ORIGINALS

A. Cenedese Æ Z. Del Prete Æ M. Miozzi Æ G. Querzoli

A laboratory investigation of the flow in the left ventricleof a human heart with prosthetic, tilting-disk valves

Received: 1 October 2004 / Revised: 22 April 2005 / Accepted: 22 April 2005 / Published online: 31 May 2005� Springer-Verlag 2005

Abstract The understanding of the phenomena involvedin ventricular flow is becoming more and more impor-tant because of two main reasons: the continuousimprovements in the field of diagnostic techniques andthe increasing popularity of prosthetic devices. On onehand, more accurate investigation techniques gives thechance to better diagnose diseases before they becomedangerous to the health of the patient. On the otherhand, the diffusion of prosthetic devices requires verydetailed assessment of the modifications that theyintroduce in the functioning of the heart. The presentwork is focussed on the experimental investigation of theflow in the left ventricle of the human heart with thepresence of a tilting-disk valve in the mitral position, asthis kind of valve is known to change deeply the struc-ture of such a flow. A laboratory model has been builtup, which consists of a cavity able to change its volume,representing the ventricle, on which two prostheticvalves are mounted. The facility is designed to be able toreproduce any arbitrarily assigned law of variation ofthe ventricular volume with time. In the present experi-ment, a physiologically shaped curve has been used.Velocity was measured using a feature-tracking (FT)algorithm; as a consequence, the particle trajectories areknown. The flow has been studied by changing both thebeat rate and the stroke volume. The flow was studiedboth kinematically, examining velocity and vorticityfields, and dynamically, evaluating turbulent and viscous

shear stresses, and inertial forces exerted on fluidelements. The analysis of the results allows the identifi-cation of the main features of the ventricular flow,generated by a mitral, tilting-disk valve, during thewhole cardiac cycle and its dependence on the frequencyand the stroke volume.

1 Introduction

In the past few years, the study of the fluid dynamics ofthe left ventricle of the heart has arisen to the attentionof researchers due to two main factors: the possibility ofapplication in the diagnostics and in the assessment ofprosthetic valves.

The improvement in the echo-cardiographic tech-niques, such as the development of 2D and 3D probesand the increase in the acquisition rate, permits, nowa-days, a relatively good investigation of ventricular flow.This fact suggested the idea that some modifications inthe ventricular flow could be effective early indicators ofcardiac diseases (Litwin and Grossman 1993; Urheimet al. 2003). On the other hand, echo-cardiography fur-nishes only one component of the velocity and the res-olution is still quite low both in time and space.

The investigations performed in vivo can be suffi-ciently detailed to capture the modification of the flow,but they are not precise enough in the physical com-prehension of the phenomena that lead to such a mod-ification. This is why laboratory simulations can be veryuseful: on one hand, laboratory models are often sche-matic in the reproduction of the geometry and of themechanical characteristics of the cardiac apparatus; onthe other hand, the flow can be investigated in detail,using techniques that are not feasible in vivo (Chew et al.2001). The information furnished by the models can beused for the interpretation of features of the flow thatare observed directly in the patient by means of echo-cardiography.

A. Cenedese Æ M. MiozziDipartimento di Idraulica Trasporti e Strade,Universita degli studi di Roma ‘‘La Sapienza’’, Rome, Italy

Z. Del PreteDipartimento di Meccanica e Aeronautica,Universita degli studi di Roma ‘‘La Sapienza’’, Rome, Italy

G. Querzoli (&)Dipartimento di Ingegneria del Territorio,Universita degli studi di Cagliari, Cagliari, ItalyE-mail: querzoli@unica.itTel.: +39-70-6755322Fax: +39-70-6755322

Experiments in Fluids (2005) 39: 322–335DOI 10.1007/s00348-005-1006-4

The second element of interest in the study of ven-tricular flow stems from the wide diffusion thatreplacement of heart valves have reached during the lastdecades. The continuous improvement of surgical tech-niques and of prosthetic valve design encourage valvereplacement in a wider and wider range of cases; how-ever, this is feasible only if all the effects of the substi-tution are carefully assessed. Many problems related tothe implant of a heart valve are related to the blood flow,among them, haemolysis, high losses of energy andthrombo-embolism. The physical phenomena that caninduce these problems are not only related to the generalfeatures of the mean flow, but also to its vortical and/orturbulent structure. Therefore, only accurate measure-ments that can be performed in laboratory models givethe required detail in the description of the flow.

Finally, a laboratory model can be used as a test casefor numerical simulations, since both the velocity field andthe initial and boundary conditions are known with greataccuracy (Baccani et al. 2002; Vierendeels et al. 2002).

A number of laboratory models of the left heart hasbeen built up in the last few decades. Some of themreproduce only the flow though an orifice, without aventricle (Kini et al. 2001), while others emulate the car-diac functionality by a flexible ventricle. The motionaround the ventricle is achieved by varying the pressure ofthe chamber that contains the ventricle or changing itsvolume (Reul et al. 1981; Verdonk et al. 1992; Morsi andSakhaeimanesh 2000; Vlachos et al. 2001; Akutsu andMasuda 2003). These models have been used mainly togenerate physiological-like conditions, with the goal ofanalysis of the flow through particular valves (both mitraland aortic ones) and assessment of their fluid-dynamiceffectiveness (Kohler et al. 1981; Scotten et al. 1983;Verdonck et al. 1997), or in vitro demonstration of theoccurrence of particular phenomena, such as cavitationand embolism (Rambod et al. 1999; Milo et al. 2003).Other studies have been devoted to the investigation ofthe intra-ventricular flow itself, in order to betterunderstand the features that have been observed byecho-cardiography in vivo. Steen and Steen (1994) (byecho-Doppler) and Cooke et al. (2004) (comparingparticle image velocimetry (PIV) and echo-Doppler)investigated the dynamics of the vortices generatedthrough the mitral during the diastole by studying thefree flow through a fixed orifice. Vlachos et al. (2001) usedPIV in a flexible, ventricle model with bi-leaflet mitraland aortic valves to depict the flow in terms of meanquantities and turbulence. Ventricular flow has beenstudied also by means of numerical models both takinginto account the fluid–wall interaction (Lemmon andYoganathan 2000) or using the wall motion (possiblymeasured in vivo) as a boundary condition (Nakamuraet al. 2003; Merrifield et al. 2004). At the inlet and outletorifices, velocities were imposed, simulating the presenceof natural valves (Baccani et al. 2002). Finally, thedynamics of the vortices developing during the diastolewas quantitatively analysed in vivo using the magneticresonance velocity mapping (Kim et al. 1995).

Ventricular flow is highly dependent on the charac-teristics of the heart valves, in particular, the mitral valve.Tilting-disk valves, in particular, induce strong modifi-cations in the whole flow. As far as the authors know, noinvestigations have been performed on the general struc-ture of the intra-ventricular flow generated by tilting-diskvalves in the mitral position using image velocimetry.

The present work focusses on the flow features andvortex dynamics in the left ventricle containing tilting-disk valves. With the aim of investigating the mainfeatures of such a flow, a laboratory model was built. Itconsists of a cavity capable of being deformed, alongwith two prosthetic valves mounted on its top andconnected to a constant head vessel. The cardiac cycleis simulated by changing the volume of the cavity inaccordance to a physiological-like law. The flow hasbeen investigated at different frequencies and strokevolumes by means of a velocimetry technique based onFT.

2 Similarity

For a laboratory model to be representative of realconditions, we need to match the non-dimensionalnumbers that control the phenomenon. The case underinvestigation is particularly complex since the dynamicsof the valve leaflets, the fluids and that of the ventricularwalls are relevant simultaneously. As a consequence,only a 1:1 scale experiment would be exactly represen-tative. Nevertheless, in the model used in the presentstudy, the dynamics of the walls is assumed to be com-pletely controlled by imposing the variation of the ven-tricular volume, and the dynamics of the valve isthought to be like the real one since the scale is notchanged too much, as will be shown below. With theseassumptions, the dimensional analysis suggests that thecharacteristics of the pulsating flow are controlled bytwo parameters:

– The Reynolds number: Re ¼ UDm ;

– The Womersley number: Wo ¼ffiffiffiffi

D2

T m

q

;

where v is the kinematic viscosity of blood and T is theperiod of the cardiac cycle. The length scale D waschosen to be the diameter of the orifice of the mitralvalve, and the velocity scale U to be the peak velocitythrough the mitral. Since, in the present experiment, thegeometrical scale was 1:1, and the working fluid waswater, which has a kinematic viscosity of about onethird of blood, the similarity is fulfilled by using physi-ological stroke volumes and increasing the duration ofthe cycle of three times.

3 Ventricle model

A schema of the experimental setup is shown in Fig. 1.The main elements of the in vitro ventricular simulatorare:

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– a Plexiglas chamber (dimensions 180·180·300 mm)filled with distilled water, where a transparent ven-tricle pouch has been immersed

– a linear motor, which drives a piston with a diameterof 100 mm into the Plexiglas chamber, able toreproduce any arbitrary law to change the volume ofthe immersed ventricle pouch

– a computerised controller, which calculates the vol-ume change laws and drives the actuator

The ventricle pouch has been designed and realisedad hoc for this study with a transparent silicon rubber(Sylgard 184, Dow Corning, left of Fig. 2). The ventricleprototype has been designed following the indications ofSchenech and Bronzino (1998) so as to obtain a semi-ellipsoidal shape whose major and minor radii areRVo=52 mm and RHo=26 mm, respectively. We wereable to obtain a pouch with a thickness of around 1 mm.

The linear motor (Baldor Actuator) was servo-controlled on the velocity, so as to simulate a physio-logical-like volume change law for the ventricle pouch(Fung 1997). The controller allowed for different dia-stolic volumes, different cardiac frequencies and differ-ent total number of cycles. The image acquisitionhardware comprises an infrared laser (12 W in power)

and a high-speed digital camera (Photron FastCameraUltima APX), capable of acquiring images at a rate ofup to 100,000 frames/s. During the test on the ventricle,the video camera recording rate was set to 500 frame/s,and the image resolution was 512·512 pixels. The laserilluminates the vertical mid-plane of the ventricle thatintersects the two centres of the valves. Two componentsof the velocity field are measured on that plane.

Fluid circulation into the ventricle has been keptcompletely separated from the chamber and has beenseeded with pollen particles about 50 lm in diameter.Two artificial cardiac valves have been applied at theinflow and at the outflow hoses of the ventricle pouch. Inthis study, we utilised tilting-disk valves; the mitral valvehad an external nominal diameter of 30 mm and a diskof 22 mm, while the aortic valve had an external nomi-nal diameter of 20 mm and a disk of 14 mm (right ofFig. 2). The main advantage of the design of the modelis that the ventricle volume, i.e. one of the fundamentalparameters of the blood dynamics, is controlled (almost)directly. On the other hand, the model has some limi-tations: the mechanical properties of the ventricularwalls and their active role during the cycle are notreproduced; therefore, the fluid–wall interaction cannotbe reproduced—the wall motion is an externallyimposed boundary condition. The ventricle shape israther schematic and affected by the rigid, circular plateon which the valves are mounted: first, its diameter doesnot change during the cycle; second, it is planar, andmust be large enough to hold the two valves; therefore, itis large compared to the valves. In particular, the ratiobetween the maximum ventricle diameter and thedimension of the mitral valve is at the upper bound ofthe realistic values.

4 Data acquisition

Five series of experiments have been performed in orderto evaluate separately the effect of the variation of thebeat frequency and stroke volume. First, the frequencywas set to 0.39 Hz, which, from the point of view of thedimensional analysis, corresponds to 70 beats/min if theworking fluid were blood. At the same time, the strokevolume was varied from 43 ml to 93 ml, which isequivalent to cardiac outputs ranging from 3.0 l/min to6.5 l/min in the case of blood. Hence, the parameters ofthis set of tests were in accordance to an internationalFDA laboratory comparison protocol (Simenauer1986). Then, the stroke volume was kept constantlyequal to the middle value of the above series (64 ml),and the frequency was changed from 0.33 Hz to 0.44 Hz,thus, simulating beat rates from 60 bpm to 80 bpm (ifthe fluid were blood). As a consequence, the Reynoldsnumber ranged form 7,000 to 15,100, and the Womers-ley number from 12.1 to 14.0. In Table 1, for eachexperiment are reported the dimensional parameters, thenon-dimensional numbers, the cardiac output and thebeat rate they would represent in the case of blood

Fig. 1 Sketch of the experimental setup. (A) Ventricle chamber, (B)infrared laser, (C) linear motor, (D) cylinder/piston, (E) mirror, (F)constant head tank, (G) high-speed camera and (H) ventricle model

Fig. 2 The model of the ventricle (left) and of the tilting-disk valves(right)

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(indicated as ‘‘equivalent’’ quantities) and the number ofcycles acquired. Compared to the anatomy of a typicalhealthy adult, the size of the mitral valve is a littlereduced, thus, the inlet velocity and, consequently, theReynolds number are a little increased. Therefore,physiological values correspond to the lower bound ofthe explored Reynolds number range.

In Fig. 3, the law of variation of the ventricularvolume, non-dimensionalised with the stroke volume,DV(t)=[ V(t)�V(0)]/SV is plotted as a function of time,made dimensionless by means of the period T of thecardiac cycle. The curve also represents the law ofmotion of the piston. The curve was rescaled in ampli-tude and time in order to realise the conditions describedin Table 1. In addition, the time derivative of DV, madedimensionless by the period, T, was plotted. It representsthe flow rate through the valves during the cycle

(namely, the flow is through the mitral valve during thediastole (0.00T–0.66T) and through the aortic valveduring the systole (0.66T–1.00T)). The derivative wasmultiplied by 10 in order to make the graphic clearer.The averaged time evolution of the shape of the ventricleis described in Fig. 4, where the vertical and horizontalmaximum radii of the ventricle are drawn as a functionof time for three different stroke volumes (43, 64 and93 ml) at the same frequency (0.39 Hz). Comparison ofthe presented curves shows that the ratio between thehorizontal and vertical radius, RH/RV, remains at0.5±0.01 during the whole cycle, and that the curvesrescale at the parameter changes without meaningfulchanges of their shape. Results from the other testconditions (not shown here) confirm the above describedbehaviour. Due to the rigid plate on which the valves aremounted, the maximum diameter moves towards the

Table 1 Parameters of theexperiments Stroke

volume (ml)Frequency inthe model (Hz)

U (m/s) Equivalent beatrate (bpm)

Equivalent cardiacoutput (l/min)

Re Wo Numberof cycles

43 0.39 0.333 70 3.0 7,000 13.1 9064 0.33 0.425 60 3.8 8,900 12.1 564 0.39 0.496 70 4.5 10,400 13.1 2464 0.44 0.567 80 5.1 11,900 14.0 793 0.39 0.721 70 6.5 15,100 13.1 6

Fig. 3 Time law of variation ofventricular volume

Fig. 4 Evolution of thehorizontal (upper plot) andvertical (lower plot) radius ofthe ventricle. The radii are non-dimensionalised with the valueat the beginning of the diastoleand time with the duration ofthe cycle. Curves withincreasing values correspond tostroke volumes of 43, 64 and93 ml, respectively, at the samefrequency (0.39 Hz)

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apex as far as the ventricle increases its volume. Thiseffect is shown in Fig. 5, where two pictures of theventricle at its minimum and maximum volumes aresuperimposed for the reference conditions (0.39-Hz fre-quency and 64-ml stroke volume).

The images were acquired only once a dynamicsteady state had been reached, i.e. only after severalcardiac cycles since the beginning of the experiment. Thenumber of acquired cycles was limited by the memory ofthe fast camera that, at a resolution of 512·512 pixelsand a sampling rate of 500 Hz, could record for about16 s, thus, producing 2 GB of image data. In one case(64 ml and 0.39 Hz; Re=10,400; Wo=13.1), theexperiment was repeated four times, whereas in another,it was repeated 15 times in order to have a large enoughstatistical set for evaluating the turbulence fluctuationsand accelerations (43 ml and 0.39 Hz; Re=7,000;Wo=13.1).

5 Methods of data analysis

Images taken from the camera embed information aboutthe fluid-dynamic behaviour of the flow: the data analysischain proposed here starts by providing Lagrangian datavelocity extraction from the image sequence and is fol-lowed by their transformation on a regular Eulerian grid.The former step is performed by means of an FT tool:gridding of the Lagrangian data is then obtained using adistance-weighted, bilinear interpolation method.

The described experimental setup is able to reproducethe flow inside an elastic cavity by means of the controlof a few parameters: different runs of the linear motorcan lead to slightly different fluid-dynamic behaviours.

Each cycle produces a sequence of images that repre-sents a sample of the whole ensemble of runs. In order todetect main flow characters, data analysis has beencarried out on the basis of the proper orthogonaldecomposition (POD). As a matter of fact, due to asmall number of samples that was possible to beacquired for each condition, the phase average itself wasnot representative of the phenomenon. To prove thisfact, for the experiment at 70 bpm and 64-ml strokevolume that, hereafter, will be considered as a referencecondition, the analysis was performed using five cyclesand then repeated, increasing the number of cycles up to24. It was observed that the description in terms ofprincipal components does not change too much. Thismeans that it is reliable also with the small number ofcycles acquired in the other experimental conditions.Conversely, five cycles were observed to be not enoughto obtain a meaningful description in terms of phaseaverage. In addition, POD provides an indication of theflow repeatability condition at each time and furnishes adimensional reduction for the velocity field against time.

5.1 Feature tracking

Feature tracking (FT) is an algorithm that extracts localregions of interest (features) from images and identifiesthe corresponding features in the next image of thesequence. For small time intervals between the frames ofa sequence, it is reasonable to assume that a conserva-tion law for the image intensity, I(x, y), holds:

DI x; yð ÞDt

¼ @I@tþ @I@x

ux þ@I@y

uy ¼ 0

This is only one equation with two unknowns and,due to noise and deformation, it is not exactly fulfilled ineach point of the field. Therefore, a cost function isdefined over an interrogation area, W centred in (x, y):

e ¼Z

W ðx;yÞ

DIDt

� �2

dW

and the velocity in (x, y) is computed as the displace-ment (ux, uy) that minimises e by imposing that itsderivatives vanish.

Thus, a closed set of linear of equations, with un-knowns ux and uy, results (Tomasi and Kanade 1991) in:

Z

W x;yð Þ

@I@x

� �2 @I@x@I@y

@I@x@I@y

@I@y

� �2

0

@

1

A

ux

uy

� �

dW ¼Z

W x;yð Þ

@I@x

@I@t

@I@y

@I@t

!

dW

This can be considered as a PIV-like technique basedon a distance measure minimisation problem (Lucas andKanade 1981). This well known algorithm can be clas-sified as a local (tracking), differential and optical flowmethod. The discretisation of the derivatives, on whichthis method is based, requires that displacements are

Fig. 5 Ventricle shape at its minimum and maximum size during arun at 0.39 Hz with 64 ml stroke volume

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small (on the order of 1 pixel); thus, larger displace-ments are resolved using a pyramidal strategy (Burt andAdelson 1983). It is possible to demonstrate that asolution of the FT set of equations exists and is wellconditioned if the system itself is solved in well texturedpoints, called ‘‘good features to track’’ (Shi and Tomasi1994). These are points where the eigenvalues of the FTset of equations are large enough, that is, where highintensity variation in both x and y directions are present.Their definition is a consequence of the FT equationset’s solvability condition: no a priori consideration hasbeen made about the object to tracked.

As in each tracking procedure, FT flow velocitydescription consists of a sparse data set: in order topermit data comparison between different runs of theexperiment, and with the aim to provide estimations ofthe other quantities (e.g. vorticity), a resampling algo-rithm is then applied to sparse Lagrangian data.

5.2 Data resampling

Among the great number of available resampling tech-niques, in this work, a fully automatic distance-weightedinterpolation has been adopted (Chetverikov 2003).

Given a point Pg on a regular grid, the set {Pj} (j=1,2..., J) of the positions of tracked features lying within acertain distance from Pg is built. Each point Pj has avelocity vector uj. Let Dj

2 be the squared distance from Pj

to Pg, D2 be the mean of Dj2 over all j, and the following

coefficients:

aj ¼ exp �D2

j

D2

!

and bj ¼aj

PJj¼1 aj

;

where 0<(aj, bj) £ 1 andPJ

j¼1 bj ¼ 1 is a partition ofunity. The interpolated velocity vector in Pg is thencomputed as:

ug ¼X

J

j¼1bjuj

.This method requires the definition of a minimum

number of points belonging to the set Pj and a maximumvalue for the search distance. In this implementation, thesearch for the Pj set starts in a small circular region R(radius of 3 pixels) surrounding the grid point: if thenumber, J, of the found features is lower than a fixedminimum value (3), the radius of R is increased by1 pixel and a new search starts. If the maximum valuefor the radius of R is reached (20 pixels), no value isassigned to the grid point, otherwise, the describedresampling procedure is applied.

5.3 Proper orthogonal decomposition

Proper orthogonal decomposition (POD), also calledprincipal component analysis (PCA) or Karhunen–

Loeve decomposition, is a powerful and elegant methodof data analysis aimed at obtaining low-dimensionalapproximate descriptions of high-dimensional processes.

POD analysis yields a set of empirical eigenfunctionsthat describes the dominant behaviour or dynamics of agiven problem. This technique was introduced in theturbulence studies by Lumley (1967): it can be used for avariety of applications, including the derivation ofreduced-order dynamical models (Holmes et al. 1996),steady analysis and design of inviscid airfoils (LeGresleyand Alonso 2001), image processing (Sirovich and Kirby1987) and pattern recognition (Fukanaga 1972).

Data analysis using POD is often conducted to ex-tract the most correlated shapes (modes) or basis func-tions from experimental data or detailed simulations ofhigh-dimensional systems, for subsequent use in Galer-kin projections that yields low-dimensional dynamicalmodels. In this paper, the POD technique is used in twodifferent ways related to the raw data processing.

Suppose that, after the resampling, the velocity isknown in the N nodes of the regular grid, and that Kcycles have been acquired. Finally, let P be the numberof snapshots acquired during one cycle. In order toperform POD analysis, the data were organised in anarray describing the state of the flow at the phase p ofthe kth cycle as follows:

w n; k; pð Þ ¼ ux n; k; pð Þ n ¼ 1; . . . ;Nw n; k; pð Þ ¼ uy n� N ; k; pð Þ n ¼ N þ 1; . . . ; 2N

where ux(i, k, p) and uy(i, k, p) indicate the two com-ponents of the velocity at the ith node of the N nodes ofthe grid.

5.3.1 POD for robust estimation

The first application stems from the fact that the numberof cycles acquired in each condition is not large enoughto permit a reliable evaluation of the phase average.Therefore, in order to describe the characteristicbehaviour of the flow, POD was used to extract the mostcorrelated part of the signal between the differentsnapshots of the velocity field acquired at the samephase. At each phase p, one has a set of K independentsamples S={w(n, 1, p),..., w(n, K, p)}, so the PODmethod was used to decompose the flow field into aweighted linear sum of orthogonal eigenfunctions as:

w n; k; pð Þ ¼X

K

i¼1ai n; pð Þui k; pð Þ:

The correlated part of the signal is described bymeans of the eigenfunctions, /i of the correlation tensor:

R k; k0; pð Þ ¼X

2N

n¼1w n; k; pð Þ � w n; k0; pð Þ

that also represents the principal directions in the phasespace at each p and approximate typical members of a

327

data ensemble, S, better than representations of thesame dimension in terms of any other bases (Holmeset al. 1996; Sirovich 1987).

The dimension of the obtained eigenvalue problem isequal to the number of the realisation of the experiment,K, which is typically much lower than the dimension (Por 2N) of the eigenvalue problems considered in the nextparagraphs. The most important property of POD isthat the eigenvalues of R are equal to the signal energyassociated with each eigenfunction and the total signalenergy is given by the sum of the eigenvalues. Thisproperty will be extensively used in the data analysisshown in this paper.

The procedure applied to reject the non-correlatedpart of the signal consists of the evaluation, at eachphase, of the eigenfunctions /i(k, p) and the eigenvalueski(p) of the correlation tensor, R(k, k¢, p). The coeffi-cients ai were computed by projecting the velocity fieldon the eigenfunctions:

ai n; pð Þ ¼X

K

k¼1w n; k; pð Þui k; pð Þ:

The number of modes required to capture at least85% of the kinetic energy (KE) of the system,

KE pð Þ ¼ 12

P

2N

n¼1

P

K

k¼1w2 n; k; pð Þ

� �

; was computed by look-

ing for the first k* satisfying the following inequality:

X

k� pð Þ

i¼1ki pð Þ>0:85

X

K

k¼1ki pð Þ ¼ 0:85 �KE pð Þ:

Finally, the ‘‘robustly representative’’ flow field, v(n,p), was obtained by averaging the fields rebuilt usingonly the k* (p) modes:

v� n; pð Þ ¼ 1

K

X

K

k¼1

X

k� pð Þ

i¼1ai n; pð Þui k; pð Þ:

The number of modes required to represent a givenfraction of the total energy of the system can be regardedas a description of its number of degrees of freedom at aspecific phase p or, in other words, as the correlationdegree between the members of the event ensemble atthat time. The higher the number of modes, the lessrepeatable the topological configuration and vice versa.Figure 6 clarifies this point: the time evolution of thenumber of modes required to capture at least the 85% ofthe total energy of the 24 events at the reference condi-tions is sketched as a normalised value, together with thetime evolution of the KE. It is clearly observed that,when the KE increases, the number of degrees of free-dom decreases and vice versa. This fact can be explainedby observing that, if a strong external source of energy isapplied to the system (during the two diastolic andsystolic phases), the topology of the flow inside thecavity is forced to assume the same configuration. Inabsence of an external source of energy, such as that

between the two diastolic peaks, the topology of the flowinside the cavity can be quite different between events,due to the sensitivity to the initial conditions.

To cut out the remaining part of noise from the flowfield obtained from the above-described procedure, thePOD decomposition was applied to the v(n, p) velocityfield. The aim of the procedure is to reject the part of thesignal that is not correlated in time, therefore, the fol-lowing time correlation tensor was considered:

~R p; p0ð Þ ¼X

2N

n¼1v� n; pð Þ � v� n; p0ð Þ;

and then the above procedure was repeated symmetri-cally.

Its eigenfunctions, wi, and eigenvalues, li, werecomputed; the number of obtained modes is equal to thenumber of snapshots acquired during a single cycle, P,(about 1,280 for the reference case). The POD coeffi-cients, bi(n), were obtained by projecting the velocityfield v(n, p) onto the eigenfunctions:

bi nð Þ ¼X

NT

p¼1v� n; pð Þ � wi pð Þ:

Finally the field was rebuilt using the first 20 modesthat retain more than 98% of the total energy of thesystem (as shown in Fig. 7):

v n; pð Þ ¼X

20

i¼1bi nð Þwi pð Þ:

This reconstruction has the property to reject the partof the signal that is less correlated in time. As a conse-quence, POD acts as a filter that cuts out the noise, and,thus, the velocity in each node of the grid is filtered in avery efficient way. The effect of such a filter on the datais shown in Figs. 8 and 9, where the vertical and

Fig. 6 Time evolution of the number, D, of modes required torebuild the system with at least 85% of the total energy normalisedby the total number of modes (red line) and KE during the differentcycles at the reference conditions (other lines)

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horizontal components of the velocity, at a grid point infront of the mitral valve, are plotted, respectively. Thegreen line represents raw data, whereas the red linerepresents filtered data.

6 Results

6.1 Velocity and vorticity

An overall description of the time evolution of thephenomenon is depicted in Fig. 10 for the so-calledreference case: stroke volume of 64 ml and an equivalent

beat rate of 70 bpm (Re=10,400; Wo=13.1). The ninemore meaningful phases have been chosen: on the top ofeach plot, the non-dimensional time is reported and thecorresponding point in the flow rate graph is representedby a red dot on the lower left corner of the graphs. Thevelocity vectors are superimposed onto a colour map ofthe vorticity, non-dimensionalised by the period, T, ofthe cardiac cycle.

The first field (t/T=0.07) is just after the beginning ofthe filling of the ventricle during the early diastole. Onthe top of the plot, in correspondence to the mitralvalve, two jets are generated on the two sides of thetilting disk: the left one is the strongest, both because itcomes from the largest portion of the inlet and because itis deflected by the curvature of the disk, as sketched inthe white circle on the same plot. At that time, thevorticity is still low.

At t/T=0.110, just before the end of the accelerationof the early diastole, the structure of the flow is basicallythe same as before, except for the fact that the jets hadtime to propagate to the middle of the ventricle and theirintensity is increased. Four peaks in the vorticity map(A, B, C, D) suggest the presence of two vortex rings(Green 1995), one for each jet, that are moving diago-nally through the ventricle. The inclination of the jets isnot due to the orientation of the valve (vertical axis inthe graph), but to the inclination and curvature of thedisk, which seems to play an essential role in increasingthe velocity of the left (primary) jet. To the extent of theauthors’ knowledge, the evolution of the two vortexrings inside the ventricle is observed for the first time,whereas many investigators described the developmentof a single vortex ring: in vivo with natural valves (e.g.Bot et al. 1990; Kim et al. 1995); in vitro with openorifice (e.g. Steen and Steen 1994; Cooke et al. 2004); invitro with a bi-leaflet mitral valve (e.g. Akutsu andMasuda 2003); numerically with an inlet conditionsimulating natural valves (e.g. Baccani et al. 2002‹;Nakamura et al. 2003).

Comparison with these results suggests that thepresence of two vortex rings is due to the flow-splittingeffect of the curved, tilted disk. Since the large, singleleaflet is curved also in the direction of the flow, and isnot completely orthogonal to the valve plane whenopened, it tends to generate two separate jets that havedifferent initial directions. On the contrary, the config-urations investigated by the other authors included openorifices and natural valves, that have no leaflets splittingthe flow, or bi-leaflet valves, that have leaflets which arenot curved in the direction of the motion and orthogonalto the valve plane when completely open. Therefore, theperturbation that they induce in the flow is much lowerand is not sufficient to generate two separate vortexrings.

The vortex ring on the right (generated by the sec-ondary jet) is asymmetric because of the interaction withthe primary jet. This asymmetry is emphasised at thepeak of the early diastolic wave (E-wave) (t/T=0.148),when only the right side of the secondary vortex ring can

Fig. 8 Original (green) and filtered (red) data (vertical velocitycomponent)

Fig. 9 Original (green) and filtered (red) data (horizontal velocitycomponent)

Fig. 7 Cumulative energy for the first 50 modes. The 98% energylevel is reported as a solid line

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be recognised (C). The primary vortex ring has reachedthe opposite wall of the ventricle and will begin toproceed parallel to the wall of the ventricle while dissi-pating and breaking into smaller structures. At this time,the measured vorticity is a maximum in the region of thetrailing jet due to the high inlet velocity.

At the end of the E-wave (t/T=0.272), the velocityfield mainly consists of two rotating structures, one onthe left, with negative vorticity, which originates fromthe left part of the primary vortex ring (A), and one onthe right, which is generated by the coalescence of theright parts of the secondary vortex rings (C) and ofthe right part of the primary one (B) that, after reachingthe opposite side of the ventricle, moved upwards,

parallel to the wall. At that stage, the vortices are widerbut less intense than before. During the later stages ofevolution, the vortex rings probably assume a complex3D shape, due to the interaction with the ventricle wallsand with other vortices. Regardless, they can be clearlyrecognised in the 2D view that the acquisitions give,suggesting that it remains a plane of symmetry.

Then, the volume of the ventricle remains constantfor an interval during which the structures generatedduring the E-wave vanish almost completely, as is shownby the plot at t/T=0.450, where neither velocity norvorticity patterns can be discerned. At the second dia-stolic wave (A-wave) that originates from the contrac-tion of the atrium, a similar, but less intense,phenomenon is observed. At the peak, t/T=0.537,though the inlet velocity is a relative maximum, the twovortex rings had not yet time to develop, so they are seen

Fig. 10 Non-dimensional velocity (vectors) and vorticity (colour)field at different times, with stroke volume 64 ml and 0.39 Hz

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completely developed only at the end of the A-wave (t/T=0.604). At this time, there is the same structure of thethree vortices that are found after the first diastolic peak.The vortices have reached the opposite side of the ven-tricle, but the jet is not present any more, since the atrialcontraction is finished. Therefore, the sheet of vorticityin the trail of the primary vortex ring is missing and thevortices are more isolated. This is also the instant beforethe beginning of the systole, during which, every struc-ture of the field is destroyed, and an outward jet isgenerated.

At the systolic peak (t/T=0.833), there is still somememory of the diastolic flow, since it is possible to rec-ognise a general circulation upwards along the walls andthen towards the aortic valve. A smooth peak of positivevorticity is placed on the bottom right. That memory iscompletely lost at the end of the systole (t/T=0.937),when the only visible pattern is the one due to the out-flow through the aortic valve. This is also what remainsof the flow when the diastole starts again at the nextcycle.

In order to point out the effect of the variation of thecontrolling parameters on the structure of the flow, thebehaviour of the jet during the first phase of the filling is

examined. In Fig. 11, the velocity and vorticity fields areplotted at two characteristic instants of the E-wave (themaximum and the end) for three different conditions.First, the beat rate was reduced at constant stroke vol-ume (0.33 Hz and 64 ml; Re=8,900; Wo=12.1). Thestructure of the flow is about the same as in the abovedescribed case (plotted in Fig. 10) at the maximum(upper left of Fig. 10), but at the end (lower left ofFig. 11), the three vortices (indicated in Fig. 10 with A,B and C) are weaker and not clearly defined. The reasonseems to be the increase of the effect of the viscosity (asthe decreased Reynolds number demonstrates). Also, onincreasing the beat rate with constant stroke volume(80 bpm; 64 ml; Re=11,900; Wo=14.0; centre ofFig. 11), the structure of the flow maintains its mainfeatures, suggesting that the first phase of propagationof the vortex rings scales with the inflow velocity(Shusser and Gharib 2000). The observation of the flowfield at the end of the first filling (lower centre of Fig. 11)supports the interpretation given above: increasing theReynolds number produces vortices that are well definedand strong, since the effect of viscosity is less intense. Inthe third case, the beat rate was kept the same as in thereference case and the stroke volume was increased to93 ml; therefore, the Reynolds number increased to15,100, whereas the Womersley number remained at13.1. At the maximum of the E-wave (upper right ofFig. 11), the vortices had already gone further way thanbefore. The flow pattern is similar to the previous oneonly at the end of the E-wave. This is consistent with thefact that the velocity of the vortices scales with the pis-ton velocity, whereas the time scale, T, does not;

Fig. 11 Non-dimensional velocity (vectors) and vorticity (colour)fields at the diastolic peak (t/T=0.148, top) and at the end of theearly filling phase (t/T=0.272) for different stroke volumes andfrequencies (SV=64 ml and 0.33 Hz, on the left; SV=64 ml and0.44 Hz in the centre; SV=93 ml and 0.39 Hz on the right). Thered arrow here, and in the next figures, represents the unit non-dimensional velocity. All the vector plots use the same scale for thedimensional velocity

331

therefore, the vortices move, in the non-dimensionaltime description, about one and half times faster than inthe previous cases.

Moreover, though the effects of the viscosity shouldbe lower, no vortices can be recognised at the end of theE-wave (lower right of Fig. 11). Experiments on startingjets through orifices suggest that the development of avortex ring stops when the so-called formation time,T*=4DV(t)/(pD3), attains a threshold value of about 4.At that time, the vortex ring stops accumulating circu-lation and, if the inflow continues, the remaining vor-ticity is distributed along the border of a trailing jet(Gharib et al. 1998). Assuming that this is alsoapproximately true in our (more complicated) configu-ration permits the estimation of the time when thedeveloping of the vortex ends and the one of the trailingjet begins. In the case of a stroke volume of 64 ml, theformation time reaches nearly 4 almost at the early peak(t/T=0.147). In the case of larger stroke volume (93 ml),the critical value of the formation time is reached earlier(t/T=0.127), therefore, a larger fraction of the flux ofvorticity entering the ventricle during the E-wave fromthe mitral does not contribute to the straightening of thevortex ring, but pertains to the trailing jet, and, there-fore, interferes with the vortex ring. This is why thevortical structure is strongly deformed already at theearly peak (upper right of Fig. 11), and any coherence inthe field is lost at the end of the E-wave. Since the fieldsthat are shown here are not averages but the ‘‘morecorrelated’’ part of the flow, no structures are visible inthe plot at the end of the E-wave in this case.

It is worth mentioning that the length scale control-ling the propagation and the successive evolution of thestructures is the size of the ventricle. Consequently, theabove described behaviour, such as degeneration toturbulence and interaction between the jet and theopposite wall, could be influenced by the diameter of theventricle model that is large compared to a typical adultconfiguration. On the contrary, the initial generation ofthe two vortex rings is controlled by the mitral geometry,and the explored range of Reynolds numbers, based onMitral diameter, includes values corresponding to nor-mal physiological conditions. Therefore, the resultsregarding this phase of the flow can be considered asrepresentative also of physiological conditions for anormal adult.

6.2 Viscous and turbulent shear stresses

It is known that forces exerted on the blood elements area possible cause of thrombus formation (Stein andSabbah 1974), as well as haemolysis (Blackshear 1972).Moreover, it has been recognised that shear stresses cancause platelet activation (Belval et al. 1984). Therefore,when evaluating the blood flow, the description of theforces that the flow induces on its constituent cells is alsoneeded. The investigation of the mechanisms that pro-duce cell damage is not easy. Many different explana-

tions have been proposed; in any case, there is agreementon the fact that the tangential component is the mostrelevant. Obviously, it depends on the orientation of thesurface that one considers, but the surface of a blood cellis closed, so it assumes all orientations. Therefore, onecan be sure that, somewhere, the surface is exposed tothe maximum possible shear stress. Thus, this will be thequantity used to describe the tensional state of the fluid.Since in the present experiment only two components ofvelocity have been measured on a plane of symmetry, the2D case will be considered in the following, but similarresults can be obtained also in 3D (if three componentsare known) (Grigioni et al. 2002). Using the constitutiveequation for Newtonian fluids (such as can be consid-ered for blood flowing in large vessels), the shear stress,st, on a surface with normal n=(cos h, sin h)T is:

st hð Þ ¼ s1&00&s2

� �

coshsinh

� ��

�sinhcosh

� �

¼ s2 � s1ð Þ2

sin 2hð Þ;

where si and ei are the eigenvalues of the stress tensorand the strain velocity tensor, respectively, and theexpression holds in their common, principal referenceframe. st is a maximum for h=p/4, where it is:

stmax ¼s2 � s1ð Þ

2¼ l e2 � e1ð Þ:

This expression was used for the evaluation of ssmax:first, the gradient of the velocity is computed from thegridded velocity at each node; second, the eigenvalues ofits symmetric part (i.e. the strain rate tensor) are com-puted; finally, the maximum shear stress was computedand made dimensionless by means of the scale qU2.

In Fig. 12, the non-dimensional, maximum shearstress is plotted at the three more meaningful instants ofthe cycle for the case with stroke volume 43 ml andfrequency 0.39 Hz (Re=7,000; Wo=13.1): the end ofthe acceleration of the early diastole, the first diastolicpeak and the systolic peak. These values are obtainedfrom the average of all the 90 cycles acquired in theseconditions. The standard error in the estimation of theaverage where the stress is a maximum is 6%. Duringthe first phase of development of the jet, just before themaximum of the E-wave (plot on the left), the highestvalues of shear stresses are localised at the sides of thejets. This is observable both on the primary andthe secondary jets. At the first diastolic peak (plot in thecentre), the flow is well developed through the wholeventricle and, at the sides of the primary jet, the valuesreach their maximum over the whole cycle. In addition,a new zone of elevated shear stress is found, where theprimary jet impinges on the ventricular wall, and in theadjacent area, where the flow is parallel to the wall,generating a shear layer. In the subsequent phases (notshown), the shear stresses do not reach values compa-rable to the ones described above. Significant values arefound again during the crucial phase of the systole (plot

332

on the right), when the flow is almost at rest everywhere,except for a sudden acceleration in the proximity of theoutlet. The strong gradients found in that zone createalso high values of shear stresses. Their magnitude is onthe order of one half of the global maximum reached atthe diastolic peak.

Using the same formulation as above with the Rey-nolds stress tensor, the maximum turbulent shear stres-ses have been computed at the same non-dimensionaltimes (Fig. 13). The range of values is several orders ofmagnitude higher than before, but the overall structureis about the same. High turbulent shear stresses arefound on the sides of the two jets during the E-wave andin correspondence with the shear layer generated by themain jet when it is deflected along the ventricular wall.At the systolic peak, the higher values are found justupstream of the aortic valve, as in the case of viscousstresses, but, in this case, the value is lower if comparedwith the absolute maximum (about 1/8 rather than 1/2).

6.3 Accelerations

As far as the so-called ‘‘good features to track’’ coincidewith seeding particles, the FT furnishes the particle

trajectories, and seeding particles, if chosen correctly,can be assumed to behave as fluid elements. Therefore,FT applied to images of a seeded fluid yields aLagrangian description of the velocity field. The analysisof these trajectories can describe a type of force actingon the blood elements that can only be measured in thisway: the inertial force. Since this is proportional to theacceleration through the mass, hereafter, the accelera-tions will be presented. They have been computed fromparticle trajectories using centred discrete derivativesand made dimensionless by means of the gravitationalacceleration, g. A map of the intensity of the accelera-tions, averaged over the whole cycle, in the 90 runs withstroke volume 43 ml and frequency 0.39 Hz, is presentedin Fig. 14. The two jets generated by the mitral valveduring the diastolic phase are apparent as two zones ofhigh acceleration. This is due to the entrainment ofexternal (slow) fluid into the (fast) jet, with a corre-sponding sudden increase of the velocity of entrainedparticles. The higher values are located at the beginningof the primary jet, just close to the mitral, and arearound 5g. Also, in front of the aortic valve, there is azone of fairly high values (on the order of 3 g). This isdue to the acceleration of fluid particles as theyapproach the outlet during the systole.

Keeping in mind that these are values averaged overthe whole cycle, it easy to predict that the instantaneousvalue can be much higher. To demonstrate this, thefraction, F, of particles that exceed a given accelerationis plotted in Fig. 15. From the plot, it is apparent thatthere is a meaningful fraction of the fluid elements that

Fig. 12 Non-dimensional maximum viscous stresses during theseries at stroke volume 43 ml and frequency 0.39 Hz (Re=7,000;Wo=13.10). Values on the right bar are multiplied by 103

Fig. 13 Non-dimensional maximum Reynolds stresses at differentpoints of the cycle at stroke volume 43 ml and frequency 0.39 Hz(Re=7,000; Wo=13.10)

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are subject to quite high accelerations: one particle in300 exceeds 10g and one in 500 exceeds 15g.

7 Conclusions

The laboratory model of the ventricle that was designedand realised permitted a detailed investigation of theventricular flow during the whole cardiac cycle. Themodel permits optical access to the flow; hence, veloci-metry based on image analysis could be used (namely, thefeature tracking (FT) method). This provided a completespatial description of the flow every 2 ms. The flowwas investigated at different beat rates and stroke vol-umes, and, for each condition, ‘‘robust representative’’was extracted by means of the proper orthogonal

decomposition (POD). The analysis of the velocity andvorticity fields demonstrated that two jets are generatedat each diastolic peak on the sides of the opened disk ofthe valve. At the leading edge of the developing jets, twovortex rings are generated that proceed diagonallythrough the ventricle, impinging on its opposite side,close to the apex. They are seen to interact with eachother and with the walls. The factors that are observed tocontribute to destroy the vortices are mainly the viscosityand the perturbations induced by the vorticity that per-tain to the trailing jet. The former is measured by theReynolds number, whereas the latter by the formationtime that is reached at the end of the early diastolic wave.As a consequence, the persistence of the vortices is in-creased by an increase in the Reynolds number and by adecrease in the final formation time. Finally, the analysisof the particle trajectories permitted the evaluation of thespatial distribution and the amplitude of the inertialforces that act on fluid elements.

Acknowledgements The authors wish to thank Fabiana Giorgi forher contribution in the realisation of the silicon rubber model of theventricle and the data acquisition. This work was funded by theMinistero dell’Istruzione, dell’Universita e della Ricerca (MIUR),Italy (FIRB project no. BAU01Z44F).

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