Biomedical Physics & Engineering

Express

PAPER

Computational simulation of an artery narrowed by plaque using 3D FSI

method: influence of the plaque angle, non-Newtonian properties of the
blood flow and the hyperelastic artery models
To cite this article: Masoud Ahmadi and Reza Ansari 2019 Biomed. Phys. Eng. Express 5 045037

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 Biomed. Phys. Eng. Express 5 (2019) 045037 https://doi.org/10.1088/2057-1976/ab323f

PAPER

Computational simulation of an artery narrowed by plaque using 3D

RECEIVED
14 March 2019
FSI method: influence of the plaque angle, non-Newtonian
properties of the blood flow and the hyperelastic artery models
REVISED
30 June 2019
ACCEPTED FOR PUBLICATION
15 July 2019
PUBLISHED
Masoud Ahmadi1 and Reza Ansari1
23 July 2019 Department of Mechanical Engineering, University of Guilan, Rasht, Iran
1
Author to whom any correspondence should be addressed.
E-mail: masoud_ahmadi_pr@yahoo.com and r_ansari@guilan.ac.ir

Keywords: atheromatous plaque, fluid-structure interaction, blood flow, non-Newtonian fluid, hyperelastic materials

Abstract
Currently, cardiovascular diseases related to atherosclerosis are among the universal primary reasons
for mortality. Artery stenosis is the narrowing of the artery, due to the formation of atheromatous
plaque in the arterial tunica intima, a region of the blood vessel located between the endothelium and
the tunica media. In the present paper, the blood flow is modeled in a stenosed artery, and the dynamic
behavior of the atherosclerosis phenomenon is studied using computational analyses. To accomplish
this aim, 3D finite element method (FEM) for modeling structural sections (including plaque and
artery) and computational fluid dynamics (CFD) for modeling fluid part (blood flow) are coupled
through a fluid-structure interaction (FSI). It is shown that the results of present FSI analysis are in
good agreement with the available data in the literature. The influences of plaque geometry, in
particular, plaque angle and percentage of stenosis, are investigated. Also, the effects of Newtonian and
non-Newtonian properties of the blood flow and different hyperelastic artery models, including
Ogden and Polynomial, are analyzed. The simulations show that with an increasing angle of the
plaque, the stress of the artery increases, while the velocity of the blood flow decreases.

1. Introduction it is important to expand atherosclerotic plaques,

parameters WSS and flow disturbances [5, 6]. Also, it
Nowadays, cardiovascular diseases are principal causes should be noted that the WSS can be considered as an
of death in the developed world. They caused about important parameter in plaque formation and rupture
7,249,000 deaths (12.7% of all global deaths) in 2008, mechanism [7]. Moreover, the peak circumferential
more exactly, 29.7% of total deaths in Europe and stress is the most essential biomechanical parameter
Central Asia, 16.9% of total deaths in Middle East and causing the rupture of atherosclerotic plaque mech-
North Africa, 13.6% of total deaths in South Asia and anism [8].
10.9% of total deaths in Latin America [1]. Coronary The computational modeling of the human cardi-
artery disease (CAD) is one of the most common types ovascular system plays an important role in studying
of heart disease. The formation, development, and cardiovascular diseases. Recently, increasing demand
rupture of atheromatous plaques are the major causes from the medical community for accurate quantitative
of CAD [2, 3]. The plaque deforms the shape of the studies on vascular diseases has given a motivation for
vessel by narrowing its lumen. The high-wall shear developing mathematical and computational models
stress (WSS) increases at the stenotic throat, and of the human cardiovascular mechanism [9]. The
decreases at the distal segment. Also, in severe stenosis, human cardiovascular mechanism is highly complex,
due to the sharp decrease of blood flow, the velocity and several computational approaches have been used
consequently decreases. It should be noted that blood for its simulation. Since the computational cost of full
flow rate is not a constant. The examination of local three-dimensional modeling of the entire human car-
changes in the vessel WSS localizes the atherosclerotic diovascular mechanism is very high, its segments are
plaque in the artery [4]. In the study of the formation, studied [10]. In large and medium vessels, blood flow

© 2019 IOP Publishing Ltd

Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

is usually considered as a Newtonian fluid. Several The constitutive model of the arteries is one of the
researchers have simulated the blood flow as a New- important issues which needs to be considered in such
tonian fluid model inside a rigid artery (e.g. [11] and biomechanics simulations. Accurate modeling of the
[12]). Verfürth [13] and Conca et al [14] considered mechanical behavior of arteries is so difficult [24].
Stokes and Navier–Stokes equations for the blood Yeleswarapu et al [25] studied experimentally three
flow. Blood is a concentrated suspension of formed constant generalized Oldroyd-B models for describing
cellular elements (erythrocytes, leukocytes, and blood. Their study includes the shear-thinning beha-
thrombocytes), which are suspended in plasma, con- vior of blood flow over a large range of shear rates.
taining electrolytes and organic molecules. This parti- Anand and Rajagopal [26] developed a constitutive
culate nature of blood leads to non-Newtonian model in the context of the general thermodynamic
behavior [9, 10]. While most hemodynamic simula- framework of [25, 27]. Their model has a good agree-
tions assumed the blood as a Newtonian fluid, several ment with experimental observations [28]. Watton
authors have suggested that appropriate nonlinear et al [29] used a realistic constitutive fluid–solid-
viscosity models should be taken into account growth model of the arterial wall for the evolution of a
[9, 10, 15]. saccular cerebral aneurysm on the internal carotid
Physiologically, arterial walls are elastic, whose sinus artery. Holzapfel et al [30] identified the layer
deformation in turn, affects the blood. Therefore, elas- properties of the human coronary artery: the intima is
tic and deformable arterial wall models are preferred. the stiffest layer and the media is the softest layer.
In this regard, the computational modeling of fluid- Moreover, to model the human coronary arteries lay-
structure interaction (FSI) [16, 17], can be a useful ers, a strain energy function was suggested in their
approach in modeling cardiovascular diseases. Theor- work. Furthermore, the effects of variability in mat-
etical modeling of blood flow-vessel wall is extremely erial properties of the media and plaque obtained from
complex due to nonlinear governing equations and direct material tests on stress and stretch conditions
the irregularity of the movement of the blood-vessel were investigated by Yuan et al [31]. Lee et al [32]
investigated the hemodynamic behavior of athero-
interface [10]. FSI modeling is an appropriate method
sclerotic plaque rupture with various conditions using
to model blood flow inside the arteries, which pro-
FSI simulation. They found that the large WSS took
vides a clear comprehension of the biomechanical
place on both sides of the branch. Also, plaque rupture
behavior of the diseased arterial. Janela et al [10] stu-
might be predicted to occur according to athero-
died a three-dimensional healthy artery model by cou-
sclerotic cap thickness, geometric properties, plaque
pling the blood flow as a non-Newtonian fluid with an
characteristics, and wall inflation pressures. By using
elastic structural vessel. They used a three-dimen-
3D FSI models with ex vivo and in vivo material prop-
sional FSI model in a compliant vessel. Their three-
erties and various axial and circumferential shrink
dimensional blood flow was described through a
combinations, Guo et al [33] investigated the material
shear-thinning generalized Newtonian model. Li et al
stiffness impact on the plaque stress and strain. Tang
[18] studied a 2D nonlinear time-dependent FSI
et al [34] used a patient-specific MRI-PET/CT-based
model of blood and stenotic carotid artery. They modeling approach to develop 3D FSI models. They
investigated the conditions of the plaque rupture. investigated the impact of inflammation on plaque
They reported that the presence of moderate carotid stress and strain conditions for better plaque assess-
stenosis with a thin fibrous cap indicates a high risk for ment. Wang et al [35] employed 3D FSI models to
plaque rupture. With the physiological velocity and investigate both structural stress/strain and also fluid
pressure waveforms, Torii et al [19] compared the biomechanical conditions.
WSS computed with the rigid and compliant wall for a To the best of authors’ knowledge, the effects of
stenotic right coronary artery. An FSI model consider- different plaque geometries (in particular, plaque
ing the non-Newtonian feature of the blood flow was angle) on the atheromatous simulations have not yet
proposed by El Khatib et al [20] to study the recircula- been studied. Hence, in the present paper, 3D compu-
tions downstream. Huo et al [21] investigated the tational analyses are conducted to evaluate the effects
influence of the compliance on the WSS computation of the plaque angle, non-Newtonian properties of the
for a porcine right coronary artery as compared to the blood, and the hyperelastic arterial wall, on the simu-
rigid wall and FSI model. Huo and Kassab [22] imple- lation results of a narrowed arterial model with plaque.
mented a hemodynamic analysis in the entire cor- To this end, 3D finite element method for modeling
onary arterial tree of diastolically arrested, vasodilated structural sections and computational fluid dynamics
pig heart that takes into account vessel compliance and for modeling fluid part are coupled through FSI using
blood viscosity. Boujena et al [23] used an FSI an arbitrary Lagrangian-Eulerian methodology.
approach and presented an analytical study for a non- Before the investigation, the validity of the present FSI
Newtonian blood flow based on Carreau’s law. They simulations is verified. It should be noted that more
investigated the effect of the wall motion on the local sophisticated models other than the present model
blood displacement, stresses and strains in the athero- exist in the literature, which can predict some other
sclerotic plaque. phenomena [36–40] including anisotropic behavior of

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Figure 1. FE model for the structural parts and the CFD model for the blood section. (a) isotropic view (b) side view.

the vessel wall, variations in plaque material property and 0.38 mm, respectively [41]. The plaque is formu-
due to calcification, the effect of helical flow at the lated by cosine functions as:
inlet, damage accumulation within the atherosclerotic S = A cos (2px / l) (1)
vessel, and pulsation of the artery causing by the heart
where A and l are amplitude and length of the stenosis,
beating and Lumen erosion. Moreover, we have sim-
respectively. Stenosis has a length equal to 10 mm, and
plified the geometry of the model in our study. The
A depends on the percentage of stenosis. Also, the
sharpness of the angles in the enneadecagon may
values of the percentage of arterial stenosis and the
influence the hemodynamic parameters, especially on values of the plaque angle are taken as variable
WSS. Also, the eccentricity of stenosis and the tortuos- parameters. The blood flow is modeled by cutting the
ity of the artery are not taken into account. Despite structural geometry (including vessel and plaque)
these simplifications, the presented model captured from a cylinder with the size of the vessel part. Figure 1
many physiological aspects of blood flow in athero- shows the FE model for the structural parts (artery and
sclerosis. Some issues are lacking in the literature, plaque) and the CFD model for the blood section in
which this paper aims to address. The investigation which the angle α represents the plaque angle.
assists the understanding of the behavior of the blood In the present study, the structural parts, including
flow in a stenosed artery and provides insightful guide- vessel and plaque, are considered as hyperelastic mate-
lines for a better simulation in the future. rials, and they are modeled by the third order Ogden
and the polynomial constitutive models. The stress-
strain curve for hyperelastic materials is non-linear,
2. Simulation which is derived from a strain energy function. The
artery, including three layers (intima, media, and
To model the structural parts, 2-dimensional sketches adventitia) is described by the third-order Ogden
are drawn first. Afterward, they are rotated along the hyperelastic model [42]. The third-order Ogden
revolution axis. Accordingly, the artery is modeled as a hyperelastic strain energy function is given as follows
cylindrical part with the length equal to 20 mm, the 3
2mi ai
inner diameter equal to 4 mm and the thickness of W= å a 2i
(l1 + la2 i + la3 i - 3)
1 mm. The shape of the artery is a full cylindrical tube i=1
3
with circular (without deviation) cross section. It has 1
+å (J - 1)2i (2)
three layers including intima, media, and adventitia i=1 Di
(from inside to outside). The thickness of intima,
media, and adventitia layers are 0.27 mm, 0.35 mm,

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Table 1. Values of the Ogden model parameters for the artery layers (Intima, Media and Adventitia) and the Hypocellular plaque [42, 43].

Material Density (kg/m3) μ1 (MPa) μ2 (MPa) μ3 (MPa) α1 α2 α3 D1

Intima 1070 −7.04 4.23 2.85 24.48 25.00 23.54 8.95e-7

Media 1070 −1.23 0.88 0.45 16.59 16.65 16.50 5.31e-6
Adventitia 1070 −1.28 0.85 0.44 24.63 25.00 23.74 4.67e-6
Hypocellular Plaque 1450 0.093 — — 8.17 — — 4.30e-7

Table 2. Values of the polynomial model parameters for the artery layers (Intima, Media and Adventitia) and the Hypocellular
plaque [42, 43].

Material Density (kg m−3) C10 (MPa) C20 (MPa) C30 (MPa) C40 (MPa) C50 (MPa) C60 (MPa)

Intima 1070 6.79e-3 0.54 −1.11 10.65 −7.27 1.63

Media 1070 6.52e-3 4.89e-2 9.26e-3 0.76 −0.43 0.087
Adventitia 1070 8.27e-3 1.20e-2 0.52 −5.63 21.44 —
Hypocellular Plaque 1450 2.38e-3 0.19 0.39 3.73 −2.54 0.57

where J. is the volumetric stretch and λi parameters Table 3. Parameters values

of the Carreau’s model for
represent the stretches in the principal directions. The the blood flow [45].
symbols μi, αi, and Di present the Ogden parameters.
μi and αi are associated with the shear modulus. Also, Parameter Value
Di parameters show the compressibility of the hyper-
μ0 0.0456 Pa.s
elastic material. Table 1 represents the model para- m¥ 0.0032 Pa.s
meters for the arteries with three layers and λ 10.03 s
Hypocellular Plaque [42, 43]. By considering infinite- n 0.344
simal values for D1 and zero value for D2 and D3, the
artery and the plaque materials are assumed to be
almost incompressible.
The incompressible polynomial strain energy
function is described as follows between the blood instance surfaces to the intima sur-
faces and the plaque surfaces. For the boundary condi-
W = C10 (I1 - 3) + C20 (I1 - 3)2 + C30 (I1 - 3)3 tions, the longitudinal displacements at both ends of
+ C 40 (I1 - 3)4 + C50 (I1 - 3)5 + C60 (I1 - 3)6 the artery are constrained, whereas the radial displace-
(3) ment is allowed. The amplitude of the inlet pressure is
where C10–C60 parameters denote the polynomial described by a function of time through the following
model parameters, and I1 is the first stretch invariant. relation [23]. The inlet flow pressure of the above
Table 2 gives the values of polynomial model para- function is shown in figure 2.
meters for artery layers and the hypocellular pla- The simulations are performed using an arbitrary
que [43, 44]. Lagrangian-Eulerian (ALE) methodology for the fluid
The fluid is considered as an incompressible liquid flow, and some portion of the fluid domain is
described by Navier–Stokes equations. Also, New- deformed consistent with a boundary motion, and the
tonian and non-Newtonian fluid models of the blood analyses are conducted with an implicit solver. The
are applied for the comparison purpose. The viscosity artery and plaque are meshed by a combination of 10-
value of the blood for the Newtonian model is taken as node quadratic tetrahedron, hybrid, constant pressure
μ=0.0035 Pa.s. Moreover, a generalized Newtonian elements (C3D10H) and 20-node quadratic brick,
fluid based on Carreau’s law is used for non-New- hybrid, linear pressure, reduced integration elements
tonian fluid modeling. Carreau’s model assumes that (C3D20RH). Also, the blood instance is meshed by
the viscosity function of the blood is expressed as 4-node linear fluid tetrahedron (FC3D4). Figure 3
shows the meshed FE and CFD models. The FE model
m = m¥ + (m 0 - m¥)(1 + (lg )2)(n - 1) / 2 (4) of the structural parts is coupled with the CFD model
where m¥, μ0 and g . respectively stand for viscosity at of the fluid by an FSI analysis. Mesh convergence study
highest shear rates, viscosity at lowest shear rates and is performed by gradually decreasing the size of ele-
the shear rate. Also, λ and n are Carreau’s model ments and verifying the artery behavior to ensure
parameters. Carreau’s model parameters for the blood convergence of the numerical solution. The fining
flow are given in table 3 [45]. Also, the blood density is mesh is performed to reach a change smaller than
taken as d=1060 kg m−3 for both Newtonian and 0.1% in the solution. Table 4 shows the mesh sensitiv-
non-Newtonian fluid models. ity analysis of used elements. As can be seen, the aver-
The surfaces of the plaque and the intima layer of age change in the results is smaller than 0.1% using
the artery are tied together, while FSI was applied 45828 tetrahedron and 7638 brick elements for solid

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Figure 2. Inlet flow pressure of the blood.

Figure 3. (a) The meshed FEM model and (b) the meshed CFD model.

Table 4. Mesh convergence study in the case of stenosis=50%, angle=180.

FE model
Mesh convergence study step CFD model
Tetrahedron Brick Tetrahedron Total number of elements Average change in the results

Step 1 21624 2403 13872 37899 —

Step 2 40365 4805 30623 75793 5.42%
Step 3 44189 5665 49991 99845 1.97%
Step 4 51534 7157 61743 120434 0.41%
Step 5 48575 7360 75899 131834 0.27%
Step 6 45828 7638 89760 143226 0.06%

parts, and 89760 tetrahedron elements for the 1 mm. The shape of the artery is a full cylindrical tube
fluid part. with a circular (without deviation) cross section. The
plaque is formulated by cosine functions where
stenosis has a length equal to 10 mm, and the percent-
3. Results and discussion age of arterial stenosis and the plaque angle are taken
as 70% and 360° respectively [23]. The peak velocities
Firstly, a comparison is made between the results of and peak pressures of the whole blood domain at
present FSI simulation and those of Boujena et al [23]
different times are presented in table 5. As can be seen,
to verify the validity of the proposed model. For the
the differences between the peak velocities and peak
comparison purpose, the model used in [23] is
pressures obtained by the present model and those of
considered as follows: the artery is modeled as a
Boujena et al [23] are about 1.69% and 0.67%,
cylindrical part with the length equal to 20 mm, the
respectively.
inner diameter equal to 3 mm and the thickness of

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Table 5. Comparison of the peak velocities and peak pressures for models. The comparison of non-Newtonian and
present model and Boujena et al [23] work at different time step.
Newtonian fluid models reveals that there is no sig-
Peak velocities (m/s) Peak pressures (Pa) nificant difference between the results obtained by
Time (s) Present Boujena et al Present Boujena et al
considering the blood flow as non-Newtonian or
Newtonian models. For example, maximum values of
0.01 0.238 0.246 145.3 145.3 the von-Misses stress, shear stress, and velocity of the
0.05 0.551 0.564 137.4 136.6 non-Newtonian and Newtonian models, respectively,
0.1 0.519 0.523 129.5 127.8
have insignificant differences of 0.13%, 0.47% and 0%
0.8 0.641 0.644 193.6 192.1
for healthy artery. Besides, it is worth mentioning that
Liu et al [46] performed CFD simulations on the geo-
metric models from the baseline computed tomo-
Comparisons of the maximum values of the von-
graphy (CT) angiography, and showed that
Misses stress and the maximum values of the shear
Newtonian and non-Newtonian models have limited
stress of the healthy artery for non-Newtonian and
differences in high-WSS area, but with large (over
Newtonian fluid models are illustrated in figures 4(a) 50%) in low-WSS areas.
and (b), respectively. Figure 5 shows the comparison Comparisons of the Ogden hyperelastic model
of the maximum values of the velocity of the blood and the Polynomial hyperelastic model for the max-
inside the healthy artery for non-Newtonian and New- imum values of the von-Misses stress, the maximum
tonian fluid models. Also, the maximum values of the values of the shear stress of the artery and the max-
von-Misses stress, the maximum values of the shear imum values of the velocity of the blood inside the
stress of the artery and the maximum values of the artery are illustrated in figures 6, 7, and table 7.
velocity of the blood inside the artery for non-New- Figures 6 and 7 represent the results for a healthy
tonian and Newtonian fluid model are compared in artery, while table 7 shows the results for different per-
table 6 for different percentages stenosis and plaque centages of stenosis and plaque angles of the model. In
angles. In the fluid model comparison, the structural the comparison of hyperelastic models, the blood is
parts are simulated using the Ogden hyperelastic considered as a Newtonian fluid. The comparison

Figure 4. Comparison of the maximum values of the (a) von-Misses stress and the (b) Shear stress of the healthy artery for non-
Newtonian and Newtonian blood.

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Figure 5. Comparison of the maximum values of the blood velocity flow inside the healthy artery for non-Newtonian and Newtonian
blood.

Table 6. Comparison of the maximum values of the von-Misses stress, Shear stress and the velocity for non-
Newtonian and Newtonian fluid model.

Newtonian Non-Newtonian

Model von-M Shear Velocity von-M Shear Velocity

stenosis=0%, angle=0 132.34 74.26 0.420 132.51 74.61 0.420

stenosis=50%, angle=180 142.27 77.76 0.331 141.99 77.68 0.327
stenosis=70%, angle=360 153.69 85.50 0.231 153.13 85.44 0.226

Figure 6. Comparison of the maximum values of the (a) von-Misses stress and the (b) Shear stress of the healthy artery for the Ogden
hyperelastic model and the Polynomial hyperelastic model.

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Figure 7. Comparison of the maximum values of the blood velocity flow inside the healthy artery for the Ogden hyperelastic model
and the Polynomial hyperelastic model.

Table 7. Comparison of the maximum values of the von-Misses stress, Shear stress and the velocity for Ogden
and Polynomial models.

Ogden Polynomial

Model von-M Shear Velocity von-M Shear Velocity

stenosis=0%, angle=0 132.34 74.26 0.420 58.95 33.85 0.427

stenosis=50%, angle=180 142.27 77.76 0.331 65.30 35.01 0.338
stenosis=70%, angle=360 153.69 85.50 0.231 69.98 41.12 0.230

Figure 8. Maximum value of von-Misses stress of the artery versus the plaque angle for different values of percentage of stenosis.

Figure 9. Maximum value of Shear stress of the artery versus the plaque angle for different values of percentage of stenosis.

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Figure 10. Maximum value of velocity of the blood flow versus the plaque angle for different values of percentage of stenosis.

Figure 11. Vector plot of blood velocity in the artery for (a) Newtonian fluid and (b) non-Newtonian fluid.

results of the hyperelastic models reveal that there is a stenosis. It is observed that increasing the plaque angle
remarkable difference between the stresses of the generally leads to increasing the von-Misses stress of
hyperelastic models. However, the difference between the artery in all of the considered percentage of ste-
the velocities is insignificant. The values of the Ogden nosis. But, from 0° to 45°, the stress does not experi-
hyperelastic model for healthy artery are 76.7% and ence significant variation. The increase of von-Misses
74.8% different from the Polynomial hyperelastic stress has a direct relationship with the percentage of
model for von-Misses and shear, respectively. stenosis of the artery. For example, changing the angle
Now, parametric studies of the percentage of ste- from 0° to 180° leads to increasing the von-Misses
nosis and the plaque angle are conducted considering stress values by 3.52%, 7.51% and 11.60% for 30%,
the structural parts as an Ogden hyperplastic model 50%, and 70% stenosis, respectively. Also, from 0° to
and the fluid as a Newtonian model. Four values for 360° results 10.46%, 13.33% and 16.14% increase for
the percentage of arterial stenosis are considered: 0%, 30%, 50% and 70% stenosis, respectively. Figure 9
30%, 50%, and 70%. Also, the values of the plaque shows the diagram of the maximum value of shear
angle are taken as 45°, 90°, 180°, 270°, and 360°. stress of the atheromatous artery versus the plaque
Figure 8 depicts the variation of the maximum value of angle for three different values of the percentage of ste-
von-Misses stress of the atheromatous artery versus nosis. According to the results from the figure, the
the plaque angle for three values of the percentage of increase of the plaque angle and artery percentage of

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

Figure 12. Contour plots of von-Misses stress and Shear stress in a healthy artery for (a) Ogden model and (b) Polynomial model.

stenosis generally leads to increasing the shear stress of Newtonian fluid exhibits a little less velocity magni-
the artery, just like what observed in the von-Misses tude than those of non-Newtonian. The shear stress
stress graphs. Increasing the values of the artery shear contours of the artery are almost similar, with the
stress from 0° to 180° for 30%, 50%, and 70% stenosis maximum shear stresses located on the intima layer,
are 1.45%, 4.71% and 9.32%, respectively. Also, from while the von-Misses stress contours on the artery are
0° to 360°, the increases are 9.05%, 12.04%, and not so similar. The maximum shear stresses are loca-
15.14%, respectively. The increase of shear stress ted on the intima layer for both Ogden and Poly-
values with the plaque angle and artery the percentage nomial hyperelastic models. The maximum von-
of stenosis is a little less than the increase of the von- Misses stresses are respectively located on the intima
Misses stress. layer and adventitia layer for Polynomial and Ogden
The influence of the plaque angle on the max- hyperelastic models.
imum value of the velocity of the blood in the artery is
displayed in figure 10 for three different values of the
percentage of stenosis. The results demonstrate that 4. Conclusion
the increase of the plaque angle leads to decreasing the
velocity of the blood. The decreases are more sig- In this work, the influence of the plaque angle formed
nificant in the higher values of the artery percentage of in the intima section of the blood on the dynamic
stenosis. For example, increasing the angle from 0° to behavior of the narrowed artery was investigated by
180° leads to decreasing the velocity values by 16.79%, coupling FEM and CFD through an FSI analysis.
21.23% and 27.39% for 30%, 50% and 70% stenosis of Newtonian and non-Newtonian properties of the
the artery, respectively. Also, from 0° to 360° results blood flow and also Ogden and Polynomial hyperelas-
21.85%, 30.65% and 44.99% decrease for 30%, 50% tic models of the artery-plaque system were taken into
and 70% stenosis of the artery, respectively. account. The validity of FSI simulations was validated
Vector plots of blood velocity in the artery for by comparing the present results with those available
Newtonian non-Newtonian fluids are shown in in the literature. The main findings of the research can
figure 11. Figure 12 shows the contour plots of von-
be summarized as follows:
Misses stress and shear stress in a healthy artery for
Ogden and Polynomial hyperelastic models. For both
• There is no significant difference between the non-
figures, the blood flow is inside the artery from the
Newtonian and Newtonian blood flow of arteries
right side, and also the figures are given in 0.7 s time.
with atheromatous plaques.
The vector plots of blood velocity are almost similar,
with the maximum velocities mostly located in the • The Ogden hyperelastic model of the artery-plaque
middle of the artery outlet. The model with system exhibits more von-Misses and shear stress

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Biomed. Phys. Eng. Express 5 (2019) 045037 M Ahmadi et al

magnitudes as compared to the Polynomial hyper- [10] João J, Moura A and Sequeira A 2010 Absorbing boundary
elastic model. conditions for a 3D non-Newtonian fluid–structure
interaction model for blood flow in arteries Int. J. Eng. Sci. 48
• The Ogden and Polynomial hyperelastic models of 1332–49
the artery-plaque system have no significant differ- [11] Temam R 1984 Navier–Stokes Equations: Theory and
Numerical Analysis 343 (Bloomington: AMS Chelsea
ences in the values of the blood velocity. Publishing) (https://doi.org/10.1090/chel/343)
[12] Ladyzhenskaya O A 1964 The Mathematical Theory of Viscous
• Increasing the plaque angle generally leads to Incompressible Flow 17 2 (Sterrebeek: Physics Today) (https://
increasing the von-Misses stress and shear stress of doi.org/10.1063/1.3051412)
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