A fundamental interpretation of ultrasonic Doppler velocimeters

Ultrasound in Med. & Biol., Vol. 2, pp. 107-111. Pergamon Press, 1976.Primed in Great Britain.

A FUNDAMENTAL INTERPRETATION OF ULTRASONIC DOPPLER VELOCIMETERS P E T E R ATKINSON Departmentof MedicalPhysics, BristolGeneralHospital,GuineaStreet, BristolBS1 6SY, U.K.

(First received 10 February 1975;and in final form 19 August 1975) Abstract--The basic Doppler equation is not entirely suitable when describing practical ultrasonic Doppler velocimeters. This article examines the operating principles of more useful devices such as continuous-waveand pulsed-Dopplersystems. The analysis is based on intuitiveconcepts, illustratedby waveformand spectral plots and supported by short mathematicalsequences. It is discoveredthat an uncertaintyprinciplelimitsthe velocity-spatial resolution product of any practical ultrasonic velocimeter. Key words: Acoustics, Ultrasonics, Doppler flowmeters. l. INTRODUCTION

Ultrasonic Doppler velocimeters are now becoming popular in both the medical and industrial fields. The Doppler principle says quite simply that if a target moves in the direction of an ultrasonic source then the echo reflected back to that source will be shifted in frequency. The magnitude of the frequency shift will depend, amongst other things, on the velocity of the target and the frequency of the incident ultrasound. As it stands, this description of the Doppler effect, although correct, is nevertheless of rather limited value when it comes to analysing a practical Doppler velocimeter. For example what is meant by "the frequency" of ultrasound in the case of a pulsed-Doppler system (Baker, 1970) where only a short burst of ultrasound (extending over a wide frequency spectrum) is transmitted? Or what happens if "the target" is a random distribution of point scatterers, e.g. red blood cells suspended in plasma? How should the basic Doppler equation be modified to suit these commonly encountered conditions? In this article the form and frequency content of the echo returning from targets moving through the sample volume (defined later) of an ultrasonic velocimeter will be studied in detail• It will be shown that the Doppler shift need not be the only modification to the frequency content of the ultrasonic echo, and that the dimensions of the sample volume play a major part in determining the shape of the Doppler-shift frequency spectrum. The analysis will not be entirely mathematical as this approach has been partially dealt with by Newhouse (1974). Instead, by making use of illustrations, an intuitive description of Doppler signal analysis will be constructed, In this way it is hoped to provide a more basic understanding of practical ultrasonic velocimeters. The paper is arranged as follows: in Section 2 the basic Doppler equation is examined to discover the precise configuration to which it applies if it is to be the only source of frequency perturbation. This section also introduces the less mathematical approach to Doppler signal analysis which will be adopted in this article, and which makes use of waveform and spectral plots to illustrate the frequency content of the ultrasonic and electrical signals. In order to create a gradual progression to the analysis of more useful devices, continuous wave and pulsed-Doppler velocimeters are then investigated in Sections 3 and 4 respectively. Up until this stage it will

have been assumed that the target is a single point scatterer. In Section 5 the analysis is extended to predict the behaviour of an ultrasonic velocimeter when the target is a random distribution of point scatterers (e.g. red blood cells suspended in plasma). 2. IMPLICATIONS OF THE BASIC DOPPLER EQUATION

The fundamental Doppler equation (which is usually quoted throughout the relevant literature) states that the echo back-scattered from a target P moving with a velocity v in the direction of an ultrasonic source, will be shifted in frequency from the transmitted frequency Jo by an amount [~

2v

=TI,~,

(l)

where c is the propagation velocity of ultrasound in the intervening medium. This process is illustrated in Fig. 1 which shows the form and frequency content of the transmitted, received and Doppler shift signals. Notice that the Doppler shift frequency spectrum (Sff) in Fig. lg) is, in accordance with equation (1), a single spectral line located at fd. However, a single frequency Doppler shift signal can only be produced if a plane target is moving at a constant velocity through a monochromatic ultrasonic field which extends over an infinitely wide beamwidth. If any of these conditions (illustrated in Fig. la) are not ultrasonic configuration

I

ult¢osoni¢

transmitted woveform

transmitted spectrum

t~.~10-6 s=d T(f)

(o) ceceived

received speclrum

wave form

t~'tO-6s "~

• time

(b)

to

~ime

Doppler difference signol

~R(f )

I

(a)

I

~. ~I0-3s . . ~

t= Doppler difference spectrum

s(O

I: I ~ (,)

f

time (t)

(g)

Fig. 1. An interpretationof the basic Dopplerequation (I). 107

108

P. ATKINSON

satisfied then the Doppler frequency spectrum cannot be a single spectral line. For example, if the target is not a plane surface or if the beamwidth is not infinitely wide, then movement of the target will eventually cause variations in the amplitude of the reflected echo. This amplitude modulation immediately broadens the Doppler shift spectrum. Similarly, if the target does not move at a constant velocity or if the transmitted ultrasound is not monochromatic, then the Doppler shift signal must contain more than one frequency component. Once again the corresponding Doppler shift spectrum deviates from a single spectral line. Thus the basic Doppler equation (1), if applied in isolation, imposes extreme restrictions on the Doppler configuration. This expression must be modified if it is to be of use in describing practical velocimeters. 3. CONTINUOUS-WAVE DEVICE

In this section two of the restrictions imposed by equation (1) will be removed. This will allow the analysis of the system conventionally known as the continuouswave Doppler velocimeter. Consider the situation illustrated in Fig. 2(a) where a single point target P, moving at a constant velocity, is interrogated by a narrow beam of monochromatic ultrasound. The velocity of the target can be resolved into component directions, parallel and perpendicular to the beam axis. Movement along the beam produces Dopplershifted components in the returning echo whereas movement across the beam results in amplitude modulation of the reflected signals as the target passes through the ultrasonic field. This amplitude modulation also appears on the Doppler-difference signal (see Fig. 2f) since this has been derived directly from the received signal by coherent detection using the transmitted signal as reference (see, e.g. Atkinson and Follett, 1975). What effect does this amplitude modulation have on the Doppler shift spectrum? Fourier analysis predicts that an amplitude-modulated sine-wave must contain more than one frequency component. The modulating function determines the shape of the corresponding frequency spectrum and, in particular, the more rapid the modulation the wider the spectrum spreads. In addition the frequency of the enveloped sine-wave fixes the centre of the frequency band. Applying these ideas to the Doppler system leads to the conclusion that, for a constant target velocity across the beam, the frequency spread of the Doppler shifted tronsmitted w~veform

ultrasonic configuration

j 1/l)

(a) received

received

waveform

spectrum

~d)

spectrum

ultrosonic b~m \

transducer

I

transmitted

,,m.__~..

fo+~d

(e)

(b)

(c)

Doppler dlfference signa~

Doppler difference spectrum

;

td

CO

I

Fig. 2. The continuous-wavedevice.

components will be inversely proportional to the ultrasonic beamwidth. For example, the narrower the beam becomes, the shorter will be the traverse time, the faster the amplitude modulation will be and the wider the Doppler spectrum will spread. Notice that the particle, even though it is travelling at a constant velocity through a monochromatic ultrasonic field, has still managed to produce a complete spectrum of Doppler shift frequencies. This is the direct result of the point scatterer moving through the narrow ultrasonic beam. How does this spectral broadening affect the performance of the velocimeter? At this stage let it be sufficient to say that, in the presence of electrical noise, the width of the Doppler shift spectrum must be related directly to the precision with which the velocity of the target can be estimated. (A more exact mathematical analysis will be developed in the next section.) Since it has already been shown that the width of the Doppler spectrum is inversely proportional to the ultrasonic beamwidth, and because the lateral resolution is directly proportional to the beamwidth, the velocity resolution-spatial resolution product must remain constant. This is a formulation of the familiar uncertainty principle which occurs throughout physical systems. If it is to be obeyed then it is only possible to trade velocity resolution for spatial resolution or vice versa. In the next section it will be shown that the mathematical relationship between these two parameters can be defined more readily when the spatial resolution is in the same direction (i.e. axial) as the velocity component causing the Doppler shifted frequencies. 4. THE PULSED-DOPPLERVELOCIMETER The continuous-wave system described in the previous section has the disadvantage of not being able to determine the range of the target. It can only detect whether or not the target is within the ultrasonic beam. The most convenient method of enabling axial resolution is to transmit a short burst of ultrasound and time the delay of the returning echoes. However, the transmitted wave is then no longer monochromatic since the short pulse of ultrasound is made up of a complete spectrum of frequencies. This section, therefore, examines the performance of the system, conventionally known as the pulsed-Doppler velocimeter, where the restriction of monochromatic ultrasound has been removed. Instead of repeating the analysis described in the previous section, a frequency domain approach will be developed here to illustrate an alternative method of interpreting the cause of spectral broadening in Doppler velocimeters. Consider the situation illustrated in Fig. 3 where the moving point target is interrogated by a short, narrow pulse of ultrasound. A time-gating system is used to Doppler process only those echoes which are backscattered from a preselected range. This means that the sample volume is now a compact region in space (Fig. 3a) defined axially by the transmitted pulse and laterally by the beamwidth. As was mentioned above, it can be shown that the short ultrasonic burst occupies a wide frequency band. The transmitted ultrasonic frequency cannot be defined precisely although, intuitively at least, it would seem that the mean transmitted frequency (fo in Fig. 3c) should be used to predict the Doppler shift. What then of the accompanying frequency components? They cannot be ignored since they form an integral part of the short transmitted pulse. It is suggested that these components once again act to broaden the Doppler-shift spectrum into a distributed band of frequencies. In fact Newhouse et al.

109

A fundamentalinterpretation of ultrasonicDopplervelocimeters transmitted wavefo~rn

ultrasonic configuration

transmitted spectrum

,r(f) :~fo:

transducer

I ti

~vlO-6sM

lime

(b)

mca~ioCionO6ppletdifference signal ~:~r~G~c~d~

wove{acre receiv~ ~ i l t ~ from point target .

(c)

fo

f

D6ppler difference spectrum

F,,~lO'3sb4

" l ~V~O"l6-s~l m e

1 ~14vlO-3s~ i

i t ~me• (f)

(g) ( e ~

Fig. 3. The pulsed-Doppler velocimeter. Note that the frequency scale in (g)is expandedby a factor2c / v relativeto (c). (1974) have shown that, if the target moves axially at a constant velocity through the sample volume, then the Doppler-shift spectrum S(/) is the same shape as the transmitted ultrasonic spectrum T(f) but is scaled down in frequency by a factor v/2c. This is illustrated in Fig. 3(c) and (g). The more exact Doppler expression can then be written:

Sff)dI=K.

\--~, / ~--~-/

(2)

to be the distance between the points where the echo amplitude has fallen to one half of its maximum value (see Fig. 3b). The widths of the frequency spectra Af0 and Af, are defined in exactly the same way (Fig. 3c and g). In a noise-free device it would not matter if the Doppler spectrum were spread over a band of frequencies. So long as the impulse response of the system were known, the exact Doppler-shift frequency (and, consequently, the precise target velocity) could he determined. However. in any real system where electrical noise must be present, it is usually assumed that the full width between the half-maximum power points gives some estimate of the frequency resolution. In other words, adjoining spectra (caused in this case by slightly different target velocities) cannot be distinguished unless the peak-amplitude frequency separation is greater than the frequency resolution. Returning to the analysis, the spectral spread Af~ of the Doppler shift signal is inversely proportional to AT and so L'

,~k :, a---~

(4)

In addition, the velocity uncertainty Av can approximately be linked to the spectral width by the Doppler equation 2Av. Afd = ~ Jo

(5)

where the constant of proportionality K allows for normalisation and variations in scattering coefficient. Equation (2) says that the power contained between frequencies f and i f + dr) in the Doppler spectrum is directly proportional to the power contained between frequencies 2cf/v and 2c(f+df)/v in the transmitted ultrasonic spectrum. This means that the width of the Doppler spectrum Afu divided by the mean Doppler frequency/d is in exactly the same ratio as the width of the transmitted spectrum A/,, divided by the mean ultrasonic frequency/o. This is reasonable when considered from an intuitive viewpoint since, by Fourier transform theory, the ultrasonic pulse can be broken down into a series of sine waves. The amplitude and frequency of each sine wave determines the amplitude and frequency of the corresponding Doppler-shift component. The frequency domain approach was adopted above to illustrate an alternative way of interpreting the cause of spectral broadening. It would have been equally valid to repeat the analysis developed in Section 3, but this time applied to movement in the axial direction. It can easily be seen that the Doppler output signal must vary in amplitude as the target moves axially through the sample volume. Once again this modulation causes broadening of the Doppler spectrum. This time, however, it is worthwhile examining the situation in more detail by introducing a short mathematical analysis. If a target is moving at an axial velocity o through the sample volume then the transit time AT is given by

which, by substituting for Afa from equation (4) can be rewritten

~ T = F~

5. PERFORMANCEWITHA BLOODTARGET The analyses developed in the previous two sections have, for the sake of simplicity, assumed that the target is a single point scatterer. This is rather restricting since it is more useful to predict the behaviour of the velocimeter with blood as the moving target.

(3)

where Ax is the axial length of the sample region. Because the edges of the sample region are not well defined, the axial length Ax (i.e. the range resolution) has been taken

Av c A --'Ax ~c--=v 2fo 2"

(6)

This is an interesting expression which says that the fractional accuracy with which the target velocity can be estimated, multiplied by the precision with which target position can be determined is dependent only on the mean wavelength A of the incident ultrasound. If velocity resolution is improved then spatial resolution is degraded and vice versa. Notice that, although the arguments developed above deal specifically with a pulsed-Doppler system, the "sample volume" could equally well have been defined more generally as that region which, when occupied by targets leads to an output signal and, conversely, when not occupied leads to no output. The conclusions would then apply to Doppler systems using the various other forms of transmission coding. These include the pulsed-random signal and correlation techniques described by Bendick and Newhouse (1974), and the frequency modulation process described by McCarty and Woodcock (1974). The limitations imposed on the simultaneous estimation of velocity and resolution must apply to all ultrasonic Doppler systems. Incidentally the claim made by Jethwa et al. (1975) that a pulsed-random Doppler signal system is capable of improving the velocity-spatial resolution product seems to be erroneous.

110

P. ATKINSON

The statistical diffraction theory describing the scattering of ultrasound by blood has been developed by Atkinson and Berry (1974). The blood is taken to be a random distribution of point scatterers. In brief, the report makes the observation that, following the transmission of an ultrasonic pulse, the echo backscattered by blood fluctuates as a function of time delay (from transmission) and lateral displacement of the ultrasonic transducer. This phenomenon is illustrated in Fig. 4 where (a) shows the fluctuations along the returning echo and (b) shows how the amplitude of the echo from a fixed range fluctuates as the transducer is moved sideways, perpendicular to the beam axis. The interesting discovery is that the dimensions of the sample volume determine the scale of fluctuation detected. For instance, if the axial length of the sample volume is reduced (i.e. the transmitted pulse is shortened) then the amplitude fluctuations along the returning echo occur more frequently. Similarly, narrowing the beamwidth leads to more fluctuations (at a fixed range) per unit of traverse of the transducer. It was suggested that this "granular echo" is not due to any special structure in the blood on the scale observed but probably arises from fluctuation scattering by the random distribution of red cells. Returning to the Doppler velocimeter, suppose the blood is moving axially at a constant velocity through the sample volume. In addition to being Doppler shifted, the returning echo will also be amplitude modulated because of the fluctuations in scattering power as different distributions of corpuscles pass through the sample volume. As before, the rate of modulation determines the width of the frequency spread and consequently the velocity resolution capability. However, it has already been mentioned above that the fluctuation rate is inversely proportional to the size of the sample volume. Since the latter defines the spatial resolution capability of the system, it can be seen, once again, that the velocity-spatial resolution product remains constant. A short mathematical analysis quickly shows how the two resolution capabilities are linked. As before the spatial resolution Ax is determined by the duration ~" of the ultrasonic pulse so that C'f

Ax = - - .

(7)

(Notice that for a pulse-echo system, the resolution cell is only half as long as the duration of the transmitted ultrasonic pulse might suggest). Also it can be shown [see Atkinson and Berry equation (21a)] that the number of mean-level crossings NT per unit of time along the returning echo is given by Nr =--.0'32

(8)

The corresponding average modulation frequency is approx. NT/2 (i.e. on the average there are two zero crossings per cycle). If the axial velocity of the blood through the sample volume is v then the spectral broadening A/caused by the amplitude fluctuations when they appear on the output Doppler signal will be 0.32r Af = - CT

(9)

Once again the spectral broadening must be linked (by the

(o) blood celis

returning

7?,IL,.L

Z transducer OOO ~.,

•

•

stofionory

(b)

T ~

_ , '1

~

time

~10" 6s

_~.

echo amplitude = from fixed range

blood cells

I~20rnm ....

..:..

)

I=':'

3tacement

Fig. 4. The "granular" structure in the echo diffractedby blood. Doppler equation) to the precision with which the velocity can be estimated. Using equation (1) it turns out that At, 0'32 l;

(I0)

f0~" "

From equations (7 and 10) the uncertainty relation with blood as the target can be written At,,.vAx =0.32).

(11)

Once again the fractional velocity resolution multiplied by the spatial resolution capability is dependent solely on the dominant wavelength of the interrogating ultrasound. The constant of proportionality in equation (11) is probably inaccurate due to the crude manner with which typical frequency has been linked with the zero-crossing rate. It is believed that a more precise analysis would reveal exactly the same uncertainty relationship for either a single target or blood, or, in fact, any target configuration between these two extremes. 6. CONCLUSION The most basic, widely quoted, Doppler equation is inadequate for describing useful ultrasonic Doppler velocimeters. A more exact analysis shows that introducing a spatial resolution capability to the system immediately degrades the velocity resolution. In essence, this is because the velocity can now only be estimated during the finite time interval that the target occupies the sample volume. The velocity-spatial resolution product is limited by the dominant wavelength of the interrogating ultrasound. It is not thought likely that the concepts developed in this article will cause drastic quantitative errors to be uncovered. In any real system the velocity profile across the sample volume will probably lead to a wider spectral spread than that introduced by transit-time broadening. Nevertheless, it is felt that the ideas developed here provide an insight into the operation of Doppler velocimeters. Acknowledgements--I would like to thank Dr. P.N.T. Wells for his interest and encouragementand the Medical ResearchCouncil for financial support.

A fundamental interpretation of ultrasonic Doppler velocimeters REFERENCES

Atkinson, P. and Follett, D. H. (1975) Problems of signal extraction in ultrasonic Doppler systems. To be published in Clinical Blood Flow Measurement (Edited by Woodcock, J. P.). Sector, London. Atkinson, P. and Berry, M. V. (1974) Random noise in ultrasonic echoes diffracted by blood. J. Phys. A 7, 1293-1302. Baker, D. W. (1970) Pulsed ultrasonic Doppler blood-flow sensing. LE.E.E. Trans. Sonics Ultrasonics 17, 170-185. Bendick, P. J. and Newhouse, V. L. (1974) Ultrasonic randomsignal flow measurement system. J. acoust. Soc. Am. 56, 860--865. Jethwa. C. P., Kaveh, M., Cooper, R. C. and Saggio, F. (1975)

111

Blood flow measurement using ultrasonic pulsed random signal Doppler system. LE.E.E. Trans. Sonics Ultrasonics 22, 1-I1. McCarty. K. and Woodcock, J. P. (1975) A new ultrasonic flowmeter for the measurement of volume flo~, direction and velocity profile, in blood vessels. To be published in: Clinical Blood Flow Measurement (Edited by Woodcock. J. P.I. Sector. London. Newhouse, V. L. (19741 Transit time broadening in ultrasonic Doppler flow measurement systems. Based on Internal report of Purdue University (dated 9 November 19731. Newhouse, V. L., Bendick. P. J. and Warner. L. W. 11974) Analysis of transit time effects on Doppler flo~ measurement. Based on Internal report of Purdue University.