Characterization of Hemodynamic Forces Induced by Mechanical Heart Valves: Reynolds vs. Viscous...

Characterization of Hemodynamic Forces Induced by Mechanical Heart

Valves: Reynolds vs. Viscous Stresses

LIANG GE,1,2 LAKSHMI P. DASI,3 FOTIS SOTIROPOULOS,1 and AJIT P. YOGANATHAN3

1St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55414, USA;2Department of Surgery, University of California, San Francisco, CA 94143, USA; and 3Department of Biomedical Engineering,

Georgia Institute of Technology, Atlanta, GA 30332, USA

(Received 14 August 2007; accepted 16 November 2007)

Abstract—Bileaflet mechanical heart valves (BMHV) arewidely used to replace diseased heart valves. Implantation ofBMHV, however, has been linked with major complications,which are generally considered to be caused by mechanicallyinduced damage of blood cells resulting from the non-physiological hemodynamics environment induced byBMHV, including regions of recirculating flow and elevatedReynolds (turbulence) shear stress levels. In this article, weanalyze the results of 2D high-resolution velocity measure-ments and full 3D numerical simulation for pulsatile flowthrough a BMHV mounted in a model axisymmetric aortato investigate the mechanical environment experienced byblood elements under physiologic conditions. We show thatthe so-called Reynolds shear stresses neither directly con-tribute to the mechanical load on blood cells nor is a propermeasurement of the mechanical load experienced by bloodcells. We also show that the overall levels of the viscousstresses, which comprise the actual flow environment expe-rienced by cells, are apparently too low to induce damage tored blood cells, but could potentially damage platelets. Themaximum instantaneous viscous shear stress observedthroughout a cardiac cycle is <15 N/m2. Our analysis isrestricted to the flow downstream of the valve leaflets andthus does not address other areas within the BMHV wherepotentially hemodynamically hazardous levels of viscousstresses could still occur (such as in the hinge gaps andleakage jets).

Keywords—Reynolds shear stress, Viscous shear stress,

Blood cell damage, Prosthetic heart valves, Turbulence,

Particle image velocimetry (PIV), Computational fluid

dynamics (CFD).

BACKGROUND

Bileaflet mechanical heart valves (BMHV) arewidely used to replace diseased heart valves. All cur-rent designs provide satisfactory overall hemodynamicperformance and are very durable. Implantation of

BMHV, however, has been linked with major compli-cations, including increased occurrence of blood clotsand embolism events, anti-coagulant related hemolysis,and tissue overgrowth.53 Two of these major compli-cations, namely hemolysis and blood clots, are directlyrelated to mechanical damage of red blood cells(RBCs) and platelets, which are collectively referred toas blood cells. Over the years a great deal of efforts hasbeen dedicated to understanding the pathway from theBMHV induced hemodynamic stresses to blood celldamage. These efforts can be loosely categorized intostudies aimed at: (1) understanding blood cell damageunder controlled mechanical force environment; and(2) investigating the details of the mechanical envi-ronment induced by BMHV.

Mechanically induced blood cell damage researchstarted back in the 1960s, with the earliest effortsfocusing on RBC damage. Blood cells transportedthrough medical devices experience pressure and shearforces as well as impact and friction forces from theirinteraction with solid surfaces. It has been shown thatthe shear force could be a major source of mechani-cally induced RBC damage21 and a great deal of re-search has focused on the understanding of thisdamage mechanism. RBCs have been shown to changefrom the biconcave shape under no flow condition toellipsoidal shape when exposed to simple shear flows15

undergoing a fluid-drop like transition.44 Furtherincrease of the shear force level could damage the cell’smembrane and cause it to rupture leading to therelease of the cell’s internal contents into the sur-rounding fluid environment.5 Typically shear-inducedRBC damage studies have been carried out by expos-ing RBCs to controlled shear-inducing flow fields,usually generated by rotational viscometers21 or sub-merged jets,11,43 and the degree of RBC damage hasbeen monitored by measuring the concentration ofplasma free hemoglobin.43 It has been found that thereexists a shear stress threshold under which there is no

Address correspondence to Liang Ge, Department of Surgery,

University of California, San Francisco, CA 94143, USA. Electronic

mail: [email protected]

Annals of Biomedical Engineering (� 2007)

DOI: 10.1007/s10439-007-9411-x

� 2007 Biomedical Engineering Society

detectable RBC damage. This threshold is reported tobe here between 150 and 400 N/m2 under long exposuretime (~minutes). When this threshold is exceeded thedegree of RBC damage is found to be a function ofshear stress and exposure time.21,30 Early findings withregard to RBC damage under shear conditions weresummarized in Leverett et al.30 and later experimentscarried out by Sallam and Hwang43 and Paul et al.40

added some new insights into the shear-induced RBCdamage. The pathway starting from mechanical stim-ulation on platelets to the final formation of blood clotsis much more complicated than that of RBC lysis. Itinvolves the activation of platelets, aggregation, andadhesion, and is known to be dependent on the level ofshear stress and exposure time, surface property, as wellas the chemical property of surrounding blood plasma(for example, the existence of platelet agonists such asADP or epinephrine is known to reduce the shear stressthreshold).20 Despite these complexities, efforts havebeen made to understand the response of platelets toshear force in controlled shear experimental facilities,similar to those used for the RBC damage experiments.It has been found that the platelets are much moresensitive to shear than RBCs and shear stress greaterthan 10 N/m2 has been shown to be able to causeplatelet lysis. Similar as RBCs, there also exists athreshold shear stress level under which no detectableplatelet lysis can be measured and the level of shearstress threshold is dependent on exposure time.20,47

Following the illustration of the strong linkagebetween mechanical force and blood cell damages, alarge body of investigations have been carried out toinvestigate the shear force environment induced byBMHVs. Early investigations used experimental mea-surements to study the flow field both in vitro4,10,32,54,55

and in vivo.38,39,50 Experimental techniques used formeasuring the flow field include laser Doppler velo-cimeter (LDV)54 and PIV.32 In more recent years,numerical simulations are also emerging as a powerfultool for investigating BMHV flow fields.3,7,12,26 A de-tail review of the related findings is provided in Yog-anathan et al.53

A large number of these studies focused on the ef-fects of turbulence, which are known to be detrimentalto blood cells.46 The mechanical load imparted by thefluid flow on blood elements is typically quantified interms of the components of the shear stress tensor. Inlaminar flow the concept of shear stress is straight-forward to define as the term refers uniquely to theviscous stress tensor, which depends on the spatialderivatives of the instantaneous velocity field. In tur-bulent flows, however, the concept of shear stress needsto be defined more carefully as the term could refer toeither the instantaneous viscous stress or the statisticalstress-like quantity known as the Reynolds stress

tensor. This duality in the concept of shear stress inturbulent flows has often been the source of confusionand misunderstanding in the blood damage relatedliterature. For example, Forstrom11 investigated RBCdamage by exposing blood cells to a submerged jetflow and reported the shear stress threshold for RBCsto be equal to 5,000 N/m2. In this study, the shearstress threshold was quantified in terms of the Rey-nolds shear stress. Sutera and Mehrjardi48 measuredthe threshold of RBC damage using a Couette vis-cometer and reported a threshold of about 150 N/m2.In this work, however, the shear stress is measured interms of the viscous shear stress acting at the cylindersurface. Sallam and Hwang43 studied the relationshipbetween Reynolds shear stress and RBC damage usinga submerged jet and reported the threshold Reynoldsshear stress of about 400 N/m2 (this value was subse-quently considered to be under evaluated17,34). Insummarizing the results of these early studies, Leverettet al.30 collected all available shear stress thresholdsinto one figure lumping together both the Reynoldsshear stress and viscous shear stress values under thecommon name ‘‘shear stress.’’ They concluded that thethreshold shear stress level is strongly dependent onexposure time and that the RBCs can sustain very highlevels of shear stress (these very high levels of shearstress being those obtained from experiments using theReynold shear stress as a measure) if the exposure timeis very short. Similar practices unifying the two shearstress measures with no distinction have also beenadopted in studies focusing on shear-induced plateletdamage. For example, Bernstein et al.2 used an appa-ratus similar to that one used by Forstrom11 to gen-erate a shear flow field and monitored the shear-induced platelet damage. A shear stress threshold forplatelet damage is reported at the level of 105 N/m2 interms of the Reynolds shear stress. A collection ofavailable shear stress thresholds for platelet was re-ported in Hellums.20 The conclusion of this article isthat platelets, similar to RBCs, can survive exposure tovery high shear stress levels if the exposure time isextremely short (in the order of 10-5 s). Once again, theshear stress here is the Reynolds shear stress.

The Reynolds shear stress has also been frequentlyused as equivalent to the viscous shear stress innumerical simulations aimed at estimating blood celldamage potential. An empirical equation based on thedata obtained in Wurzinger et al.52 links the RBCdamage to shear stress and exposure time was devel-oped in Giersiepen et al.14:

LRBC ¼ 3:62� 10�5t0:785exp s2:416 ð1Þ

where the RBC damage index LRBC is measured asplasma free hemoglobin level, texp is exposure time (s)

GE et al.

and s shear stress (N/m2). The platelet damage is afunction of exposure time and viscous shear stress as

Lpl ¼ 3:66� 10�6t0:77exp s3:075 ð2Þ

where Lpl is the level of cytoplasm enzyme (LDH) re-leased by platelets. The shear stress term s in the aboveequations includes both the viscous and Reynoldsshear stress.16 Mathematical modeling of blood celldamage using Reynolds shear stress is widely used inongoing research and can be found both from experi-mental measurements32 and numerical simula-tions3—see Goubergrits16 for a recent review.

Reynolds shear stress is often the main focus ofBMHV hemodynamics investigations with the inten-tion to understand the potential blood cell damagescaused by the valve induced flow turbulence. Forexample, Yoganathan et al.54 measured the flow fielddownstream of Bjork-Shiley disc valve and investi-gated the total shear stress, defined as s ¼ sl þ st wheresl is the viscous shear stress and st is the Reynoldsshear stress. Figlio and Mueller10 used LDV to mea-sure the velocity profile and Reynolds stress in thevicinity of four different types of disc or caged ballvalves. Chandran et al.4 compared the Reynolds nor-mal stress induced by caged ball and tilting disc valvesunder both pulsatile and steady flow conditions.Yoganathan et al.55 employed 2D LDV to measureReynolds shear stress distribution near St. Jude Med-ical bileaflet valve as well as Medtronic-Hall andBjork-Shiley valves and reported the maximum Rey-nolds shear stress ranging from 120 to 200 N/m2.Nygaard et al.39 carried out in vivo measurements ofReynolds shear stress distributions downstream threedifferent MHVs implanted in the aortic position,including SJM, CarboMedics and Starr-Edwardsvalves and reported maximum normal Reynolds shearstress of 63 N/m2 for SJM, 72 N/m2 for the Carbo-Medics and 56 N/m2 for Starr-Edwards valve, respec-tively. Lim et al.31 used PIV to measure the Reynoldsshear stress distribution downstream of four differentprosthetic heart valve, including a Starr-Edwardscaged ball valve, a Bjork-Shiley Convexo–Concavetilting disc valve, a St. Vincent porcine tissue and a St.Vincent Meditech tilting disc valve under steadyincoming flow conditions. The same technique waslater applied to measure the Reynolds shear stressdistribution downstream of a St. Vincent GDII porcinexenograft valve under pulsatile flow conditions.32 Inthese investigations, the measured Reynolds stresslevels were compared with the reported Reynolds shearstress threshold to discuss possible damage on bloodcells. Recently, when clinical trials of the MedtronicParallel BMHV revealed a propensity for thrombusformation in the hinge region, researchers turned their

attention to the understanding of turbulence within thehinge region9 as well as in the near field of the leakagejet.45,50,51 Maximum Reynolds shear stress levels ashigh as 600 N/m2 were reported in these regions.9 Thetypical observed Reynolds shear stress levels in thesestudies is rather close to, albeit less than, the reportedthreshold for RBC damage measured by Sallam andHwang.43 It was thus concluded that stress levels in thehinge and leakage jet regions could cause subleathaldamage to RBCs and platelets. Following these stud-ies, Reynolds shear stress has been established as animportant metric for evaluating the performance ofvarious MHV designs18,27 as well as the performanceof a given MHV design with regard to implantationorientation.29 Reynolds shear stress is in fact beingconsidered today to be the key quantity that needs tobe minimized in order to optimize MHV designs.53

As we alluded to above, the main difficulty inquantifying the mechanical environment experiencedby blood elements in terms of either viscous or Rey-nolds shear stresses stems from the fundamentallydifferent nature of these two quantities. The viscousshear stress measures the viscous shear force per unitarea experienced by fluid element. In this sense, theviscous shear stress measures the strength of the ‘‘real’’physical force exert on blood cells from the deformedblood plasma. The Reynolds shear stress, on the otherhand, is a statistical quantity that has no direct link toany physical forces. In this sense, the Reynolds stress isa pseudo-stress like term23 but not a ‘‘real’’ quantity8;it is rather a mathematical artifact, which is quiteeffective in the statistical description of a turbulentflow field. This important point has been previouslyrecognized by Jones25 and discussed as recently as inQuinlan and Dooley41 and Quinlan.42 To facilitate ourdiscussion, let us emphasize again this subtle differencebetween Reynolds and viscous shear stresses by con-sidering the equations governing blood flow. Assumingthat blood can be considered to be a Newtonian fluid,its motion, regardless of whether it occurs under lam-inar or turbulent conditions, is governed by theinstantaneous, incompressible Navier–Stokes equa-tions, which read in tensor notation as follows:

@ui@tþ @uiuj

@xj¼ � 1

q@p

@xiþ 1

q@sij@xj

ð3Þ

where ui (i ¼ 1; 2; 3) are the Cartesian components ofthe instantaneous velocity field and p is the instanta-neous pressure. sij is the viscous stress tensor defined as:

sij ¼ l@ui@xjþ @ui@xj

� �:

As it is evident in the above equation, the only viscousforce term experienced at any instant in time by fluid

Reynolds vs. Viscous Stresses

elements per unit fluid volume is due to the gradientsof viscous shear stress tensor (or equivalently, viscousshear force). The Reynolds shear stress term does notappear in the governing equations until one applies theReynolds decomposition for the velocity and pressurefields and statistically averages Eq. (3). The resultingequations are known as the Reynolds-averaged Na-vier–Stokes (RANS)24 and read in Cartesian tensornotation as follows:

@ui@tþ @uiuj

@xj¼ � 1

q@p

@xiþ 1

q@

@xjsij � qu0iu

0j

h ið4Þ

where ð�Þ represents the Reynolds averaging operation(time-averaging, ensemble-averaging, etc.), ui and p arethe Reynolds-averaged velocity components and pres-sure, respectively, and u0i are the fluctuating compo-nents above the averaged quantities. The term qu0iu

0j

represents the components of the Reynolds shear stresstensor. As stated above this term is a mathematicalartifact because it did not exist in the original governingequations but arose as the result of Reynolds-averagingprocedure described above. The terms of the Reynoldsstress tensor quantify in a statistical sense the transportof turbulent fluctuating momentum by turbulentvelocity fluctuations—qu0iu

0j for instance is the transport

of the ith component of fluctuating momentum per unitvolume ðqu0iÞ by velocity fluctuations of the jth velocitycomponent. These terms have the dimension of forceper unit area and as such can be used to provide astatistical measure of the influence on the averagedvelocity field due to turbulent fluctuations at a givenposition in space. Therefore, these terms do not quan-tify any real force. Furthermore, the numerical valuesof the Reynolds shear stress terms are strongly depen-dent upon the averaging method, i.e., time-averaging,ensemble-averaging, phase-averaging, etc. Indeed, in-stead of using time-averaging, one can apply spatialaveraging (more frequently referred as filtering in theLarge Eddy Simulation literature) to Eq. (3), whichleads to the filtered Navier–Stokes equations. TheReynolds shear stress terms do not appear in the filteredNavier–Stokes equation; instead, a new term referredto as the subgrid scale stress (SGS) appears on theequation. Regardless of the averaging procedure used,however, neither Reynolds nor subgrid scale stresseshave any link to physical forces. Therefore, re-empha-sizing and somewhat rephrasing the discussions inJones25 and Quinlan,42 the following significant con-clusion emerges form the above discussion: sincemechanically induced blood cell damage is initiated bymechanical forces and Reynolds shear stress is not re-lated to any physical forces experienced by cells, theReynolds shear stress should not be a parameter that isused to directly quantify mechanically induced bloodcell damage.

The above conclusion naturally raises the followingvery important question: what is the appropriatemeasure that quantifies the effect of mechanical forcesdue to turbulence on blood cells? To help answer thisquestion we should point out that blood cells are inessence suspended particles traveling with blood plas-ma. The mechanical force environment experienced bysuspended particles in a turbulent flow field is knownto be very complex and some useful quantitative in-sights can be obtained using the theory developed byHinze.22 According to this theory, the mechanical forceexerted by the turbulent flow on suspended blood cellsconsists of contributions from: (1) instantaneous vis-cous shear force; and (2) dynamic pressure. Theinstantaneous viscous shear force results from the fluidviscosity and the straining and deformation of fluidelements by the instantaneous flow, while the dynamicpressure contribution is the result of the fluctuatingturbulent eddies. The viscous shear force can be cal-culated in a straightforward manner from the viscousshear stress tensor. The dynamic pressure force on theother can be estimated according to the Hinze model asfollows:

P ¼ quðdÞ2 ð5Þ

where u(d) is the velocity difference over a character-istic length scale, d, of the blood cell. It is apparentfrom this equation that the dynamic pressure termdepends not only on the intensity of turbulent fluctu-ations in the vicinity of the blood cell, but also on therelationship between the size of blood cell and the so-called Kolmogorov scale—i.e., the scale of the smallestturbulence eddies possible in a given flow or equiva-lently the scale at which the molecular viscosity of thefluid becomes effective in dissipating energy. Unlessthe size of the blood cell is comparable to or largerthan the Kolmogorov scale, the dynamic pressure isnot likely to have a major role in damaging the sus-pended particles. Such Kolmogorov scale dependentcell damage in turbulent flow fields have been previ-ously observed experimentally in Croughan et al.,6

Kunas and Papoutsakis,28 and McQueen et al.36 TheKolmogorov scale in BMHV flows has been previouslyestimated in experimental works and has been found tobe larger than both the diameter of red blood cells(about 10 lm) and platelets (~2 lm). For example, Liuet al.33 reported the Kolmogorov scale is larger than20 lm; Ellis et al.9 estimated the Kolmogorov scale tobe about 7 lm; a follow-up investigation carried outby Travis et al.,50 however, showed that the valuereported by Ellis et al. underestimated the Kolmogo-rov scale due to cycle-to-cycle flow variation andreported values of 20–70 lm when the cyclic variationeffects are removed. It is, therefore, reasonable toassume that the dynamic pressure force arising from

GE et al.

turbulent fluctuations is not important when consid-ering potential cell damage and the dominantmechanical force controlling the damage of blood cellshere should be the viscous shear force.

This article seeks to carry out a comprehensiveevaluation of the viscous and Reynolds stresses in theflow field downstream of a clinical quality BMHV inorder to: (1) shed new insights into the real mechanicalforce environment relevant to blood cell damage in theflow field downstream of the leaflets of a BMHV; and(2) elucidate the source of Reynolds shear stresses inBMHV flows. The flow fields that are analyzed areobtained by high resolution 2D in vitro PIV measure-ments and fully 3D CFD simulations. The article isorganized as follows. In the second section, we presenta brief overview of the most important flow physics inthe wake region of a St. Jude Medical 23 mm Regentbileaflet valve. This is followed by in depth discussionof the issue of Reynolds and viscous stress componentsevaluation from measured and simulated velocity field.In the third section, we provide detail comparisons andin depth analysis of the difference between Reynoldsand viscous stress fields and illustrate the importanceof instantaneous viscous stress with regard to theconsideration of mechanical force environment in-duced by medical devices such as BMHVs. Finally, inthe last section of the article we summarize the majorfindings in this article and discuss future explorationwith regard to flow environment induced by BMHVs.

OVERVIEW OF FLOW PHYSICS AND

METHODS

We have recently carried out a detailed experimentaland computational investigation of the flow physicsinduced by a BMHV.7 A St. Jude Medical 23 mmRegent bileaflet valve is used for the investigation. Thevalve is inserted into a modeled straight aorta geome-try, which employs an axisymmetric expansion tomodel sinus geometry. Both phase-locked digital par-ticle image velocimetry (DPIV) measurement and CFDsimulations were used to investigate the complex vor-tical dynamics.

In the experiment, a physiological flow waveformshown as the solid line in Fig. 1 is fed into the flowloop (see Dasi et al.7 for details). The cycle duration isset to 860 ms. The mean flow rate was adjusted to4.5 l/min with a peak flow rate of 24.6 l/min and aforward flow duration of 340 ms. The blood is mod-eled as a Newtonian fluid with a mixture of sodiumiodide, glycerin, and water that yielded a solutionwhose kinematic viscosity of 3:5� 10�6 m2/s. The flowwas seeded using fluorescent polymeric particles(diameters 1–20 lm). The PIV system consisted of a

pair of pulsed Nd:YAG lasers. The PIV images werecaptured using a CCD camera (Model 1101MRPO,Lavision, Germany) of 1600� 1200 pixel resolution,with a spatial resolution of 16.6 lm/pixel. The mea-surements were phase-locked to 43 time instancesregularly spaced by 20 ms over the entire cardiac cycle.For each phase, 250 measurement samples was ac-quired. Laser pulse separation was optimized for eachof the 43 phases to yield sufficient pixel displacementsto maintain the error in velocity vectors at <5%error.7 The standard error in the mean velocity wasfound to be below 1% indicating good statisticalconvergence of the mean.

A full 3D CFD simulation was also carried out intandem with the experiment (see Dasi et al.7 for de-tails). The geometry of the model aorta and BMHVleaflets used in the simulation are identical to thoseused in the experiment. The only geometrical featurethat is not included in the current CFD simulation isthe hinge mechanism connecting the leaflets to valvehousing. The numerical method employed to simulatethe flow is the recently developed sharp-interfaceCurvilinear/Immersed-boundary (CURVIB) methodof Ge and Sotiropoulos.13 The basic idea of themethod is, as shown in Fig. 2, to discretize the aortalumen with a body-fitted curvilinear mesh, while thecomplicated geometry of valve housing as well asleaflets are discretized with unstructured triangularmesh. The valve housing as well as the moving leafletsare handled as sharp-interface immersed boundaries. Agrid system consisting of about 10 million grid nodes(201� 201� 241 in the two spanwise and streamwisedirections, respectively) is used to discretize the aorticlumen. The grid is clustered near the valve, where richvortical dynamics was observed in the experiments.The valve housing and leaflet surfaces are discretizedwith triangular meshes consisting of about 2100 nodes.

time (ms)

Flo

w (

liter

/ m

in)

0 200 400 600 800

0

10

20

LaminaTurbulence Turbulence

EA

MA

D

P

FIGURE 1. Incoming flow waveform. Shaded area indicatesthe part of cardiac cycle when the flow is laminar. EA:early acceleration; MA: middle acceleration; P: peak; D:deceleration.


In the simulation, the incoming flow waveform as wellas the leaflet kinematics observed in the experiment arespecified as boundary conditions. The leakage flow isnot accounted for in the simulation. Each cardiac cycleis discretized with 1290 time steps, corresponding to aphysical time step of 0.67 ms. The numerical simula-tion has been carried out for 10 cardiac cycles.

To facilitate discussions in the current article, webriefly summarize here the most important flow phys-ics, which were derived by juxtaposing the measure-ments with the CFD results. For more detail withregard to the observed vorticity dynamics, as well asexperimental and numerical methods used in theinvestigation, readers are referred to Dasi et al.7 andGe and Sotiropoulos.13

1. At the end of each cardiac cycle (before the onset ofthe systolic phase), the flow downstream of thevalve is dominated by the presence of weak and

decaying ‘‘turbulence’’ characterized by a numberof chaotic and seemingly random vortices. Thesevortices are the residual turbulence produced by thevalve from the previous cardiac cycle.

2. At the onset of the systolic phase, the incoming flowrapidly washes out all this weak ‘‘turbulence,’’which indicates that every cardiac cycle is a freshstart in terms of vorticity dynamics.

3. During most of the acceleration phase and beforethe peak flow condition is reached, the flow down-stream to the valve is dominated by two types oforganized vortical structures induced by the valvegeometry: (1) vortices shaped like and shed from theleaflet trailing edges; and (2) a vortex ring inducedby the roll up of aortic wall vorticity upstream ofthe valve into the sinus. Throughout this phase, theflow is clearly well organized, laminar and essen-tially repeatable from cycle to cycle. As the flowcontinues to accelerate, however, these coherentvortical structures are stretched by the flow, inten-sify, undergo complex, 3D instabilities, and give riseto a very complex, albeit laminar and organized,flow.

4. At a time instant close to the peak flow condition,the complex laminar flow transitions almostinstantaneously and in an explosive manner into aturbulent state characterized by multiple small-scalevortical structures. Such a chaotic turbulent flowstatus persists with no signs of laminarizationthroughout the remaining of the systolic and dia-stolic phases until the onset of a new cardiac cycle,with the strength of the chaotic vortices graduallydiminishing during the deceleration phase. Based onthese findings and as illustrated in Fig. 1, we divideherein the cardiac cycle into two regimes: the lam-inar regime and the turbulence regime. Althoughthe exact point when the flow transition occurscannot be determined with precision, it occurssomewhere very close to the peak flow condition.Thus, the laminar flow regime covers most part ofthe acceleration phase while the turbulent regimerefers to the deceleration and diastolic phases.

5. Starting from the second half of the acceleratingphase and throughout the rest of the cardiac cycle,the flow downstream of the leaflets and in the sinuswas found to exhibit strong cycle-to-cycle variation.The sources of such cycle-to-cycle variations wasattributed to the variation of leaflet kinematics (i.e.,the opening and closing of the leaflets are not quiterepeatable from cycle-to-cycle) and the chaoticnature of vortex shedding in the wake area.

In the current article, we analyze the same data setreported in Dasi et al.7 to elucidate the Reynolds andviscous shear stress fields in the wake of the valve

FIGURE 2. Illustration of the CURVIB method used to simu-late the flow past through a BMHV. The background aorta isdiscretized with a curvilinear grid system and the surfaces ofleaflets and valve housing are discretized with triangularmesh. For clarity purpose only every three grid lines areshown here.

GE et al.

leaflets. Note that, as discussed earlier in this sectionthe CFD simulation has only been carried out for 10cardiac cycles and as such statistical analysis of suchsmall ensemble size is not meaningful. Therefore, in thecurrent article all discussions with regard to Reynoldsshear stress is based on data acquired from the phase-locked DPIV measurements. Another issue that wewant to stress here is that, in both the experiments andCFD simulations, the leakage flow passing through thehinge region upon leaflets closing was not investigatedand is not considered in the current paper. Theselimitations, however, will not affect the main conclu-sions of this paper as our focus here is to discuss andcompare the various types of shear stresses vis-a-vizblood cell damage.

Due to the pulsatile unsteady flow nature, the dis-tribution of viscous shear stress and Reynolds shearstress changes rapidly in time throughout a cardiaccycle. To illustrate and analyze the detailed distribu-tion of these two terms, we select in this paper fourtime instants within a cardiac cycle denoted by thesolid dots in Fig. 1. Time instants EA and MA standfor early acceleration and middle acceleration, respec-tively. At both time instants, the wake flow is laminar.Time instant P is the near peak flow point, while timeinstants D is for deceleration phase. For all these laterthree time instants, the wake flow is in the aforemen-tioned turbulent state.

Computation of Reynolds and Viscous Stresses

The Reynolds and viscous stresses need to be com-puted by post-processing either experimental or com-putational data. In this section we discuss the methodswe employ to calculate the various shear stress termsfrom the velocity field and emphasize their overallaccuracy. The Reynolds shear stresses are quantifiedform the experimental data alone while the instanta-neous viscous stresses are computed both from exper-imental and numerical flow fields.

Ensemble-Averaging and Calculation of ReynoldsStresses

Phased-locked DPIV measurements are carried outto obtain the 2D velocity field at the center planeperpendicular to the leaflet axis. We consider 43 phasepoints and at each phase point we acquire 250 samplesto obtain a very detailed description of the 2Dinstantaneous velocity field. To facilitate our sub-sequent discussion we define the coordinate system asshown in Fig. 3, where the x-axis is defined as thestreamwise direction and y-axis is defined as thetransverse direction. The corresponding velocity com-ponents are defined as u,v along the x and y axes,

respectively. We also introduce the notation ul;km;n; vl;km;n

to indicate the lth (l = 1,…250) sample of the phase-locked DPIV velocity measurements at phase k(k ¼ 1; . . . 43) at the m,n spatial location.

At each measured flow phase, each instantaneousvelocity components can be decomposed as follows(for simplicity the grid indices m and n are dropped):

ul;k ¼ hUl;ki þ u0l;k ð6Þ

In the above equation, the ensemble average of themeasured velocity field is defined as:

hUl;ki ¼ 1

N

XNl¼1

ul;k ð7Þ

where N is the total number of samples ðN ¼ 250Þ, u0l;kis the velocity fluctuation about the ensemble-averagedvalue. The Reynolds shear stress terms can be obtainedby taking the correlation of velocity fluctuations. Forexample, qu0u0 is calculated as

qu0u0k ¼ 1

N

XNl¼1

u0l;ku0

l;k ð8Þ

The remaining components of the Reynolds shearstress tensor can be readily obtained via the straight-forward extension of the above procedure.

An important issue that is particular to BMHVflows and is of extreme importance for interpreting thenature of the velocity fluctuations introduced in thedecomposition given by Eq. (8) stems from the factthat BMHV are known to be plagued by cycle-to-cyclevariations. By that we imply fluctuations that are overand above those induced by the chaotic, non-repeat-able nature of the turbulent flow and are primarilylinked to variations in the incoming pulsatile flow33,49

and other causes (see subsequent discussion). In the

FIGURE 3. The onset of Karman-like vortex structure in thewake to the valve. The color shows out-of-plane vorticitycontours. D is the distance between two Karman-like vortexcores and K is the center point of a vortex core.


decomposition given by Eq. (6) above the turbulentand other cycle-to-cycle fluctuations have been lumpedtogether in a single term, u0l;k. An alternative approachthat has been suggested in the literature49 is to separatethe effects of the various contributions to velocityfluctuations by employing the following triple-decom-position:

ul;k ¼ hUl;ki þ ful;k þ u00l;k ð9Þ

where ful;k denotes fluctuations due to cycle-to-cyclevariations while u00l;k represents the fluctuation com-ponent arising from flow turbulence. By comparing theabove equation with Eq. (6) it can be readily seen that:

u0l;k ¼ ful;k þ u00

l;k ð10Þ

In terms of the triple decomposition, the Reynoldsshear stress can be obtained via the ensemble-averagingprocedure shown in Eq. (8) as follows:

qu0u0k ¼ qeueuk þ qu00u00

k ð11Þ

The above decomposition provides a straightforwarddescription of the two different contributions to thetotal Reynolds stresses. Namely those due to cycle-to-cycle variations in the flow and those due to the chaoticnature of turbulence.

Typically the triple decomposition shown in Eq. (9)is accomplished in the frequency domain. A cut-offfrequency, selected based on power-spectrum distri-bution of the measured instantaneous velocity data,can be applied to divide the time series signal into alow-frequency and a high frequency parts (such adivision in frequency domain can be accomplishedeither by using FFT or low-pass filters). The low fre-quency part is then considered to contain the fluctua-tions arising from the flow pulsatility and cycle-to-cycle fluctuation (hUl;ki and ful;k in Eq. (9), respec-tively); the high-frequency part, on the other hand, isconsidered to be due to turbulence fluctuations. Inwhat follows we will illustrate that such a cut-off fre-quency is not applicable for flow problems as compli-cated as BMHV flows.

In Dasi et al.,7 we have identified three sources ofcycle-to-cycle variations in BMHV flows: (1) the cycle-to-cycle variation of leaflet kinematics; (2) the ran-domly distributed vortical structures at the end of eachcardiac cycle; and (3) large-scale organized unsteadi-ness due to coherent vortex shedding downstream ofthe leaflets. As illustrated in Dasi et al.,7 the variationsarising from leaflet kinematics as well as the remainingvortical structure from previous cardiac cycle appearto be quite random, thus cannot be easily categorizedas low or high frequency contributions. The frequencyof the large-scale vortex shedding, on the other hand,can be roughly estimated as follows. As shown in

Fig. 3, the distance D between the two vortex cores isabout 5.8 mm. The instantaneous streamwise velocityat the vortex core uK, measured at point K as shown inFig. 3, is about 0.65 m/s. The frequency of the Kar-man-like vortex street can thus be estimated asf � uK=D ’ 118 Hz. It is practically impossible to tellwith certainty whether such high frequency fluctua-tions arise from turbulence or other sources based onthe frequency information alone. Moreover, the typicalcut-off frequency used before is about 25 Hz.33,49

Using this cut-off frequency value, the variation arisingfrom coherent vortex shedding would be considered tobe the result of flow turbulence. Yet as shown in Fig. 3,the instantaneous flow field at this time instant isclearly laminar. Based on the above analysis, it is notdifficult to conclude that for flows as complicated asthat in a BMHV it is almost impossible to clearlyseparate the turbulence fluctuations from cycle-to-cy-cle variations. Such failure in separating these twovariations is also clearly evident in the data presentedin Tiederman.49 For that reason in this work, we willanalyze the Reynolds shear stress based on the simpleensemble-averaging approach shown in Eq. 8.

To illustrate the varying contribution and relativesignificance of the stresses due to cycle-to-cycle vari-ations to the total Reynolds stresses at various in-stants during the cardiac cycle, we plot in Figs. 4–7the measured Reynolds stresses at the four previouslydefined time instants. At time instant EA (Fig. 4),which is in the early acceleration phase, pockets oflarge levels of the two normal stresses, namely qu0u0

and qv0v0; are observed just downstream of the twoleaflets. The peak value observed at this time instantis about 150 N/m2 for qu0u0 and 70 N/m2 for qu0u0,respectively. The shear stress qu0v0, on the other hand,is very small with a maximum magnitude in the wakeregion about 6 N/m2. At time instant MA much lar-ger overall values of Reynolds stresses are observed inthe wake of the leaflets and furthermore the size ofthe pockets of high stress levels has grown consider-ably in size. The peak values for qu0u0, qv0v0, and qu0v0

at time instant MA are 500, 400, and 40 N/m2,respectively. For the sake of our subsequent discus-sion, it is important to emphasize herein that at boththe EA and MA time instants the flow is clearlylaminar. As such, the observed Reynolds stressescannot possible arise from small-scale turbulent fluc-tuations.

The measured Reynolds stress distributions at thethree time instants in the turbulent regime are plottedin Figs. 6 and 7. At peak systole (Fig. 6), the stressmaxima are about 420, 320, and 50 N/m2 for qu0u0,qv0v0, and qu0v0, respectively. As the flow enters thedecelerating phase of systole, however, the peak stresslevels for all components are seen to decrease with the

GE et al.

decreasing flow rate. At time instant D (Fig. 7) theyare 320, 240, and 35 N/m2, respectively.

A number of striking and novel conclusions can bederived from these results. First, it is clear that duringthe laminar phase of the cardiac cycle pockets of in-creased Reynolds stresses are observed in the wake ofthe leaflets within which the peak stress levels are ashigh or even higher than those observed at time in-stants during the turbulent phase of the cycle. As

small-scale turbulent fluctuations are not present dur-ing the laminar phase, the entire production of Rey-nolds stresses must be due to the so-called cycle-to-cycle variations. At time instant EA, when large-scalevortex shedding has not commenced yet, the high levelsof normal stresses should be entirely due to cycle-to-cycle variations of the leaflet kinematics. At instantMA, on the other hand, the flow in the wake of theleaflets is dominated by large-scale, vortex shedding,

FIGURE 4. Distribution of Reynolds stress components,qu0u0;qv 0v 0; qu0v 0; obtained from DPIV measurements at timeinstant EA. Unit for each Reynolds stress is N/m2.

FIGURE 5. Distribution of Reynolds stress components,qu0u0;qv 0v 0;qu0v 0; obtained from DPIV measurements at timeinstant MA. Unit for each Reynolds stress is N/m2.


which along with cycle-to-cycle variations of the leaf-lets kinematics should also contribute to the totalReynolds stress production. Therefore, our findingsclearly underscore the significant pitfalls associatedwith trying to estimate blood damage potential usingthe peak levels of the Reynolds stresses as the solemetric. The mechanism that generates these stressesand the spatial scales of the eddies that contribute totheir production should also be taken into account.

This issue will be further reinforced in the next sectionof the article when we estimate the Kolmogorov scaleof the flow in the wake of the leaflets.

Another important conclusion is with regard to theimportance of cycle-to-cycle variations in leaflet kine-matics as a mechanism for producing Reynolds stres-ses. As shown in Figs. 4 and 5, for both the earlyand middle acceleration phase, the largest value ofReynolds shear stress is observed in the immediate

FIGURE 6. Distribution of Reynolds stress components,qu0u0;qv 0v 0; qu0v 0, obtained from DPIV measurements at timeinstant P. Unit for each Reynolds stress is N/m2.

FIGURE 7. Distribution of Reynolds stress components,qu0u0;qv 0v 0;qu0v 0; obtained from DPIV measurements at timeinstant D. Unit for each Reynolds stress is N/m2.

GE et al.

vicinity of the two leaflets. As we already pointed out,during early acceleration the flow is still stable and novortex shedding is present in the wake. Clearly cycle-to-cycle variations of leaflet kinematics is the onlymechanism producing Reynolds stresses. Even at themiddle acceleration phase, however, when vortexshedding has commenced the overall levels of Rey-nolds stresses are seen to be substantially higher in thenear wake of the leaflets. It is thus reasonable to con-clude that even in the presence of vortex shedding cy-cle-to-cycle variations of leaflet kinematics stillcontribute significantly in the production of Reynoldsstresses. This finding has important implications inso-far as the numerical prediction of the phase-averagedstatistics of the flow is concerned. It evident that unlessthe computational methods can reproduce the signifi-cant cycle-to-cycle variations of leaflet kinematics it ishighly unlikely that they will be able to capture thecorrect levels of the measured Reynolds stresses duringthe early systolic phase. For this to be possible, cou-pled fluid-structure-interaction (FSI) algorithms needto be developed and validated.

Finally, it is important to point out a fundamentaldifference between the Reynolds stress distributionsduring the laminar and turbulent regimes. Even thoughlarge levels of stresses are present in the laminar regimethese stresses are confined in pockets of relatively smallspatial extent. During the turbulent flow phase, how-ever, the pockets of increased Reynolds stresses arespread over a much wider area in the wake of theleaflets. This feature of the flow is the hallmark of itsturbulent nature and is due to the highly augmentedability of a turbulent flow relative to a laminar flow totransport momentum via intense small-scale turbulentfluctuations.

Viscous Stresses

Viscous stresses are calculated using three-pointcentral finite differencing scheme to compute the spa-tial derivatives of the velocity field obtained eitherfrom the DPIV measurements or the CFD. Forexample, sxy at grid node (m,n) is calculated as follows:

sxyjðm;nÞ ¼ lvðmþ1;nÞ � vðm�1;nÞxðmþ1;nÞ � xðm�1;nÞ

þuðm;nþ1Þ � uðm;n�1Þyðm;nþ1Þ � yðm;n�1Þ

� �

ð12Þ

The above equation, when applied to the CFD velocityfield, provides a second-order accurate approximationof the viscous shear stress field. When applying thesame formula to the velocity field obtained from DPIVmeasurements, however, large errors could be intro-duced due to inherent uncertainties in the measuredvelocity field. For example, as shown in Luff et al.,35

using central differencing to compute the vorticity fieldfrom PIV velocity data could yield a vorticity field witherrors as high as 70%. In order, therefore, to accu-rately compute velocity derivatives from experimentaldata an advanced smoothing procedure should be ap-plied to the velocity field prior to the computation ofthe derivatives. In this article, we use the 3� 3Gaussian filter suggested in Luff et al.,35 which asshown in Luff et al.35 can reduce the maximum error inthe spatial derivative from the early mentioned 70% toabout 5%. This spatial filter reads as follows:

Ui;j ¼P3

a¼�3P3

b¼�3 wða; bÞui�a;j�b� �W

ð13Þ

Vi;j ¼P3

a¼�3P3

b¼�3 wða; bÞvi�a;j�b� �W

ð14Þ

where w(a, b) is the filter kernel defined as:

wða; bÞ ¼ exp�2ða2 þ b2Þ

p2

� �ð15Þ

and the weight factor W is

W ¼X3a¼�3

X3b¼�3

wða; bÞ ð16Þ

In the above equations, u and v denote the rawexperimental data while U and V are the filteredvelocity components. Following Luff et al.,35 the con-stant p in our current article is set equal to 3.

A major issue related to the accurate computationof velocity derivatives either from experimental orCFD data stems from certain spatial resolutionrequirements, which become especially strict when theflow is turbulent (as is the case throughout the past-peak-systole portion of the cardiac cycle). The mini-mum size of the CFD or measurement grid in this caseis set by the size of the energy dissipating eddies, theso-called Kolmogorov scale g.35,37 The Kolmogorovscale in BMHV flows has been previously studied by anumber of researchers. For example, Baldwin et al.1

reported the Kolmogorov scale to be 52 lm for theforward flow phase and 6 lm for the backward flowphase; Ellis et al.9 reported that the Kolmogorov scaleto be about 7 lm for the leakage jet; Liu et al.33 re-ported the Kolmogorov scale at various time instantsthroughout a cardiac cycle and the estimated Kol-mogorov scale is typically larger than 20 lm. It isworth noting here that the concept of Kolmogorovscale is developed based on fully developed homoge-neous isotropic turbulence where it can be shown to beproportional to ðm3=eÞ1=4, where m is the kinetic vis-cosity of the fluid and e is the dissipation rate of tur-bulence kinetic energy (TKE). Therefore, the task of


estimating the Kolmogorov scale is equivalent tomeasuring the dissipation rate of TKE, e. By definition,

e ¼ 2m sijsij� �

where sij is the strain rate tensor of the turbulentfluctuation velocity field. The direct measurement of einvolves the measurement of all components of thestrain rate tensor, which is far from trivial undertaking.A much simpler approach for estimating e is based onisotropic homogeneous turbulence theory33,50:

e ¼ 15mu2rms

k2� ðu3rms=lÞ ð17Þ

where urms is the root mean square of turbulencevelocity fluctuations, k is the Taylor micro-scale and lis the characteristic length of largest turbulence eddies.It is not difficult to see from the above two equations(see also Travis et al.51) that the estimation of e isstrongly dependent on the estimation of the RMS ofthe velocity fluctuations. As we just illustrated in‘‘Ensemble-Averaging and Calculation of ReynoldsStresses’’ section, however, for BMHV flows themeasured velocity fluctuations inevitably contain con-tributions arising both from cycle-to-cycle variationsand turbulence. Separating the two effects is difficult ifnot impossible, which makes it very difficult to accu-rately quantify the true velocity fluctuations due toturbulence and estimate e. Therefore, the estimation ofthe Kolmogorov scale using the measured urms is ofvery limited value insofar as estimating the true size ofthe energy dissipating eddies is concerned.

To demonstrate the influence of the cycle-to-cyclevariations on the estimation of the Kolmogorov scale,let us analyze the experimental data at time instantsMA and P. As shown in Fig. 5, downstream of thevalve, the maximum urms at time instant B is about0.55 m/s. Following Eq. (17) and selecting the radiusof the upstream aorta as the characteristic length,l(0.0127 m), one can estimate e ¼ u3rms=l ¼ 6:55 m2/s3

which yields a Kolmogorov scale of about 42 lm. Thisestimated Kolmogorov scale is somewhere in betweenthe value reported in Liu et al.33 and Baldwin et al.1

Remember that as we already discussed earlier, at timeinstant MA, the flow downstream to the valve is lam-inar and so the entire concept of the Kolmogorov scaleis meaningless. Nevertheless, by directly applying thetheories derived from fully developed turbulent flowwe can estimate a Kolmogorov scale that turns out tobe quite small.

At time instant P, the maximum measured velocityfluctuation is about 0.48 m/s (see Fig. 6), which leadsto an estimated Kolmogorov scale of about 47 lm.Recall that as we discussed in ‘‘Ensemble-Averagingand Calculation of Reynolds Stresses’’ section, the

measured velocity fluctuations observed at this timeinstant also contain contributions from the cycle-to-cycle variation. As such the measured fluctuationsoverestimate the true turbulent fluctuations, which,according to Eqs. (11) and (17), would result in theoverestimation of e and the underestimation of theKolmogorov scale. It is, therefore, reasonable to con-clude that the spatial resolution requirement based onthe experimentally estimated Kolmogorov scale isprobably too restrictive. Furthermore, estimating thetrue Kolmogorov scale from experimental measure-ments of BMHV flows is far from trivial task. In factwithout an approach for separating cycle-to-cyclevariations from actual turbulence the accurate esti-mation of g is impossible.

Because of these difficulties, we propose herein analternative approach for evaluating the spatial resolu-tion of the measurements which is inspired from thegrid sensitivity study idea typically carried out todemonstrate the accuracy of numerical simulations.The grid sensitivity study is carried out by evaluatingthe viscous shear stress at each measurement pointwith different levels of grid spacing following the samediscretization scheme as shown in Eq. (12). Let h bethe grid spacing for DPIV measurement, the viscousstress sh, s2h, and s4h are evaluated as

skhxyjðm;nÞ ¼ lvðmþk;nÞ � vðm�k;nÞ

2khþuðm;nþkÞ � uðm;n�kÞ

2kh

:

ð18Þ

By comparing the results among the various grids weseek to demonstrate that the velocity gradient termsare not sensitive to further grid refinement (i.e., finerexperimental resolution) and indirectly show that ourmeasurements capture essentially all relevant scales ofmotions. Since we have already showed in Dasi et al.7

that the current CFD simulation captures all measuredflow features with good accuracy, this measurement-grid-resolution study also shows that the grid resolu-tion employed in the CFD is sufficiently fine for gridinsensitive solutions. In the DPIV measurements, thegrid spacing at the object plane is about h = 135 lm.In the CFD, however, we use an aorta-fitted curvilin-ear grid and the grid lines are clustered in the vicinityof the valve resulting to a variable grid resolutionthroughout the flow domain. In the near vicinity to thevalve, which is the area of interest in this work, thecomputational grid resolution is about 125 lm.

We carry out the grid sensitivity study at the peakflow phase when the shear stress magnitude is thelargest within the entire cycle and the measuredinstantaneous flow is dominated by multiple smallscale structures. In Fig. 8, we show the instantaneousshear stress contours obtained on the three grid levels

GE et al.

(h, 2h, and 4h). As can be seen in the figure, the twofinest meshes yield results that are essentially identical.Significant discrepancies are observed, however, be-tween the stress field computed on the 4h mesh and thetwo finer meshes. To illustrate the differences betweenthe grid levels more clearly, we superimposing Fig. 9the spanwise shear stress profiles at x ¼ 30 (see Fig. 3for axes definition) obtained on all three grid levels. Itis evident that the 2h mesh captures essentially all finesubtleties of the stress distribution on the finest mesh

both qualitatively and quantitatively. The shear stressobtained from the coarsest grid spacing level, however,misses many of the finer features observed on the twofine meshes. To quantify the difference levels betweenthe three grids, we plot in Fig. 10 contours of thepercentage of the error between the stress levels

FIGURE 8. Comparison of the distribution of sxy calculatedfrom instantaneous velocity field with different grid spacinglevels. Unit is N/m2.

Y

τ xy(

N/m

2 )

-10 -5 0 5 10

-2

-1

0

1τxy

4h

τxyh

τxy2h

FIGURE 9. Comparisons of spanwise sxy profile calculatedfrom different grid spacing levels.

FIGURE 10. Distribution of PE (as defined in Eq. 19). Re-gions of red color are the area where PE > 5%.


calculated on a given coarse mesh (2h or 4h) and thoseobtained on the actual measurement grid (the h mesh).For example the error at the 2h level is defined asfollows:

PE2h ¼ 100� s2h � sh

shmax

ð19Þ

where sh denotes the stress computed on the h-size grid.The contour maps shown in Fig. 10 show the 5% errorlevel. It is evident that the difference between the h and2h grids is mostly smaller than 5%, while the coarsestgrid level shows errors larger than 5% in a large por-tion of the flow domain.

Due to the nature of DPIV measurement, themeasured velocity at each node is not local velocity;rather, it represents the velocity spatially averaged overa small interrogation window surrounding the node.Typically PIV measurements use overlap interrogationwindow and in the current measurement uses a 50%overlap is used. Such an overlap makes the vorticitycalculation at the two finer grid levels (h and 2h) notcompletely independent from each other. This is thebest illustrated in Fig. 11, which shows the five hori-zontally neighboring grid nodes (ðm; nÞ; ðm� 1; nÞ andðm� 2; nÞ) and their corresponding interrogationwindows. Due to the 50% interrogation windowoverlap, each interrogation window can be divided intotwo halves: one uniquely belong to the node and oneoverlap with the next neighbor. Therefore, the fiveinterrogation windows can be divided into six regionsand as shown in the figure we denote themR�; O�; L�; Lþ; Oþ; Rþ, respectively. The interroga-tion window for node ðmþ 1; nÞ, for example, consistsof Lþ and Oþ and the PIV measured velocity at thesetwo nodes can be described as

vmþ1;n ¼ SðLþ þ OþÞðvÞ ð20Þ

where S stands for a spatial averaging function, whoseexact form is determined by the DPIV algorithm. Tosimplify our discussion, we assume for the time being it

is a linear average function. The above equation thencan be reformulated as

vmþ1;n ¼ 0:5� ðSðLþÞðvÞ þ SðOþÞðvÞÞ ð21Þ

and SðLþÞðvÞ can be considered as DPIV measuredvelocity at (m + 0.5, n). We denote this asvmþ1

2;n¼ SðLþÞðvÞ. By applying similar denotation to

all neighboring nodes, it can be shown that the calcu-lation of shx;y is equivalent to

shxy �0:5�1

2

vmþ12;n� vm�1

2;n

hþ 3

2

vmþ32;n� vm�3

2;n

3h

� �þ � � �

¼ 1

4s0:5hxy þ

3

4s1:5hxy

where the dots imply terms in the other dimension.Similarly we have

s2hxy �5

8s2:5hxy þ

3

8s1:5hxy ð22Þ

The above analysis illustrates that the grid-sensitivitystudy, which compares sh against s2h, essentially makesthe comparison between 5

8 s2:5hxy and ð38 s1:5hxy þ 14 s0:5hxy Þ and

it can be concluded here that by illustrating closeresemblance between sh and s2h we indeed show gridindependence. Due to the nonlinearity of the functionS, however, the above analysis is not rigorous. Butgiven the complexity of a turbulent flow field, thenonlinearity is not likely to make the comparison be-tween sh and s2h better than a linear S. Considering theclose resemblance between these two finer grid levels,and the small grid spacing we used in the DPIV mea-surement, it is therefore reasonable to assume that theexperimental grid we have employed is sufficiently finefor capturing most relevant scales of motion in theflow. Of course our analysis does not provide a rigor-ous answer to the question of the accurate estimationof the Kolmogorov scale in BMHV flows.

COMPARISON OF REYNOLDS AND VISCOUS

STRESS FIELDS

The major objective of this article is to illustrate thedifferences in magnitude between the Reynolds andviscous stresses in BMHV flows. Having established inthe previous section of the article our ability to com-pute these terms accurately, we shall now discuss anappropriate framework for comparing the variousstress terms. Both the Reynolds and viscous stressescomprise second-order tensors and we thus need toestablish a coordinate-independent metric for com-paring their relative magnitudes. Similar as in Grigioniet al.19 and Travis et al.,51 we employ the concept ofthe local maximum stress, which is defined as follows.

FIGURE 11. Interrogation windows for five neighboring gridnodes ðm;nÞ; ðm � 1;nÞ; ðm � 2;nÞ. The interrogation windowoverlap is illustrated by the shaded area near nodes (m + 1,n) and (m + 2, n).

GE et al.

The principal stresses of a given stress tensor sij can beobtained by solving the following equation:

r3 � I1r2 þ I2r� I3 ¼ 0 ð23Þ

where I1, I2, and I3 are the first, second, and thirdinvariants of the tensor (repeated indices imply sum-mation):

I1 ¼ sii ð24Þ

I2 ¼ sijsij ð25Þ

I3 ¼ sijsjkski ð26Þ

Assuming that rmax and rmin are the maximum andminimum principle stresses of the tensor, then acoordinate invariant local maximum stress can be de-fined as

smax ¼1

2rmax � rminð Þ ð27Þ

The above definition is equally applicable to the Rey-nolds and viscous stress tensors. To facilitate oursubsequent discussion in this section let us introducethe following notation. RSmax and VSmax denote thelocal maximum value of the Reynolds and viscousstresses, respectively. When these values are computedfrom experimental measurements, the tensors involvedare 2D ð2� 2Þ as the PIV data provide only the twovelocity components at a plane—the Reynolds andviscous stress tensor in two dimensions each have onlythree independent components, qu0u0; qv0v0; and qu0v0,and sxx, syy, and sxy, respectively. When CFD data areused, on the other hand, these local maxima can becomputed both the 2D and 3D tensors—recall that inthe present study only the viscous stresses are availablefrom the CFD data.

In what follows we present the spatial distributionsof RSmax and VSmax at the selected four time instantsduring the cardiac cycle. We compute (using Eq. 27with the appropriate tensor components) and plot thefollowing quantities: (1) RSmax based on the measured2D Reynolds stress tensor; (2) VSmaxjexp;inst based onthe instantaneous viscous stress tensor from theexperiments; (3) VSmaxjexp;avg based on the componentsof the measured ensemble-averaged viscous stresstensor; (4) VSmaxjCFD;2D based on the 2D instanta-neous stress tensor obtained from the CFD by omit-ting the out-of-plane velocity component and thederivatives of the other two velocity components alongthe plane-normal direction; and (5) VSmaxjCFD;3D basedon the full 3D instantaneous viscous stress tensor fromCFD.

These data are presented here and compared witheach other to illustrate that: (a) there are drastic dif-ferences between the distributions of Reynolds and

viscous stresses and consequently using the former toquantify the mechanical force environment experi-enced by blood elements is not appropriate (compari-sons between RSmax and VSmaxjexp;inst); (b) averagingprocedures eliminate much of the essential informationwith regard to the stress distribution and for that oneneeds to look at the instantaneous flow field foraccurate representation of the mechanical force envi-ronment (comparisons between VSmaxjexp;inst andVSmaxjexp;avg); and (c) the three dimensionality of theflow plays a very important role in quantifying themechanical force environment and needs to be takeninto account in order to develop meaningful metrics ofblood damage (comparisons of VSmaxjCFD;2D andVSmaxjCFD;3D).

The first three quantities, RSmax;VSmaxjexp;inst andVSmaxjexp;avg are all from experimental data and theircontour plots are shown in left column (i.e., (a), (c) and(e)) of Figs. 12–15, with each figure shows the corre-sponding data at the four selected time instants, EA,MA, P, D, respectively. The most obvious feature thatfollows from these figures is the stark contrast betweenthe overall levels of RSmax and VSmax with the formerbeing nearly two orders of magnitude higher than thelatter. At the early acceleration phase, VSmax (Fig. 12a)is uniformly small (except for two very small pocketsnear the leaflets) VSmax<1 N/m2 and the peak valueobserved is smaller than 2 N/m2. The RSmax field(Fig. 12e), on the other hand, is very large with peakvalues reaching at about 60 N/m2. As the strength ofincoming flow increases, the magnitude of both RSmax

and VSmax increases. The growth of RSmax, however, ismuch larger in magnitude than that of VSmax causingthe discrepancy between the Reynolds and viscousstress metric to grow further. As seen in Fig. 13, whichshows the data at time instant MA, pockets of rela-tively higher VSmax (Fig. 13a) are observed in the shearlayer region following the two leaflets and the edge ofthe valve housing. The overall magnitude, however, isstill quite low, with the peak value not exceeding about4 N/m2. The peak magnitude of RSmax (Fig. 13e), onthe other hand, is much higher reaching about 250 N/m2.At the later time instants the flow transitions to tur-bulence and the distribution of VSmax is clearly morechaotic and disorganized compared with the earlierlaminar flow stage. The maximum value of VSmax,however, does not differ significantly from the earlierstages. At the peak flow condition, the measuredVSmax (Fig. 14a) is about 12 N/m2 (which, by the way,is only observed in very small pockets), and the relativelarger values of VSmax are only observed in the fourshear layer generated by the two leaflets as well as theedges of the valve housing. Due to the turbulent natureof the flow field, the instantaneous distribution ofVSmax does not convey all the information of the


VSmax in time. As indicated earlier, for every timeinstant measured in the experiments, we obtained 250different realizations and for all instantaneous mea-surements the VSmax observed is <15 N/m2. Thesituation is similar at the time instants D. The differ-ence between the distribution of RSmax and VSmax

remains large with the former always exceeding by farthe latter as we observed in the earlier laminar flowstages. More specifically at time instants P, and D therespective maxima of RSmax are 180, and 120 N/m2.Overall the observed levels of RSmax are comparable towhat has been reported before in the literature.

The main conclusions derived from the above datacan be summarized as follows:

1. Reynolds stresses in the flow arise from temporalvariations of the velocity field, which in BMHVarise both from cycle-to-cycle variations and tur-bulence fluctuations. These stresses are statisticalquantities that on average quantify the transport of

fluctuating momentum by fluctuating velocitycomponents, but are not representative of the truemechanical environment experienced by blood ele-ments. For example, and as we extensively discussedin the introduction of this article, in laminar flowthe mechanical environment experienced by cells iswell understood in the literature to be due to vis-cous stresses. Yet we have shown above thatthroughout the laminar portion of the cardiac cyclethe wake of leaflets exhibits large regions of veryhigh levels of Reynolds stresses, which arise notfrom turbulent fluctuations but from cycle-to-cyclevariations. That is, regions of very high Reynoldsstresses are found in the wake of the leafletsthroughout the entire cardiac cycle from the lami-nar early systolic phase to the fully turbulent past-peak-systole phase. As such, the Reynolds stressmagnitude could not possibly provide a meaningfulindicator for quantifying mechanical forces that arepotentially harmful to cells.

FIGURE 12. Comparisons of (a) VSmaxjexp;inst, (b) VSmaxjCFD;2D, (c) VSmaxjexp;avg; (d) VSmaxjCFD;3D; and (e) RSmax at time instant EA.Unit is N/m2.

GE et al.

2. There are large discrepancies between the magni-tudes of the observed Reynolds and viscousstresses. Following the standard argument in theliterature that Reynolds stresses contribute towardcell damages, this finding could lead to the con-clusion that the wake flow is potentially lethal forblood cells. Recall that at all time instants weobserved peak values of RSmax in the neighbor-hood of 100 N/m2 with the maximum valuereaching as high as 250 N/m2. The real mechani-cal environment, which is measured by viscousshear stress, is very different. Throughout thecardiac cycle in the wake of the leaflets the max-imum shear stress level is <15 N/m2, a level thatcould be generally considered to be safe for RBCs(as suggested by a number of previous studies,RBCs begin to break up when exposed to shearstress levels of about 150 N/m2) while is poten-tially damaging to platelets. Such a finding isactually consistent with the clinical finding that

the occurrence of blood clotting is far more fre-quent than hemolysis. It is, however, important tore-iterate herein that in this work we focus entirelyon the bulk wake flow. With regard to mechani-cally induced cell damage, the backward flowstage, when the flow passes through the small gapsbetween the leaflets and valve housing, is poten-tially more dangerous and the implications of ourfindings with regard to the impact of this flowstage are not clear at this point. Further explo-ration with regard to the gap flow is currentlyunderway and results will be presented in futurepublications. The key conclusion of this work,however, that viscous stresses comprise a far moreappropriate metric than Reynolds stresses forquantifying potential for blood damage should bevalid regardless.

Also shown in these figures are the comparisonsbetween VSmaxjexp;inst and VSmaxjexp;avg. As seen in

FIGURE 13. Comparisons of (a) VSmaxjexp;inst, (b) VSmaxjCFD;2D, and (c) VSmaxjexp;avg, (d) VSmaxjCFD;3D, and (e) RSmax at time instantMA. Unit is N/m2.


Figs. 12a and 12c, at the early acceleration phase, thesetwo fields closely resemble each other. At the middleacceleration (Fig. 13), the features observed from thesetwo set of data are still quite similar to each otherexcept for the far wake region behind the two leaflets.As shown in Fig. 3, this is the area where the Karman-like vortex structures appear and phase-averaging actsto smooth out all the oscillatory vortical structures,which in turn leads to very small phase-averaged stresslevels. The same small-scale smoothing effect in the farwake region at peak flow condition (Fig. 14) also leadsto similar differences between VSmaxjexp;inst andVSmaxjexp;avg with the discrepancy growing even moreat instants D, as shown in Fig. 15. These results clearlysuggest that the instantaneous field rather than thephase-averaged field should be used to develop metricsfor quantifying blood damage in BMHV flows andother biomedical devices.

Finally, in the right column of Figs. 12–15 we showthe distribution of VSmaxjCFD;2D and VSmaxjCFD;3D

obtained from the CFD at the four selected time in-stants. These figures clearly highlight the effects of flowthree dimensionality. At the laminar time instants EA(Fig. 12) and MA (Fig. 13), the distribution ofVSmaxjCFD;2D (Figs. 12b and 13b) and VSmaxjCFD;3D(Figs. 12d and 13d) are very similar with each other.Once the flow transitions to turbulence (see Figs. 14and 15) any similarity between the two quantities dis-appears. For example, at peak flow conditions pocketsof VSmaxjCFD;3D >3 N/m2 (Fig. 14d) are clearly widerthan those of with similar levels of VSmaxjCFD;2D(Fig. 14b). These results clearly show that the three-dimensionality of the flow should be carefully consid-ered in order to quantify the true mechanical forceenvironment experienced by cells. Such conclusionfurther underscores the enormous importance of highresolution CFD simulations, which provide the onlymeans for obtaining accurate 3D descriptions of theflow to supplement the 2D data we typically obtainfrom experiments.

FIGURE 14. Comparisons of (a) VSmaxjexp;inst, (b) VSmaxjCFD;2D, (c) VSmaxjexp;avg, (d) VSmaxjCFD;3D, and (e) RSmax at time instant P.Unit is N/m2.

GE et al.

CONCLUSIONS

The main focus of this article is to investigate thecharacterization of the mechanical forces experiencedby blood cells in a turbulent flow field, such as the flowdownstream to BMHVs. We first conducted thoroughliterature review and show that currently Reynoldsstress is often used to characterize the mechanicalforces induced by turbulent flow. We presented theo-retical analysis to illustrate that Reynolds stress is in-deed not directly related to mechanical forcesexperienced by blood cells and the instantaneous vis-cous stress provides the proper measurement of themechanical forces in BMHV induced turbulent flowfield. We then carried out detail evaluation of viscousand Reynolds stress in the flow downstream to a typ-ical BMHV to (a) elucidate the actual mechanicalforce; and (b) illustrate the difference between thedistribution of viscous and Reynolds stress. The valveused is a 23 mm St. Jude Medical Regent valve and the

evaluation of both stress uses velocity data obtainedfrom high resolution DPIV measurements and CFDsimulations. The evaluation of both stress is verychallenging and it is especially so for the instantaneousviscous stress. The viscous stress is a function of spatialderivative of velocity and, due to the existence of smallscale motions in a turbulent flow field, its’ evaluationrequires a grid resolution that is fine enough to catchthe smallest scale motions. We indirectly illustratedthat such a condition is practically satisfied by showinggrid independency using velocity data obtained fromDPIV measurement. The calculated instantaneousviscous stress is shown to reveal the mechanical forcein the turbulent flow field downstream to a BMHV. Tothe best of our knowledge, this is the first time that theinstantaneous viscous stress in a BMHV flow is eval-uated and the results clearly suggest for a major par-adigm shift of hemodynamics study related toturbulence on BMHV or similar prosthetic medicaldevices from the traditional approach focusing on the

FIGURE 15. Comparisons of (a) VSmaxjexp;inst, (b) VSmaxjCFD;2D; (c) VSmaxjexp;avg, (d) VSmaxjCFD;3D, and (e) RSmax at time instant D.Unit is N/m2.


statistical quantities of turbulence (Reynolds stress) toan approach that can accurately quantify the instan-taneous viscous stress.

The major finding of our work is the stark contrastin the distribution of Reynolds and viscous stresses inthe leaflets wake. Since both stress entities are tensors,we quantify their relative magnitude in terms of thelocal maximum stress, a scalar, coordinate-invariantquantity that is proportional to the difference of themaximum and minimum principle stresses of eachtensor. We showed that the overall levels of maximumReynolds stress are one to two orders of magnitudegreater than maximum viscous stress throughout theentire cardiac cycle. The levels of the maximum Rey-nolds stress are found to be within 100–250 N/m2

while the corresponding range for the maximum vis-cous stress is between 5 and 15 N/m2. If we are toadopt the prevailing wisdom in the literature thatReynolds stresses determine blood cell and plateletdamage potential, the observed Reynolds stress levelsare sufficiently high to readily conclude that themechanical force environment in the wake of theleaflets could be hazardous to blood elements. Yet suchconclusion would be incorrect for two main reasons.First, the Reynolds stresses are not representative ofthe actual mechanical environment experienced byblood elements, but are rather a mere statistical artifactused to quantify on average the transport of fluctuat-ing momentum by fluctuating velocities. Second, con-sidering only the magnitude of the Reynolds stressesfails to account for the actual mechanism that pro-duces them and the underlying scales of the Reynolds-stress-producing motions. Our results clearly show thathigh levels of maximum Reynolds stress are observedeven during the laminar phase of the cardiac cyclewhen the flow is organized and void of small scaleturbulent eddies. Cycle-to-cycle variations of the leafletkinematics and coherent vortex shedding from theleaflet wakes were identified as the primary mecha-nisms for producing Reynolds stresses during theaccelerating phase. The flow structures that arise as aresult of the laminar vortex shedding from the leaflets,however, are several orders of magnitude larger thanthe typical blood cell size and as such they cannotpossibly inflict any damage on blood elements.

The true mechanical environment in BMHV flowsshould be quantified by the instantaneous viscousstresses. The observed levels of maximum viscousstress were found to be well below the range requiredto inflict damage to RBC, but borderline the experi-mentally determined threshold of approximately 10 N/m2 for platelet damage.20,30 This finding is entirelyconsistent with clinical results, which suggest thatblood clotting rather than hemolysis is the most likelycomplication in implanted BMHVs.

Our work also underscored the significance of thethree-dimensionality of the flow for accurate quantifi-cation of the viscous stress field in the wake of theleaflets. As such, we argue that any attempt to developmetrics for blood damage in BMHV flows shouldintegrate laboratory measurements, which are typically2D, with high resolution full 3D numerical simula-tions.

Finally, we should emphasize once again that ourwork so far was restricted to the flow in the wake of theleaflets and the sinus region. It is well known, however,that during the backward flow phase regions of veryhigh shear could develop in the tiny gaps in the valvehinges and the leakage jet emanating between theclosed leaflets. In these regions it is reasonable toanticipate very high levels of viscous stresses, whichcould even reach the threshold for inducing RBCdamage. Detailed quantitative investigation of the flowin these regions, however, is beyond the scope of thiswork as it would require very careful measurementscoupled with hinge and gap-resolving, multi-scalenumerical simulations. Such work is currently underway and will be reported in a future publication.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financialsupport from Tom and Shirley Curley’s and the Na-tional Heart, Lung, and Blood Institute (HL 720621).L. G. and F. S. are grateful to the University ofMinnesota Supercomputing Institute for assistancewith the computations.

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Characterization of Hemodynamic Forces Induced by Mechanical Heart Valves: Reynolds vs. Viscous...

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