Pulsatility creates a cyclically varying region of flow
separation and reversal at those locations which
change markedly in amplitude throughout the cycle.
Bifurcation regions, where these instabilities occur, are sites of large
fluctuations in WSS magnitude and are consistent with the location where
aneurysms are predicted to develop.Unequal flow between left-right A1 segments produces a
change in the WSS pattern as compared to symmetrical flow conditions. Both an
increase in WSS amplitude at the impingement point and a double amplitude peak
at the opposite bifurcation region are observed due to local recirculation
effects. The impingement sites of the incoming parent artery flow also
shift toward the apex of the bifurcation region
along the A2-ACOM adjoining wall where the median muscular layer of the
arterial wall is least developed [1]. This leads to greater hemodynamic stresses at
locations that are least able to withstand them and a possible mechanism for
mechanical deformation from cyclic fatigue stresses. Figures 3–6 also
demonstrate that as the impingement site from the dominant A1 shifts toward the
contralateral ACA with unequal A1 flow rates, the flow impinges at a smaller
impingement zone with a more acute angle with respect to the normal of the
impingement surface. In addition, the impingement region tends to oscillate
along the A2-ACOM adjoining wall with shifting toward the contralateral ACA
during systole and vice versa during diastole. By contrast, the case with equal
flow rates demonstrates (see Figure 3) that the
incoming streams hit at a more oblique angle with respect to the impingement
region and oscillate less. Cebral et al.
have found that aneurysms that had smaller and
changing impingement regions and disturbed flow patterns were associated with
rupture. These related findings further strengthen the association of unstable
progression of aneurysm growth with large temporal gradient WSS.Figures4–6 demonstrate large spatial WSS
gradients at the left impingement site where the higher flow parent stream
impinges and bifurcates. Spatial gradients from disturbed laminar shear stress
from regional flow disturbances have been found to be important local
modulators of endothelial cell expression of various growth factors [16]. The plots of the right
bifurcation region as seen in Figures 4–6, and 12 demonstrate that
the unsteady effects of flow separation region in unequal flow rates
are subject to a double peak in the temporal
evolution of WSS. While the magnitude of WSS is decidedly less in the unequal
A1 flow rate condition as compared to equal flow rate conditions, it has been
shown that low-amplitude oscillating shear stresses in unsteady flows
are an important determining factor in endothelial
cell turnover and may be more important than WSS amplitude [17, 18]. Another interesting
observation from Figure 12 is that the frequency between the two peaks
increases as the imbalance between the incoming A1 inflows is greater. The
increased frequency is about 4% of the total pulsation cycle between the 2 : 1
and 3 : 1 case and increases to 8% between the 2 : 1 and 4 : 1 studies. These
differences reflect the time during the pulsation cycle at which flow separation
occurs as the cross-flow navigates through the ACOM segment for the various
imbalances in A1 flow. Thus, unequal A1-ACA flow rates
induce a change in pattern of hemodynamic stresses
which may affect endothelial cells by mechanotransduction in addition to
mechanical deformation stresses.In some cases, flow separation has been shown to be
sufficient to cause activation of platelets and to cause freely floating
aggregates [19–21]. This suggests that certain
regions of unsteady flow may possibly contribute to thromboembolic phenomena.
Thus, the ACA-ACOM junction may serve as key centers of platelet aggregates
which may release constituents and become a source of emboli. Spontaneous
thrombosis and extension of thrombi to bilateral ACAs from an unruptured ACOM
aneurysm have been reported in literature and
demonstrate that hemodynamic phenomenon in
susceptible geometry can lead to cerebrovascular accidents [22].It is interesting to note that the alteration in
hemodynamic stress changes occurs despite the
overall reduced flow rate through the anterior cerebral circulation. Unequal
flow through the ACAs is a situation which may occur from a number of
pathological clinical situations as well as iatrogenic interventions.
Intracranial hemorrhage after carotid stenting is a known fatal complication in
the postprocedure period [23]. Indeed, the hemodynamic implications of endovascular
interventions such as stenotic vessel angioplasty and stenting deserve further
investigation. The findings presented may provide some insight into the
etiology of wall vessel weakening and perhaps assist in clinical management of
patients with ACA-ACOM aneurysms.The current study is concentrated on the effect of
unequal parent ACA flow rates on the hemodynamics of ACOM. By employing a 2D
symmetric model, other geometrical factors could be controlled and quick
simulation results could be obtained. Even though 2D simulations could be
anticipated to reasonably resemble the main characteristics of 3D flows,
significant differences between the 2D and 3D simulations
are also anticipated to appear in cases of complex
geometrical models. The model used in this study employed a high degree of
symmetry in an attempt to minimize such 3D confounding factors. More realistic
patient-specific 3D models are the subject of
future studies.


## 5. Conclusions

Unequal pulsatile flow rates through the A1-ACA
arteries result in bifurcation and unstable cross-flow through the ACOM from
the higher flow parent artery. There is an increase in WSS magnitude at the
impingement site on the same side as the parent A1 segment and a change in
position of the maximum WSS related to the left to right flow ratios. The
increased hemodynamic stress appears to occur where
aneurysms are expected to develop. At the contralateral A2-ACOM adjoining wall,
a double peak in the WSS amplitude occurs from local unsteady effects arising
from flow separation as cross-flow negotiates the ACOM segment. These increased
hemodynamic stresses at the bifurcation regions may be a contributing factor to
vessel wall weakening where aneurysms often occur.



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