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  1. 1. PART (2): PROPERTIES OF SOUND A sound wave is a mechanical disturbance in a gas, liquid or solid that travels outward from the source with some definite velocity. The sound vibrations in air cause local increases and decreases in pressure relative to the atmospheric pressure (Fig.1). These pressure increases, called compressions, and decreases called rarefactions, spread outward as a longitudinal wave. Figure1. Schematic representation of a longitudinal sound wave at one instant. 'Particles' move back and forth about fixed mean positions, being alternately compressed and pushed forward and stretched and pulled back. The pressure variations are passed from one layer of particles to the next at the speed of sound c. 1
  2. 2. Wherever the density or compressibility of tissue changes in the path of an ultrasonic wave, echoes are sent back to the ultrasound probe. These may be weak reflections from the interfaces between different tissues, or even weaker scatter from the numerous small scale structures within tissue. In most applications, the diagnostic information from an ultrasound scan comes from scattered echoes rather than from echoes reflected from larger interfaces. Fortunately, ultrasound pulses travel at a fairly constant speed along narrow pencil beams, so that the direction and range of echo sources can be measured and plotted. However, distortions and artifacts do occur, and some understanding of the basic physics of sound propagation and of the techniques used in scanning equipment is necessary, if high quality scans are to be produced and their limitations appreciated. Speed of sound The speed with which the pressure disturbances (both positive and negative) travel away from the source is known as the speed of sound (c). The speed of sound is a constant for any medium and is completely determined by the density () and compressibility of the medium. It does not, therefore, depend on the frequency of the wave. The relevant measure of compressibility is the bulk modulus of elasticity (k), which is the ratio of the pressure applied to a fixed mass of medium to the fractional 2
  3. 3. change in volume. It is high for relatively incompressible media such as solids, water or tissues, but low for compressible media such as gases. The speed of sound may be expressed in terms of k and p by: In practice, tissues differ much more in compressibility than in density, so that bone (very incompressible, high k) has a higher speed of sound than muscle, despite the fact that it is more dense. The mean value of the speed of sound in soft tissue is generally taken to be 1540 m s-1. Energy, power and intensity The source does work and gives energy to the first layer of particles of the medium as it pushes and pulls them. This energy is passed from particle to particle as the wave propagates, eventually being absorbed as heat. A single ultrasound pulse from a diagnostic scanner leaving the probe might typically carry with it a few microjoules (µJ) of energy. Over any specified time period, any source of ultrasound will transmit a certain amount of energy. Power is the rate at which energy is transferred, its unit being the watt (W), where 1 watt equals 1 joule per second. The rate of working by the source, and hence the transmitted acoustic power, 3
  4. 4. varies from instant to instant. The instantaneous acoustic power is zero when the source momentarily stops and changes direction, and reaches its peak when it is pushing the adjacent medium forwards, or pulling it backwards, at maximum speed. The temporal average acoustic output power of the source is the total energy transferred from the source to the medium in every second. This may be up to a few hundred mW for a medical diagnostic scanner, which might typically transmit a few thousand pulses per second. A quantity that is often of more interest than power is acoustic intensity. This is a measure of the local concentration of power, and is defined as the energy flow per unit area per second, or the power per unit area (assuming the area considered is perpendicular to the direction of travel of the wave). Although defined in terms of an area, intensity describes the situation at a point. It equals the power that would be measured passing through a tiny area centred on the point, divided by that area. Strictly the SI unit of intensity is a watt per square meter (W m-2), but ultrasound intensities are usually quoted in W cm-2 or mW cm-2. The intensity (I) of a sound wave is the energy passing through unit area in unit time, i.e. 4
  5. 5. For a plane wave I is given by: Where  is the density of the medium; v is the velocity of sound; f is the frequency;  is the angular frequency, which is equal to 2πf; A is the amplitude of the wave or the maximum displacement of the molecules from the equilibrium position; and Z, which equals  v, is the acoustic impedance of the medium. Some typical values of , v, and Z are given in Table 1 . The intensity can also be expressed as: Where Po is the maximum acoustic pressure. Table1. Values of , v, and Z for various substances 5
  6. 6. The Decibels Scale : A special unit, the bel, has been developed for comparing the intensities of two sound waves (I2/I1), two powers or two energies. This unit was named after Alexander Graham Bell, who invented the telephone and did research in sound and hearing. The intensity ratio in bels is equal to log10 (I2/I1) or equal to 10 log10 (I2/I1) in decibels, (one bel =10 decibels 'dB'). Number of dB = 10 log10 Since I is proportional to P, that is, I2/I1 = hence, the pressure ratio between two sound levels can be expressed as Number of dB = 10 log10 = 20 log10 This Expression can lie used to compare any two sound pressures in the same medium. For hearing test, it convenient to use a reference sound intensity (or sound pressure) to which other sound intensities (or sound pressures) can be compared. The reference sound intensity Io is 10-12 W/m2 and the reference sound pressure pQ is 2 x 10-5 N/m2. A 1000 Hz note of this intensity is barely audible to person with good hearing. 6
  7. 7. If a sound intensity is given in decibels with no reference to any other sound intensity, you can assume that Io is the reference intensity. Decibels are also used to express the ratio of the amplitudes of two waves or two electronic signals. As stated above, decibels are for use with energy or power quantities, so it is actually the ratio of the powers associated with the two amplitudes that is given in decibels. It is therefore necessary to square the two amplitudes before taking the logarithm, since the power associated with an ultrasonic wave is proportional to the square of the pressure and the electrical power associated with an electrical voltage is proportional to the square of the voltage. A mathematically equivalent alternative to squaring the amplitudes is to use 20 instead of 10 in the dB formula. Thus, if A1 and A2 represent the two amplitudes Number of dB = 10 log10 = 20 log10 Pulse waves, energy spectra and bandwidth The pulses used in medical ultrasound generally have a length of only about 2 cycles (figure 2(a)). Typically, peak positive and negative pressures are up to about a megapascal (MPa) or so. 7
  8. 8. The peak back and forth displacements of particles are inversely proportional to frequency. Strictly, only a continuous wave can be characterized by a single frequency. A plot of amplitude versus frequency is known as the amplitude spectrum of the pulse. Since energy is proportional to the square of amplitude, the energy spectrum (figure 2(b)) of the pulse, showing the relative energy at each frequency, is given by squaring amplitude in the amplitude spectrum. Figure 2. Pressure-time waveform of a typical ultrasonic pulse and its energy spectrum. The pulse centre frequency (f) and the pulse bandwidth are indicated. 8
  9. 9. Two useful characteristics of the energy spectrum are the centre frequency (fc), at which the spectrum has its maximum height, and the pulse bandwidth, which is defined as the width of the energy spectrum at half its maximum height. An important rule is that pulse bandwidth increases as pulse length decreases. In fact: pulse bandwidth (MHz) = . Thus a continuous wave, which might be considered to be a pulse of constant amplitude and infinite length, has an infinitely narrow bandwidth (i.e. it has a spectrum consisting of a single line). For a typical two-cycle imaging pulse, the bandwidth is about 50% of fc. Thus, a '3 MHz' imaging pulse really means a pulse with a centre frequency of 3 MHz, but containing substantial energy at frequencies between about 2.2 MHz and 3.8 MHz. The Propagation of Ultrasound waves in Tissue Wave attenuation is the reduction of intensity with distance from the source. For a wave travelling through the body the causes of attenuation include divergence of the beam, partial reflection and rarefaction at tissue interfaces, and absorption and scattering within individual tissues. 9
  10. 10. Reflection: Wherever an ultrasound wave meets an interface where the characteristic acoustic impedance changes, a reflected wave is produced which carries with it a fraction of the power of the original wave. If the interface is smooth (on the scale of a wavelength) it is said to be a specular reflector, and behaves in the same way that a mirror (or partial mirror) reflects light waves. In particular, the angle of reflection equals the angle of incidence (figure 4(c)). This has important practical consequences since it means that where the source of the ultrasound is also the receiver, as in medical ultrasonic scanning, the wave reflected from a smooth surface can only be detected if the incident wave is perpendicular to the surface (figure 4(b)). For many tissue boundaries, small surface irregularities produce weak scattered waves over a very wide range of angles. Such boundaries are described as diffuse reflectors by analogy with the way that a matt surface or ground glass plate produces diffuse reflection of light. The echoes they produce on an ultrasound image are weaker than those from a specular reflector, but they are much more likely to be registered, as they do not require that the interface is perpendicular to the incident wave direction. 10
  11. 11. When a sound wave hits the body, part of the wave is reflected and part is transmitted into the body. The ratio of the reflected pressure amplitude (Pr) to the incident pressure amplitude (Pi) is given by: Where Z1 and Z2 are the acoustic impedances of medium 1 and 2 respectively. If Z1=Z2 , there is no reflected wave and transmission to the second medium is complete. Figure 3. A sound wave of amplitude pressure. Pi. Incident upon the body. Part of the wave, of amplitude pressure Pr. is reflected and part, of amplitude pressure Pt is transmitted The ratio of the transmitted pressure amplitude Pt. to the incident wave amplitude Pi is given by 11
  12. 12. The last two equations are for sound waves striking perpendicular to the surface. The ratio of reflected intensity to the total intensity is given by: The ratio of transmitted intensity to the total intensity is given by: It is clear that when, the acoustic impedances of the two media are similar almost all the sound is transmitted into the second medium. Choosing materials with similar acoustic impedances is called impedance matching. Getting sound energy into the body requires impedance matching. Refraction of Sound Waves : If an ultrasound wave meets, at an oblique angle, a boundary between two media having different speeds of sound, the transmitted wave will be deflected. This is known as refraction, and is illustrated in figure 4(a). The effect is analogous to that of a 12
  13. 13. light beam meeting a glass or water interface. In common with optics, Snell's law applies: Here c1 and c2 are the speeds of sound in the first and second media respectively, and angles are measured from a line (normal) perpendicular to the boundary. The law shows that the transmitted beam is deflected further away from the normal when c2 > c1 or towards the normal (as in figure 4(a)) when c2 < c1. If c2 = c1 or if the beam strikes the boundary at right angles (regardless of the values of c2 and c1) then no refraction takes place. In soft tissues, because variations in the speed of sound are small, beam deviations are generally only slight, but they are often sufficient to degrade the image quality and produce image artifacts. Where the speed changes from a lower to a higher value at an interface, and the angle of incidence is large, it is possible for the sine of the angle of transmission, as calculated from Snell's law, to be greater than 1. Since the sine of a real angle cannot be more than 1, this means there can be no transmission. The surface then acts as a complete reflector and the beam undergoes total internal reflection back into the first medium (4(c)). 13
  14. 14. Figure 4. (a) Partial reflection occurs when an ultrasound beam meets the boundary between two media of different characteristic impedances. If the speed of sound is different in the two media, as assumed here, the transmitted beam is refracted (ө1≠ ө2). (b) Perpendicular incidence is assumed in the definition of reflection coefficient, (c) Total internal reflection occurs if sin ө2 xc2/c1 > 1. Absorption When a sound wave passes through tissue, there is some loss of energy due to frictional effect. The absorption of energy in the tissue causes a reduction in the amplitude and the intensity of the incident sound wave. The amplitude (A) at a depth x cm in a medium is related to the initial amplitude Ao (at x=0) by the exponential equation: Where α, in cm-1 is the absorption coefficient for the medium at a particular frequency. Since the intensity is proportional to the square of the amplitude, its dependence with depth is 14
  15. 15. Where Io is the incident intensity at x = 0 and I is the intensity at depth x in the absorber. Since the absorption coefficient in the last equation is 2α therefore the intensity decreases more rapidly than the amplitude with depth. The half-value thickness (HVT) is the tissue thickness needed to decrease Io to Io/2 . Table2 gives typical HVTs for different tissues. Note the high absorption in the human skull and that the absorption increases as the frequency of the sound increases Table 2. Absorption Coefficients and Half-Value Thicknesses for various substances. 15
  16. 16. The Stethoscope : Many sounds from the chest region can be useful in the diagnosis of disease. The stethoscope is a simple "hearing aide" permits a physician to listen to sounds made inside the body, primarily in the heart and lungs. The act of listening to these sounds with a stethoscope is called mediate auscultation or usually just auscultation. The main parts of a modern stethoscope are the bell, which is either open or closed by a thin diaphragm, the tubing, and the earpieces (Fig. 5). The open bell is an impedance matcher between the skin and the air and accumulates sounds from the contracted area. The skin under the open bell behaves like a diaphragm. The skin diaphragm has a natural resonant frequency at which it most effectively transmits sounds; the factors controlling the resonant frequency are its tension and the diameter of the bell. The tighter the skin is pulled, the higher its resonant frequency. The larger the bell diameter, the lower the skin's resonant frequency. Thus it is possible to enhance the sound range of interest by changing the bell size and varying the pressure of the bell against the skin and thus the skin tension A low frequency heart murmur will appear to go away if the stethoscope is pressed hard against the skin. 16
  17. 17. A closed bell is a bell with a diaphragm of known resonant frequency, usually high, that tunes out low frequency sounds. Its resonant frequency is controlled by the same factors that mentioned above. The closed-bell stethoscope is primarily used for listening to lung sounds, Which are of higher frequency than heart sounds. Figure 4. Most of the heart sounds are of low frequency in the region where the sensitivity of the ear is poor. Lung sounds generally have higher frequencies. The curve represents the threshold of hearing for a good ear. Some of the heart and lung sounds are below this threshold For the best shape for the bell, it is desirable to have a bell with as small a volume as possible. The smaller the volume of air, the greater the pressure change for a given movement of the diaphragm at the end of the bell. The volume of the tubes should also be small, and should be little frictional loss of sound to the walls of the tubes. The small 17
  18. 18. volume restriction suggests short, small diameter tubes, while the low friction restriction suggests large diameter tubes. If the diameter of the tube is too small, frictional losses occur, and if it is too large, the moving air volume is too great; in both cases the efficiency is reduced. A compromise is a tube with a length of about 25 cm and a diameter of 0.3 cm. The earpieces should fit snugly in the ear because air leaks reduce the sounds heard, the lower the frequency, the more significant the leak. Leaks also allow background noise to enter the ear. 18