Numerical Simulation of Blood Flow Through the System of ... · Coronary Arteries with diseased Left Anterior Descending Pearanat Chuchard, Thitikom Puapansawat, Thanongchai Siriapisith,

Numerical Simulation of Blood Flow Through the System ofCoronary Arteries with diseased Left Anterior Descending

Pearanat Chuchard, Thitikom Puapansawat, Thanongchai Siriapisith, Yong Hong Wu and BenchawanWiwatanapataphee

Abstract—This paper aims to study the arterialstenosis effect on blood flow problem in the systemof coronary arteries. Blood is assumed to be non-Newtonian incompressible fluid. The system of coro-nary arteries with diseased Left Anterior Descending(LAD) is considered. Governing equations are theNavier-Stokes equations and continuity equation sub-jected to the time-dependent pulsatile boundary con-ditions. Based on finite element method, the solutionof the governing equations is solved numerically. Dis-turbances of blood flow through the diseased LAD forthe restrictions of 25%, 50% and 75% are investigated.Flow characteristics, wall pressure and wall shear ratehave been studied in detail. Numerical studies showthat blood flow with high speed and pressure rapidlydrops in the area supplied by the stenosed artery. Asthe degree of coronary-artery stenosis increases, themaximal coronary flow decreases.

Index Terms—blood flow, stenosed coronaryartery, pulsatile boundary condition , finite elementmethod.

I. INTRODUCTIONThe most common form of heart disease is

the coronary artery disease (CAD). Over 7 millionpeople in developed and developing countries suffer

This work was supported by the Development and Promotionof Science and Technology Talents Project (DPST), the Office ofthe Higher Education Commission, and the Thailand ResearchFund

P. Chuchard is with Department of Mathematics, Facultyof Science, Mahidol University, 272 Rama 6 Road, Ra-jthevee, Bangkok 10400 THAILAND (corresponding author; e-mail:[email protected]).

T. Puapansawat is with Department of Mathematics, Facultyof Science, Mahidol University, 272 Rama 6 Road, Rajthevee,Bangkok 10400 THAILAND (e-mail:[email protected]).

T. Siriapisith is with Department of Radiology, Faculty ofMedicine Siriraj Hospital, Mahidol University, Bangkok, THAI-LAND (e-mail:[email protected]).

Y.H. Wu is with Department of Mathematics and Statistics,Curtin University of Technology, Perth, WA 6845 AUSTRALIA(e-mail:[email protected]).

B. Wiwatanapataphee is with Department of Mathematics,Faculty of Science, Mahidol University, 272 Rama 6 Road,Rajthevee, Bangkok 10400 THAILAND (corresponding author;e-mail:[email protected]).

from it. CAD is an occlusion of the coronary arteriesresulting in insufficient supply of blood and oxygendeprivation to the heart muscle. When the blockageof an artery is complete, it results in a heart attack,or in very severe cases, may result in death of thepatient. To overcome the problem, detailed knowl-edge associated with the disordered flow patternsin stenosis is then important for the detection oflocalized arterial disease in its early stages. In orderto understand the pathogenesis of coronary diseases,a number of in-vivo and vitro experiments have beenconducted using animal models [12, 14, 17, 25].Uren et al. (1994) studied the relation between theseverity of stenosis in a coronary artery and thedegree of impairment of myocardial blood flow inlaboratory animals. It is reported that the increasingdegree of coronary-artery stenosis increases gives themaximal coronary flow. Due to the difficulty in de-termining the critical flow conditions for the in-vivoand vitro experiments, the exact mechanism involvesis not well understood. On the other hand, mathemat-ical modeling and numerical simulation can lead tobetter understanding of the phenomena involved invascular disease [1, 5, 6, 8, 9, 11, 13, 16, 19, 20,21, 22, 23]. Over the last two decades, a number ofmathematical models and numerical techniques hasbeen proposed to describe blood flow in a stenosedcoronary artery [7, 10, 12, 15, 18]. Wiwatanapat-aphee (2008) [4] studied the effect of stenosed ar-teries in a stenosed tube with non-Newtonian modelof blood. It is indicated that blood pressure dropssignificantly around the stenosis region. The bloodpressure continuously drops as the degree of stenosisincreases. Chaniotis et al. (2010) [2] studied the pul-satile blood flow in the computational geometry ofcoronary. The Newtonian model and non-Newtonianmodel were used to compare on wall shear stressdistribution. Their results showed that the effect ofnon-Newtonian model was significant. Cheung etal. (2010) [3] studied blood flow through stenosedbifurcation vessel using experimental and numericalmodels. The experimental study was based on PIVmeasurements, while the numerical study was basedon CFD simulation. They concluded that numerical

INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATION

Issue 4, Volume 5, 2011 334

simulation based on an realistic geometry of humancoronary arteries provided accurate result against theexperiment.

Due to the fact that construction of a realisticgeometry of the coronary artery system is a diffi-cult task, a unrealistic vessel geometry including astraight tube or curve tube with branches and withno branch are used in the above mentioned studies.It has been accepted that the results that obtainedfrom unrealistic geometry may not be appropriate forclinical use. In spite of extensive modeling studieson blood flow in the stenosed coronary artery, littlework has been done to solve the blood flow problemin the system of the coronary arteries. Most studiesuse either the right coronary artery (RCA) or the leftcoronary artery (LCA) without the aorta.

In this study, we simulated the three-dimensionalunsteady non-Newtonian blood flow in the systemof human coronary artery with diseased LAD. Thepulsatile boundary condition is considered. Effectof different degree of stenosis on the blood flowis investigated. The rest of the paper is organizedas follows: Section II concerns mathematical modeland finite element formulation. Numerical exampleis presented in section III. Conclusion is given insection IV.

II. MATHEMATICAL MODELS

A. Governing EquationsIn this study, the motion of blood flow is gov-

erned by the continuity equation and Navier-Stokeequations:

∇ · u = 0, (1)

∂u

∂t+ (u · ∇)u =

1

ρ∇ · σ. (2)

where u is blood velocity, ρ is blood density, σ is thestress-deformation rate relation which is described by

σ = [−pI + η(γ)(∇u+ (∇u)T )] (3)

where p is blood pressure and η(γ) denotes theviscosity of incompressible non-Newtonian blooddetermined by Carreau model [24]:

η(γ) = n∞ + (n0 − n∞)[1 + (λγ)2](n−1)/2, (4)

where n∞, n0, λ, n are parameters and the shear rateγ defined by

γ =

√2tr(

1

2(∇u+ (∇u)T )2. (5)

To specify the pulsatile behavior of blood flow,the cyclic nature of the heart pump is considered.In this study, we assume that the cardiac cycle

has no difference in time variation, the pulsatilepressure and the flow rate of blood can be expressedby the periodic functions p(t) = p(t + nT ) andQ(t) = Q(t + nT ) for n = 0, 1, 2, ... and T is thecardiac period. Mathematically, the pulsatile pressurep(t) and flow rate Q(t) can be written in the formof the truncated Fourier series:

p(t) = p+

4∑k=1

αpk cos kωt+ βpk sin kωt,

Q(t) = Q+

4∑k=1

αQk cos kωt+ βQk sin kωt,

(6)

where p denotes the mean pressure, Q is the meanflow rate and ω = 2π/T is the angular frequencywith cardiac period T .

On the inflow surface Γaorta of the aorta, weenforce the pulsatile velocity condition:

u(t) =Q(t)

A, (7)

where A is the area of the inlet surface. On the out-flow surface, we enforce the corresponding pulsatilepressure condition:

p = p0(t), η(∇u+ (∇u)T ) = 0. (8)

In summary, the blood flow in the coronaryartery system is governed by the following boundaryvalue problem (BVP):

BVP: Find u and p such that equation (1)-(6)and all boundary conditions (7)-(8) are satisfied.

TABLE IVALUES OF THE PARAMETERS αQ

k , βQk , αp

k AND βpk USED IN

THE FORMULA 6

k 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

B. Finite Element FormulationThis section concerns the finite element formula.

Starting with the total weighted residual of thesystem of equation (1-2), we obtain the variationalstatement corresponding to the BVP:



Fig. 1. The computational domain of the system of coronaryarteries

Find u and p ∈ H1(Ω) such that for all wu and wp

∈ H10 (Ω), all boundary conditions are satisfied and

(∇ · u, wp) = 0, (9)

(∂u

∂t,wu) + ((u · ∇)u,wu) =

1

ρ(∇ · [−pI + η(∇u+ (∇u)T )],wu).

(10)

where H1(Ω) is the Sobolev space W 1,2(Ω) withnorm ∥ · ∥1,2,Ω, H1

0 (Ω) = v ∈ H1(Ω)|v = 0 onthe Dirichlet type boundary and (·, ·) denoted theinner product on the square integrable function spaceL2(Ω), defined by.

(u, v) =

∫Ω

u · v dΩ. (11)

The second order derivatives in equations (9)and (10) are reduced to the first order using Green’sformula. We then have

(∇ · u, wp) = 0, (12)

(∂u

∂t,wu) + ((u · ∇)u,wu)

− 1

ρ(∇wu, [pI − η(∇u+ (∇u)T )])

− 1

ρ[a(wu, σ · n)] = 0 (13)

To solve the above problem using the GalerkinFinite Element method. For u and wu, we choose anL-dimensional subspace HL of H1(Ω) and a basisfunctions ϕLi=1 of HL. We then obtain

u(x, t) ≈ uL =

L∑α=1

uα(t)ϕα(x), (14)

wu ≈ (wu)L =

L∑β=1

wuβ(t)ϕβ(x). (15)

Choosing an M-dimensional subspace HM of H1(Ω)with a basis functions φMi=1 for p and wp. We thenobtain

p(x, t) ≈ pM =∑M

γ=1 pγ(t)φγ(x), (16)

wp ≈ (wp)M =∑M

δ=1wpδ (t)φδ(x), (17)

From equations (14)-(17), replacing u, p, wu andwp in equations (12)-(13) for arbitrary wu, wp andapplying boundary condition 8, we then have

L∑α=1

(ψδ,∂ϕα∂xi

)ujα = 0, (18)

L∑α=1

(ϕβ, ϕα)uiα +

L∑α=1

(ϕβ, uj∂ϕα∂xj

)uiα

− 1

ρ

M∑γ=1

(ψγ ,∂ϕβ∂xi

)pγ +1

ρ

L∑α=1

(∂ϕβ∂xj

, η∂ϕα∂xj

)uiα

+1

ρ

L∑α=1

(∂ϕβ∂xj

, η∂ϕα∂xi

)ujα = 0 (19)

The system of equations (18) and (19) can bewritten in matrix from as

CT1 U1 + CT

2 U2 + CT3 U3 = 0 (20)

M

U1

U2

U3

+A

U1

U2

U3

−

C1

C2

C3

P = 0 (21)



where

M =

mβα 0 00 mβα 00 0 mβα

with mβα = (ρϕβ, ϕα)

A =

B +K11 K12 K13

K21 B +K22 K23

K31 K32 B +K33

with

B = D +K11 +K22 +K33,

Kij = (Kijβα) with Kijβα = (η∂ϕβ∂xj

,∂ϕα∂xi

),

D = (Dβα) with Dβα = (ρuj∂ϕα∂xj

, ϕβ),

Ci = (Ciβα) with Ciβα = (ψγ ,∂ϕβ∂xi

).

The system of equations (20)-(21) can be writtenas

CT1 U1 + CT

2 U2 + CT3 U3 = 0 (22)

MU +AU − CP = 0 (23)

By using the backward Euler difference scheme fortypical time step (tr → tr+1), the system can bereduced to:

CT1r+1U1 + CT

2r+1U2 + CT3r+1U3 = 0 (24)

(M +trA)Ur+1 +trCPr+1 =MUr. (25)

Since A in equation (25) depends on Ur+1, thesystem is then nonlinear which can be solved byusing iterative updating scheme, i.e.,

CTi+11r+1U1 + CTi+1

2r+1U2 + CTi+13r+1U3 = 0 (26)

(M +trAir+1)U

i+1r+1 +trCP i+1

r+1 =MU ir. (27)

where i is the ith iteration step. U1 and P1 can bedetermined by U0

1 = U0, P 01 = P0, and repeatedly

solving the above system until ∥Ur+11 − Ur

1∥ < ϵuand ∥Pr+1

1 −Pr1∥ < ϵp.

III. NUMERICAL RESULTSIn this work, we study the flow phenomena of bloodin the system of coronary arteries with diseasedLeft Anterior Descending. The characteristic of thesystem of coronary arteries is shown in Fig.1. Theexample under investigation is a coronary systemconsisting of the base of the aorta, the RCA andthe LCA with LAD and LCX. Total volume of thecoronary system is 31.1 cm3. The area of aorta inletsurface is 6.712 cm2.

The computational domains used in this studyare constructed using a number of CT-images andMimic software. A three-dimensional system ofcoronary arteries in stereo-lithography (STL) formathas very rough surface which is not suitable forcomputational fluid dynamic. To smooth its surface,the cubic B-spline interpolation is required.

We simulate the blood flow in the coronarysystem with different degree of stenosis, 25%, 50%and 75% stenosed artery. The position of the stenosisis at 2.5 cm from the LCA inlet surface. Otherparameters are as follows. The density of bloodρ = 1.06 g/cm3, the infinite shear rate viscosityη∞ = 0.0345 (dyne/cm2) · s, the zero shear rateviscosity η0 = 0.56 (dyne/cm2) · s, two modelparameters λ = 3.313 s and n = 0.3568, the meanpressures p of the aorta, the RCA and the LCA are97, 65, 65 mmHg respectively, and the mean flowrate in the aorta Q = 0.09537 cm3s−1, and thecardiac period T = 0.8 s.

The boundary value problem presented in sec-tion 2 is solved using the Bubnov-Galerkin finiteelement method via COMSOL Multiphysics. In orderto investigate the effect of stenosis on the bloodflow as well as on the pressure field and shearrate, three computational domains are constructed forthree coronary systems of different degree stenosis.Fig. 2 shows a finite element mesh with a numberof Lagrange order two elements.

Fig. 3 shows the vector plot of the velocity fieldand the sub-domain plot of the pressure of blood inthe coronary system having 75% stenosed LAD atthe peak of systole. It is noted that blood flows intothe system of cornary arteries from the inlet aortato outlet aorta, the RCA and the LCA. High bloodspeed and a pressure drop are present around thestenosis arterial section.

To investigate the effect of stenosis on thepressure field, shear rate and mean velocity in thestenosed artery, we consider the numerical results ofthe pressure field and shear rate at the peak of systolealong the LCA axis as shown in Fig.5 and results ofmean velocity on the LCA outlet surfaces as shownin Fig.8. The numerical solutions of pressure fieldand shear rate along the LCA axis and the meanvelocity of blood at outlet surfaces of the stenosedLCA are presented in Figs. 6,7 and 9. The resultsindicate that stenosis of different degrees, namely25%, 50% and 75%, has a considerable effect onthe flow field of blood. It is apparent that a higherdegree of stenosis generates a higher pressure and ahigher shear rate in the stenosed artery. The meanvelocity of blood at three outlet surfaces of the leftcoronary artery as shown in Fig. 8 is presented. Itis observed that the variation of the stenosis degree



reasonably affects the mean velocity instantaneously.The higher the stenosis degree is, the lower meanvelocity at the outlet surface of the stenosed vesselis. By increasing the stenosis degree from 25% to75%, the magnitude of the mean velocity decreasesby a remarkable amount, from 22.38 cm s−1 to19.50 cm s−1 at ΓLAD

1 , from 18.53 cm s−1 to 12.67cm s−1 at ΓLAD

s and from 25.8 cm s−1 to 20.83cm s−1 at ΓLCX , as shown in Fig.9.

IV. CONCLUDING REMARKSA three-dimensional domain of the system of

coronary arteries is constructed using MIMIC soft-ware and a mathematical model has been developedto study the blood flow in the system of coronaryarteries taking into account the pulsatile flow rateand pulse pressure on the inlet and outlet boundaries.The results obtained clearly show that a high bloodspeed and a pressure drop are found to present in theregion next to the throat of stenosis region. With anincrease of stenosis degree, the pressure increasesand blood velocity deceases. Due to a decrease inblood flow velocity, the blood flow decreases causingnot enough blood supply to the heart muscle.

Acknowledgments: The first author would like tothank the Development and Promotion of Scienceand Technology Talents Project (DPST).The lastauthor gratefully acknowledge the support of theOffice of the Higher Education Commission and theThailand Research Fund.

Fig. 2. Domain mesh of a system of coronary arteries.

Fig. 3. Velocity and pressure fields at the peak of systoleobtained from the model with 75% stenosed LAD

Fig. 4. The 75% stenosed LAD and velocity fields around thisarea



Fig. 5. The system of coronary artery with the investigatedarterial axis

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Fig. 6. Pressure profile along the investigated axis obtainedfrom the model with different degree of stenosis : (a) 25%; (b)50% and (c) 75%.



Fig. 7. Shear rate along the investigated axis obtained fromthe model with different degree of stenosis : 25% (dotted line);50% (dashed line) and 75% (solid line).

Fig. 8. The system of coronary artery with investigated outletsurfaces ΓLAD

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(a)

(b)

(c)

Fig. 9. Mean velocity obtained from the model with differentdegree of stenosis : 25% (dotted line); 50% (dashed line) and75% (solid line) along the diameter of three LCA outlet surfaces:(a) ΓLAD

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Numerical Simulation of Blood Flow Through the System of ... · Coronary Arteries with diseased Left Anterior Descending Pearanat Chuchard, Thitikom Puapansawat, Thanongchai Siriapisith,