The basis of all ultrasound imaging methods is the propagation of sound waves through tissue. Sound waves are mechanical waves. Two common types of mechanical waves are transverse waves and longitudinal waves. For a transverse wave, particle motion is perpendicular to the direction of wave propagation. For a longitudinal wave, particle motion is parallel to the direction of wave propagation. In tissue, transverse waves are attenuated in short distances, converting all of the wave energy into heat. However, ultrasound does not travel through tissue as a transverse wave; it passes through tissue as a longitudinal wave, with regions of tissue compression and regions of tissue decompression. The pressure increase and decrease as an ultrasound wave passes can be several times the atmospheric pressure. Although the motion of the tissue as the ultrasound wave passes is just a few nanometers, the maximum molecular velocities are near 1 cm/s, and the accelerations are nearly 1000 times the acceleration of gravity.

The physical principles of ultrasound transmission can be derived from Newton’s equations using a model of masses (representing molecules) and springs (representing chemical bonds) (Fig. 7.1):

Force = mass × acceleration, and Force = stiffness × compression

When these equations are applied to molecules in a material, the compression force is set equal to the acceleration force:

Molecular force = density × molecular acceleration = stiffness × (compression change with distance)

Ultrasound wave mechanics are derived from this equation. If u is the distance of each molecule from its resting position, y is distance along the direction that the wave is traveling, and t is time, the previous equations become

Molecular force = density × d²u/dt² = stiffness × d²u/dy²

where the second derivatives d²u/dt² and d²u/dy² represent tissue acceleration and tissue distortion, respectively. These equations can be solved by assuming that molecular displacement u is dependent on the variable group (y + C × t) having units of centimeters; C has units of centimeters per second (cm/s). The previous derivatives can then be expressed as

and the equation becomes

Therefore, for any waveshape of u, the equation is correct if C² = (stiffness/density). In this equation, C² is positive, but the value of C can be either positive or negative, and it is the speed of the ultrasound wave.

The meaning of C is further understood by looking at Figure 7.2. The wave has different locations at different times. As time advances (from top to bottom), the wave moves from left to right. The line marked “wave speed” shows a place on the wave near a crest where the value (y + C × t) is constant. In Figure 7.2, C is negative; thus, as time increases, y must also increase to keep (y + C × t) constant. The expression (y + C × t) provides the relationship between advancing time and advancing location, which is speed. The wave period T (the time it takes to change from one peak to the next) and the wavelength λ (the distance it takes to change from one peak to the next) are related by C, the wave speed:

C = (wave length)/(wave period) = λ/T

Wave period (T) is the inverse of frequency (F), so that

C = λ × F

The following equation is also an important relationship to remember for any longitudinal mechanical (sound) wave traveling through a material:

where κ is stiffness and ρ is density. Of course, both stiffness and density depend on temperature; thus, C will vary with temperature.

The speed of sound in typical soft tissue is 1540 m/s or 1.54 mm/µs, whereas the speed of sound in air is 331 m/s. These are determined by the density and stiffness of the materials. Typical sound and ultrasound wavelengths are presented in Table 7.1.

Frequency is measured in hertz, which is an expression of cycles of compression and decompression per second. Medical ultrasound frequencies are measured in millions of cycles per second, or megahertz (MHz). Wavelength is related to the spatial resolution of an ultrasound image. Because your ears are about 15 cm apart, you cannot tell where a 20-Hz sound (15-m wavelength in air) is coming from, but you can identify where a 2-kHz sound (15-cm wavelength in air) is coming from. By using 5-MHz ultrasound in tissue (0.3-mm wavelength), it is possible to resolve objects that are a few millimeters apart.

As a sound pulse (or wave) travels through tissue, zones of high pressure and low pressure are created. The high-pressure regions occur where the molecules are squeezed together, and the low-pressure regions occur where the molecules are spread apart. The pressure elevation or depression from atmospheric pressure is called pressure fluctuation (p). Pressure fluctuation is equal to the stiffness (κ) times du/dy. As a sound pulse travels through tissue, the molecules oscillate in the direction of the sound wave. The instantaneous molecular velocity (v) of the oscillating molecules is du/dt, and because u is dependent on (y + C × t), du/dt = C × du/dy. Therefore, v/C = p/κ, or

Z is called the acoustic impedance of the tissue. In physical terms, impedance is the ratio between the pressure fluctuation that the tissue feels as a wave passes and the molecular velocity of the molecules as the wave passes. Be sure to avoid confusing the molecular velocity of the molecules (v) with the wave speed (C). The molecular velocity (v) oscillates in the positive and negative directions during ultrasound wave passage at the ultrasound frequency; v is greater when the wave intensity is greater. C is the speed at which the wave passes and does not change with wave intensity, as long as the intensity is low. If the intensity is high, the “linear” model of springs and masses in Figure 7.1 does not hold. The “nonlinear” waves at higher intensities cause harmonics, which are the basis of harmonic imaging (discussed later in this chapter).

It is important to remember that impedance (Z), like wave speed (C), is dependent on the density and the stiffness (1/elasticity) of the tissue. Just as wave speed changes with temperature, impedance changes with temperature because temperature affects both stiffness and density.

In addition to the wavelength (in units of distance) and period (in units of time), each wave has two other properties: the amplitude and the phase (Fig. 7.3). All of the information in a wave is encoded in the amplitude and phase. Ultrasound B-mode imaging displays the echo amplitude, and Doppler velocity information is acquired from the phase. These two kinds of information are independent: one can change while the other remains constant.

The amplitude of a sound wave can be measured in many different ways, but each method measures properties experienced by the molecules of the material through which the sound is traveling. Each molecule experiences a pressure fluctuation, which is a displacement back and forth along the direction of wave travel associated with a molecular velocity and acceleration. The displacement, velocity, and acceleration are like the displacement, velocity, and acceleration of a swing or a pendulum.

Frequency and phase are closely related. When a 5-MHz Doppler looks at an approaching blood velocity of 75 cm/s, the Doppler echo has a frequency of 5.005 MHz. The frequency is increased by 0.1% or 1/1000 of the transmit frequency. Another way to think of this is that the phase becomes more advanced with every cycle: for every 1000 cycles, the phase has advanced 1 cycle. After the first 10 cycles, the phase has advanced 1/100 of a cycle. Because one cycle is 360 degrees, after 10 cycles, the phase has advanced 3.6 degrees, and after 100 cycles, the phase has advanced 36 degrees. It can be convenient to think about the Doppler shift as a continuing change in phase rather than a frequency shift.

Molecular velocity, displacement, acceleration, and pressure are related to ultrasound intensity. These four measured values are ways to look at the mechanical “shaking” that the molecules experience as an ultrasound wave passes through the tissue. First, ultrasound intensity is related to energy:

Energy = force × distance

Power = energy/time = force × distance/time

Intensity = power/area = force/area × distance/time = pressure fluctuation × molecular velocity

As discussed earlier, the pressure fluctuation is the fluctuation of the pressure in the tissue due to the passing of an ultrasound wave; the molecular velocity is the velocity of molecules in the tissue due to the passing of the ultrasound wave.

As shown previously, in the solution to Newton’s equations, the ratio of tissue pressure fluctuation to molecular velocity fluctuation represents the tissue impedance (Z):

Z = (pressure fluctuation)/(molecular velocity)

By substitution:

Intensity = (pressure fluctuation)²/Z = Z × (molecular velocity)²

If the ultrasound is continuous wave (CW), the average intensity is the average of the sine wave amplitude squared. That average is half of the squared maximum value:

Average intensity = (maximum pressure fluctuation)²/2 × Z = Z/2 × (molecular velocity)²

This is also true within an ultrasound pulse. The temporal peak intensity is computed from this same expression.

A graph of the pressure fluctuations versus CW intensity (Fig. 7.4) indicates that a physical problem occurs within the range of diagnostic ultrasound intensities: at temporal peak intensities greater than 320 mW/cm², the pressure fluctuation becomes greater than 1 atm. Therefore, the minimum pressure theoretically becomes negative. This is impossible according to the physical laws of thermodynamics. Thus, the linear equation for compressibility, which is based on thermodynamics, does not apply for these conditions. The wave becomes nonlinear (Fig. 7.5) and gives rise to harmonics.

With nonlinear mechanics, the linear equations presented earlier no longer apply. The relationship between pressure and displacement, which is based on the nature of molecular bonds, does not hold for large fluctuations in pressure.

The nonlinear relationship between compression and force is shown in Figure 7.6. In the central region, stiffness is linear, but when compression becomes too great or too small, the effective stiffness changes. When the wave intensity is very large, during compression, the stiffness is increased, causing an increase in the wave speed and an increase in the wave impedance. Likewise, during decompression, the stiffness is decreased, causing decreases in both the wave “speed” and “impedance.” Speed and impedance are in quotation marks in these cases because their definitions become less useful in these nonlinear conditions than in the linear conditions.

The flattened valleys and the enhanced peaks of the waveshape in Figure 7.5 (right) can be represented as a combination of sine waves (Fig. 7.7). In this example, a sine wave at the original frequency and a sine wave at two times the original frequency are shown. The wave at the original frequency is called the fundamental; the wave at two times the frequency is called a harmonic. The wave at two times the fundamental frequency is called by different names: it is called the “first harmonic” by some people and the “second harmonic” by others. Both groups agree on the name “first overtone” for that frequency. Harmonics occur at two times, three times, or any integer multiple of the fundamental. Any periodic (repeating) waveshape can be formed from a series of sine wave harmonics of the fundamental by selecting the phase and the amplitude of each harmonic. This is the Fourier theorem and is the basis of the Fourier transform. The lower curve in Figure 7.7 shows the combination of the fundamental and the harmonic.

Harmonics are present in all diagnostic ultrasound echoes. If the transmitted intensity is increased, the intensity of the harmonics increases as more of the power from the fundamental frequency is converted to the harmonics. The attenuation of ultrasound is proportional to frequency; thus, the harmonics are attenuated more rapidly than the fundamental. Some of that attenuation is due to absorption (conversion to heat), so that if the ultrasound intensity is doubled, the amount of ultrasound converted to heat more than doubles because of the conversion to harmonics and the subsequent conversion to heat.

The subject of harmonics has been recognized for decades but was not often discussed until recently, and harmonic displays are relatively recent additions to diagnostic ultrasound instruments. Interest in the display of harmonics is primarily a result of a general desire to improve detection of ultrasound contrast agents. When ultrasound contrast agents were introduced, they were designed to increase the strength of the echo signals in ultrasound images. However, they did not produce the strong echoes on images that were hoped for. This led to the development of methods to display harmonics. The bubbles in ultrasound contrast agents will change the linear portions of the line in Figure 7.6 and cause the generation of harmonics. Thus, displaying harmonic echoes was expected to show contrast agents more prominently. However, tissues without contrast agents also reflect harmonic echoes back to the transducer (Fig. 7.8). In general, tissue harmonic imaging improves the lateral resolution and image contrast.

To generate a harmonic image using a 3-MHz transducer, it is logical to transmit at 3 MHz and receive at 6 MHz, but this cannot be done because a transducer is not sensitive at even multiples of the frequency. The 3-MHz transducer must have a “damping” material to make it “broadband,” so that it will operate between 1.5 and 4.5 MHz. Then, for harmonic imaging, it is possible to reduce the transmit frequency to 2 MHz and increase the receive frequency to select 4-MHz echoes to generate the harmonic image.

There is a great deal of confusion in ultrasound physics literature about power and intensity. Here is the reason for the confusion. Power is a measure of energy per time and has units of watts. As an ultrasonic wave passes through tissue, the power
is distributed over the cross-sectional area of the ultrasound beam pattern. Power divided by the cross-sectional area is equal to the intensity. The intensity is easily measured with an ultrasound transducer called a hydrophone and determines whether the ultrasound propagation is linear or nonlinear. Intensity is related to pressure fluctuation and is therefore the quantity usually discussed in ultrasound physics. Intensity is dependent on both the ultrasound power in the beam pattern and the cross-sectional area of the beam pattern. Changes in intensity due to changes in cross-sectional area of the beam pattern can easily be confused with changes in intensity due to changes in ultrasound power. It is important to keep the two factors separate.

In tissue, ultrasound power decreases with distance as the wave propagates. The decrease in power is due to attenuation in the tissue. Attenuation has two factors: (1) conversion of the ultrasound power to heat (absorption) and (2) scattering of the ultrasound power in directions other than the direction of the ultrasound beam. The attenuation (both absorption and scattering) of ultrasound in tissue is dependent on the tissue type and the ultrasound frequency.

As an ultrasound wave passes farther and farther into tissue, the energy in the ultrasound pulse decreases because of the conversion of that energy into other forms of energy, including heat, and into scattered ultrasound. Conversion can also occur from one ultrasound frequency into another, forming harmonics of the fundamental frequency. Attenuation computations can be easily understood by using the concept of the half-value layer (or half-energy layer). A half-value layer is a layer of tissue thick enough to convert half of the energy in an incoming ultrasound pulse into heat and scattered ultrasound, leaving the remaining half of the ultrasound energy in the pulse as it passes out of the layer. All other attenuation computations can be derived from this concept. Each tissue type has an attenuation rate that can be expressed as a half-value layer (Table 7.4). For most tissues, attenuation increases with ultrasound frequency, so the half-value layer for 2-MHz ultrasound is half as thick as a half-value layer for 1-MHz ultrasound. The wavelength of 2-MHz ultrasound is half as long as the wavelength of 1-MHz ultrasound; therefore, it is most convenient to express the thickness of the half-value layer in wavelengths.