DOPPLER1 (Раздаточные материалы), страница 3

(Notice that N- 1 has been replaced by N and that this is allowed because the number of scatterers in the blood volume is effectively infinite.)

Returning to the rather arbitrary definition of a as an "excluded volume", it is interesting to examine the implications of Eqn (1.18) in more detail. In particular when an is unity the reflected intensity given by this expression is zero. Since fluctuation scattering can only vanish if the medium is homogeneous, an would only be unity when the whole volume of "blood" was packed full with corpuscles. This begins to resolve the problem of identifying a: it is the effective volume t of an average corpuscle, at least for dense distributions.

The mean value <p_env> of the pressure envelope follows from combining Eqns (1.19) and (1.13). In order to account for the transmitted energy the efficiency of the transducer and antenna gain, it is convenient to normalize the expression for <p_env> by comparing it with the peak envelope p_env^ref received from a perfect reflector placed at a range z = ct/2. The resulting ratio P is given by

If a well-damped cylindrical piston transducer (as used in medical pulse-echo ultrasonics) is shock excited with a voltage impulse then experimental investigation shows that to a good approximation,

where T_L, is a measure of the pulse length and R_B, a measure of the beam radius at
z = ct/2 (see Figs 1.7a and 1.10). Thus Pbecomes, finally

The value P² relates to the backscattered power which is a factor limiting the inherent signal-to-noise capability of an ultrasonic flowmeter. The practical implications of Eqn (1.22) will be considered in detail later. The analysis will now proceed to characterize the shape of the diffraction pattern backscattered by blood.

As the time delay t and position of the transducer R₀ vary, the echo envelope p_env fluctuates about its mean value <p_env> given by Eqns (1.19) and (1.13). Typical fluctuations in p_env with t and R₀ are shown in Figs 1.7b, 1.7c, The simplest quantities characterising the fluctuations can be defined by two fading rates N_t, and N_R ^. N_t is the average number of crossings of p_env through its mean value <p_env> during unit interval of t with the source-receiver position R₀ held fixed. Similarly N_R is the mean number of crossings of p_env through the value <p_env> at a fixed time delay during a unit lateral displacement of the source-receiver. Thus N_t, and N_R, characterize the fluctuation of the echo envelope in directions along and perpendicular to the ultrasonic beam,

To calculate N_t, and N_R requires that the spatial and temporal statistical distribution of the echo p_c (t, R₀) be known. The echo given by Eqn (1.12) is formed by the combination of a very large number of statistically independent contributions from each individual point target. This number is equal to the number of red blood cells in the sample volume, that is roughly pnR_B²cT_L. The echo waveform p_c(p, R₀) will therefore be gaussian random in form and can be analysed using standard gaussian noise theory first developed by Rice (1944, 1945) and Longuet Higgins (1956) and subsequently summarized by Berry (1973). This theory shows that the fading rates N_t, and N_R which refer to the envelope may be calculated from the autocorrelation function C(t, R) of the complex echo pressure p_c, multiplied by exp(iwt) to remove the non-random time factor in Eqn (1.10). This autocorrelation function is defined as

which is the normalized ensemble average of the echo originating at point (t₁,R₀) multiplied by the echo originating at a point displaced by (t, R) from the first. Using Eqns (4.38) and (4.l9) from the article by berry (l973) gives the fading rates:

The procedure for calculating the averages in Eqn (1.23) is precisely similar to the method used to derive Eqn (1.19) from Eqn (1.14), giving

Tedious but elementary analysis then gives

Thus the fading rates characterising the shape of the echo backscattered by blood can be calculated by combining Eqns (1.26) and (1.24) so long as the pulse envelope and beamwidth functions are known. In particular, if the simple forms for a(t) and b(R) given by Eqn (1.21) are valid for a typical clinical pulse-echo transducer, then by calculating the integrals in Eqn (1.26)

(ii) Experimental Studies

These predictions were verified experimentally in the article by Alkinson and Berry (l974) from which most of the above analysis was adapted, The parameters characterizing the shape of the ultrasonic pulse were measured first by reflecting the pulse from a plane target and recording the shape of the echo to define a(t) and then, using a method described by Wells (1969), by tracking a single point target laterally across the beam and recording the backscattered power to define p. The transducer acting as both transmitter and receiver of ultrasonic waves was then immersed in a container full of blood. Short pulses of ultrasound were transmitted at intervals of about 10^-2s which was long enough to allow the reverberation to decay away between pulses but short enough to enable a stable echo wavetrain to be displayed on an oscilloscope tube. The weak echo returning from the suspended corpuscles was envelope detected using a conventional rectifier and smoothing network. The experimental value of N_t was measured by counting the frequency at which the returning echo envelope crossed its own mean level per unit time delay. It was found to agree quite well with the predicted value. The lateral fading rate N_R was measured by sampling the envelope at a fixed time delay from each transmission (i.e. range gating) and then moving the transducer sideways. The number of times the sampled envelope crossed its own mean level per unit traverse of the transducer agreed well with the predicted value of N_R. It therefore seems that the shape of the diffraction pattern is dependent on the characteristics of the interrogating ultrasonic pulse and the only role played by the blood is to provide a continuous spectrum of fluctuations in scattering power producing random noise in the diffracted ultrasound.

The formula (Eqn (1.20)) predicting the mean reflected relative envelope II does involve properties of the blood: p, a, c, r, r^-, K, K^- and t. Atkinson and Berry discovered an order of magnitude discrepancy between the predicted and experimentally measured values for P but did not believe that this disagreement invalidates the proposed mechanisms of fluctuation scattering for a number of reasons, mainly concerned with assumptions made to simplify the theory and the practical problems of preventing clotting and rouleaux formation in blood. A more important discrepancy arises when the predicted variation in scattering power with haematocrit is compared with the experimental data collected by Shung et al. (1976). The predicted peak scattering power given by Eqn (1.21) should occur at an = 1/2 corresponding to a haematocrit of about 50%.. However, Shung reports that the measured peak occurs at a haematocrit of about 25-30%. dependent on frequency, as shown in Fig. 1.11. The theory developed by Sigelmann and Reid (1973) and applied by Shung correctly describes the observed variation in scattering power with haematocrit over a range of frequencies as shown by the dashed curves in Fig. 1.11. This is probably because this analysis recognizes that as the red cell concentration increases the particles can no longer be considered independent of one another. In normal blood (haematocrit = 45%), it can be calculated that the average distance between two red cells is about 10%. of their diameter. Under these conditions the value for a which has been associated with the excluded volume can no longer be approximated simply by the volume of a single corpuscle t. To deal with this problem Shung applied to good effect the heuristic "hole" approach suggested by Beard (1967) to describe the dense concentration of particles. Shung also verified experimentally that the scattering power from blood obeys Rayleigh's scattering theory and increases with frequency to the fourth power. This is in agreement with Eqn (1.8).

1.4 SUMMARY

The analysis developed above describes how ultrasound interacts with blood. However the mathematical detail sometimes obscures the purpose and meaning of the analysis.

This summary will therefore examine the scattering process in a less rigorous manner. This will also provide an opportunity for interpreting some of the practical implications of the theory and comparing the scattering process with another random noise phenomenon.

The most important feature characterizing the structure of blood and the form of the backscattered echo is the spatial autocorrelation function. This is very similar to the pair correlation function introduced in Eqn (1.16) and illustrated in Fig. 1.9a. The value of spatial correlation function considered along one dimension, predicts the probability that the centres of two red cells will be separated by a particular distance. Because two cells cannot interpenetrate the value of the spatial correlation function is zero close to the origin but, as the separation increases, rapidly rises to a constant value. This indicates that there is no preferred separation distance of scattering centres in blood as there would be for instance in a crystal lattice. On the scale of an ultrasonic wavelength, the red cells appear randomly distributed and so the blood itself cannot contribute to the shape of the backscattered diffraction pattern. Instead the scattering characteristics are defined by the shape of the interrogating ultrasonic pulse. For instance, Eqn (1.27b) indicates that the

beamwidth R_B, defines N_R, the lateral fading rate. If the beamwidth were made wider then the transducer would have to move further before it was interrogating a completely different column of red cells. Each column of red cells produces a characteristic independent echo amplitude. It is therefore reasonable to suppose that as the beam becomes wider the transducer will have to move further before the returning echo amplitude changes significantly. The lateral fading rate is therefore slower as predicted by the analysis.

It is also interesting to compare the axial fading with bandpass filtering of white noise. The two processes are shown diagramatically in Fig.1.12. White noise is bandpass filtered to produce a waveform centred on the fitter frequency and fluctuating in amplitude. The rate at which these fluctuations can occur is limited by the bandwidth of the filter: the wider the bandwidth, the faster the fluctuation allowed. In blood, the transducer shock excitation pulse provides a wideband source of input which is re-duplicated in time by scattering from the random distribution of point targets spaced in range. The effect is to produce an unfiltered echo very similar to a white noise signal. However, in practice, the resonant properties of the transmit-receive transducer bandwidth limits the frequency extent of the echo. The order of processing is not important if the system is linear and it can be seen and also shown experimentally that both processes produce similar looking output waveforms. In fact, if the filter bandwidths are the same then the two fading rates will be identical.

In conclusion, this chapter has attempted to introduce those concepts of sound-wave theory referred to during the remainder of this text. lnteraction of sound at a plane surface relates to reflection at vessel walls and possibly heart valve leaflets (see Chapter 7). Scattering of sound by blood is fundamental to the description of Doppler blood flowmeters and will be referred to many times during the remainder of this book.