Physics of Ultrasound

Wavelength and Frequency

Sound is the result of mechanical energy traveling through matter as a wave producing alternating compression and rarefaction. Pressure waves are propagated by limited physical displacement of the material through which the sound is being transmitted. A plot of these changes in pressure is a sinusoidal waveform ( Fig. 1.1 ), in which the Y axis indicates the pressure at a given point and the X axis indicates time. Changes in pressure with time define the basic units of measurement for sound. The distance between corresponding points on the time-pressure curve is defined as the wavelength (λ), and the time (T) to complete a single cycle is called the period. The number of complete cycles in a unit of time is the frequency (f) of the sound. Frequency and period are inversely related. If the period (T) is expressed in seconds, f = 1/T, or f = T × sec ⁻¹ . The unit of acoustic frequency is the hertz (Hz); 1 Hz = 1 cycle per second. High frequencies are expressed in kilohertz (kHz; 1 kHz = 1000 Hz) or megahertz (MHz; 1 MHz = 1,000,000 Hz).

In nature, acoustic frequencies span a range from less than 1 Hz to more than 100,000 Hz (100 kHz). Human hearing is limited to the lower part of this range, extending from 20 to 20,000 Hz. Ultrasound differs from audible sound only in its frequency, and it is 500 to 1000 times higher than the sound we normally hear. Sound frequencies used for diagnostic applications typically range from 2 to 15 MHz, although frequencies as high as 50 to 60 MHz are under investigation for certain specialized imaging applications. In general, the frequencies used for ultrasound imaging are higher than those used for Doppler. Regardless of the frequency, the same basic principles of acoustics apply.

Propagation of Sound

In most clinical applications of ultrasound, brief bursts or pulses of energy are transmitted into the body and propagated through tissue. Acoustic pressure waves can travel in a direction perpendicular to the direction of the particles being displaced ( transverse waves), but in tissue and fluids, sound propagation is primarily along the direction of particle movement ( longitudinal waves). Longitudinal waves are important in conventional ultrasound imaging and Doppler, while transverse waves are measured in shear wave elastography. The speed at which pressure waves move through tissue varies greatly and is affected by the physical properties of the tissue. Propagation velocity is largely determined by the resistance of the medium to compression, which in turn is influenced by the density of the medium and its stiffness or elasticity. Propagation velocity is increased by increasing stiffness and reduced by decreasing density. In the body, propagation velocity of longitudinal waves may be regarded as constant for a given tissue and is not affected by the frequency or wavelength of the sound. This is in contrast to transverse (shear) waves for which the velocity is determined by Young modulus, a measure of tissue stiffness or elasticity.

Fig. 1.2 shows typical longitudinal propagation velocities for a variety of materials. In the body the propagation velocity of sound is assumed to be 1540 meters per second (m/sec). This value is the average of measurements obtained from normal soft tissue. Although this value represents most soft tissues, such tissues as aerated lung and fat have propagation velocities significantly less than 1540 m/sec, whereas tissues such as bone have greater velocities. Because a few normal tissues have propagation values significantly different from the average value assumed by the ultrasound scanner, the display of such tissues may be subject to measurement errors or artifacts ( Fig. 1.3 ). The propagation velocity of sound (c) is related to frequency and wavelength by the following simple equation:

Thus a frequency of 5 MHz can be shown to have a wavelength of 0.308 mm in tissue: λ = c/ f = 1540 m/sec × 5,000,000 sec ⁻¹ = 0.000308 m = 0.308 mm. Wavelength is an important determinant of spatial resolution in ultrasound imaging, and selection of transducer frequency for a given application is a key user decision.

Distance Measurement

Propagation velocity is a particularly important value in clinical ultrasound and is critical in determining the distance of a reflecting interface from the transducer. Much of the information used to generate an ultrasound scan is based on the precise measurement of time and employs the principles of echo-ranging ( Fig. 1.4 ). If an ultrasound pulse is transmitted into the body and the time until an echo returns is measured, it is simple to calculate the depth of the interface that generated the echo, provided the propagation velocity of sound for the tissue is known. For example, if the time from the transmission of a pulse until the return of an echo is 0.000145 seconds and the velocity of sound is 1540 m/sec, the distance that the sound has traveled must be 22.33 cm (1540 m/sec × 100 cm/m × 0.000145 sec = 22.33 cm). Because the time measured includes the time for sound to travel to the interface and then return along the same path to the transducer, the distance from the transducer to the reflecting interface is 22.33 cm/2 = 11.165 cm. By rapidly repeating this process, a two-dimensional (2-D) map of reflecting interfaces is created to form the ultrasound image. The accuracy of this measurement is therefore highly influenced by how closely the presumed velocity of sound corresponds to the true velocity in the tissue being observed (see Figs. 1.2 and 1.3 ), as well as by the important assumption that the sound pulse travels in a straight path to and from the reflecting interface.

Acoustic Impedance

Current diagnostic ultrasound scanners rely on the detection and display of reflected sound or echoes. Imaging based on transmission of ultrasound is also possible, but this is not used clinically at present. To produce an echo, a reflecting interface must be present. Sound passing through a totally homogeneous medium encounters no interfaces to reflect sound, and the medium appears anechoic or cystic. The junction of tissues or materials with different physical properties produces an acoustic interface. These interfaces are responsible for the reflection of variable amounts of the incident sound energy. Thus when ultrasound passes from one tissue to another or encounters a vessel wall or circulating blood cells, some of the incident sound energy is reflected. The amount of reflection or backscatter is determined by the difference in the acoustic impedances of the materials forming the interface.

Acoustic impedance (Z) is determined by product of the density (ρ) of the medium propagating the sound and the propagation velocity (c) of sound in that medium (Z = ρc). Interfaces with large acoustic impedance differences, such as interfaces of tissue with air or bone, reflect almost all the incident energy. Interfaces composed of substances with smaller differences in acoustic impedance, such as a muscle and fat interface, reflect only part of the incident energy, permitting the remainder to continue onward. As with propagation velocity, acoustic impedance is determined by the properties of the tissues involved and is independent of frequency.

Reflection

The way ultrasound is reflected when it strikes an acoustic interface is determined by the size and surface features of the interface ( Fig. 1.5 ). If large and relatively smooth, the interface reflects sound much as a mirror reflects light. Such interfaces are called specular reflectors because they behave as “mirrors for sound.” The amount of energy reflected by an acoustic interface can be expressed as a fraction of the incident energy; this is termed the reflection coefficient (R). If a specular reflector is perpendicular to the incident sound beam, the amount of energy reflected is determined by the following relationship:

where Z ₁ and Z ₂ are the acoustic impedances of the media forming the interface.

Because ultrasound scanners only detect reflections that return to the transducer, display of specular interfaces is highly dependent on the angle of insonation (exposure to ultrasound waves). Specular reflectors will return echoes to the transducer only if the sound beam is perpendicular to the interface. If the interface is not at a near 90-degree angle to the sound beam, it will be reflected away from the transducer, and the echo will not be detected (see Fig. 1.5A ).

Most echoes in the body do not arise from specular reflectors but rather from much smaller interfaces within solid organs. In this case the acoustic interfaces involve structures with individual dimensions much smaller than the wavelength of the incident sound. The echoes from these interfaces are scattered in all directions. Such reflectors are called diffuse reflectors and account for the echoes that form the characteristic echo patterns seen in solid organs and tissues (see Fig. 1.5B ). The constructive and destructive interference of sound scattered by diffuse reflectors results in the production of ultrasound speckle, a feature of tissue texture of sonograms of solid organs ( Fig. 1.6 ). For some diagnostic applications, the nature of the reflecting structures creates important conflicts. For example, most vessel walls behave as specular reflectors that require insonation at a 90-degree angle for best imaging, whereas Doppler imaging requires an angle of less than 90 degrees between the sound beam and the vessel.

Refraction

When sound passes from a tissue with one acoustic propagation velocity to a tissue with a higher or lower sound velocity, there is a change in the direction of the sound wave. This change in direction of propagation is called refraction and is governed by Snell law:

where θ ₁ is the angle of incidence of the sound approaching the interface, θ ₂ is the angle of refraction, and c ₁ and c ₂ are the propagation velocities of sound in the media forming the interface ( Fig. 1.7 ). Refraction is important because it is one cause of misregistration of a structure in an ultrasound image ( Fig. 1.8 ). When an ultrasound scanner detects an echo, it assumes that the source of the echo is along a fixed line of sight from the transducer. If the sound has been refracted, the echo detected may be coming from a different depth or location than the image shown in the display. If this is suspected, increasing the scan angle so that it is perpendicular to the interface minimizes the artifact.

Attenuation

As the acoustic energy moves through a uniform medium, work is performed and energy is ultimately transferred to the transmitting medium as heat. The capacity to perform work is determined by the quantity of acoustic energy produced. Acoustic power, expressed in watts (W) or milliwatts (mW), describes the amount of acoustic energy produced in a unit of time. Although measurement of power provides an indication of the energy as it relates to time, it does not take into account the spatial distribution of the energy. Intensity (I) is used to describe the spatial distribution of power and is calculated by dividing the power by the area over which the power is distributed, as follows:

The attenuation of sound energy as it passes through tissue is of great clinical importance because it influences the depth in tissue from which useful information can be obtained. This in turn affects transducer selection and a number of operator-controlled instrument settings, including time (or depth) gain compensation, power output attenuation, and system gain levels. Attenuation is measured in relative rather than absolute units. The decibel (dB) notation is generally used to compare different levels of ultrasound power or intensity. This value is 10 times the log ₁₀ of the ratio of the power or intensity values being compared. For example, if the intensity measured at one point in tissues is 10 mW/cm ² and at a deeper point is 0.01 mW/cm ² , the difference in intensity is as follows:

As it passes through tissue, sound loses energy, and the pressure waves decrease in amplitude as they travel farther from their source. Contributing to the attenuation of sound are the transfer of energy to tissue, resulting in heating (absorption), and the removal of energy by reflection and scattering. Attenuation is therefore the result of the combined effects of absorption, scattering, and reflection. Attenuation depends on the insonating frequency as well as the nature of the attenuating medium. High frequencies are attenuated more rapidly than lower frequencies, and transducer frequency is a major determinant of the useful depth from which information can be obtained with ultrasound. Attenuation determines the efficiency with which ultrasound penetrates a specific tissue and varies considerably in normal tissues ( Fig. 1.9 ).

Transmitter

Most clinical applications use pulsed ultrasound, in which brief bursts of acoustic energy are transmitted into the body. The source of these pulses, the ultrasound transducer, is energized by application of precisely timed, high-amplitude voltage. The maximum voltage that may be applied to the transducer is limited by federal regulations that restrict the acoustic output of diagnostic scanners. Most scanners provide a control that permits attenuation of the output voltage. Because the use of maximum output results in higher exposure of the patient to ultrasound energy, prudent use dictates use of the output attenuation controls to reduce power levels to the lowest levels consistent with the diagnostic problem.

The transmitter also controls the rate of pulses emitted by the transducer, or the pulse repetition frequency (PRF). The PRF determines the time interval between ultrasound pulses and is important in determining the depth from which unambiguous data can be obtained both in imaging and Doppler modes. The ultrasound pulses must be spaced with enough time between the pulses to permit the sound to travel to the depth of interest and return before the next pulse is sent. For imaging, PRFs from 1 to 10 kHz are used, resulting in an interval of 0.1 to 1 ms between pulses. Thus a PRF of 5 kHz permits an echo to travel and return from a depth of 15.4 cm before the next pulse is sent.

Transducer

A transducer is any device that converts one form of energy to another. In ultrasound the transducer converts electric energy to mechanical energy, and vice versa. In diagnostic ultrasound systems the transducer serves two functions: (1) converting the electric energy provided by the transmitter to the acoustic pulses directed into the patient and (2) serving as the receiver of reflected echoes, converting weak pressure changes into electric signals for processing.

Ultrasound transducers use piezoelectricity, a principle discovered by Pierre and Jacques Curie in 1880. Piezoelectric materials have the unique ability to respond to the action of an electric field by changing shape. They also have the property of generating electric potentials when compressed. Changing the polarity of a voltage applied to the transducer changes the thickness of the transducer, which expands and contracts as the polarity changes. This results in the generation of mechanical pressure waves that can be transmitted into the body. The piezoelectric effect also results in the generation of small potentials across the transducer when the transducer is struck by returning echoes. Positive pressures cause a small polarity to develop across the transducer; negative pressure during the rarefaction portion of the acoustic wave produces the opposite polarity across the transducer. These tiny polarity changes and the associated voltages are the source of all the information processed to generate an ultrasound image or Doppler display.

When stimulated by the application of a voltage difference across its thickness, the transducer vibrates. The frequency of vibration is determined by the transducer material. When the transducer is electrically stimulated, a range or band of frequencies results. The preferential frequency produced by a transducer is determined by the propagation speed of the transducer material and its thickness. In the pulsed wave operating modes used for most clinical ultrasound applications, the ultrasound pulses contain additional frequencies that are both higher and lower than the preferential frequency. The range of frequencies produced by a given transducer is termed its bandwidth. Generally, the shorter the pulse of ultrasound produced by the transducer, the greater is the bandwidth.

Most modern digital ultrasound systems employ broad-bandwidth technology. Ultrasound bandwidth refers to the range of frequencies produced and detected by the ultrasound system. This is important because each tissue in the body has a characteristic response to ultrasound of a given frequency, and different tissues respond differently to different frequencies. The range of frequencies arising from a tissue exposed to ultrasound is referred to as the frequency spectrum bandwidth of the tissue, or tissue signature. Broad-bandwidth technology provides a means to capture the frequency spectrum of insonated tissues, preserving acoustic information and tissue signature. Broad-bandwidth beam formers reduce speckle artifact by a process of frequency compounding. This is possible because speckle patterns at different frequencies are independent of one another, and combining data from multiple frequency bands (i.e., compounding) results in a reduction of speckle in the final image, leading to improved contrast resolution.

The length of an ultrasound pulse is determined by the number of alternating voltage changes applied to the transducer. For continuous wave (CW) ultrasound devices, a constant alternating current is applied to the transducer, and the alternating polarity produces a continuous ultrasound wave. For imaging, a single, brief voltage change is applied to the transducer, causing it to vibrate at its preferential frequency. Because the transducer continues to vibrate or “ring” for a short time after it is stimulated by the voltage change, the ultrasound pulse will be several cycles long. The number of cycles of sound in each pulse determines the pulse length. For imaging, short pulse lengths are desirable because longer pulses result in poorer axial resolution. To reduce the pulse length, damping materials are used in the construction of the transducer. In clinical imaging applications, very short pulses are applied to the transducer, and the transducers have highly efficient damping. This results in very short pulses of ultrasound, generally consisting of only two or three cycles of sound.

The ultrasound pulse generated by a transducer must be propagated in tissue to provide clinical information. Special transducer coatings and ultrasound coupling gels are necessary to allow efficient transfer of energy from the transducer to the body. Once in the body, the ultrasound pulses are propagated, reflected, refracted, and absorbed, in accordance with the basic acoustic principles summarized earlier.

The ultrasound pulses produced by the transducer result in a series of wavefronts that form a three-dimensional (3-D) beam of ultrasound. The features of this beam are influenced by constructive and destructive interference of the pressure waves, the curvature of the transducer, and acoustic lenses used to shape the beam. Interference of pressure waves results in an area near the transducer where the pressure amplitude varies greatly. This region is termed the near field, or Fresnel zone. Farther from the transducer, at a distance determined by the radius of the transducer and the frequency, the sound field begins to diverge, and the pressure amplitude decreases at a steady rate with increasing distance from the transducer. This region is called the far field, or Fraunhofer zone. In modern multielement transducer arrays, precise timing of the firing of elements allows correction of this divergence of the ultrasound beam and focusing at selected depths. Only reflections of pulses that return to the transducer are capable of stimulating the transducer with small pressure changes, which are converted into the voltage changes that are detected, amplified, and processed to build an image based on the echo information.

Receiver

When returning echoes strike the transducer face, minute voltages are produced across the piezoelectric elements. The receiver detects and amplifies these weak signals. The receiver also provides a means for compensating for the differences in echo strength, which result from attenuation by different tissue thickness by control of time gain compensation (TGC) or depth gain compensation (DGC).

Sound is attenuated as it passes into the body, and additional energy is removed as echoes return through tissue to the transducer. The attenuation of sound is proportional to the frequency and is constant for specific tissues. Because echoes returning from deeper tissues are weaker than those returning from more superficial structures, they must be amplified more by the receiver to produce a uniform tissue echo appearance ( Fig. 1.10 ). This adjustment is accomplished by TGC controls that permit the user to selectively amplify the signals from deeper structures or to suppress the signals from superficial tissues, compensating for tissue attenuation. Although many newer machines provide for some means of automatic TGC, the manual adjustment of this control is one of the most important user controls and may have a profound effect on the quality of the ultrasound image provided for interpretation.

Another important function of the receiver is the compression of the wide range of amplitudes returning to the transducer into a range that can be displayed to the user. The ratio of the highest to the lowest amplitudes that can be displayed may be expressed in decibels and is referred to as the dynamic range. In a typical clinical application, the range of reflected signals may vary by a factor of as much as 1 : 10 ¹² , resulting in a dynamic range of up to 120 dB. Although the amplifiers used in ultrasound machines are capable of handling this range of voltages, gray-scale displays are limited to display a signal intensity range of only 35 to 40 dB. Compression and remapping of the data are required to adapt the dynamic range of the backscattered signal intensity to the dynamic range of the display ( Fig. 1.11 ). Compression is performed in the receiver by selective amplification of weaker signals. Additional manual postprocessing controls permit the user to map selectively the returning signal to the display. These controls affect the brightness of different echo levels in the image and therefore determine the image contrast.

Image Display

Ultrasound signals may be displayed in several ways. Over the years, imaging has evolved from simple A-mode (amplitude-mode) and bistable display to high-resolution, real-time, gray-scale imaging. The earliest A-mode devices displayed the voltage produced across the transducer by the backscattered echo as a vertical deflection on the face of an oscilloscope. The horizontal time sweep of the oscilloscope was calibrated to indicate the distance from the transducer to the reflecting surface. In this form of display, the strength or amplitude of the reflected sound is indicated by the height of the vertical deflection displayed on the oscilloscope. With A-mode ultrasound, only the position and strength of a reflecting structure are recorded.

Another simple form of imaging, M-mode (motion-mode) ultrasound, displays echo amplitude and shows the position of moving reflectors ( Fig. 1.12 ). M-mode imaging uses the brightness of the display to indicate the intensity of the reflected signal. The time base of the display can be adjusted to allow for varying degrees of temporal resolution, as dictated by clinical application. M-mode ultrasound is interpreted by assessing motion patterns of specific reflectors and determining anatomic relationships from characteristic patterns of motion. Currently, the major application of M-mode display is evaluation of embryonic and fetal heart rates, as well as in echocardiography, the rapid motion of cardiac valves and of cardiac chamber and vessel walls. M-mode imaging may play a future role in measurement of subtle changes in vessel wall elasticity accompanying atherogenesis.

The mainstay of imaging with ultrasound is provided by real-time, gray-scale, B-mode display, in which variations in display intensity or brightness are used to indicate reflected signals of differing amplitude. To generate a 2-D image, multiple ultrasound pulses are sent down a series of successive scan lines ( Fig. 1.13 ), building a 2-D representation of echoes arising from the object being scanned. When an ultrasound image is displayed on a black background, signals of greatest intensity appear as white; absence of signal is shown as black; and signals of intermediate intensity appear as shades of gray. If the ultrasound beam is moved with respect to the object being examined and the position of the reflected signal is stored, the brightest portions of the resulting 2-D image indicate structures reflecting more of the transmitted sound energy back to the transducer.

In most modern instruments, digital memory is used to store values that correspond to the echo intensities originating from corresponding positions in the patient. At least 2 ⁸ , or 256, shades of gray are possible for each pixel, in accord with the amplitude of the echo being represented. The image stored in memory in this manner can then be sent to a monitor for display.

Because B-mode display relates the strength of a backscattered signal to a brightness level on the display device, it is important that the operator understand how the amplitude information in the ultrasound signal is translated into a brightness scale in the image display. Each ultrasound manufacturer offers several options for the way the dynamic range of the target is compressed for display, as well as the transfer function that assigns a given signal amplitude to a shade of gray. Although these technical details vary among machines, the way the operator uses them may greatly affect the clinical value of the final image. In general, it is desirable to display as wide a dynamic range as possible, to identify subtle differences in tissue echogenicity (see Fig. 1.11 ).

Real-time ultrasound produces the impression of motion by generating a series of individual 2-D images at rates of 15 to 60 frames per second. Real-time, 2-D, B-mode ultrasound is the major method for ultrasound imaging throughout the body and is the most common form of B-mode display. Real-time ultrasound permits assessment of both anatomy and motion. When images are acquired and displayed at rates of several times per second, the effect is dynamic, and because the image reflects the state and motion of the organ at the time it is examined, the information is regarded as being shown in real time. In cardiac applications the terms 2-D echocardiography and 2-D echo are used to describe real-time, B-mode imaging; in most other applications the term real-time ultrasound is used.

Transducers used for real-time imaging may be classified by the method used to steer the beam in rapidly generating each individual image, keeping in mind that as many as 30 to 60 complete images must be generated per second for real-time applications. Beam steering may be done through mechanical rotation or oscillation of the transducer or by electronic means ( Fig. 1.14 ). Electronic beam steering is used in linear array and phased array transducers and permits a variety of image display formats. Most electronically steered transducers currently in use also provide electronic focusing that is adjustable for depth. Mechanical beam steering may use single-element transducers with a fixed focus or may use annular arrays of elements with electronically controlled focusing. For real-time imaging, transducers using mechanical or electronic beam steering generate displays in a rectangular or pie-shaped format. For obstetric, small parts, and peripheral vascular examinations, linear array transducers with a rectangular image format are often used. The rectangular image display has the advantage of a larger field of view near the surface but requires a large surface area for transducer contact. Sector scanners with either mechanical or electronic steering require only a small surface area for contact and are better suited for examinations in which access is limited.

Mechanical Sector Scanners

Early ultrasound scanners used transducers consisting of a single piezoelectric element. To generate real-time images with these transducers, mechanical devices were required to move the transducer in a linear or circular motion. Mechanical sector scanners using one or more single-element transducers do not allow variable focusing. This problem is overcome by using annular array transducers. Although important in the early days of real-time imaging, mechanical sector scanners with fixed-focus, single-element transducers are not presently in common use.

Arrays

Current technology uses a transducer composed of multiple elements, usually produced by precise slicing of a piece of piezoelectric material into numerous small units, each with its own electrodes. Such transducer arrays may be formed in a variety of configurations. Typically, these are linear, curved, phased, or annular arrays. High-density 2-D arrays have also been developed. By precise timing of the firing of combinations of elements in these arrays, interference of the wavefronts generated by the individual elements can be exploited to change the direction of the ultrasound beam, and this can be used to provide a steerable beam for the generation of real-time images in a linear or sector format.