Journal of Biomechanics 67 (2018) 114–122

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Journal of Biomechanics
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www.JBiomech.com

Simulation of Low Density Lipoprotein (LDL) permeation into multilayer

coronary arterial wall: Interactive effects of wall shear stress
and fluid-structure interaction in hypertension
Mehrdad Roustaei, Mohammad Reza Nikmaneshi, Bahar Firoozabadi ⇑
Center of Excellence in Energy Conversion, School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history: Due to increased atherosclerosis-caused mortality, identification of its genesis and development is of
Accepted 26 November 2017 great importance. Although, key factors of the origin of the disease is still unknown, it is widely believed
that cholesterol particle penetration and accumulation in arterial wall is mainly responsible for further
wall thickening and decreased rate of blood flow during a gradual progression. To date, various effective
Keywords: components are recognized whose simultaneous consideration would lead to a more accurate approxi-
LDL mation of Low Density Lipoprotein (LDL) distribution within the wall. In this research, a multilayer
Atherosclerosis
Fluid-Structure Interaction (FSI) model is studied to simulate the penetration of LDL into the arterial wall.
FSI
WSS
Distention impact on wall properties is taken into account by considering FSI and Wall Shear Stress (WSS)
Coronary dependent endothelium properties. The results show intensified permeation of LDL whilst the FSI
approach is applied. In addition, luminal distension prompted by FSI reduces WSS along lumen/wall
interface, especially in hypertension. This effect leads to a lowered endothelial resistance against LDL per-
meation, comparing to the case in which WSS effect is overlooked. The results are in an acceptable con-
sistency with the clinical researches on WSS effect on atherosclerosis development.
Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Many researches are conducted to more accurately predict the

LDL distribution across the wall and also to identify effective ele-
Fatality rate caused by heart attacks and brain strokes, has ments, mostly concentrated on enhancement of the wall parame-
experienced a dramatic rise in recent century (Go et al., 2014). ters and modeling assumptions. The simplest model for the
Cardiovascular diseases (CVDs) are nowadays the most death- arterial wall (wall-free model) considering it as a boundary condi-
causing ones among the others in the world (Laslett et al., 2012; tion, has been used in numerous investigations to find the LDL con-
Mendis et al., 2011). Only 80 million people of adult Americans centration across the lumen (Wada and Karino, 2000). The fluid-
are suffering CVDs, according to a recent report (Go et al., 2014). wall model, on the other hand, describes the wall as a single layer
In order to prevent this disease from escalating any further, many homogeneous porous media (Stangeby and Ethier, 2002). Multi-
researches are carried out. Even though the main cause of CVDs is layer models, which comprise several distinct layers, are the most
still unknown, LDL permeation into inner layers of arterial wall is complex model and can approximate the LDL distribution much
believed to play an important role in the pathogenesis. LDL accu- better (Ai and Vafai, 2006; Kenjereš and de Loor, 2014). Dabagh
mulation within inner layers, especially intima, results in wall stiff- et al. (2009) introduced another model based on microscopic struc-
ness and thickening as well. In the next stages of progression, ture of layers, assuming blood pressure dependent properties and
plaque formation inside the wall and the eventual stenosed artery thicknesses for endothelium and intima, respectively.
reduces the blood flow. It is of paramount priority to develop a Experimental studies (Graf and Barras, 1979; Meyer and Tedgui,
comprehensive model which can explain atherogenesis and effec- 1996) have shown that wall properties varies with hydrodynamic
tive factors. parameters including blood pressure. Meyer and Tedgui (1996)
studied the pressure effects on the rabbit aortic wall with and
without the presence of endothelium. They obtained LDL and albu-
⇑ Corresponding author. min distribution profiles across the wall and plasma filtration
E-mail addresses: mehrdad_roustaei@mech.sharif.ir (M. Roustaei), mr_nikmane- velocity in different blood pressures. Biological investigations have
shi@mech.sharif.edu (M.R. Nikmaneshi), firoozabadi@sharif.edu (B. Firoozabadi). also shown endothelial cell shape and properties adjustment with

https://doi.org/10.1016/j.jbiomech.2017.11.029
0021-9290/Ó 2017 Elsevier Ltd. All rights reserved.
 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122 115

Nomenclature

C LDL concentration e circumferential strain

r reflection coefficient w half width of endothelium leaky junctions
K permeability / area fraction of leaky junctions
D diffusion coefficient Rcell endothelial cell radius
u radial velocity #LC number of leaky cells per surface area
v axial velocity #MC number of mitotic cells per surface area
P pressure SI shape index
l viscosity WSS wall shear stress
ul the velocity of plasma within arterial wall Sh Sherwood number
q density K sh mass transport coefficient on wall/endothelium inter-
ss stress tensor face
Lnj hydraulic conductivity Jr LDL flux on wall/endothelium interface
t end endothelium thickness

respect to WSS (Levesque et al., 1986; Lin et al., 1989). The imple- and only circumferential strain’s influence is considered.
mentation of this effect in computational models (Olgac et al., Deyranlou et al. (2015) on the other hand, applied both circumfer-
2008; Sun et al., 2009) has indicated that in a stenosed artery, ential strain and endothelium thickness effects simultaneously in
low WSS regions are more likely to develop atherosclerosis, which carotid artery whose elasticity is way lower than that of coronary.
is compatible with clinical results (van der Giessen et al., 2007). Hereby, we have added the WSS dependency of endothelium prop-
Alimohammadi et al. (2014) employed this method in a FSI model erties into the FSI model to more accurately anticipate hyperten-
of realistic artery. They did not include property dependency of sion impact on biological characteristics of the wall. The direct
endothelium on FSI model which was suggested by Chung and relation between endothelium thickness and normal junction per-
Vafai (2012), based on permeability variation of endothelium with meability is presented by Liu et al. (2011) for carotid arteries in a
its circumferential strain. Deyranlou et al. (2015) added the model with constant endothelium properties. But to the best of
endothelium thickness on normal junction permeability (Liu the author’s knowledge, there has not been a similar relation for
et al., 2011) to the work of Chung and Vafai (2012) with non- coronary arteries as well. In the present research, the effect of
Newtonian assumption for blood and plasma. Their studies and endothelium thickness variation on its normal junction permeabil-
the work of Iasiello et al. (2016a) showed more evident effect of ity is derived for coronary arteries. The main purpose of the present
non-Newtonian viscosity in higher blood pressures. Nematollahi study is to capture the interactive effects of WSS and FSI on LDL
et al. (2015) used a non-Newtonian model to capture the stenosis concentration in a multilayer arterial wall, and since in a rigid wall
effect on LDL retention in arterial wall. They incorporated the model, the WSS along the artery and at different transmural pres-
impact of Red Blood Cells’ rotation on LDL diffusion coefficient in sures would be the same, all explored cases in the present study
a monolayer carotid artery. Analytical solutions are also suggested include the FSI dependent property variation of the endothelium.
to compute LDL concentration in multilayer models. Hong et al. The interactive effects of coupled WSS and FSI on LDL concentra-
(2012) analytically solved the flow and concentration governing tion, as the innovation of the present study, are taken into account
equations within the wall to capture effects of non-Newtonian in a multilayer coronary artery.
plasma and pulsatile flow on concentration trends. According to
their results, although the filtration velocity varies during a cycle, 2. Mathematical modeling
concentration fields remain constant across the wall. The non-
Newtonian assumption is also implemented in patient-specific A five-layer axisymmetric arterial wall comprising of
geometries of thoracic arteries by Mpairaktaris et al. (2017) in endothelium, intima, IEL, media and adventitia is employed as
which WSS effect on LDL concentration in a monolayer arterial Fig. 1. Glycocalyx layer is not included due to its negligible
was is evaluated. Wang and Vafai (2015) also performed an analyt- thickness and effect on LDL distribution (Liu et al., 2011).
ical investigation on artery to elucidate the role of the secondary According to the coronary artery anatomy, the initial lumen
flow in curved geometries. Hyperthermia effects on LDL diffusion radius is taken R0 ¼ 1:85mm (Dodge et al., 1992; Kolandavel
have also been analyzed in some recent researches (Chung and et al., 2006; Olgac et al., 2008) and thicknesses of wall layers
Vafai, 2014; Iasiello et al., 2016b, 2015). are listed in Table 2 (Chung and Vafai, 2012). This table repre-
Taking the above review into consideration and for summariza- sents the properties of different layers of arterial wall and the
tion purposes for different modeling ideas, hereby in Table 1, we correspondingly Newtonian plasma flow across. It is noteworthy
have gathered typical features of the most important investiga- that fluid equations are not solved in adventitia and therefore
tions, so that the researchers can easily detect the interrelated only solid stress-strain equations with proper boundary condi-
content. tions are applied in there.
In this study, the multifactorial dependency of endothelium Considering negligible pulsation impact on LDL permeation into
properties on WSS, circumferential strain and endothelium thick- the wall, the steady state equations are utilized for both fluid and
ness in a multi-layer wall model is simultaneously applied to structural parts of the solution (Yang and Vafai, 2006). In lumen
account the conjunct impacts. Luminal distention induced by region, Navier-Stokes and continuity equations of blood (as incom-
hypertension reduces WSS and consequently raises the number pressible Newtonian fluid) are invoked to calculate the flow field.
of leaky endothelial cells. High blood pressure, aside from the Besides, convection-diffusion equation is manipulated to solve
increasing filtration velocity and luminal radius, has a secondary for LDL concentration. These governing equations are respectively
reverse effect on WSS magnitude and endothelium properties as as below:
well. Endothelium thickness influence in the study of Chung and
Vafai (2012) and Roustaei et al. (2017), was not taken into account qðu:ruÞ þ rp  lr2 u ¼ 0 ð1Þ
 116 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122

r:u ¼ 0 ð2Þ
Bifurcation

u:rC ¼ Dr2 C ð3Þ

where u, p, l and D represent velocity flow field, pressure, blood
–
–
–
–

–
–
–
–

–
+

+
viscosity and LDL luminal diffusion coefficient respectively. All wall
layers are assumed to be as homogeneous porous medium for
boundary

which Brinkman and continuity equations are solved. Convection-

Moving

diffusion-consumption equation for LDL concentration (C) transport

–
–
–
–

–
–

–
–
–

–
+

+
within the wall should also be invoked. These equations can be
written as
EGL1

l l 2
–
–

–
–

–
–
–
–

–
–

–
+

ul þ r p  r ul ¼ 0 ð4Þ
e
Stenosis

r:ul ¼ 0 ð5Þ
–

–
–

–
–

–
–

–
–

–
+

ð1  rÞul :rC ¼ Dr2 C  kC ð6Þ

Pressure dependent

in which ul is the volume-averaged velocity, ll the viscosity of

the plasma, K the LDL permeability through the wall, e porosity
properties

and k illustrates the consumption rate due to LDL absorption

on the surface of the smooth muscle Cells. r in Eq. (6) refers
to a sieving or reflection coefficient. Cauchy stress tensor, sym-
–
–
–

–
–

–
–

–
–

–
+

+
+

bolized by s, is employed in the lumen/wall interface in elastody-

Newtonian

namic equation to carry out the FSI method in the following

equation.
Non-

qs€rs ¼ rss þ f s ð7Þ

–
–
–
–

–
–
–
–

–
–

–
+

€r and qs respectively are wall radial acceleration and density and f s

is the body force. Regarding steady state conditions used in this
WSS dependent

study and absence of external body force, first and third terms are
properties

eliminated. Therefore this equation reduces to roots of gradient of

the stress tensor as
rss ¼ 0 ð8Þ
–
–
–
–

–
+

By solving this equation with the below-explored boundary

conditions, circumferential strain which is required for calculating
FSI dependent

the FSI-dependent properties of the wall will be obtained. The

properties

internal boundary stress is taken from the fluid part and media/
adventitia pressure is applied as the external boundary condition
–
–
–

–
–

–
–
–
–

–
–

–
+

as it is also used in fluid flow calculations within the wall (2-

way). It is noteworthy that the circumferential strain in the above
FSI

–
–
–

–
–

–
–
–
–

–
+

equation is a function of wall layers’ elasticity and the applied

boundary conditions.
dependent
Time

2.1. Boundary conditions

–
–
–

–
–

–
+

For the lumen inlet, a fully developed velocity field with Re =

300, equivalent to 30 cm/s of average velocity (Iliceto et al.,
layers
Wall

1991), is assumed as illustrated in Fig. 1. Constant pressure B.C.

4
4
5
5

4
4

4
5
2

4
2

4
0

is employed at the lumen outlet and along the media/adventitia

Dimension

interface. For media/adventitia interface, a 30 mmHg extravascular

A review on modeling methods and assumptions.

pressure B.C. is used (Ai and Vafai, 2006) and the artery outlet B.C.
is changed in different cases of the simulation. The pressure differ-
2
2
2
2

2
1

1
2
2
3

2
3

ence between these sections, artery outlet and media/adventitia

Analytical

interface, which is known as transmural pressure, is taken as a

Endothelial Glycocalyx Layer.

variable parameter (70, 120 and 150 mmHg) in this study specify-
ing the artery outlet pressure. Zero velocity B.C. is also applied for
–
–
–
–

–
–

–
–
–
–

–
+

inlet of the wall layers considering negligible axial velocity within

Yang and Vafai (2006)

Iasiello et al. (2016b)

Jesionek et al. (2016)
Iasiello et al. (2016a)

Alimohammadi et al.
Dabagh et al. (2009)

them (Olgac et al., 2008).

Jesionek and Kostur
Ai and Vafai (2006)

Olgac et al. (2008)

Hong et al. (2012)

Kolandavel et al.
Chung and Vafai

In order to obtain the relative concentration across the wall, the

Liu et al. (2011)

Deyranlou et al.

lumen inlet concentration (C 0 ) is set to unity and zero concentra-

tion B.C. is applied in media/adventitia interface. Zero-gradient
(2012)

(2015)

(2017)

(2015)
(2006)

concentration is also used at artery outlet. Staverman filtration

Table 1

coefficient is invoked to satisfy the concentration transport conti-

1

nuity equation among inner porous wall layers as

 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122 117

= 30
Adventitia
=0

Media
=0 IIEL
=0
=0
Intima Endothelium
E

=
= =0
=0
=
Lumen

Symmetry

Fig. 1. Fluid-Solid computational zone for a five-layer coronary arterial wall.

Table 2
Properties of different layers of the artery parts and the fluid within them (Chung and Vafai, 2012).

Lumen Endothelium Intima IEL Media Adventitia

Density (kg=m )3 1057 1070 1070 1070 1070 1070

Viscosity (Pa  s) 3:7  103 7:2  104 7:2  104 7:2  104 7:2  104
Thickness (lm) 1850 a 2 10 2 200 100
Permeability (m2 ) 3:22  1021 2  1016 4:392  1019 2  1018
Porosity (1) 5  104 0:983 0:002 0:258
Diffusivity (m2 =s) 2:87  1011 5:7061  1018 5:4  1012 3:18  1015 5  1014
Elasticity (Mpa) 2 2 2 3b 3b
Reflection coefficient (1) 0:9888 0:8272 0:9827 0:8836
Reaction coefficient (s1 ) 0 0 0 0 3:197  104
a
Olgac et al. (2008).
b
Gow and Hadfield (1979).

   
@c @c where Lnj is the hydraulic conductivity, t end signifies endothelium
ð1  rÞuc  D ¼ ð1  rÞuc  D ð9Þ
@r  @r þ thickness and ll is plasma viscosity. Hereby, Lnj is modified in order
to reach coronary compatible value of permeability
Fixed structural B.C. in both sides of the artery is used for lumen
(K nj ¼ 3:09  1021 m2 ) according to Jesionek et al. (2016), the
as well as the wall. Fluid stresses along lumen/wall interface are
exerted to calculate the FSI. Similar to the fluid part, the media/ modified Lnj ¼ 1:576  109 ms1 mmHg 1 (which results in the
adventitia B.C. is maintained at the pressure of 30 mmHg. mentioned permeability) is employed in the present study.
Based on Chung and Vafai (2012) method linking h-direction
2.2. Physiological parameters strain, e, to the half width of endothelium leaky junctions (w),
and to diffusion coefficient, permeability and reflection coefficient
In Table 2, parameters and properties used in each region of the can be modified. The fully-coupled equations of leaky junction half
computational zone are defined. Those parameters marked with width, permeability, diffusion and reflection coefficients are
stars, can vary in accordance with FSI results and WSS. These rela- respectively shown in Eq. (12)-(17)
tions are precisely elaborated in the following.
1 þ blj e
w ¼ w70mmHg ð12Þ
1 þ blj e70mmHg
2.3. Endothelium characteristics
The value of w70mmHg is presumed to be 14.343 nm (Chung and
Endothelium porosity is presumed to be constant in different Vafai, 2012). blj is taken as 10 as considered in earlier investiga-
pressures. But, its other properties are rather variable. Endothe- tions (Chung and Vafai, 2012; Deyranlou et al., 2015). From pore
lium overall permeability, K end , is calculated based on two different theory, the relation between K lj and w can be described as Eq. (13)
kinds of cell junctions permeability as
w2 4w/
K end ¼ K nj þ K lj ð10Þ K lj ¼ ð13Þ
3 Rcell
in which K nj is normal junction permeability and K lj is that of leaky
junction. In this study both of these parameters are inconstant. The where Rcell is the cell radius and / characterizes the surface fraction
permeability of normal junction in can be considered as a thickness of leaky junctions, that is a function of WSS as stated by Olgac et al.
dependent variable (Liu et al., 2011) (2008). Endothelium normal junction’s diffusivity is zero due to its
lower half width compared to average LDL radius, and then these
K nj ¼ Lnj ll t end ð11Þ particles can only pass through leaky junctions. Hence, the LDL
118 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122

diffusion coefficient across endothelium, using pore theorem can be to the large size difference between lumen radius and its adjacent
described as wall layers, structured quadrilateral mesh with boundary layer ele-
  4w/ ments is employed within the whole computational zone and
Dend ¼ D1 ð1  alj Þ 1  1:004alj þ 0:418a3lj  0:169a5lj ð14Þ mesh element numbers are shown in Table 3.
Rcell
In this equation, alj is LDL radius (r LDL ¼ 11nm) to w ratio and
4. Results and discussion
D1 represents the diffusion coefficient in a free media which is
defined as the following equation.
In order to validate the results, LDL concentration across the
kb T arterial wall, at 70 mmHg transmural pressure, is compared to
D1 ¼ ð15Þ
6pll r LDL the results of FSI model, analytical solution and non-Newtonian
flow cases presented by Chung and Vafai (2012), Iasiello et al.
in which kb is the Boltzmann constant (1:38  1023 J=K), T is the (2016b) and Iasiello et al. (2016a), respectively, in Fig. 2. This figure
plasma temperature (310.15 K), and ll is the plasma viscosity. shows acceptable consistency and similar trends of concentration
Overall endothelium reflection coefficient, can be calculated as in different layers.
follows Fig. 3(a–d) displays the WSS along the arterial wall and the fil-
tration velocity, permeability and the reflection coefficient at dif-
rnj K nj þ rlj K lj
rend ¼ ð16Þ ferent blood pressures. Due to distended lumen, which is
K nj þ K lj
prompted by hypertension, WSS is reduced. The blood pressure
In which rnj is unity according to impossibility of LDL particles has increased the lumen radius, and consequently, has led to a
for passing through normal junctions. From pore theorem, leaky reduction in WSS (Fig. 3(a)); however, high blood pressure, which
junction permeability is described as directly affects the brinkman equation (Eq. (4) inside the wall, has
  a stronger effect than WSS impact on filtration velocity, and as a
3 1 1
rlj ¼ 1  1  a2lj þ a3lj ð1  a2lj Þ ð17Þ result, hypertension enhances the filtration process in overall. This
2 2 3 effect has consequently caused the slight increase in filtration
The fraction of leaky junctions along endothelium is WSS velocity (between WSS dependent and independent cases), illus-
dependent. The number of leaky cells is related to that of mitotic trated in Fig. 3(b), which also shows the dominant influence of
cells as below (Olgac et al., 2008) the blood pressure. Effect of WSS on overall endothelium perme-
ability is shown in Fig. 3(c) which is nearly constant in radial direc-
#LC ¼ 0:307 þ 0:805:ð#MCÞ ð18Þ tion due to the negligible variation of circumferential strain. It is
in which #LC and #MC refers to the number of leaky and mitotic noteworthy that the higher the transmural pressure, the faster rate
cells respectively. According to work of Chien (2003), the number of mitosis will be and additional leaky junctions will be formed.
of mitotic cells in a unit area of 64mm2 is a function of the shape Reduction of endothelium resistance is also evident from the smal-
index (SI) of the endothelial cells as below ler obtained endothelial reflection coefficient as the pressure
increases (Fig. 3(d)). Reflection coefficient is also constant in radial
#MC ¼ 0:003797e14:75SI ð19Þ direction.
Fig. 4 describes the circumferential effective proportion, relative
where the shape index can be defined as
thickness variation of the layers, endothelial normal and leaky
4p  area junction contribution in permeability and von misses stress across
SI ¼ 2
ð20Þ
ðperimeterÞ the arterial wall. In Fig. 4(a), the relation of endothelium circumfer-
ential strain and transmural pressure is found to be nearly direct
The shape of the cells varies with the value of local WSS (Levesque and secondary effects of WSS and FSI on wall properties are not
et al., 1986) as notable. The relative radial changes of thickness of the layers are
SI ¼ 0:38e0:79WSS þ 0:225e0:043WSS ð21Þ shown in Fig. 4(b). Variation of the first three layers whose elastic-
ity is comparatively low and on which higher pressure difference is
These relations affect the definition formulas of endothelium applied is more significant. Lumen change of thickness, which is
properties like permeability (Eq. (13)), diffusivity (Eq. (14)) and positive because of its increase in radius, equals to the thickness
reflection coefficient (Eq. (16)). The / parameter can be calculated variation of every single layer plus their overall displacement.
using the following equation Although the outer structural stress boundary condition is applied
on media/adventitia interface, adventitia’s elasticity is of impor-
#LC  pR2cell
/¼ ð22Þ tance due to the circular geometry of the assumed artery. From
unit area endothelium toward adventitia, the dimensionless thickness varia-
tion shows a slight decrease in absolute value. This is explainable
3. Solution procedure by considering the negative structural gradient of the solid stress
in cylindrical geometries (Shigley, 2011).
A commercial FEM code, COMSOL Multiphysics Version 5.2 is Generally, only a minority of the endothelial cells are leaky,
utilized for implementation of the model. This commercial soft- which can be inferred from their surface ratio, / ffi 5  104 , and
ware uses a finite element method code to solve the governing therefore, leaky junction permeability is relatively low in compar-
equations. Grid study is performed to ensure that the cells in any ison with that of normal junctions, and most of the plasma passes
layer are small enough so the correct solutions are achieved. Due through normal junctions. Accordingly, the normal junctions form

Table 3
The number of elements used in lumen and arterial wall layers.

Region Lumen Endothelium Intima IEL Media Adventitia

Cell number 45,000 18,000 9000 18,000 30,000 6000
 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122 119

Fig. 2. Comparison of concentration across (a) intima and IEL (b) media of the present study, Iasiello et al. (2016b), Iasiello et al. (2016a) and Chung and Vafai (2012) at 70
mmHg.

2.4 5.0
a ΔP=150mmHg b with WSS effect
ΔP=120mmHg 4.5 without WSS effect
2.38
ΔP=70mmHg

4.0
2.36
Vfiltration(×108m/s)

3.5
WSS (Pa)

= , ,
2.34
3.0

2.32
2.5

2.3
2.0

2.28 1.5
0 1 2 3 4 5 6 0 1 2 3 4 5 6
x(cm) x(cm)

1
3.34 c with WSS effect d with WSS effect
without WSS effect without WSS effect
3.32
0.99
Kendothelium(×10 m )

3.3
2

= , ,
21

= , ,
3.28
σendothelium

0.98

3.26

3.24 0.97

3.22

3.2 0.96
0.5 1 1.5 2 0 0.5 1 1.5 2
r-rlumen(μm) r-rlumen(μm)

Fig. 3. (a) WSS along the artery wall and Endothelium (b) thickness, (c) permeability and (d) reflection coefficient for different transmural pressures.

the greater portion of the overall permeability (Fig. 4(c)). This fig- endothelial pores. On the other hand, the reduction in normal junc-
ure shows the intensified permeability of leaky junction at hyper- tion permeability is negligible due to small thickness variation of
tension, which is caused by the effect of reduced WSS on number endothelium. Von Mises Stress across the wall, illustrated in
of leaky junctions and also, the FSI impact on widening of the Fig. 4(d), presents the media and adventitia as the main stress
120 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122

a 0.04
1.02 b
0.03
(1+εθ) / (1+ε70)

lumen
1.01 0.02

Δt / t0
endothelium
intima
IEL
0.01 media
adventitia
1
0

-0.01
0.99
45 60 75 90 105 120 135 150 40 60 80 100 120 140 160
P (mmHg) P (mmHg)

160
c d ΔP=70 mmHg
3 ΔP=120mmHg
140 ΔP=150mmHg
2.5
Klj
Knj
120
σVon Mises (kPa)
K (×10 m )
2

2
21

1.5 100

1 80

0.5
60

0
40 60 80 100 120 140 160 0 50 100 150 200 250 300
P (mmHg) r-rlumen

Fig. 4. (a) Endothelium circumferential strain, (b) dimensionless thickness variation of different regions, (c) normal and leaky junction permeability and (d) Von Mises stress
across the wall, for various transmural pressures.

bearers. The Von Mises stress in these layers decreases towards number, Sh, along the lumen/wall interface are presented in
artery external shell. Fig. 5 Nondimensional Sherwood number, representing the ratio
Here, we have also compared the effects of convection and dif- of convection to diffusion mass transfer, is defined as below
fusion transport mechanism on the LDL transport. For this aim,
K sh L
both LDL diffusivity across the endothelium and the Sherwood Sh ¼ ð23Þ
D

15 1200

14
a With WSS effect b With WSS effect
Without WSS effect
Without WSS effect
1000
13
= , ,
12
Dendothelium(×10 m /s)

800
2

11 = , ,
18

Sh

10 600

9
400
8

7
200
6

5 0
0.5 1 1.5 2 0 1 2 3 4 5 6
r-rendothelium(μm) x(cm)

Fig. 5. (a) LDL diffusion coefficient across endothelium and (b) Sherwood number along the wall/lumen interface at different transmural pressures.
 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122 121

where L is a characteristic length (endothelium thickness) and D is WSS on LDL accumulation in all wall layers, especially in hyperten-
LDL diffusion coefficient. K sh signifies the convective mass transfer sion. WSS impact on accumulation of LDL within intima, where
coefficient, which is defined as the following equation for stiffening of the wall begins, is more evident in Fig. 6(b). This effect
endothelium: is mostly induced by the elevation of endothelium permeability,
and consequently, decrease in endothelium reflection coefficient.
Jr
J r ¼ K sh  ðc  c0 Þ ! K sh ¼ ð24Þ This intensified permeation is caused by the lower WSS in the cur-
c  c0
rent study comparing to those from which / ¼ 5  104 is obtained
In which J r is the LDL flux in radial direction and c0 is the free (Lin et al., 1988; Truskey et al., 1992). Higher WSS caused by
flow concentration which is equal to 1 in here. As it is illustrated increased inlet velocity or decreased lumen diameter would lead
in Fig. 5(a), The WSS effect on diffusion coefficient is negligible. to lower LDL trends within the wall and consequently lower
This result implies that the major influence of WSS on mass trans- atherosclerosis possibility. Therefore, the low velocity regions
port across the wall must be through the convection transport and recirculation zones, for instance, would be more likely to
mechanism. Therefore, diffusivity of particles in endothelium is develop atherosclerosis.
mostly affected by FSI dependency assumption. This can be clearly The LDL concentration on wall/lumen interface, on the other
seen in Fig. 5(b), displaying the Sherwood number, where a com- hand, is rarely dependent on FSI and WSS and varies only with
parison is performed between the WSS dependent and indepen- transmural pressure. Hence Fig. 6(d) suggests a LDL concentration
dent cases. independency from lumen radius along the interface.
Since the diffusive transport remains constant for WSS depen-
dent and independent cases in a specific transmural pressure, the
higher value of Sherwood number with WSS effect is caused by 5. Conclusion
the intensified convection mass transport. As a result of polariza-
tion (the increase in lumen/wall interface concentration along In the present work, a comprehensive 5-layer wall model of
the artery), the concentration difference along the lumen/wall axisymmetric coronary artery incorporating cumulative FSI and
interface decreases toward the artery outlet and this reduces the WSS influence on endothelium properties is simulated. Our pur-
Sherwood number as well (Fig. 5(b)). pose is to accurately determine how coronary wall distension
Concentration distribution within the artery wall layers is and WSS affect LDL transport and retention within the wall. The
shown in Fig. 6 in both considered and overlooked WSS effect con- results underline the necessity of considering coupled WSS and
ditions. Comparison of these plots implies the additive effect of FSI effect on arterial wall properties in hypertension. As a

1
with WSS effect
1 b without WSS effect
a 0.8
0.8 = , ,

0.6
0.6
C/C0

C/C0

0.4
0.4

0.2 = , ,
0.2 with WSS effect
without WSS effect

0 0
1.5 1.6 1.7 1.8 1.9 2.0 0 2 4 6 8 10 12
r-rlumen (μm) r-rendothelium/intima (μm)
1.04
0.2 c with WSS effect
d with WSS effect
without WSS effect without WSS effect

1.03
0.15
= , ,
C/C0

1.02
C/C0

0.1 = , ,

1.01
0.05

0 1
0 20 40 60 80 100 0 1 2 3 4 5 6
r-rIEL/mdia (μm) x(cm)

Fig. 6. Concentration of LDL within (a) endothelium, (b) intima and IEL, (c) media and (d) the polarization of the artery in different transmural pressures.
122 M. Roustaei et al. / Journal of Biomechanics 67 (2018) 114–122

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Conflict of interest statement
Kolandavel, M.K., Fruend, E.-T., Ringgaard, S., Walker, P.G., 2006. The effects of time
varying curvature on species transport in coronary arteries. Ann. Biomed. Eng.
No conflict of interest exists. This paper has only been sent to 34, 1820–1832.
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