Numerical Simulation of Dispersed Particle-Blood Flow in...

Research ArticleNumerical Simulation of Dispersed Particle-Blood Flow inthe Stenosed Coronary Arteries

Mongkol Kaewbumrung 1 Somsak Orankitjaroen 12 Pichit Boonkrong3

Buraskorn Nuntadilok 4 and BenchawanWiwatanapataphee 5

1Department of Mathematics Faculty of Science Mahidol University Bangkok 10400 Thailand2Centre of Excellence in Mathematics Commission of Higher Education (CHE) Bangkok 10400 Thailand3Department of Mathematics College of Information and Communication Technology Rangsit UniversityPathumThani 12000 Thailand4Department of Mathematics Faculty of Science Maejo University Chiang Mai 50290 Thailand5School of Electrical Engineering Computing and Mathematical Sciences Curtin University Perth WA 6845 Australia

Correspondence should be addressed to Somsak Orankitjaroen somsakoramahidolacth

Received 14 March 2018 Revised 22 May 2018 Accepted 4 June 2018 Published 1 August 2018

Academic Editor Peiguang Wang

Copyright copy 2018 Mongkol Kaewbumrung et alThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

A mathematical model of dispersed bioparticle-blood flow through the stenosed coronary artery under the pulsatile boundaryconditions is proposed Blood is assumed to be an incompressible non-Newtonian fluid and its flow is considered as turbulencedescribed by the Reynolds-averaged Navier-Stokes equations Bioparticles are assumed to be spherical shape with the same densityas blood and their translation and rotational motions are governed by Newtonian equations Impact of particle movement onthe blood velocity the pressure distribution and the wall shear stress distribution in three different severity degrees of stenosisincluding 25 50 and 75 are investigated through the numerical simulation using ANSYS 182 Increasing degree of stenosisseverity results in higher values of the pressure drop and wall shear stresses The higher level of bioparticle motion directly varieswith the pressure drop and wall shear stressThe area of coronary artery with higher density of bioparticles also presents the higherwall shear stress

1 Introduction

Atherosclerosis is a disease narrowing a coronary artery dueto plaque buildup Generally there is no symptom until itseverely narrows the artery causing serious problems includ-ing heart attack stroke or even death Critical informationof blood flow in the stenotic coronary arteries is a principlefactor of the development and progression of atherosclerosisFigure 1 presents an angiogram of a critical proximal leftanterior descending artery (LAD) in a patient with Wellensrsquosyndrome Atherosclerosis is often associated with someforms of abnormal blood flow in the blocked coronaryarteriesDealingwith the pathogenesis of coronary artery dis-eases (CAD) various practical treatment of CAD includingdrug delivery stent replacement and coronary artery bypassgrafting (CABG) have been developed through a number ofin vivo and in vitro experiments Due to the high rate of stent

and graft failures development of vascular drug delivery oneof the key rubrics of targeted therapeutics and nanodevicesbecomes more and more important [1] Recently severaldrug delivery approaches are undergoing clinical testing andmedical industry development

Over decades many researchers have carried out experi-mental models and computational simulations to explore theflow phenomena in the stenotic arteries in order to optimizemedical methods of treatment Due to the difficulty andlimitation in determining the critical flow conditions forboth in vivo and in vitro experiments the exact mechanismsinvolving these treatments are not well understood Thusmathematical modelling and numerical simulation are cho-sen to be a better alternative to analyze the problem Complexphenomena of blood flow in arteries subject to variousphysiological conditions has been extensively analyzed usingvarious mathematical models [2ndash12] The flow phenomena

HindawiInternational Journal of Differential EquationsVolume 2018 Article ID 2593425 16 pageshttpsdoiorg10115520182593425

2 International Journal of Differential Equations

Figure 1 Tight severe stenosis (95) of the proximal LAD in apatient with Wellensrsquo warning

includes asymmetric flow unsteady laminar-to-turbulentflow [13ndash16] The governing equations include the Navier-Stoke equations and the continuity equations subjected to theselected inlet velocity no-slip condition at the arterywall andstress-free condition at the outflow surfaceTheunsteady flowalso is characterized by a high pressure drop and high wallshear stress (WSS) in the stenotic artery [1 17ndash19]

For the requirement of simulation the constant densityof blood is assumed to be equal to 1055 kg mminus3 The vesselwall may be considered as rigid or elastic and the Power lawnon-Newtonian model or Carreau-Yasuda non-Newtonianmodel are normally applied to describe the viscosity of bloodConsidering the blood flow as turbulent the two-equationturbulence model so-called the standard kndash120576 model or thekndash120596 model has been commonly employed for the analysisYoung (1973) studied blood flow through an occluded tubeunder a pulsatile pressure gradient Mazumdar et al (1996)investigated the unsteady Newtonian blood flow througha stenosed artery and observed that the pressure gradientattains the maximum at the throat of stenosis and decreaseswith an increase of hematocrit parameter Sanyal and Maiti(1998) proposed a mathematical model of blood flow in theartery with mild stenosis They reported that the pressuregradient increases with an increase in hematocrit value wherethere is higher value in systolic and lower value in diastolicpressure Deplano and Siouffi (1999) performed experimentaland numerical study of pulsatile flows through a 75 severitystenosis to determine the wall shear stress temporal evolutiondownstream from the stenosis The result shows high wallshear stress values of about 120 Pa (or dyn cmminus2) duringthe cardiac cycle at the throat and low values downstreamfrom the stenosis of about -25 Pa (or dyn cmminus2) Lee andXu (2002) studied blood flow through a rigid mild stenosedtube Wiwatanapataphee and Wu (2012) investigated theunsteady non-Newtonian blood flow through the real rightcoronary artery bypass graft system under the real pulsatileconditionThey reported that the existence and intensity of astenosis in the artery have significant effect on the blood flow

behaviour Gupta and Agrawal (2015) simulated the bloodflow passing through an irregular stenotic descending aortausing a Finite Volume method The results demonstrate thatthe formation of wall shear stress in the stenotic region bythe irregular stenosis model is much complex than by regularstenosis High oscillation of wall shear stress appears behindthe irregular stenosis in which the Reynold number (Re) isbetween 130 and 540

As themechanical interaction between blood and arterialwall has an important role in the propagation of pressurewave from the heart to the whole body many researchershave investigated the fluid-structural response to pulsatileblood flow through a stenosed vessel subject to variousphysiological conditions Chan (2006) investigated the fluid-structural response to the pulsatile non-Newtonian bloodflow through an axisymmetric stenosed vessel using ANSYSThe solid model was set to have isotropic elastic propertiesThe Fluid-Structural Interaction (FSI) coupling was two-way and iterative It is found that interaction between vesselwall and the blood gives reasonable results Due to the wallexpansion the axial velocity decreases and the recirculationeffect of the flow increases Torii et al (2009) studied bloodflow in the deformable cerebral artery using an FSI modeland reported that the maximum wall shear stress tends todecrease when the blood flow impinges strongly on the wall

The consideration of a non-Newtonian behaviour ofblood in small arteries gives more relevant results andprediction and understanding of pressure distribution andwall shear stress have significant importance in diseasediagnosis and surgical planning The models with full three-dimensional FSI problem increase the computational workHence development of the computational framework builtupon image-based CFD and discrete particle dynamics mod-elling is a big challenge Bernad et al (2013) investigated theparticle motion in coronary serial stenoses to analyze thehemodynamic significance of three serial stenoses namedST1 ST2 and ST3 in the right coronary artery (RCA) con-structed from multislice computerized tomography imagesBlood was assumed to be an incompressible Newtonian fluidand the artery walls was rigid and impermeable Resultsillustrate that pressure drop increases with an increase ofpercentage stenosis During the systolic phase the pressuredrop is higher about 3284 mmHg and 3678 mmHg for thestenosis ST1 and ST3 during the systolic phase while it islower about 462 mmHg and 481 mmHg during the diastolicphase For stenosis ST2 the pressure drop is not significantduring the systolic and diastolic phases They reported thatwall shear stress distribution has a close reflection of theoutline of the stenosis and the formation of recirculationzone Range of wall shear stress varies from 7 to 262 PaThree intense regions of wall shear stress appear downstreamat each stenosis and its value is low in the recirculationzone Mukherjee and Shadden (2017) studied embolic par-ticle dynamics and transport through swirling chaotic flowstructures of various vasculature beds The results show thecomplex interplay of particle inertia fluid-particle densityratio and wall collisions with chaotic flow structures whichrender the overall motion of the particles to be nontriviallydispersive in nature These researches motivate the present

International Journal of Differential Equations 3

study to deal with the particle motion in a pulsatile bloodflow through a stenotic artery However a few attempts havebeenmade to study the flow phenomena in the stenotic arteryin which the fluid-particle interaction the particle-particlecollision and the particle-wall collision are considered

The aim of this study is to investigate the hemodynamicparameter the pressure distribution and wall shear stressin a stenosed artery using Reynolds-averaged Navier-Stokesequations for turbulence fluid modelling and Newtonianequations for the translation and rotational motion of thebioparticles Using the pulsatile boundary conditions basedon a physiological waveforms of flow velocity and blood pres-sure the results are compared with those obtained from themodel with no particle to highlight the role of particlemotionin the turbulence model of blood flow in the left coro-nary artery (LCA) connecting to the stenosed left anteriordescending artery (LAD) (Figure 16) and the normal leftcircumflex coronary artery (LCX) (Figure 15)The rest of thispaper is organized as follows In the following sections amathematical model describing the turbulence flow of fluidand movement of dispersed phase under pulsatile conditionis taken into presentation The governing equations of dis-persed particle-blood flow are theoretically presented in Sec-tion 2 Section 3 concerns numerical investigations to analyzevelocity field pressure distribution and wall shear stressdistribution along two investigated axial lines with respect todifferent degrees of stenosis severity At the end of this papersome discussion and conclusion are given in Section 4

2 Mathematical Model

The computational domain is modelled by using two coor-dinate systems ie an Eulerian frame Ω(X1X2X3) for thefluid flow and a Lagrangian frame Ω119871(x1x2x3) for particlemovement The carrier fluid is assumed to behave as a non-Newtonian incompressible fluid Both phases of dispersedparticles and blood exchange momentum are allowed tohave the fluid-particle interaction In this study we focuson two-way coupling model in which fluid phase influencesparticulate phase via drag and turbulence and particulatephase influences fluid phase via source terms of mass andmomentum The dispersion of particles due to turbulencein the fluid phase is predicted using the stochastic tracking(random walk) model including the effect of instantaneousturbulent velocity fluctuations on the particle trajectoriesthrough the use of stochastic methods [20] The fluid flowinfluences the particle trajectories and the dispersed particleswith the particle-particle collision and the particle-wall colli-sion have a significant effect on the flow turbulence

21 Laminar Flow of Non-Newtonian Incompressible FluidThe flow of a non-Newtonian incompressible fluid is gov-erned by the continuity equation and the Navier-Stokesequations as follows120597119906119894120597119883119894 = 0 (1)

120588119891 120597119906119894120597119905 + 120588119891119906119895 120597119906119894120597119883119895 = minus 120597120597119883119894119901 + 120597120597119883119895 (120583120597119906119894120597119883119895) + 119865ext

119894 (2)

where 119906119894 and 119865ext119894 denote respectively velocity component of

fluid and the external force 119901 is the fluid pressure 120588119891 is thefluid density and120583 is the viscosity of the non-Newtonian fluidbased on the Carreau model ie

120583 = 120583infin + (1205830 minus 120583infin) [[1 + (120582(radic2sum119894119895

1198782119894119895)119886

)(119899minus1)119886]

] (3)

where 1205830 and 120583infin represent zero shear viscosity and infiniteshear viscosity 119886 is constant shape parameter 119899 is consistentindex 120582 is time constant and 119878119894119895 = (120597119906119894120597119883119895 + 120597119906119895120597119883119894)222 Turbulence Flow of Non-Newtonian Incompressible FluidThe turbulence flowof a non-Newtonian incompressible fluidwhich is formulated on amoving reference frame is governedby the mean continuity equation the Reynolds-averagedNavier-Stokes equations and Menterrsquos SST 119896ndash120596 model [21]as follows

120597119906119894120597119883119894 = 0 (4)

120588119891119863119906119894119863119905 = minus 120597119901120597119883119894 +120597120591119894119895120597119883119895 + 119865ext

119894 (5)

120588119891119863119896119863119905 = minus120591119894119895 120597119906119894120597119883119895 minus 120588119891120573lowast120596119896+ 120597120597119883119895 ((120583 + 120583119905120590119896) 120597119896120597119883119895)

(6)

120588119891119863120596119863119905 = minus 120572120583119905 120591119894119895120597119906119894120597119883119895 minus 1205881198911205731205962

+ 120597120597119883119895 ((120583 + 120583119905120590120596) 120597120596120597119883119895)+ 2120588119891 (1 minus 1198911) 1205901205962120596 120597119896120597119883119895

120597120596120597119883119895 (7)

with the operator

119863119863119905 equiv 120597120597119905 + 119906119895 120597120597119883119895 (8)

where 119906119894 denote mean velocity component of fluid and 120583 isthe fluid viscosity based on the Carreau model (3) In (5)

120591119894119895 = 2120583(119878119894119895 minus 13120575119894119895 120597119906119896120597119883119896) + 120591119894119895 (9)

where 120591119894119895 is the Reynolds stress term According to theconcept of Reynolds decomposition the individual instan-taneous velocity component 119906119894 119894 = 1 2 3 (the dependentvariables) of the continuity equation are decomposed intoNavier-Stokes equations Consequently the mean part 119906119894 andthe fluctuation part 1199061015840119894 are defined as

119906119894 = 119906119894 + 1199061015840119894 (10)


where 1199061015840119894 (119894 = 1 2 3) are constant functions of time obtainedfrom stochastic model and their random value is kept con-stant over an time interval given by the characteristic lifetimeof the eddies

In (9) 120591119894119895 representing the nonlinear convective transportdue to turbulent velocity fluctuations is defined by

120591119894119895 = 2120583119905 (119878119894119895 minus 13120575119894119895 120597119906119896120597119883119896) minus 23120588119891119896120575119894119895 (11)

where 119906119896 is component of themean velocity 119878119894119895 = (120597119906119894120597119883119895+120597119906119895120597119883119894)2 and 120583119905 is eddy viscosity defined by

120583119905 = 120588119891119896120596 1max 1120572lowast (1198781198912) (1198861120596) (12)

for120572lowast = 120572lowastinfin(120572lowast0 +Re119905119877119896)(1+Re119905119877119896)minus1 Re119905 = 120588119891119896(120583120596)minus1 and119877119896 = 6 120572lowast0 = 1205731198943 120573119894 = 0072 1198861 = 031 and 119878 equiv (2119878119894119895119878119894119895)12is the modulus of the mean rate-of-strain tensor

Two blending functions1198911 in (7) and1198912 in (12) are definedby

1198911 = tanh (12060141) 1206011 = minmax radic119896120573lowastinfin120596119910

5001205831205881199102120596 41205881198961205901205962119863+1205961199102 (13)

1198912 = tanh (12060122) 1206012 = max2 radic119896120573lowastinfin1205961199105001205831205881199102120596 (14)

with119863+120596 = max2120588120590minus11205962120596minus1(120597119896120597119883119895)(120597120596120597119883119895) 10minus10 and 119910 isthe distance to the next surface In (6)

120573lowast = 120573lowastinfin [[415 + (Re119905119877120573)41 + (Re119905119877120573)4 ]

] (15)

where 120573lowastinfin = 009 and 119877120573 = 8 Other four parameters 120572 120573 120590119896and 120590120596 in the SST 119896-120596model are determined by

120572 = 11989111205721 + (1 minus 1198911) 1205722120590119896 = 11989111205901198961 + (1 minus 1198911) 1205901198962 (16)

120573 = 11989111205731 + (1 minus 1198911) 1205732120590120596 = 11989111205901205961 + (1 minus 1198911) 1205901205962 (17)

which are the relationships that transform the constants of theoriginal 119896-120596model into the constants of the SST 119896-120596model

The external force 119865ext119894 in (5) is formulated by examining

the change in momentum of the solid particle passingthrough each control volume computed as a function of theparticle mass flow rate 119901 and the time step Δ119905 ie

119865ext119894 = 119899119901sum119901=1

(18120583119862119863ReV241205882119901 (119906119894 minus 119906119894119901)) 119901Δ119905 (18)

where 120588119901 is the particle density and Re] is the dimensionlessrelative particle Reynolds number defined by

ReV = 12058811989111988911990110038171003817100381710038171003817u minus k119901

10038171003817100381710038171003817120583 (19)

The velocity component of the 119901th particle V119894119901 is the arith-metic average nodal values on the face based onGreenndashGaussnode-based method defined by

V119894119901 = 1119873face

119873facesum119901=1

V119894119901 (20)

where 119873face represents number of nodes on face and V119894119901 isnodal value on the face

The drag coefficient 119862119863 is based on the spherical draglaw [22 23] ie

119862119863 = 1205721 + 1205722ReV

+ 1205723Re3V

(21)

for

1205721 1205722 1205723

=

0 24 0 00 le ReV lt 013690 2273 00903 01 le ReV lt 11222 291667 minus38889 10 le ReV lt 1006167 4650 minus11667 10 le ReV lt 10003644 9833 minus 2778 100 le ReV lt 10000357 14862 minus47500 1000 le ReV lt 5000046 minus490546 578700 5000 le ReV lt 1000005191 minus16625 5416700 ReV ge 10000

(22)

To be more specific the remaining constant model parame-ters introduced in this subsection are declared as follows

1205721 = 59 1205731 = 340

1205901198961 = 0851205901205961 = 051205722 = 0441205732 = 00828

1205901198962 = 11205901205962 = 0856

(23)

23 Movement of Dispersed Particle Phase For the dispersedparticle phase we assume that all particles are spherical andheat and mass transfer are omitted All particles are treated


as point masses and inert typeThe translation and rotationalmotions of the 119901th particle 119901 = 1 119899119901 are governed by thefollowing Newtonian equations

119889x119901119889119905 = k119901 (24)

119898119901 119889k119901119889119905 = F119901119891 + F119901119888 (25)

119868119901 119889Ω119901119889119905 = 1205881199012 (1198891199012 )5 119862120596Υ (26)

where x119901 denotes the particle position vector k119901 andΩ119901 arerespectively Lagrangian velocity and angular velocity of the119901 particle 119898119901 is the particle mass and 119868119901 the moment ofrotational inertia given by 119868119901 = (12058760)1205881199011198895119901 for the sphereparticle with diameter of 119889119901 and density of 120588119901 The rightside term of (26) is the resulting torque depending on therotational drag coefficient 119862120596 and the relative particle-fluidangular velocity Υ defined by

Υ = 12nabla times 119906 minus Ω119901 (27)

F119901119891 are the sum of various forces including drag force F119863Saffman force F119878 and the Magnus or rotational lift force F119877119871acting on the particle by carrier fluid

F119863 = u minus k119901120591119901 (28)

F119904 = 2120581radic120592120588119891119889119894119895120588119901119889119901 (119889119897119896119889119896119897)14 (u minus k119901) (29)

F119877119871 = 1205881198912 119860 119894119901119862119877119871 VΥ (V times Υ) (30)

where 120591119901 = 1205881199011198892119901(18120583)minus1 sdot 24(119862119863119877119890V)minus1 120581 = 2594 and 119889119895119895 isthe deformation tensor along the path of particles defined by

119889119894119895 = 12 ( 120597V119894120597119909119895 +120597V119895120597119909119894) (31)

Regarding (30) 119860 119894119901 is the 119894th projected particle surface areaV is the relative fluid-particle velocity and 119862119877119871 denotes therotational lift coefficient Taking into account the effect of therotational Reynolds number Re120596 and the particle Reynoldnumber Re] the coefficients 119862119877119871 and 119862120596 are assigned as

119862119877119871 = 045+ (Re120596

ReVminus 045) exp (minus005684Re04120596 Re03V ) (32)

119862120596 = 645radicRe120596+ 321Re120596

Re120596 = 120588119891 Ω 11988921199014120583

(33)

The term F119901119888 in (25) represents the contact forces due to theparticle-particle collision and the particle-wall collision ieF119901119888 = F119901119902 + F119901119908 Based on the Spring-Dashpot Collision Law[24] F119901119902 is given by

F119901119902 = (119870120575 minus 120574 (k119901119902 sdot e119901119902)) e119901119902 (34)

where K is the elastic collision coefficient 120575 is the overlapof any two particles 120574 is the damping coefficient k119901119902 is therelative velocity and e119901119902 is a unit vector For the positionvectors x119901 x119902 and the radii 119903119901 and 119903119902 of the particles 119901 and119902 we have

120575 = 10038171003817100381710038171003817x119902 minus x11990110038171003817100381710038171003817 minus (119903119901 + 119903119902) (35)

k119901119902 = k119902 minus k119901 (36)

e119901119902 = (x119902 minus x119901)10038171003817100381710038171003817x119902 minus x11990110038171003817100381710038171003817 (37)

The damping coefficient 120574 in (34) depending on the mass lossin the collision process119898119901119902 and the collision time scale 119905119888 aredefined by

120574 = 2119898119901119902 ln 120578119905119888 (38)

119898119901119902 = 119898119901119898119902119898119901 + 119898119902 (39)

119905119888 = radic(1205872 + ln2120578) 119898119901119902119870 (40)

where 120578119898119901 and 119898119902 denote the damper restriction coeffi-cient themass of particle119901 and themass of particle 119902 respec-tively The contact force due to the particle-wall collision F119901119908is calculated in the same way as F119901119902

24 Initial and Boundary Conditions Initialization of theparticle properties in the domain is not required and there isno interface boundary condition for point-mass particles Tocomplete defining the boundary value problem the followingboundary conditions are required In this study the pulsatilemass flow inlet and the pulse pressure outlet are used Thewave forms of the mass flow rate 119876(119905) and the pressure 119901(t)are calculated by the following Fourier series

119876 (119905) = 119876 + 4sum119899=1

119886119876119899 cos (119899119908119905) + 119887119876119899 sin (119899119908119905) (41)

119901 (119905) = 119901 + 4sum119899=1

119886119901119899 cos (119899119908119905) + 119887119901119899 sin (119899119908119905) (42)

where119908 denotes the angular frequency defined by119908 = 2120587119879with a cardiac period 119879 and all values of the parametersare given by Wiwatanapataphee et al [16] On the inflowboundary we have the pulsatile velocity as

u = u = 119880119899 (119905) k119901 = 119880119899 (119905 = 02119904) (43)


for

119880119899 (119905) = 119876 (119905)119860 (44)

where 119876(119905) is the pulsatile flow rate and 119860 is the inflowsurface area The turbulence kinetic energy and the specificdissipation rate at the inflow are

119896 = 32 (119880119899 (119905) 120589)2 (45)

120596 = 0164311989615119871 (46)

with the percentage of turbulence intensity 120589 =016(120588119880119899(119905)119863120583)18 and turbulence length scale 119871 = 007119863for119863 = 11311986012

On the outflow boundaries ΓLCA including ΓLCA1 ΓLCA2and ΓLCA3 the boundary conditions are set to 120590 sdot n = 119901(119905)nand the normal pressure gradient field is corresponding to120597119901120597n = 0 For an inert and point-mass particle interfacecondition between fluid and solid particle is not requiredFurthermore it is also assumed that the wall has an infiniteradius and zero velocity and no-slip condition is applied onthe arterial wall

3 Numerical Investigation

This section presents numerical simulation of particle move-ment and turbulence non-Newtonian fluid flow The simu-lation of discrete particles is carried out using the discreteelement method With this method particle trajectories arecalculated through the simulation domain in Lagrangianreference To reduce time-consuming simulation a limitednumber of representative trajectories is calculated The sim-ulation of the continuous phase is carried out using theFinite Volume method By this method the mean continuityequation the Reynolds-averaged Navier-Stokes equationsand the Menterrsquos SST 119896-120596 model are solved in Eulerianreference frame

31 Validation Study Studying the turbulence fluid-solid(two-phase) flow in the coronary arterywith stenosis requiresa reliable model that can fully describe the complex phe-nomena occurring in the artery with nonlinear responseThefirst task is undertaken for evaluating the suitability of themathematical model using ANSYS 182The simulation of theturbulence two-phase flow in the normal curved tube and75 stenosis-curved tube was setup using the discrete modelcoupled with the mean continuity equation the Reynolds-averaged Navier-Stokes equations and Menterrsquos SST 119896-120596model for a turbulent viscous incompressible non-NewtonianfluidThe computational domains of both tubes are displayedin Figure 2 Five complete pulses of pressure and flow velocitywere used in each simulationThe results as shown in Figures3 and 4 indicate that pressure drop presents in the tubewith restricted areaThemodel with particle movement givesvariation of pressure in the area occupied by the particleThese results show that our proposed model can captureimportant phenomena in the flow channel without and withrestricted area

2725 mm

75 of stenosis

Figure 2 Computational domain of validation study

Table 1 Mesh information

Stenosis Number of elements0 31415025 31889750 35026575 444764

32 Fluid-Particle Flow through the Human Left CoronaryArterywith Stenosis Tobegin the numerical simulation a 3Dcomputational domain of a human left coronary artery withits branches including the LAD and LCX is firstly constructedby replicating the multislice computerized tomography (CT)image as shown in Figure 5 Taking into account the flowdirection there are a single inlet at the beginning of LCA andthree outlets at the tail ends of LAD and LCX

Next the effect of coronary stenosis on the flow of fluid(blood) and discrete particles (bioparticles) is taken intoinvestigation In this simulation 414 particles are trackedto determine the behaviour of the dispersed phase Thedistribution of the dispersed phase bioparticles in the midstof blood inside the arterial vessel is closely focused whenthere exists a stenosis Three different degrees of stenosisseverity including 25 50 and 75 are assigned at theproximal part of LAD as can be seen in Figure 6(a) After gridindependent test we obtained suitable domain mesh with 10boundary layers with the size of the first cell 5 times 10minus3 mm forfour cases including the normal artery and the artery with25 50 and 75 stenosis details as shown in Table 1

The flow pattern the pressure distribution and the wallshear stress (WSS) distribution are analyzed To investigatethe effect of particle motion on pressure distribution and wallshear stress distribution 500 bioparticles are injected intothe LCA inflow surface at the same speed of the pulsatilevelocity in the first cardiac cycle t = 02 s Assigning values ofmodel parameters as in Table 2 and running the simulationcorresponding to various numerical options as in Table 3 thenumerical results along the left coronary artery connecting tothe critical LADwith various-degree stenosis and the normalLCX simulated using ANSYS 182 for the turbulent dispersedparticle-fluid flow are obtained in Tables 2 and 3

Focusing on the proximal part of LCA as shown inFigure 7(a) the results obtained from the turbulent flow


60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(a) 119905 = 065 119904

60

80

100

120

140

Pres

sure

[mm

Hg]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904

Figure 3 Blood pressure obtained from the model with no particle (solid line) and the model with particles (dotted line) in a normal curvedtube at two different peak systoles (a) 119905 = 065 119904 (b) 119905 = 386s

10 20 30 40 50 60 700Arc-length [mm]

60

80

100

120

140

Pres

sure

[mm

Hg]

(a) 119905 = 065 119904

60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904

Figure 4 Blood pressure obtained from the model with no particle (solid line) and the model with particles (dotted line) in the 75 stenosistube at two different peak systoles (a) t = 065 s (b) t = 386s

Inlet

LCX Outlet 1

LAD Outlet 2

LAD Outlet 3

LCA

LCX

LAD

Figure 5 Computational domain of the LCA and its branchesincluding the LAD and the LCX

Table 2 Model parameters

Parameters value unitsBlood density (120588) 1050 kg mminus3

Zero shear viscosity (1205830) 056 g cmminus1 sminus1

Infinite shear viscosity (120583infin) 00345 g cmminus1 sminus1

Time constant (120582) 33130 sConsistency index (n) 03568 -Shape parameter (a) 2 -Particle diameter (dp) 100 120583mParticle density (120588p) 1050 kg mminus3

Particle mass flow rate (119901) 550 x 10minus8 kg sminus1

Particle time step (Δ119905) 1 times 10minus6 s

model and the turbulent dispersed particle-fluid flow modeldemonstrate the effect of turbulence blood flow on theparticle trajectories and the impact of particle motion with


25

75

50

(a) LAD with stenosis (b) The LCA-LAD axial line (c) The LCA-LCX axial line

000

2000

4000

6000

8000

10000

12000

14000

Pres

sure

[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]

Time 065 s Time 386 s

(d) Pulsatile pressure at the LCA inflow surface

Figure 6 Stenosis conditions at the proximal LAD two investigated lines along the LCA connecting to the LAD and the LCX and pulsatilepressure with two investigated times at the peak systole in the first and the fifth cardiac cycles

Table 3 Numerical setting

Numerical Parameters SettingSolver Pressure - basedPressure ndash Velocity Coupling CoupledMultigrid AMGUnder ndash relaxation factors 075 - momentum

075 - pressure080 - turbulent kinetic energy080 - specific dissipation rate090 - discrete phase source

Spatial discretization Second order upwind - momentumSecond order upwind - turbulent kinetic energySecond order upwind - specific dissipation rateSecond order upwind - transient formulation

Gradient Green - Gauss node - based

the particle-particle and the particle-wall collisions on theblood flow pattern

In the fluid flow model small turbulent flow appears inthe transition area connecting the LCA with its branches asshown in Figures 7(b) and 7(c) In the dispersed particle-fluidflow model high turbulent flow appears in the LCA regionand the particles continue to proportionally flow from LCAto LAD and LCX as displayed in Figure 7(d) Considering theblood velocity inside LCA LAD and LCX the vector plots

in three different planes are exhibited in Figure 8 The higherdegree stenosis reduces number of particles flowing throughthe downstream LAD and increases number of particlesflowing into the downstream LCXThe model with 0 2550 and 75 degree of stenosis severity on the proximalLAD allows 120 64 34 and 1 of all particles flowingthrough the downstream LAD respectively Particles arealmost completely blocked in the LAD with critical 75stenosis


(a) A part of LCA (b) Streamline of blood flow

(c) Velocity field of blood flow (d) Particle deposition patterns

Figure 7 Dispersed phase flow in the LCA connecting to the LAD and LCX

Distribution of systolic pressure along two axial linesincluding the LCA-LAD line in as shown in Figure 6(b) andthe LCA-LCX line as shown in Figure 6(c) is investigatedin the first and the fifth cardiac cycle Regarding the fluidflowmodel and the dispersed particle-fluid flowmodel bloodpressure at the peak systole at t = 065 s and t = 386 s alongthe two axial lines with three different degrees of stenosisseverity are plotted in Figures 11 and 12 respectively Theresults indicate that a pressure drop across the 75 stenosisis significant at the peak systole t = 065 s In the fluid flowmodel at the peak systole in a cardiac cycle the pressuredrop is about 32 mmHg In the dispersed particle fluid flowmodel the pressure drop at the peak systole in the first

and the fifth cardiac cycles is significantly different due toparticle deposition patterns in the stenotic area Higher levelof particle motion makes more pressure drop At t = 065 s apressure drop is about 55 mmHg and 40 mmHg at t = 386 srespectively

The level of particle motion varies with time The highestlevel is at the peak systole of the first cardiac cycle t = 065s To show the effect of particle motion on the blood pressureand thewall shear stress the results at the peak systole t = 386s obtained from the turbulent flow model and the turbulentdispersed particle-fluid flow model are compared in Figures9ndash14The results indicate that the coronary arterywith critical75 stenosis generates a sudden drop of pressure with high


3

1

2

(a) Cutting plane

(a)

(b)

(c)

(b) Blood velocity in each plane

Figure 8 Vector plot of blood velocity at the peak systole at the time t = 065 s in three different planes

(a) Model with no particle (b) Model with particle motion

Figure 9 Systolic pressure at t = 065 s in the artery model with 75 intensity of stenosis at the proximal LAD obtained from two differentmodels (a) model with no particle (b) model with particle motion

Time = 386 [s]Pressure [mmHg]

(a) Pressure distribution with particle deposition patterns

Time = 386 [s]

(b) Wall shear stress distribution with particle deposition patterns

Figure 10 Pressure distribution and wall shear stress distribution with particle deposition patterns in the artery model with 75 intensity ofstenosis at the proximal LAD at the peak systole in the fifth cardiac cycle at t = 386 s


20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75

Figure 11 Blood pressure at the peak systole t = 065 s and t = 386 s along the LCA-LAD axial line obtained from the model with no particle(solid line) and the model with particles (dotted line) obtained from the domain with different intensity of stenosis at the proximal LAD (a)25 (b) 50 (c) 75

wall shear stress around the stenosis cite Higher degree ofstenosis gives higher values of the pressure drop and wallshear stresses Pressure distribution and wall shear stressesare plotted when the flow is at a maximum at the peak systole

in the first and the fifth cardiac cycles (t =065 s and t =386 s)The dispersed particle-fluid flow model gives high variationof pressure and wall shear stress especially in the LCA to theproximal LAD Figures 11 and 12 describe the effect of particle


20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75

Figure 12 Blood pressure at the peak systole t = 065 s and t = 386 s along the LCA-LCX axial line obtained from the model with no particle(solid line) and the model with particles (dotted line) obtained from the domain with different intensity of stenosis at the proximal LAD (a)25 (b) 50 (c) 75

motion on the pressure distribution in two investigated linesthe LCA-LAD axial line and the LCA-LCX axial line as shownin Figures 6(b) and 6(c)

Wall shear stress along the first half of the LCA connect-ing to the LAD is higher as shown in Figure 13 At the firstcardiac cycle t = 065 s the maximum wall shear stress in the


20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75

Figure 13Wall shear stress at the peak systole t = 065 s and t = 386 s along the LCA-LAD axial line obtained from themodel with no particle(solid line) and the model with particles (dotted line) obtained from the domain with different intensity of stenosis at the proximal LAD (a)25 (b) 50 (c) 75

proximal 75 stenosis of the LAD is about 225 Pa in the fluidflow model and 275 Pa in the dispersed particle-fluid flowmodel In addition at the fifth cardiac cycle t = 386 s thismaximum wall shear stress is about 240 Pa in the fluid flow

model and 250Pa and the dispersed particle-fluid flowmodelIn Figure 14 presents variation of wall shear stress along theLCA connecting to the normal LCX in the first and the fifthcardiac cycle It indicates that particle motion in the carrier


20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75

Figure 14Wall shear stress at the peak systole t = 065 s and t = 386 s along the LCA-LCX axial line obtained from themodel with no particle(solid line) and the model with particles (dotted line) obtained from the domain with different intensity of stenosis at the proximal LAD (a)25 (b) 50 (c) 75


Figure 15The system of coronary arteries including the base of theaorta and the normal left and the normal right arteries

Figure 16The system of coronary arteries including the base of theaorta and the 75 stenosed left artery and the normal right artery

fluid as shown in Figure 10 has significant effect on the wallshear stress High wall shear stress occurs in the area withhigh particle concentration (particle cluster)

4 Discussion and Conclusion

This paper presents the mathematical model of the dispersedbioparticle-blood flow in the left coronary artery (LCA)with its branches including the LAD and LCX The com-bination of the mean continuity equation the Reynolds-averaged Navier-Stokes equations and the Menterrsquos SST kndash120596models is employed to investigate the turbulence flow ofblood the non-Newtonian incompressible fluid Describingthe movement of dispersed particle phase the Newtonianequations are used to examine the translation and rotationalmotion of bioparticles Running the simulation of two-phaseflow inside LCA the 3D computational domain togetherwith initial and boundary conditions are necessary Thedispersed phase flow the pressure distribution and the wallshear stress distribution are analyzed corresponding to three

different stenosis intensities of 25 50 and 75 at theproximal LAD The results demonstrate a significant effectof turbulence blood flow on the particle trajectories anda high impact of particle motion with the particle-particleand the particle-wall collisions on the blood flow patternThe coronary artery with critical 75 stenosis generates asudden drop of pressure with high wall shear stress aroundthe stenosis cite Higher degree of stenosis gives highervalues of the pressure drop and wall shear stresses Pressuredistribution and wall shear stresses are plotted when the flowis at a maximum at the peak systole in the first and the fifthcardiac cycle Pressure drop is significantly different due toparticle deposition patterns in the stenotic area at the peaksystole Higher level of particle motion makes more pressuredrop and has significant effect on the wall shear stress thatoccurring in the area with higher particle concentration

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Supplementary Materials

The symptoms of coronary heart disease depend on theseverity of the blockage in the 2 main coronary arteries theleft main and the right coronary arteries Understandingblood flow around the blockage is thus necessary for a bypasssurgery Two supplementary figures show the coronary sys-temof human arteries thatwas constructed using 1000 imagesof computed tomography scans of the human coronary sys-tem The system consists of the base of the aorta connectingwith the normal right coronary artery (RCA) and the leftcoronary (LCA) with appearance of a LAD stenosis locatedat 5mm from the aorta-LCA connectionOne supplementaryfigure presents the normal coronary system Another one isthe system with LAD stenosis of 75 In this study we con-sidered only the left main coronary artery with two branchesthe left anterior descending artery (LAD) and the circumflexartery (LCX) to investigate the effect of the severity ofcoronary artery stenosis (Supplementary Materials)

References

[1] S I Bernad E S Bernad T Barbat V Albulescu and R Susan-Resiga ldquoEffects of different types of input waveforms in patient-specific right coronary atherosclerosis hemodynamics analysisrdquoInternational Journal of Design amp Nature and Ecodynamics vol5 no 2 pp 1ndash18 2010

[2] B Wiwatanapataphee ldquoModelling of non-Newtonian bloodflow through stenosed coronary arteriesrdquo Dynamics of Contin-uous Discrete and Impulsive Systems Series B Applications andAlgorithms vol 15 no 5 pp 619ndash634 2008

[3] W Y Chan Simulation of arterial stenosis incorporating flu-idstructural interaction and non-Newtonian blood flow [masterthesis] RMIT University Melbourne Australia 2006


[4] V Deplano and M Siouffi ldquoExperimental and numerical studyof pulsatile flows through stenosis wall shear stress analysisrdquoJournal of Biomechanics vol 32 no 10 pp 1081ndash1090 1999

[5] K W Lee and X Y Xu ldquoModelling of flow and wall behaviourin a mildly stenosed tuberdquo Medical Engineering amp Physics vol24 no 9 pp 575ndash586 2002

[6] P K Mandal ldquoAn unsteady analysis of non-Newtonian bloodflow through tapered arteries with a stenosisrdquo InternationalJournal of Non-Linear Mechanics vol 40 no 1 pp 151ndash1642005

[7] H P Mazumdar U N Ganguly S Ghorai and D C DalalldquoOn the distributions of axial velocity and pressure gradient ina pulsatile flow of blood through a constricted arteryrdquo IndianJournal of Pure andAppliedMathematics vol 27 no 11 pp 1137ndash1150 1996

[8] D C Sanyal and A KMaiti ldquoOn steady and pulsatile motion ofbloodrdquo Czechoslovak Journal of Physics vol 48 no 3 pp 347ndash354 1998

[9] D F Young and F Y Tsai ldquoFlow characteristics in models ofarterial stenoses mdash II Unsteady flowrdquo Journal of Biomechanicsvol 6 no 5 pp 547ndash559 1973

[10] B Wiwattanapataphee S Amornsamankul Y Hong Wu WShi and Y Lenbury ldquoNon-Newtonian blood flow throughstenosed coronary arteriesrdquo in Proceedings of the 2nd WSEASInt Conference on Applied and Theoretical Mechanics pp 259ndash264 Venice Italy 2006

[11] S Amornsamankul B Wiwattanapataphee Y Hong Wu andY Lenbury ldquoEffect of non-Newtonian behaviour of blood onpulsatile flows in stenotic arteriesrdquo World Academy of ScienceEngineering andTechnology International Journal ofMedical andHealth Science vol 1 no 2 pp 108ndash112 2007

[12] B Wiwatanapataphee Y H Wu T Siriapisith and B Nun-tadilok ldquoEffect of branchings on blood flow in the system ofhuman coronary arteriesrdquo Mathematical Biosciences and Engi-neering vol 9 no 1 pp 199ndash214 2012

[13] J R Buchanan and C Kleinstreuer ldquoSimulation of particle-hemodynamics in a partially occluded artery segment withimplications to the initiation of microemboli and secondarystenosesrdquo Journal of Biomechanical Engineering vol 120 no 4pp 446ndash454 1998

[14] J-J Chiu D LWang S Usami S Chien and R Skalak ldquoEffectsof disturbed flow on endothelial cellsrdquo Journal of BiomechanicalEngineering vol 120 no 1 pp 2ndash8 1998

[15] I Marshall S Zhao P Papathanasopoulou P Hoskins and XY Xu ldquoMRI and CFD studies of pulsatile flow in healthy andstenosed carotid bifurcation modelsrdquo Journal of Biomechanicsvol 37 no 5 pp 679ndash687 2004

[16] B Wiwatanapataphee and Y H Wu ldquoMathematical study ofblood flow in the real model of the right coronary arterymdashbypass graft systemrdquo Dynamics of Continuous Discrete andImpulsive Systems Series B Applications and Algorithms vol 19no 4-5 pp 621ndash635 2012

[17] S I Bernad and E S Bernad ldquoCoronary venous bypass graftfailure hemodynamic parameters investigationrdquo in Proc ofthe IASTED International Conference Biomedical Engineering(BioMed 2012) pp 509ndash515 Innsbruck Austria February 2012

[18] T Frauenfelder E Boutsianis T Schertler et al ldquoIn-vivo flowsimulation in coronary arteries based on computed tomographydatasets Feasibility and initial resultsrdquo European Radiology vol17 no 5 pp 1291ndash1300 2007

[19] N Sun R Torii N B Wood A D Hughes S A MThom andX Y Xu ldquoComputational modeling of LDL and albumin

transport in an in vivo CT image-based human right coronaryarteryrdquo Journal of Biomechanical Engineering vol 131 no 2 pp1ndash9 2009

[20] D J Thomson and J D Wilson ldquoLagrangian modeling ofthe atmosphere geophysical monograph series 200rdquo AmericanGeophysical Union 2012

[21] F R Menter ldquoTwo-equation eddy-viscosity turbulence modelsfor engineering applicationsrdquo AIAA Journal vol 32 no 8 pp1598ndash1605 1994

[22] ANSYS Inc Userrsquos manual R182 Theory guide 2017[23] S AMorsi andA J Alexander ldquoAn investigation of particle tra-

jectories in two-phase flow systemsrdquo Journal of FluidMechanicsvol 55 no 2 pp 193ndash208 1972

[24] L Zhou L Zhang L Bai et al ldquoExperimental study and tran-sient CFDDEMsimulation in a fluidized bed based on differentdragmodelsrdquoRSCAdvances vol 7 no 21 pp 12764ndash12774 2017

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Figure 1 Tight severe stenosis (95) of the proximal LAD in apatient with Wellensrsquo warning

includes asymmetric flow unsteady laminar-to-turbulentflow [13ndash16] The governing equations include the Navier-Stoke equations and the continuity equations subjected to theselected inlet velocity no-slip condition at the arterywall andstress-free condition at the outflow surfaceTheunsteady flowalso is characterized by a high pressure drop and high wallshear stress (WSS) in the stenotic artery [1 17ndash19]

For the requirement of simulation the constant densityof blood is assumed to be equal to 1055 kg mminus3 The vesselwall may be considered as rigid or elastic and the Power lawnon-Newtonian model or Carreau-Yasuda non-Newtonianmodel are normally applied to describe the viscosity of bloodConsidering the blood flow as turbulent the two-equationturbulence model so-called the standard kndash120576 model or thekndash120596 model has been commonly employed for the analysisYoung (1973) studied blood flow through an occluded tubeunder a pulsatile pressure gradient Mazumdar et al (1996)investigated the unsteady Newtonian blood flow througha stenosed artery and observed that the pressure gradientattains the maximum at the throat of stenosis and decreaseswith an increase of hematocrit parameter Sanyal and Maiti(1998) proposed a mathematical model of blood flow in theartery with mild stenosis They reported that the pressuregradient increases with an increase in hematocrit value wherethere is higher value in systolic and lower value in diastolicpressure Deplano and Siouffi (1999) performed experimentaland numerical study of pulsatile flows through a 75 severitystenosis to determine the wall shear stress temporal evolutiondownstream from the stenosis The result shows high wallshear stress values of about 120 Pa (or dyn cmminus2) duringthe cardiac cycle at the throat and low values downstreamfrom the stenosis of about -25 Pa (or dyn cmminus2) Lee andXu (2002) studied blood flow through a rigid mild stenosedtube Wiwatanapataphee and Wu (2012) investigated theunsteady non-Newtonian blood flow through the real rightcoronary artery bypass graft system under the real pulsatileconditionThey reported that the existence and intensity of astenosis in the artery have significant effect on the blood flow

behaviour Gupta and Agrawal (2015) simulated the bloodflow passing through an irregular stenotic descending aortausing a Finite Volume method The results demonstrate thatthe formation of wall shear stress in the stenotic region bythe irregular stenosis model is much complex than by regularstenosis High oscillation of wall shear stress appears behindthe irregular stenosis in which the Reynold number (Re) isbetween 130 and 540

As themechanical interaction between blood and arterialwall has an important role in the propagation of pressurewave from the heart to the whole body many researchershave investigated the fluid-structural response to pulsatileblood flow through a stenosed vessel subject to variousphysiological conditions Chan (2006) investigated the fluid-structural response to the pulsatile non-Newtonian bloodflow through an axisymmetric stenosed vessel using ANSYSThe solid model was set to have isotropic elastic propertiesThe Fluid-Structural Interaction (FSI) coupling was two-way and iterative It is found that interaction between vesselwall and the blood gives reasonable results Due to the wallexpansion the axial velocity decreases and the recirculationeffect of the flow increases Torii et al (2009) studied bloodflow in the deformable cerebral artery using an FSI modeland reported that the maximum wall shear stress tends todecrease when the blood flow impinges strongly on the wall

The consideration of a non-Newtonian behaviour ofblood in small arteries gives more relevant results andprediction and understanding of pressure distribution andwall shear stress have significant importance in diseasediagnosis and surgical planning The models with full three-dimensional FSI problem increase the computational workHence development of the computational framework builtupon image-based CFD and discrete particle dynamics mod-elling is a big challenge Bernad et al (2013) investigated theparticle motion in coronary serial stenoses to analyze thehemodynamic significance of three serial stenoses namedST1 ST2 and ST3 in the right coronary artery (RCA) con-structed from multislice computerized tomography imagesBlood was assumed to be an incompressible Newtonian fluidand the artery walls was rigid and impermeable Resultsillustrate that pressure drop increases with an increase ofpercentage stenosis During the systolic phase the pressuredrop is higher about 3284 mmHg and 3678 mmHg for thestenosis ST1 and ST3 during the systolic phase while it islower about 462 mmHg and 481 mmHg during the diastolicphase For stenosis ST2 the pressure drop is not significantduring the systolic and diastolic phases They reported thatwall shear stress distribution has a close reflection of theoutline of the stenosis and the formation of recirculationzone Range of wall shear stress varies from 7 to 262 PaThree intense regions of wall shear stress appear downstreamat each stenosis and its value is low in the recirculationzone Mukherjee and Shadden (2017) studied embolic par-ticle dynamics and transport through swirling chaotic flowstructures of various vasculature beds The results show thecomplex interplay of particle inertia fluid-particle densityratio and wall collisions with chaotic flow structures whichrender the overall motion of the particles to be nontriviallydispersive in nature These researches motivate the present







120588119891 120597119906119894120597119905 + 120588119891119906119895 120597119906119894120597119883119895 = minus 120597120597119883119894119901 + 120597120597119883119895 (120583120597119906119894120597119883119895) + 119865ext

119894 (2)




1198782119894119895)119886

)(119899minus1)119886]

] (3)


120597119906119894120597119883119894 = 0 (4)

120588119891119863119906119894119863119905 = minus 120597119901120597119883119894 +120597120591119894119895120597119883119895 + 119865ext

119894 (5)


(6)

120588119891119863120596119863119905 = minus 120572120583119905 120591119894119895120597119906119894120597119883119895 minus 1205881198911205731205962

+ 120597120597119883119895 ((120583 + 120583119905120590120596) 120597120596120597119883119895)+ 2120588119891 (1 minus 1198911) 1205901205962120596 120597119896120597119883119895

120597120596120597119883119895 (7)

with the operator

119863119863119905 equiv 120597120597119905 + 119906119895 120597120597119883119895 (8)


120591119894119895 = 2120583(119878119894119895 minus 13120575119894119895 120597119906119896120597119883119896) + 120591119894119895 (9)


119906119894 = 119906119894 + 1199061015840119894 (10)




120591119894119895 = 2120583119905 (119878119894119895 minus 13120575119894119895 120597119906119896120597119883119896) minus 23120588119891119896120575119894119895 (11)


120583119905 = 120588119891119896120596 1max 1120572lowast (1198781198912) (1198861120596) (12)




5001205831205881199102120596 41205881198961205901205962119863+1205961199102 (13)




] (15)


120572 = 11989111205721 + (1 minus 1198911) 1205722120590119896 = 11989111205901198961 + (1 minus 1198911) 1205901198962 (16)

120573 = 11989111205731 + (1 minus 1198911) 1205732120590120596 = 11989111205901205961 + (1 minus 1198911) 1205901205962 (17)




119865ext119894 = 119899119901sum119901=1

(18120583119862119863ReV241205882119901 (119906119894 minus 119906119894119901)) 119901Δ119905 (18)


ReV = 12058811989111988911990110038171003817100381710038171003817u minus k119901

10038171003817100381710038171003817120583 (19)


V119894119901 = 1119873face

119873facesum119901=1

V119894119901 (20)



119862119863 = 1205721 + 1205722ReV

+ 1205723Re3V

(21)

for

1205721 1205722 1205723

=


(22)


1205721 = 59 1205731 = 340

1205901198961 = 0851205901205961 = 051205722 = 0441205732 = 00828

1205901198962 = 11205901205962 = 0856

(23)




119889x119901119889119905 = k119901 (24)

119898119901 119889k119901119889119905 = F119901119891 + F119901119888 (25)

119868119901 119889Ω119901119889119905 = 1205881199012 (1198891199012 )5 119862120596Υ (26)




F119863 = u minus k119901120591119901 (28)

F119904 = 2120581radic120592120588119891119889119894119895120588119901119889119901 (119889119897119896119889119896119897)14 (u minus k119901) (29)

F119877119871 = 1205881198912 119860 119894119901119862119877119871 VΥ (V times Υ) (30)


119889119894119895 = 12 ( 120597V119894120597119909119895 +120597V119895120597119909119894) (31)


119862119877119871 = 045+ (Re120596


119862120596 = 645radicRe120596+ 321Re120596

Re120596 = 120588119891 Ω 11988921199014120583

(33)


F119901119902 = (119870120575 minus 120574 (k119901119902 sdot e119901119902)) e119901119902 (34)


120575 = 10038171003817100381710038171003817x119902 minus x11990110038171003817100381710038171003817 minus (119903119901 + 119903119902) (35)

k119901119902 = k119902 minus k119901 (36)



120574 = 2119898119901119902 ln 120578119905119888 (38)

119898119901119902 = 119898119901119898119902119898119901 + 119898119902 (39)

119905119888 = radic(1205872 + ln2120578) 119898119901119902119870 (40)



119876 (119905) = 119876 + 4sum119899=1

119886119876119899 cos (119899119908119905) + 119887119876119899 sin (119899119908119905) (41)

119901 (119905) = 119901 + 4sum119899=1

119886119901119899 cos (119899119908119905) + 119887119901119899 sin (119899119908119905) (42)


u = u = 119880119899 (119905) k119901 = 119880119899 (119905 = 02119904) (43)


for

119880119899 (119905) = 119876 (119905)119860 (44)


119896 = 32 (119880119899 (119905) 120589)2 (45)

120596 = 0164311989615119871 (46)






2725 mm

75 of stenosis









60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(a) 119905 = 065 119904

60

80

100

120

140

Pres

sure

[mm

Hg]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


10 20 30 40 50 60 700Arc-length [mm]

60

80

100

120

140

Pres

sure

[mm

Hg]

(a) 119905 = 065 119904

60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


Inlet

LCX Outlet 1

LAD Outlet 2

LAD Outlet 3

LCA

LCX

LAD











25

75

50


000

2000

4000

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12000

14000

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sure

[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]




















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

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sure

[mm

Hg]

60

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100

120

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160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

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100

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sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

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Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

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sure

[mm

Hg]

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[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

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160

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sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

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160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

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160

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sure

[mm

Hg]

60

80

100

120

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160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





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20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

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]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

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20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

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]

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(a) 25

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all S

hear

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Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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Wal

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tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018














120588119891 120597119906119894120597119905 + 120588119891119906119895 120597119906119894120597119883119895 = minus 120597120597119883119894119901 + 120597120597119883119895 (120583120597119906119894120597119883119895) + 119865ext

119894 (2)




1198782119894119895)119886

)(119899minus1)119886]

] (3)


120597119906119894120597119883119894 = 0 (4)

120588119891119863119906119894119863119905 = minus 120597119901120597119883119894 +120597120591119894119895120597119883119895 + 119865ext

119894 (5)


(6)

120588119891119863120596119863119905 = minus 120572120583119905 120591119894119895120597119906119894120597119883119895 minus 1205881198911205731205962

+ 120597120597119883119895 ((120583 + 120583119905120590120596) 120597120596120597119883119895)+ 2120588119891 (1 minus 1198911) 1205901205962120596 120597119896120597119883119895

120597120596120597119883119895 (7)

with the operator

119863119863119905 equiv 120597120597119905 + 119906119895 120597120597119883119895 (8)


120591119894119895 = 2120583(119878119894119895 minus 13120575119894119895 120597119906119896120597119883119896) + 120591119894119895 (9)


119906119894 = 119906119894 + 1199061015840119894 (10)




120591119894119895 = 2120583119905 (119878119894119895 minus 13120575119894119895 120597119906119896120597119883119896) minus 23120588119891119896120575119894119895 (11)


120583119905 = 120588119891119896120596 1max 1120572lowast (1198781198912) (1198861120596) (12)




5001205831205881199102120596 41205881198961205901205962119863+1205961199102 (13)




] (15)


120572 = 11989111205721 + (1 minus 1198911) 1205722120590119896 = 11989111205901198961 + (1 minus 1198911) 1205901198962 (16)

120573 = 11989111205731 + (1 minus 1198911) 1205732120590120596 = 11989111205901205961 + (1 minus 1198911) 1205901205962 (17)




119865ext119894 = 119899119901sum119901=1

(18120583119862119863ReV241205882119901 (119906119894 minus 119906119894119901)) 119901Δ119905 (18)


ReV = 12058811989111988911990110038171003817100381710038171003817u minus k119901

10038171003817100381710038171003817120583 (19)


V119894119901 = 1119873face

119873facesum119901=1

V119894119901 (20)



119862119863 = 1205721 + 1205722ReV

+ 1205723Re3V

(21)

for

1205721 1205722 1205723

=


(22)


1205721 = 59 1205731 = 340

1205901198961 = 0851205901205961 = 051205722 = 0441205732 = 00828

1205901198962 = 11205901205962 = 0856

(23)




119889x119901119889119905 = k119901 (24)

119898119901 119889k119901119889119905 = F119901119891 + F119901119888 (25)

119868119901 119889Ω119901119889119905 = 1205881199012 (1198891199012 )5 119862120596Υ (26)




F119863 = u minus k119901120591119901 (28)

F119904 = 2120581radic120592120588119891119889119894119895120588119901119889119901 (119889119897119896119889119896119897)14 (u minus k119901) (29)

F119877119871 = 1205881198912 119860 119894119901119862119877119871 VΥ (V times Υ) (30)


119889119894119895 = 12 ( 120597V119894120597119909119895 +120597V119895120597119909119894) (31)


119862119877119871 = 045+ (Re120596


119862120596 = 645radicRe120596+ 321Re120596

Re120596 = 120588119891 Ω 11988921199014120583

(33)


F119901119902 = (119870120575 minus 120574 (k119901119902 sdot e119901119902)) e119901119902 (34)


120575 = 10038171003817100381710038171003817x119902 minus x11990110038171003817100381710038171003817 minus (119903119901 + 119903119902) (35)

k119901119902 = k119902 minus k119901 (36)



120574 = 2119898119901119902 ln 120578119905119888 (38)

119898119901119902 = 119898119901119898119902119898119901 + 119898119902 (39)

119905119888 = radic(1205872 + ln2120578) 119898119901119902119870 (40)



119876 (119905) = 119876 + 4sum119899=1

119886119876119899 cos (119899119908119905) + 119887119876119899 sin (119899119908119905) (41)

119901 (119905) = 119901 + 4sum119899=1

119886119901119899 cos (119899119908119905) + 119887119901119899 sin (119899119908119905) (42)


u = u = 119880119899 (119905) k119901 = 119880119899 (119905 = 02119904) (43)


for

119880119899 (119905) = 119876 (119905)119860 (44)


119896 = 32 (119880119899 (119905) 120589)2 (45)

120596 = 0164311989615119871 (46)






2725 mm

75 of stenosis









60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(a) 119905 = 065 119904

60

80

100

120

140

Pres

sure

[mm

Hg]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


10 20 30 40 50 60 700Arc-length [mm]

60

80

100

120

140

Pres

sure

[mm

Hg]

(a) 119905 = 065 119904

60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


Inlet

LCX Outlet 1

LAD Outlet 2

LAD Outlet 3

LCA

LCX

LAD











25

75

50


000

2000

4000

6000

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10000

12000

14000

Pres

sure

[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]




















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

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]

0

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s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

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Wal

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20 40 60 80 100 1200Arc-length [mm]

0

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]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

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20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

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20 40 60 80 100 1200Arc-length [mm]

0

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Wal

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t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

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0

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all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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Wal

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]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018











120591119894119895 = 2120583119905 (119878119894119895 minus 13120575119894119895 120597119906119896120597119883119896) minus 23120588119891119896120575119894119895 (11)


120583119905 = 120588119891119896120596 1max 1120572lowast (1198781198912) (1198861120596) (12)




5001205831205881199102120596 41205881198961205901205962119863+1205961199102 (13)




] (15)


120572 = 11989111205721 + (1 minus 1198911) 1205722120590119896 = 11989111205901198961 + (1 minus 1198911) 1205901198962 (16)

120573 = 11989111205731 + (1 minus 1198911) 1205732120590120596 = 11989111205901205961 + (1 minus 1198911) 1205901205962 (17)




119865ext119894 = 119899119901sum119901=1

(18120583119862119863ReV241205882119901 (119906119894 minus 119906119894119901)) 119901Δ119905 (18)


ReV = 12058811989111988911990110038171003817100381710038171003817u minus k119901

10038171003817100381710038171003817120583 (19)


V119894119901 = 1119873face

119873facesum119901=1

V119894119901 (20)



119862119863 = 1205721 + 1205722ReV

+ 1205723Re3V

(21)

for

1205721 1205722 1205723

=


(22)


1205721 = 59 1205731 = 340

1205901198961 = 0851205901205961 = 051205722 = 0441205732 = 00828

1205901198962 = 11205901205962 = 0856

(23)




119889x119901119889119905 = k119901 (24)

119898119901 119889k119901119889119905 = F119901119891 + F119901119888 (25)

119868119901 119889Ω119901119889119905 = 1205881199012 (1198891199012 )5 119862120596Υ (26)




F119863 = u minus k119901120591119901 (28)

F119904 = 2120581radic120592120588119891119889119894119895120588119901119889119901 (119889119897119896119889119896119897)14 (u minus k119901) (29)

F119877119871 = 1205881198912 119860 119894119901119862119877119871 VΥ (V times Υ) (30)


119889119894119895 = 12 ( 120597V119894120597119909119895 +120597V119895120597119909119894) (31)


119862119877119871 = 045+ (Re120596


119862120596 = 645radicRe120596+ 321Re120596

Re120596 = 120588119891 Ω 11988921199014120583

(33)


F119901119902 = (119870120575 minus 120574 (k119901119902 sdot e119901119902)) e119901119902 (34)


120575 = 10038171003817100381710038171003817x119902 minus x11990110038171003817100381710038171003817 minus (119903119901 + 119903119902) (35)

k119901119902 = k119902 minus k119901 (36)



120574 = 2119898119901119902 ln 120578119905119888 (38)

119898119901119902 = 119898119901119898119902119898119901 + 119898119902 (39)

119905119888 = radic(1205872 + ln2120578) 119898119901119902119870 (40)



119876 (119905) = 119876 + 4sum119899=1

119886119876119899 cos (119899119908119905) + 119887119876119899 sin (119899119908119905) (41)

119901 (119905) = 119901 + 4sum119899=1

119886119901119899 cos (119899119908119905) + 119887119901119899 sin (119899119908119905) (42)


u = u = 119880119899 (119905) k119901 = 119880119899 (119905 = 02119904) (43)


for

119880119899 (119905) = 119876 (119905)119860 (44)


119896 = 32 (119880119899 (119905) 120589)2 (45)

120596 = 0164311989615119871 (46)






2725 mm

75 of stenosis









60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(a) 119905 = 065 119904

60

80

100

120

140

Pres

sure

[mm

Hg]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


10 20 30 40 50 60 700Arc-length [mm]

60

80

100

120

140

Pres

sure

[mm

Hg]

(a) 119905 = 065 119904

60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


Inlet

LCX Outlet 1

LAD Outlet 2

LAD Outlet 3

LCA

LCX

LAD











25

75

50


000

2000

4000

6000

8000

10000

12000

14000

Pres

sure

[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]




















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

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450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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450

Wal

l She

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]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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200

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450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

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Wal

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s [Pa

]

0

50

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Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

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Wal

l She

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s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

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350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

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450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

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450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

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450

Wal

l She

ar S

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s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

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200

250

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350

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450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018










119889x119901119889119905 = k119901 (24)

119898119901 119889k119901119889119905 = F119901119891 + F119901119888 (25)

119868119901 119889Ω119901119889119905 = 1205881199012 (1198891199012 )5 119862120596Υ (26)




F119863 = u minus k119901120591119901 (28)

F119904 = 2120581radic120592120588119891119889119894119895120588119901119889119901 (119889119897119896119889119896119897)14 (u minus k119901) (29)

F119877119871 = 1205881198912 119860 119894119901119862119877119871 VΥ (V times Υ) (30)


119889119894119895 = 12 ( 120597V119894120597119909119895 +120597V119895120597119909119894) (31)


119862119877119871 = 045+ (Re120596


119862120596 = 645radicRe120596+ 321Re120596

Re120596 = 120588119891 Ω 11988921199014120583

(33)


F119901119902 = (119870120575 minus 120574 (k119901119902 sdot e119901119902)) e119901119902 (34)


120575 = 10038171003817100381710038171003817x119902 minus x11990110038171003817100381710038171003817 minus (119903119901 + 119903119902) (35)

k119901119902 = k119902 minus k119901 (36)



120574 = 2119898119901119902 ln 120578119905119888 (38)

119898119901119902 = 119898119901119898119902119898119901 + 119898119902 (39)

119905119888 = radic(1205872 + ln2120578) 119898119901119902119870 (40)



119876 (119905) = 119876 + 4sum119899=1

119886119876119899 cos (119899119908119905) + 119887119876119899 sin (119899119908119905) (41)

119901 (119905) = 119901 + 4sum119899=1

119886119901119899 cos (119899119908119905) + 119887119901119899 sin (119899119908119905) (42)


u = u = 119880119899 (119905) k119901 = 119880119899 (119905 = 02119904) (43)


for

119880119899 (119905) = 119876 (119905)119860 (44)


119896 = 32 (119880119899 (119905) 120589)2 (45)

120596 = 0164311989615119871 (46)






2725 mm

75 of stenosis









60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(a) 119905 = 065 119904

60

80

100

120

140

Pres

sure

[mm

Hg]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


10 20 30 40 50 60 700Arc-length [mm]

60

80

100

120

140

Pres

sure

[mm

Hg]

(a) 119905 = 065 119904

60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


Inlet

LCX Outlet 1

LAD Outlet 2

LAD Outlet 3

LCA

LCX

LAD











25

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Pres

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[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]




















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

80

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Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

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160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

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160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

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Pres

sure

[mm

Hg]

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80

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160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

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160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

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160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

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160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

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160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

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Wal

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]

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Wal

l She

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20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

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Wal

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t = 065 s t = 386 s

(b) 50

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Wal

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20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

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20 40 60 80 100 1200Arc-length [mm]

0

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Wal

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]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

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Wal

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]

0

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all S

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Str

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Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

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Wal

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]

20 40 60 80 100 1200Arc-length [mm]

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450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









for

119880119899 (119905) = 119876 (119905)119860 (44)


119896 = 32 (119880119899 (119905) 120589)2 (45)

120596 = 0164311989615119871 (46)






2725 mm

75 of stenosis









60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(a) 119905 = 065 119904

60

80

100

120

140

Pres

sure

[mm

Hg]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


10 20 30 40 50 60 700Arc-length [mm]

60

80

100

120

140

Pres

sure

[mm

Hg]

(a) 119905 = 065 119904

60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


Inlet

LCX Outlet 1

LAD Outlet 2

LAD Outlet 3

LCA

LCX

LAD











25

75

50


000

2000

4000

6000

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12000

14000

Pres

sure

[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]




















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

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Pres

sure

[mm

Hg]

60

80

100

120

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160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

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160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

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160

Pres

sure

[mm

Hg]

60

80

100

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160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

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140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

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160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

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Wal

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]

0

50

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Wal

l She

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tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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Wal

l She

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]

20 40 60 80 100 1200Arc-length [mm]

0

50

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Wal

l She

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tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

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Wal

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ar S

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s [Pa

]

0

50

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Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

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Wal

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]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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200

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450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

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Wal

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s [Pa

]

0

50

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150

200

250

300

350

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450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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Wal

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s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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250

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350

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450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(a) 119905 = 065 119904

60

80

100

120

140

Pres

sure

[mm

Hg]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


10 20 30 40 50 60 700Arc-length [mm]

60

80

100

120

140

Pres

sure

[mm

Hg]

(a) 119905 = 065 119904

60

80

100

120

140Pr

essu

re [m

mH

g]

10 20 30 40 50 60 700Arc-length [mm]

(b) 119905 = 386 119904


Inlet

LCX Outlet 1

LAD Outlet 2

LAD Outlet 3

LCA

LCX

LAD











25

75

50


000

2000

4000

6000

8000

10000

12000

14000

Pres

sure

[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]




















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

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200

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Wal

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]

0

50

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Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

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Wal

l She

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s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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Wal

l She

ar S

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]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

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Wal

l She

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]

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50

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Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

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100

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Wal

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]

20 40 60 80 100 1200Arc-length [mm]

0

50

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450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

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Wal

l She

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]

0

50

100

150

200

250

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450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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450

Wal

l She

ar S

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s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

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200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









25

75

50


000

2000

4000

6000

8000

10000

12000

14000

Pres

sure

[mm

Hg]

050 100 150 200 250 300 350 400 450000Time [s]




















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018
















3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









3

1

2

(a) Cutting plane

(a)

(b)

(c)







Time = 386 [s]




20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

20 40 60 80 100 1200Arc-length [mm]

60

80

100

120

140

160

Pres

sure

[mm

Hg]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(c) 75





20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018









20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(a) 25

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

0

50

100

150

200

250

300

350

400

450W

all S

hear

Str

ess [

Pa]

20 40 60 80 100 1200Arc-length [mm]

t = 065 s t = 386 s

(b) 50

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

20 40 60 80 100 1200Arc-length [mm]

0

50

100

150

200

250

300

350

400

450

Wal

l She

ar S

tres

s [Pa

]

t = 065 s t = 386 s

(c) 75









Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018















Data Availability






References


































Journal of












Journal of







Volume 2018






Volume 2018






































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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