PHYSICAL PRINCIPLES OF ULTRASOUND

ULTRASOUND AS A WAVE

Sound is propagated through a medium (e.g. air) as a mechanical vibration of the particles of that medium and in simple terms may be categorised by its loudness and pitch or frequency. “Ultra” means beyond, ultrasound is sound with a frequency beyond that of human perception (i.e. >20 kHz), and has the same physical properties as “audio” sound. Most clinical diagnostic applications of ultrasound employ frequencies in the range 2 - 10 MHz.

Ultrasonic energy travels through a medium in the form of a wave. Although a number of different wave modes are possible, in almost all diagnostic applications, ultrasound propagates in the form of a longitudinal wave, where the particles of the medium oscillate in the direction of propagation of the sound. Energy is transferred through the medium in a direction parallel to that of the oscillations of the particles. The particles themselves do not move through the medium. They simply vibrate to and fro about their mean position.

The vibrations of individual particles may be complex For simplicity consider the movement of a single particle excited by pure sinusoidal continuous wave.


The graph above shows the displacement of the particle about its mean position plotted against time. The time taken to execute one complete cycle, T, is called the period. The maximum displacement, a, is known as the amplitude. If the frequency of the wave is f (Hz) i.e. it executes f complete cycles per second. The time taken to execute one complete cycle, T, is given by

Period = 1/f seconds

It is often useful to think of the source of ultrasound, the transducer, as a vibrating piston. As it moves it displaces the adjacent particles of the adjoining medium. These in turn displace more particles throughout the medium. Since the particles are not rigidly fixed to each other, they do not all move together. There is a delay between the movement of adjacent particles (analogous to a series of balls connected by springs). At a particular time there will be some regions where the particles are closer together and the pressure and density of the medium is increased (regions of compression) and areas where the particles are further apart and the pressure and density of the medium is decreased (regions of ). These regions of compression or rarefaction move through the medium as a wave.



compression rarefraction

Consider again the case of excitation by a simple sinusoidal waveform. At a given time, the displacement of the particles (or the pressure or density of the particles) plotted against distance is shown below.


The wavelength, lambda, is defined as the distance in the medium between points of equal value (displacement, pressure or density). For a sound wave of wavelength  and frequency f, the distance travelled by the wave per second (i.e. its velocity, c) will be the number of cycles passing a given point in unit time multiplied by the length of each wave.

i.e. c = f.lambda

The velocity of a sound wave in a medium is determined by the delay which occurs between the movements of neighbouring particles. This depends on the elasticity and density of the medium. It is a property of the medium and is essentially independent of f, the frequency of the ultrasound. Rigid materials have higher wave velocities than compressible materials like gases.

MATERIAL Velocity c (m s-1)

Air 330
Lung 650
Water 1480
Aluminium 6400
Bone 3500
Brain 1540
Blood 1570
Fat 1460
Muscle 1580
Soft tissue (average) 1540

The Table above shows that the values for the speed of sound in different tissues are very similar. It is assumed that a value for wave velocity of 1540 m/s is a reasonable approximation in most clinical applications. This is of fundamental importance in diagnostic use of ultrasound.

INTENSITY


When an ultrasound wave is generated in tissue, energy, in the form of kinetic energy of motion of the particles, passes through the tissue. The intensity, I, of an ultrasound field is the quantity of energy flowing through unit area in unit time. i.e.
where E is the total energy in J, A is the area, t = time
Intensity is usually measured in units of Webers per square meter (W m-2) or mW cm-2

Intensity is not normally measured directly, calibrated transducers known as hydrophones are usually used to measure pressure amplitudes within ultrasound beams. It can be shown that

where P is the pressure amplitude and  is the density

Intensity is proportional to the square of the pressure.

For continuous wave fields, provided the measurement is carried out over many cycles, the intensity will always have the same value. However for pulsed fields, intensity figures are usually quoted in terms of temporal average, pulse average or temporal peak. The temporal average intensity ITA is the measurement obtained after averaging over many cycles. The pulse average intensity IPA is the value obtained by averaging only during the duration of a pulse and not during the 'off-time'. The temporal peak intensity lTP is the maximum instantaneous value measurable and corresponds to the peak value of the pulse.

Spacial variation must be considered. Within the beam there will be areas of high intensity and areas of low intensity. The regions of maximum intensity might for example be at the focus of a focused system. This maximum may extend for only a few mm in any direction. If the intensity is measured over that small region, the value will be high. On the other hand if it is averaged over the whole cross-section a much smaller value will result. It has become conventional to refer to spatial peak and spatial average intensities. The spatial peak intensity ISP value is the maximum intensity sampled over a very small distance found anywhere in the beam. The spatial average intensity ISA value is the average value across the beam at some distance from the transducer.

Combining the above concepts we get a whole range of intensity parameters. It is ISPTA (Spatial Peak Temporal Average) value which is most critical as it relates to the local heating effect in tissue.


DECIBEL NOTATION

An absolute measurement of intensity is difficult and often inappropriate. Usually, we are more interested in knowing the ratio of intensities. particularly if the level of one of these is taken as a reference for comparison (e.g. ratio of energy reflected at a different tissue boundaries). Expressing such ratios as logarithms provides a simple method of expressing numbers which extend over many orders of magnitude.
The relative intensity in decibels (dB) where I1 and I0 are the intensities
or
The relative amplitude in decibels (dB) where A1 and A0 are the wave amplitudes
Where I1 > I0, dB values are +ve, where I1 < I0, dB values are -ve.


DECIBEL NOTATION FOR INTENSITIES
dB I1/I0 dB I1/I0
0 1 0 1
+3 1.995 -3 0.501
+10 10 (101) -10 0.1 (10-1)
+20 100 (102) -20 0.01 (10-2)
+30 103 -30 10-3
+40 104 -40 10-4


DECIBEL NOTATION FOR AMPLITUDES
dB A1/A0 dB A1/A0
0 1 0 1
+3 1.413 -3 0.708
+6 1.995 -6 0.501
+10 3.162 -10 0.320
+20 10 (101) -20 0.1 (10-1)
+40 102 -40 10-2
+60 103 -60 10-3

The existence of these two separate expressions can lead to confusion. The first term is used when comparing intensity or power, the second term is used when comparing pressure amplitude or voltage.

In ultrasound systems, the intensity or power of an ultrasound transducer is changed by varying the excitation voltage. Since power is proportional to the voltage squared, V2, 10 log10 V2 = 20 log10 V. Similarly, intensity is measured by hydrophones which measure pressure amplitude and intensity is proportional to the pressure squared. Provided the correct definition is used there is a complete equivalence between the decibel relative level for both terms. For example, if the voltage gain of an output amplifier driving an ultrasound transducer is increased by +3dB, the intensity or power of the ultrasound wave will also increase by +3dB.


INTERFERENCE


When two (or more) waves are transmitted into a medium the resultant particle motion is obtained by adding the motion due to one wave to the motion due to the other. This phenomenon is known as interference.

When two ultrasound pressure waves with the same frequency and in step (in phase) overlap (as in (a)), they reinforce each other and the resulting waveform has an increased amplitude. The interference has resulted in a wave with an increased intensity - this process is known as constructive interference.

Conversely, when two waves with the same frequency but out of step (1800 out of phase) combine (as in (b)), the resultant amplitude would be small since the summation of the wave motion would tend to cancel each other out. This is know as destructive interference.

For interference to occur the waves have to be coherent. The phase relationship must hold over many cycles and the frequencies must be equal, or very nearly so.


PROPAGATION OF ULTRASOUND THROUGH TISSUE

As an ultrasound wave propagates through tissue, its intensity is attenuated by a number of mechanisms. The ultrasound beam will diverge due to the difficulty of generating a parallel beam (see later) and the refraction, reflection and scattering of the ultrasound wave. Furthermore, the mechanical energy of the ultrasound beam will be converted to heat by absorption. As a rough rule of thumb, the total attenuation of soft tissue is approximately 1 dB cm-1 MHz-1.

Refraction

When a wave meets a boundary between two media at normal incidence (90º), it is propagated without deviation into the second medium. At oblique incidence the wave is bent by refraction. The amount determined by Snell's law:


where c1 = wave velocity in medium 1 and c2 = wave velocity in medium 2.

Ultrasound refraction is normally insignificant in most areas of medical ultrasound apart from the eye at the interface between aqueous and vitreous humour or if trying to scan through bone. It can b significant when using a test phantom.

Reflection

When an ultrasound wave meets a boundary between two different media, where the size of the boundary is large compared with the wavelength of ultrasound and the roughness of the boundary is small compared with the wavelength, a proportion of the ultrasound energy is reflected. This specular reflection is similar to optical reflection i.e. *i = *r. In normal incidence the reflected beam will return to the transducer along the same path. This returned “echo” forms the basis of pulse echo ultrasound imaging.


The proportion of the incident energy reflected by the boundary is also important and depends on the acoustic impedance (Z) of each medium.
Z * *c where * is the density of the material.

In normal incidence the fraction of the wave reflected is given by

where

Ir = intensity of reflected ultrasound
Ii = intensity of incident ultrasound
Z1¬ = acoustic impedance in medium 1
Z2 = acoustic impedance in medium 2


Hence, it is the difference between the acoustic impedance of the two structures that determines the proportion of the incident energy that is reflected. Examples of values of Z are given below together with examples of the proportion of energy reflected at typical boundaries


MATERIAL Z (106 kg m-2 s-1)

AIR 0.0004
LUNG 0.26 - 0.46
BONE 3.75 - 7.38
WATER 1.52
LIVER 1.65
KIDNEY 1.62
BLOOD 1.61
FAT 1.38
TISSUE 1.35 - 1.68



BOUNDARY % REFLECTED
at normal incidence

MUSCLE/BLOOD 0.07%
FAT/MUSCLE 1.08%
SOFT TISSUE/WATER 0.23%
SOFT TISSUE/AIR 99.9%
SOFT TISSUE/BONE 41.2%


When Z1 = Z2, all the energy is transmitted across the boundary and there is no reflected echo. However, when the difference in acoustic impedance between medium 1 and 2 is large, (e.g. between a tissue/air or tissue/bone interface) most of the ultrasound energy is reflected and very little is transmitted. Hence it is difficult to visualise through bone or through air in the lungs and bowel. In order to exclude air between the transducer and skin surface a coupling gel is used to ensure adequate penetration of ultrasound into the tissues.


Scattering

When an ultrasound wave strikes targets which are small or rough compared to the wavelength (e.g. within soft tissues, organs and blood), these targets re-radiate (scatter) the ultrasound energy in many directions. Where there are many scattering targets, multiple scattering occurs.



These scatterers act as secondary sources of ultrasound.
A proportion of this scattered ultrasound energy will return in the direction of the source (back-scattered). The contribution which scattering makes to the total attenuation is frequency dependent. When scatterer size << *, scattered intensity * f4. However the scattered intensity is small in comparison to the energy reflected from major tissue boundaries. For example, compared to the reflected intensity from a fat/muscle boundary, the approximate scattered intensities from a number of different structure are given below:


Placenta -20dB (10-2)
Liver -30dB (10-3)
Kidney -40dB (10-4)
Blood -60dB (10-6)

Between the strong directional echoes from specular reflection at boundaries and the weak multi-directional echoes from scattering targets, a range of intermediate echoes are received. Any ultrasound system thus has to be capable of processing a wide dynamic range of echoes.

Absorption

Absorption is the process by which some of the mechanical energy of the ultrasound is converted into heat in the tissues. In soft tissue, absorption account for over 90% of the total attenuation of the ultrasound beam. Absorption falls off exponentially with distance, the same fraction of the incoming energy is lost in each unit distance travelled.


The intensity, I, at a distance is given by:

where is the initial intensity at x = 0,
and is the intensity absorption coefficient.
The absorption coefficient depends on the characteristics of the medium and is also approximately proportional to ultrasound frequency. Hence in order to achieve greater penetration (less attenuation) lower frequencies are used.

Instead of giving values of , it more convenient to define the half-value thickness as the thickness of material required to reduce the intensity of an ultrasound beam by half, or -3dB.

HALF VALUE THICKNESS (cm)

MATERIAL 2MHz 5MHz

Air 0.06 0.01
Bone 0.1 0.04
Water 340 54
Soft Tissue 2.1 0.86
Blood 8.5 3.0
Liver 1.5 0.5

Air and bone have high values of . As well as strongly reflecting ultrasound at any interface with tissue as mentioned earlier, they attenuate the small proportion transmitted. Problems of scanning through the head or imaging through the lungs or bowel are compounded. Water and other body fluids have low attenuation, having a full bladder is a standard technique to get good views of the uterus.


GENERATION AND DETECTION OF ULTRASOUND


Ultrasound is generated and detected by a transducer, which converts electrical energy into mechanical vibrations and vice versa. Materials which generate a potential difference across their surface when their shape is changed by an applied pressure wave by the piezoelectric effect, are used as transducers. These materials also change their shape when a voltage is applied (the inverse piezoelectric effect) and are therefore used as both transmitters and receivers of ultrasound. There are many naturally occurring piezoelectric materials such as quartz, but it is normal to use synthetic materials such as a ceramic - lead zirconate/titanate (PZT) or a plastic - polyvinyldifluoride (PVDF) transducer.

If a piezoelectric transducer is excited with a continuous sinusoidal electrical signal, it will oscillate and generate an ultrasonic wave at the same frequency as the excitation frequency. Transducers display a natural frequncy where resonance occurs and generation of ultrasound waves is particularly efficient.



When a sinusoidal electrical signal is applied to a piezoelectric material the walls will vibrate. Some of the enegry will travels into any adjoining medium. A wave is also reflected inside the transducer and be reflected to and fro. If the time taken for this internal wave to travel from one side to the other and then back again is the same as the period of the applied signal, the internally reflected waves interfere constructively, and the resultant ultrasound wave is enhanced. This first resonance occurs where the thickness of the material, t = /2,

therefore the resonant frequency


Often only this lowest resonant frequencies carries significant energy. If a transducer is designed so that its thickness is equal to half the wavelength corresponding to the required frequency of operation the transducer is operating at its fundamental resonant frequency giving maximum efficiency in transmission and reception.


If the applied frequency is varied, the displacement of the transducer, and hence the energy of the resulting ultrasound wave will decrease as shown. The width of the peak, delta f, where the amplitude has fallen by -3dB (A1/A0 delta 0.7) is known as the transducer bandwidth. The bandwidth of commercial transducers may extend over several MHz. The Q-factor, Q, of a transducer is defined as

Q = fr /delta f .


Transducer design depends on the mode of operation. In a simple transducer each face of the piezoelectric element is coated with a thin metallic layer to act as an electrode, and the complete assembly is housed in a metal cylinder. In continuous wave applications and in order to obtain high efficiency, the rear face of the piezoelectric material is backed by air. This helps to reflect energy from the rear face back into the material to reinforce the wave from the front face as described above. However most applications use pulsed excitation in order to obtain a short burst or pulse of ultrasound. In early instruments a voltage pulse was applied to the transducer forcing it to oscillate at its natural resonant frequency. Modern pulsed systems apply a few cycles of a sinusoidal waveform. In both cases it is required to produce a short pulse of ultrasound. Highly efficient air backed transducers are unsuitable since the internally reflected signal would continue to produce ultrasound after the applied electrical signal had stopped. This ringing can be reduced by a backing material which reduces reflections at the back face. Short pulse outputs of ultrasound can be achieved at the expense of a less efficient, higher bandwidth (lower Q) transducer.


ULTRASONIC FIELD

The ultrasonic field of a transducer describes the spatial distribution of its radiated energy. The field during transmission is identical to the sensitivity distribution of the transducer when used as a receiver.

The ultrasound beam shape produced by transducer is complex. In an idealised situation of a circular transducer generating a continuous wave of ultrasound, the beam shape can be considered to have two distinct regions.


In the near field (or Fresnel region) the ultrasound beam is approximately cylindrical with a diameter roughly equal to the transducer diameter. The near field extends for a distance of D2/4* from the transducer face, where D is the transducer diameter and * is the wavelength of the ultrasound. In the far field (or Fraunhofer region) the beam diverges with an angle given by sin* = 1.22*/D.

However, within these two regions the beam intensity is not uniform and becomes even more complex when rectangular, focused and pulsed transducers are used.


A more accurate estimation of the field of a transducer can be obtained by considering the surface of the transducer to be an array of separate elements each radiating spherical waves. By ascertaining the points where the waves maxima and minima meet, points of constructive and destructive interference can be established, and the ultrasonic field estimated. This 2-dimensional example gives some idea of the complexity of an ultrasonic field compared with the simple “text book” shape.


The theoretical field for a circular transducer is shown above. Moving along the central axis of the beam away from the transducer in the near zone, the intensity shows successive axial maxima and minima which become further apart away from the transducer. There are also several maxima across the beam diameter. The last axial maximum occurs at the end of the near zone (at a distance of D2/4). Beyond this in the far zone the central axis intensity decreases and the beam diverges. Rectangular transducers and pulsed ultrasound complicate these fields.

It is normal to operate ultrasound systems in the near field in order to have a narrow beamwidth (good lateral resolution) with little divergence. Since ultrasound can only penetrate a limited depth, transducers are usually designed so that the end of the near field corresponds to the limit of penetration.

Smaller diameter crystals produce a narrower beam but at the expense of a shorter near field and greater divergence in the far field. Higher operating frequencies give a longer near field, but unfortunately higher frequencies have a higher attenuation so the penetration is less. Transducer design is therefore a compromise.