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J. Biomedical Science and Engineering, 2013, 6, 1109-1116 JBiSE
http://dx.doi.org/10.4236/jbise.2013.612139 Published Online December 2013 (http://www.scirp.org/journal/jbise/)
Mass deposition and fluid flow in stenotic arteries:
Rectangular and half-circular models
Dipak Kumar Mandal1, Somnath Chakrabarti2
1Department of Mechanical Engineering, College of Engineering and Management, Kolaghat, India
2Department of Mechanical Engineering, Bengal Engineering and Science University, Shibpur, India
Email: [email protected], [email protected]
Received 7 September 2013; revised 15 October 2013; accepted 29 October 2013
Copyright © 2013 Dipak Kumar Mandal, Somnath Chakrabarti. This is an open access article distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the
owner of the intellectual property Dipak Kumar Mandal, Somnath Chakrabarti. All Copyright © 2013 are guarded by law and by
SCIRP as a guardian.
ABSTRACT
Mass deposition inside the artery wall may play a
significant role in the development of the disease
atherosclerosis. Locally elevated concentrations of
LDL in the arterial wall are considered to be the ini-
tiator of atherosclerotic plaque formation. In this
study, an attempt has been made to study initially the
effect of fluid dyna mic parameters on the disease and
finally proposed a concept, from the idea of basic flow
characteristics in constricted arteries, for the assess-
ment of mass deposition in the arterial wall to some
extent for rectangular as well as half circular ste-
nosed models. Reynolds numbers are chosen as 100,
200, 300 and 400 and percentage of restrictions as
30%, 50%, 70% and 90% respectively. The govern-
ing Navier-Stokes and continuity equations are solved
in the artery lumen with the commercial CFD code
ANSYS 12.1. The pressure-velocity coupling equa-
tions are solved by SIMPLE (Semi-Implicit Method
for Pressure-Linked Equations) algorithm. The stud-
ies on pressure drop at stenosis zone and flow separa-
tion zone reveal that the effect of percentage of re-
striction is more dominant than Reynolds number on
the progression of the disease, atherosclerosis for any
shaped restriction. The mass deposition results of
rectangular and half circular stenotic models moti-
vate to conclude that the effect of percentage of re-
striction is more prone to the disease than that of
Reynolds number. Half circular stenotic shape insists
for the less chance of mass deposition in the arterial
wall compared to rectangular shaped restriction.
Keywords: Atherosclerosis; Deposition; Pressure Drop;
Velocity Contour
1. INTRODUCTION
Atherosclerosis is a disease of the coronary, carotid, and
other proximal arteries that involve a distinctive accu-
mulation of large molecules such as low-density lipopro-
tein (LDL) and other lipid-bearing materials due to mass
transport in the arterial wall. This deposition of low den-
sity lipoprotein (LDL) has received considerable interest
in the recent years due to its significant work in the early
stages of atherosclerosis. It is a progressive disease char-
acterized by localized plaques that form within the artery
wall. As the disease progresses, these plaques enlarge
and, either directly or indirectly, lead to impairment of
blood flow in the concerned arterial system or artery.
This in turn can have serious consequences, such as
blockage of the coronary arteries (leading to myocardial
infarction), femoral arteries and carotid arteries (leading
to strokes as plaques detach and occlude the cerebral
vasculature). Fluid dynamics of blood (hemodynamics)
is considered to play a major role in the initiation and
progression of atherosclerosis. The hemodynamic be-
haviour of the blood flow in arterial stenoses bears some
important aspects due to engineering interest as well as
feasible medical applications.
Stangeby and Ethier [1] have reported the fluid flow
and mass transfer of LDL in a stenotic artery. The results
show an elevated LDL concentration at the downstream
side of the stenosis. The mass transport on symmetric
and non-symmetric stenotic artery models was studied by
Kaazempur-Mofrad et al. [2]. The complex flow field
due to the non-symmetric stenosis affects the mass trans-
fer patterns substantially different from those exhibited
by the symmetric stenosis. Chakravarty and Sen [3] have
performed the study of the blood flow and convection-
dominated diffusion processes in a model bifurcated ar-
OPEN ACCESS
D. K. Mandal, S. Chakrabarti / J. Biomedical Science and Engineering 6 (2013) 1109-1116
1110
tery under stenotic conditions. In their work, they have
studied the effects of constricted flow characteristics and
the wall motion on the wall shear stress, the concentra-
tion profile and the mass transfer. The influences of WSS
on LDL transport by modeling the blood flow and solute
transport in the lumen and arterial wall are investigated
by Nanfeng Sun et al. [4]. The unsteady non-Newtonian
blood flow and mass transfer in symmetric and non-
symmetric stenotic arteries are numerically simulated by
Valencia and Villanueva [5], where they have considered
the fluid-structure interaction in their simulation. Valeria
et al. [6] have made a simple model for the plaque
growth rate based on the accumulation of oxidized LDL
in the artery wall, including a dependence of the endo-
thelial permeability on the shear stresses, and on the
LDL blood concentration. Sun et al. [7] have studied the
influence of pulsatile flow on LDL transport and exam-
ine the validity of steady flow assumption. Yang and
Vafai [8,9] have developed a robust multi-layer porous
model for the description of the mass transport in the
arterial wall coupled with the mass transport in the arte-
rial lumen. From their study, they have concluded that
the LDL transport in the arterial wall is one-dimensional
when a straight circular geometry is considered. Olgac et
al. [10] have studied the effects of the local arterial wall
shear stress distribution on the endothelial cell layer in
order to accurately calculate LDL transport. They have
chosen a realistic 3-D coronary artery segment with a
single-layer porous arterial wall. Khakpour and Vafai [11]
have critically assessed the different arterial transport
models as far as the recent research activities on this is-
sue are concerned with the help of the relevant governing
equations in the study of fluid flow and mass transfer
within the arteries by giving emphasis on the role of po-
rous media.
From the available literature, it is noted that very little
work has been done on the issue of quantifying the mass
deposition of the plaque in the arterial wall. Therefore, in
the present study, an attempt has been made to study
initially the effect of fluid dynamic parameters on the
disease, atherosclerosis and finally proposes a concept,
from the idea of basic flow characteristics in constricted
arteries, for the assessment of mass deposition in the
arterial wall to some extent. The effect of Reynolds
number and percentage of restriction on pressure drop
through stenosis, velocity contour, velocity vector, and
mass deposition in the wall is studied for the rectangular
as well as half circular restrictions. Reynolds numbers
are chosen as 100, 200, 300 and 400 and percentages of
restrictions as 30%, 50%, 70% and 90% respectively.
2. METHOD
The governing Navier-Stokes and continuity equations
are solved in the artery lumen with the commercial CFD
code ANSYS 12.1. The pressure-velocity coupling equa-
tions are solved by SIMPLE (Semi-Implicit Method for
Pressure-Linked Equations) algorithm. Least square cell
based gradient, pressure: standard, momentum: first or-
der momentum, are chosen for spatial discretization. The
under relaxation factors for pressure as 0.3, density as
1.0 and momentum as 0.7 are considered during simula-
tion. The flow under consideration has been assumed to
be steady, two-dimensional, laminar and axisymmetric,
and fluid is considered to be Newtonian and incom-
pressible. Uniform velocity at inlet, no slip condition at
wall and zero pressure at exit are used in our simulation.
Since coronary artery is an important artery for the dis-
ease of atherosclerosis, therefore the coronary artery (dia
= 4 mm) is taken for simulation. The density of blood
and viscosity are considered as 1056 Kg/m3 and 0.0035
Pas respectively. The schematic diagrams of the compu-
tational domain selected for our study are illustrated in
Figures 1(a) and (b) respectively. Total length of the
computational domain is taken as 0.2 m. Restriction is
placed at the middle of the computational domain.
The results are generated for different Reynolds num-
bers of 100, 200, 300 and 400 and percentage of restric-
tion of 30%, 50%, 70% and 90%. The details of nodes
and elements, considered during our study, are as follows
in Table 1.
3. RESULT AND DISCUSSIONS
One of the most serious consequences of an arterial
stenosis is the large pressure loss which may develop
across a severe stenosis. The reduced pressure distal to
the stenosis significantly alters the blood flow to the pe-
ripheral blood supplied by the artery. Atheromatuos
plaques appear in the regions of low pressure because a
suction action exerted on the surface endothelium even-
tually causes the layer to be selectively separated from
adjacent tissue. This tearing action is thought to cause
damage, in turn, to the endothelium and adjacent wall
layers, with subsequent thickening of the intima and
eventual plaque development [12]. The pressure loss is
primarily dependent on the flow rate and the geometry of
the stenosis due to relatively constant fluid properties of
density and apparent viscosity. Moreover, the initiation
and progression of atherosclerosis are dependent on the
accumulation of LDL in the artery wall. One of the bio-
mechanical forces of the chance of the deposition is de-
pending on transmural flux. Moreover, the recirculation
zone in the post stenotic region is considered to be an
important phenomenon for fluid flux [13]. Since the fluid
flux also depends on pressure of the blood, therefore, the
pressure of blood at any section of the concerned artery
and the pressure drop across the restricted zone may be
thought to offer an idea to some extent regarding the
formation and propagation of atherosclerosis. The physi-
Copyright © 2013 SciRes. OPEN ACCESS
D. K. Mandal, S. Chakrabarti / J. Biomedical Science and Engineering 6 (2013) 1109-1116 1111
0.002 m
0.001 m
Centre line
Rectangular Restriction
x
y
Inlet
0.1 m
Artery wall
Exit
0.1 m
0.002 m
21
1 2
(a)
0
.
001 m
0.002 m
Centre line
Half circular Restriction
x
y
Inlet
0.1 m
Artery wall
Exit
0.1 m
0.002 m
21
1 2
(b)
Figure 1. (a) Computational domain for 50% rectangular re-
striction. (b) Computational domain for 50% half circular re-
striction.
Table 1. Details of nodes and elements of the computational
domain.
PR = 30% PR = 50% PR = 70% PR = 90%
Nodes 32,033 32,961 34,473 35,818
Elements 152,419 157,389 165,269 172,545
ological significance of the recirculation zone is that the
bloodstream stagnates locally in this area and allows
platelets and fibrin to form a mesh at the inner wall in
which lipid particles become trapped and eventually
coalesce to form atheromatous plaque, this may tend to
accumulate to cause a more severe stenosis [14]. After
the recirculation zone, the blood reattaches on the arterial
wall. This point of reattachment having the significance
on the formation and propagation of atherosclerosis. The
high cell turnover rate takes place near the reattachment
point due to high cell division and cell density near that
region. For this, a leaky junction may develop which is
considered to be the possible pathway for transport of
LDL through the arterial wall [15]. This phenomenon
may be thought to assist the deposition. The recirculation
zone can also be conceived with the use of velocity con-
tour and velocity vectors. The quantification of mass
deposition of macromolecules inside the arterial wall
may be considered to be an important parameter in as-
sessing the extent of disease, atherosclerosis. In the pre-
sent numerical work, the effect of Reynolds number and
percentage of restriction on pressure drop through steno-
sis, velocity contour, velocity vector, and mass deposi-
tion in the wall is studied for our considered rectangular
as well as half circular restrictions. Reynolds numbers
are chosen as 100, 200, 300 and 400 and percentage of
restrictions as 30%, 50%, 70% and 90% respectively. In
this work, the mass deposition is computed from the size
of recirculation zone and finally the formulation, used for
the computation is as, deposition = 2/3 (stenosis height
(stenosis length + reattachment length) π Diameter of
artery density of plaque material. Where 2/3 is assumed
to present the volume covered under the considered cases,
and the magnitude of density of the deposited material is
taken as 1.04 gm/ml, which is the density of LDL.
Figure 2 shows the effect of Reynolds number and
percentage of restriction on the pressure drop in the
stenosis zone of a rectangular restriction. In case of Fig-
ure 2(a), the percentage of restriction has been fixed as
50%. It is noted from the figure that as Reynolds number
increases, the pressure drop across the stenosis increases.
Figure 2(b) presents the variation of the pressure drop
across the rectangular stenosis with percentage of restric-
tion at fixed Reynolds number of 100. From the figure, it
is observed that initially upto 50%, the increase in the
pressure drop is less, then from 50% to 70%, the pressure
drop increases moderately, after that the pressure drop
enhances drastically. Comparing Figures 2(a) and (b), it
is also observed that the magnitude of pressure drop in-
crease is more prominent in case of increase in the per-
centage of restriction compared to Reynolds number.
Figure 3 shows the variation of pressure drop with
Reynolds number and percentage of restriction in case of
half circular stenotic model. Here also for Figure 3(a),
percentage of restriction has been considered to be 50%,
and for Figure 3(b), Reynolds number has been consid-
ered as 100. The nature of both the Figures 3(a) and (b)
are more or less same as observed in case of Figures 2(a)
and (b). In case of this model the magnitude of pressure
drop for each case is observed to be significantly less
than that of the corresponding each case of rectangular
stenotic model. This is expected because the flow of
fluid in poststenotic region is smoother in case of half
circular stenosis model due to its favorable geometric
feature than that of rectangular one.
From this study, it is concluded that the effect of per-
centage of restriction is more dominant than Reynolds
number on the progression of the disease, atherosclerosis
for any shaped restriction. Moreover, the rectangular
shaped restriction is noted to be more prone to this dis-
ease.
Copyright © 2013 SciRes. OPEN ACCESS
D. K. Mandal, S. Chakrabarti / J. Biomedical Science and Engineering 6 (2013) 1109-1116
1112
PR=50%
0
200
400
600
800
1000
1200
1400
1600
0100 200300 400
Re
Pressure drop
Pr es s ur e
dr o p
(a)
Re=100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
30% 50%70% 90%
PR
Pressure drop
Pressure
dr op
(b)
Figure 2. Effect of Re and PR on pressure
drop for rectangular restriction.
Figure 4 represents the velocity contour and velocity
vector for rectangular shaped restriction. From the figure,
it is noted that the negative velocity zone increases with
both Reynolds number and percentage of restriction.
This indicates that with the increase in Reynolds number
and percentage of restriction, there is more possibility of
mass influx into arterial wall leading to higher mass
deposition in the artery. Figure 5 shows the velocity
contour and velocity vector for half circular shaped re-
striction. The nature of flow separation is similar as ob-
served in case of rectangular stenotic model. Here the
magnitude of recirculating zone at the post stenotic re-
gion is less.
Therefore from the Figures 4 and 5, it is revealed that
the physiological effect of recirculation zone on the dis-
ease is less in case of half circular stenotic model. From
the figures, it is expected that the magnitude of the mass
PR=50%
0
0.5
1
1.5
2
2.5
0100 200300 400
Re
Pressure drop
Pressure
dr op
1.5
Pressure drop
(a)
Re=100
0
10
20
30
40
50
60
70
80
30% 50%70% 90%
PR
Pressure drop
Pressure
dr op
(b)
Figure 3. Effect of Re and PR on pressure
drop for half circular restriction.
deposition will increase with the increase either Rey-
nolds number or percentage of restriction for both the
considered stenotic models. The chance of deposition is
expected to be less for half circular stenotic model than
that of rectangular one.
As per our expectation, we have observed that the
magnitude of deposition in the artery wall increases with
increase in Reynolds number or percentage of restriction
in case of rectangular shaped restriction. This observa-
tion has been depicted in Figure 6. The effect of Rey-
nolds number and the effect of percentage of restriction
on mass deposition have been presented in Figures 6(a)
and (b) respectively. The rate of increase in deposition
with increase in percentage of restriction is observed to
be more than that of increase in deposition rate with in-
crease in Reynolds number for the said stenotic model.
These results highlight that te effect of percentage of h
Copyright © 2013 SciRes. OPEN ACCESS
D. K. Mandal, S. Chakrabarti / J. Biomedical Science and Engineering 6 (2013) 1109-1116
Copyright © 2013 SciRes.
111 3
Re = 100
Re = 200
Re = 300
Re = 400
Re = 10
0
Re = 20
0
Re = 30
0
Re = 40
0
(a) (b)
PR = 30
PR = 50
PR = 70
PR = 90
PR = 3
0
PR = 5
0
PR = 7
0
PR = 9
0
(c) (d)
(a) Velocity contours at 50% restriction. (b) Velocity vectors at 50% restriction. (c) Velocity contours at Re = 100. (d) Ve-
locity vectors at Re = 100.
Figure 4. Effect of Re and PR on velocity contour and velocity vector for rectangular restriction.
restriction is more than that the effect of Reynolds num-
ber for the progression of atherosclerosis, which substan-
tiates our earlier observations from pressure drop, veloc-
ity contour and velocity vector.
OPEN ACCESS
Figure 7 highlights the variation of deposition of the
plaques in the arterial wall for the half circular stenotic
model. Figure 7(a) represents the effect of Reynolds
number on the said variation and Figure 7(b) represents
the effect of percentage of restriction. From the figures,
the similar nature of increasing trend of mass deposition
is seen for this considered model except the magnitude of
deposition compared to the case of rectangular shaped
model.
Comparing the computed mass deposition results for
rectangular and half circular stenotic models, it may be
stated that the effect of percentage of restriction is more
prone to the disease than that of Reynolds number, and
half circular stenotic shape insists the less chance of
mass deposition in the arterial wall.
4. CONCLUSION
In the present numerical work, the effect of Reynolds
number and percentage of restriction on pressure drop
through stenosis, velocity contour, velocity vector, and
finally by proposing a concept, from the idea of basic
flow characteristics in constricted arteries, for the assess-
ment of mass deposition in the wall, and the effect of
Reynolds number and percentage of restriction on mass
deposition have been studied for our considered rectan-
gular as well as half circular restrictions. Reynolds num-
bers are chosen as 100, 200, 300 and 400 and percentages
of restrictions as 30%, 50%, 70% and 90% respectively.
From this study of variation of pressure drop, it is also
D. K. Mandal, S. Chakrabarti / J. Biomedical Science and Engineering 6 (2013) 1109-1116
1114
Re = 10
0
Re = 20
0
Re = 30
0
Re = 40
0
Re = 10
0
Re = 20
0
Re = 30
0
Re = 40
0
(a) (b)
PR = 3
0
PR = 5
0
PR = 7
0
PR = 9
0
PR = 30
PR = 50
PR = 70
PR = 90
(c) (d)
(a) Velocity contours at 50% restriction. (b) Velocity vectors at 50% restriction. (c) Velocity contours at Re = 100. (d)
Velocity vectors at Re = 100.
Figure 5. Effect of Re and PR on velocity contour and velocity vector for half circular restriction.
PR=50%
0
1
2
3
4
5
6
7
8
0100 200300 400
Re
Deposition
Deposition
5
4
3
De
p
osition
Re=100
0
2
4
6
8
10
12
14
16
18
20
30% 50%70% 90%
PR
Depo sitio n
Deposition
(a) (b)
Figure 6. Effect of Re and PR on deposition for rectangular restriction.
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D. K. Mandal, S. Chakrabarti / J. Biomedical Science and Engineering 6 (2013) 1109-1116 1115
PR=50%
0
1
2
3
4
5
6
7
8
0100 200300 400
Re
Deposition
Deposition
5
4
3
De
osition
Re=100
0
2
4
6
8
10
12
14
16
18
20
30% 50%70% 90%
PR
Depo sit ion
Depos ition
(a) (b)
Figure 7. Effect of deposition on deposition Re and PR for half circular restriction.
revealed that the magnitude of pressure drop increase is
more prominent in case of increase in the percentage of
restriction compared to Reynolds number. The study of
flow separation zone focuses on the effect of percentage
of restriction that is more dominant than Reynolds num-
ber on the progression of the disease, atherosclerosis for
any shaped restriction. Moreover, the rectangular shaped
restriction is noted to be more prone to this disease. The
mass deposition results of rectangular and half circular
stenotic models motivate to conclude that the effect of
percentage of restriction is more prone to the disease
than that of Reynolds number, and half circular stenotic
shape insists for the less chance of mass deposition in the
arterial wall.
5. ACKNOWLEDGEMENTS
This work is supported by AICTE, RPS Scheme, (1023/POR/RID/RPS-
101/2009/10).
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NOMENCLATURES
A: Area at any section, [m2]
D: Dia of the artery, [m]
R: Radius of the artery, [m]
L: Total length of computational domain, [m]
p: Static pressure, [Nm−2]
Re: Reynolds number
U: Average velocity in r-direction at inlet, [ms−1]
r: Density, [kg·m−3]
µ: Dynamic viscosity, [Pas]
PR: Percentage of restriction (by diameter)
Copyright © 2013 SciRes. OPEN ACCESS