Numerical Simulation of Blood Flow in the System of HumanCoronary Arteries with Stenosis

PEARANAT CHUCHARD1∗ , BENCHAWAN WIWATANAPATAPHEE1∗ , THITIKOM PUAPANSAWAT1,THANONGCHAI SIRIAPISITH2

1Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand2Department of Radiology, Faculty of Medicine Siriraj Hospital, Mahidol University, Bangkok, Thailand

∗Corresponding author: g5237100@student.mahidol.ac.th,scbww@mahidol.ac.th

Abstract: This paper aims to present three-dimensional simulation of blood behavior in the system of humancoronary arteries. The mathematical model is a set of partial differential equations including continuity equationand Navier-Stokes equations. The pulsatile conditions due to the heart pump during a cardiac cycle is imposed onthe boundaries. Computational domain consists of the base of aorta, the left and the right coronary arteries. Finiteelement method is applied for the solution of the mathematical model Blood flow and temperature distribution incoronary system with normal arteries and stenosed arteries are computed. The results show that the appearance ofstenosis reduces blood flow rate in the stenosed artery.

Key–Words: numerical simulation, stenosis, coronary artery, image reconstruction, blood flow.

1 IntroductionA system of human coronary arteries consists of theaorta, the right coronary artery and the left coronaryartery. The left coronary artery (LCA) divides into leftanterior descending (LAD) and circumflex branches(LCX). The right and the left coronary arteries provideblood supply to the heart muscle. When blood flow incoronary arteries is subjected to severe stenosis, criti-cal conditions occur such as high blood pressure andinadequate blood supply to the heart. To understandthe blood flow problem in the stenosed coronary ar-teries, extensive researches have been carried out tostudy blood flow problems in either the right coronaryartery or the left coronary artery, including experimen-tal, analytical and numerical studies over the last two[2, 4, 5, 8, 9, 10, 12, 13, 14, 15].

It is well established that the fluid-structure inter-action determines the behavior of blood flow througharteries [11]. Recently, various studies have focusedon the coupled fluid flow and arterial wall deforma-tion problem. Basombrio et al. (2002) constructednumerical experiments for non-trivial flow, close torealistic situations in hemodynamics [7]. Boutsia-nis et al. (2004) studied non-Newtonian blood flowin human right coronary arteries with transient sim-ulation [6]. The model was constructed from a CTscan. Wiwatanapataphee (2008) investigated the ef-fect of stenosis on non-Newtonian blood flow througha stenosed tube [3]. The results obtained from the

domains having 25%, 50% and 75%-area severityshow that blood pressure drops dramatically aroundthe stenosis site and creates a jet flow at the throat ofthe stenosis. Chaniotis et al. studied computationalblood flow in curve and bifurcation geometries [1].Comparison with the Newtonian model, their resultshowed the significance of the non-Newtonian modelon the shear stress distribution.

Due to a difficult task to construct realistic geom-etry of the coronary artery, the studies as mentionedabove used unrealistic geometry of blood vessel suchas a straight tube or curve tube with branches and withno branch. It has been recognized that the results ob-tained from unrealistic domain may not be applicablefor clinical use.

In this study, we simulate non-Newtonian pul-satile blood flow in a system of human coronary ar-teries with stenosed left coronary artery. Influence ofstenosis occurring in the middle part of the LCA onblood flow problem is investigated.

2 Governing equations and Bound-ary conditions

In this study, blood is modeled as an incompressiblenon-Newtonian fluid. Based on the Carreau model,blood viscosity is a function of shear rate and is deter-

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ISBN: 978-960-474-298-1 59

mined by

η(γ̇) = η∞ + (η0 − η∞)[1 + (λγ̇)2](n−1)/2, (1)

where η∞, η0, λ and n are parameters. The govern-ing equations of blood flow including the continuityequation and Navier-Stokes equations are as follows:

ui,i = 0, (2)

ρ(∂ui∂t

+ ujui,j) = −p,i + (ηγ̇(ui,j + uj,i)),j ,(3)

where ui is blood velocity in the i-direction for i =1, 2, 3; p denotes blood pressure and ρ is blood densityof 1.06 g/cm3. The shear rate γ̇ in equation (1) isdefined by

γ̇ =

√2tr(

1

2(ui,j + uj,i))2. (4)

To specify the boundary conditions for the bloodflow, we consider the blood flow mechanism due tothe heart pump. The cyclic nature of the heart pumpcreates pulsatile conditions in the coronary arteries. Inthis study, we ignore the variation in different cardiaccycle, the pulsatile pressure and flow rate can be ex-pressed by p(t) = p(t+ nT ) and Q(t) = Q(t+ nT )for n = 0, 1, 2, 3, and T is the cardiac period. Math-ematically, the pulsatile pressure and flow rate can bewritten in the form of the truncated Fourier series:

p(t) = p̄+

4∑k=1

αpk cos (kωt) + βp

k sin (kωt), (5)

Q(t) = Q̄+

4∑k=1

αQk cos (kωt) + βQ

k sin (kωt), (6)

where Q̄ denotes the mean flo rate and p̄ is the meanpresure, ω = 2π/T is the angular frequency with pe-riod T .

On the inflow surface of the aorta Γaorta, we im-pose the pulsatile velocity

u(t) =Q(t)

A, (7)

where A is the cross-sectional area of the inflow sur-face of the aorta which is 6.7287 cm2. On the out-flow surfaces {Γaorta

1 , ΓRCA1 , ΓRCA

2 , ΓRCA3 , ΓRCA

4 ,ΓLCA1 , ΓLCA

2 , ΓLCA3 }, we impose corresponding pul-

satile pressure condition, i.e., for i, j = 1, 2, 3

p = p0(t), (η(ui,j + uj,i)) · n = 0. (8)

(a)

(b)

Figure 1: The computational domain with (a) LCAaxis, (b) inflow and outflow surfaces of the system ofcoronary arteries

Figure 2 presents the flow rate waveform and cor-responding pressure in the aorta. Values of the meanflow rate on inflow surface of the aorta and mean pres-sure on outflow surface of the aorta are 5.7222 l/minand 97.2222 mmHg, respectively. Other values of in(5) and (6) are given in Table 1.

Table 1: Values of theparameters αQk , β

Qk , α

pk and βp

k

η αQk βQ

k αpk βp

k

1 1.7048 -7.5836 8.1269 -12.41562 -6.7035 -2.1714 -6.1510 -1.10723 -2.6389 2.6462 -1.3330 -0.38494 0.7198 0.2687 -2.9473 1.1603

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Figure 2: The flow rate wave form and corresponding,pressure in the aorta

In summary, the blood flow in the coronary arterysystem is governed by the following boundary valueproblem (BVP):

BVP : Find u and p such that equations (1) - (4)and all boundary conditions are satisfied.

3 Numerical ExampleWe have simulated the 3-D blood flow through thesystem of coronary arteries with normal arteries andwith stenosis at the middle part of left coronary artery(LCA) using COMSOL multipysics. The computa-tion domain is shown in Figure 3

We constructed two domain finite element meshesof the system with normal arteries and with 75%stenosed LCA consisting of 20,192 tetrahedral ele-ments with 115,514 degrees of freedom and 21,446tetrahedral elements with 121,194 degrees of free-dom, respectively.

To investigate the effect of stenosis on the bloodflow behavior in the system of coronary arteries, wesimulate the blood flow from the models with nor-mal arteries and with 75% stenosed LCA. The pul-satile pressure along the LCA axis (see Figure 1(a))and velocity field on the outflow surface ΓLAD

1 of theend of LAD are investigated. The results as shownin Figures 4 indicate the stenosis has a significant ef-fect on the blood flow problem. Compared with theresults obtained from the model with normal arteries,the pressure drop significantly in the model with 75%stenosed LCA. Figure 5 illustrates the surface plot ofthe velocity field on the outflow ΓLAD

1 of the LAD.It is noted that the maximum velocity at the peak ofsystole and diastole in the system with 75% stenosed

LCA are respectively 22.811 cm/s and 15.840 cm/swhile the normal system has more maximum velocity:29.179 cm/s at the peak of systole; 20.015 cm/s atthe peak of diastole. This indicates that when stenosisis present, it reduces the flow rate.

(a)

(b)

Figure 3: The element mesh of the computational do-main with stenosed LCA

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ISBN: 978-960-474-298-1 61

Figure 4: Pressure along the LCA axis at the peak of systole and the peak of diastole obtain fromthe normal system (solid line) and the system with 75 % stenosed LCA (dash line).

(a) A system with normal arteries

(b) A system with 75% stenosed LCA

Figure 5: The surface plot of velocity field on outflow ΓLAD1 of LAD at the peak of systole and the peak of

diastole in two systems: (a) a system with normal arteries; (b) a system with 75% stenosed LCA.

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4 ConclusionIn this work, we developed three-dimensional math-ematical model of blood flow taking into accountthe pulsatile boundary condition. The computationaldomains are two coronary systems with normal ar-teries and stenosed arteries. The solutions of theproblem were solved by finite element method us-ing COMSOL multiphysics. Effect of the systemwith the stenosed arteries on the blood flow problemwas investigated. The result showed that the systemwith stenosed arteries provided lower blood pressurearound the stenosed region and lower blood speed onthe outflow of stenosed vessel than those in the systemwith normal arteries. This implies that the stenosedarteries cause an inadequate blood supply to the heart.

Acknowledgements : The authors are grateful to theThailand Research Fund (TRF) for support of this re-search and also wish to thank the Development andPromotion of Science and Technology Talents Project(DPST).

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