The Wall Shear Stress of a Pulsatile Blood Flow in a Patient Specific Stenotic Right Coronary Artery

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Engineering, 2013, 5, 396-399

http://dx.doi.org/10.4236/eng.2013.510B080 Published Online October 2013 (http://www.scirp.org/journal/eng)

Copyright © 2013 SciRes. ENG

The Wall Shear Stress of a Pulsatile Blood Flow in a

Patient S pecific Stenotic Rig ht Coronary Artery

Biyue Liu

Department of Mathematics Monmouth University West Long Branch, NJ, USA

Email: [email protected]

Received 2013

ABSTRACT

A computer simulation of the blood flow in a patient specific atherosclerotic right coronary artery is carried out to study

the blood flow pattern and the wall shear stress (WSS) distribution in the artery. Both temporal and special distr ibution

patterns of the WSS of the non-Newtonian blood flow are presented and the regions on the lumen surface where the

WSS is constantly lower than 1 N/m2 are identified.

Keywords: Wall Shear Stress; Stenotic Right Coronary Artery; Non-Newtonian

1. Introduction

Atherosclerosis is the chief cause of death in the United

States. It is a disease which involves complex interac-

tions of many factors. It is widely believed that the wall

shear stress (WSS) is one of the important local biome-

chanical factors responsible for the initiation and the pro-

gression of atherosclerosis [1-7]. A detailed hemodyna-

mic evaluation of the spatial and temporal WSS distrib u-

tions may give additional insight to understanding the

progression of the disease and may have useful clinical

value.

In the past few decades, numerous clinical researches

and computer simulations have been performed to study

the flow phenomena in human atherosclerotic arteries

and to investigate the correlation between the WSS and

the intima-media thickness [1-10]. Caro et al. [1] sug-

gested that the distribution of fatty streaking in human

aorta might be coincident with the regions in which the

shear rate at the arterial wall is locally reduced. Friedman

et al. [2] and Nerem et al. [6] performed experiments that

showed an accelerated occurrence of atherosclerosis in

human subjects with a coronary geometry. Zarins et al.

[7] and Ku et al. [4] estimated shear stress with laser-

Doppler anemometry and found that intimal thickening

bears an inverse relationship to both maximum shear

stress and minimum shear stress. Gibson et al. [3] inves-

tigated the relationship between the vessel wall shear

stress and the r ate of atherosclerosis progression by means

of quantitative angiography. They found a significant

correlation between the low shear stress and an increased

rate of atherosclerosis progression. Johnston et al. [10]

compared models of Newtonian and Non-Newtonian flow

in healthy right coronary arteries with no sign of athero-

ma. Although the blood flow in stenotic arteries has been

extensively studied, there is much to be done both in un-

derstanding the disease process itself and also in under-

standing the role of hemodynamics and the associated

WSS as an influence factor on the initiation and the pro-

gression of athe ros c l erosis .

The objectives of the current work are to simulate the

blood flow in a stenotic human right coronary artery and

to investigate the spatial and temporal WSS distribution

patterns during a cardiac cycle.

2. Mathematical Model

This study assumes that the fluid is Laminar, Non-New-

tonian, viscous and incompressible and the artery wall is

rigid. These assumptions have been shown to be ade-

quate for the pulsatile blo od flow simulation in th e artery

models under physiological flow conditions by many

investigators. The computational domain is a patient spe-

cific atherosclerotic right coronary artery shown in Fig-

ure 1(a), re-constructed based on the in vivo intravascu-

lar ultrasound (IVUS) image of a patient [9]. The lumen

cross section area fro m the inlet to the outlet is plotted in

Figure 1(b), where the horizontal axis is the normalized

axial length with inlet z = 0 and the outlet z = 1. The

minimum lumen cross section area at the neck of the

stenosis is 0.042 cm2. The maximum area of the cross

section proximal to the stenosis is 0.130 cm2. The r educ-

tion of the lumen area at the neck of stenosis is approx-

imately 68%. It is a severely stenotic right coronary ar-

tery segment.

The time dependent three dimensional Navier Stokes

B. Y. LIU ET AL.

Copyright © 2013 SciRes. ENG

397

(a)

(b)

Figure 1. (a) Geometry of the computational domain; (b)

Lumen cross section area from the inlet to the outlet. Neck

of the stenosis at z = 0.6.

equations are used as the governing equations. The inlet

boundary is imposed with a fully developed flow with a

physiological human right coronary waveform (Figure

2), scaled to yield a time averaged flow velocity of 0.166

m/s at the centre-line. A no-slip condition is applied to

the velocities at the wall boundary, treated to be inelastic

and impermeable. The outlet boundary is treated as an

open boundary with the zero normal stress:

((() ))

T

p

η

−+∇ +∇=I uu0n

(1)

where n = (n1, n2, n3) is the outward normal unit vector at

the outlet boundary. The initial cond itions for the veloci-

ty and the pressure are obtained by solving the system of

steady state Navier Stokes equations. The blood is treated

as a non-Newtonian fluid obeying the Carreau model with

the viscosity-shear rate relation:

1

22

0

()[1()]

n

η ηηηλγ

−

∞∞

=+−+



(2)

where η0 = 0.056 Pa∙s is the zero shear rate viscosity,

η

∞

= 0.00345 Pa ·s is the infinite sh ear rate viscos ity, λ =

3.313s is a parameter, and n = 0.3568 is a dimensionless

parameter [7,10]. The blood density

ρ

is assumed to be

constant at 1050 kg/m3.

3. Observations and Discussion

The Navier Stokes equations are solved numerically us-

ing the finite element method with piecewise quadratic

functions for velocity and piecewise linear functions for

pressure over a tetrahedral mesh. Four consecutive car-

diac cycles are simulated to ensure that the flow is truly

periodic and the computations are repeated over different

meshes to ensure that the numerical solutions are mesh

independent. The numerical computations are performed

using COMSOL 4.2.

Figure 3 presents the contour plots of the WSS (N/m2)

Figure 2. Pulsatile coronary velocity waveform at the inlet.

Figure 3. Spatial distribution of the WSS (N/m2) along the

artery wall when (a) t/ tp = 0.1, (b) t/tp = 0.35, (c) t/tp = 0.85,

respectively.

(a) during the deceleration in systole (t/tp = 0.1), (b) at

the peak in systole (t/tp = 0.35), and (c) at the maximum

flow rate (t/tp = 0.85), respectively. Here the inlet is on

the top, and the outlet is on the bottom. Figure 3 shows

the spatial distribution pattern of the WSS on the lumen

surface. The negative WSS means that the flow moves

backwards. The overall negative WSS along the artery

wall showing in Figure 3(a) is due to the factor that it is

during the reverse flow in the systole (see the first doted

point on the velocity waveform in Figure 2). The nega-

tive WSS showing in Figure 3(c) indicates that flow re-

circulation appeared in these regions with a negative

WSS. F ig ures 3(b) and (c) show a basically similar pat-

tern of the spatial distribution of the WSS along the ar-

tery at the peaks of the systole and the diastole. The ma-

ximum WSS appears at the neck of the stenosis and the

WSS is overall high in stenotic region. There is another

region with a relatively high WSS downstream where the

artery is slightly narrowed. The minimum WSS o ccurs at

the inner wall in the post stenotic region and in the region

proximal to the stenosis where the artery expands. The

plot of the contour of the WSS at any other time during

the forw ard flow has the same pattern as that showing in

Figure 3(b).

-0.01

0.04

0.09

0.14

00.2 0.4 0.6 0.81

Lumen Area (cm2)

Normalized Axial Length

-0.05

0

0.05

0.1

0.15

0.2

0.25

00.2 0.4 0.6 0.81

Velocity (m/s)

t/t

p

B. Y. LIU ET AL.

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398

To quantitatively examine the v ariation of the WSS on

the lumen surface, the WSS is averaged on the wall

boundary of each lumen cross section and the plots at

some representing times are included in Figure 4 (t/tp =

0.1, 0.25, 0.35 and 0.85, respectively, which are doted

times in Figure 2). The horizontal axis is the normalized

axial length. The inlet is at z = 0 and the outlet is at z = 1.

The neck of the stenosis is at z = 0.6. Figure 4 confirms

the above observation on the special distribution of the

WSS from Figure 3. The maximum average WSS occurs

at the neck of the stenosis.

Figure 5 plots the WSS in an entire cardiac cycle at

various points on the ar tery wall: (a) at the proximal side

of the stenosis and at the neck of the stenosis, (b) in the

post stenosis and the downstream. It demonstrates the va-

riation of the local WSS during an entire cardiac cycle at

the four locations of most interest: two around the neck

of stenosis and two in the post-stenosis region. From

Figure 5 we can see that other than the area on the inner

wall in the post-stenosis region, the local WSS at any

point basically experiences the same fluctuation as that

of the flow waveform at the inlet: a dip with negative

WSS value caused by the reverse flow in systole follow-

ed by two small peaks and a large elevation in diastole.

However, the purple curve (labeled PostS) in Figure 5(b)

shows that the WSS is constantly low in a whole cardiac

cycle on the inner wall in the post stenotic region, with-

out a dominantly high peak when t/tp = 0.85. These be-

haviors are consistent with the observation made by the

author previously to the simulation based on a simplified

geometry with idealized circular cross sections [7].

Many studies [1-5,7,9] have attempted to correlate the

WSS with the intimal thickness, suggesting that overall it

is in low shear regions that intimal thickness will be

greater. Low local velocity, resulting low WSS values,

increases residence time and interaction between the

blood lipoproteins and vessel endothelium. Long resi-

dence time with endothe lium results in an increased lipo-

protein intake, which might cause the thickening of arte-

rial wall. Clinical studies suggest that intimal thickening

Figure 4. WSS averaged on each lumen cross section when

t/tp = 0.1, t/tp = 0.25, t/tp = 0.35, t/tp = 0.85.

Figure 5. Temporal variation of the WSS at various points

on the inner wall (a) at the proximal side of the stenosis and

at the neck of the stenosis, (b) in the post stenosis and the

downstream of the stenosis.

likely occurs when the average WSS is below 1 N/m2 (=

10 dynes/cm2), which presents an inverse hyperplasia

with respect to the shear stress [5]. Therefore, it is of

special interest to know the location and the size of the

region on the lumen surface where the WSS is lower than

1 N/m2 (including the negative WSS resulted from the

reverse flow or flow recir culati on ) an d the duration of the

low WSS area in a cardiac cycle.

At each point on the lumen surface, we calculated the

total length of the time when the location experiences a

low WSS < N/m2. Figure 6 presents the contour plot of

the duration of the WSS under N/m2 in a cardiac cycle.

Figures 6(a)-(c) are the views of the artery from the side,

the inner wall and the outer wall, respectively. A point

with a value of 1.0 on the lumen surface indicates that

the WSS at this location is lower than 1 N/m2 anytime

during the entire cardiac cycle. The accumulated duration

time of the WSS lower than 1 N/m2 in a cycle has the

minimum value of 0.17 in the region of stenosis neck.

From the length of the interval where the grey curve (la-

beled Neck) is below the horizontal axis line setting at

1.0 level in Figure 5(a), we can tell that this duration of

the low WSS occurs approximately during the systole

between t/tp = 0.03 and t/tp = 0.2 in a cardiac cycle. It is

also apparent that the duration value is 1.0 in the area on

the inner wa ll in the post -stenosis region and the area on

the inner wall in the region proximal to the stenosis

where the lumen is expanded. This indicates that these

B. Y. LIU ET AL.

Copyright © 2013 SciRes. ENG

399

Figure 6. The duration of the WSS under 1 N/m2 (a) the side

view of the artery, (b) the inner wall of the artery, (c) the

outer wall of the artery.

areas experience a lower WSS during the whole cardiac

cycle. The purple curve (labeled PostS) in Figure 5(b)

also demonstrates this, where the whole curve is below

the horizontal axis line setting at 1.0 level. Comparing

Figures 6(b) with (c) we can see that the low WSS re-

gions mostly occur on the inner wall of the stenotic right

coronary artery. The WSS on the outer wall is relatively

higher, which may prevent the deposition of the particles

and the further intimal thickening along this side of the

artery.

4. Conclusion

In this work, a computer simulation of the blood flow in

a patient specific stenotic right coronary artery has been

performed to investigate the phasic variation and the spa-

tial distribution pattern of the wall shear stress on the

lumen surface. Based on the computational results pre-

sented, we can see that the WSS distribution is highly

non-uniform both temporally and spatially. The WSS ele-

vates in the stenotic region, reaches the maximum at the

neck of the stenosis, and drops sharply in the post-steno-

sis region. The areas on the inner wall in the post-steno-

sis region and in the region proximal to the stenosis are

subject to a low WSS less than N/m2 during the entire

cardiac cycle.

5. Acknowledgements

This work was partially supported by a grant from the

Simons Foundation (#210082 to Biyue Liu) and a sab-

batic al gra nt f rom Monm o uth Uni ve rsi ty. The author thanks

Dr. Dalin Tang for providing the stenotic right coronary

artery data for re-constructing the computational domain.

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