Performance of time-frequency representation techniques to measure blood flow turbulence with...

PII: S0301-5629(00)00361-6

● Original Contribution

PERFORMANCE OF TIME-FREQUENCY REPRESENTATIONTECHNIQUES TO MEASURE BLOOD FLOW TURBULENCE WITH

PULSED-WAVE DOPPLER ULTRASOUND

GUY CLOUTIER,*†‡ DANMIN CHEN,* L OUIS-GILLES DURAND**Laboratory of Biomedical Engineering, Institut de recherches cliniques de Montreál; †Laboratory of Biorheology

and Medical Ultrasonics, Centre Hospitalier de l’Universite´ de Montreal; and‡Department of Radiology, Universite´de Montreal, Montreal, Quebec, Canada

(Received12 July 2000; in final form 20 November2000)

Abstract—The current processing performed by commercial instruments to obtain the time-frequency repre-sentation (TFR) of pulsed-wave Doppler signals may not be adequate to characterize turbulent flow motions. Theassessment of the intensity of turbulence is of high clinical importance and measuring high-frequency (small-scale) flow motions, using Doppler ultrasound (US), is a difficult problem that has been studied very little. Theobjective was to optimize the performance of the spectrogram (SPEC), autoregressive modeling (AR), Choi–Williams distribution (CWD), Choi–Williams reduced interference distribution (CW-RID), Bessel distribution(BD), and matching pursuit method (MP) for mean velocity waveform estimation and turbulence detection. Theintensity of turbulence was measured from the fluctuations of the Doppler mean velocity obtained from asimulation model under pulsatile flow. The Kolmogorov spectrum, which is used to determine the frequency ofthe fluctuations and, thus, the scale of the turbulent motions, was also computed for each method. The best setof parameters for each TFR method was determined by minimizing the error of the absolute frequencyfluctuations and Kolmogorov spectral bandwidth measured from the simulated and computed Doppler spectra.The results showed that different parameters must be used for each method to minimize the velocity variance ofthe estimator, to optimize the detection of the turbulent frequency fluctuations, and to estimate the Kolmogorovspectrum. To minimize the variance and to measure the absolute turbulent frequency fluctuations, four methodsprovided similar results: SPEC (10-ms sine-cosine windows), AR (10-ms rectangular windows, model order5 8),CWD (wN and wM 5 10-ms rectangular windows,s 5 0.01), and BD (wN 5 10-ms rectangular windows,a 5 16).The velocity variance in the absence of turbulence was on the order of 0.04 m/s (coefficient of variation rangingfrom 8.0% to 14.5%, depending on the method). With these spectral techniques, the peak of the turbulenceintensity was adequately estimated (velocity bias< 0.01 m/s). To track the frequency of turbulence, the bestmethod was BD (wN 5 2-ms rectangular windows,a 5 2). The bias in the estimate of the210 dB bandwidth ofthe Kolmogorov spectrum was 3546 51 Hz in the absence of turbulence (the true bandwidth should be 0 Hz),and 21936 371 Hz with turbulence (the simulated210-dB bandwidth was estimated at 1256 Hz instead of 1449Hz). In conclusion, several TFR methods can be used to measure the magnitude of the turbulent fluctuations. Totrack eddies ranging from large vortex to small turbulent fluctuations (wide Kolmogorov spectrum), the Besseldistribution with appropriate set of parameters is recommended. (E-mail: [email protected]) © 2001 WorldFederation for Ultrasound in Medicine & Biology.

Key Words:Ultrasonography, Echocardiography, Simulation, Spectral methods, Heart valve disease, Insuffi-ciency, Regurgitation, Stenosis, Turbulence, Reynolds stress.

INTRODUCTION

Turbulence is an irregular eddying motion in time andspace characterized by velocity and pressure fluctuationsabout their mean values. These turbulent fluctuations are

made of eddies of different sizes within larger eddies.Pathophysiologically, at a given flow rate, turbulenceincreases the flow resistance and the wall shear ratecompared to laminar flow (Milnor 1989). In the cardio-vascular system, turbulence has been associated withpoststenotic dilatation, aneurysms, atherogenesis andthrombosis (Nichols and O’Rourke 1990). A region ofthe cardiovascular system where turbulence plays a sig-nificant role is the ascending thoracic aorta distal to the

Address correspondence to: Dr. Cloutier, Laboratory of Biomed-ical Engineering, Institut de recherches cliniques de Montreál, 110avenue des Pins Ouest, Montreál, Quebec, Canada, H2W 1R7. E-mail:[email protected]

Ultrasound in Med. & Biol., Vol. 27, No. 4, pp. 535–550, 2001Copyright © 2001 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/01/$–see front matter

535

aortic heart valve. Thrombogenic complications afterreplacement of diseased heart valves with mechanicalprostheses have been shown to be related to the hemo-dynamics in the vicinity of the valve, namely flow stasisand turbulent high shear stresses (Butchart 1998; Chese-bro and Fuster 1998). One mechanism by which turbu-lence promotes thrombogenic complications in patientswith mechanical heart valve substitutes is the effect ofturbulent Reynolds stresses on the rupture of erythro-cytes (Sallam and Hwang 1984) and platelets (Hung etal. 1976) and, possibly, on the function of endothelialcells (Stein et al. 1977). The release of adenosine diphos-phate by ruptured blood cells or simply the shearingeffect can promote platelet aggregation, adhesion to theendothelium and thrombosis (Butchart 1998; Kroll et al.1996).

According to the literature, very few studies haveinvestigated the accuracy of Doppler ultrasound (US) totrack turbulent velocity motions. Giddens and Khalifa(1982) proposed a phase-lock loop frequency-trackingmethod to measure the time fluctuations of the Dopplermean velocity. This approach was applied to study post-stenotic flow disturbances in dogs (Talukder et al. 1986).The time resolution of their instrumentation was greaterthan 5 ms. Zero-crossing detectors, providing a timeresolution on the order of 2.5 ms, were proposed to mapturbulent velocity fluctuations within the aorta of patientswith mechanical heart valves (Nygaard et al. 1994b). Thesystem developed by this group uses a cuff containingfive Doppler probes in contact with surgically exposedaortas (Nygaard et al. 1994a). Today, although bettertime resolutions can be achieved with commercial USinstruments, the variance of the time-frequency represen-tation (TFR) method used to compute the real-timeDoppler spectrum may not allow turbulent motions to beefficiently detected.

To our knowledge, the detection of random turbu-lent fluctuations covering a wide range of eddy sizes hasnot been addressed in the US literature. For the purposeof measuring turbulence produced by prosthetic heartvalvesin vivo, this is an important objective to achievebecause blood cell hemolysis and thrombosis may notonly be related to the turbulence intensity and duration ofexposure, but also to the scale of the turbulent fluctua-tions. Clinical studies aiming at evaluating the relation-ship between turbulence stresses and thrombogenic com-plications in patients after heart valve replacement can-not be performed with the current technologies. Only UScan provide such a tool for noninvasive studies in pa-tients. The magnetic resonance imaging (MRI) technol-ogy is currently under development and may eventuallyrepresent an alternative to US (Fontaine et al. 1996;Walker et al. 1995).

Two basic criteria have to be optimized for obtain-

ing an optimum mean frequency computed from theDoppler spectrum. First, the variance of the TFR methodhas to be low to avoid an overestimation of the turbu-lence intensity, and second, it is mandatory to have atechnique with a good time-frequency resolution to allowthe detection of fast turbulent fluctuations. Of course, asignal x(n) cannot simultaneously be time-limited andbandlimited. The selection of the optimum TFR methodshould be based on a compromise between time andfrequency resolutions. In the current study, the varianceand accuracy of six TFR methods to track turbulentrandom fluctuations were evaluated with a simulationmodel of the Doppler cardiac signal.

METHODS

Time-frequency representation methods tested

The Spectrogram.The spectrogram is a traditionalmethod of analyzing a signal in the joint time-frequencydomain. It provides the TFR of the signalx(n), at thediscrete timen, by computing the power spectrum of asmall segment of the signal aroundn. It is computed byusing the discrete Fourier transform (DFT) as follow:

SPECx(n,k)5 U Ot52`

1`

e2j2pkt/Nx(t)w(t 2 n)U 2

, (1)

wheren and k correspond to the discrete time and fre-quency variables, respectively,j is the imaginary num-ber, N is the number of samples of the signalx(n), andw(t) is a window function.

Autoregressive modeling.This method is similar tothe spectrogram, except that the DFT of each segment ofthe signalx(n) is replaced by an autoregressive (AR)model given by:

ARx~n,k! 5dp

2~n!

U 1 1 Om51

p

a~m, n!expS2j2pk

NmDU 2 ,

(2)

wheredp2(n) is the variance of the modeling error signal

corresponding to the model orderp. The complex time-varying coefficientsa(m,n) are computed by using theYule–Walker equations together with the Levinson–Durbin algorithm for an efficient recursive solution. Thetime windoww(t) is applied to the signalx(n), at the timen, to extract a small segment before computing the co-efficients a(m). More details on this algorithm can befound in Kay and Marple (1981)).

536 Ultrasound in Medicine and Biology Volume 27, Number 4, 2001

The Choi–Williams distribution.Choi and Williams(1989) introduced a bilinear distribution by using anexponential kernelF(j, t) 5 e2(j2t2)/s. It is a member ofa group of time-frequency distributions (Cohen group)individually characterized by a cross-term suppressingtime-frequency smoothing kernel function. The Choi–Williams distribution is given by (Jeong and Williams1992a):

CWDx~n,k!

5 2 Ot52`

1`

wN~t!e2j2pkt /NF Om52`

1`

wM~m!1

Î4pt2/sexp

S2~2m 1 t!2

16t2/s Dx~n 1 m 1 t! x* ~n 1 m!G , (3)

where wN(t) is a symmetric window having nonzerovalues within the range of2N/2 # t # N/2, wM(m) is arectangular window that has a value of 1 between2M/2 # m # M/2, the parameters is used to trade-offauto-term resolution for cross-term suppression, and *indicates the complex conjugate.

The Choi–Williams reduced interference distribu-tion. Jeong and Williams (1992b) defined a class ofTFRs called the reduced interference distributions (RID).The discrete-time Choi–WilliamsRID is:

CW2 RIDx~n,k!

5 2 Ot52`

1`

wN~t!e2j2pkt /NF Om5~2t2utu!/ 2

~2t1utu!/ 2 1

Î4pt2/sexp

S2~2m 1 s!2

16t2/s Dx~n 1 m 1 t! x* ~n 1 m!G . (4)

The Bessel distribution.Guo et al. (1994a) intro-duced the Bessel distribution (BD) based on the RIDconcept. The discrete-timeBD is expressed as:

BDx~n,k! 5 2 Ot52`

1`

wN~t!e2j2pkt /NF Om5~2t22autu!/ 2

~2t12autu!/ 2 2

pautu

Î1 2 S2m 1 t

2at D 2

x~n 1 m 1 t! x* ~n 1 m!G , (5)

wherea . 0 is a scaling factor. For all time-frequencymethods described above (SPEC, AR, CWD, CW-RIDand BD), the lag between each time windoww(t) orwN(t) was selected to 0.2 ms (time resolution).

The matching pursuit method (Gabor wavelet trans-formation).The matching pursuit method (Mallat andZhang 1993) is based on a dictionary that contains afamily of wavelet functions called time-frequency atoms.The decomposition of a signal is performed by projectingit over the function dictionary iteratively and by selectingthe atoms or wavelets that can best match the localwaveform of the signal. The matching pursuit methodrepresents a discrete signalx(n) with N samples as:

x~n! 5 Oi50

1`

a ihi~n!, (6)

with

hi~n! 5 b igi~n!ui~n!, (7)

gi~n! 5 gSn 2 pi

siD , (8)

and

ui~n! 5 cos(2pkin/N 1 ui). (9)

In the above equations,ai is the expansion coefficientsthat represent, when squared, the part of the signal en-ergy associated with the atomhi(n). In eqn (7), the atomhi(n) is given by the product of the wavelet envelopefunctiongi(n) with the sinusoidui(n). The scale factorssi

are used to control the width of the envelope ofhi(n), andpi give the temporal positions of each atom. The param-etersbi are normalizing factors to keep the norm ofhi(n)equal to 1. The parameterski and ui are the discretefrequency and phase of the cosine discrete functionui(n),respectively. In our application,

gi~n! 5 21/4e2pSn2pi

siD2

(10)

is a Gabor function (Mallat and Zhang 1993).Mallat and Zhang (1993) proposed to use the sum of

the Wigner distributions of all the individual atoms com-posing a signal to represent its energy distribution in thetime-frequency plane, which they calledMP-basedWigner distribution. TheMP-based Wigner distributionof x(n) in eqn (6) was represented as:

MPx~n,k! 5 Oi51

L

a i2 z WFhiSn 2 pi

si, 2psi~k 2 ki!DG ,

(11)

whereL is the number of atoms,ai, pi, si andki are theparameters of theith time-frequency atom, and

Performance of TFR techniques to measure turbulence● G. CLOUTIER et al. 537

W@h~n,k!# 5 2e22p~n21k2! (12)

is the Wigner distribution of the atomsh(n,k). In eqn(11), L and si are input parameters given to the model,whereasai, pi, andki are output parameters characteriz-ing each atom. To reduce the computational burden, thescale is generally limited tosi 5 2l, wherel is the octaveof the scalesi, which varies between 0 and log2N. Hence,only the scalessi that satisfy this relation and lie in [1, N]are selected from the dictionary. A predetermined max-imum number of atoms is used to stop the iterativeprocess. This reduces the computational burden and lim-its undesirable modeling of the background noise. It isimportant to note that the matching pursuit method findsthe time-frequency atoms in a decreasing energy order.The higher-energy components of the signal are ex-tracted first. Consequently, to track low-energy turbulentfluctuations, a high number of atoms may be necessary.For the current application, the position of each atom onthe time scale was adjusted to allow a time resolution of0.2 ms, which is similar to the value set for the otherTFRs described above.

Simulation of Doppler signals under turbulent flowThe Doppler signals(n), sampled at 14 kHz, was

simulated by a sum of sinusoids (Mo and Cobbold 1989),whose frequenciesfm subdivide the frequency range [0,fmmax1 ftmax] in M bins of equal widthDf (see below thedefinition of the time-varying functionsfmmax andftmax).Because turbulence is an irregular eddying motion oc-curring about the mean velocities, a random frequencycomponentft(n) was used to generates(n).

s~n! 5 Om51

M

gm~n!cos@2p~ fm 1 ft~n!!n 1 fm#, (13)

where

gm~n! 5 Î2S~n,k!Df ym (14)

is a Rayleigh distributed random variable, and

fm 5 ~m 2 0.5!Df. (15)

The random phasefm is uniformly distributed between[0, 2p]; S(n,k) is the time-varying theoretical powerspectral density function of the turbulence-free signal,defined over the range [0,fmmax]; Df 5 fmmax /M is anincremental frequency domain step;ym is a x2 randomvariable with two degrees of freedom; andM is thenumber of sinusoids used to simulate the Doppler signalover the signal durationT. In the present study, a valueof M 5 4185 was chosen to respect the criterion defined

by Mo and Cobbold (1989). Based on this criterion, thesimulated Doppler signals were assumed to be Gaussian-distributed in amplitude.

The theoretical power spectral density of thepulsed-wave Doppler signal as a function of time, for acase without turbulence, is given by (Mo and Cobbold1989):

S~n,k! 5 A~ fmmax 2 fk!2 exp@2B~ fmmax 2 fk!

2#, (16)

wherefmmax, A, andB are time-varying parameters rep-resenting the maximum frequency of the sinusoidsfm, thepower scaling factor, and the bandwidth factor, respec-tively. The parameterfk is the frequency correspondingto the spectral indexk. In eqn (16), the greater the valueof B, the narrower is the spectrum. The total power of thetime-varying spectraS(n,k)were normalized to unity byvarying the value ofA.

In the current study, a Doppler signal of 293-msduration was simulated containing positive frequenciesduring systole. The Doppler signal was not simulatedduring diastole because turbulence is not generated in theascending aorta when the aortic valve is closed. Themaximum time-varying frequency,fmmax, was definedby:

fmmax~n! 5 3.5 sin~2pf1n! 1 1 (17)

wheref1 5 1.79 Hz, andfmmax is in kHz. A(n) andB(n)in eqn (16) were chosen to values obtained by the pro-cedure described by Guo et al. (1994b).

Because the maximum turbulence intensity underpulsatile flow usually occurs during the deceleration ofthe fluid (Shen 1961), the random frequenciesft, whichare Gaussian-distributed, were bounded between6ftmax(n) with the maximum occurring during late systole.The time varying parameterftmax was defined as a trian-gle, as represented by the dashed line in Fig. 1. Theturbulent velocity fluctuations start at 84 ms, reach themaximum intensity at 168 ms, and decay to reach zero at293 ms. Nygaard et. al. (1992) observed, for aortic me-chanical heart valves implanted in humans, that the max-imum turbulent Reynolds normal stress (RNS5 ru’ 2

n ),measured with hot-film anemometry, could reach 120N/m2, which corresponds to a velocity variance in theaxial direction of the flowu’ 2

n , of 0.11 m2/s2 whenconsidering a density of bloodr at 1.093 g/cm3. For atypical transmitted ultrasound frequencyfo of 5 MHz, aspeed of soundc in blood at 1570 m/s, a maximumturbulent velocity fluctuationu’n at 0.33 m/s and a Dopp-ler angleu of 60°, the Doppler frequency shift

fd 52foun’ cosu

c5 1056 Hz. (18)


In the present study, the maximum frequency valueftmax at 168 ms was chosen to be 1 kHz (0.314 m/s) formost evaluations. A higher maximum turbulence inten-sity of 2 kHz (0.628 m/s) was also used to test thestability of the TFR methods to an increase in turbulencelevel. Fig. 1 summarizes the basic time-frequency func-tions of the parameters used to generate the Dopplersignal s(n) with eqn (13). Fig. 2 shows examples ofsimulated time-varying Doppler spectra without turbu-lence (i.e., ft 5 0 Hz), and with a maximum turbulenceintensity bounded betweenftmax(n) 5 6 1 kHz (6 0.314m/s). Because the maximum frequency of the functionfmmax(n) defined by eqn (17) was 4.5 kHz, the simulatedconditions can correspond to a maximum blood flowvelocity in the ascending aorta of 1.4 m/s, when consid-ering the Doppler parameters defined above. In thisstudy, although both the in-phase and quadrature com-ponents of the Doppler signals(n) were generated, onlythe in-phase signal was used. This choice is justified bythe fact that the statistical characteristics of the in-phaseand quadrature signals are the same, and that reverseflow was absent from the simulations.

Optimization of the parameters affecting the perfor-mance of the TFR methods

Table 1 presents the range of parameters tested tooptimize the performance of each TFR technique. Theparameters investigated were the window type, windowlength, model order, parameters, parametera, the oc-tave parameterl, and the maximum number of atomsL.

The types of window tested are rectangular, hanning,hamming, and sine-cosine. For the sine-cosine window,the first 10% of the samples were weighted by the sinefunction (computed between 0 and 90°), and the last 10%by the cosine also defined between 0 and 90°.

Assessment of the performance of the TFR methodsTo track the turbulent velocity fluctuations, the in-

stantaneous mean frequency waveform was estimatedfrom the Doppler power spectra according to the follow-ing equation:

fmean~n! 5

Ok5Kbwlow

Kbwhigh

fkS@n,k#

Ok5Kbwlow

Kbwhigh

S@n,k#

, (19)

whereKbwlow andKbwhigh are the samples correspondingto the lower and higher frequencies of the25-dB band-

Fig. 1. Basic time-frequency functions of the parameters usedto generate the Doppler signals(n)of eqn (13). For each instantwithin the cycle,M sinusoids at frequenciesfm1ft are summedto simulates(n). The maximum offm [i.e., fmmax(n)] corre-sponds to the parabola (dotted line). The maximum offt [i.e.,ftmax(n)] is given by the dashed triangular curve. The solid linerepresents an example of the Gaussian frequency fluctuations,

ft, simulating flow turbulence.

Fig. 2. Examples of simulated Doppler TFR,S(n,k), (a) withoutturbulence and (b) with maximum turbulent frequency fluctu-ations bounded betweenftmax(n) 5 6 1 kHz ( 6 0.314 m/s).


width of the dominant frequency peak,fk is the frequencycorresponding to the spectral indexk, and S representsthe TFR of each method tested on the Doppler signals(n). To assess the performance of each method,fmean

was compared to the theoretical mean frequency wave-form, fmean, evaluated from the theoretical time-varyingpower spectral density. More specifically,fmeanwas com-puted as:

fmean~n! 5 fmmax~n! 1 ft~n! 22

ÎpB~n!, (20)

wheren is the discrete time,fmmax(n) is defined in eqn(17), ft(n) is the Gaussian-distributed random frequencysimulating turbulence, andB(n) is the bandwidth factor.As described by Guo et al. (1994b), the theoretical com-putation of fmean(n) can be found analytically. Figure 3shows the theoretical mean frequency waveformfmean(n)for no turbulence (i.e., ft(n) 5 0) and a case with amaximum turbulent frequency fluctuationftmax(n) 5 6 1kHz (60.314 m/s).

Three procedures were used to test the accuracy ofeach TFR method to track turbulent velocity fluctuations.It is important to remember that, in the absence ofturbulence, the fluctuations of the Doppler mean veloc-ities are attributed to the variance of the spectral estima-tor.

Procedure 1.The accuracy of each method for mea-suring the absolute turbulent frequency fluctuations wasdetermined by computing the following difference equa-tion:

Mean_Abs_Freq_Dif51

NcO

n5N1

N2

$u f t~n!u2uft~n!u%, (21)

where Nc is the number of samples between the timeindexesN1 5 84 ms andN2 5 293 ms (period overwhich turbulence was simulated),ft(n) is the zero-meanturbulent frequency fluctuations measured with each

TFR method, andft(n) corresponds to the simulatedturbulent Doppler signal fluctuations (Fig. 1). The indexMean_Abs_Freq_Dif, expressed in Hz, allows assess-ment of the accuracy of the frequency fluctuations, in-dependently of the phase of the signal (because absolutefrequencies are computed). A value of zero for this index

Table 1. Parameters of the time-frequency representation techniques and their ranges of values

Time-frequencytechnique Window type

Window length(ms) Other parameters

SPEC rectangular, hanning, hamming, sine-cosine 1, 2, 5, 10 –AR rectangular, hanning, hamming, sine-cosine 1, 2, 5, 10 model order5 8, 10, 12, 14, 16CWD wN 5 rectangular, sine-cosine wN 5 1, 2, 5, 10 s 5 0.01, 0.05, 0.1, 0.2

wM 5 rectangular wM 5 1, 2, 5, 10 –CW-RID wN 5 rectangular, sine-cosine wN 5 1, 2, 5, 10 s 5 0.01, 0.05, 0.1, 0.2BD wN 5 rectangular, sine-cosine wN 5 1, 2, 5, 10 a 5 1, 2, 5, 10, 16MP Octave parameter (l ): 7, 8, 9, 10, 11

– Maximum number of atoms (L): 100, 150, 300,500, 1000

Fig. 3. Theoretical mean frequency waveformfmean(n) for (a)no turbulence and (b) a case with maximum turbulent fre-

quency fluctuationsftmax(n) 5 6 1 kHz ( 6 0.314 m/s).


means a good accuracy of the TFR technique to track themagnitude of the Doppler turbulent fluctuations; a valueabove zero indicates an overestimation, whereas a valuebelow zero means an underestimation of the magnitudeof the fluctuations.

Procedure 2.By definition, the intensity of turbu-lence, in m/s, is given by the r.m.s. of the turbulentvelocity fluctuations. The turbulence intensity was com-puted, betweenN1 5 0 ms andN2 5 293 ms, with thefollowing equation:

Turb_Int ~n! 5 Î 1

NcycleOm51

Ncycle

nt~n,m!2, (22)

where n is the time increment,m is the cardiac cycleconsidered,Ncycle 5 30 is the number of cycles simu-lated, andnt is the velocity of the turbulent fluctuationsobtained by converting the frequency fluctuationsft(n)into velocity fluctuationsnt(n) by applying the classicalDoppler equation. For comparison, eqn (22) was alsocomputed by using the theoretical turbulent fluctuationsft(n).

Procedure 3.Turbulent blood flow produces abroad range of eddy sizes, each having a different ve-locity fluctuation when moving in the mainstream at afinite mean velocity. The Kolmogorov power spectrum(Hinze 1975; Nygaard 1994), which can be displayed inthe form of a log-log plot of the turbulent power as afunction of frequency, is commonly used to characterizethe size of turbulent eddies (scale of turbulence). Typi-cally, the Kolmogorov spectrum is computed from theturbulent fluctuationsft(n) or ft(n). The low frequenciesof the Kolmogorov spectrum correspond to large perma-nent eddies (typically frequencies, 20 Hz), whereasmicroscale turbulent motions are in the upper frequencyrange of the spectrum (typically. 1000 Hz).

As reviewed by Hinze (1975), the rate of increase ordecay of the spectral density as a function of frequencydetermines the proportion of each eddy size in the vol-ume of interest (large permanent eddies, energy-contain-ing eddies, Kolmogorov inertial subrange, and mi-croscale turbulent motions). For isotropic turbulence, theslopes of the log-log spectrum, in decades of power perdecades of frequency, have been theoretically derived foreach of these frequency ranges. Values are between 4and 1 for the low-frequency permanent eddies, between1 and25/3 for energy-containing eddies (Von Ka´rman’spower law),25/3 for the inertial subrange (Kolmogor-ov’s power law), and27 for microscale eddies (Heisen-berg’s power law). These slopes were obtained for highReynolds numbers where viscosity effects are negligible

(Hinze 1975). Figure 4, described later, shows the sim-ulated Kolmogorov spectrum used in this study.

Computing the following equation assessed the ac-curacy of each TFR method to track the whole range ofeddy sizes:

BW_Dif 5 BW210 dB~K@ f t# 2 K@ ft#!, (23)

whereBW_Dif, in Hz, is the difference of the maximumfrequencies that correspond to the bandwidth at210 dB(BW210 dB) of the Kolmogorov spectrum of the estimatedfluctuations obtained from each TFR method,K[f t], andof the simulated frequency fluctuations,K[ft]. To com-pute the Kolmogorov spectrum, nonoverlapping 50-msrectangular windows were applied, betweenN1 5 84 msandN2 5 293 ms, to the signalsft(n) andft(n) sampled ata frequency of 5 kHz (the sampling frequency corre-sponds to the time resolution of the TFR methods, whichis 0.2 ms). A power spectrum was evaluated for each50-ms window by using the discrete Fourier transformand a reliable estimate of the Kolmogorov spectrum wasobtained by averaging the power spectra over 1 or 30cardiac cycles (Welch 1967). Only 1 cycle was consid-ered when determining the optimal parameters of eachTFR method, whereas 30 cycles were used when per-forming the final comparison of the TFR techniques.

Fig. 4. Kolmogorov power spectrum averaged over 30 flowcycles for turbulent frequency fluctuationsft(n) bounded be-tweenftmax(n) 5 6 1 kHz ( 6 0.314 m/s). For each flow cycle,the mean spectrum was computed between 84 and 293 ms byapplying the Welch spectral method (Welch 1967) over 50-msnonoverlapping rectangular windows. The theoretical slopesfor the different frequency ranges considered, in decades ofpower per decades of frequency, are indicated on the graph(Hinze 1975). The210-dB bandwidth of this spectrum (pa-

rameterBW210 dB of eqn (23) is 1449 Hz.


Tested conditions.The best parameters of each TFRmethod were selected from one typical flow cycle byminimizing BW_Dif and Mean_Abs_Freq_Dif. The per-formance assessment was determined for the cases of noturbulence, and for a maximum turbulent frequency fluc-tuationftmax(n) 5 6 1 kHz ( 6 0.314 m/s). After havingobtained the best set of parameters for each TFR tech-nique, comparison of the methods was performed over30 independent simulated cardiac cycles without turbu-lence, and with maximum turbulent fluctuationsftmax(n)5 6 1 kHz ( 6 0.314 m/s) and6 2 kHz ( 6 0.628 m/s).The selection of the best method was based on a statis-tical analysis of variance on the parametersBW_Dif andMean_Abs_Freq_Dif. The turbulence intensityTurb_Int(n) and the Kolmogorov spectrum were com-puted for the best methods.

RESULTS

The Kolmogorov spectrum offt(n) was evaluatedover 30 flow cycles generated with independent randomsequences. Figure 4 presents the power spectrum for amaximum turbulent frequency fluctuationftmax(n) of 6 1kHz ( 6 0.314 m/s). Because only the magnitude of thefluctuations changed for simulations withftmax(n) 5 6 2kHz, the Kolmogorov mean spectrum for this case isvery similar. According to the mean spectrum of Fig. 4,the whole scale of turbulent fluctuations was simulated(large vortex to small eddy).

Optimization of the parameters for each TFR methodThe criteria allowing selection of the best set of

parameters are the following: 1. The TFR methodshould be able to track rapid turbulent fluctuations.This means that the bandwidth at210 dB of theKolmogorov spectrum should be as close as possibleto that of the simulated fluctuations (BW_Dif should

tend toward 0 Hz). 2. The TFR method should be ableto estimate the magnitude of the absolute fre-quency fluctuations. This criterion meansthatMean_Abs_Freq_Dif should converge toward 0 Hz forthe simulations performed with turbulence. Finally, 3.the variance of the TFR algorithm should be minimum.In other words, Mean_Abs_Freq_Dif also should tendtoward 0 Hz for the simulations performed without tur-bulence.

As shown in Fig. 5, few differences were foundbetween the rectangular and sine-cosine windows, andbetween the hanning and hamming windows for thespectrogram. When there was turbulence, the Kolmog-orov spectrum was better modeled when the windowlength was reduced from 10 to 1 ms, for all windowtypes. On the other hand, the parameterMean_Abs_Freq_Dif was generally reduced as the win-dow length was increased for both turbulent and laminarflows. According to the selection criteria defined above,the 1-ms hamming or sine-cosine windows are goodselections to model the Kolmogorov spectrum. Both the10-ms rectangular and sine-cosine windows minimizedthe parameter Mean_Abs_Freq_Dif for turbulent andlaminar flows within the range of window durationsconsidered.

Although several model orders (8 to 16) weretested for autoregressive modeling, this parameter hadno significant effect on the results. Thus, the data inFig. 6 are presented for a fixed model order of 8. Thevariations ofBW_Dif and Mean_Abs_Freq_Dif as afunction of the window length and type were verysimilar to those obtained with the spectrogram. How-ever, for no turbulence, the variance of theAR esti-mator was slightly larger than the spectrogram. Ac-cording to Fig. 6, the 1-ms sine-cosine window is thebest selection to minimize the bias ofBW_Dif. On the

Fig. 5. Optimization of the spectral parameters for the spectrogram. ParametersBW_Dif, eqn (23), andMean_Abs_Freq_Dif, eqn (21), are plotted as a function of the window duration for simulations with and withoutturbulence. In the case of turbulence, the maximum frequency fluctuationsftmax(n) 5 6 1 kHz ( 6 0.314 m/s). The

legend gives the window type that corresponds to each curve.


other hand, the 10-ms rectangular window providedthe best estimate of Mean_Abs_Freq_Dif for bothturbulent and laminar flows.

The results obtained with the Choi–Williams distri-bution were better when using a rectangular windowinstead of the sine-cosine function forwN (results notshown). Effectively, for any given window durationwM

and wN, and any values of the parameters, the band-width of the Kolmogorov spectrum was always moreseverely underestimated with the sine-cosine windowwN. On the other hand, Mean_Abs_Freq_Dif was notsignificantly affected by the window typewN. The effectof varying the length ofwN between 1 and 10 ms wasas follows (results not shown): changing the lengthhad very little effect on the Kolmogorov spectralband-width; Mean_Abs_Freq_Dif was almost unaffected bythis parameter for the case of turbulence; andMean_Abs_Freq_Dif slightly decreased as the windowlengthwN was increased when there was no turbulence.

According to these observations, the results in Fig. 7 arepresented forwN being a 10-ms rectangular window. Asseen on this figure, the bandwidth of the Kolmogorovspectrum increased (BW_Dif converged toward 0 Hz) asthe window durationwM was reduced, and that wasobserved for any values of the parameters. The ten-dency for the effect ofwM on Mean_Abs_Freq_Dif is theopposite, reducing the window length increased the errorin the estimate of the absolute turbulent fluctuations andvariance of the TFR estimator. According to Fig. 7, thebest set of parameters to reduceBW_Dif is wN 5 10-msrectangular window,wM 5 1-ms rectangular window,ands 5 0.1. For Mean_Abs_Freq_Dif, the best param-eters arewN andwM 5 10-ms rectangular windows, ands 5 0.01 for both turbulent and laminar flows.

Figure 8 presents the results obtained with theCW-RID method andwN 5 a sine-cosine window. The resultsfor the rectangular window are not given because thevalues of Mean_Abs_Freq_Dif were at least 2 times

Fig. 6. Optimization of the spectral parameters for autoregressive modeling (model order5 8). ParametersBW_Dif, eqn(23), and Mean_Abs_Freq_Dif, eqn (21), are plotted as a function of the window duration for simulations with andwithout turbulence. In the case of turbulence, the maximum frequency fluctuationsftmax(n) 5 6 1 kHz ( 6 0.314 m/s).

The legend gives the window type that corresponds to each curve.

Fig. 7. Optimization of the spectral parameters for the Choi–Williams distribution (wN 5 10-ms rectangular window,wM 5 rectangular). ParametersBW_Dif, eqn (23), and Mean_Abs_Freq_Dif, eqn (21), are plotted as a function of thewindow durationwM for simulations with and without turbulence. In the case of turbulence, the maximum frequencyfluctuationsftmax(n) 5 6 1 kHz (6 0.314 m/s). The legend gives the value of the parameters (controls of the auto-term

resolution for cross-term suppression) that corresponds to each curve.


larger for any given values of the window length andparameters. With this TFR method,s had no effect onBW_Dif and Mean_Abs_Freq_Dif (the variance of theresults was within6 1 Hz), consequently the results ofFig. 8 are presented fors 5 0.1 only. The optimalwindow duration for this TFR was 5 ms when modelingthe Kolmogorov spectrum. To estimate the absolute tur-bulent fluctuations and to minimize the variance in theabsence of turbulence, the 10-ms window is better.

The results for the Bessel distribution are presentedin Fig. 9. Both the sine-cosine and rectangular windowswN provided similar satisfactorily results. Generally,Mean_Abs_Freq_Dif was slightly smaller when usingthe sine-cosine window (results not shown). On the otherhand, the bandwidth of the Kolmogorov spectrum was

better estimated with the rectangular window. Conse-quently, this window was used for Fig. 9. According tothis figure, the best compromise forBW_Dif is ob-tained with the 2-ms window anda 5 2. To estimateMean_Abs_Freq_Dif, the best results were obtainedwith the 10-ms rectangular window anda 5 16.

As seen in Fig. 10, the matching pursuit did notprovide satisfactory results to estimate the Kolmogorovbandwidth. However, by properly selecting the numberof atoms and the octave parameter, similar results tothe other TFR techniques could be obtained forMean_Abs_Freq_Dif. The best parameters to estimateBW_Dif and Mean_Abs_Freq_Dif are the octave pa-rameterl 5 10 and the number of atomsL of 150.

Selection of the best techniques to model BW_Dif andMean_Abs_Freq_Dif

According to the above results, more than one TFRtechnique and more than one set of parameters can beselected to track turbulent motions. For a given tech-nique, it is also clear that different parameters must beused to optimally determine the Kolmogorov spectrum,minimize the variance of the estimator, and measure theabsolute turbulent frequency fluctuations. To help com-pare each technique, Table 2 showsBW_Dif computedover 30 flow cycles for the optimal parameters selectedearlier. The first three columns present results for noturbulence, and turbulent fluctuationsftmax(n) 5 6 1 kHz( 6 0.314 m/s) and6 2 kHz (6 0.628 m/s), respectively.The absolute values ofBW_Dif also were computed foreach column and averaged to give the mean bias of eachtechnique (the bias is given in the fourth column and it isnamed the mean of the absoluteBW_Dif; absolute valueswere used becauseBW_Dif could be either positive ornegative). To determine the best technique, an analysisof variance (one-way ANOVA, SigmaStat statisticalsoftware, SPSS, San Rafael, CA, version 2.03) was per-

Fig. 8. Optimization of the spectral parameters for the Choi–Williams reduced interference distribution (wN 5 sine-cosineand s 5 0.1). ParametersBW_Dif, eqn (23), andMean_Abs_Freq_Dif, eqn (21), are plotted as a function of thewindow durationwN for simulations with and without turbu-lence. In the case of turbulence, the maximum frequency fluc-

tuationsftmax(n) 5 6 1 kHz ( 6 0.314 m/s).

Fig. 9. Optimization of the spectral parameters for the Bessel distribution (wN 5 rectangular). ParametersBW_Dif, eqn(23), and Mean_Abs_Freq_Dif, eqn (21), are plotted as a function of the window durationwN for simulations with andwithout turbulence. In the case of turbulence, the maximum frequency fluctuationsftmax(n) 5 6 1 kHz ( 6 0.314 m/s).

The legend gives the value of the parametera (scaling factor) that corresponds to each curve.


formed on the means of the absoluteBW_Dif. FromTable 2, the Bessel distribution withwN 5 2 ms rectan-gular windows anda 5 2 provided the smallest bias ofthe Kolmogorov mean spectra with a value for the band-width at210 dB of 3556 139 Hz. Although the biasesfor the spectrogram,AR modeling, and theCW-RIDdistribution were higher, these values were not statisti-cally different from those obtained with the Bessel dis-tribution.

In theory,BW_Dif should be 0 Hz when there is noturbulence. As noted in the first column of Table 2, thevariance of all TFR estimators produced fluctuationscharacterized by finite bandwidths. Among the best fouralgorithms (SPEC, AR, CW-RIDandBD), the frequencyof these fluctuations was minimum for the Bessel distri-bution (BW_Dif 5 354 6 51 Hz). In the absence ofturbulence, the worse results (fastest fluctuations thatcould be interpreted as fast turbulent eddies) were ob-tained with the spectrogram. On the other hand, thistechnique provided the best estimate of the Kolmogorov

mean spectrum when there was turbulence (BW_Dif 585 6 108 Hz and 506 149 Hz forftmax(n) 5 6 1 and62 kHz, respectively). According to this discussion, theBessel distribution is recommended if one wants to re-spect the criterion of the Kolmogorov spectral band-width.

Table 3 shows the best techniques to minimize thevariance in the absence of turbulence, and to estimatethe absolute turbulent fluctuations (parameterMean_Abs_Freq_Dif). The fourth column gives themean of the absolute values of Mean_Abs_Freq_Difcomputed over the first three columns. Based on ananalysis of variance, the mean of the absoluteMean_Abs_Freq_Dif was optimum for the spectrogram,AR modeling, the Choi–Williams distribution, and theBessel distribution. All four methods provided meanvalues close to 60 Hz. In the absence of turbulence, thevariance of the TFR estimators was minimum for theChoi–Williams distribution (Mean_Abs_Freq_Dif579 6 10 Hz). For all methods, the magnitude of the

Fig. 10. Optimization of the spectral parameters for the matching pursuit method. ParametersBW_Dif, eqn (23), andMean_Abs_Freq_Dif, eqn (21), are plotted as a function of the octave parameterl for simulations with and withoutturbulence. In the case of turbulence, the maximum frequency fluctuationsftmax(n) 5 6 1 kHz ( 6 0.314 m/s). The

legend gives the number of atoms (L) used for the computation of the TFR that correspond to each curve.

Table 2. Optimal parameters to model the Kolmogorov spectral bandwidth (parameterBW_Dif) for each TFRtechnique

Time-frequency techniqueNo

turbulence (Hz)

Frequency fluctuations(Hz)

(ftmax(n) 5 6 1 kHz)


(ftmax(n) 5 6 2 kHz)

Mean of theabsoluteBW_Dif

(Hz) p

SPEC, 1 ms hamming 10596 263 856 108 506 149 4346 94 –AR, 1 ms sine-cosine; model order5 8 6996 200 21176 281 21786 308 4306 102 –CWD, wN 5 10 ms rect.;wM 5 1 ms rect.;s 5 0.1 7506 176 25666 206 26146 136 6446 72 *CW-RID, wN 5 5 ms sine-cosine;s 5 0.1 4146 71 23426 422 24366 342 4416 139 –BD, wN 5 2 ms rect.;a 5 2 3546 51 21936 371 22556 364 3556 139 –MP, l 5 10; L 5 150 3486 57 29746 112 29626 106 7626 48 *

The results are presented for no turbulence and for maximum frequency fluctuations of6 1 kHz (6 0.314 m/s) and6 2 kHz (6 0.628 m/s). Theabsolute values ofBW_Dif were averaged over the data of the first three columns, and results are presented in the fourth column (mean of the absoluteBW_Dif). All results were averaged over 30 flow cycles (mean6 one SD).SPEC, AR, CW-RIDandBD represent the best techniques as confirmedby an analysis of variance on the means of the absoluteBW_Dif.

See text for the definition of the terms; *p , 0.001.


turbulent frequency fluctuations was overestimated forthe smallest fluctuations (6 1 kHz), whereas it wasunderestimated or closer to the true simulated value forthe highest value of6 2 kHz.

Turbulent time-intensity variations and Kolmogorovmean spectrum

According to Tables 2 and 3, three TFR methods,each with a different set of parameters, can optimize theKolmogorov mean spectrum, the variance of the estima-tor, and the absolute turbulent frequency fluctuations(SPEC, ARand BD). Arbitrarily, this section showsresults obtained with the spectrogram and the Besseldistribution. Figure 11 presents Turb_Int(n), in m/s, forthe case of no turbulence, and a maximum turbulentfluctuation ftmax(n) 5 6 1 kHz. The theoretical time-varying values of this parameter are presented for com-parison. As expected from Table 3, both techniquesestimated velocity fluctuations in the absence of turbu-lence. The mean intensity was on the order of 0.04 m/sfor both TFR methods. When computed as a relativeindex :

1/N On50

N

f t~n! 3 100/fmean~n!, (24)

the coefficient of variation was 12.4% forSPEC and8.0% for BD. For AR, the coefficient of variation was14.5% (results not shown). The intensity of the fluctua-tions was overestimated when compared to the fluctua-tions measured from the theoretical TFR, for the case ofturbulence. Neither the spectrogram, the Bessel distribu-tion nor any other methods (AR, CWD, CW-RID, MP)could accurately estimate the triangular shape of theturbulent intensity. At the peak of the simulated turbulent

Table 3. Optimal parameters to minimize the variance of the spectral estimator and to estimate the absoluteturbulent frequency fluctuations (parameter Mean_Abs_Freq_Dif) for each TFR technique

Time-frequency technique

Noturbulence

(Hz)


(fmax(n) 5 6 1 kHz)


(ftmax(n) 5 6 2kHz)

Mean ofthe absolute

Mean_Abs_Freq_Dif(Hz) p

SPEC, 10-ms sine-cosine 956 14 656 18 2216 21 616 9AR, 10-ms rectangular; model order5 8 956 12 626 16 2266 21 626 8CWD, wN 5 10 ms rect.;wM 5 10 ms rect.;s 5 0.01 796 10 866 18 56 24 626 8CW-RID, wN 5 10 ms sine-cos;s 5 0.1 1166 11 1356 13 516 17 1016 9 *BD, wN 5 10 ms rect.;a 5 16 856 15 736 21 2116 28 616 9MP, l 5 10; L 5 150 1166 19 1656 27 986 46 1266 20 *

The results are presented for no turbulence and for maximum turbulent frequency fluctuations of6 1 kHz (6 0.314 m/s) and6 2 kHz (6 0.628m/s). The absolute values of Mean_Abs_Freq_Dif were averaged over the data of the first three columns, and results are presented in the fourth column(mean of the absolute Mean_Abs_Freq_Dif). All results were averaged over 30 flow cycles (mean6 one SD).SPEC, AR, CWD, andBD representthe best techniques, as confirmed by an analysis of variance on the means of the absolute Mean_Abs_Freq_Dif.

See text for the definition of the terms; *p , 0.001.

Fig. 11. Intensity of the velocity fluctuations in m/s (i.e.,parameter Turb_Int, eqn (22)), as a function of the timingwithin the flow cycle for measurements performed with thespectrogram (SPEC) and the Bessel distribution (BD). Theparameters used were 10-ms sine-cosine windows forSPEC,andwN 5 10-ms rectangular windows anda 5 16 for BD. Atotal of 30 flow cycles were used to compute the parameterTurb_Int(n). The results are given for no turbulence, and max-imum turbulent frequency fluctuationsftmax(n) 5 6 1 kHz ( 60.314 m/s). The expected simulated velocity fluctuations areindicated on the figure (theory). Note that the maximum ofTurb_Int is about one third offtmax(n), which is 0.314 m/s. Bydefinition (eqn (22)), Turb_Int is the r.m.s. of the simulatedvelocity fluctuations, which is equivalent to the standard devi-ation (SD). Because 99.7% of the values are within the mean63 SD for a Gaussian statistical distribution, this explains the

smaller values noted in this figure.


intensity, the bias of bothSPECand BD methods wasbelow 0.01 m/s.

Figure 12 presents, for one typical flow cycle, themean velocity as a function of time for a maximumturbulent fluctuation of6 1 kHz (6 0.314 m/s). The leftpanels show the mean waveforms for parameters opti-mized to estimate the Kolmogorov spectrum, whereasthe right panels give the results for parameters selected toreduce the velocity variance and to measure the absoluteturbulent frequency fluctuations. The Kolmogorov powerspectra corresponding to the cases of Fig. 12 are shownin Fig. 13. As noted earlier, the spectrogram with 1-mshamming windows is a good selection to estimate thefrequency of the turbulent fluctuations. However, thesensitivity of discrimination between turbulent and noturbulent fluctuations is limited because the spectra aresimilar for both cases. For the Bessel distribution withwN 5 2-ms rectangular windows anda 5 2, the band-widths of the Kolmogorov spectra are smaller than thesimulated one. However, the difference is emphasizedbetween the case of turbulence and no turbulence. Theright panels show the results for parameters optimized tomeasure the magnitude of the turbulent fluctuations. As

seen, the Kolmogorov bandwidths are much smaller forthese cases.

DISCUSSION

In an article by our group (Cloutier et al. 1996),blood flow turbulence downstream of a stenosis wasmeasured by using pulsed-wave Doppler US. No mea-surement of the Kolmogorov spectrum was performed inthat study. The velocity fluctuations were assessed atseveral locations after the stenosis and hot-film anemom-etry was used to validate the spatial distribution of theturbulence intensity. Although the relative changes inturbulence and the position of the peak downstream ofthe stenosis were correctly mapped, significant overesti-mation of the intensity (more than 5 times) was notedwhen comparing the results to hot-film anemometry. Itwas suggested that velocity-independent instrumentnoise, the relatively large size of the Doppler samplevolume, and instability of the flow in the model couldexplain these results. In this study, the Doppler spectrawere processed by usingARmodeling over 2-ms rectan-gular windows (theAR model was the same as the one

Fig. 12. Typical mean velocity waveforms over one flow cycleobtained with the spectrogram (SPEC) and the Bessel distribu-tion (BD), for maximum turbulent frequency fluctuationsft-max(n) 5 6 1 kHz (6 0.314 m/s). Optimally, these waveformsshould depict the theoretical variations noted in Fig. 3b. Theparameters used for each spectral method are indicated on thefigure. The left panels represent the mean waveforms for pa-rameters optimized to model the Kolmogorov spectrum,whereas the right panels give the results for parameters selectedto reduce the velocity variance and to measure the magnitude of

the turbulent frequency fluctuations.

Fig. 13. Kolmogorov power spectrum averaged over 30 flowcycles for turbulent frequency fluctuations bounded betweenftmax(n) 5 6 1 kHz ( 6 0.314 m/s), and for the case of noturbulence. The spectrum obtained from the simulations is alsopresented for comparison (theory). The parameters used for thespectrogram (SPEC) and the Bessel distribution (BD) are indi-cated on the figure. The left panels present the spectra forparameters optimized to track the velocity fluctuations, whereasthe right panels give the results for parameters selected toreduce the velocity variance and to measure the magnitude of

the turbulent frequency fluctuations.


used here). According to Fig. 6, this window length is notoptimum to minimize the variance of the spectral esti-mator and to measure the magnitude of the turbulentfluctuations. Thus, the intrinsic variance of theARspec-tral estimator could have contributed to the unexpectedhigh turbulence intensities measured in our previousstudy (Cloutier et al. 1996). The use of a 10-ms rectan-gular window instead of the 2-ms window would haveimproved the accuracy of the results.

In other previous studies (Giddens and Khalifa1982; Nygaard et al. 1994a), the approaches used tomeasurein vivo the velocity fluctuations with US dif-fered significantly from the methods tested here. Thecomparison of our results is, thus, difficult in this con-text. On the other hand, some investigators reportedsimulations aimed at evaluating the stability of TFRestimators to track the Doppler mean frequency.Kaluzynski and Palko (1993) tested the variance of theDoppler mean frequency estimated with two autoregres-sive algorithms as a function of the model order andwindow length. Sava et al. (1999) performed similarcomputations for a broader range of short-time spectralparametric methods (twoARmodels, the Prony method,and the Steiglitz–McBride autoregressive moving aver-age method). In both studies, the results were comparedto the standard short-time Fourier method. As noted inthese studies, the variance increased when reducing thewindow length from 10 to 1 ms. Similar findings werefound in the present study for the case of no turbulence(Figs. 5 to 9). In Kaluzynski and Palko (1993) and Savaet al. (1999), the selection of theAR model order hadlittle effect for values in the range of 10 to 40. By varyingthe model order between 8 to 16, no significant effect onMean_Abs_Freq_Dif was also noted in the present study.

As shown in Table 3, four algorithms providedsimilar biases to estimate Mean_Abs_Freq_Dif whenthere was no turbulence (SPEC, AR, CWDand BD).Although no statistical test was performed in their study,Sava et al. (1999) found that the method with the mini-mum variance, for a signal-to-noise (S:N) ratio greaterthan 10 decibels and window lengths varying between 1and 10 ms, wasAR (Levinson–Durbin algorithm andmaximum entropy method) followed by the spectrogram.These results are consistent with those reported in Table3. Guo et al. (1994b) computed the variance (normalizedr.m.s. error) of the Doppler mean frequency for theshort-time Fourier method,ARmodeling, the Bessel dis-tribution, the Choi–Williams distribution, and the Choi–Williams reduced interference distribution. The smallesterrors, for S:N ratios of 10 decibels and higher, wereobtained with the Bessel distribution for the optimumspectral parameters considered in that study (wN 5 4-mssine-cosine windows anda 5 16). By using a 10-msrectangular window anda 5 16, the Bessel distribution

also was considered a good method to minimize thevariance of the Doppler mean frequency (Table 3). Asreported in the Results section, it was found that thesine-cosine window was slightly better than the rectan-gular window when the parameter Mean_Abs_Freq_Difis considered. By using a 10-ms sine-cosine window,slightly better results could have been obtained by Guo etal. (1994b).

Some investigators suggested strategies to detectdisturbed flow with Doppler ultrasound (Gaupp et al.1999; Talhami and Kitney 1988; Wang and Fish 1996).These strategies were usually proposed as an alternativeto the short-time Fourier algorithm. The maximum-like-lihood Kalman model (an autoregressive moving averagemethod) was used to track ramp-type variations andlow-frequency sinusoidal oscillations (period of 4 ms,typically) of the Doppler mean frequency (Talhami andKitney 1988). Although not specified in the study, thesinusoidal oscillations simulated during flow decelera-tion of the time-varying signal may correspond to large-scale vortex shedding. The performance of the Kalmanmodel was better than the Fourier method for the situa-tions considered. The detection of very large-scale de-terministic vortices (period of 15 ms), as found duringthe deceleration of the flow in the carotid artery, wasinvestigated with the short-time Fourier transform and anew FFT-based stationarizing algorithm, theAR covari-ance model, the pseudo-Wigner–Ville method, and theChoi–Williams algorithm (Wang and Fish 1996). For thespectral parameters considered, the stationarizing FFT-based algorithm provided the best detection of the largevortices. Moreover, the mean frequency estimated withthe Choi–Williams distribution was shown to be insen-sitive to the choice ofs for values between 0.05 and 10.As noted in Fig. 7, a similar behavior was found forsvarying between 0.05 and 0.2. Fors 5 0.01, we noted aneffect on Mean_Abs_Freq_Dif as the window length wasreduced. Recently, the same group characterized vortexshedding in vascular anastomosis models with the short-time Fourier transform (Gaupp et al. 1999). Determinis-tic vortices were detected with a time resolution of 2.5ms.

As discussed earlier, the detection of random tur-bulent fluctuations covering a wide range of eddy sizeshas not been addressed so far in the US literature. Thepresent study may provide basic guidelines for selectingthe optimal method and set of parameters for this pur-pose. According to Hinze (1975), the upper size limit ofthe eddies that can be detected is mainly determined bythe size of the apparatus, whereas the lower limit isdetermined by viscosity effects and decreases with in-creasing velocity of the average flow. Consequently, forin vivo clinical assessment of the scale of turbulentmotions with US, a key parameter to optimize is the size


of the Doppler sample volume. To our knowledge, thedetermination of the effect of the US sample volume sizeon the detectability of microscale turbulent motions hasnot been addressed so far in the literature. With theexception of the velocity vector perpendicular to the USbeam, the Doppler modality is sensitive to the 3-D mo-tion of the flow. However, more work should be con-ducted to understand the contribution of 3-D vorticessmaller than the sample volume size on the Dopplerspectrum. The advancement in high-frequency US andnew beam-forming strategies may improve the detect-ability of microscale vortices by reducing the size of thesample volume. For cardiac applications, transesopha-geal echography may allow the use of high-frequencytransducers.

CONCLUSION

An exhaustive evaluation of the performance oftime-frequency representation techniques has been pre-sented for applications related to the measurement ofblood flow turbulence with US. For each technique, theparameters expected to provide adequate mapping of themagnitude and scale of the turbulent fluctuations weretested. To minimize the variance of the estimator and tomeasure the magnitude of the turbulent frequency fluc-tuations, four methods provided similar results:SPEC(10-ms sine-cosine windows),AR (10-ms rectangularwindows, model order5 8), CWD(wN andwM 5 10-msrectangular windows,s 5 0.01), andBD (wN 5 10-msrectangular windows,a 5 16). To measure the scale ofthe turbulent motions (Kolmogorov spectrum), theBessel distribution gave the best results (wN 5 2-msrectangular windows,a 5 2).

Because of the lack of information in the literature,no clear guideline could be used to select the optimal setof parameters for the application in hand (characteriza-tion of large vortices to microscale turbulent motions). Inthe current study, the range of window lengths has beenlimited to 1 to 10 ms. The shortest windows were used tooptimize time resolution, whereas the upper limit of 10ms was selected to respect the stationarity criterion of theDoppler blood flow signal (Guo et al. 1993). From theresults of Figs. 5 to 9, the longest windows may also bea good selection to minimize the variance of the estima-tor and to measure the intensity of turbulence. The per-formance for longest windows still needs to be deter-mined. It is clear that the conclusions drawn from thisstudy may depend on the simulation model used. Furthervalidation with in vitro and in vivo data would be nec-essary. The recognition of Doppler US as a precisemeans of measuring blood flow turbulence noninvasivelywill require further validation with “gold standards” suchas hot-film or laser Doppler anemometry.

Acknowledgement—This work was supported by a research scholarshipfrom the Fonds de la Recherche en Sante´ du Quebec, and by a grantfrom the Natural Sciences and Engineering Research Council of Can-ada (grant #OGPIN 335). The authors acknowledge Drs. S. G. Mallatand Z. Zhang for providing the matching pursuit software.

REFERENCES

Butchart EG. Prosthetic heart valves. In: Verstraete M, Fuster V, TopolEJ, eds. Cardiovascular thrombosis: Thrombocardiology andthromboneurology. Philadelphia, New York: Lippincott - Raven,1998:395–414.

Chesebro JH, Fuster V. Valvular heart disease and prosthetic heartvalves. In: Verstraete M, Fuster V, Topol EJ, eds. Cardiovascularthrombosis: Thrombocardiology and thromboneurology. Philadel-phia, New York: Lippincott-Raven, 1998:365–394.

Choi HI, Williams WJ. Improved time-frequency representation ofmulticomponent signals using exponential kernels. IEEE TransAcoust Speech Signal Processing 1989;37:862–871.

Cloutier G, Allard L, Durand LG. Characterization of blood flowturbulence with pulsed-wave and power Doppler ultrasound imag-ing. J Biomech Eng 1996;118:318–325.

Fontaine AA, Heinrich RS, Walker PG, Pedersen EM, ScheideggerMB, Boesiger P, Walton SP, Yoganathan AP. Comparison ofmagnetic resonance imaging and laser Doppler anemometry veloc-ity measurements downstream of replacement heart valves: Impli-cations for in vivo assessment of prosthetic valve function. J HeartValve Dis 1996;5:66–73.

Gaupp S, Wang Y, How TV, Fish PJ. Characterization of vortexshedding in vascular anstomosis models using pulsed Dopplerultrasound. J Biomech 1999;32:639–645.

Giddens DP, Khalifa AMA. Turbulence measurements with pulsedDoppler ultrasound employing a frequency tracking method. Ultra-sound Med Biol 1982;8:427–437.

Guo Z, Durand LG, Allard L, Cloutier G, Lee HC, Langlois YE.Cardiac Doppler blood flow signal analysis. Part I: Evaluation ofthe normality and stationarity of the temporal signal. Med Biol EngComput 1993;31:237–241.

Guo Z, Durand LG, Lee HC. The time-frequency distributions ofnonstationary signals based on a Bessel Kernel. IEEE Trans SignalProcessing 1994a;42:1700–1707.

Guo Z, Durand LG, Lee HC. Comparison of time-frequency distribu-tion techniques for analysis of simulated Doppler ultrasound signalsof the femoral artery. IEEE Trans Biomed Eng 1994b;41:332–342.

Hinze JO. Turbulence. New York: McGraw-Hill, 1975.Hung TC, Hochmuth RM, Joist JH, Suteras P. Shear-induced aggre-

gation and lysis of platelets. Trans Am Soc Artif Intern Organs1976;22:285–291.

Jeong J, Williams WJ. Alias-free generalized discrete-time time-fre-quency distributions. IEEE Trans Signal Processing 1992a;40:2757–2765.

Jeong J, Williams WJ. Kernel design for reduced interference distri-butions. IEEE Trans Signal Processing 1992b;40:402–412.

Kaluzynski K, Palko T. Effect of method and parameters of spectralanalysis on selected indices of simulated Doppler spectra. Med BiolEng Comput 1993;31:249–256.

Kay SM, Marple SL Jr. Spectrum analysis—A modern perspective.Proc IEEE 1981;69:1380–1419.

Kroll MH, Hellums JD, McIntire LV, Schafer AI, Moake JL. Plateletsand shear stress. Blood 1996;88:1525–1541.

Mallat SG, Zhang Z. Matching pursuits with time-frequency dictionar-ies. IEEE Trans Signal Processing 1993;41:3397–3415.

Milnor WR. Hemodynamics. Baltimore, Hong Kong, London, Sydney:Williams & Wilkins, 1989.

Mo LYL, Cobbold RSC. A nonstationary signal simulation model forcontinuous wave and pulsed Doppler ultrasound. IEEE Trans Ul-trason Ferroelec Freq Cont 1989;36:522–530.

Nichols WW, O’Rourke MF. McDonald’s blood flow in arteries. The-oretical, experimental and clinical principles. Philadelphia, Lon-don: Lea & Febiger, 1990.

Nygaard H. Quantitative evaluation of turbulence downstream of aortic


valves with special reference to patients with stenotic and prostheticaortic valves. Aarhus University Hospital, Ph.D. Thesis, 1994.

Nygaard H, Hasenkam JM, Pedersen EM, Kim WY, Paulsen PK. Anew perivascular multi-element pulsed Doppler ultrasound systemfor in vivo studies of velocity fields and turbulent stresses in largevessels. Med Biol Eng Comput 1994a;32:55–62.

Nygaard H, Paulsen PK, Hasenkam JM, Kromann-Hansen O, PedersenEM, Rovsing PE. Quantitation of the turbulent stress distributiondownstream of normal, diseased and artificial aortic valves inhumans. Eur J Cardio-thorac Surg 1992;6:609–617.

Nygaard H, Paulsen PK, Hasenkam JM, Pedersen EM, Rovsing PE.Turbulent stresses downstream of three mechanical aortic valveprostheses in human beings. J Thorac Cardiovasc Surg 1994b;107:438–446.

Sallam AM, Hwang NHC. Human red blood cell hemolysis in aturbulent shear flow: Contribution of Reynolds shear stresses. Bio-rheology 1984;21:783–797.

Sava H, Durand LG, Cloutier G. Performance of short-time spectralparametric methods for reducing the variance of the Doppler ultra-sound mean instantaneous frequency estimation. Med Biol EngComput 1999;37:291–297.

Shen SF. Some considerations on the laminar stability of time-depen-dent basic flows. J Aerospace Sci 1961;28:397–404, 417.

Stein PD, Sabbah HN, Pitha JV. Continuing disease process of calcificaortic stenosis. Role of microthrombi and turbulent flow. Am JCardiol 1977;39:159–163.

Talhami HE, Kitney RI. Maximum likelihood frequency tracking of theaudio pulsed Doppler ultrasound signal using a Kalman filter.Ultrasound Med Biol 1988;14:599–609.

Talukder N, Fulenwider JT, Mabon RF, Giddens DP. Poststenotic flowdisturbance in the dog aorta as measured with pulsed Dopplerultrasound. J Biomech Eng 1986;108:259–265.

Walker PG, Pedersen EM, Oyre S, Flepp L, Ringaard S, Heinrich RS,Walton SP, Hasenkam JM, Jorgensen HS, Yoganathan AP. Mag-netic resonance velocity imaging: A new method for prostheticheart valve study. J Heart Valve Dis 1995;4:296–307.

Wang Y, Fish PJ. Comparison of Doppler signal analysis techniquesfor velocity waveform, turbulence and vortex measurement: Asimulation study. Ultrasound Med Biol 1996;22:635–649.

Welch PD. The use of fast Fourier transform for the estimation ofpower spectra: A method based on time averaging over short,modified periodograms. IEEE Trans Audio Electroacoust 1967;15:70–73.


Performance of time-frequency representation techniques to measure blood flow turbulence with...

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