# Principles of Ultrasound

## Key Points

• Ultrasound is mechanical energy in the form of vibrations that propagate through a medium such as tissue.

• Ultrasound interacts with tissue by undergoing absorption, reflection, refraction, and scattering and produces an image representative of tissue structure.

• Imaging artifacts can be recognized and understood based on a knowledge of the principles of ultrasound.

## Basic Ultrasound Physics

Sound is mechanical energy in the form of vibrations that propagate through a medium such as air, water, or tissue. The frequency of audible sound ranges from 20 to 20,000 Hz (cycles per second). Ultrasound involves a frequency spectrum that is greater than 20,000 Hz. Medical applications use frequencies in the range of 1,000,000 to 50,000,000 Hz (1 to 50 MHz). The propagation of ultrasound results from the displacement and oscillation of molecules from their average position and the subsequent displacement and oscillations of molecules along the direction of propagation of the ultrasound wave.

Ultrasound waves can be described using the common properties of waves. Fig. 1.1 is an illustration of a sinusoidal wave with the pressure amplitude along the y -axis and the time or distance along the x -axis. Fig. 1.1 is referred to in the following sections to introduce the basic properties of waves.

## Wavelength, Frequency, and Velocity

The wavelength is the distance in the propagating medium that includes one complete cycle (see Fig. 1.1 ). The wavelength (λ) is dependent on the frequency (f) of the oscillations and the velocity (c) of propagation in the medium. The relationship of wavelength, frequency, and velocity is given in Eq. 1.1 .

<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='c=fλ’>?=??c=fλ
c=fλ

The frequency of a wave is the number of oscillations per unit of time. Typically in ultrasound, this is stated in terms of cycles per second or Hertz (1 c/s = 1 Hz). The period of a wave (τ) is the inverse of the frequency and represents the time required to complete one cycle. The relationship between frequency and period is given in Eq. 1.2 .

<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='c=1τ’>?=1?c=1τ
c=1τ

The velocity of propagation depends on the physical properties of the medium in which the wave is propagating. The primary physical properties governing the velocity of propagation are the density and compressibility of the medium.

## Density, Compressibility, and Bulk Modulus

The density ( <SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml='ρ’>?ρ
ρ
) of a medium is the mass per unit volume of that medium (kg/m 3 in SI units). The compressibility (K) of a medium is a property that reflects the relationship between the fractional decrease in volume and the pressure applied to a medium. For example, air has high compressibility (a small amount of pressure applied to a volume of air will result in a large fractional decrease in volume), whereas bone has relatively low compressibility (a large amount of pressure applied to a volume of bone will result in a small fractional decrease in volume). Finally, the bulk modulus (β), which is the inverse of the compressibility, is the negative ratio of pressure applied to a medium and the fractional change in volume of the medium and reflects the stiffness of the medium.

The acoustic velocity (c) of a medium can be determined once the density ( <SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='ρ’>?ρ
ρ
) and the compressibility (K), or bulk modulus (β), are known. Eq. 1.3 demonstrates the relationship of the three physical properties.

Density, compressibility, and bulk modulus are not independent of one another. Typically, as density increases, compressibility decreases and bulk modulus increases. However, compressibility and bulk modulus typically vary more rapidly than does density, and they dominate in Eq. 1.3 .

<SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='c=1Kρ=βρ’>?=1??=??c=1Kρ=βρ
c=1Kρ=βρ

The acoustic velocity in different media can be determined by applying the equations to practice. For example, water at 30°C has a density of 996 kg/m 3 and a bulk modulus of 2.27 × 10 9 N/m 2 . Inserting these values into Eq. 1.3 yields an acoustic velocity of 1509 m/s in water. Values for density and bulk modulus have been characterized extensively and can be found in the literature. A summary of relevant tissue properties is given in Table 1.1 . The acoustic velocity is not dependent on the frequency of the propagating wave (i.e., acoustic waves of different frequencies all propagate with the same acoustic velocity within the same medium).

TABLE 1.1

Physical Properties of Tissue

Adapted from Duck FA. Physical Properties of Tissue . London: Academic Press; 1990.

Tissue or Fluid Density (kg/m 3 ) Bulk Modulus (×10 9 N/m 2 ) Acoustic Velocity (m/s)
Water (30°C) 996 2.27 1509
Blood 1050–1075 2.65 1590
Pancreas (pig) 1040–1050 2.63 1591
Liver 1050–1070 2.62 1578
Bone, cortical 1063–2017 28.13 3760

## Ultrasound Interactions in Tissue

Ultrasound imaging of tissue is achieved by transmitting short pulses of ultrasound energy into tissue and receiving reflected signals. The reflected signals that return to the transducer represent the interactions of a propagating ultrasound wave with tissue. A propagating ultrasound wave can interact with tissue, and the results are reflection, refraction, scattering, and absorption.

## Reflection

Specular reflections of ultrasound occur at relative large interfaces (greater than one wavelength) between two media of differing acoustical impedances. At this point, it is important to introduce the concept of acoustic impedance. The acoustic impedance (Z) of a medium represents the resistance to sound propagating through the medium and is the product of the density ( <SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='ρ’>?ρ
ρ
) and the velocity (c):

<SPAN role=presentation tabIndex=0 id=MathJax-Element-7-Frame class=MathJax style="POSITION: relative" data-mathml='Z=ρc’>?=??Z=ρc
Z=ρc

Sound will continue to propagate through a medium until an interface is reached where the acoustic impedance of the medium in which the sound is propagating differs from the medium that it encounters. At an interface where an acoustic impedance difference is encountered, a proportion of the ultrasound wave will be reflected back toward the transducer and the rest will be transmitted into the second medium. The simplest case of reflection and transmission occurs when the propagating ultrasound wave is perpendicular (90 degrees) to the interface ( Fig. 1.2 ). In this case the percentage of the incident beam that is reflected is as follows:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-8-Frame class=MathJax style="POSITION: relative" data-mathml='%reflected=(Z2−Z1Z2+Z1)2×100′>%reflected=(?2?1?2+?1)2×100%reflected=(Z2−Z1Z2+Z1)2×100
%reflected=(Z2−Z1Z2+Z1)2×100

The percentage of the incident beam that is transmitted is as follows:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-9-Frame class=MathJax style="POSITION: relative" data-mathml='%transmitted=100−%reflected’>%transmitted=100%reflected%transmitted=100−%reflected
%transmitted=100−%reflected

## Refraction

When the incident beam arrives at the interface at an angle other than 90 degrees, the transmitted beam path diverges from the incident beam path because of refraction ( Fig. 1.3 ). The angle at which the transmitted beam propagates is determined by Snell’s law:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-10-Frame class=MathJax style="POSITION: relative" data-mathml='sinφ1sinφ2=c1c2′>sin?1sin?2=?1?2sinφ1sinφ2=c1c2
sinφ1sinφ2=c1c2