DOPPLER2 (Раздаточные материалы), страница 3

Mathematically, the Doppler frequency limit ¦_max is Nyquist-related to one half the pulse repetition frequency ¦_p by

Substituting v_max for ¦_max in the basic Doppler expression (Eqn (2.15)) gives

indicating that the maximum velocity limit is equivalent to a l₀/4 travel per pulse repetition period.

(vii) Range-velocity Limitations

In addition to a velocity limitation, pulse-Doppler systems are subject to maximum range restrictions. Because it is necessary to wait for returns from the most distant target before transmitting another burst, the maximum range z_max which can unambiguously be determined is given by

Combining Eqns(2.23) and (2.24) gives the relationship

which shows that the product of maximum observable velocity and range is limited for conventional pulse-Doppler systems. Figure 2.11 illustrates this maximum range-velocity limitation for ultrasound propagating in human tissue for a range of ultrasonic frequencies.

Typical range-velocity products for various blood vessels are also indicated and it can be seen that in some cases, the pulse-Doppler flowmeter would have difficulty in determining flow in, for example, the renal artery or aorta. The fact that a vessel lies outside the range-velocity limit means simply that in the time it takes the ultrasound pulse to travel to and from the vessel, the blood within the vessel has moved more than one quarter of wavelength, thus making it impossible to track the changing phase of the Doppler shifted echo. Later it wilt be shown how more complex transmission coding combined with correlation receivers can help overcome this fundamental problem. First, however, it is worthwhile examining how pulsing the transmission and imposing restraints on the axial sensitivity has affected the basic Doppler equation,

(viii) Analysis of AxiaI Restraints

Suppose that blood is moving through the fixed sample-volume of a pulse-Doppler system at an axial velocity p towards the transducer shown in Fig. 2.12a. As well as being Doppler shifted, the returning echo pulses will also be amplitude modulated due to the fluctuations in scattering power as different distributions of corpuscles pass through the sample volume (see Chapter 1). By repeating the analysis developed above for the CW case it would be possible to describe the Doppler spectral broadening in terms of the transit time of the target through the sample volume. However, this time it is interesting and perhaps more useful to consider an alternative frequency-domain interpretation of Doppler spectral broadening, as follows.

Fourier analysis of the short transmitted pulse (Fig. 2. 12b) reveals that it occupies a wide frequency bandwidth (c). A complete spectrum of frequencies is transmitted and although, intuitively at least, it might seem that the mean or centre frequency should be used to predict the Doppler shift, the accompanying frequency components cannot be ignored. Newhouse et al. (1976) have shown that it is these components which broaden the Doppler shift spectrum making it a scaled-down version of the transmitted ultrasonic spectrum. For the simple case illustrated in Fig. 2.12, the basic Doppler equation should be replaced by the more exact expression

where the constant of proportionality K takes account of normalization and variations in scattering coefficient. This relation indicates that the power contained in the Doppler shift spectrum S(¦_d) between frequencies ¦_d and (¦_d + d¦_d) is directly proportional to the power contained in the transmitted ultrasonic spectrum between the Doppler-related frequencies 2(¦_d + d¦_d)c¦_d/v and 2c(¦_d + d¦_d)/v. Thus the fractional width D¦_d/¦_d of the Doppler spectrum in Fig. 2.12g is the same ratio as the fractional width D¦₀/¦₀ of the transmitted spectrum and in general these two spectra are the same shape.

It is also useful to develop a mathematical analysis of axial motion similar to that describing movement across the beam. If Dz is the axial length of the sample volume then the transit time Dt of a particle moving at velocity v is given by

and the Doppler shift bandwidth D¦_d (= 1/Dt) is

(Notice that, because the fixed sample volume has slowly decaying boundaries rather than sharply defined edges, its characteristic axial length is arbitrarily taken to be the distance between the points where the echo amplitude has fallen to 1/Ö2 of its maximum value. The width of the frequency spectra are similarly defined using this "3db-down" bandwidth.)

In addition to Eqn (2.28), the Doppler spectral width can also be related to the velocity resolution Dv by the Doppler expression

which, substituting for D¦_d from Eqn (2.28) and rearranging becomes

This expression means that for axial movement through the sample volume, the fractional accuracy with which target velocity can be estimated multiplied by the precision with which the target can be located is dependent only on the mean wavelength l₀ of the incident ultrasound. If velocity resolution is improved then spatial resolution is degraded and vice versa. This uncertainty-type relationship is a direct effect of imposing restraints on the axial sensitivity of the ultrasonic system. It arises because the velocity of the target can be estimated only during the short time interval that it occupies the fixed sample volume.

It also becomes apparent that it is not possible to formulate a relationship similar to Eqn (2.30) but which describes movement across the beam simply because the Doppler shift frequencies are generated by the axial movement along the beam. In fact Newhouse et al. (l976) point out that transit-time effects can be taken rigorously into account only if they are due to the properties of the transmitted signal rather than the geometry of the ultrasound beam.

2.3b Frequency Modulation

The pulse-Doppler is, and probably always will be, the simplest and most popular type of range-discriminating ultrasonic flowmeter for medical applications. However, because it uses such a simple method of transmission coding, the performance of the system is always limited by the range-velocity ambiguity. In addition the transmitted energy is concentrated into short pulses of ultrasound so that the ratio of the peak to the mean transmitted power is relatively large. Since the mean intensity ultimately defines the sensitivity of the system and because there are some indications (see for example, Wells, 1974) that high intensity ultrasound can damage living tissue, the signal-to-noise capability of a pulse-Doppler system is ultimately limited by patient safety considerations. It has therefore been suggested (McCarty and Woodcock, 1974) that frequency modulation (FM) could provide a useful alternative method of transmission coding which would allow the peak-to-mean intensity ratio to be reduced.

In this section, the range and velocity resolving abilities of FM systems will be examined and compared with the performance of pulse-amplitude modulated devices operating over similar ultrasonic bandwidths. The problems of extracting Doppler signals will also be mentioned.

(i) Basic Principles

An FM device operates by transmitting a constantly changing ultrasonic frequency and then measuring the time it takes the echo from a target to return at each particular (coded) frequency. The simplest method of coding the frequency is to ramp it linearly in time as shown in Fig. 2.13a. This type of transmitted output can be generated by applying a triangular or sawtooth waveform to the voltage controlled oscillator (VCO) illustrated in the block diagram of Fig. 2.14. The echoes received back at the transducer are amplified and coherently demodulated in the mixer (and lowpass filter) using the output of the VCO as the reference waveform. Figure 2.13 illustrates the effect in both the frequency (a) and time (b) domains when there is a single stationary target in the ultrasound beam.

In the frequency domain, ultrasound transmitted at frequency ¦₁ at time t₁ travels towards the target, is reflected and arrives back at the source at t₂. By this time the transmitted frequency has increased to ¦₁ and the difference in frequency (¦₁ - ¦₂) between the received and currently-transmitted ultrasonic components is related to the time of flight (t₂-t₁) by the expression

where g is by definition the frequency-versus-time gradient of the transmitted waveform. For a constant ultrasonic propagation velocity c the expression for the beat frequency
¦_b (=¦₂ - ¦₁ ) can be rewritten

where z is the range of the target. Thus, so long as the transmission frequency increases linearly with time, the difference between the transmitted and received frequencies will be constant and proportional to the range of the target.

In the time domain (Fig. 2.13b) the ultrasonic path to and from the target effectively shifts the received waveform backwards in time so that it becomes a delayed version of the transmitted signal. When multiplied together in the mixer, these two waveforms combine to produce a constant difference (or "beat") frequency proportional to the time delay and also the range of the target.

(ii) Range Resolution

In any practical situation it is apparent that the transmission frequency cannot increase indefinitely since it is limited by the bandwidth characteristics of the transducer element. The modulating waveform has therefore either to begin decreasing to produce a triangular modulation or to be reset producing a sawtooth modulation. In either case the difference frequency is discontinuous and is no longer related unambiguously to target range. To see in detail how these discontinuities affect the performance, it is necessary to introduce a short mathematical description of the frequency modulated system.

Suppose B is the maximum ultrasonic bandwidth available for transmission. If the sweep rate is g then the sweep reaches its limits and must therefore reverse or reset at intervals T_R = B/g. The mixer output will be bursts of sine waves of frequency ¦_b given by Eqn (2.32) and lasting for intervals up to T_R. The discontinuities in the modulating waveform during flyback or reset will generate high frequency beats which interrupt the continuity of the mixer output. The effect in the frequency domain is shown in Fig. 2.15. In order to separate two closely spaced targets the output waveform from the mixer is frequency analysed. However, because the output is constant only for the time interval T_R the best frequency resolution possible D¦_b, is limited to 1/T_R that is, the reciprocal of the burst duration. Thus by replacing ¦_b by D¦_b, in Eqn(2.32) the limiting range resolution Dz is given by the relationship

or, substituting D¦_b = T_R = g/B and rearranging,