Pulsatile flow in a coronary artery using multiphase kinetic theory

ARTICLE IN PRESS

Journal of Biomechanics 42 (2009) 743–754

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Journal of Biomechanics

0021-92

doi:10.1

� Corr

E-m

www.JBiomech.com

Pulsatile flow in a coronary artery using multiphase kinetic theory

Jing Huang a, Robert W. Lyczkowski b,�, Dimitri Gidaspow a

a Illinois Institute of Technology, Department of Chemical and Biological Engineering, Chicago, IL 60616, USAb Argonne National Laboratory, Energy Systems Division, 9700 S. Cass Avenue, Argonne, IL 60439-4815, USA

a r t i c l e i n f o

Article history:

Accepted 9 January 2009Pulsatile flow in a model of a right coronary artery (RCA) was previously modeled as a single-phase fluid

and as a two-phase fluid using experimental rheological data for blood as a function of hematocrit and

Keywords:

Computational fluid dynamics

Blood viscosity

Fahraeus–Lindqvist effect

Wall shear stress

Wall shear stress gradients

90/$ - see front matter & 2009 Elsevier Ltd. A

016/j.jbiomech.2009.01.038

esponding author. Tel.: +1630 252 5923; fax

ail address: [email protected] (R.W. Lyczko

a b s t r a c t

shear rate. Here we present a multiphase kinetic theory model which has been shown to compute

correctly the viscosity of red blood cells (RBCs) and their migration away from vessel walls: the

Fahraeus–Lindqvist effect. The computed RBC viscosity decreases with shear rate and vessel size,

consistent with measurements. The pulsatile computations were performed using a typical cardiac

waveform until a limit cycle was well established. The RBC volume fractions, shear stresses, shear stress

gradients, granular temperatures, viscosities, and phase velocities varied with time and position during

each cardiac cycle. Steady-state computations were also performed and were found to compare well

with time-averaged transient results. The wall shear stress and wall shear stress gradients (both spatial

and temporal) were found to be highest on the inside area of maximum curvature. Potential

atherosclerosis sites are identified using these computational results.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Disorders of the vascular system are the leading causes ofdeath and disability in the western world, with most heart attacksand strokes resulting from the development of arterial athero-sclerotic lesions. The distribution of atherosclerotic lesions,observed to occur typically at arterial bifurcations, bends andnear side branches in the arterial system, is heterogeneous.(Asakura and Karino, 1990; Ojha et al., 2001; Debakey et al., 1985;Thubrikar and Robicsek, 1995). Local hemodynamic factors arebelieved to play an important role in the atherosclerosis location,initiation and progression (Caro et al., 1971; Fry, 1968; Berger andJou, 2000; Nerem, 1993; Stone et al., 2003). In coronary diseases,this hypothesis should be corroborated by determination ofquantitative information on local hemodynamic parameters, suchas pressure, velocity, wall shear stress (WSS), and wall shear stressgradients (WSSG).

The existence of cell depletion and shear thinning has beendetermined experimentally for straight narrow tubes (Reinkeet al., 1987; Stadler et al., 1990; Haynes, 1960). In recent years,single-phase computational fluid dynamics (CFD) techniques,combined with noninvasive medical imaging, have been usedincreasingly by researchers seeking to better understand hemo-dynamic factors (Berthier et al., 2002; Steinman, 2002; Zeng et al.,2003; Gijsen et al., 1999; He and Ku, 1996; Hoi et al., 2004;

ll rights reserved.

: +1630 2521342.

wski).

Santamarina et al., 1998; Boutsianis et al., 2004; Politis et al., 2007,2008).

The two-phase hemodynamic model consists of a continuousplasma phase and a dispersed RBC phase which is treated as acontinuum. Various two-phase flow models were discussed at the1974 Round Table Discussion (Gidaspow, 1974). They were appliedto modeling of various phenomena, such as combustion ofgranular propellants (Markatos and Kirkcaldy, 1983; Markatos,1986) and fluidization with heat transfer (Gidaspow, 1986). This isthe model used by Jung et al. (2006a, 2006b) to simulate pulsatileflow in a three-dimensional curved tube and a model of realisticRCA, respectively, using rheological data for blood as a function ofhematocrit and shear rate fit to the modified Carreau–Yasudaviscosity model. They compared these two-phase model resultswith a single-phase model.

The kinetic theory of granular flow, invented by Savage (1983),reviewed by Gidaspow (1994), explains shear-induced diffusion.The random oscillations of particles and shear rate gives rise tothe random kinetic energy, called the granular temperature, toincrease near vessel walls. This granular temperature is thedriving force for particle migration, analogous to thermal diffusionfor gases (Hsiau and Hunt, 1996) and gives rise to theFahraeus–Lindqvist effect. Bishop et al. (2002) measured thegranular temperature as the standard deviation of velocity toexplore its effect on RBC aggregation and shear rate. Gidaspowand Huang (2009) computed the Fahraeus–Lindqvist effect andshear-induced migration of RBCs by using a multiphase kinetictheory two-phase model. The fact that they successfully predictedthe experimental decrease of apparent viscosity caused by the

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ARTICLE IN PRESS

Table 1Simulation conditions and material properties.

Plasma density, rplasma 1003 kg/m3

Plasma viscosity, mplasma 0.001 kg/m s

RBC diameter, dRBC 8mm

RBC density, rRBC 1096 kg/m3

Restitution coefficient, e 0.99999

Wall restitution coefficient, ew 0.9999

Specularity coefficient, f 0.60

RBC packing limit, es,max 0.70

Time step size 1�10�4 s

J. Huang et al. / Journal of Biomechanics 42 (2009) 743–754744

Fahraeus–Lindqvist effect for over a two order range of capillarydiameters (and therefore shear rates) also demonstrates theshear-thinning effect as shown in this analysis.

The granular temperature is computed in the multiphasekinetic theory model. The basic equation derivations are given inthe standard texts (Gidaspow, 1994; Jackson, 2000) and have beenprogrammed into commercial CFD codes, such as FLUENT ANSYS,Inc. (2006). Gidaspow et al. (2004) have shown how the kinetictheory model compares to gas–solid flow experiments.

The multiphase kinetic theory two-phase model developed byGidaspow and Huang (2009) for steady-state fully-developed flowin channels is used in this study to model hemodynamics in arealistic RCA as used by Jung et al. (2006b). The multiphase kinetictheory model described the spatial and temporal distributions ofRBCs and their depletion at vessel walls: the Fahraeus–Lindqvisteffect. Potential atherosclerotic lesion locations are identified andthe mechanism for their development is explained by theoscillating shear index (OSI), WSS and WSSG distributions.

Inlet

Plane 1

Plane 2

Plane 3

Plane 4

Plane 5

Plane 6

Plane 7Plane 8

Plane 9Plane 10

Plane 11

Plane 12

Plane 13

Plane 14

Side BSide A

Z X

Y

Side B Side A

000000000

Inle

t vel

ocity

(m

/s)

0.4496

0.4498

0.4500

0.4502

0.4504

0.4506

0.4508

0.0

Cardia Cycle Tim

RB

Cs

Vol

ume

Frac

tion

2.0 4.0 6.0 8.0

Fig. 1. (A) 3-D computational mesh for a realistic RCA model generated using Cooper m

2006). Side A: inner wall, Side B: outer wall. The inset shows the inlet velocity waveform

Point (2): peak of the systole. Point (3): end of the systole or the beginning of diastole.

cycles at a point near Side B in plane 10.

2. Methods

2.1. Multiphase granular theory hemodynamic model

The basic mass and momentum balances for the plasma and the red blood cells

are as follows (f ¼ plasma, s ¼ RBC).

Outlet

0.1.09.08.07.06.05.04.03.02.01

00 0.2 0.4 0.6 0.8

Cardiac Cycle Time (s)

(2)

(1) (3)

(4)

Pulsatile inlet Velocity

e (s)

10.0 12.0 14.0 16.0

esh generation algorithm and GAMBIT software (Berthier et al., 2002; ANSYS, Inc.,

used in the simulations. Point (1): end of the diastole or the beginning of systole.

Point (4): peak of the diastole. (B) RBC volume fraction as a function of 21 cardiac

ARTICLE IN PRESS

J. Huang et al. / Journal of Biomechanics 42 (2009) 743–754 745

Fluid phase, f, mass balance

@ðrf �f Þ

@tþr � ðrf �f~uf Þ ¼ 0 (1)

Side A

Side B

inlet

z = 0.0 mm z = 2.5 mm z = 5.0 mm

Side A

Side B

inlet

z = 0.0 mm z = 2.5 mm z = 5.0 mm

Side A

Side B

inlet

z = 0.0 mm z = 2.5 mm z = 5.0 mm

Fig. 2. Computed boundary layer development of (A) RBCs volume fraction (VF), (B) gra

at the beginning of the 21st cardiac cycle (14.7 s).

Solids phase, s, mass balance

@ðrs�sÞ

@tþr � ðrs�s~usÞ ¼ 0 (2)

where r is density, e is volume fraction, t is time, and ~u is velocity.

VF0.4550

0.4525

0.4500

0.4475

0.4450

0.4425

0.4400

0.4375

0.4350

z = 10 mm z = 15 mm

z = 10 mm z = 15 mm

z = 10 mm z = 15 mm

GT (m2/s2)0.0018

0.0016

0.0014

0.0012

0.0009

0.0007

0.0005

0.0003

0.0001

Viscosity (kg/m-s)

0.0015

0.0014

0.0013

0.0012

0.0011

0.0010

0.0009

0.0008

0.0007

nular temperature (GT), and (C) mixture viscosity in the entrance region of the RCA

ARTICLE IN PRESS


Fluid phase, f, momentum balance

@ðrf �f~uf Þ

@tþr � ðrf �f~uf~uf Þ ¼ �frf

~g � �frP þr � ¯̄tf þ bð~us �~uf Þ (3)

Solids phase, s, momentum balance

@ðrs�s~usÞ

@tþr � ðrs�s~us~usÞ ¼ �srs

~g � �srP �rPs þr � ¯̄ts þ bð~uf �~usÞ (4)

where P is pressure, ~g is gravity, ¯̄t is the stress tensor, and b is the interface

momentum exchange coefficient.

The volume fractions for each phase sum to one as

�f þ �s ¼ 1 (5)

The granular temperature equation for RBCs is expressed as

3

2

@ðrs�syÞ@t

þr � ðrs�sy~usÞ

� �¼ ð�Ps

¯̄I þ ¯̄tsÞ : r~us þr � ðksryÞ � g (6)

where y is granular temperature, ks is granular conductivity, g is collisional energy

dissipation, and ¯̄I is the unit tensor.

Several constitutive equations are required to close the set of Eqs. (1)–(6).

The stress tensor for each phase is given by a Newtonian-type approximation,

as

¯̄tf ¼ �fmf ðr~uf þr~uTf Þ �

2

3�fmfr �~uf I (7)

¯̄ts ¼ msðr~us þr~uTs Þ þ xs �

2

3ms

� �r �~us

~~I (8)

where Ps is RBC pressure, ms is shear viscosity, and xs is bulk viscosity expressed as

functions of granular temperature (Gidaspow, 1994)

Ps ¼ �srsyþ 2rsð1þ eÞ�2s g0y (9)

ms ¼4

5�2

s rsdpg0ð1þ eÞyp

� �1=2

þ10rsdp

ffiffiffiffiffiffiypp

96ð1þ eÞg01þ

4

5g0�sð1þ eÞ

� �2

(10)

xs ¼4

3�2

s rsdpg0ð1þ eÞyp

� �1=2

(11)

Time: 14.75 s

0.00

0.01

0.02

0.03

0.04

0.05

0.06

-1

Dimensionless radial position

Axi

al v

eloc

ity (

m/s

)

z = 0.0 mm z = 2.5 mmz = 5.0 mm z = 10.0 mmz = 15.0 mm

Side A Side B

Time: 15.04 s

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Axi

al v

eloc

ity (

m/s

)


-0.5 0 0.5 1

-1

Dimensionless radial positionSide A Side B

-0.5 0 0.5 1

Fig. 3. Development of RBC axial velocity profiles in the entrance region at (1): begin

(15.04 s) and (4): peak of diastole (15.075 s). Note some backflow near the wall at the

where dp is the RBC diameter, e is the restitution coefficient, and g0 is the radial

distribution function given by

g0 ¼ 1��s

�s;max

� �1=3" #�1

(12)

es,max is the solids volume fraction at maximum packing.

The granular conductivity, ks, is given by (Gidaspow, 1994)

ks ¼150rsdp

ffiffiffiffiffiffiypp

384ð1þ eÞg0

1þ6

5�sg0ð1þ eÞ

� �2

þ 2rs�2s dpð1þ eÞg0

ffiffiffiffiyp

r(13)

The energy dissipation, g, due to inelastic collision of particles, first evaluated by

Savage (1983), is given by

g ¼ 12ð1þ eÞg0

dpffiffiffiffipp rs�

2s y

3=2 (14)

The interphase momentum exchange coefficient, b, is given by (Gidaspow, 1994)

b ¼ 150�sð1� �f Þmf

�f d2p

þ 1:75rf �sj~us �~uf j

dp�f p0:8 (15)

b ¼3

4CD

�s�frf j~us �~uf j

dp��2:65

f �f 40:8 (16)

The drag coefficient, CD, is given by

CD ¼24

�f Rep½1þ 0:15ð�f RepÞ

0:687� (17)

Rep is the Reynolds number given by

Rep ¼rldpj~us �~uf j

mf

(18)

2.2. RCA geometry

Fig. 1(A) shows the three-dimensional (3-D) geometric model of a realistic

human RCA used for the computations. It is the same as used by Jung et al.

Time: 14.84 s

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Axi

al v

eloc

ity (

m/s

)


Time: 15.075 s

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Axi

al v

eloc

ity (

m/s

)


-1


-0.5 0 0.5 1

-1


-0.5 0 0.5 1

ning of systole (14.75 s), (2): peak of systole (14.84 s), (3): beginning of diastole

beginning of diastole (15.04 s).

ARTICLE IN PRESS

Time : 14.75 s Time : 14.84 s

Time : 15.04 s Time : 15.075 s

Time : 14.75 s Time : 14.84 s

Time : 15.04 s Time : 15.075 s

VF0.4550

0.4525

0.4500

0.4475

0.4450

0.4425

0.4400

0.4375

0.4350

GT (m2/s2)0.0450

0.0394

0.0338

0.0282

0.0226

0.0169

0.0113

0.0057

0.0001

Fig. 4. Instantaneous (A) RBC volume fraction (VF) and (B) RBC granular

temperature (GT) contour plots at (1): beginning of the systole (14.75 s), (2): peak

of the systole (14.84 s), (3): beginning of the diastole (15.04 s) and (4): peak of the

diastole (15.075 s). Note: the increase in granular temperature due to decreased

flow area past the area of maximum curvature and at the end.


(2006b), constructed by Berthier et al. (2002). The total length of the RCA segment

is approximately 0.13 m with no bifurcation or side branches. The inlet area is

15 mm2 and the outlet area is 2.2 mm2. Three boundary meshes of total thickness

0.182 mm were used to evaluate more accurately the WSS and spatial WSSG

(WSSGs). The total number of nodes is 40,734. The total number of mesh elements

is 36,418 with 278 cross sections of 131 faces per cross section.

This number of meshes is somewhat lower than the 57,850 used by Myers

et al. (2001) to simulate a similar RCA model which was shown to be in excellent

agreement with meshes of 167,607 and 357,643 using a well-validated in-house

Navier–Stokes solver. Similar conclusions were arrived at by Zeng et al. (2003)

using 42,517, 112,175, and 211,526 meshes. Jung et al.’s (2006a) idealized RCA

model having 46,339 meshes was shown to be essentially grid independent. Thus

we have high confidence that our results are essentially grid size independent.

2.3. Simulation conditions

The partial differential equations given in Section 2.1 were solved using the

commercial code FLUENT 6.3. (2006). The numerical solution method in FLUENT

6.3 uses a finite-volume, unstructured-mesh, staggered-grid arrangement. The

pressure-based solver was used. This solver uses a co-located scheme that stores

the pressure and velocity at the cell centers. The momentum equations are solved

using a staggered mesh, while the continuity equations are solved using a donor-

cell method. Table 1 gives the simulation conditions and the properties of plasma

and RBCs, which were assumed to be rigid spherical particles. For initial

conditions, the entire domain is filled with plasma and RBCs (having a uniform

volume fraction of 45% and a granular temperature of 0.0001 m2 s�2) having zero

velocities.

The inset in Fig. 1(A) shows the pulsatile inlet velocity waveform used for both

RBCs and plasma, having a cardiac period of 0.735 s. This waveform is consistent

with in vivo pressure and flow rate measurements in the ascending aorta (Cholley

et al., 1995; Poppas et al., 1997) and an RCA (Nichols and O’Rourke, 1998). The inlet

velocity is uniform across the inlet. Myers et al. (2001) studied the effect of the

inlet velocity waveform (flat, fully developed, and Dean type) for an RCA and found

its effect on time-averaged secondary flow and wall shear stress to be negligible

after about 0.2L or less where L is approximately 6.6 cm. Since our RCA model

geometry is about 13 cm in length, the effect of assuming a flat profile is even more

localized. A straight cylindrical entrance region having a length of 20 mm

minimizes the effect of the inlet boundary condition. The exit was set to

continuous outflow boundary conditions for both phases. The walls are assumed to

be rigid. The reference pressure was set to 111,323 Pa (75 mmHg above atmo-

spheric). At the vessel walls, no-slip velocity boundary conditions were used for

the plasma phase, and Johnson and Jackson (1987) boundary conditions were used

for the RBC phase velocity given by

us;w ¼ �6ms�s;maxffiffiffi

3p

pfrs�sg0

ffiffiffiyp

@us

@n

��w

(19)

The boundary condition for the granular temperature at the vessel walls is given

by (Sinclair and Jackson, 1989)

yw ¼ �kygw

@yw

@n

��w

þ

ffiffiffi3p

pfrs�su2s;slipg0y

3=2

6�s;maxgw

(20)

where us,slip is the slip velocity parallel to the wall, and

gw ¼

ffiffiffi3p

pð1� e2wÞ�srsg0y

3=2

4�s;max(21)

n is the normal component to the wall, us,w is the RBC velocity in the direction

parallel to the wall, gw is the energy dissipation due to inelastic collisions between

particles and the wall, ew is the wall restitution coefficient, yw is the granular

temperature at the wall, and f is the specularity coefficient.

The simulation was run for 15.435 s (21 cardiac cycles) with the constant time

step of 10�4 s. This was sufficient time to establish a limit cycle behavior as shown

in Fig. 1(B). The Boutsianis et al. (2004) model of a realistic LCA and the

Santamarina et al. (1998) idealized curved tube model of a coronary artery used

100 and 1000 time steps per cardiac cycle, respectively, both of which they

considered to be numerically sufficient. We used approximately 7000 time steps

per cardiac cycle. Thus we have high confidence that our results are time-step

independent. The residuals of mass, momentum, and granular temperature for

both phases were checked for convergence with the criteria of 10�3 at each time

step. The average running time per cardiac cycle for the transient case was 19 h on

a personal computer having a 2.8 GHz processor and 2–3/4 h for the steady-state

case on a laptop computer having a 1.8 GHz processor.

Computations performed with the single-phase model used the modified

Carreau–Yasuda viscosity correlation developed by Jung et al. (2006a, b) using

blood rheological data of Brooks et al. (1970) and Wojnarowski (2001) given by

mblood ¼ m0 ½1þ ð0:11_gÞ2�ðn�1Þ=2 (22)

where _g is shear rate (1/s), m0 ¼ 9.109�10�3 kg/m s, and n ¼ 0.749 at an RBC

volume fraction of 0.45 (hematocrit ¼ 45%).

The following were used to determine potential atherosclerotic sites. The

oscillating shear index (OSI) accounting for the departure of instantaneous WSS

from its time-averaged value is defined as (Buchanan and Kleinstreuer, 1998)

OSI ¼1

21�jR T

0¯̄tjdtR T

0 j ¯̄tjdt

!(23)

where T is the cardiac period and ¯̄t is the instantaneous WSS.

The temporal and spatial WSS gradients, WSSGt and WSSGs, are defined as

WSSGt ¼@ ¯̄t@t

�� (24)

WSSGs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@tm

@m

� �2

þ@tn

@n

� �2s

(25)

where tm and tn are the normal and tangential components of the time-averaged

WSS in the local m-n-l coordinate system.

ARTICLE IN PRESS


3. Results

All results are presented herein are for the 21st cardiac cycle.Fig. 2 clearly shows the computed boundary layer development inthe entrance region for RBC volume fraction, granular tempera-ture, and viscosity. There is a little more depletion at the top (sideA) than at the bottom (side B) due to gravitational effects. Fig. 3shows that flattened RBC velocity profiles developed toward theend of the entrance region during the cardiac cycle similar to the

0.444

0.445

0.446

0.447

0.448

0.449

0.450

0.451

0.452

-1

Dimensionless Radial Position

RB

Cs

Vol

ume

Frac

tion

Plane 1 Plane 2Plane 3 Plane 4Plane 5

Side A Side B

0.441

0.443

0.445

0.447

0.449

0.452

0.454

RB

C V

olum

e Fr

actio

n

Plane Plane

Plane 1 Plane 2 Plane 3

Plane 8Plane 7Plane 6

Plane 11 Plane 12 Plan

-0.5 0 0.5 1

-1

Dimensionle Side A

-0.5

Fig. 5. (A) Time-averaged RBC volume fraction (VF) contour plots showing the Fahraeus

Time-averaged RBC volume fraction plots from Side A to Side B shown in Fig. 1 for 14

results of Jung et al. (2006b). The maximum values of velocitiesoccur at the peak of the systole (14.84 s). Note a slight backflownear the walls at the beginning of the diastole (15.04 s).

After the entrance region, computed local hemodynamicfactors vary considerably both spatially and temporally alongthe RCA proper, which has non-uniform cross-sectional area. Fig. 4shows contour plots of instantaneous RBC volume fraction andgranular temperature. Corresponding mixture viscosity plots areshown in Fig. S1. Note the decrease in RBC volume fraction in the

0.439

0.441

0.443

0.445

0.447

0.449

0.452

0.454

RB

C V

olum

e Fr

actio

n


11 Plane 1213 Plane 14

Plane 4 Plane 5

Plane 10Plane 9

e 13 Plane 14

VF

0.4550

0.4525

0.4500

0.4475

0.4450

0.4425

0.4400

0.4375

0.4350

ss Radial PositionSide B

0 0.5 1

-1

Dimensionless Radial Position Side A Side B

-0.5 0 0.5 1

–Lindqvist effect. Velocity magnitude vector plots are shown for planes 6 to 10. (B)

planes.

ARTICLE IN PRESS

Time : 14.75 s

Time : 14.84 s

Time : 15.075 s

Time : 15.04 s

Plane 1 Plane 2 Plane 3 Plane 4 Plane 5 Plane 6 Plane 7 Plane 8 Plane 9 Plane 10 Plane 11 Plane 12 Plane 13 Plane 14GT (m2/s2)

0.0400

0.0350

0.0300

0.0250

0.0201

0.0151

0.0101

0.0051

0.0001

Fig. 6. Instantaneous RBC granular temperature (GT) contour plots in planes 1–14 at (1): beginning of the systole (14.75 s), (2): peak of the systole (14.84 s), (3): beginning

of the diastole (15.04 s) and (4): peak of the diastole (15.075 s).

0.0015

0.0017

0.0019

0.0021

0.0023

0.0025

-1


Mix

ture

Vis

cosi

ty (

kg/m

-s)

Plane 1 Plane 2 Plane 3Plane 4 Plane 5

0.0015

0.0018

0.0021

0.0024

0.0027

0.0030

0.0033

0.0036

Mix

ture

Vis

cosi

ty (

kg/m

-s) Plane 6 Plane 7

Plane 8 Plane 9Plane 10

0.0024

0.0027

0.0030

0.0033

0.0036

0.0039

Mix

ture

Vis

cosi

ty (

kg/m

-s)

Plane 11 Plane 12Plane 13 Plane 14

0.00240

0.00245

0.00250

0.00255

0.00260

21.3

Shear Rate (1/s)

Mix

ture

Vis

cosi

ty (

kg/m

-s) H = 0.449843

H = 0.449796

-0.5 0 0.5 1Side BSide A

-1



-1



21.4 21.5 21.6 21.7 21.8 21.9

Fig. 7. (A) Time-averaged mixture viscosity distributions from Side A to Side B shown in Fig. 1 for 14 planes. (B) Computed shear thinning effect plotted for two points near

the center of plane 10 between 14.7 and 14.75 s when the RBC hematocrits (H) are almost equal.


ARTICLE IN PRESS


area of maximum curvature from the beginning of the systole(14.75 s) to the peak of the systole (14.84 s) caused by increasedmigration due to increased granular temperature, Fig. 4(B). TheRBC volume fraction and granular temperature then increase tothe end of the RCA because of decreased flow area.

Fig. 1 shows the 14 planes selected to analyze the computa-tional results in detail. The RCA is characterized by threesegments, (1) before maximum curvature (planes 1–5), (2) inthe vicinity of maximum curvature (planes 6–10), and (3) aftermaximum curvature (planes 11–14). The cross-sectional area isincreasing around planes 4 and 14 and decreasing around planes 6through 13.

0.0

0.1

0.2

0.3

0.4

0.5

-1


RB

Cs

shea

r st

ress

(Pa

)


Side A Side B

0.0

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0.6

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RB

Cs

Shea

r St

ress

(Pa

)

Plane 11

Plane 13

Side A (Inner wall)

0.00

1.00

2.00

3.00

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5.00

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Time (s)

Wal

l She

ar S

tres

s (P

a)

Multiphase kinetic theory model

Single phase shear thinning model

Multiphase shear thinning model

-0.5 0 0.5 1

-1

Dimensionless RSide A

-0.5 0

14.8 15 15.2 15.4 15.6

Fig. 8. (A) Time-averaged RBC shear stress distributions from Side A to Side B shown in

kinetic theory model and the multiphase shear thinning model (recomputed using the m

the area of maximum curvature.

Fig. 5 shows time-averaged contour plots and distributions ofRBC volume fractions for planes 1–14 and velocity magnitudeplots for planes 6–10. Fig. 6 shows the RBC granular temperature.Fig. S2 shows time-averaged RBC granular temperature distribu-tions, Fig. 7(A) shows time-averaged mixture viscosity distribu-tions, and Fig. S3 shows instantaneous mixture viscosity contourplots. Fig. 7(B) demonstrates the shear-thinning effect.

The time-averaged granular pressure distributions shown inFig. S4 are similar to the time-averaged granular temperaturedistributions shown in Fig. S2. In addition to the shear stressesshown in Fig. 8(A), it is the granular pressure in the radialcomponent of the RBC momentum balance, which is the driving

0.0

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RB

Cs

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r St

ress

(Pa

)


Plane 12

Plane 14

Side B (Outer wall)

0.00

0.40

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1.60

2.00

Wal

l She

ar S

tres

s (P

a)

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Single phase shear thinning model

Multiphase shear thinning model

adial Position Side B

0.5 1

-1

Dimensionless Radial PositionSide A Side B

-0.5 0 0.5 1

14.6

Time (s)

14.8 15 15.2 15.4 15.6

Fig. 1 for 14 planes. (B) Comparison of mixture WSS calculated from the multiphase

odel of Jung et al., 2006b), and the single-phase shear-thinning model for plane 7 in

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force for RBC migration giving rise to the Fahraeus–Lindqvisteffect. Both of them are functions of RBC granular temperatureand volume fraction. The time-averaged shear stresses are a directfunction of the shear rate shown in Fig. S5(A). The patterns arevery similar since the radial variation of RBC viscosity is relativelyflat. The steady-state shear rates are shown in Fig. S5(B). The time-averaged RBC shear rates are consistent with the shapes of thetime-averaged RBC velocity plots shown in Fig. S6.

Three parameters were employed to determine potentialatherosclerotic lesion sites: OSI, WSSGt, and WSSGs. The resultsare given in Figs. 8–10 and discussed below.

4. Discussion

Fig. 2(A) shows that RBCs migrate away from the wall inthe entrance region resulting in their depletion, the Fahraeus–

0.00

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1Plan

OSI

plane 4

0

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40

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Time (s)

WSS

Gt (

Pa/s

)

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Side B (outer wall)

plane 10

0

20

40

60

80

100

120

WSS

Gt (

Pa/s

)

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Side B (outer wall)

2 3 4 5 6 7

14.8 15 15.2 15.4 15.6

14.6

Time (s)

14.8 15 15.2 15.4 15.6

Fig. 9. (A) Oscillating shear index (OSI) for planes 1–14. (B) Instantaneous t

Lindqvist effect, as demonstrated experimentally by Taylor (1955)and Bugliarello and Hayden (1962). The source of the migration isshear in the multiphase granular theory model as explained byGidaspow and Huang (2009). The amount of depletion is muchless than that in small, say 0.1 mm diameter capillaries, because oflow shear in the 4.37 mm diameter inlet region. The RBC viscosityin Fig. 2(C) is seen to depend more on the granular temperature,Fig. 2(B).

Fig. 5 shows that the time-averaged RBCs are quite non-uniformly distributed in all 14 RCA planes. The migration of RBCsaway from the vessel walls, the Fahraeus–Lindqvist effect, isstronger in the RCA than in the entrance region because ofhigher granular temperature and varies considerably from planeto plane and is asymmetric because of variations in cross-sectional areas and curvatures. The existence of secondary,swirling flows is indicated by the velocity vector plots in planes6 through 10 in Fig. 5(A). Such swirling flows have been computed

e

plane 7

0

20

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WSS

Gt (

Pa/s

)

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Side B (outer wall)

plane 13

0102030405060708090

100

WSS

Gt (

Pa/s

)

Side A (Inner wall)

Side B (outer wall)

8 9 10 11 12 13 14

14.6

Time (s)

14.8 15 15.2 15.4 15.6

14.6Time (s)

14.8 15 15.2 15.4 15.6

Side ASide B

emporal wall shear stress gradients (WSSGt) for planes 4, 7, 10, and 13.

ARTICLE IN PRESS

Plane 4

0.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

14.7Time (s)

WSS

Gs

(Pa/

m)

Side A (Inner wall)

Side B (Outer wall)

Plane 7

0.0E+001.0E+042.0E+043.0E+044.0E+045.0E+046.0E+047.0E+048.0E+04

WSS

Gs

(Pa/

m) Side A (Inner wall)

Side B (Outer wall)

Plane 10

0.0E+00

5.0E+04

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1.5E+05

2.0E+05

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3.0E+05

WSS

Gs

(Pa/

m)

Side A (Inner wall)

Side B (Outer wall)

Plane 13

0.0E+00

1.0E+04

2.0E+04

3.0E+04

4.0E+04

5.0E+04

6.0E+04

7.0E+04

WSS

Gs

(Pa/

m)

Side A (Inner wall)

Side B (Outer wall)

14.8 14.9 15.0 15.1 15.2 15.3 15.4

14.7Time (s)

14.8 14.9 15.0 15.1 15.2 15.3 15.4 14.7Time (s)

14.8 14.9 15.0 15.1 15.2 15.3 15.4

14.7Time (s)

14.8 14.9 15.0 15.1 15.2 15.3 15.4

Fig. 10. Instantaneous spatial wall shear stress gradients (WSSGs) for planes 4, 7, 10, and 13.


in a cast of a similar diseased RCA (Langenhove et al., 2000). Thedepletion of RBCs at both the inside and outside vessel walls, SidesA and B, due to the Fahraeus–Lindqvist effect is clearly seen for all14 planes. It is strongest on the inside curvature, Side A, in thearea of maximum curvature for planes 6 through 10 as shown inFig. 5(B). The smaller Fahraeus–Lindqvist effect in planes 4, 5 and14 shown in Fig. 5(A), is due to the larger cross-sectional area.

Figs. 6 and S2 show that the instantaneous and time-averagedRBC granular temperatures, respectively, in planes 1 through 5 arelow with slight increases at the RCA vessel walls since this is anarea of relatively low curvature. Slight variations exist from planeto plane because of changes in cross-sectional area increasing inplanes 4 through 6. The granular temperatures begins to increasein planes 6 through 10, the area of maximum curvature, and havehigher values on the inside curvature, side A shown in Fig. S2.However, granular temperatures for planes 11 through 14 havehigher values on the outside curvature, side B, due in part by thenarrowing cross-sectional area, indicating a redistribution of theshear caused by swirling flows. The instantaneous granulartemperatures are highest during the peak of systole (14.84 s)and show that greatest spatial variation.

The time-averaged mixture viscosity distributions shown inFig. 7(A) and their instantaneous contour plots shown in Fig. S3bear a strong resemblance to the granular temperatures shown incorresponding Figs. S2 and 6. This is because the RBC viscosity,given by Eq. (10), depends much more strongly upon the granulartemperature than the RBC volume fraction. The temporal andspatial trends for mixture viscosity closely follow those for thegranular temperature.

The multiphase kinetic theory model for RBC viscosity does notexplicitly depend upon shear rate. Fig. 7(B) plots the instanta-neous mixture viscosity dependency upon shear rate for twopoints near the center of plane 10 where the RBC volume fractionsdiffer by less than 0.0002 percent showing that the multiphasekinetic theory model does display the shear-thinning effect. TheRBC viscosity depends explicitly upon RBC volume fraction andtherefore the mixture viscosity increases with hematocrit asobserved experimentally as shown by Gidaspow and Huang(2009).

Many researchers have attempted to develop consistentcorrelations for atherosclerotic lesion locations to WSS (low orhigh), WSSGt, and WSSGs with little success (Caro et al., 1971; Fry,1968; Zeng et al., 2003; Friedman and Ehrlich, 1975; Qiu andTarbell, 2000; Ku et al., 1985; DePaola, et al., 1992). Resultscomputed from the multiphase kinetic theory model are used toidentify such potential locations.

Fig. 8(B) compares the WSS predicted from the single-phasemodel, the shear-thinning multiphase model of Jung et al.(2006b),and the multiphase kinetic theory model. The single-phase modelused Eq. (22) for blood viscosity. All three models predictedvery similar instantaneous WSS waveforms during the cardiaccycle in the area of maximum curvature and closely resemble theinput velocity waveform shown in the inset of Fig. 1(A). Slightdifferences result because of spatial and temporal RBC volumefraction variations. The WSS is consistently higher on the insidecurvature. This is opposite to the idealized RCA WSS computed byJung et al. (2006a). However, before and after the area ofmaximum curvature, lower WSS occurs on the inside curvatures.It is for reasons like this that researchers have found inconsistentcorrelations of atherosclerotic sites with WSS, Wentzel et al.(2003) for example. The multiphase kinetic theory modelpredicts somewhat lower WSS due to the presence of theFahraeus–Lindqvist effect, absent in the other two models.

High temporal and spatial wall shear gradients have beenlinked to intimal thickening by enhancing the endothelialpermeability by macromolecules, such as low-density lipoprotein(LDL) (Lei et al., 1995). The temporal variation of WSS can becharacterized by the OSI given by Eq. (23). The theoreticalmaximum value of OSI is 0.5 (Glor et al., 2005). Fig. 9(A) showsthat the OSI value increases dramatically in plane 4 where the RCAcross-sectional area abruptly increases and less so in plane 5resulting from an oscillatory recirculation. The OSI is significantlylower in the others 12 planes and is generally higher on the insidecurvature. Ku et al. (1985) correlated intimal thickening with highOSI and Buchanan et al. (1999) observed it in regions where theendothelial cells do not align with the flow direction.

Figs. 9(B) and 10 present plots of instantaneous WSSGt andWSSGs, respectively, computed from Eqs. (24) and (25). The

ARTICLE IN PRESS


WSSGt are seen to spike in Fig. 9(B) around the beginning of thesystole (14.75 s) and diastole (15.04 s) where there is a suddenchange in slope of the cardiac waveform for inflow velocityshown in Fig. 1(A). WSSGt are highest in planes 7 and 10 andWSSGs shown in Fig. 10 are highest in plane 10, on the insidecurvature, in the area of maximum curvature. The waveforms forWSSGs differ significantly from those of the WSSGt with multiplepeaks in plane 10. The waveforms in plane 13 resemble thewaveform for the prescribed inflow velocity except for somespikes on Side B. It is proposed that the regions of highest andmost rapid increase in WSSGt and WSSGs are candidates forpossible atherosclerotic lesion formation.

The time-averaged mixture WSS, OSI, WSSGt and WSSGs aretabulated in Table S1 for planes 1 through 14 at both the insideand outside of the RCA.

5. Conclusions

1.
The multiphase kinetic theory model predicts the depletion ofRBCs in the vicinity of the RCA vessel walls: the Fahraeus–Lindqvist effect.
2.
The computed RBC granular temperature, granular pressure,viscosity, volume fraction, and WSS varied temporally andspatially during the cardiac cycle.
3.
Computed mixture viscosities are in reasonable agreementwith experimental data, increase with hematocrit, and displaythe shear-thinning effect.
4.
The computed mixture WSS using the multiphase kinetictheory model is in close agreement with the single-phaseshear-thinning model and multiphase shear-thinning model ofJung et al. (2006b).
5.
Potential locations of atherosclerotic lesion formation havebeen identified using computed OSI. WSS, WSSGs, and WSSGt.
Conflict of interest statement

There is no conflict of interest.

Acknowledgment

The award of an Illinois Institute of Technology FieldhouseFellowship to the first author was instrumental in enabling theresearch presented herein.

Appendix A. Supporting Information

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.jbiomech.2009.01.038.

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Pulsatile flow in a coronary artery using multiphase kinetic theory