Relative Residence Time and Oscillatory Shear Index of Non

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Abstract Four molecular Non-Newtonian viscosity modelsplus the Newtonian one were analyzed for the normal humanaorta under oscillating flow via: molecular viscosity, timeAverage Wall Shear Stress (AWSS), Oscillatory Shear Index(OSI) and Relative Residence Time (RRT). The capabilities ofthe applied non-Newtonian law models appear at low strainrates. The Newtonian blood flow treatment is considered to be agood approximation at mid-and high-strain rates. All bloodflow models yield a consistent aorta pattern. High RRT valuesdevelop in the concave part of the aortic arch downstream toleft subclavian artery. In this region the molecular viscosity iselevated, the WSS is low and the OSI is high. Concave aortaparts are prone to exhibit elevated RRT. The non-NewtonianPower Law blood flow model approximates the molecularviscosity, WSS, OSI and particularly the RRT in a moresatisfactory way. High RRT distribution is emerging as anappropriate tool for identifying the possible regions ofatheromatic concentrations.

I. I NTRODUCTION hypothesis, that an increased plasma viscosity may be

a link between cardiovascular risk factors and coronaryheart disease, is supported [1]. The role of viscosity in

the development and progression of coronary/arterial heartdisease is important [2] and [3]. Atherosclerosis shows

preference localization at sites where flow is either slow ordisturbed and where Wall Shear Stress (WSS) as well astheir gradients are low [4]. According to non-Newtonian

behavior, viscosity is velocity gradient-dependent andsubsequently varies along the course of the vessel.Henceforth, the variation depends on the applied flowconditions, vascular geometry, flow particularities and localflow composition. Capable blood viscosity models areneeded to capture the main flow characteristics at low strainrates [5].

Besides molecular viscosity distribution, the temporalvariation during the cardiac cycle of the hemodynamicmechanical parameters has been proposed as an atherogenicfactor [6]. Complex configuration of the vessel promotesflow disturbances with low time-averaged WSS and highshear stress temporal oscillations during the cardiac cycle

Manuscript submitted on the 10 th of June 2011.J. V. Soulis is, is with Fluid Mechanics Division, Faculty of Engineering,

Democrition University of Thrace, 67100 Xanthi, GREECE. (e-mail:[email protected] ).

O. P. Lampri is with Fluid Mechanics Division, Faculty of Engineering,Democrition University of Thrace, 67100 Xanthi, GREECE. (e-mail:[email protected] ).

D. K. Fytanidis, is with Fluid Mechanics Division, Faculty ofEngineering, Democrition University of Thrace, 67100 Xanthi, GREECE.(Phone: ++302310260011, Fax: ++302310994837, e-mail:[email protected] ).

G. D. Giannoglou 1st Cardiology Department, CardiovascularEngineering and Atherosclerosis Laboratory, AHEPA General Hospital,Aristotelion University of Thessaloniki, GREECE. ( [email protected] ).

[6]. The application of unsteady flow constitutes achallenging issue, since it demands computational powerand analysis time. The Oscillatory Shear Index (OSI) and theRelative Residence Time (RRT), which indicates the time ofresidence the molecules spent at endothelium, are emergingas appropriate tools for atherosclerosis localization, [7].However, little research has focused on the exact role oflocal viscosity differentiation within the aortic arch and itsimplications to atherogenesis under oscillating flows.

The choice and application of an appropriate non- Newtonian model for the cardiovascular flow analysis,which would put emphasis in low strain rates, is crucial inachieving acceptable results. In the current research work,four non-Newtonian models plus the Newtonian one arecompared in a normal human aortic arch under oscillating

flow conditions. These models are: Power Law [8], Carreau[8], Casson [9], Non-Newtonian Power Law [8] and the

Newtonian one. The time-Average Wall Shear Stress(AWSS), Molecular Viscosity, Oscillatory Shear Index(OSI) and Relative Residence Time (RRT) are analyzed.

II. MATERIALS AND METHODS

A. Geometry ReconstructionThe first step is to generate the aortic arch centerline.

Thereafter, the daughter vessels centerlines are generatedand placed at the appropriate aortic arch centerline locations.Finally, various vessel cross-sections, taken every 1.0 mm,are set perpendicular to ascending-descending aortic archand daughter vessel centerlines at the appropriate locations.These data are processed and resulted into a 3D geometrymodel of the aortic arch, Fig.1. The computational modelincludes the ascending aorta, brachiocephalic artery, leftcommon carotid artery, left subclavian artery anddescending aorta. The lumen diameter of the ascending aortaat entrance measures 3.477 cm. The descending aorta outletdiameter is 3.076 cm, while the corresponding outletdiameters of the brachiocephalic artery, left common carotidartery and left subclavian artery segments are 1.435 cm,1.273 cm and 1.433 cm, respectively. The computationalgrid consists of 797000 cells and 318000 grid nodes.

B. Non-Newtonian modelsIn total, four different non-Newtonian computational

analyses are performed using the underneath blood viscositymodels (plus the Newtonian model),

Newtonian model

sm / kg 00345.0 = (1)

Relative Residence Time and Oscillatory Shear Index of Non-Newtonian Flow Models in Aorta

Johannes V. Soulis, Olga P. Lampri, Dimitrios K. Fytanidis, and George D. Giannoglou

A

978-1-4577-0554-0/11/$26.00 2011 IEEE


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Fig. 1 Aortic arch geometry

Power Law model

( ) 1n0= , sm / kg 035.00 = , 6 .0n = (2)

Carreau model

( ) ( )[ ]( ) 2 / 1n20 1

++= (3)

s313.3= , 3568.0n = , sm / kg 056 .00 = (4)

sm / kg 00345.0 = and 1/s is the strain rateCasson model

( ) 2 / 1222 / 1 y2 / 14 / 1

22 J 2 J

+= (5)

2 J 2= , ( )3 y H 625.01.0= (6)

( ) 5.20 1 = H , sm / kg 0012.00 = and 37 .0 H = (7)

Non-Newtonian Power Law model

According to this law the fluid shear stress, denoted by (N/m 2), is calculated as,

(8)

( m / skg 2n ), n=power-k =consistency index

law index, 0T = reference temperature (k),

7.0=n , k 310T 0 = , m / skg .k n 2 _ 006220= (9)

C. Flow EquationsThe assumptions made about the nature of the flow are that

it is: 3D, steady, laminar, isothermal, with no external forcesapplied on it. The arterial wall is comprised from non-elasticand permeable material. In their generality the flowgoverning equations are,

( ) 0 =+ ut (10)

(kg/m 3) is the density, t (sec) is the time, u (m/s) is thevelocity vector. The conservation of momentum (in thegeneral form) is,

( ) ( ) g ) ( puu u t

+=++

(11)

= (12)Here, p (N/m 2) is the static pressure; (N/m 2) is the shear

stress tensor and g (N/m 3) the gravitational body force. Theshear stress tensor is,

( )T u u = +

(13)

is the molecular viscosity.

D. Boundary ConditionsThe applied velocity inlet pulse wave is shown in Fig. 2.

The pulse period of this waveform is 800.0 msec. Theoutflow discharges were calculated using a slightly modifiedversion of the Murrays Law (the power index was set equalto 2.4), [10]. The applied flow discharges are as follows:ascending aorta (100.0 %), brachiocephalic artery (3.8 %),left common carotid artery (6.8 %) and left subclavian artery(8.8 %) and descending aorta (80.6 %).

Fig. 2 The applied blood waveform at the aortic arch inlet

E. Wall Shear Stress and Oscillating Shear IndexThe components of the WSS possibly have different effects

upon endothelial cells. The actual shear stress, given byEq.12, is applied for the wall (Wall Shear Stress, WSS). Thetime-Averaged WSS magnitude (AWSS) is defined as,

=

0

1 AWSS WSS dt T

(14)

S S W is the instantaneous WSS (N/m 2) magnitude and T

(sec) is the pulse period.Another transient flow property, related with the wall, is

the magnitude of time-Averaged WSS vector (AWSSV)defined as,

=

0

1 AWSSV WSSdt

T

(15)

Another mechanical factor related to flow oscillation is theOSI, which monitors the differences between AWSS andAWSSV values. Using these values, the OSI clarifies theWSS vector deflection from blood flow predominantdirection, during cardiac cycle. Thus, OSI is calculated,

=

0

0

WSSdt OSI 0.5 1.0

W SS dt

(16)

The OSI value can vary from 0.0, for no-cyclic variation ofWSS vector, to 0.5, for 180.0 o deflection of WSS direction.

F. Relative Residence Time (RRT)The Relative Residence Time (RRT) is introduced [10] as,

ji

j i

uu x x

= +

T on 1

T ke

= ( ) =


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[ ] 1( 1 2.0 OSI ) AWSS RRT ~ (17) The proportionality of the above equation is of relativeimportance since the results are presented as normalizedones. The OSI modifies the AWSS effects on the RRT at agiven region of the endothelium. Henceforth, the RRT

parameter includes the effects of both OSI and AWSS.Convergence is achieved when all mass, velocity

component and energy changes, from iteration to iteration,achieved less than 10 -6 accuracy. The ANSYS commercialcode is utilized, [11]

III. RESULTS AND DISCUSSIONThe detailed results of only two blood flow models,

namely the Carreau and the Non-Newtonian Power Law are presented here.

A. Time Average Wall Shear StressThe AWSS (N/m 2) contours over the entire aortic

surfaces for Carreau flow model and Non-Newtonian PowerLaw flow model are shown in Figs. 3a,b, respectively. High

AWSS is encountered at the convex parts of the curved flowregions. Low AWSS develops at the concave parts of thecurved flow regions. Furthermore, the flow within the aortic

branches is strongly dependent upon the geometry of the branch and its particular placement within the aorta.

a) b)

Fig. 3 Time-average Wall Shear Stress (N/m 2) magnitude distributions fora) Carreau flow model, b) Non-Newtonian Power Law flow model

B. Molecular ViscosityFor the Carreau flow model, the molecular viscosity

values for the entire aorta, typically at the very end of thecardiac cycle, range from 0.0083 kg/m-s to 0.052 kg/m-s,Fig. 4a. The range between maximum and minimummolecular viscosity values using non-Newtonian Power Lawis 0.005-0.05 kg/m-s, the range for Power Law is 0.0005-0.0048 kg/m-s and the range for the Casson Law is 0.0045-

0.0085, for the same time instant. Figure 4b shows themolecular viscosity for the non-Newtonian Power Lawmodel. High molecular viscosity values develop in theconcave part of the aortic arch downstream to left subclavianartery.

C. Oscillatory Shear IndexThe OSI reveals the overall WSS vector oscillation during

the cardiac pulse wave. The OSI contours of the aortic archare shown in Figs. 5a,b, respectively. All models predictsimilar behavior. High OSI values, reaching the 0.44, are

encountered at brachiocephalic artery, left common carotidartery and left subclavian artery. Furthermore, high OSIvalues develop at the convex part of the ascending aorta aswell as at the concave aortic part downstream to leftsubclavian artery.a) b)

Fig. 4 Molecular Viscosity (kg/m-s) distribution at the very end of thecardiac cycle for a) Carreau flow model and b) Non-Newtonian Power Law

flow model

a)

b)

Fig. 5 Oscillatory Shear Index distribution fora) Carreau flow model and b) Non-Newtonian Power Law flow model

D. Residence TimeContours of the RRT are shown in Figs. 6a,b for Carreauflow model and Non-Newtonian Power Law flow model,respectively. High RRT values, reaching more than 15 for


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Carreau Law and more than 18 for Non-Newtonian PowerLaw, develop in the concave part of the aortic archdownstream to left subclavian artery. All models predictsimilar RRT distribution. However, the Power Law model

predicts high RRT value reaching 55.

E. Average Wall Shear Stress versus Oscillatory Shear Index

Typical relationship between AWSS and OSI over theentire aortic surfaces for Carreau flow model and Non-

Newtonian Power Law flow model are shown in Figs. 7a,b,respectively. It is evident that increasing OSI values co-existwith decreasing WSS. Similar relationships hold for the othernon-Newtonian blood flow models, not shown.

a)

b)

Fig. 6 Relative Residence Time distribution fora) Carreau flow model and b) Non-Newtonian Power Law flow model

IV. CONCLUSIONFour molecular viscosity models plus the Newtonian one

were analyzed for the normal human aorta under oscillatingflow. All blood flow models yield a consistent aorta patternfor all tested biomechanical parameters. High RRT valuesdevelop in the concave part of the aortic arch downstream toleft subclavian artery. In this region the molecular viscosityis elevated, the WSS is low and the OSI is high. Concaveaorta parts are prone to exhibit elevated RRT. The non-

Newtonian Power Law blood flow model approximates themolecular viscosity, WSS, OSI and particularly the RRT in amore satisfactory way, particularly at low strain rates. RRTdistribution (high values) emerges as an appropriate tool for

identifying the possible regions of atheromatic plaquelocalization.

a)

b)

Fig. 7 Time-average Wall Shear Stress (N/m 2) versus Oscillatory ShearIndex for a) Carreau flow model and b) Non-Newtonian Power Law flowmodel

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Relative Residence Time and Oscillatory Shear Index of Non

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