Blood flow dynamics and arterial wall interaction in a saccular
aneurysm model of the basilar artery
Alvaro Valencia *, Francisco Solis
Department of Mechanical Engineering, Universidad de Chile, Castilla Santiago, Chile
Abstract
Blood flow dynamics under physiologically realistic pulsatile conditions plays an important role in the growth, rupture and surgical
treatment of intracranial aneurysms. This paper describes the flow dynamics and arterial wall interaction in a representative model of a
terminal aneurysm of the basilar artery, and compares its wall shear stress, pressure, effective stress and wall deformation with those of a
healthy basilar artery. The arterial wall was assumed to be elastic or hyperelastic, isotropic, incompressible and homogeneous. The flow
was assumed to be laminar, Newtonian, and incompressible. The fully coupled fluid and structure models were solved with the finite
elements package ADINA. The intra-aneurysmal pulsatile flow shows single recirculation region during both systole and diastole.
The pressure and shear stress on the aneurysm wall exhibit large temporal and spatial variations. The wall thickness, the Young’s modulus in the elastic wall model and the hyperelastic Mooney–Rivlin wall model affect the aneurysm deformation and effective stress in the
wall especially at systole.
Keywords: CFD; FSI; Intracranial aneurysm; WSS
1. Introduction
Intracranial aneurysms are lesions of the arterial wall
commonly located at branching points of the major arteries
coursing through the subarachnoid space, predominantly
at the circle of Willis in the base of the brain. Aneurysms
of the posterior circulation of the circle of Willis are most
frequently located at the bifurcation of the basilar artery or
at the junction of a vertebral artery and the posterior inferior cerebellar artery. The most common form of intracranial aneurysms is the saccular type [1].
The rupture of an intracranial aneurysm produces a subarachnoid hemorrhage (SAH) associated with high mortality and morbidity rates, as SAH accounts for 1/4 of
cerebrovascular deaths. The rupture rate of asymptomatic
small aneurysms is only 0.05% per annum in patients with
*
Corresponding author. Tel.: +056 2 978 4386; fax: +056 2 698 8453.
E-mail address: [email protected] (A. Valencia).
no prior SAH, but increases to 0.5% for large (>10 mm
diameter) aneurysms [2]. Aneurysms may be treated by surgical clipping or by interventional neuroradiology. Surgical
clipping of asymptomatic aneurysms has morbidity and
mortality rates of 10.9 and 3.8%, respectively, [2]. Treatment of aneurysms by Guglielmi coils carries 4% morbidity
and 1% mortality, but only achieves complete aneurysm
occlusion in 52–78% of the cases. These treatments promote blood coagulation inside the aneurysm avoiding
impinging blood flow on the vessel wall and, thus, impeding its rupture [2].
The pathogenesis and causes for expansion and eventual
rupture of saccular aneurysms are not clear. It is generally
accepted that unique structural features of the cerebral vasculature contribute to the genesis of these aneurysms. A
typical saccular aneurysm has a very thin tunica media
and the internal elastic lamina is absent in most cases [1].
Thus, the arterial wall is composed of only intima and
adventitia. Consequently, hemodynamics factors, such as
blood velocity, wall shear stress, pressure, particle
A. Valencia, F. Solis
Nomenclature
a
ci, D1,
d
E
FSI
f
h
M–R
Re
p
u
Um
Ui
WSS
basilar artery radius
D2 Mooney–Rivlin material constants
basilar artery diameter
Young’s modulus
fluid structure interaction
frequency
arterial thickness
Mooney–Rivlin model
Reynolds number = qUmd/l
pressure
velocity
mean velocity at inlet
translation
wall shear stress
residence time and flow impingement, play important roles
in the pathogenesis of aneurysms and thrombosis. Aneurysm hemodynamics is contingent on the aneurysm geometry and its relation to the parent vessel, its volume and
aspect ratio (depth/neck width) [3]. For this reason, studies
of hemodynamics inside models of saccular aneurysms with
arterial wall interaction are important to obtain quantitative criteria for their treatments.
Liou and Liou [4] presented a review of in vitro studies
of hemodynamics characteristics in terminal and lateral
aneurysm models. They reported that with uneven branch
flow, the flow dynamics inside a terminal aneurysm and
the shear stresses acting on the intra-aneurysmal wall
increase proportionally with the bifurcation angle. Imbesi
and Kerber [5] have used in vitro models to study the
flow dynamics in a wide-necked basilar artery aneurysm.
They investigated the flow after placement of a stent
across the aneurysm neck and after placement of Guglielmi detachable coils inside the aneurysm sac through
the stent.
Lieber et al. [6] used particle image velocimetry to study
the influence of stent design on the flow in a lateral saccular aneurysm model, showing that stents can induce favorable changes in the intra-aneurysmal hemodynamics.
Tateshima et al. [7] studied the intra-aneurysmal flow
dynamics in an acrylic model obtained using three-dimensional computerized tomography angiography. They
showed that the velocity structures were dynamically
altered throughout the cardiac cycle, particularly at the
aneurysm neck.
Although aneurysm rupture is thought to be associated
with a significant change in aneurysm size, there is still
great controversy regarding the size at which rupture
occurs. The relationship between geometric features and
rupture is closely associated with low flow conditions.
The stagnation of blood flow in large aneurysms is commonly observed with the use of cerebral angiography and
is related to intra-aneurysmal thrombosis [8]. Aneurysm
Greek symbols
a
Womersley number = a(2pf/m)1/2
b
angle; tilt of the aneurysm dome with respect to
the parent vessel
d
arterial displacement
e
strain
k
stretch ratio
l
fluid molecular viscosity
m
fluid kinematic viscosity = l/q
q
density
r
stress
s
shear stress
geometry often dictates the success of coil embolization.
Jou et al. [9] presented a compartment model for the estimate of intra-aneurysmal flow and hemodynamic forces
that can be used to predict the outcome of coil embolization or possible aneurysm recurrence.
Foutrakis et al. [10] report on two-dimensional simulations of fluid flow in curved arteries and arterial bifurcations and the relationship to aneurysm formation and
expansion. They suggest that the shear stress and pressure
developed along the outer wall of a curved artery and at the
apex of an arterial bifurcation create a hemodynamics state
that can promote aneurysm formation. Recently, Steinman
et al. [11] presented image-based computational simulations of the flow dynamics in a giant anatomically realistic
human intracranial aneurysm. Their analysis revealed high
speed flow entering the aneurysm at the proximal and distal
ends of the neck, promoting the formation of persistent
and transient vortices within the aneurysm sac. Cebral
et al. [12] developed a methodology for modeling patientspecific blood flows in cerebral aneurysms that combines
medical image analysis and computational fluid dynamics
(CFD). The flow analysis in six intracranial aneurysms
revealed complex unsteady flows patterns.
Arterial compliance contributes to the mechanical loading and progressive expansion of a blood vessel [13]. Bathe
and Kamm [14] performed fluid–structure interaction (FSI)
finite element analysis of pulsatile flow through compliant
axisymmetric stenotic arteries and found that severe stenosis causes artery compression, negative flow pressure and
high flow shear stress. This important effect has been considered in few reported studies of hemodynamics in arteries
with aneurysms. Finol et al. [15] compare wall stress distributions in an abdominal aortic aneurysm obtained with
time dependent fluid–structure interaction and with static
walls. Low et al. [16] examined in models of lateral aneurysms the effects of distensible arterial walls, they found
out that the increase and decrease of the flow velocity
reflect the expansion and contraction of the aneurysm wall
A. Valencia, F. Solis
where the maximal wall displacement during systolic acceleration is about 6% of the aneurysm diameter. Humphrey
and Canham [17] modeled a saccular aneurysm as spherical
hyperelastic membrane surrounded by a viscous cerebrospinal fluid; they showed that the aneurysm model does
not exhibit dynamic instabilities in response to periodic
loads.
The compliant aneurysm wall, which must withstand
arterial blood pressure, is composed of layered collagen.
Wall strength is related to both collagen fiber strength
and orientation. The main characteristic of the aneurysm
wall is its multidirectional collagen fibers; with physiological pressures, they become straight and thereby govern the
overall stiffness of the lesion [18]. Fluid shear stress modulates endothelial cell remodeling via realignment and elongation, and the time variation of wall shear stress affects
significantly the rates at which endothelial cells remodel
[19]. The influence of non-Newtonian properties of blood
in an idealized arterial bifurcation model with a saccular
aneurysm has been investigated in [20], in the regions with
relatively low velocities there is no essential difference in the
results with both fluid models.
Ma et al. [21] have recently performed a three-dimensional geometrical characterization of cerebral aneurysms
from computed tomography angiography data, reconstructing the geometry of non-ruptured human cerebral
aneurysms. This geometry can be classified as hemisphere,
ellipsoid, or sphere. Parlea et al. [22] presented a relatively
simple approach to characterizing the simple-lobed aneurysm shape and size using angiographic tracings. Their
measurements characterize the range of shapes and sizes
assumed by these lesions, providing useful guidelines for
the development of models for numerical investigation of
hemodynamics in cerebral aneurysms.
In this work, we present detailed numerical simulations
of three-dimensional unsteady flow with arterial wall interaction in a representative saccular aneurysm model and
one healthy model of the basilar artery. The computational
geometry of the saccular aneurysm was developed with representative dimensions as reported in [22]. The purpose of
this paper is to report the effects of arterial wall thickness,
the Young’s modulus in the elastic wall model and the
hyperelastic Mooney–Rivlin aneurysm wall model on pressure, wall shear stress, effective stress in the wall and aneurysm deformation and to establish comparisons of these
variables with a healthy basilar artery. The uniaxial
stress–stretch response of human cerebral arteries can be
reasonably modeled on an individual test basis using an
exponential form, [23]. The effects of model the wall artery
as hyperelastic material, such as Mooney–Rivlin model, are
investigated in detail.
The fully-coupled fluid–structure interaction reported in
this work for a saccular aneurysm of the basilar artery has
not been reported previously in the literature and it can
provide valuable insight in the study of statistically representative aneurysms subject to physiologically realistic pulsatile loads.
2. Mathematical and numerical modeling
2.1. Fluid and solid models
For the fluid model, the flow was assumed to be laminar,
Newtonian, and incompressible. The incompressible
Navier–Stokes equations with arbitrary Lagrangian–Eulerian (ALE) formulation were used as the governing equations which are suitable for problems with fluid structure
interaction (FSI) and frequent mesh adjustments. Flow
velocity at the flow–vessel interface was set to move with
vessel wall.
Continuity:
ru¼0
ð1Þ
Equation of Navier–Stokes:
qf ðou=ot þ ððu ug Þ rÞuÞ ¼ rp þ lr2 u
ð2Þ
where qf is the fluid density, p is the pressure, u is the fluid
velocity vector, and ug is the moving coordinate velocity,
respectively. In the ALE formulation, (u ug) is the relative velocity of the fluid with respect to the moving coordinate velocity. Blood is modeled to have a density
qf = 1050 kg/m3 and a dynamic viscosity l = 0.00319 kg/
ms. The governing equation for the solid domain is the
momentum conservation equation given by Eq. (3). In contrast to the ALE formulation of the fluid equations, a
Lagrangian coordinate system is adopted:
r rs ¼ qs u_ g
ð3Þ
where qs is the solid density, rs is the solid stress tensor,
and u_ g is the local acceleration of the solid. The artery wall
was assumed to be elastic or hyperelastic, isotropic, incompressible, and homogeneous with a density qs = 1050 kg/
m3 and a Poisson’s ratio t = 0.45, [15,17].
The applied boundary conditions on the fluid domain
are: (i) a time dependent uniform velocity at inlet and (ii)
a zero normal traction at outlets. These are presented in
mathematical form in Eqs. (4) and (5):
ujinlet ¼ UðtÞ
rnn joutlet ¼ 0
ð4Þ
ð5Þ
The boundary conditions on the FSI interfaces state that:
(i) displacements of the fluid and solid domain must be
compatible, (ii) tractions at these boundaries must be at
equilibrium and (iii) fluid obeys the no-slip condition.
These conditions are given in Eqs. (6)–(8):
ds ¼ df
ð6Þ
rs ^ns ¼ rf ^nf
ð7Þ
u ¼ ug
ð8Þ
where d, r and n^ are displacements, stress tensors and
boundary normals with the subscripts f and s indicating a
property of the fluid and solid, respectively. An external
boundary condition of pressure is also applied:
A. Valencia, F. Solis
Fig. 1. Model geometries (all dimensions in mm): (a) healthy basilar artery model, (b) aneurysm model.
ðrs ^ns Þ ^
ns ¼ P c
ð9Þ
where Pc is the intracranial pressure produced by the cerebral spinal fluid.
The 3D elastic model and the hyperelastic Mooney–
Rivlin (M–R) model were used to describe the material
properties of the vessel wall, [24]. The strain energy function is given in the M–R model by:
W ¼ c1 ðI 1 3Þ þ c2 ðI 2 3Þ þ c3 ðI 1 3Þ2 þ c4 ðI 1 3ÞðI 2 3Þ þ c5 ðI 2 3Þ2
þ c6 ðI 1 3Þ2 þ c7 ðI 1 3Þ2 ðI 2 3Þ þ c8 ðI 1 3ÞðI 2 3Þ2
2
þ c9 ðI 2 3Þ þ D1 ½expðD2 ðI 1 3ÞÞ 1
I 1 ¼ C kk I 2 ¼ 1=2 I 21 Cij Cij
ð10Þ
ð11Þ
where I1 and I2 are the first and second strain invariants of
the Cauchy–Green deformation tensor Cij, Cij = 2eij + dij,
where dij is the Kronecker delta; c1 to c9 and D1 to D2
are material constants, [24]. The 3D stress/strain relations
can be obtained by finding various partial derivatives of
the strain energy function with respect to proper variables
(strain or stretch components). In particular assuming:
k1 k2 k3 ¼ 1;
k2 ¼ k3 ;
k ¼ k1
ð12Þ
where k1, k2, and k3 are stretch ratios in (x, y, z) directions
respectively, the uniaxial stress/stretch relation for an isotropic material is obtained from Eq. (10), in a material with
ci = 0,
A. Valencia, F. Solis
r ¼ oW =ok ¼ D1 D2 ½2k 2k2 exp½D2 ðk2 þ 2=k 3Þ
ð13Þ
1.6
A
1.4
1.2
1.0
U(t) [m/s]
In this work, the following values were chosen to match the
Mooney–Rivlin uniaxial stress/stretch relation with experimental data obtained in [23] for a normal artery, ci = 0,
D1 = 555560. Pa and D2 = 1.5. With these values the M–
R model fits average uniaxial behavior of normal arteries
stretched in the experiments. The equivalent Young’s modulus with these values is E = 5 MPa for small stretch, calculated with the formula E = 6(D1D2), [24].
0.8
0.6
0.4
C
2.2. Model geometry and boundary conditions
Fig. 1 shows the geometry of our basilar artery models
in the cases of a healthy bifurcation and with a saccular
aneurysm. Aneurysm model is a saccular asymmetric aneurysm with b = 23.2. The angle b indicates the tilt of the
aneurysm dome with respect to the parent vessel. Physiologically, the bifurcation geometry and aneurysm location
exhibit a great amount of irregularity and asymmetry.
Average values of neck width, dome height, dome diameter, semi-axis height, basilar artery diameter, and a typical
b angle of saccular aneurysm of the basilar artery were
obtained from the measurements reported by Parlea et al.
[22]. The bifurcation angle between the basilar and the
posterior cerebral arteries was taken as 140, also used by
Liou et al. [4]. The lengths of the arteries are 6.5 and 7.1
times the respective artery diameters for the basilar and
posterior cerebral arteries, so that the inlet velocity boundary condition and the zero normal traction boundary condition at outflows were imposed distant from the aneurysm
location.
The blood flow waveform in arteries of the anterior circulation of the circle of Willis has been studied in detail.
Holdsworth et al. [25] have characterized the blood flow
waveform of the common carotid artery in normal human
subjects. However, a detailed measurement of the blood
flow waveform in the basilar artery is not available. It is
difficult to perform Doppler ultrasound measurements in
this region of the posterior circulation of the circle of
Willis. The mean blood flow in the basilar artery is
195 mL/min as reported in [26] for healthy human subjects.
We have used the blood flow waveform given in [25]
multiplied by the mean blood flow in the basilar artery
and divided by the mean blood flow in the common carotid
artery. Fig. 2 shows the representative physiological waveform of the velocity at the inlet of the basilar artery as
used in the present work. The mean blood velocity at
the inlet of the basilar artery is 39.7 cm/s. We also consider
a reduction of diameter for the posterior cerebral arteries
as 0.75 times the basilar artery diameter. With this reduction the mean flow velocity in our posterior cerebral
arteries model is 35.6 cm/s; this value is very similar to
the mean velocity of 36.0 cm/s reported in [27] for the posterior cerebral artery in normal patients. The time dependency of the inflow velocity is imposed by a Fourier
0.2
B
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
t [s]
Fig. 2. Physiological waveform of inlet velocity.
representation of order 12 and the natural frequency of
the pulsatile flow is set to f = 1 Hz. The Womersley number, which characterizes the flow frequency, the geometry
of the model and the fluid viscous properties is a = 2.32.
Peak systolic flow occurs at t = 0.18 s and the time-averaged Reynolds number is Rem = 422. The peak Reynolds
number at systole is Remax = 1620, this Reynolds number
is lower than 2300, so that the inflow can be considered
laminar.
Fig. 3 shows the boundary conditions for the wall artery
modeled as 3D structure, where Ui represents the translations. The displacements are restricted in some directions
as shown in Fig. 3(a). These restrictions are applied far from
the saccular aneurysm, and therefore they do not affect the
aneurysm deformation and effective stress in the aneurysmal wall. The external pressure produced by the cerebral
spinal fluid was assumed to be constant Pc = 400 Pa, [17],
and was applied as external boundary condition in our
models.
Tables 1 and 2 show the characteristics for the 6 cases of
the healthy basilar bifurcation and for the 8 cases of the
basilar bifurcation with a saccular aneurysm studied in this
work, respectively. The wall thicknesses were selected similar to experimental values of wall thickness of intracranial
aneurysm reported in [18]. Experimental results of elasticity
studies in an aneurysm show that the Young’s modulus is
as high as 300 MPa compared to 1 MPa for a normal
artery. Measurements of E for pure collagen range between
300 MPa and 2500 MPa, compared with 0.3 MPa for elastin, suggesting that the aneurysm wall is composed primarily of collagen, [28]. We have varied E between 5 MPa and
300 MPa, to report the influence of different wall compositions on effective wall stress and deformation under pulsatile load. The arteries and the saccular aneurysm are
modeled using the same properties, due to the displacements of the normal arteries are very low compared with
the displacements of the aneurysm, as our preliminary tests
have shown.
A. Valencia, F. Solis
Fig. 3. (a) Grid and boundary conditions used in the aneurysm solid model, (b) details of the grid used in the healthy solid model, and (c) details of the
grid and characteristic points used in the aneurysm solid model.
Table 2
Properties of cases for aneurysm model
Table 1
Properties of cases for healthy basilar artery model
Healthy geometry
Wall thickness h (lm)
Solid model E (MPa)
Case
Case
Case
Case
Case
Case
200
200
300
300
300
–
Elastic E = 5
Elastic E = 10
Elastic E = 5
Elastic E = 10
M–R
Rigid wall
1
2
3
4
5
6
Aneurysm model
Wall thickness h (lm)
Solid model E (MPa)
Case
Case
Case
Case
Case
Case
Case
Case
200
200
200
300
300
300
300
–
Elastic E = 10
Elastic E = 100
Elastic E = 300
Elastic E = 5
Elastic E = 10
Elastic E = 100
M–R
Rigid wall
1
2
3
4
5
6
7
8
A. Valencia, F. Solis
2.3. Numerical method
Systole [1.18 s]
5000
2
3
4500
4
5
4000
p [Pa]
The fully coupled fluid and structure models were solved
by a commercial finite element package ADINA (v8.1,
ADINA R&D, Inc., Cambridge, MA) which has been
tested by several applications, [24]. The finite element
method (FEM) is used to solve the governing equations.
The FEM discretizes the computational domain into finite
elements that are interconnected by element nodal points.
The fluid domain employs special flow-condition-basedinterpolation (FCBI) tetrahedral elements, [29].
The unstructured grids were composed of tetrahedral
with four-node elements in the fluid and hexahedral with
eight-node elements in the solid, Fig. 3 shows the solid
grids used for the healthy basilar bifurcation and for the
aneurysm model. Unstructured grid of 70 250 tetrahedral
elements was used for the fluid and 46 980 hexahedral elements for the solid in the healthy model, with terminal
aneurysm the fluid was meshed with 80 400 tetrahedral elements and the wall artery with 28 600 hexahedral elements,
in the cases with wall thickness of 300 lm.
We have used the formulation with large displacements
and small strains in the FSI calculation available in
ADINA. As iteration we used full Newton method with
a maximum of 200 iterations in each time step of 0.01 s.
A sparse matrix solver based on Gaussian elimination is
used for solving the system. The relative tolerance for all
degrees of freedom was set to 0.001.
To verify grid independence, numerical simulations
based on the aneurysm model with rigid walls (case 8) were
performed on five grids with 42 308, 60 202, 68 449, 79 166
and 84 471 tetrahedral elements in the fluid. Fig. 4 shows
the convergence of pressure in four points for the five used
grid sizes in the aneurysm model at systolic time. Between
the two finer grid sizes the predictions of pressure are similar, so that we use a grid with 80 000 tetrahedral elements
in the fluid for the simulation of flow in our basilar artery
model with saccular aneurysm.
The workstation used to perform the simulations in this
work is based on an Intel Pentium IV processor of 2.8 GHz
clock speed, 1.5 Gb RAM memory, and running on the
3500
3000
2500
2000
30000
40000
50000
60000
70000
80000
90000
N
Fig. 4. Relative pressure versus grid size in four characteristic points of
the aneurysm model, case 8.
Linux Redhat v8.0 operating system. The simulation time
for one FSI case based on 2 consecutive pulsatile flow
cycles employing 200 time iterations was approximately
72 CPU hours.
3. Results and discussion
3.1. Flow dynamics
The relative pressure and wall shear stress (WSS) distributions at systole for the healthy basilar bifurcation are
shown in Fig. 5. Evidently, the wall pressure is maximum
around the stagnation point 1, see Fig. 3(b), downstream
the bifurcation it develops higher WSS than in the basilar
artery due higher velocity gradients produced in the smaller
posterior cerebral arteries.
Velocity vectors for the aneurysm model are shown in
Fig. 6 at peak systole, in planes parallel to the basilar artery
longitudinal axis (Y–Z plane) and perpendicular to the basilar artery longitudinal axis (X–Z plane) respectively. The
asymmetry of the dome yields a high velocity jet through
the inflow region and high shear in a large region of the
aneurysmal wall. The asymmetry of the geometry affects
Fig. 5. (a) Relative pressure and (b) WSS distributions in the healthy basilar model at systolic time, case 5.
A. Valencia, F. Solis
Fig. 6. Velocity vectors in aneurysm model at systolic time, (a) Y–Z plane, (b) X–Z plane, case 7.
the intra-aneurysmal flow by yielding a single recirculation
region in the dome.
In our asymmetric aneurysm model, the jet of fluid flows
through the low part of neck of the aneurysm (i.e., inflow
region). This finding is important for the endovascular
treatment of cerebral aneurysms, since controlling blood
flow at the inflow region is a critical step in achieving permanent occlusion of an aneurysm. The long-term anatomical durability of coil embolization in cerebral aneurysms
by means of microcoil technology depends on the aneurysm shape and the blood flow dynamics at the aneurysm
neck. Ideally, the coil must be inserted through the inflow
region to avoid flow recanalization. Therefore, careful consideration of the aneurysm asymmetry with respect to the
parent vessel must be taken into account during these
endovascular procedures in relation to the resulting flow
dynamics at the inflow region.
The characteristics of the pressure and wall shear stress
(WSS) are shown in Fig. 7 for the aneurysm model, in
instantaneous representations for systolic time. As shown
in Fig. 7, the highest pressure and WSS are located asymmetrically with respect to a centerline in X–Z plane in a
region where the flow is lead by the asymmetry of the
geometry, with fluid induced stresses diminishing closer
to the centerline. The flow in the aneurysm does not have
stagnation point, and therefore it exists a more extended
region with high pressure compared with the pressure distribution in the healthy bifurcation, see Fig. 5. The maxima
values of WSS in the aneurysm are around 50% lower as in
the healthy basilar bifurcation.
The temporal variation of both WSS and pressure at
precise location of the aneurysm is an important variable
for obtaining information on the cause of aneurysm wall
rupture with respect to the intensity of the stresses and their
temporal evolution in sensitive regions. Fig. 8(a) and (b)
show the temporal evolution of WSS and pressure in the
characteristic point 5, see Fig. 3(c), of aneurysm model
for the eight cases. The distribution of pressure closely
resembles the inlet velocity waveform, the effects of the
elastic or hyperelastic wall model and aneurysm wall thickness are not present in the time dependence of pressure.
The time dependence of WSS shows the effect of aneurysm
wall thickness, the three cases with 200 lm show higher
WSS as the cases with 300 lm independently of wall model.
The differences on prediction can reach 43% at peak
systole.
Fig. 7. (a) Relative pressure and (b) WSS distributions in the aneurysm model at systolic time, case 7.
A. Valencia, F. Solis
5.5
of an artery with saccular aneurysm measured in [18] is
around 0.7 MPa, therefore, it can be concluded that in a
healthy basilar bifurcation the genesis and growth of saccular aneurysm can not be explained only by mechanical
loads of the arteries. It is necessary to consider the arterial
wall remodeling mechanism produced by high WSS, as it
will be discussed later. The effects of wall model and thickness in temporal variation of displacement and effective
stress at point 1 of the bifurcation are shown in Fig. 10.
a
5
4.5
Case 1
4
Case 2
WSS [Pa]
3.5
Case 3
3
Case 4
2.5
Case 5
Case 6
2
Case 7
1.5
Case 8
1
0.5
0.000035
a
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00003
t [s]
0.000025
3500
b
Case 1
[m]
3000
2500
Case 2
Case 3
Case 4
0.000015
Case 1
Case 5
Case2
2000
p [Pa]
0.00002
0.00001
Case 3
Case 4
1500
0.000005
Case 5
Case 6
1000
0
Case 7
0
Case 8
500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
0
30000
0.1
0.2
-500
0.3
0.4
0.5
0.6
0.7
0.8
0.9
b
1
t [s]
Fig. 8. Time dependence of (a) WSS and (b) pressure in point 5 for the
eight cases of the aneurysm model.
3.2. Solid dynamics
The displacement magnitude and the effective stress of
the arterial walls induced by the unsteady blood flow in
the healthy basilar bifurcation for the case five are shown
in Fig. 9. The flow induced displacement is larger around
the bifurcation to the posterior cerebral arteries. The maximum displacement at systole is only around 10% of artery
thickness. The maximum effective stress into the basilar
bifurcation is around 2.5 · 104 Pa, the breaking strength
25000
Effective stress [Pa]
0
20000
Case 1
Case 2
Case 3
15000
Case 4
Case 5
10000
5000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Fig. 10. Time dependence of (a) displacement and (b) effective stress in
point 1 for the five cases of healthy basilar model.
Fig. 9. (a) Displacement and (b) effective stress distributions on the healthy basilar solid model at systolic time, case 5.
A. Valencia, F. Solis
Fig. 11. (a) Displacement and (b) effective stress distributions on the aneurysm solid model at systolic time, case 7.
0.0004
a
0.00035
0.0003
Case 1
[m]
Case 3
A cerebral wall artery consists in three layers: the tunica
intima, tunica media and tunica adventitia, see Fig. 13. The
intima is a single layer of endothelial cells on the innermost
section of the vessel wall. Media refers to the middle section of the vessel wall and consists of smooth muscle cells
surrounded by collagen and elastic tissue. Adventitia, the
outermost covering of the vessel wall, consists of a mixture
of collagen with admixed elastin, nerves, fibroblasts, and
the vasa vasorum, [30].
Case 4
0.0002
Case 5
0.00015
Case 6
Case 7
0.0001
0.00005
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
50000
b
40000
Case 1
30000
Case 2
Case 3
Case 4
20000
Case 5
Case 6
10000
Case 7
0
0
-10000
3.3. Discussion
Case 2
0.00025
Effective stress [Pa]
As expected, the displacement is larger in the cases with
lower Young’s modulus and wall thickness, however the
effective stress variation depends only of the wall thickness
and a variation of Young’s modulus or wall model did not
make difference.
The distributions of wall displacement and effective
stress in the saccular aneurysm showed in Fig. 11 indicate
that the wall displacement increases from the neck to the
dome of the aneurysm, and the displacements in the basilar
and posterior cerebral arteries are practically zero. The
maximum wall displacement at systolic time is practically
equal to the wall thickness. The maximum effective stress
in the aneurysmal wall is around 5 · 104 Pa at systole and
it is around 8% of the breaking strength reported in [18].
The influence of the hyperelastic Mooney–Rivlin model
on displacement in point 4 of the aneurysm is clearly seen
in Fig. 12. The case 4 with E = 5 MPa shows the maximum
values, and the displacement predicted with the M–R
model at systole is 25% lower. If the aneurysmal wall is
composed principally of collagen, represented by cases 2,
3 and 6, the aneurysmal wall can be practically considered
as rigid. The distribution of effective stress closely resembles the inlet velocity waveform for the 7 cases, and only
at systole it exists important differences of effective wall
stress. The values of effective stress with the wall thickness
of 200 lm are similar independently of Young’s modulus,
and the effective stress predicted with the M–R model at
systole shows the lowest value.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Fig. 12. Time dependence of (a) displacement and (b) effective stress in
point 4 for the seven cases of aneurysm model.
There are no previously published reliable predictions of
WSS, pressure, effective stress in the artery and wall displacement distributions in a representative geometry of
saccular aneurysm in the bifurcation of the basilar artery,
although several authors have reported on their importance with respect to the pathogenesis and evolution of
cerebral aneurysms. High WSS is regarded as a major
A. Valencia, F. Solis
the importance of utilizing flow waveform measured in
the respective artery of the circle of Willis by means of
pulsed Doppler ultrasound, if patient-specific models of
terminal aneurysm are used to describe their three-dimensional unsteady flow with arterial wall interaction.
4. Conclusions
Fig. 13. Schematic cross-section of a typical cerebral artery.
factor in the development and growth of cerebral aneurysms. The endothelial cells are sensitive to changes in
WSS and regulate local vascular tone by releasing vasodilator and vasoconstrictor substances [7]. Consequently, it
has been suggested that WSS plays a major role in the
adaptive dilation of arteries to a sustained increase in blood
flow. Localized stagnation of blood flow is known to result
in the aggregation of red blood cells, as well as the accumulation and adhesion of both platelets and leukocytes along
the intimal surface. This occurs due to dysfunction of flowinduced nitric oxide, which is usually released by mechanical stimulation through increased shear stress. These
factors may cause intimal damage, lead to small thrombus
formation and infiltration of white blood cells and fibrin
inside the aneurysm wall [8]. Damage to the endothelial
cells is typically hypothesized as a contributing cause to
aneurysm growth and rupture. Once the integrity of this
layer of cells lining the lumen is breached, subsequent deterioration of the structural fibers of the vessel may occur.
Flow and solid dynamics are involved in the process leading to vessel remodeling due to wall injury. It is assumed
that a WSS of 2 Pa is suitable for maintaining the structure of the aneurysmal wall and a lower WSS will degenerate endothelial cells, [31]. Our results show values lower
than 2 Pa in large regions of the aneurysmal wall over
one cardiac cycle. This low WSS may be one important factor underlying the degeneration, indicating the structural
fragility of the aneurysmal wall. Our results suggest that
potential endothelial damage occurs due to high amplitude
stresses with fatigue playing a key role as a consequence of
flow-induced wall stress gradients.
The zero normal traction boundary condition used in
the outlet produces that the pressure in this region is
approximately zero. From physiological point of view the
calculated pressure difference and WSS are correct. The
pressure inside the aneurysm is however underestimated,
and therefore the reported wall stress and displacement
are also underestimated. Another relevant outcome of this
investigation is the underlying relationship between the
temporal distributions of stress and displacement of the
aneurysmal wall with respect to the flow waveform
imposed at the model inlet. This is illustrated by the almost
constant WSS, pressure, effective stress and displacement
obtained during diastole for the healthy basilar artery
and in the aneurysm model. This observation emphasizes
This paper presents a numerical investigation of the
fluid solid interaction in a saccular aneurysm model
and in a healthy model of the basilar artery. The effects
of wall model and wall thickness are studied in detail with
respect to the spatial and temporal distributions of pressure, wall shear stress, effective wall stress and aneurysm
displacement.
The flow dynamics in the aneurysm model shows an
unsteady vortex structure. Large spatial and temporal variations of pressure and shear stress are obtained on the
aneurysmal wall. The aneurysmal wall displacement and
effective stress depend on wall thickness and wall model
especially at systole. Further studies are necessary to investigate the effects of arterial compliance on the development,
growth, and rupture of patient-specific intracranial aneurysms of the basilar artery.
Acknowledgement
The financial support received from FONDECYT
Chile under grant number 1030679 is recognized and
appreciated.
References
[1] Schievink WI. Intracranial aneurysms. New England J Med
1997;336:28–40.
[2] Wardlaw JM, White PM. The detection and management of
unruptured intracranial aneurysms. Brain 2000;123:205–21.
[3] Weir B, Amidei C, Kongable G, Findlay JM, Kassell NF, et al. The
aspect ratio (dome/neck) of ruptured and unruptured aneurysms. J
Neurosurg 2003;99:447–51.
[4] Liou TM, Liou SN. A review on in vitro studies of hemodynamic
characteristics in terminal and lateral aneurysm models. Proc of
National Science Council, Republic of China, Part B, Life Sciences
1999;23:133–48.
[5] Imbesi SG, Kerber CW. Analysis of slipstream flow in a wide-necked
basilar artery aneurysm: Evaluation of potential treatment regimens.
Am J Neuroradiol 2001;22:721–4.
[6] Lieber BB, Livescu V, Hopkins LN, Wakhloo AK. Particle image
velocimetry assessment of stent design influence on intra-aneurysmal
flow. Ann Biomed Eng 2002;30:768–77.
[7] Tateshima S, Murayama Y, Villablanca JP, Morino T, Takahashi H,
Yamauchi T, et al. Intraaneurysmal flow dynamics study featuring
an acrylic aneurysm model manufactured using a computerized
tomography angiogram as a mold. J Neurosurg 2001;95:1020–7.
[8] Ujiie H, Tachibana H, Hiramatsu O, Hazel AL, Matsumoto T,
Ogasawara Y, et al. Effects of size and shape (aspect ratio) on the
hemodynamics of saccular aneurysms: a possible index for surgical
treatment of intracranial aneurysms. Neurosurgery 1999;45:119–30.
[9] Jou LD, Saloner D, Higashida R. Determining intra-aneurysmal flow
for coiled cerebral aneurysms with digital fluoroscopy. Biomed Eng –
Appl, Basis Commun 2004;16:43–8.
A. Valencia, F. Solis
[10] Foutrakis G, Yonas H, Sclabassi R. Saccular aneurysm formation in
curved and bifurcating arteries. Am J Neuroradiol 1999;20:1309–17.
[11] Steinman DA, Milner JS, Norley CJ, Lownie SP, Holdsworth DW.
Image-based computational simulation of flow dynamics in a giant
intracranial aneurysm. Am J Neuroradiol 2003;24:559–66.
[12] Cebral JR, Hernández M, Frangi AF. Computational analysis of
blood flow dynamics in cerebral aneurysms from CTA and 3D
rotational angiography image data. In: Proc of International Congress on Computational Bioengineering, Spain, 2003.
[13] Ho PC, Melbin J, Nesto RW. Scholarly review of geometry and
compliance: Biomechanical perspectives on vascular injury and
healing. Am Soc Artif Internal Organs J 2002;48:337–45.
[14] Bathe M, Kamm RD. A fluid–structure interaction finite element
analysis of pulsatile blood flow through a compliant stenotic artery.
ASME J Biomech Eng 1999;121:361–9.
[15] Finol EA, Di Martino ES, Vorp DA, Amon CH. Fluid structure
interaction and structural analyses of an aneurysm model, 2003
Summer ASME Bioengineering Conference, ASME 2003 BED-54,
2003.
[16] Low M, Perktold K, Raunig R. Hemodynamics in rigid and
distensible saccular aneurysms: a numerical study of pulsatile flow
characteristics. Biorheology 1993;30:287–98.
[17] Humphrey JD, Canham PB. Structure, mechanical properties, and
mechanics of intracranial saccular aneurysms. J Elasticity
2000;61:49–81.
[18] MacDonald DJ, Finlay HM, Canham PB. Directional wall strength
in saccular brain aneurysms from polarized light microscopy. Ann
Biomed Eng 2000;28:533–42.
[19] Hsiai TK, Cho SK, Honda HM, Hama S, Navab M, Demer LL,
et al. Endothelial cell dynamics under pulsating flows: Significance of
high versus low shear stress slew rates (os/ot). Ann Biomed Eng
2002;30:646–56.
[20] Perktold K, Peter R, Resch M. Pulsatile non-Newtonian blood flow
simulation through a bifurcation with an aneurysm. Biorheology
1989;26:1011–30.
[21] Ma B, Harbaugh RE, Raghavan ML. Three-dimensional geometrical
characterization of Cerebral aneurysms. Ann Biomed Eng
2004;32:264–73.
[22] Parlea L, Fahrig R, Holdsworth DW, Lownie SP. An analysis of the
geometry of saccular intracranial aneurysms. Am J Neuroradiol
1999;20:1079–89.
[23] Monson K. Mechanical and failure properties of human cerebral
blood vessels. Ph.D. Thesis. University of California, Berkeley, USA,
2001.
[24] ADINA Theory and Modeling Guide. Volume I: ADINA. ADINA
R& D, Inc. Watertown, MA, USA, 2004.
[25] Holdsworth DW, Norley CJD, Frayne R, Steinmam DA, Rutt BK.
Characterization of common carotid artery blood-flow waveforms in
normal human subjects. Physiol Meas 1999;20:219–40.
[26] Guppy KH, Charbel FT, Corsten LA, Zhao M, Debrum G.
Hemodynamic evaluation of basilar and vertebral artery angioplasty.
Neurosurgery 2002;51:327–33.
[27] Owega A, Klingelhöfer J, Sabri O, Kunert HJ, Albers M, Saß H.
Cerebral blood flow velocity in acute schizophrenic patients: A
transcranial Doppler ultrasonography study. Stroke 1998;29:1149–54.
[28] Drangova M, Holdsworth DW, Boyd CJ, Dunmore PJ, Roach MR,
Fenster A. Elasticity and geometry measurements of vascular
specimens using a high-resolution laboratory CT scanner. Physiol
Meas 1993;14:277–90.
[29] Bathe KJ, Zhang H. Finite element developments for general fluid
flows with structural interactions. Int J Numer Meth Eng
2004;60:213–32.
[30] Humphrey JD. Cardiovascular Solid Mechanics. New York: Springer-Verlag; 2002. Chapter 7, pp. 249–364.
[31] Shojima M, Oshima M, Takagi K, Torii R, Hayakawa M, Katada K,
et al. Magnitude and role of wall shear stress on cerebral aneurysm
computational fluid dynamic study of 20 middle cerebral artery
aneurysms. Stroke 2004;35:2500–5.