1. INTRODUCTION 17

doppler ultrasound

and its use in

clinical measurement

1. INTRODUCTiON

This book describes the principles and clinical use of the many types of ultrasonic Doppler flowmeters, ranging from simple continuous wave devices through pulse-Dopplers to more complex random signal sequence systems. All these instruments rely ultimately on the interaction between an incident sound wave and a moving target. The purpose of this first chapter is to introduce some of the basic concepts of sound-wave theory, The analysis starts with reflection and refraction at a plane surface and progresses to the backscatter from random distributions of point targets. It is stressed that these analyses will not be developed from first principles since this approach would be of limited use to the medical physicist and of even less value to the clinical ultrasonologist. Instead, commonly accepted formulae will be quoted, interpreted in practical terms and then used as a basis from which, in later chapters, the theoretical description of Doppler flowmeters can be developed, References are included, when appropriate, to provide more detailed information for the more specialized reader.

The mathematically disinclined are advised that Section 1.4 presents an intuitive description of the scattering of ultrasound by blood and interprets equations previously derived in more practical terms. Every attempt will be made to adopt this combined approach throughout the book: the mathematical derivations will wherever possible be accompanied by less formal descriptions which should help clarify the practical implications of the analysis.

1.1 SOME SOUND-WAVE THEORY

Sound-wave theory describes the propagation of mechanical vibrational disturbances. These vibrations travel through media in the form of waves, usually sinusoidal (see Fig. 1.1a), which obey the laws of reflection, refraction, diffraction and dispersion. The medium itself is considered to consist of a series of particles (Fig. 1.1b). The vibrations can be characterized in terms of either the displacement of these particles or more usually the pressure causing the particle displacement. The most common mode of vibration produced in human tissue is a longitudinal wave where the particle oscillates in the direction of wave propagation as shown in Fig. 1.1c. Sideways oscillations are known as transverse or shear waves (Fig. 1.1d) and can only be adequately supported in solids such as skeletal bone.

By way of an introduction some general aspects of sound-wave propagation will be described. This will also provide a useful opportunity to develop the basis of the mathematical notation used later.

1.1a Notation

Consider a sound wave propagating along the direction shown in Fig. 1.2. A system of Cartesian coordinates is chosen so that the sound wave travels from the origin in a direction parallel to the z-axis. The y-axis lies vertically in the plane of the paper and the
x-axis is perpendicular to it pointing outwards. This choice of coordinates locates the source of ultrasound at the origin, transmitting along the z-axis. As time progresses the pressure wave moves en bloc from left to right along the z-axis, continuously emerging from the transmitting element. It is difficult to illustrate the propagation of a sound wave in a totally meaningful way. A realistic representation of a pressure waveform would require a five-dimensional display: three spatial dimensions, one dimension for time and one for pressure. Since five-dimensional space is unavailable, any illustration of a sound wave has to be simplified not only by showing the pressure wave at one instant in time but also by using the height of the curve above the z-axis to represent the instantaneous pressure.

In mathematical terms the pressure p(z, t) at any point (0, 0, z ) on the z-axis at time t is given by the equation:

where p₀ is the peak pressure, k = 2p/l where l is the wavelength and w = 2p¦ where ¦ is the frequency of the sound wave. Although the pressure p is a real function of position and time, it is often more convenient to express pressure as a complex quantity p_c, whose real part is p and where

Also, in systems exhibiting cylindrical symmetry (such as diffraction fields from the cylindrical transducers commonly used in Doppler flowmeters) the analysis can be simplified by choosing a system of cylindrical polar coordinates as illustrated in Fig. 1.3. Each point in the diffraction action field can then be described by the parameters (R, z) where R is the two-dimensional vector in the x, y plane.

1.1b Interaction at an Interface

Sound waves travel in straight lines until they meet interfaces which reflect, refract or diffract them. An interface is defined as the boundary between two media with differing acoustic impedance. The acoustic impedance Z ( = rc) is a mechanical property of a medium defined as the product of the density r and the propagation velocity of sound c. Consider the interaction shown in Fig. 1.4 where a sound wave of peak pressure p_i is incident at an angle q_i to the surface normal of an interface between two media of acoustic impedance Z₁, and Z₂. By imposing limitations (or boundary conditions) so that the particle position and velocity must be continuous across the interface (a restriction which ensures that the two media stay in physical contact and do not fly apart) it can be shown (see for example Wells, 1969) that part of the wave is reflected and part is transmitted. The reflected component has a peak pressure amplitude p_r, where

and travels off at an angle q_r (=q_i) to the normal. This is known as "specular reflection". The transmitted component of amplitude p_t , is given by

Thus the detailed interaction between a sound wave and a plane interface can be fully characterized by the change in acoustic impedance (or acoustic mismatch) across the interface and the angle of the incident beam. An important point to notice is that the amplitude of the reflected component increases with the mismatch in acoustic impedance. Furthermore, the reflected wave will return along its approach path (and back to the source-receiver) only when the angle of incidence is normal to the surface. In practice, human tissue interfaces are rarely perfectly flat and surface roughness tends to produce a degree of reflection over a range of incident angles. Also the sound beam has been represented by a single line indicating the path taken by a single "ray" during the interaction. In any real situation the incident pressure wave will occupy a finite beamwidth which is continually diverging due to the diffraction effects of the source. This beam contains many rays propagating over a range of incident angles. Equations (1.3) to (1.5) should really be applied to each ray in turn to obtain a more accurate representation of the interaction. Diffraction effects tend to complicate even the most simple of situations.

1.2 THE STRUCTURE OF BLOOD

Before formulating the theory describing the scattering of ultrasound by blood it is first necessary to study the small-scale structure and constituents of this fluid medium.

Human blood is composed of a liquid plasma in which are suspended erythrocytes (red blood corpuscles), leukocytes (white blood cells) and platelets. The erythrocyte is a flexible biconcave disc with an average diameter of 7x10^-6 microns (1 micron = 10^-6 m) and an average thickness of 2 microns. The mean corpuscular volume is about 90 cubic microns and there are approximately 5 x 10⁶ erythrocytes per cubic millimetre. This concentration corresponds to a haematocrit (the volume fraction of cells in whole blood) of about 45%. It is generally presumed that these erythrocytes are the major source of scattering in blood because white cells, although much larger than red cells, are relatively few in number (7·5x10³ mm^-3). Platelets, which are quite densely distributed
(3·5x10³ mm^-3), are much smaller than erythrocytes. Figure 1.5 shows a red blood corpuscle in more detail although it will be shown in Section 1.3b that at clinically useful ultrasonic frequencies the shape of the cell plays no part in defining its scattering characteristics. This is because the cell is much smaller than the ultrasonic wavelength making it effectively a point target. The scattering coefficient then depends only on the volume of the corpuscle and the acoustic mismatch between it and the suspending plasma.

The erythrocytes can be separated from the less dense plasma by centrifuging so that the physical parameters of the two components can be measured. Of particular interest are the adiabatic compressibility K and bulk density r since these parameters are important for calculating the scattering cross-section (see Eqn (1.6)). Using a propagation velocity method, Urick (1947) has measured values of K for both packed red cells and plasma and found that K_plasma = 40 ^. 9x 10¹² cm²/dyne and
K_erythrocyte = 34^.1 x 10¹² cm²/dyne at 20°С. Straightforward density measurement shows that erythrocytes are slightly more dense (r_erythrocyte = 1^.092 gm cm³) than plasma (r_plasma= 1^.092 gm cm³).

Blood is normally a homogeneous fluid medium with well defined characteristics. Abnormal features that can occur due to pathological conditions include minor variations in haematocrit and more rarely aggregations of erythrocytes into multicellular clumps known as "rouleaux". It is unlikely that either of these conditions would significantly affect the observed backscattering power of blood in vivo. However it is important to be aware that, unless handled carefully, clotting and rouleaux formation of blood samples in vitro can cause large variations in backscattering coefficients measured during experimental studies.

1.3 THE SCATTERING OF SOUND BY BLOOD

1.3a Scattering of Sound by a Point Target (Red Corpuscle)

The theory describing the scattering and diffraction of sound at a small flexible target, such as a red corpuscle, has been developed by Rschevkin (1963) This analysis is based on the situation shown in Fig.1.6 where a plane sound beam of peak pressure p_i is incident on small flexible target P. If there is an acoustic impedance mismatch between the target and its surroundings then the incident wave will be diffracted at the target surface producing a scattered wave of amplitude p_s, By imposing- the boundary conditions that the pressure and normal velocity component must be continuous at the sphere surface, Rschevkin was able to show that the amplitude of the wave scattered to an angle q_s, is given by

where t is the target volume and c, r and c, r are the adiabatic bulk moduli and mass densities of the target and its surroundings respectively. For the particular case where
q_s, = 0. the amplitude backward scattering coefficient A_s ( = p_s/p_i) is given by