Principles of Ultrasound Physics: Sound Waves, Parameters, and Pulsed Imaging

Nature and Propagation of Sound Waves

In diagnostic ultrasonography, sound pulses are generated by a transducer to travel through biologic tissue, or media. These sound waves reflect off boundaries between different structures within the body and return to the transducer, where they are processed into anatomical images. All waves, including heat, sound, magnetic, and light, serve as a means to carry energy from one location to another. Sound is specifically a mechanical wave that requires a medium to travel through; it cannot exist in a vacuum. As sound propagates, the particles or molecules in the medium vibrate back and forth from a fixed position, creating alternating regions of compression, where molecules are squeezed together, and rarefaction, where they are stretched apart. Sound travels strictly in a straight line and is categorized as a longitudinal wave.

The relationship between a sound wave and the medium it traverses is described by two distinct concepts. Acoustic propagation properties refer to the effects that the medium has upon the sound wave as it passes through. Conversely, biologic effects describe the impact that the sound wave has upon the biologic tissue through which it propagates.

Acoustic Variables and Wave Identification

Acoustic variables are used to identify sound waves by their rhythmic oscillations. There are three primary acoustic variables: pressure, density, and distance. Pressure is defined as the concentration of force within a specific area and is measured in units of pascals ($Pa$). Density refers to the concentration of mass within a volume, reported in units such as $kg/cm^3$. Distance represents a measure of particle motion and is reported in units of length, including $cm$, $mm$, feet, or miles. If any of these three variables oscillates rhythmically, the wave is classified as a sound wave, also known as an acoustic wave. If a different variable oscillates, the wave is not a sound wave.

Wave Directions and Phase Relationships

Sound waves are characterized by the direction of particle motion relative to the direction of wave propagation. In a transverse wave, particles move in a direction that is perpendicular, or at right angles, to the direction of wave propagation. An example of this is shaking a string up and down; while the string moves vertically, the energy moves horizontally away from the hand. In contrast, sound is a longitudinal wave, where particles move in the same direction that the wave propagates. The particles move back and forth along the same path as the energy of the wave.

Waves can also interact based on their phase relationship. A pair of waves is considered in-phase when their peaks, or maximum values, and their troughs, or minimum values, occur at the same time and at the same location. These waves are said to be in step, much like a synchronized marching band. Out-of-phase waves are out of step, meaning their peaks and troughs occur at different times. When more than one sound beam travels through a medium and arrives at the same location simultaneously, they undergo interference. The individual waves combine to form a single wave.

Constructive interference occurs when in-phase waves combine to create a single wave with a greater amplitude than either of the original components. Destructive interference occurs when out-of-phase waves combine to form a single wave with a lesser amplitude than at least one of its components. If two out-of-phase waves have equal amplitude, they may undergo complete destructive interference and cancel each other out. If waves of different frequencies interfere, they will be in-phase at some moments and out-of-phase at others, resulting in a pattern of both constructive and destructive interference.

Fundamental Acoustic Parameters: Period and Frequency

Seven acoustic parameters are used to describe the characteristics of a sound wave: period, frequency, amplitude, power, intensity, wavelength, and propagation speed. These parameters are determined either by the sound source, which is the ultrasound system and transducer, or by the medium through which the sound travels. Some parameters can be adjusted by the sonographer, while others are fixed.

Period is the time required for a wave to complete a single cycle, or the duration from the start of one cycle to the start of the next. It is reported in units of time, typically microseconds ($\mu s$), seconds, hours, or days. In diagnostic ultrasound, the typical value for period ranges from $0.06$ to $0.5\,\mu s$, which can be written as $6 \times 10^{-8}$ to $5 \times 10^{-7}$ seconds. Period is determined solely by the sound source and is not adjustable by the sonographer using a basic transducer.

Frequency is the number of cycles that occur in one second. It is reported in units of per second, hertz ($Hz$), kilohertz ($kHz$), or megahertz ($MHz$). One hertz is equal to one cycle per second. In clinical imaging, the frequency ranges from approximately $2\,MHz$ to $15\,MHz$, or $2$ million to $15$ million cycles per second. Frequency is determined only by the sound source and is not adjustable. Sound is classified by its frequency relative to human hearing: infrasound is less than $20\,Hz$ and is inaudible; audible sound ranges from $20\,Hz$ to $20,000\,Hz$; and ultrasound exceeds $20,000\,Hz$ ($20\,kHz$). Frequency is vital because it affects both penetration depth and image quality.

Period and frequency are inversely related and are reciprocals of one another. As frequency increases, the period decreases, and as frequency decreases, the period increases. When multiplied together, their product is always $1$: $\text{Period} \times \text{Frequency} = 1$. For example, if a wave has a frequency of $8\,Hz$, its period is $\frac{1}{8}$ of a second. It is important to use complementary units, such as seconds and hertz, or milliseconds and kilohertz, when calculating these values.

Bigness Parameters: Amplitude, Power, and Intensity

Three parameters describe the magnitude or strength of a sound wave: amplitude, power, and intensity. Amplitude is the difference between the maximum or minimum value and the average, undisturbed value of an acoustic variable. It can be expressed in units of pressure ($Pa$), density ($g/cm^3$), or distance ($cm$, inches), as well as in decibels ($dB$). In clinical imaging, pressure amplitude ranges from $1$ million pascals ($1\,MPa$) to $3$ million pascals ($3\,MPa$). Peak-to-peak amplitude is the total difference between maximum and minimum values, making it twice the value of the standard amplitude. Initially determined by the sound source, amplitude decreases as sound propagates, a rate dependent on the wave and the medium. It is an adjustable parameter.

Power is the rate of energy transfer or the rate at which work is performed, measured in watts ($W$). In clinical imaging, typical power values range from $0.004$ to $0.090\,W$ ($4$ to $90\,mW$). Like amplitude, initial power is determined by the source and decreases during propagation. It is adjustable by the sonographer. Power is mathematically proportional to the square of the amplitude: $\text{power} \propto \text{amplitude}^2$. For instance, if amplitude is tripled, power increases ninefold ($3 \times 3 = 9$). If amplitude is halved, power decreases to one-fourth ($25\%$) of its original value.

Intensity is the concentration of energy in a sound beam and is calculated by dividing the power of the beam by its cross-sectional area: $\text{Intensity} = \frac{\text{Power}}{\text{Area}}$. Units are watts per square centimeter ($W/cm^2$). Typical clinical values range from $0.01$ to $300\,W/cm^2$. Initial intensity is source-determined and adjustable, but it changes as sound travels through the body. Intensity is proportional to power and to the square of amplitude: $\text{intensity} \propto \text{power}$ and $\text{intensity} \propto \text{amplitude}^2$. If power is doubled, intensity is doubled; if amplitude is doubled, intensity increases fourfold.

Wavelength and Propagation Speed

Wavelength is the distance or length of a single complete cycle, similar to the length of a single boxcar in a train. It is measured in units of length ($mm$, meters). In soft tissue, typical wavelengths range from $0.1$ to $0.8\,mm$. Wavelength is unique as the only parameter determined by both the sound source and the medium. It cannot be adjusted by the sonographer. Period and wavelength are often confused; however, wavelength is a measure of distance while period is a measure of time. Wavelength and frequency are inversely related; in a single medium, higher frequencies result in shorter wavelengths. In soft tissue, the wavelength of $1\,MHz$ sound is exactly $1.54\,mm$. To find the wavelength of any sound in soft tissue, the formula is $\text{wavelength (mm)} = \frac{1.54\,mm/\mu s}{\text{frequency (MHz)}}$. Shorter wavelengths, created by higher frequencies, produce higher quality images with more detail.

Propagation speed is the rate at which a sound wave travels through a medium, measured in meters per second ($m/s$) or millimeters per microsecond ($mm/\mu s$). In the body, speed ranges from $500\,m/s$ to $4,000\,m/s$. Speed is determined solely by the medium and is not affected by the sound wave's frequency. In any specific medium, all sound waves travel at the same speed. The average speed of sound in soft tissue is $1,540\,m/s$ ($1.54\,mm/\mu s$ or $1.54\,km/s$). Typical speeds in biologic media include: Lung ($500\,m/s$), Fat ($1,450\,m/s$), Liver ($1,560\,m/s$), Blood ($1,560\,m/s$), Muscle ($1,600\,m/s$), Tendon ($1,700\,m/s$), and Bone ($3,500\,m/s$). In other materials, air is $330\,m/s$, water is $1,480\,m/s$, and metals range from $2,000$ to $7,000\,m/s$. Generally, sound travels fastest in solids, slower in liquids, and slowest in gases. Speed is calculated as $\text{speed (m/s)} = \text{frequency (Hz)} \times \text{wavelength (m)}$.

Speed is determined by the medium's stiffness and density. Stiffness, or bulk modulus, is the ability of an object to resist compression. Stiffness and speed are directly related; as stiffness increases, speed increases. Density describes the relative weight of a material. Density and speed are inversely related; as density increases, speed decreases. Although both affect speed, stiffness has a much greater influence. Materials that are stiff but not dense, like bone, have the fastest speeds, while materials that are not stiff and highly dense have the slowest speeds.

Characteristics of Pulsed Ultrasound

Imaging systems use pulsed ultrasound, which consists of short bursts or pulses of acoustic energy, because continuous wave sound cannot create anatomic images. A pulse is a collection of cycles that travel together as a single unit, with a clear beginning and end. Each pulse has a transmit, talking, or "on" time, and a receive, listening, or "off" time.

Pulse duration is the actual time from the start of a pulse to its end, representing a single transmit time. It is measured in microseconds ($\mu s$), with typical values ranging from $0.3$ to $2.0\,\mu s$. Determined by the sound source only and not adjustable, pulse duration is calculated as \text{pulse duration (\mu s)} = \#\text{cycles} \times \text{period (\mu s)} or \text{pulse duration (\mu s)} = \frac{\#\text{cycles}}{\text{frequency (MHz)}}. Pulses typically contain $2$ to $4$ cycles. Shorter duration pulses are more desirable because they provide greater image accuracy.

Spatial pulse length is the distance that a pulse occupies in space, measured in millimeters ($mm$). In soft tissue, it ranges from $0.1$ to $1.0\,mm$. It is determined by both the source and the medium and is calculated as $\text{spatial pulse length (mm)} = \#\text{cycles} \times \text{wavelength (mm)}$. Short pulse lengths, created by fewer cycles or cycles with shorter wavelengths, are preferred for more accurate imaging.

Pulse repetition period (PRP) is the time from the start of one pulse to the start of the next, including both the transmit and listening time. It ranges from $100\,\mu s$ to $1\,ms$ and is determined by the sound source and the imaging depth selected by the sonographer. PRP is directly related to depth; as imaging depth increases, PRP increases. The sonographer can only adjust the listening time portion of the PRP by changing the depth. Pulse repetition frequency (PRF) is the number of pulses transmitted into the body per second, measured in hertz ($Hz$). Typical values are $1,000$ to $10,000\,Hz$. PRF is inversely related to depth; as depth increases, PRF decreases.

Sound Beam Intensities and Bioeffects

Intensity in a sound beam is not uniform; it varies across space and time. Spatial refers to location or space, with spatial peak ($I_{sp}$) being the intensity at the maximum location and spatial average ($I_{sa}$) being the average intensity across the beam's cross-sectional area. Spatial peak intensity is always higher than spatial average intensity. Temporal refers to time, accounting for transmit and receive cycles. Temporal peak ($I_p$) is the intensity at the instant of its maximal value. The maximum intensity averaged over the most intense half-cycle is called $I_{max}$ or $I_m$. Pulse average intensity ($I_{pa}$) is the average during the transmit time, while temporal average ($I_{ta}$) is the average over the entire pulse repetition period. Temporal intensities, from largest to smallest, are $I_{tp}$, $I_{max}$, $I_{pa}$, and $I_{ta}$.

Combining these factors yields six methods to report intensity: $SPTP$ (spatial peak, temporal peak), $SATP$ (spatial average, temporal peak), $SPTA$ (spatial peak, temporal average), $SATA$ (spatial average, temporal average), $SPPA$ (spatial peak, pulse average), and $SAPA$ (spatial average, pulse average). All intensities use units of $W/cm^2$. The Rank of intensities from largest to smallest is: $SPTP \rightarrow I_m \rightarrow SPPA \rightarrow SPTA \rightarrow SATA$. $SPTA$ is the most relevant intensity regarding tissue heating and the study of bioeffects.

A few supplemental rules govern the behavior of these intensities. The beam uniformity coefficient, or SP/SA factor, describes the spread of a beam in space and is a unitless number of $1$ or greater. The duty factor is a unitless number between $0$ and $1$ describing the temporal relationship of intensities. For continuous wave ultrasound, where the beam is always "on," the pulse average and temporal average intensities are identical, meaning $SPTA = SPPA$ and $SATA = SAPA$. If pulsed and continuous waves have the same $SPTP$ or $SATP$ intensities, the continuous wave will have the higher $SPTA$ or $SATA$ intensity, respectively.", "title": "Principles of Ultrasound Physics: Sound Waves, Parameters, and Pulsed Imaging"}