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Numerical Simulations of Time-Dependent non-Newtonian Blood Flow through Typical Human Arterial Bypass Grafts.

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Dev.Chem. Eng. Mineral Process., 7(1/2),pp. 179-2O0, 1999.
Numerical Simulations of Time-Dependent,
non-Newtonian Blood Flow through Typical
Human Arterial Bypass Grafts
J.S. Cole*, M.A. Gillan, S. Raghunathan
Department of Aeronautical Engineering, The Queen ’s University of
Berfast, Belfast, BT9 5AG, Northern Ireland, UK
M.J.G. O’Reilly
Vascular Unit, Belfast City Hospital, Belfast, BT9 7A3, Northern
Ireland, UK
The development of the detrimental feature of intimal hyperplasia at the locations
where bypass grajis are surgicaIIy attached to host arteries is believed to be promoted
by haemodynamic factors. Thus, it is of interest to model the flow of blood through
typical arterial bypass configurations in order to identi& which features of the flow
$eId encourage the progression of the disease.
Computational Fluid Dynamics (CFD) simulations of the non-Newtonian flow of
blood through typical three-dimensional human femoral artery bypass graji models
have been performed.
The complete bypass configuration, rather than just the
proximal or distal junction in isolation, has been analysed. Steady flow studies
verx$ed the strong dependence of the anastomoticflowfield on the problem geometry.
Flow disturbances were minimised at low anastomotic angles. A pulsatile flow
computation, using a realistic femoral artery jlow pulse, demonstrated noteworthy
temporal and spatial variations in the jlow $el& at the proximal and distal
anastomoses during the cardiac cycle. Due to the oscillations in direction of thejlow
at the distal anastomosis, and given the persistent zones of low momentum
recirculating fluid, it is concluded that jluid particle residence times in the
neighbourhood of the distal anastomosis are high. This feature may be of
signrfcance with regard to haemodynamic mechanifmsfor intimal hyperplasia.
*Authorfor correspondence
J.S. Cole et al.
The bypassing of critically occluded arteries using biological or prosthetic grafts is a
common surgical technique for alleviating life-threatening angina or limb-threatening
However, the detrimental feature of intimal hyperplasia is largely
responsible for the early and medium term failure of bypass grafts.
hyperplasia is manifested by the progressive thickening of the innermost layer of the
arterial wall, with the resultant narrowing of the vessel lumen. The disease is
observed to have a focal nature, occurring exclusively at the suture line where the
graft is surgically attached to the host artery, and on the floor of the host artery
opposite the junction [11. Furthermore, the disease is much more likely to arise at the
distal anastomosis rather than at the proximal junction. Figure 1 displays a typical
arterial bypass configuration and indicates the sites of the preferential development of
intimal hyperplasia.
intimal hyperplasia
.. .. ..
. . .
floor I bed
Figure 1. Arterial Bypass Configuration and Observed Distribution of Intimal
Numerical simulations of non-Newtonian blood flow
The exact mechanisms for the development of the disease are unclear. In addition
to biological and chemical factors, it is believed that haemodynamic and mechanical
influences exist [2].
In vivo studies suggest that the compliance mismatch between the graft and host
artery promotes anastomotic intimal hyperplasia [3, 41. The ratio of graft to host
artery diameter may also be an important factor with regard to future graft patency
[4-61, while a small anastomotic angle is necessary for minimising flow disturbance at
the distal anastomosis 171.
Many in vitro investigations have been concerned with predicting the flow field in
separate models of the proximal and distal anastomoses. Flow visualisation studies
[S-1 11 have confirmed that the anastomotic flow patterns are strongIy dependent on
the model geometry and flow conditions.
The wall shear stress is thought to be a prominent haemodynamic variable
associated with intimal hyperplasia.
Ojha et a/. [12, 131 have observed a
complicated, time-dependent shew stress distribution in the vicinity of the
anastomosis and suggest that the movement of the distal host artery floor stagnation
point during the flow cycle may be significant. Intimal hyperplasia may be the
adaptive response of the arterial wall to the unphysiological flow patterns which
occur at the bypass graft anastomoses [I]. Arteries are known to adjust their diameter
in order to maintain a wall shear stress in the range 1-2 Pa [14].
With recent advances in computer technology and in numerical methods, it has
become possible to conduct numerical simulations of blood flow through models of
arterial structures. Typical investigations [ 15, 161 have highlighted the complex flow
patterns in the anastomotic region with secondary motions and zones of recirculating
fluid being detected. The flow field is strongly determined by the anastomotic angle.
The modelling of arterial blood flow is a challenging problem since that
phenomenon exhibits many special properties [17-191. In general, the flow field is
three-dimensional and laminar, with turbulence being detected only in the aorta as
blood leaves the heart. The pulsatile motion of the blood is due to the pumping action
of the heart. Blood is a unique fluid, being a suspension of cells and platelets in a
J.S. Cole et al.
plasma. It is described as non-Newtonian pseudoplastic with its viscosity decreasing
with increasing shear rate. Another elaborate feature is the distensibility of the
The arterial walls are inhomogeneous, anisotropic, non-linear and
viscoelastic. It is agreed [20] that in a study of arterial blood flow, it is of primary
importance to employ a realistic geometry and appropriate flow parameters, and to
account for flow pulsatility . Wall compliance and the non-Newtonian rheology of
blood are believed to have secondary influences on the haemodynamics in medium
and large arteries.
The current investigation uses the method of Computational Fluid Dynamics
(CFD) to perform numerical simulations of the three-dimensional, non-Newtonian
flow of blood through typical models of a human femoral artery bypass. Both steady
and pulsatile computations are described. A detailed analysis of the bypass flow field
is outlined and the dependence of the flow features on the anastornotic angle is
discussed. In contrast to previous studies, flow through the complete bypass
configuration has been computed, rather than that at either the proximal or distal
anastomosis in isolation.
Model Geometry
The symmetry plane of one of the rigid femoral artery bypass models is illustrated in
Figure 2. The host artery, representative of a human femoral artery, has a diameter of
8 mm and is assumed to be fully occluded. The bypass graft is a circular tube of
diameter 6 mm, symbolising a synthetic graft. For the model depicted in Figure 2,
the graft is attached at an angle of 20" to the host artery at both the proximal and
distal anastomoses. It is felt that, due to the obvious physical constraints, this is the
smallest possible angle at which the surgeon can suture the graft to the artery. The
graft is of length 15 cm and it possesses smooth curvature in the symmetry plane.
Two similar models were constructed having anastomotic angles of 30" and 45".
Since the occluded section of the host artery is assumed to have the same length in
each case, the graft length increases with increasing anastomotic angle.
Numerical simulations of non-Newtonian bloodflow
outer / far wall
host artery
bypass graft outer wall
inner / near wall
Figure 2. Symmetry Plane of the Bypass Model. Letters A and B indicate
cross-sectionalplanes where results are plotted.
It is claimed [21, 221 that the idealisation of rigid arterial walls can be made when
only local flow patterns in short segments of large arteries are of interest. These
conditions prevail in the current study.
Numerical Model and Flow Conditions
In this study, blood is assumed to be an incompressible, non-Newtonian fluid of
density 1050 kg m-3and viscosity, bf,given by the power law:
where k = 0.042 Pa s" and n = 0.61. Blood flow through the femoral artery bypass
has the attributes of three-dimensional, time-dependent, incompressible, isothermal,
laminar flow. The governing equations for such a flow are:
v - u= 0
continuity equation
J.S. Cole et al.
p -+ p( u V)u = -vp + V
. (pd vu)
Navier-Stokes equations ...(3)
The commercial CFD flow solver FLUENT [23], based on the finite volume
method [24], was used to obtain a numerical solution to the governing equations. The
structured physical mesh overlaying the model is presented in Figure 3.
Figure 3. Structured Physical Grid.
Steady flow computations were performed in each of the three geometrical models
at an inflow Reynolds number (based on host artery diameter, average inlet velocity,
and a reference blood viscosity' of 0.0035 kg m-' s-') of 408. (* This is the viscosity
of blood at high shear rates when its behaviour approaches that of a Newtonian fluid).
For the 20" bypass model, pulsatile flow was also simulated. A time-dependent
inflow velocity, calculated fi-om the representative femoral artery flow pulse [25]
(Figure 4), was specified at the inlet boundary. The mean flow rate of 2.25 ml s-'
corresponds to a mean inflow velocity of 4.48 cm s-' or a mean inflow Reynolds
Numerical simulations of non-Newtonian blood flow
number of 107. The maximum Reynolds number during the cycle is 830. The pulse
frequency is 75 cycles per minute. Each pulse cycle was discretised into 160 time
time (s)
Figure 4. Femoral Artery Volume Flow Pulse.
For all computations, FLUENT’S OUTLET condition was stipulated at the
outflow boundary, enforcing fully developed flow at that location. In this work, the
term “fully developed” implies that the flow velocity profile is unchanging in the
flow direction.
In order to validate the numerical model, steady and pulsatile, Newtonian and
non-Newtonian flows of blood through a generic carotid artery bifurcation model
have been performed [26, 271 with excellent agreement being attained with results
published in the literature [28-301.
J.S. Cole et al.
Results and Discussion
A. Steady Flow
As illustrated in Figure 5a, the flow exhibits a fully developed nature on approach to
the proximal anastomosis. Separation takes place from the floor of the host artery
opposite the entrance to the graft (label SW), with most of the fluid being directed
into the bypass. The fluid in the graft is accelerated, the graft cross-sectional area
being smaller than that of the artery. Just upstream from the occlusion, there is low
speed, recirculating fluid (label SX). It is observed that, after travelling upstream
along the outer wall of the host artery, these fluid particles are pulled above and
below the symmetry plane, across the artery, and into the bypass, alongside its outer
As a result of centrifugal forces acting on the blood on its passage through the
graft, the fluid possessing high momentum is forced towards the bypass outer wall so
that, on arrival at the distal anastomosis (Figure 5b), the velocity profiles in the graft
are skewed. Secondary motions are present within the downstream half of the graft.
On exiting the graft, the fluid is forced across the host artery with separation
taking place at the distal toe (label SY). A stagnation point (label SZ) is identified on
the floor opposite the toe, but within the host artery upstream of the distal
anastomosis, the recirculation region contains almost stagnant fluid.
Figure 5c shows that at Site A in the distal host artery, there is secondary motion;
at the centre of the lumen, fluid is moving towards the outer wall of the artery, while
fluid particles at the outer wall follow circumferential paths towards the opposite
wall. The secondary flow decays with increasing distance downstream (Figure 5d).
The combination of axial and secondary flows implies that the blood in the distal host
artery travels downstream along a pair of counter-rotating helical paths, one above
and one below the symmetry plane.
Figures 6 and 7 depict the steady flow fields in the 30" and 45" bypass models
respectively. At the entrance to the graft, the velocity profiles become more skewed
towards the bypass inner wall with increasing anastornotic angle.
differences in the flow patterns at the distal anastomosis are clear. For the 30" model,
Numerical simulations of non-Newtonian blood jlow
(a) Proximal Anastomosis
40 cmd'
(b) Distal Anastomosis
. . . . . .
. . . . . ........
40 cmd'
Figures S(a) and (6). Velocity Vectors in the 20 Bypass Model Symmetry Plane and
in Cross-sectional Planes A and B. (In this and subsequentfigures, the arrow length
is directly proportional to the fluid speed)
J.S. Cole et al.
(d) Site B
(c) Site A
\++ ++ ++ ++ ++ ++ ++ ++ ++ +G +Y +./.
Figures S(c) and (d).
there is a jet of high momentum fluid alongside the host artery's outer wall, while a
large separation region containing some reversed flow develops just downstream of
the toe. These features are enhanced in the 45' model. The floor stagnation point
moves slightly upstream with increasing anastomotic angle as the fluid leaving the
graft impinges more directly on the host artery floor. Figures 6c and 7c confm that
secondary flow in the distal host artery is greatly augmented in the larger angled
The general characteristics of the computed flow fields correspond with those
attained under similar flow conditions, but employing simpler geometries [ 161. The
observation that flow disturbances are increased when the anastomotic angle is larger
is in accord with the conclusion of Staalsen et al. [7].
Numerical simulations of non-Newtonian blood flow
B. Pulsatile Flow
During the cardiac cycle, the flow patterns at the proximal and distal anastomoses
exhibit very complicated temporal and spatial variations. Since intimal hyperplasia is
much more likely to develop at the distal anastomosis, this work concentrates on
reporting the flow features at the distal junction.
In the early part of the pulse cycle, when flow acceleration takes place in the
femoral artery, relatively undisturbed flow patterns are observed at the distal
anastomosis (Figure 8a).
Separation does not occur at the toe.
A region of
recirculating fluid possessing low momentum exists between the junction and the
occlusion (label PW).
At the maximum flow rate, it is clear that within the graft, the higher momentum
fluid has been forced towards the outer wall (Figure 8b). On returning to the host
artery, fluid particles are directed across the lumen, with separation just taking place
at the toe (label PX). The floor stagnation point has moved downstream (label PY).
The flow field at the distal anastomosis is altered dramatically as the femoral artery
flow decelerates sharply (Figure 8c). The zone of recirculating fluid close to the heel
of the distal junction has extended downstream and is now centred opposite the
anastomosis. Some fluid particles leave this region, being pulled upstream along the
inner wall of the bypass. The separation region downstream of the junction is greatly
augmented (label PZ). Downstream fluid motion is now confined to a narrow jet
traversing alongside the far wall of the host artery.
As the femoral artery flow rate is reduced further, flow reversal in the bypass, and
downstream of the distal anastomosis, is enhanced. The vortex of fluid particles
rotating anticlockwise has moved towards the toe (Figure 8d).
During the rest of the cycle, the flow velocities are much lower than those detected in
the earlier stages. Within the graft, and downstream of the junction, the fluid
oscillates in direction as small accelerations and decelerations take place upstream in
the host artery. The size and location of the recirculation region in the proximity of
the distal heel also continue to vary during this latter part of the cycle.
J.S. Cole et al.
(a) Proximal Anastomosis
(b) Distal Anastomosis
. .. .. . . ,
. . .
. . .
. .. .. .. ,
. . .
Figures 6(a) and (b). Velocity Vectors in the 30 Bypass Model Symmetry Plane and
in Cross-sectional Planes A and B.
Numerical simulations of non-Newtonianblood flow
(c) Site A
(d) Site B
Figures 6(c) and (d).
J.S. Cole et al.
(a) Proximal Anastomosis
(b) Distal Anastomosis
Figures 7(a) and (6). Velocity Vectors in the 45 O Bypass Model Symmetry Plane and
in Cross-sectional Planes A and B.
Numerical simulations of non-Newtonianblood flow
(c) Site A
(d) Site B
Figures 7(c) and (d).
J.S. Cole et al.
The existence of secondary flow in the host artery downstream of the distal
anastomosis is also dependent on time, During the initial acceleration phase of the
pulse, the flow streamlines in the distal host artery are axially aligned (Figure 9a).
However, as deceleration takes place in the femoral artery, secondary motions
downstream of the distal junction become very significant (Figure 9b).
particles close to the artery’s symmetry plane travel across the channel towards the far
wall, while at the side walls, the fluid follows curved paths in the opposite direction.
Given the oscillations in direction of the fluid flow at the distal anastomosis, and
the persistent recirculation regions in that locality, it is clear that fluid particle
residence times will be higher than those to be expected under normal physiological
This feature of enhanced residence times, and the adverse, time-
dependent wall shear stress distribution at the anastomosis, have been mentioned
[2, 8, 121 with regard to possible haemodynamic mechanisms for the progression of
distal anastomotic intimal hyperplasia.
Studies such as this one, which provide a detailed description of the anastomotic
flow field, are important as attempts continue to determine which aspects of the
haemodynamics promote the progression of the disease. If these detrimental flow
conditions are identified, then efforts can be concentrated on modifying the bypass
flow field, possibly by improving bypass graft design, so that the anastomotic regions
will be less susceptible to disease. If successful, then the consequences for the
patient, in terms of a better quality of life, are obvious.
Numerical simulations of non-Newtonian bloodjow
(a) Time 0.08s
(b) Time 0.12 s
Figures 8(a) and (6). Velocity Vectors in the 20 O Bypass Model Symmetry Plane at
Various Times During the Pulse Cycle.
J.S. Cole et al.
( c ) Time 0.2 s
(d) Time 0.24 s
40 cms-'
Figures 8(c) and (d).
Numerical simulations of non-Newtonian blood flow
(a) Time 0.08 s
(b) Time 0.2 s
Figure 9. Seccm d u y Flow at Site A at Vurious Times During the Pu lse Cycle.
J.S. Cole et al.
Steady flow simulations have demonstrated that the flow patterns at the anastomoses
are very strongly influenced by the configuration geometry. At larger anastornotic
angles, flow disturbances are augmented.
The time-dependent computation
established that significant temporal and spatial variations occur in the anastomotic
flow fields.
The importance of modelling the complete bypass configuration, rather than
simply the proximal or distal anastomosis in isolation, was confirmed since the flow
within the graft does not attain a fully developed nature. Secondary motions were
observed as the blood passed through the graft.
The results indicate the tremendous potential for the application of CFD
simulations to the investigation of problems in medicine and in the medical device
and other related industries.
non-Newtonian power law parameter
non-Newtonian power law parameter
velocity vector
Greek letters
shear rate
dynamic viscosity
( m s-' )
Numerical simulations of non-Newtonian blood flow
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