DOPPLER3 (Раздаточные материалы), страница 8

D₂(t) = -Asin(w_At+f_A) - Bsin(w_Bt+f_B), (3.42b)

Differentiation of Channel 1 gives:

D'₁(t) = -w_AAsin(w_At+f_A) -w_B Bsin(w_Bt+f_B), (3.43)

The differentiated and demodulated Channel 1 signal D'₁(t) is then multiplied by the demodulated Channel 2 signal D₂(t) to give the numerator X(t),

X(t) = D'₁(t)^.D₂(t), (3.44)

or, substituting Eqns (3.43) and (3.42b)

The demodulated Channel 2 signal is multiplied by itself to give the denominator Y(t)

Since the two components A and B are uncorrelated in time, the "cross terms" involving both A and B in Eqns (3.45) and (3.46) will average out to zero. The low-pass filters with cut-offs well below the Doppler shift frequencies will therefore yield the outputs |X(t)| and |Y(t)| given by

|X(t)| = w_A A² + w_BB² (3.47a)

and

|Y(t)| = A² + B² (3.47b)

After dividing the numerator |X(t)| by the denominator |Y(t)| in the analogue divider, the output quotient Q(t) is

which is the normalized first moment of the input Doppler power spectrum. This is the mean Doppler frequency from which ^-v^- can be calculated using Eqn(3.38).

By assuming circuit linearity it can be seen that this network will similarly process more complex combinations of sine waves to give the mean frequency of the input spectrum. Practical implementation of this method requires the accurate alignment of the multipliers and dividers. DC level drift and non-linearities can introduce computing errors which are especially troublesome at lower Doppler frequencies. In addition it has been found useful to incorporate an automatic gain stage to help compensate for the limited dynamic range of the multipliers and dividers. A more detailed circuit for mean frequency computing has been described by Skidmore (1979).

The comparison between clinical Doppler diagnosis based on analysis of peak or mean frequency has not yet been established. Peak frequency has historically been the parameter of choice, mainly due to its ease of measurement, and most definitions of abnormality are based on observations derived from peak frequency/time waveforms. However mean frequency computers should provide less noisy and therefore more sensitive indication of flow dynamics and perhaps ought increasingly to be considered for use in the clinical situation.

3.3d Velocity Profilometers

This chapter has until now assumed that clinically useful information lies only in the peak or mean velocity versus time waveform. The spatial variation of flow across the vessel, that is the velocity profile, has been largely ignored. However the shape of the velocity profile can not only provide a basis for computing volume flow but can also be a potentially useful indicator of arterial disease. At low flow rates streamline flow will create a parabolic profile in which indentations on the inner walls of arteries produce downstream perturbations, sensitively indicating the presence of a wall lesion. This section will describe the techniques devised to provide on-line display of velocity profiles.

(i) Multichannel Pulse-Doppler

By sweeping the sample volume slowly across the vesseI, a conventional single-channel pulse-Doppler could be adapted for use as a velocity profilometer. However during pulsatile flow this method would be time consuming and unreliable and so multichannel pulse-Doppler devices have been devised to perform real-time velocity profilometry (see for example, Peronneau et al. (1974), McLeod (1974)). The multiple range gates are located at equal intervals across the vessel and the velocity profile is reconstructed by Doppler processing the signal received at each range. The disadvantage of this method is the large component count. For real-time operation and display each channel requires a Doppler signal processor and the system can rapidly become complex and costly. However once constructed, a multichannel Doppler device is a reliable and accurate method of investigating velocity profile.

(ii) Mooing Target Indicators

Velocity profilometry can be achieved more simply than the multichannel device by using an ingenious device first described by Grandchamp (1975) and developed by Brandestini (1975, 1976). The technique is based on moving target indication (or MTI) developed for use in RADAR and used with some success in medical ultrasound to detect heart valve movement (Barnes and Thurstone, 1971) The principle of operation of phase detection profilometers is illustrated in Fig. 3.29. If a series of identical ultrasonic pulses interrogate a group of targets, then successive echo wavetrains will be identical only if all the targets are stationary. If the

targets contain moving structures the echoes will vary from pulse to pulse and furthermore the differences will be greater for faster velocities. Figure 3.29 shows two successive echo wavetrains 1 and 2 returning from targets containing two moving structures A and B. The target configuration changes, between pulses and the waveforms, differ at point A, which is an echo originating from the structure advancing rapidly towards the transducer and point B, an echo from a slowly receding structure. Suppose now that these waveforms are fed to a phase detector which has the characteristic shown in Fig. 3.30. The output of this phase detector is linearly proportional to the phase difference between the two input signals being zero when the signals are in phase. The output for the two echo inputs will be zero except at

the ranges corresponding to the moving targets because at these points the echo waveforms differ in phase. Furthermore, the rapidly advancing target A will cause a large positive deflection while the slowly receding target B will reproduce a small negative one. For equal echo amplitudes, the phase detector output is proportional to the phase difference between echoes. This in turn is related to the distance and direction the structure has moved between pulses.

Mathematically, suppose the pulse repetition period is T_p and the velocity of the target B is v_B away from the transducer. Between pulses, the target will move a distance Dz_B,

Dz_B = v_BT_p (3.49)

and this will cause the time delay echo from B to change by a time Dt_B where

since this is extra time required by the ultrasound to travel the additional (round trip) distance to the new position of the target. The phase difference f_B between the two signals is the product of the ultrasonic angular frequency and the additional time delay, and so

f_B = w₀ ^. Dt_B (3.51)

or, substituting Eqn (3.50)

Thus, at every point along the returning echo, the output of the phase comparator will be a voltage proportional to the velocity (and amplitude) of the target at the corresponding range. If blood is the moving target then the " phase comparator output represents the velocity profile of flow along the utrasonic path. By transmitting an adequately frequent number of pulses, real-time velocity profilometry can be achieved using this surprisingly simple method.

Notice however the limitations implied by Eqn (3.52) and the phase comparator characteristic. The phase difference f_B must not be allowed to exceed 180° otherwise the phase comparator will produce an ambiguous output. The maximum velocity v_max which can unambiguously be accommodated is given by setting f_B = p in Eqn (3.52) so that

or

Although simple in concept. this phase detection profilometer is relatively difficult to implement as a clinically useful device. The superimposition of successive echoes is the source of difficulty since it requires the first echo to be delayed for exactly one pulse repetition period so that it can compare accurately with the second. If the delay is not exact, then the large clutter echoes from stationary targets will not be precisely in phase and will therefore produce spurious velocity-simulating artefacts. For 40 dB suppression of stationary targets, the delay time and pulse repetition periods have to be matched to within 1% of the ultrasonic wave period over the full ultrasonic bandwidth. Brandestini (1976) has used a delay line canceller inserted before the velocity profilometer to improve the dynamic handling of stationary targets. He suggests the configuration shown in Fig. 3.31 using two matched delay

lines 1 and 2. Delay line 1 helps to cancel the echoes from stationary targets by subtracting successive echoes while delay line 2 provides the delayed input for the phase detector. In order to improve the channel matching it has been found necessary to use a bandpass filter in the undelayed path to simulate the filter characteristics of the delay line. Furthermore, the elegant concept of trigger regeneration helps solve the problem of stationary target cancellation. A portion of the transmitted pulse is fed direct to the delay and is used to retrigger the transmitter as it re-emerges at the far end of the line. The pulse repetition period is therefore locked to the signal delay and automatically follows any changes due to temperature variations or general drift.