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This expression applies only when the incident wavelength is very much larger than the typical dimensions of the target. This condition is always amply satisfied when clinically

useful ultrasonic frequencies are used to investigate red blood corpuscles. Thus the backscattering from a red cell is independent of its shape and depends only on its volume and impedance mismatch relative to its surroundings.

1.3b Scattering by a Random Distribution of Small Targets (Blood)

The scattering of waves propagating in a dense distribution of small targets is very complex. By applying the principle of superposition it ought to be possible to determine the echo structure by calculating the shape, magnitude and position of the diffraction pattern generated by each red blood corpuscle and summing all the individual contributions. However, this complete solution would require a knowledge of the position of each individual scatterer and is therefore intractable. Furthermore, such a specific solution would be of little use since it would define the echo structure for only one particular target distribution,

A more useful approach is to apply statistical techniques to compute temporally and spatially averaged parameters which characterize the backscattered echo. This approach has been adopted by several groups of workers who have examined the ultrasonic echo from blood from both a theoretical and experimental viewpoint.

Sigelmann and Reid (1973) developed a practical approximation to describe the backscattering of periodic bursts of sine waves by a volume of randomly distributed scatterers. By comparing the root mean-square (or RMS) value of the backscattered signal with that reflected from a target of known reflection coefficient they were able to define a "volumetric backscattering cross-section" for the point targets. In a subsequent article Shung et al. (1976) developed this analysis to determine the volumetric scattering cross-section of the erythrocyte over a range of ultrasonic frequencies.

In a more fundamental approach Atkinson and Berry (1974) developed a statistical diffraction theory to describe the scattering of ultrasound by blood. This analysis, described in detail here, was inspired by the experimental observation that after transmitting a short burst of ultrasound (Fig. 1.7a), the echo backscattered by blood fluctuated as a function of time delay and lateral displacement of the source-receiver. The weak echo returning from the bulk liquid showed a division of the waves into quasi-random groups as shown in Fig. 1.7b. A similar fluctuation in the echo from a fixed range
occurred if the source-receiver of ultrasound was moved sideways in a plane perpendicular to the transmission direction (Fig. 1.7c). It seems that this granular echo is not due to any special structure in the blood on the scale observed, but arises from fluctuation scattering by the random distribution of red cells. An important feature of this analysis is the experimentally verified prediction that the dimensions of the ultrasonic pulse determine the scale of fluctuation detected. This is because the pulse echo technique cannot possibly resolve individual corpuscles since the ultrasonic wavelength is much too large (at 5 MHz, l~0^.3 mm). However, if the blood cells are randomly distributed, then the numbers contained in different small volumes of the same size V will not simply be given by nV where n is the overall number density, but will fluctuate about this mean value. After an ultrasonic pulse has been transmitted the echo arriving back at the source originates from a region known as the sample volume defined axially by the pulse length and laterally by the source-receiver beam shape. If the small volume V is the sample volume of a pulse-echo system then the fluctuations in the number nV of red cells contributing to the echo will cause a continuous variation in the scattering power throughout the blood. The following analysis is based on the assumption that the blood echo arises from this "fluctuation scattering" which is analogous to Rayleigh-Tyndall scattering responsible for the blue of the sky (Van de Hulst, 1957).

(i) A Statistical Diffraction Theory

This section develops a statistical diffraction theory to describe the scattering of ultrasound by a random distribution of point targets. The analysis will be based on the scattering of short quasi-monochromatic pulses of ultrasound a few cycles long emitted from a cylindrical transducer similar to the type commonly used in clinical applications. These pulses are chosen to be short enough to allow adequate range discrimination and yet sufficiently long to enable a dominant ultrasonic "centre" frequency to be established. The diffraction theory will be formulated in terms of the complex function p, whose real part p describes the acoustic pressure of the ultrasonic wave. Both p, and p are quasi-monochromatic and it is easily shown (using for example the method of Berry (1973), Section 2.1) that the envelope p_env of the sound wave (illustrated in Fig. 1.7b) is given by

(a relation which always holds, irrespective of whether or not scattering has occurred). As before a system of coordinates is chosen so that the ultrasonic source-receiver is located at the origin pointing along the z-axis. The pressure distribution of the incident wave, that is before scattering occurs. is described by the function p_c^inc (R, z, t). The shape of this function depends on the time variation of the pulse emerging from the crystal (described by the dimensionless function a(t)exp(- iwt) shown in Fig. 1.7a), and also the diffraction beam of the ultrasonic source.

For the purposes of this analysis it will be assumed that the sidelobes of the diffraction beam can be neglected ( for a cylindrical vibrating piston the sidelobes are usually more than 20 dB down on the main lobe) and that the lateral pressure distribution can be adequately described by the cylindrically symmetrical dimensionless function b(R). In the region of interest beyond the near field of the transducer, practical investigation shows that the function b(R) can be approximated by a simple exponentially decaying function as shown later in Fig. 1.10.

The incident pressure at time t and at any point (R, z) can therefore be written as

where (R₀, 0) is the position of the transducer expressed in cylindrical polar coordinates. (For the moment the transducer is located at the origin and so R₀ = 0. Later it will be required to traverse the transducer in a direction perpendicular to the z-axis. The variable R₀ has been included in Eqn (1.8) to allow this.)

To obtain precise agreement with experimental results. the analysis should take account of ultrasonic absorption by the blood. However, for the sake of simplicity these relatively minor effects are ignored.

Equation (1.8) describes the wave incident on each red cell at its position
r_i ( = R_i, z_i) as shown in Fig. 1.8. Every red cell acts as a flexible point target exciting a scattered wave whose amplitude falls off linearly with distance. The scattered echo p_i^s(t,R₀) arriving back at the source from a single corpuscle located at r_i is given by

where A_s, is the amplitude for backward scattering from a corpuscle given in Eqn(1.6) as

where t is the volume of a single corpuscle. (Strictly speaking k² in Eqn (1.10) should be a mean value over the spectral range since the pulse is not monochromatic, but no appreciable error arises from using the centre frequincy since the wavelength is much larger than the scatters.)

Eventually the scattered waves reach the receiver where, provided no multiple scattering occurs (see later), they combine to give echo p_i^s(t,R₀) of the form

where the summation is over all the corpuscles and powers of |R_i - R₀|/z_i higher than the first have been neglected. This assumption is amply justihed when working in the far field of the transducer where z; is much larger than |R_i - R₀|

It is worth mentioning here that in order to ease the mathematics. it has been assumed that both the beam shape and pulse length are dehned only during transmission. The receiver is assumed to be "perfect" in that it possesses a wide frequency response (and therefore does not distort the received echo shape) over a wide aperture (and is therefore omnidirectiona1). Since the same (bandwidth- and beamwidth-limited) transducer is used both to transmit and to receive, part of the pulse shaping and beam forming occurs during transmission and part during reception. In a linear system it is permissible to combine the two effects and describe the complete "go-and-return" pulse shape.

Equation (1.11) can be rewritten by defining t = 0 as the instant that the maximum of the pulse is transmitted and also using the fact that a(t) is a compact function (i.e. the pulse is short) to replace all the values z_i; (which determine fall off of amplitude with distance) by the single approximate value ct/2, giving the final echo formula

Notice that the phase-determining range components exp(2ikz_i) in this equation cannot be approximated to ct/2 within the summation because otherwise the all important interference effects would be destroyed. The "scale" term outside the summation in
Eqn (1.10) describes the magnitude of backscatter (p₀ A_s) and the way in which its amplitude falls off inversely with range (ct/2). The summation itself takes account of the superposition of the individual contributions from the random distribution of point scatterers.

The returning echo has now to be characterized in some way. The best method of defining the magnitude of the backscattered echo is to calculate the mean value <P_env> of the echo envelope. It is simplest first to calculate the mean square modulus |p_c|² of the echo and then use the relation

which follows from Eqn (1.7) together with the fact that |p_c| possesses a Rayleigh distribution because p_c, is a gaussian random variable (a statement which will be justified later). Thus from Eqn (1.12) the mean square modulus of the echo at a time t is given by

where the brackets < ... > denote an ensemble average over all possible positions of the scatterers.

Equation (1.14) can be interpreted by assuming that there is a total number N of red cells randomly distributed over a volume W which is very large in comparison with the travelling pulse of ultrasound. Of the N² terms in Eqn (1.14), N have i = j while N(N-1) have i ¹ j. This summation can therefore be analysed by introducing a singlet probability distribution P₁(r_i)dr_i, defined as the probability that the i-th corpuscle lies in a small volume dr_i centred on r_i and given by

(which is simply the corpuscle-to-total volume ratio) and also by introducing the doublet probability distribution P₂(r_i, r_j), defined as the probability that the i-th corpuscle lies in a small volume dr_i centred on r_i and that the j-th corpuscle lies in a small volume dr_j centred on r_j:

where g(r_i - r_j) is the pair correlation function. When i and j are well separated g(r_i - r_j) is unity since long range interaction between two corpuscles is impossible. Alternatively
g(r_i - r_j) falls to zero when |r_i - r_j| is less than a few microns since the corpuscles cannot interpenetrate. An approximate plot of g(r_i - r_j) is shown in Fig. 1.9a. Since the

fastest variation in the function to be averaged in Eqn (1.14) occurs over distances of order l₀ and because Fig. 1.9 shows that g(r_i - r_j) differs from unity over regions a thousand times smaller than l₀, the pair correlation function can, to an excellent approximation, be rewritten as:

where d(r_i - r_j) is the Dirac delta function and a is a measure of the "excluded volume" surrounding a corpuscle which will be defined more precisely in a moment. First of all it is worthwhile examining the expression of Eqn (1.17) in more detail and interpreting it by means of the illustrations shown in Fig. 1.9. The delta function (Fig. 1.9b) is zero except when r_i = r_j when it becomes very large. The theoretical definition is that d approaches infinity as r_i approaches r_j (see Section 3.1.c. (i)). Since the excluded volume a is very small, the product a ^. d(r_i - r_j) will be approximated to unity in the region where r_i = r_j. It is now possible to work out the ensemble averages in Eqn (1.14) so that

The integral containing the oscillatory factor exp(2ikz) is completely negligible (it can be shown that its value is of order exp(-4p²m²) where m is the number of waves in the pulse). Only the integrals without oscillatory factors are significant and, by defining n = N/W as the number density of corpuscles, the mean square echo modulus can be written as

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