Movement artefact suppression in blood perfusion measurements using a multifrequency technique

PII: S0301-5629(01)00476-8

● Original Contribution

MOVEMENT ARTEFACT SUPPRESSION IN BLOOD PERFUSIONMEASUREMENTS USING A MULTIFREQUENCY TECHNIQUE

TOMAS JANSSON, HANS W. PERSSONand KJELL LINDSTROM

Department of Electrical Measurements, Lund University, Lund, Sweden

(Received 29 November 2000; in final form 13 September 2001)

Abstract—The standard way of suppressing movement artefacts in Doppler measurements is by means of ahigh-pass filter. This is because artefacts usually are of high amplitude, but have low frequencies. The immediatedrawback is, then, that low-velocity blood flow is also filtered out. In this paper, a method to reduce movementartefacts in blood perfusion measurements is proposed, using simultaneous transmission and reception ofmultiple frequencies in a continuous-wave Doppler system. It is shown that Doppler signals originating fromblood may be considered uncorrelated for a large enough frequency separation between channels, and tissuemovements are more correlated. By subtracting perfusion estimates obtained by time-domain processing,correlated signals can be suppressed. The subtraction algorithm is shown to produce a linear perfusion estimate,but with twice the standard deviation compared to an estimate obtained by simply averaging channels.Movement artefacts in both in vitro and in vivo models are shown to be reduced by the algorithm. Imbalancebetween channels does, however, cause the artefacts to be only partly reduced. The problem can be alleviated byfiltering the signals prior to subtraction, but this results in a nonlinear estimate, especially for large timeconstants in the filter. Some amount of filtering can still be desirable to suppress partly correlated artefacts, evenif identical time-domain processing units are implemented, as could be done digitally. (E-mail:[email protected]) © 2002 World Federation for Ultrasound in Medicine & Biology.

Key Words: Doppler ultrasound, Continuous-wave Doppler, Speckle, Frequency compounding, Perfusion, Clut-ter, Artefact.

INTRODUCTION

In all methods of estimating blood perfusion using ultra-sound (US), movement artefacts (i.e., involuntary move-ments of the transducer, moving vessel walls, musclevibration, etc.) are the most difficult to control (Janssonet al. 1999). The common way to reduce unwantedmotion artefacts, or clutter, is by means of a high-passfilter, because the echoes from moving tissue structuresusually have a low Doppler shift. The obvious drawbackis that low-velocity blood flow still contributing to tissueperfusion, such as capillary flow, cannot be measured.The fact that echoes originating from tissue have higheramplitude than those from blood is used for artefactsuppression in imaging systems, such as color Doppler,where color (i.e., velocity information) is not displayed ifthe echo amplitude exceeds some preset level. Clutterfilters based on singular value decomposition have also

been investigated for multiline Doppler modalities (Le-doux et al. 1997).

Currently, there is a large interest in US contrastagents for perfusion estimation (Mulvagh et al. 2000). Ingeneral, contrast agents increase the detectability ofsmall vessels because of the dramatic increase of thepower backscattered from blood vessels. Certain tech-niques, such as harmonic imaging and pulse-inversionDoppler, can further enhance the blood signal, utilizingnonlinearities of the contrast agent, whereby motion ar-tefacts can be suppressed (Schrope et al. 1992; Burns etal. 1994; Simpson et al. 1999).

Although effective, the use of contrast agents maynot be preferable in certain situations as, for instance,measurements of fetal perfusion. Further, studies haveshown bioeffects arising from US contrast agents incombination with high-level US exposure. Examples in-clude capillary ruptures, erythrocyte extravasation(Skyba et al. 1998; Ay et al. 2001), and effects on levelsand uptake of vascular endothelial growth factors(Mukherjee et al. 2000), known to stimulate angiogene-sis and dilatation. Although these results may not be

Address correspondence to: Dr. Tomas Jansson, Department ofElectrical Measurements, P. O. Box 118, Lund University, SE-221 00Lund, Sweden. E-mail: [email protected]

Ultrasound in Med. & Biol., Vol. 28, No. 1, pp. 69–79, 2002Copyright © 2002 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/02/$–see front matter

69

directly applicable to humans, even limited effects couldpotentially affect the parameter under investigation,namely, the perfusion. In addition, the preparation andinjection procedure can also be laborious, and a way ofsimply obtaining a quick perfusion estimate still robustto motion artefacts, but without needing to resort tocontrast agents, would, naturally, be desirable.

Buhrer et al. (1996) suggested a subtraction proce-dure to reduce movement artefacts. The basic idea was touse two small adjacent sample volumes for the measure-ment. Further, the signal originating from blood is sto-chastic in nature, and the movements in tissue can beconsidered small and correlated between the two samplevolumes. By subtracting the signals from the two vol-umes, the time-correlated part can be suppressed. Theinvestigators could show good results for measurementson a phantom, but artefacts were only partly suppressedin vivo. This paper proposes a way to make use of thisidea for a single sample volume using a continuous-wave(CW) US Doppler, and the use of multiple transmitfrequencies instead of multiple sample volumes.

The blood perfusion in the sample volume of a CWDoppler system may, under certain conditions, be esti-mated as the first moment of the Doppler power spec-trum, S( f ) (i.e., the spectrum obtained after mixing thereceived and transmitted signals, followed by low-passfiltering to remove the sum frequency) (Hertz 1981;Dymling 1985; Dymling et al. 1991).

The conditions mentioned above are, however, dif-ficult to meet in practice. For instance, the receivedpower is not proportional to the number of blood cells inthe sample volume, and only velocity components in thedirection of the probe are measured. Further, this maynot reflect the true capillary (nutritive) flow becausevessels may simply traverse the sample volume. Thus,only a relative measure is available, even though this stillmay be useful, as shown in previous experiments wherea perfusion decrease during smoking was observed (Dy-mling et al. 1991), as well as reactive hyperaemia (Dy-mling 1985). In the context of this paper, however, themodel will be such that our detected signal will beproportional to the mean number of particles in thesample volume multiplied by their mean (absolute) ve-locity along the direction of the probe. This is anotherway of expressing the first moment of the Doppler powerspectrum (Atkinson and Woodcock 1982).

For long observation times, it has been noted thatthe perfusion signal fluctuates significantly (Eriksson etal. 1991). Eriksson et al. (1995) showed that, by trans-mitting a number of frequencies simultaneously, virtu-ally independent perfusion estimates were obtained onthe individual channels. The estimates varied largelyover time due to speckle (Jansson et al. 1996), andforming the mean of the channels gave a more stable

result. The sensitivity for movement artefacts, however,remained the same.

A recent study has investigated the transmit fre-quency separation necessary to obtain uncorrelated re-ceived power from a random set of scatterers in a CWsystem (Jansson et al. 2001). The separation was foundto be dependent on the geometry of the transducer con-figuration, but this was verified only for a static case.

The aim of this paper was, first, to show if theDoppler signal power decorrelates vs. frequency in thesame way as the received power in a static case—at leastfor small Doppler shifts, and only for the transducerconfiguration used in these experiments. These resultswere employed second, to investigate the benefit of asubtraction algorithm in suppressing movement artefacts.Specifically, how this algorithm affects the mean andvariance of the perfusion estimate will be examined.

METHOD

Independent Doppler signalsIn the procedure proposed by Buhrer et al. (1996) the

crucial assumption was that the signals stemming fromblood flow are uncorrelated, but tissue movements are cor-related between two adjacent sample volumes. While in-vestigating the same sample volume using CW US withtwo (or more) simultaneously-transmitted separate frequen-cies, the question arose as to whether or not the sameassumption can be made here for the different channels.

In a study of frequency-dependence of speckle incontinuous-wave US, Jansson et al. (2001) showed that thepower received from a stationary set of small randomly-dispersed scatterers varies with transmitted frequency. Therate of change of received power vs. frequency can bedescribed by a covariance function C given by:

C� f0, f1� � �A1�V0

�V1

�� f0, r0��*� f1, r1�

� ��r0��*�r1��d3r0d3r1�2, (1)

where �(f, r) is the (complex) sensitivity of the system atlocation r, when transmitting frequency f. The complexsensitivity describes how the phase changes in everypoint of the active sample volume as the frequency isaltered. It is calculated as the product of the complextransmitter and receiver fields; thus, being dependent ontransducer apertures, and the geometry of the setup. Ai isa constant and ��(r0)�*(r1)� is the autocorrelation func-tion of the medium inhomogeneities. The brackets de-note expected value, and �(r) � �� (r) � i � o��(r)describes the local compressibility (�) and density (�)variations; i is a unit vector in the propagation direction

70 Ultrasound in Medicine and Biology Volume 28, Number 1, 2002

of the incident wave and o is a unit vector in the directionof the received beam; �� and �� are as defined by Morseand Ingard (1968). The integrations are performed overvolumes that include the entire support of �� and ��.Strictly, this function is only valid under the Born ap-proximation, and in the far field.

For random media that are statistically homoge-neous, ��(r0)�*(r1)� is dependent only on the differencebetween r0 and r1. Further, if the structure of the mediumis spatially uncorrelated, as if it contains small, random-ly-dispersed particles, the autocorrelation function can beset equal to the Dirac delta function. Then we can writethe covariance function:

C�� f0, f1� � �A2�V0

�� f0, r0��*� f1, r0�d3r0�2. (2)

The normalized covariance function (i.e., the correla-tion coefficient r(f0,f1)) is then a function that is onefor f1 � f0, and decreases to zero if f1 is sufficientlylarger than f0.

On the other hand, if the medium is correlated, thecovariance function may be broadened. For the limitingcase of a plane reflector in the sample volume, where��(r0)�*(r1)� is constant along the boundary, the re-ceived power would not change appreciably by changingthe transmitted frequency (at least for transducers havinga flat frequency response).

The above was shown for a static case, but is thisalso true for a moving structure? In other words, couldthe Doppler signals on two channels be considereduncorrelated if a moving random medium was in-sonated and the transmit frequencies were separatedthe amount predicted for a static case? And would theybe correlated if a plane reflector was moving in thesample volume?

This question will be answered experimentally but, fornow, we will assume that the squared amplitude of theDoppler signal phasor (from here on denoted Doppler pha-sor magnitude) will have the same decorrelation propertiesfor varying transmit frequency as the received power in thestatic case. The Doppler phasor magnitude will here bedefined as I2 � Q2, where I denotes the signal obtained aftermixing the received signal with the transmitted signal, and

Q denotes the signal obtained after mixing with the trans-mitted signal, phase-shifted by 90°.

Perfusion estimate from time-domain processingFrom the Doppler signal, the perfusion, P, can now

be estimated, as was mentioned above, with the firstmoment of the Doppler power spectrum. That is,

P � �0

�

fS� f �df. (3)

This processing of the Doppler signal can be per-formed with analog circuitry, as by Eriksson et al. (1991,1995). A theoretical motivation is also given by Eriksson(1994), but a short recapitulation will be given here forconvenience.

A block diagram of the unit is seen in Fig. 1. First,a few assumptions will have to be made about the Dopp-ler signal x(t) (e.g., the signal obtained after multiplica-tion with the transmitted frequency, followed by low-pass filtering). First, x(t) will be assumed to be a Gauss-ian random process (Mo and Cobbold 1986). Further, itwill also be assumed to be band-limited and wide-sensestationary and, thereby, also ergodic (Mo and Cobbold1986). The validity of these assumptions will be dis-cussed later.

In the analog processing unit, x(t), which has thepower spectrum Sx( f), is first transformed to y(t) via alinear filter. The filter has the frequency function:

H� f � � � fei�� f �, (4)

where �(f) is the phase function of H( f ). The powerspectrum of y(t), Sy( f ), is therefore related to Sx( f ) as:

Sy� f � � �H� f ��2Sx� f �. (5)

The integral of Sy( f ) is, thus, equal to the perfusionvalue as defined in eqn (3). This can also be expressed:

P � �Sy� f �df � ry�0�, (6)

where ry(0) is the covariance function with � � 0 (i.e.,the variance). This is equivalent to E[ y2(t)] if E[ y(t)] �0, as here (E[�] denotes ensemble average). Because x(t)is ergodic, we have:

E y2�t� � limT3�

1

2T ��T

T

y2�t�dt. (7)

Fig. 1. Block diagram of the analog processing unit.

Movement artefact suppression ● T. JANSSON et al. 71

The integral in eqn (7) is, in practice, estimated over afinite time using a low-pass filter.

Subtraction algorithm using time domain processingFor these experiments, the analog unit was modified

so that the Doppler signal was sampled directly after thefilter H( f ). This had the advantage of increasing thedynamic range, and avoiding offset errors and nonlin-earities in the analog multiplier.

Further, subtracting channels demodulated to giveonly the in-phase component does not work becausemotion artefacts are correlated only in amplitude, but notphase. Therefore, quadrature detection was employed,and each quadrature component filtered with H( f ).

To find the perfusion estimate, a slightly differentapproach was used that described above. A block dia-gram of the modified processing unit performing thesubtraction is seen in Fig. 2. In short, the variance, 2, ofthe Gaussian random variable Y � y(t) is sought. Now,the envelope of such a random variable (U and V, re-spectively, in Fig. 2) is Rayleigh-distributed with themean �/2 (Lindgren and Rootzen 1991).

Subtracting two independent random variables withdensity functions fU(u) and fV(v) will result in a newrandom variable with the density function (Gut 1995):

fW�w� � ��

�

fU� z� fV� z � w�dz. (8)

Evaluating this integral for two Rayleigh distributionsresults in a complicated expression, which is given inAppendix A. The resulting density function is, however,very similar to a Gaussian distribution with zero meanand variance s

2 � (4 � )2 (N(0, s2)), provided that

the Rayleigh distributions have the mean mentionedabove. Again, we are left with the problem of estimatingthe variance from a, this time, nearly Gaussian randomvariable. Squaring, forming the time average, as with theanalog unit, gives the perfusion estimate Psub for thesubtraction algorithm. Note how the estimate is a factor4 � smaller than the sought value 2.

The estimate Psub is then compared to what is obtainedusing the previous method, from here on called the standardmethod. Here, the envelope is first squared, time-averaged,and then the two channels are added. The square of theenvelope of a N(0, 2) distributed random process is a newrandom process that has an exponential distribution withmean m � 22 (Lindgren and Rootzen 1991). Addition oftwo such independent random variables gives a new ran-dom variable that is gamma-distributed with the mean and,thus, perfusion estimate Padd, 2m ( (2, m)) (Gut 1995). Theratio between the perfusion estimate from the standardmethod, Padd, and the subtraction algorithm, Psub, is, there-fore, 4/(4 � ) � 4.66.

Both algorithms appear to give perfusion estimatesthat only differ by a constant. However, the quality of theestimates are somewhat different. Psub is the mean of a �2

(1) (Chi square with one degree of freedom) distributedrandom variable, as a result of squaring the nearly Gaus-sian-distributed random variable. This has a SD that is�2 times its mean, �2Psub. Padd, on the other hand,being the mean of a (2, m)- distributed random variable,has the SD �2m, �2Padd/2. By normalizing the perfu-sion estimates to each other, we see that the estimatebased on the subtraction algorithm has a SD twice that ofthe estimate based on the standard method.

Theoretically, this procedure would now yield aperfusion estimate that has a doubled SD compared tothe standard method but, in return, would suppress cor-related amplitudes between the channels. In practice,however, it turns out that the signal after envelope de-tection has to be filtered for the artefact suppression towork properly. The reason is that the filters H( f ) are notidentical, which results in a ripple on the detected enve-lope. This filter is indicated as “MA-filter 1” in Fig. 2.This filter has, however, the adverse effect of makingPsub nonlinear because the random variables U and V areno longer Rayleigh-distributed after the filter.

ValidationThis section gives an outline of the experiments

performed. A detailed description of the equipment usedis found in the Materials section.

The method described above was validated using acylindrical agar phantom with randomly dispersed glassparticles. When the phantom is rotated, the particle velocitywill increase linearly with increasing radius from the rota-tional axis for a constant angular speed. If the rotational axisis placed on the z-axis as defined in Fig. 3, the powerspectrum of the received signal will be symmetric aroundthe carrier frequency. The z component of the velocity (theone that is measured by the Doppler system) will, forreasons of geometry, be constant along each line parallel tothe z-axis, so it is not crucial that the rotational axis coin-cides with the intersection of the acoustic axes of the trans-

Fig. 2. Block diagram of the modified processing unit. Boxesindicated as “MA-filter” are (uniform) moving average filters.


ducers (see Appendix B). Because the sample volume isconstant and, on average, contains the same amount ofparticles, the “perfusion” will be a linear function of theangular speed of the phantom.

To investigate what frequency separation was nec-essary to obtain uncorrelated Doppler phasor magnitudesof signals from a moving random medium, the followingprocedure was employed using the two-transducer con-figuration seen in Fig. 3. Two frequencies were transmit-ted simultaneously, and the received signal was quadra-ture-demodulated to obtain both the in-phase (I) andquadrature (Q) components. The phantom was rotated,and a time sequence of I and Q signals for both frequen-cies were recorded. This procedure was repeated for eachfrequency separation, and the covariance of the se-quences If0

2 � Qf0

2 and If1

2 � Qf1

2 was calculated. Note thatthe data in this case were sampled prior to the filter H( f).Filtering that was employed was a low-pass filter toeliminate the sum frequency after the mixer, and a largeby-pass capacitor to block offset drift from a preampli-fier.

Perfusion estimates from the subtraction algorithmwere compared with the standard method using the samedata. Measurements were performed with seven differentangular velocities of the phantom and, also, three differ-ent amplitudes of the transmitted signal, because theperfusion is expected to increase not only with speed ofthe phantom, but also with received power. In in vivomeasurements, more power is assumed to correspond tomore moving blood cells (an assumption that may not be

entirely valid, at least for larger vessels; (Shung et al.1992). Strictly, phantoms with different concentrationsof scattering objects could have been manufactured, butincreasing the transmitted power seemed to be a simplersolution. The frequencies employed were 4.8 and 4.85MHz.

One experiment was also conducted where a strip ofan overhead film was inserted between the transducerpair and the phantom. This was to simulate a tissue layer,and carefully touching the film caused large artefacts onall channels. As the film swayed back and forth, artefac-tual signals could be recorded for a few seconds. Ofcourse, this does not resemble an in vivo situation, wherecapillaries are coupled to the tissue. The model is, how-ever, useful to determine if it is possible to separatecorrelated from uncorrelated signals.

Finally, an in vivo measurement was made on thethumb pad of a 32-year-old male with no known circu-latory defects. This time, the 10-MHz two-element trans-ducer used by Eriksson et al. (1995) was used, transmit-ting two frequencies at 9.95 MHz and 10.1 MHz. Gentlymoving the finger resulted in strong artefactual signals.

MATERIALS

The manufacturing procedure of the phantom hasbeen reported elsewhere (Jansson et al. 1998, 2001), andthe medium has been proven to produce fully developedspeckle at the frequencies used here (Jansson et al.2001). The phantom had a cylindrical shape with adiameter of 34 mm and a length of 120 mm. The scat-tering particles were glass spheres with a mean diameterof 93 m (type A2429 glass, Potters Industries, ValleyForge, PA). The phantom was molded to a holder so thatit could be rotated in a water bath around its axis witha small DC motor (Dunkermotoren, Bonndorf/Schwartzwald, Germany), geared a ratio of 1:100. Thewalls of the water tank were covered with a sound-absorbing material, to avoid standing waves in the tank.

Two circular 5-MHz transducers were used to ver-ify the theoretical calculations described above. Thetransducers were manufactured in-house and had a phys-ical diameter of 6.35 mm. They were fitted in a specially-designed holder so that their acoustic axes could beangled 30° to each other. The acoustic axes intersected30 mm from the transducer face, which corresponds to7–8 mm beyond the Fresnel–Fraunhofer limit for theseparticular transducers. Fixation of the holders usingturned steel rods tapered to an angle of 30° assured theangle and distance to be accurate. The transducer pairhad a sensitivity maximum at 4.85 MHz, and the �3 dBbandwidth was approximately 4.5–5.6 MHz.

For the experiments, four frequency synthesizers(Hewlett Packard HP3325A, Andover, MA) were used to

Fig. 3. The transmitter and receiver in a Cartesian coordinatesystem, together with the cylindrical phantom. The dashed linesrepresent the acoustic axes of the transducers and the rotational

axis of the phantom.


produce the transmitted signals and their 90° references.The frequency synthesizers were phase-locked pair-wise,and the phase was adjusted on the one serving as refer-ence for the Q channel, to produce sin2ft, and thereference for the I channel gave cos2ft. The outputsignals from the two I-channel synthesizers were addedin a resistor network and fed to the transducer acting asemitter.

The receiving transducer was connected via an am-plifier with 40-dB gain over 0.5–20 MHz (Panametrics5676, Waltham, MA), to a power splitter with four 50-�outputs. These were connected to the RF inputs of themixer (SRA 1H, MiniCircuits, Brooklyn, NY), and the“sync” outputs of the synthesizers were coupled as ref-erence. After a 200-Hz low-pass filter and some 60 dBadditional gain, the I and Q signals as shown in Fig. 2were obtained.

Essentially, the filter H( f) is realized as cascadedactive high-pass filters, and the squared amplitude func-tion, or �H( f)�2, is plotted in Fig. 4. The squared ampli-tude function is plotted to better assess the linearity vs.increased perfusion. The amplitude function itself devi-ates by less than 4% from the ideal �f response at full

scale. The measurements presented in Fig. 4 were ob-tained after the experiments. Reference measurementsbefore the experiments showed somewhat better results,but problems with drift degraded the performance to thatseen in Fig. 4. The nonlinear behaviour at low frequen-cies is a result of stacking high-pass filters with differenttime constants and is, thus, inherent in the design of theanalog �H( f )�2 � f filter.

The filtered signals were then sampled using a four-channel digitizing oscilloscope (Tektronix, TDS 3014,Wilsonville, OR, USA) at 1 kHz for the experiment todetermine the correlation of Doppler power, and at 500Hz in the other experiments to give a longer time record.The memory depth of the oscilloscope was 10,000 pointsper channel, which allowed a 10 s record in the firstexperiment, and 20 s in the other experiments.

The recorded data were transferred to a standardpersonal computer via GPIB and stored to files for fur-ther processing off-line. The evaluation of the data ac-cording to the processing described above was per-formed in MATLAB (The Math Works, Natick, MA,USA).

RESULTS

The results from the investigation of the frequencyseparation necessary to obtain uncorrelated Doppler pha-sor magnitudes are seen in Fig. 5. The original timerecord was divided into 10 groups with 1000 samples ineach. The normalized covariance function was then cal-culated for each group, and the solid line in Fig. 5 is themean of these 10 graphs. The error bars indicate plus orminus 1 SE of the mean. In the same figure, the exper-

Fig. 4. Frequency and phase characteristics of the filters H(f).The top graph shows the squared amplitude of the outputsignal, for an input voltage of 1 mV peak-to-peak. Channel 1(—); channel 2 (- - -); channel 3; (. . .) and channel 4 (grey

line).

Fig. 5. Normalized covariance function of the magnitude of theDoppler signal phasor (or in other words, the sum of thesquared quadrature channels, I2 � Q2), with mean of 10 groupswith 1000 samples in each (—) � 1 SE (error bars). The dashedline is the experimental results from the static case; data fromJansson et al. (2001). The thinly dashed line is the expected

asymptotic value.


imental data reported by Jansson et al. (2001) are plottedas the dashed line. These data points were obtained usingexactly the same transducer configuration, (i.e., the sameangle between emitter and receiver, the same placementin the holder etc.,) but for a static case. The SE of thelatter measurement was on the same order of magnitudeas the Doppler measurement, but these error bars are notplotted for reasons of clarity.

How the “perfusion” of glass particles in thesample volume is estimated for the subtraction vs. thestandard algorithm is shown in Fig. 6. The perfusionvalues are taken as the mean over the entire timerecord (10,000 samples) of Psub and Padd, respectively.Measurements were made at seven different angularspeeds of the phantom (12, 18, 24, 30, 36, 42 and 48°per s), and at three peak-to-peak voltages of the trans-mitted signals (353, 500, and 612 mV). The speedsresulted in maximum detectable Doppler shifts rang-ing from 25 to 100 Hz. The amplitude levels werechosen to give detected perfusion values that had therelative levels 1, 2 and 3, which experimentally turnedout to be 2.00 (� 0.08) and 2.95 (� 0.13) for thesubtraction algorithm, and 2.01 (� 0.05) and 2.99 (�0.08) for the standard method.

The SD of Psub was predicted to be twice that ofPadd. In the experiments, the ratio was 2.17 (range 2.05 to2.25) for all speeds and power levels.

Theoretically, all common mode signals (arte-facts) should be suppressed using the subtraction al-gorithm. In practice, however, the suppression is notperfect if the envelope signals are not filtered prior tosubtraction. Figure 7 shows how increasing filterlength influences artefact suppression in the in vitroexperiment. Three lengths for the moving average

filter (MA filter 1) are shown, applied to the same data(no filter, 10 samples)(20 ms at 500 samples per s),and 20 samples (40 ms), compared with the standardmethod.

The result from the in vivo recording is seen in Fig.8. Here, filter lengths of 1, 10, 20 and 40 samples wereused (all samples weighted equally). The filter length ofMA filter 2 was 100 samples (0.2 s) for both the in vitroand in vivo cases.

MA filter 1 has the drawback of making theperfusion estimate after the subtraction algorithm non-linear. Figure 9 shows the nonlinear behaviour of Psub

compared to Padd for a filter length of 20 samples.

Fig. 6. Perfusion estimates from the addition (—) and subtrac-tion algorithms (- - -), respectively, for three different peak-to-peak voltages of the transmitted signals. On the horizontal axisis angular speed of the phantom in degrees per second. Psub has

been multiplied by a factor of 4.66.

Fig. 7. A comparison of the standard method to the subtractionalgorithm for three different filter lengths in MA filter 1. Thetop panel shows the standard method and, thereafter, the sub-straction algorithm for filter lengths of 1 (no filter), 10 and 20samples. Artefacts are introduced at t � 5 s and t � 13.5 s, bygently pushing the overhead film suspended between the trans-ducers and the phantom. Note how the film sways back and

forth, generating artefacts for some time.


How the ratio between Padd and Psub changes withangular velocity of the phantom is shown in Fig. 10 forfour different filter lengths (no filter, 10, 20 and 40samples). The error bars indicate the variation be-tween the three power levels.

DISCUSSION

It appears, from Fig. 5, that the decorrelation rate ofthe Doppler phasor magnitude closely follows what wasfound in the static case. This study does not, however, claimthat this is generally the case. For instance, one questionmay be if the decorrelation rate is different for other speedsof the phantom. Preliminary results suggested that the cor-relation coefficient was independent of the angular speed,but here, of course, the investigated speeds all resulted inDoppler shifts that were significantly lower than the widthof the correlation function. For the purpose of this investi-gation, a frequency separation of 50 kHz was chosen, basedon Fig. 5. It is important to note, however, that the corre-

Fig. 8. A comparison of the standard method with the subtrac-tion algorithm for the in vivo recording. The top panel showsthe standard method and, thereafter, the subtraction algorithmfor filter lengths of 1 (no filter), 10, 20 and 40 samples. Two

artefacts are introduced at t � 7 s and t � 14 s.

Fig. 9. Perfusion estimates from the addition (—) and subtrac-tion algorithms (- - -), respectively, for three different peak-to-peak voltages of the transmitted signals. On the horizontal axisis angular speed of the phantom in degrees per second. A filterlength of 20 samples has been employed in MA filter 1 for the

subtraction algorithm.

Fig. 10. Ratio between Padd and Psub vs. angular velocity of thephantom for four different filter lengths (no filter, 10, 20 and 40samples). The error bars indicate the variation between the

three power levels.


lation function is dependent on the absolute frequencies f0and f1, not only their difference. The measurements re-ported in Fig. 5 actually start at 4.7 MHz, and the frequen-cies used in the perfusion experiments were 4.8 and 4.85MHz. A similar measurement to that in Fig. 5, but startingat 4.8 MHz, showed, however, that the width of the corre-lation function was somewhat narrower than in Fig. 5.

Apparently, the autocovariance functions for theDoppler phasor magnitude and the static cases agree, butit is not necessary that the same applies for the perfusionestimate, as obtained by the time-domain processingdescribed above. However, because the signals to becorrelated are passed through (ideally identical) linearfilters before the squaring operation (whereby the poweris obtained), one could assume a similar response. Ameasurement of decorrelation of perfusion estimates vs.transmitted frequency revealed that the correlation func-tion became somewhat narrower (decorrelation occurredat a frequency some 20% less than for the static case).This may, however, be an effect of the nonideal filtersH( f) used in the present perfusion-processing unit (Fig.4). In any case, 50-kHz transmit frequency separation issufficient for this transducer configuration.

Figure 6 shows that the perfusion, as estimated fromthe two algorithms, agrees closely after multiplication ofPsub by the factor 4.66, as predicted by the theory. Theactual ratio is around 4.35 (Fig. 10), and this discrepancyis attributed to either Psub not being truly �1

2-distributed,or possibly the imbalance between the four filters H( f)(Fig. 4). The ratio of SDs for the two estimates alsoagrees well with theory. For the range of velocitiesinvestigated, the perfusion increases linearly with in-creasing speed (R � 0.9987 to 0.9995). Note, however,that extrapolating the estimates to zero velocity results innegative perfusion values. In reality, the estimates, ofcourse, deviate from the linear behaviour at lower veloc-ities and approach zero perfusion for zero velocity of thephantom. This is a result of the low-frequency behaviourof the filter H( f) (Fig. 4). Some clipping was noted whenacquiring the signal at the highest power level and thehighest speed. This is probably the reason for the some-what lower Psub in this case.

The results from the standard method used here differfrom what was employed by Eriksson et al. (1995) roughlyby a factor of two because here, the envelope of the Gauss-ian signal y(t) is considered, as opposed to only the in-phasecomponent by Eriksson et al. It deserves to be noted that theI and Q channels can be multiplied by a factor to correct fordifferences in the frequency response of the transducers.This was not done here, however, because the difference insensitivity was only a few %.

The subtraction algorithm clearly reduces move-ment artefacts in both the in vitro and the in vivo mea-surements. For the unfiltered envelope data, however, the

reduction is more modest. In the in vitro experiment, thesecond artefact is reduced by a factor two to four, and thefirst is only somewhat, if at all, suppressed. The reasonfor this is presumably the different response for the fourfilters H( f ) (Fig. 4). This caused, 1. the I and Q channelsnot to be exactly the quadrature components of the signal(mixing both channels with cos2ft did not produceidentical results) and, consequently, 2. the detected en-velopes at the two transmit frequencies have somewhatdifferent phase and amplitude. The moving average filter(MA filter 1) reduces this discrepancy between the twodetected envelopes, but has the adverse effect of giving anonlinear perfusion estimate, as pointed out before (Fig.9). This effect is expected because higher frequenciesare, of course, suppressed by the filtering action (Fig.10). Note also that the perfusion estimate is nonlinearonly in frequency, so that it is not sufficient to multiplythe estimate by some factor (Fig. 10). It should be notedthat the artefact level was not changed appreciably forany type of filtering action in the standard method.

Another possible explanation for the partial artefactsuppression could be that the correlated Doppler shiftson the channels may not have exactly the same frequencyand phase content, because they result from differentcarrier frequencies. After the H( f ) filters, only high-frequency components remain, and these may be ex-pected to vary more between channels. Again, due to theimbalance between the filters, it is difficult to say any-thing specific about the main cause for the suboptimalartefact suppression.

In the in vivo experiment, a filter length of 40 samplesis necessary to reduce the level of the artefacts to that of thedetected perfusion. At this level of filtering, however, it isdifficult to say anything definite about how the detectedperfusion is related to blood flow. The higher level offiltering necessary in the in vivo experiment could be aconsequence of the lesser amount of correlation that ispresent in tissue compared to the reflector-like overheadfilm. Further, in tissue the capillaries also move togetherwith the tissue, as opposed to the situation with the agarphantom/overhead film model. Presumably, this wouldmake the artefact suppression less effective. On the otherhand, this points to a possible benefit of a moderate filteringof the detected envelopes. Even if perfectly matched H( f )filters were realized, as if implemented digitally, the algo-rithm would be more robust to motion artefacts that are notperfectly correlated.

The nonlinearity caused by filtering the envelopedata could also be accounted for in the design of theH( f ) filters, so that an overall linear response is ob-tained. One should, however, need carefully to considerany trade-off effects by increasing the high-frequencycomponents after the H( f ) filters.

An alternative solution could be to subtract the


Gaussian signals xi(t), prior to any filtering and envelopedetection. This is basically the method proposed by Buhr-er et al. (1996). It would necessitate sampling x(t) di-rectly and, thereafter, implementing the filter H( f ) dig-itally. Their method, however, adds the complexity ofalso estimating the phase difference between the twochannels. Because here the variance of the time signal issought, this appears to be a somewhat unnecessary step.

By using more channels, the variance of the esti-mate from the subtraction algorithm can be substantiallyreduced. By adding just one more channel, three differ-ent Psub may be formed, which will decrease the SD ofthe estimate by a factor of �3, compared to what hasbeen presented here. Naturally, which basically amountsto the same thing, the speckle effect, as seen by Erikssonet al. (1995), will decrease by adding more channels.However, one may have to take into consideration thatsignals from moving tissue will be less correlated, thelarger the frequency difference. How much is subject tofurther investigation, but it can be assumed that a smallersample volume will give a larger range over which tissuesignals are correlated. On the other hand, the necessaryfrequency separation for blood to look uncorrelated willincrease, eqns (1) and (2) for a smaller sample volume.The number of channels that can be used is, of course,dependent on the usable bandwidth of the transducer, butthere is a trade-off between the number of channels and,thereby, deposited energy, and the sensitivity (and,thereby, bandwidth) of the transducer.

For the theoretical motivation of the time-domainperfusion estimation unit to hold, it was assumed thatx(t) was a band-limited and wide-sense stationary,ergodic, Gaussian random process. Conceivably, thisholds true for the agar phantom model, but it is natu-rally questionable in vivo. Mo and Cobbold (1986)gave an elaborate discussion on this topic, and, forlarge vessels, they concluded that these conditions aremet for time intervals �10 ms. Here, much smallervessels are considered, so the conditions may hold forlonger time intervals. Implicitly, the time interval hashere been chosen to 200 ms, by the time constant inMA filter 2. This is, however, not to suggest that thisis a suitable time constant to ensure a “ true” perfusionestimate.

Provided that the suboptimal artefact suppression is aresult of the filter imbalance, this method may be a powerfultool to measure blood perfusion in skeletal muscle, anobjective not achievable using standard methods due to thepresence of muscle tonus (Heimdal and Torp 1997). Be-cause the sample volume is practically identical for thedifferent channels, correlation may be expected from Dopp-ler signals stemming from vibrating muscles.

CONCLUSION

The subtraction of two uncorrelated perfusion sig-nals from a multifrequency CW Doppler system is po-tentially an effective method to reduce motion artefacts.The resulting perfusion estimate is linear for both in-creased velocity and number of (independent scattering)particles. The quality of the estimate is, however, re-duced compared to adding (in effect averaging) the per-fusion estimates, the SD being a factor of 2 higher.

The time-domain processing was here performedusing analog filters that were not perfectly matched be-tween channels. This was probably the main reason for areduced ability to suppress movement artefacts. The ar-tefact reduction was enhanced by filtering the signalsprior to subtraction. The resulting perfusion estimate,however, became nonlinear from this operation, the non-linearity arising from the filter reducing the high-fre-quency content in the Doppler signal.

Acknowledgements—SSF (The Swedish Foundation for Strategic Re-search, project CORTECH), TFR (The Swedish Research Council forEngineering Sciences), and the ELFA Foundation are thanked for theirfinancial support. Numerous helpful discussions with Dr. P. Wahlbergare acknowledged with pleasure, as are helpful hints from Prof. G.Lindgren and Prof. U. Holst.

APPENDIX A

Subtracting two independent Rayleigh-distributed random vari-ables with the same parameter, �, results in a new random variable withthe density function:

fx� x� �1

4�2�x�ex2

2�� x��2x2�2 �

2��x�

ex2

2��

� �2�� x2

�� x2�erf� � x2

�

�2 �� , x � 0

fx� x� �

�

2

2��, x � 0, (9)

where erf() is the error function. This distribution has zeromean, and the variance �/2(4�). It is very similar to aGaussian distribution and the square of a random variable withthe distribution described by eqn (9), has practically the samedensity function as a squared Gaussian random variable withthe right relationship between variances, as developed previ-ously (Fig. 11).

APPENDIX B

Consider a particle at a distance r from the center of the phan-tom. The particle will follow a circular trajectory as the phantom isrotated at a constant speed, with a velocity directly proportional to r(i.e., we have v � kr). Now, assume the particle to be on the z-axis, sothat vz � 0 (on the side closest to the transducer, and the phantomrotating in the indicated direction; Fig. 3). After rotating an angle �,vz � kr sin �. At this point, the particle will be a distance x0 � r sin


� from the z-axis. Comparing the expressions, we see that we can writevz � kx0, or in other words, vz is constant at each distance x0 from thez-axis. Because we have discrete particles, we need to form a temporalaverage to measure all velocity components.

Strictly, a small deformation of the sample volume may beexpected because the rotational speed of the phantom will add to thesound speed, but this effect is shown to be extremely small if the speedof the moving medium is much smaller than the sound speed (Wells etal. 1989; Kremkau 1990; Wells and Halliwell, 1990).

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Fig. 11. The upper panel shows a N(0, 1) distribution (- - -)plotted together with the density function obtained after sub-tracting two Rayleigh distributions with mean �2 (—).Squaring two such distributed random variables gives practi-

cally identical density functions (lower panel).


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Movement artefact suppression in blood perfusion measurements using a multifrequency technique