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Copyright © 1998 Biomedical Engineering Society
Annals of Biomedical Engineering, Vol. 26, pp. 975–987, 1998
Printed in the USA. All rights reserved.
Finite Element Modeling of Three-Dimensional Pulsatile Flow
in the Abdominal Aorta: Relevance to Atherosclerosis
CHARLES A. TAYLOR,*† THOMAS J. R. HUGHES,† and CHRISTOPHER K. ZARINS*
*Department of Surgery, and †Department of Mechanical Engineering, Stanford University, Stanford, CA
(Received 12 November 1997; accepted for publication 28 July 1998)
observed to be devoid of plaque. Ku et al.15 noted strong
positive correlations between intimal thickening and the
inverse of the maximum wall shear stress, the inverse of
the mean wall shear stress, and oscillations in shear characterized by an oscillatory shear index.
In the aorta, it is observed that atherosclerotic disease
develops first in the abdominal aorta, and is much more
common in the abdominal aorta ~i.e., below the diaphragm! than the segment of the aorta above the diaphragm, the thoracic aorta. Roberts et al.34 noted that the
abdominal aorta contained by far the most severe aortic
atherosclerosis with the greatest involvement occurring
below the celiac artery. Glagov et al.6 noted that abdominal aortic atherosclerosis was greater than thoracic aortic
atherosclerosis in the majority of cases for males and
females, normotensives and hypertensives. Cornhill
et al.4 in examining fatty streaks in young subjects noted
greater involvement along the lateral and posterior walls
of the abdominal aorta. Friedman et al.7 noted increased
intimal thickness in regions of low wall shear stress
along the lateral walls of the abdominal aortic bifurcation. Moore et al.26 measured intimal thickness in the
distal abdominal aorta in subjects with minimal atherosclerotic disease and noted a positive correlation between
oscillatory shear and intimal thickness and a negative
correlation between mean shear stress and intimal thickening. Clearly, to examine the relationship between vascular disease and hemodynamic conditions detailed
quantitative data on flow conditions in the abdominal
aorta are required.
The pulsatile flow in the abdominal aorta is particularly complex under normal resting conditions as a result
of the multiple branches at the level of the diaphragm
which deliver blood to the organs in the abdomen. As in
the case of carotid artery flow, idealized models representing abdominal aortic anatomy and physiologic conditions have been the basis for most of the investigations
into abdominal aortic hemodynamics. Ku et al.16 describe flow visualization techniques applied to study
steady flow in an abdominal aorta model under resting,
Abstract—The infrarenal abdominal aorta is particularly prone
to atherosclerotic plaque formation while the thoracic aorta is
relatively resistant. Localized differences in hemodynamic conditions, including differences in velocity profiles, wall shear
stress, and recirculation zones have been implicated in the
differential localization of disease in the infrarenal aorta. A
comprehensive computational framework was developed, utilizing a stabilized, time accurate, finite element method, to solve
the equations governing blood flow in a model of a normal
human abdominal aorta under simulated rest, pulsatile, flow
conditions. Flow patterns and wall shear stress were computed.
A recirculation zone was observed to form along the posterior
wall of the infrarenal aorta. Low time-averaged wall shear
stress and high shear stress temporal oscillations, as measured
by an oscillatory shear index, were present in this location,
along the posterior wall opposite the superior mesenteric artery
and along the anterior wall between the superior and inferior
mesenteric arteries. These regions were noted to coincide with
a high probability-of-occurrence of sudanophilic lesions as reported by Cornhill et al. ~Monogr. Atheroscler. 15:13–19,
1990!. This numerical investigation provides detailed quantitative data on hemodynamic conditions in the abdominal aorta
heretofore lacking in the study of the localization of atherosclerotic disease. © 1998 Biomedical Engineering Society.
@S0090-6964~98!01406-4#
Keywords—Hemodynamics, Plaque localization, Computational methods, Parallel computing, Shear stress.
INTRODUCTION
Hemodynamic factors are thought to be responsible
for the localization of vascular disease in areas of complex flow in the coronary, carotid, abdominal, and femoral arteries. These complex flow regions often occur due
to branching, bifurcations, and curvature of the arteries.
Zarins et al.42 noted that in the carotid artery, atherosclerotic lesions localize along the outer wall of the carotid
sinus region where wall shear stress is low. Conversely,
the inner wall of the carotid sinus, an area of rapid,
laminar and axial flow, and high wall shear stress, was
Address correspondence to Charles A. Taylor, Assistant Professor,
Division of Vascular Surgery, Stanford University Medical Center,
Suite H3600, Stanford, CA 94305-5450. Electronic mail:
[email protected]
975
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TAYLOR, HUGHES, and ZARINS
postprandial, and exercise conditions. Moore et al.22 describe pulsatile flow visualization in an idealized abdominal aorta model under resting, postprandial, and exercise conditions. Pedersen et al.28 describe quantitative
flow measurements made at three sites in an abdominal
aorta flow model using laser Doppler anemometry. Differing physiologic conditions were examined including
simulated rest and exercise. Moore and Ku23 describe
quantitative flow measurements, using magnetic resonance imaging, for the pulsatile flow in an idealized
abdominal aorta model under simulated resting conditions.
Noninvasive imaging techniques have also been applied in recent years to measure blood flow in the abdominal aorta in vivo. Maier et al.21 used Doppler ultrasound and magnetic resonance imaging techniques to
measure blood flow at a single location between the
renal arteries and aortic bifurcation in the abdominal
aorta of nine volunteers. Mostbeck et al.27 used MR imaging to measure blood flow patterns in nine volunteers
at four locations in the abdominal aorta: 1 cm proximal
to the celiac artery, between the superior mesenteric and
renal arteries, 1 cm below the renal arteries and 1 cm
above the aortic bifurcation. Moore et al.25 measured
blood velocity profiles in the abdominal aorta in vivo
using magnetic resonance imaging. As in the case of the
in vitro model flow studies, current noninvasive imaging
technology is limited to the extraction of velocity data at
a relatively small number of axial locations. Complete
temporal and spatial field variations of velocity and shear
stress in the abdominal aorta have not been obtained.
In recent years, computational techniques have been
used increasingly by researchers seeking to understand
vascular hemodynamics. These methods can augment the
data provided by in vitro and in vivo methods by enabling a complete characterization of hemodynamic conditions under precisely controlled conditions. Application
of these methods to flow in the carotid bifurcation and
bypass grafts has provided significant information on
vascular hemodynamics. Perktold et al.30 used a finite
element method to simulate the pulsatile flow of a Newtonian fluid in a model of a carotid artery bifurcation
using a rigid walled approximation. Detailed results on
the velocity, pressure, and wall shear stress was presented. Lei et al.18 describe the hemodynamic conditions
in a model of a rabbit aortoceliac junction and postulate
a role for the wall shear stress gradient in atherogenesis.
Numerical methods are well suited to the investigation of
phenomena difficult to describe using in vitro techniques
including wall compliance, mass transport, particle residence time, and geometric variations. In an investigation
of the effect of wall compliance on pulsatile flow in the
carotid artery bifurcation, Perktold and Rappitsch32 describe a weakly coupled fluid–structure interaction finite
element method for solving for blood flow and vessel
mechanics. Steinman and Ethier37 investigated the effect
of wall distensibility in a two-dimensional end-to-side
anastomosis. Rappitsch and Perktold33 describe the transport of albumin in a model of a stenosis. Kunov et al.17
proposed a new method for describing particle residence
time. Perktold et al.31 examine the effect of bifurcation
angle on hemodynamic conditions in carotid artery bifurcation models. Lei et al.19 describe the application of
computational methods to the design of end-to-side anastomoses.
In contrast to the relatively large number of studies of
pulsatile flow in models of the carotid bifurcation and
end-to-side anastomosis, there have been few numerical
studies of flow in the abdominal aorta. Taylor et al.38
describe the flow in an abdominal aorta model under
simulated resting and exercise steady flow conditions. It
was noted that a region of flow recirculation and low
wall shear stress develop along the posterior wall of the
infrarenal abdominal aorta under simulated resting conditions and disappears under simulated moderate and
vigorous exercise conditions. Taylor et al.39 describe,
qualitatively, the pulsatile flow in a model of an abdominal aorta under simulated resting and moderate exercise
conditions. As in the steady flow case, a flow recirculation region was noted at mid-diastole in the infrarenal
abdominal aorta under simulated resting conditions. No
attempt was made in that study to quantify the hemodynamic conditions under resting or exercise conditions.
The computational method described herein was used
to quantitatively characterize the hemodynamic conditions under simulated resting pulsatile flow conditions in
an idealized model of an abdominal aorta. To date, no
numerical results have been published detailing the hemodynamic conditions in the abdominal aorta under pulsatile flow conditions. In particular, although velocity has
been measured at a few locations in the abdominal aorta
using in vivo and in vitro methods, the spatial variations
of wall shear stress have not been reported heretofore.
This article details the mean wall shear stress and shear
stress oscillations in the abdominal aorta and compares
these findings with the observed patterns of sudanophilic
lesions as reported by Cornhill et al.4
METHOD
A geometric solid model of an idealized abdominal
aorta, shown in Fig. 1, was constructed using a custom
software system, ‘‘The Stanford Virtual Vascular Laboratory’’ developed to aid in the solution of blood flow
problems.39,40 This system was developed using a knowledge based, object oriented programming language and
combines geometric solid modeling, automatic finite element mesh generation, multiphysics finite element
methods, and scientific visualization capabilities.43–46
The anatomic dimensions of the idealized abdominal
Finite Element Modeling of Flow in the Abdominal Aorta
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FIGURE 1. Abdominal aorta model with branches identified. The model is not symmetric about the midsagittal plane. Note the
tapering of the aorta from the diaphragm to the aortic bifurcation.
aorta model were obtained primarily from Moore et al.22
In the model utilized for the present investigations, the
aorta tapers from a circular cross section with a diameter
of 2.54 cm at the diaphragm to a circular cross section
with diameter of 1.72 cm at the inferior mesenteric artery
and then tapers uniformly to an elliptical cross section
with a major axis of 1.53 cm and minor axis of 1.3 cm
at the aortic bifurcation. It should be noted that the
model constructed is not symmetric about the midsagittal
plane, but rather includes the feature that the left renal
artery is located inferior to the right renal artery. A finite
element mesh with 268,563 tetrahedral elements and
58,151 nodes was generated using an automatic mesh
generator.36 The surface of the finite element mesh above
the aortic bifurcation is displayed in Fig. 2. Note the
FIGURE 2. Closeup of abdominal aorta finite element mesh
above the aortic bifurcation. Note the greater mesh refinement used for the infrarenal aorta and the branch arteries.
refinement of the mesh in the region of the aorta below
the renal arteries and along the branch vessels.
Under resting conditions, approximately 70% of the
blood that enters the abdominal aorta is extracted by the
celiac, superior mesenteric, and renal arteries. The majority of the remaining 30% flows down the infrarenal
segment through the bifurcation into the legs. In contrast
to steady flow studies, in vitro and computational studies
of pulsatile blood flow require the specification of flow
waveforms as well as mean flow rates. The flow rates as
a function of time used in the present study for the
inflow and branch vessels are shown in Fig. 3. The flow
rate waveforms shown in Fig. 3 for the renal artery and
iliac artery are those for each of the right and left renal
and iliac arteries, respectively. The suprarenal and infrarenal flow rate time functions and the celiac, superior
mesenteric, renal, and inferior mesenteric mean flow
rates were obtained from Moore and Ku.23 The flow rate
waveforms in the celiac, superior mesenteric, renal, and
inferior mesenteric arteries were computed to conserve
mass and yield the mean flow rates given by Moore
et al.23 Note that the abdominal aorta inflow and renal
artery outflows are always positive, whereas the iliac
flow waveform exhibits the triphasic character of in vivo
measurements of infrarenal aortic flow. Based on the
volume flow waveforms shown in Fig. 3, pulsatile flow
velocity boundary conditions, derived from the Womersley theory, were prescribed for the inflow boundary and
all outflow boundaries excluding the left and right iliac
outflow boundaries, where zero pressure boundary conditions were prescribed.41 It should be noted that the
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TAYLOR, HUGHES, and ZARINS
FIGURE 3. Flow rate time functions for resting conditions. Note the triphasic nature of the flow waveforms and the different
ordinate scales for each plot.
Womersley boundary conditions are time dependent, axisymmetric velocity profiles at the inlet and outlet boundaries.
Computational modeling of blood flow requires solving, in the general case, three dimensional, transient flow
equations in deforming blood vessels. The appropriate
framework for problems of this type is the arbitrary
Lagrangian–Eulerian ~ALE! description of continuous
media in which the fluid and solid domains are allowed
to move to follow the distensible vessels and deforming
fluid domain.10
The vessel diameter change during the cardiac cycle
is observed to be approximately 5%–10% in most of the
major arteries, and, as a first approximation, the vessel
Finite Element Modeling of Flow in the Abdominal Aorta
walls are often treated as being rigid. In addition, in
diseased vessels which are often the subject of interest,
the arteries are even less compliant and wall motion is
further reduced. The assumption of zero wall motion was
utilized for the computations presented herein. Under this
assumption, the ALE description of incompressible flow
in a deforming fluid domain reduces to the Eulerian
description of a fixed spatial domain.
The strong form of the problem governing incompressible, Newtonian fluid flow in a fixed domain consists of the Navier–Stokes equations and suitable initial
and boundary conditions. Direct, analytic solutions of
these equations are not available for complex domains
and numerical methods must be used. The finite element
method has been the most widely used numerical method
for solving the equations governing blood flow.
The finite element method employed in the present
investigation is based on the theory of stabilized finite
element methods developed by Hughes and
colleagues.2,11,12 The method is described in detail by
Taylor et al. as it is implemented in the commercial
finite element program, SPECTRUM™.39 The essential features of stabilized methods in the context of incompressible flows are the simultaneous stabilization of the advection operator and the circumvention of the Babuska–
Brezzi inf–sup condition restricting the use of many
convenient interpolations, including the linear velocity,
linear pressure interpolations used in the present work.
The basic idea is to augment the Galerkin finite element
formulation with a least-squares form of the residual,
including appropriate stabilization parameters. These stabilization parameters are designed so that the method
achieves exact solutions in the case of one-dimensional
model problems involving, for example, the steady
advection-diffusion equation.
A semidiscrete formulation with a second-order accurate time-stepping algorithm is utilized resulting in a
nonlinear algebraic problem in each time step. This nonlinear problem is linearized and the resulting linear systems of equations are iteratively solved using conjugate
gradient and matrix-free GMRES solvers to reduce
memory requirements.13,35 As the flow computations
were performed on a parallel computer, the computational domain was divided into 16 subdomains using a
graph partitioning method.14 The nonlinear evolution
equations were solved for the velocity and pressure fields
over three cardiac cycles with 200 time steps per cardiac
cycle on an IBM SP-2 parallel computer. Analyses run
with a greater number of time steps showed no observable differences in the solution. The method employed
has been shown via analytical and experimental validation studies to yield accurate solutions for pulsatile flow
problems.39
In addition to the velocity field, quantitative values of
the magnitude of the surface tractions are also of interest.
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The traction vector can be computed from the stress
tensor, s, and surface normal vector, n, by t5 s n and
then the surface traction vector, ts , defined as the tangential component of the traction vector, can be computed from ts 5t2(t–n)n.
We can define the mean shear stress, t mean , a scalar
quantity, as the magnitude of the time-averaged surface
traction vector as
t mean5
UE U
1
T
T
0
ts dt ,
~1!
and define t mag , another scalar quantity, as the timeaveraged magnitude of the surface traction vector as
t mag5
1
T
E
T
0
u ts u dt.
~2!
Following He and Ku,8 we define the oscillatory shear
index ~OSI! as
OSI5
S
D
1
t mean
12
.
2
t mag
~3!
Note that in the special case of the steady, uniaxial
flow of a Newtonian fluid in a circular cylinder, the
magnitude of the surface traction reduces to the wall
shear stress, t w 54 m Q/ p r, 3 where m is the viscosity, Q
is the flow rate, and r is the lumen radius. However, due
to the complex flow fields in the vascular system, the
surface traction is a more general, and useful, measure of
surface forces as it is a vector quantity.
RESULTS
A contour slice of velocity magnitude and a vector
slice of the velocity field is displayed in Fig. 4 under
simulated resting conditions at three times in the cardiac
cycle along the midsagittal plane of the aorta. Note that
a vortex develops along the posterior wall of the aorta in
late systole and is present throughout diastole. The posterior wall of the aorta is a region with consistently low
flow velocities relative to the anterior wall. Also note the
orientation of the vector field towards the anterior wall,
observed even at peak systole, as a result of the celiac
and superior mesenteric outflows.
Figure 5 displays the mean surface traction vectors
along the anterior wall of the abdominal aorta. It is
observed that the direction of the mean surface traction
vectors is primarily in the superior to inferior direction
along most of the surface of the aorta with the exception
of the neighborhoods of the branch vessels. It is of note
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TAYLOR, HUGHES, and ZARINS
FIGURE 5. Mean surface traction vectors along the anterior
wall of abdominal aorta. Note the circumferential orientation
of the mean surface traction vectors near the branch vessels.
FIGURE 4. Midplane slice of the abdominal aorta model displaying contours of velocity magnitude and the velocity vector field at „a… peak systole, „b… end systole, and „c… middiastole. Note the low flow velocities and flow recirculation
along the posterior wall of the infrarenal aorta.
that the direction of the mean surface traction vectors is
from posterior to anterior in the neighborhood of the
ostia of the celiac and superior mesenteric arteries but
from anterior to posterior at the renal ostia. Figure 6
displays the mean surface traction vectors along the posterior wall of the abdominal aorta. It is noted that the
direction of the surface traction vectors is again primarily
from superior to inferior with the exception of the surface traction vectors on the aorta wall in the neighborhood of the branch vessels. It is noted that for the two
sites of relatively low mean wall shear stress along the
posterior wall, namely, the opposite of the celiac and
superior mesenteric arteries and distal to the renal arteries, that the direction of the mean surface traction vectors
is predominantly circumferential. Furthermore, opposite
the celiac and superior mesenteric arteries the mean surface traction vectors is from posterior to anterior whereas
the opposite is true distal to the renal arteries where the
mean surface traction vectors are from anterior to posterior. It is also of interest that the mean surface traction
vectors form concentric rings around the ostia of the
renal vessels as is apparent for the left renal artery in
Fig. 6. Figure 7 displays the mean surface traction vectors at the aortic bifurcation. Note that for the iliac vessel
wall opposite the aortic bifurcation flow divider, the direction of the mean surface traction vectors is from anterior to posterior along the anterolateral wall and from
posterior to anterior along the posterolateral wall.
Contour plots of mean wall shear stress, t mean , are
displayed in Fig. 8 from anterior and posterior views.
Note that under resting conditions a region of low mean
wall shear stress develops below the renal artery branch
and that, furthermore, shear stresses are high on the inside wall of the aortic bifurcation and low on the lateral
walls.
FIGURE 6. Mean surface traction vectors along the posterior
wall of abdominal aorta. Note the circumferential orientation
of the mean surface traction vectors along the posterior wall
of the aorta in the neighborhood of the renal arteries.
Finite Element Modeling of Flow in the Abdominal Aorta
FIGURE 7. Mean surface traction vectors at the aortic bifurcation. Note the orientation of the mean surface traction vectors along the lateral wall of the right iliac artery in the
neighborhood of the aortic bifurcation.
981
Mean wall shear stress, t mean , is plotted in Fig. 9 as
a function of arc length along the anterior wall. Note that
the gaps in the plots of shear stress correspond to the
location of the branch vessels. Observe that mean shear
stress increases significantly in the neighborhood of the
celiac, superior mesenteric and inferior mesenteric vessels, and that the level of shear stress in the distal infrarenal aorta exceeds the shear stress at the level of the
diaphragm.
Mean wall shear stress is plotted in Fig. 10 as a
function of arc length along the posterior wall. The mean
shear stress approaches zero opposite the celiac and superior mesenteric vessels and approximately 2 cm distal
to the left renal artery. The mean shear stress increases
in the distal infrarenal abdominal aorta to a level exceeding that of the diaphragm.
FIGURE 8. Mean shear stress contours along „a… the anterior wall and „b… the posterior wall. Note the regions of low shear stress
opposite the celiac and superior mesenteric arteries, in the infrarenal abdominal aorta and along the lateral wall of the aortic
bifurcation.
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TAYLOR, HUGHES, and ZARINS
FIGURE 9. Mean wall shear stress along the anterior wall
from the diaphragm to the aortic bifurcation. Note the low
mean shear stress in the infrarenal aorta between the superior and inferior mesenteric arteries „arc length of ;9 cm….
FIGURE 11. Oscillatory shear index „OSI… along the anterior
wall from the diaphragm to the aortic bifurcation. Note the
elevated OSI in the infrarenal aorta between the superior and
inferior mesenteric arteries „arc length of ;9 cm….
The OSI is plotted as a function of arc length in Figs.
11 and 12 along the anterior and posterior walls, respectively. Examination of the OSI along the anterior wall
reveals a sharp increase from zero in the suprarenal aorta
to approximately 0.3 immediately distal to the left renal
artery and then a gradual decline to a value of approximately 0.05 at the level of the aortic bifurcation. The
OSI along the posterior wall increases from zero in the
suprarenal aorta to approximately 0.4 opposite the superior mesenteric artery, decreases to less than 0.1 at the
level of the renal arteries, increases to a maximum value
FIGURE 10. Mean wall shear stress along the posterior wall
from the diaphragm to the aortic bifurcation. Note the low
mean shear stress opposite the celiac and superior mesenteric arteries „arc length of ;5.5 cm… and in the infrarenal
abdominal aorta „arc length of ;9 cm….
FIGURE 12. Oscillatory shear index „OSI… along the posterior
wall from the diaphragm to the aortic bifurcation. Note the
elevated OSI opposite the celiac and superior mesenteric
arteries „arc length of ;5.5 cm… and in the infrarenal abdominal aorta „arc length of ;9 cm….
Finite Element Modeling of Flow in the Abdominal Aorta
of 0.47 distal to the renal arteries, and then decreases to
a value of approximately 0.1 at the level of the aortic
bifurcation.
DISCUSSION
Numerical solution of pulsatile flow in the abdominal
aorta reveals complex hemodynamic patterns under
simulated resting conditions. The present approach to
characterize abdominal aorta hemodynamics enables the
extraction of complete spatial and temporal variations in
field quantities, including velocity and shear stress, for
correlation with observed patterns of vascular disease.
The significance of the findings regarding velocity field
and shear stresses is subsequently described, followed by
an examination of the assumptions employed in the
present investigation.
Two features of the velocity field in the infrarenal
abdominal aorta are worth noting. First, it is observed
that the velocity field in the infrarenal abdominal aorta is
directed preferentially towards the anterior wall. Second,
a flow recirculation region appears along the posterior
wall of the abdominal aorta in late systole and is present
throughout diastole. This recirculation region extends
from the level of the renal arteries to the inferior mesenteric artery. Moore et al.,22 using an in vitro model on
which the present numerical study was based, noted that,
under resting conditions, vortices appeared at the level of
the renal arteries and propagated through the infrarenal
aorta. Pedersen et al.,28 using laser Doppler anemometry,
observed an orientation of the velocity profile in the
posterior to anterior direction with the velocities being
highest at the anterior wall. Moore and Ku,23 using magnetic resonance imaging, noted extensive flow reversal
along the posterior wall and an orientation of the velocity profile in the posterior to anterior direction in the
infrarenal portion of an in vitro model. Complex velocity
profiles were also observed at the aortic bifurcation with
flow reversal and oscillation most prominent along the
lateral walls proximal to the bifurcation. Maier et al.,21
using Doppler ultrasound and magnetic resonance imaging techniques in humans, again noted greater anterior
flow velocity as compared to posterior velocity during
early diastole and sometimes during peak systole. Mostbeck et al.,27 using MR imaging of human subjects, observed that in the infrarenal aorta the mean duration of
retrograde flow was greatest along the posterior wall of
the aorta.
Greater flow velocity along the anterior wall relative
to that along the posterior wall and a flow recirculation
zone along the posterior wall of the infrarenal abdominal
aorta was observed in the computed velocity field. This
finding is consistent with prior in vitro and in vivo studies of flow in the abdominal aorta. It should be noted,
however, that the extent of the flow recirculation region
983
under resting conditions has not been reported previously
due to the fact that velocity measurements in the infrarenal aorta have been restricted to a single cross section.
The spatial distribution of the direction of the mean
surface traction vectors provides qualitative information
on the forces acting on the luminal surface. For most of
the abdominal aorta the direction of the mean shear
stress vectors is primarily in the superior to inferior direction, but four exceptions are noteworthy. First, in the
neighborhood of the ostia of the celiac and superior mesenteric arteries the mean shear stress vectors are radially
oriented towards the ostia. Second, in the neighborhood
of the renal arteries the shear stress vectors form concentric rings around the renal ostia. Third, for the two
sites of relatively low mean wall shear stress along the
posterior wall, opposite the celiac and superior mesenteric arteries and distal to the renal arteries, the mean
surface shear stress vectors are circumferentially oriented. Finally, the direction of the mean surface traction
vectors is from anterior to posterior along the anterolateral wall and from posterior to anterior along the posterolateral wall for the iliac vessel wall opposite the aortic
bifurcation flow divider. It should be noted that experimental methods which only measure the axial component
of velocity do not permit observation of these nonaxial
shear stress vectors.
Examination of the spatial distribution of the magnitude of the mean shear stress vector reveals several new
findings. It is noted that under resting conditions regions
of low mean wall shear stress develop along the lateral
and posterior walls opposite the celiac and superior mesenteric arteries and below the renal arteries. It is also
noted that mean shear stress increases in the distal abdominal aorta from a level above the inferior mesenteric
artery to the aortic bifurcation. This is presumably due to
the tapering of the aorta in this region. Although shear
stresses were reported by Moore et al.26 for two locations in the infrarenal abdominal aorta, the spatial distribution of shear stress magnitude has not been quantified
previously. Finally, as noted by Friedman et al.7 and
more recently by Moore et al.26 mean shear stresses are
high on the inside wall of the aortic bifurcation and low
on the lateral walls.
The plots of shear stress and oscillatory shear index
along the anterior and posterior walls provide quantitative data on the spatial variation of shear down the
length of the abdominal aorta. The mean shear stress in
the abdominal aorta at the level of the diaphragm was
calculated to be approximately 1.2 dynes/cm2. Moore
et al.26 measured the shear stress at this location to be
1.360.6 dynes/cm2 in an in vitro study with approximately the same flow conditions and model dimensions
as those used in this study. Along the anterior wall the
ostia of the celiac, superior mesenteric, and inferior mesenteric arteries are sites of relatively high mean shear
984
TAYLOR, HUGHES, and ZARINS
stress and low oscillatory shear index. As discussed subsequently, this is believed to be due to an inadequate
modeling assumption, incorporated into the idealized
model, whereby the junctures between the branch vessels
and the aorta are not smoothly tapered. The anterior wall
of the infrarenal aorta between the superior mesenteric
and inferior mesenteric arteries is a site of low shear
stress and high oscillatory shear index. Along the posterior wall, the mean shear stress is low and the oscillatory
shear index high opposite the superior mesenteric artery
as well as approximately 2 cm distal to the left renal
artery in the region noted for flow recirculation. The
distal abdominal aorta below the inferior mesenteric artery and proximal to the bifurcation is a site of relatively
high mean shear stress and moderate values of the oscillatory shear index.
As noted previously, vascular disease is more prominent in the abdominal aorta as compared with the thoracic aorta. The localization of early atherosclerotic lesions was examined by Cornhill et al.4 by staining with
Sudan IV and using image processing techniques to identify sudanophilic lesions in the cadaveric aortas of young
accident victims. It was noted that the highest
probability-of-occurrence of sudanophilia were associated with the inflow tracts of the celiac, superior mesenteric, renal, and inferior mesenteric ostia. In the present
study, along the aortic wall, these ostia were not observed to be sites of low mean shear stress, yet regions
of low mean shear stress were noted within the branch
vessels. Again, note that the model does not incorporate
the tapering of the branch vessels in the immediate vicinity of the ostia.
Cornhill et al.4 also noted a region of high
probability-of-occurrence of sudanophilia along the posterolateral wall of the infrarenal aorta and along the anterior wall between the superior and inferior mesenteric
arteries. Cornhill et al.4 state:
‘‘A major portion of areas of high probability of
sudanophilia, namely the extensive region along the
dorso-lateral aortic surface from the ductus scar to
the aorto-iliac bifurcation and the ventral surface of
the abdominal aorta midway between the renal and
inferior mesenteric ostia, does not occur in regions
expected to experience unusual hemodynamic
stresses.’’
The results presented herein clearly demonstrate otherwise. Namely, these regions experience low mean wall
shear stress and high oscillatory shear index. Finally,
Cornhill et al.4 show areas of high probability-ofoccurrence of sudanophilia along the posterior wall at the
level of the celiac and superior mesenteric arteries. As
noted previously, this is also a region of low mean wall
shear stress and high oscillatory shear index. Clearly,
direct correlative studies need to be performed between
hemodynamic conditions and plaque localization in the
abdominal aorta, but the results presented herein demonstrate that low mean wall shear stress and high oscillatory shear index occur in regions noted to have a high
probability-of-occurrence of sudanophilic lesions.
The major assumptions employed in this investigation
involve the anatomic dimensions of the model, the flow
conditions, a rigid wall approximation, and Newtonian
viscosity. The utilization of these assumptions, as for
prior in vitro investigations7,23–26,28 is believed to constitute a reasonable first approximation to the actual hemodynamic conditions in the abdominal aortas of normal,
healthy, humans at rest.
The anatomic dimensions employed in the present
study were based on those reported by Moore et al.22 and
include a tapering of the abdominal aorta and lumbar
curvature. Potential limitations of this representation include a relatively abrupt change in dimensions at the
branch ostia and the lack of curvature of the branch
vessels. Because of a lack of available anatomic data, the
tapering of the branch vessels in the neighborhood of the
ostia on the aortic wall was not incorporated in the geometric model. This lack of branch vessel taper is expected to significantly alter flow conditions in the neighborhood of the ostia and could explain the relatively high
shear stresses at these locations. The effect of this modeling assumption on shear stresses in the neighborhood
of the ostia needs to be investigated. Recent work by
Caro et al.3 argues that branch vessel tapering as well as
the nonplanar curvature of branch vessels beyond the
aortic ostia may have a significant influence on hemodynamic conditions. Curvature of the branch vessels was
also not considered in the present study, but the importance of including this feature of actual human anatomy
in aortic flow studies merits further investigation.
The physiologic conditions employed in the present
study were based on those reported by Moore et al.22 and
Moore and Ku,23 and were chosen to represent resting
abdominal aortic flow conditions, yet represent the lower
end of resting flow rates. Examination of greater resting
flow rates, while maintaining resting flow distribution,
merits further study. The physiologic conditions differ
from those reported by Moore et al.22 and Moore and
Ku23 in that the numerical method utilized herein requires the specification of the time-varying outflow for
the celiac, superior mesenteric, renal, and inferior mesenteric vessels. In contrast, only mean flow was determined
and reported by Moore et al.22 and Moore and Ku23 for
these branch vessels.
Three features of abdominal aortic flow waveforms,
consistently observed in normal subjects under resting
conditions, were used in the present study. First, the flow
rate is observed to be positive throughout the cardiac
cycle at the entrance to the abdominal aorta at the level
Finite Element Modeling of Flow in the Abdominal Aorta
of the diaphragm. It was noted by Holenstein and Ku9
that 32 of 35 young, normal subjects had no reverse flow
as measured by Doppler ultrasound approximately 2–4
cm proximal to the superior mesenteric artery. Bogren
and Buonocore1 used MR cine velocity measurements to
quantify separately antegrade and retrograde flow and
noted low retrograde and positive net flow in the distal
descending thoracic aorta. It is important to note that,
due to the nature of pulsatile flow, some retrograde flow
can appear near the wall during early diastole even in the
presence of a positive net flow. In a further study of
abdominal aortic flow, Moore et al.,25 using MR velocimetry, noted only a slight flow reversal at end systole in
the supraceliac aorta in four healthy volunteers. The second feature of abdominal aortic flow observed in normal
subjects under resting conditions is the triphasic flow
waveform in the infrarenal aorta. This has been noted in
numerous studies employing doppler ultrasound and
MRI methods for velocity quantification. Holenstein and
Ku9 noted that 34 of 35 young, normal subjects had
reverse flow at end systole as measured by Doppler ultrasound. Bogren and Buonocore1 and Moore et al.25 also
noted reverse flow at end systole using MR cine velocity
measurements in normal, healthy subjects. The third feature of abdominal aortic flow observed in normal subjects under resting conditions is a positive net renal artery flow throughout the cardiac cycle and relatively high
diastolic flow. Holenstein and Ku9 noted positive left
renal artery flow and an absence of reverse flow using
Doppler ultrasound. Bogren and Buonocore1 measured
blood flow in the abdominal aorta above and below the
renal arteries and noted a total renal artery flow of approximately 1 L/min. Pelc et al.29 used PC-MRI techniques to directly measure renal blood flow in a normal
healthy subject and observed flow rates in excess of 0.6
L/min throughout the cardiac cycle.
The determination of appropriate boundary conditions
and flow waveforms for investigations of abdominal
aorta hemodynamic conditions merits further attention.
In addition to the specification of flow waveforms, assumptions are required for the variation in velocity at the
boundaries where flow is specified. The velocity profiles
at the inlet to the abdominal aorta at the level of the
diaphragm, and the celiac, superior mesenteric, renal,
and inferior mesenteric artery outflows were specified
using Womersley theory for pulsatile flow, resulting in
an axisymmetric velocity profile. Moore et al.25 confirmed the validity of the assumption of axisymmetric
flow in the supraceliac abdominal aorta. This condition
was also utilized in the model flow studies reported by
Pedersen et al.28 and Moore and Ku.23 Regarding the
outflow velocity profiles, the length of the branch vessels
was chosen to minimize the effect of the outlet boundary
condition on the flow in the aorta. It is likely that the
assumption of planar branch vessels in the vicinity of the
985
ostia on the aorta has a greater effect on the flow conditions in the aorta than the axisymmetric outlet velocity
profile, but this was not examined in the present study.
The effect of aortic compliance was not considered in
the present investigation. Prior investigations of flow in
deformable models have shown that, in general, incorporating the effect of compliance alters the magnitude of
shear stress, but does not change the locations of low
shear and high shear regions. Duncan et al.5 examined
flow in a deformable model of an aortic bifurcation and
showed that at the outer wall of the aortic bifurcation the
shear rates were reduced in the deformable model as
compared to a rigid model. The radial deformation measured in this model flow study was on the average of
2%–4%. In an investigation of the effect of wall compliance on pulsatile flow in the carotid artery bifurcation,
Perktold and Rappitsch32 describe a weakly coupled
fluid-structure interaction finite element method for solving for blood flow and vessel deformation. They concluded that the wall shear stress magnitude decreases by
approximately 25% in the distensible model as compared
to the rigid model, yet the overall effect on the velocity
field was relatively minor. Steinman and Ethier,37 based
on a study of a compliant two-dimensional end-to-side
anastomosis, concluded that the effects of wall distensibility on the flow field are less pronounced as compared
to changes in anatomic dimensions or physiologic conditions. It should be noted that although neglecting wall
distensibility may not have a major effect on the primary
flow field, the incorporation of wall mechanics in these
studies is important for other reasons including the description of the stress environment within the vessel
walls and the interaction between deformability and mass
transport phenomena. In summary, the use of rigid models for flow studies in idealized models of the human
abdominal aorta should be viewed as a first approximation.
A Newtonian constitutive model for viscosity was
employed in the present investigation. It is generally
accepted that this is a reasonable first approximation to
the behavior of blood flow in large arteries. Perktold
et al.31 examined non-Newtonian viscosity models for
simulating pulsatile flow in carotid artery bifurcation
models. They concluded that the shear stress magnitudes
predicted using non-Newtonian viscosity models resulted
in differences on the order of 10% as compared with
Newtonian models. Future studies of abdominal aorta
hemodynamic conditions should assess the effects of
non-Newtonian rheologic models.
The accuracy of the finite element method for solving
the governing equations was not examined quantitatively
in the present study due to a lack of published data. In
other models Taylor et al.40 examined the accuracy of
this method employed for blood flow computations. Numerical results were found to be in excellent agreement
TAYLOR, HUGHES, and ZARINS
986
with Womersley theory and with laser Doppler anemometry velocity data obtained by Loth20 for steady and
pulsatile flow in a model of an end-to-side anastomosis.
It was not possible to examine further refinements in the
finite element mesh employed due to the limitation of
computational resources. However, the stabilized finite
element method employed has been shown to exhibit
excellent coarse grid accuracy.40 It is not expected that
further refinements in the mesh will result in substantial
quantitative differences in the computed solution.
CONCLUSIONS
The characterization of the temporal and spatial variations of the velocity field in an idealized model of the
abdominal aorta enables new insights into the wall shear
stresses acting on the luminal surface under resting conditions and should help resolve the question of which
hemodynamic factors correlate with the disease localization patterns observed in cadavers and in vivo. Regions
of low mean wall shear stress and high oscillatory shear
index were noted along the anterior wall between the
superior mesenteric and inferior mesenteric arteries and
along the posterior wall opposite the superior mesenteric
artery and approximately 2 cm distal to the renal arteries.
These locations were also characterized by Cornhill
et al.4 as having high probability-of-occurrence of
sudanophilic lesions. The numerical studies described
herein provide further impetus to examine abdominal
aorta hemodynamics in vivo using magnetic resonance
imaging techniques, and to assess the validity of the
assumptions made regarding vascular anatomy and
physiologic conditions, blood rheology, and vessel mechanics.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the assistance of
Jeff Buell, Mary Draney, David Parker, and Arthur Raefsky. Software was provided by Centric Engineering Systems, Inc., TechnoSoft, Inc., XOX Inc., and Rensselaer
Polytechnic Institute’s Scientific Computing Research
Center. Hewlett-Packard provided a model 755/125
workstation used in this research.
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