Ultrasound Physics and Instrumentation Notes

Waves and Types

Waves are disturbances that propagate through a medium or space. They can be classified as mechanical waves or electromagnetic waves. Mechanical waves require a medium to travel through, whereas electromagnetic waves can propagate without a material medium. In the ultrasound context, we deal with mechanical waves in tissue, collectively referred to as ultrasound when the frequency is above human hearing.

Mechanical Waves and Ultrasound

Mechanical waves include infrasound, sound, and ultrasound. Ultrasound refers to frequencies higher than the upper limit of human hearing. The propagation of these waves arises from periodic changes in pressure as vibrating molecules interact with neighboring molecules and transfer energy from the source to distant regions.

Wave Propagation

Propagation is the transit of a wave from the source to distant regions. It is the process by which the periodic pressure changes are conveyed through the medium as the wave travels. The speed and behavior of propagation depend on the properties of the medium and the wave.

Sound and Basic Properties

Sound originates from a source generating pressure variations that cause particle density fluctuations as the wave travels through a medium. These properties—amplitude, wavelength, frequency, and acoustic velocity—describe the character of the sound wave and its interaction with tissue.

Amplitude, Wavelength, Frequency, and Velocity

Amplitude describes the maximum displacement of particles in the medium. Wavelength (λ) is the spatial distance between successive identical points of the wave. Frequency (f) is the number of cycles per unit time. Acoustic velocity (c) is the speed at which the wave propagates through the medium.

Amplitude, Wavelength, and Period

The amplitude and wavelength are fundamental spatial characteristics of the wave, while the period (T) is the time for one complete cycle. The period and frequency are inverses of each other, with the relationship $T = rac{1}{f}.$ In the diagrams,

• Amplitude corresponds to the height of the wave

• Wavelength (λ) corresponds to the distance between successive peaks

• The period (T) corresponds to the time interval for one cycle

Frequency and its Meaning

Frequency is the number of cycles per second. It is the reciprocal of the period: $f = rac{1}{T}.$ This reflects how often the wave repeats at a given point per unit time.

Acoustic Velocity

The speed at which a wave propagates through a medium is termed the acoustic velocity, denoted by $c.$ It depends on the medium’s properties (density, elasticity, etc.).

Relationship: C = fλ

The fundamental relationship connecting velocity, frequency, and wavelength is $c = f \, \lambda.$ This equation implies that increasing frequency while keeping velocity constant reduces wavelength, and vice versa.

Propagation Parameters

Key propagation parameters include the Period (T), Wavelength (λ), and Acoustic Velocity (c), with the general relation $c = f\lambda.$ The wavelength is typically expressed in units such as millimeters, and the period in seconds. These parameters determine how the wave travels through a medium under varying pressure conditions (high and low pressure phases).

Interactions of Ultrasound with Tissue

Ultrasound waves interact with tissue in several ways, including reflection, transmission, scattering, absorption, and interference. These interactions influence image formation and energy deposition in tissue.

Reflection

Reflection occurs at boundaries where impedance (or acoustic properties) change between two media. At a boundary, part of the incident wave is reflected, part may be transmitted into the second medium, and part can be reflected back from subsequent interfaces. In diagrams, the incident wave approaches the interface, with reflected and transmitted waves emerging into the respective media.

Specular Reflectors

Specular reflectors are large, smooth interfaces where reflection is predominantly in a single direction. Examples include the diaphragm and pericardium. These interfaces reflect energy in a predictable, mirror-like manner and are responsible for well-defined echoes in ultrasound images.

Acoustic Impedance

Acoustic impedance is a measure of resistance to sound passing through a medium. It is analogous to electrical resistance in circuits. While not given by a single universal formula in the notes, impedance is the property that governs how much of the incident wave is reflected versus transmitted at an interface.

Impedance Mismatch

When the impedances of adjacent media differ, part of the ultrasound wave is reflected and part is transmitted at the boundary. The typical representation shows a sound source, an incident wave, a transmitted wave, and a reflected wave across two media.

The Fetal Head Sonogram (Illustrative)

The slide references a sonogram of the fetal head, illustrating how ultrasound interacts with tissue interfaces to create an image. This serves as a real-world example of reflection, transmission, and scattering producing an interpretable image.

Percent of Reflection

The percentage of reflected energy at an interface is given by the reflection coefficient (as a percentage):
$R\% = \left( \frac{Z<em>2 - Z</em>1}{Z<em>2 + Z</em>1} \right)^2 \times 100.$
Total reflection occurs when the impedance of medium 2 is much greater than that of medium 1 (i.e., $Z<em>2 \gg Z</em>1$), in which case $R\% \approx 100\%.$

Bone-Soft Tissue Interface

At a bone-soft tissue interface, the impedance difference is large, producing high reflection and low transmission. This leads to poor penetration behind bone due to the strong reflections at the interface.

Air Layer Between Transducer and Patient

Even an extremely thin air layer at the transducer-tissue interface causes nearly total reflection (air-tissue interface). This is because air has a very different impedance compared with tissue, drastically reducing transmitted energy.

Coupling Gel

Coupling gel serves two primary purposes: (1) to eliminate air bubbles at interfaces, reducing spurious reflections, and (2) to reduce friction to facilitate smooth contact between the transducer and the skin, improving coupling.

Interface Reflection Percent (Representative Values)

Interface reflection percentages (soft tissue–air, soft tissue–lung, soft tissue–bone, aqueous humor–lens, fat–liver, soft tissue–fat) are listed as:

• Soft tissue–air: 99.9%

• Soft tissue–lung: 52%

• Soft tissue–bone: 43%

• Aqueous humor–lens: 1:1

• Fat–liver: 0.79

• Soft tissue–fat: 0.69
  These values illustrate how different tissue interfaces reflect energy to varying degrees.

Scattering

Scattering refers to non-specular reflection due to internal textures within organs. It occurs at small interfaces and the interface acts as a new, separate sound source. This contributes to the texture seen in ultrasound images rather than a single mirror-like reflection.

Sound Source, Transmitted Wave, and Scattered Wave

In scattering, a point or small source can give rise to both transmitted and scattered waves. The scattered wave emanates from the interface and contributes to image formation through secondary wavefronts.

Interference

Interference involves the superposition of waves from different sources (A, B, etc.). It can produce constructive or destructive interference, altering the resultant wave intensity pattern. The figures show combinations of A and B waves and their interference outcomes.

Absorption

Absorption is the process by which sound energy is dissipated within the medium, converting energy to other forms such as heat. Absorption is a key mechanism by which ultrasound energy is attenuated as it propagates. It can have therapeutic applications in physiotherapy when controlled.

Factors Influencing Absorption

The absorption of ultrasound is influenced by:

• The beam’s frequency

• The tissue’s viscosity

• The relaxation time of the medium
  Higher frequency generally leads to greater absorption.

Attenuation: Combined Effects

Attenuation in tissue comprises scattering and absorption effects as the wave propagates. The total attenuation is the sum of the contributions from scattering and absorption.

Mathematical Description of Attenuation

A common description of attenuation with distance is:
$A(x) = A<em>0 \, e^{-\alpha x},$ where $A(x)$ is the amplitude after traveling a distance $x$, and $\alpha$ is the attenuation coefficient. The total attenuation coefficient is the sum of the individual contributions: $\alpha = \alpha</em>s + \alpha<em>a,$ where $\alpha</em>s$ is the scattering coefficient and $\alpha_a$ is the absorption coefficient.

Attenuation in Human Tissue (1 MHz)

Attenuation coefficients (in nepers per centimeter, Np/cm) at 1 MHz for various tissues are approximately:

• Blood: 0.021

• Fat: 0.069

• Kidney: 0.115

• Muscle (across fibers): 0.380

• Muscle (along fibers): 0.138

• Brain: 0.098

• Liver: 0.103

• Lung: 4.6

• Skull: 2.3
  These values illustrate substantial variation in attenuation among tissues, with lung and skull showing markedly higher attenuation at 1 MHz.

Frequency Dependence of Attenuation and Practical Imaging Implications

As frequency increases, attenuation increases. This has diagnostic implications: for imaging deep organs, use low-frequency transducers to maximize penetration; for imaging shallow structures, high-frequency probes provide better resolution due to shorter wavelengths and higher spatial fidelity.

Intensity

Intensity is the physical parameter describing the amount of energy flowing through a cross-sectional area per unit time. It quantifies the rate at which energy is transmitted by the wave through a small area and is a key metric in both diagnostic imaging and safety considerations.

Biological Effects and Intensity

The study of potential biological effects is linked to intensity. Higher-intensity ultrasound waves impart greater particle velocity and displacement, making high-intensity ultrasound more disruptive to living systems than low-intensity ultrasound. This underpins safety guidelines and regulatory considerations in clinical ultrasound.

Attenuation and Distance: A Note on Intensity Loss

As waves propagate, intensity diminishes due to attenuation (including absorption and scattering). In practical terms, energy delivered to deeper tissues is reduced, which is why the choice of frequency balances penetration depth with image resolution and safety.

Echo-Ranging Technique (Time-of-Flight)

A fundamental principle of ultrasound imaging is the time-of-flight of the pulse. If a pulse travels to a target at depth d and back to the transducer, the round-trip time is:
$t = \frac{2d}{c},$
where $c$ is the sound speed in the medium. A common tissue speed is approximately $c \approx 1540\ \mathrm{m/s}.$ The echo time is used to estimate depth and generate the image.

Example: Depth-Resolution Timing

Consider a distance of 1 cm (0.01 m) in tissue. The round-trip time is approximately:
$t = \frac{2 \times 0.01\ \mathrm{m}}{1540\ \mathrm{m/s}} \approx 1.30 \times 10^{-5}\ \mathrm{s} = 13\ \mu\mathrm{s}.$
Thus, each centimeter of one-way travel corresponds to about $6.5\ \mu\mathrm{s}$ of round-trip time in tissue, which matches the practical timing estimates used in clinical ultrasound exam timing.

Summary of Key Relationships and Concepts

• Wave types: mechanical (including ultrasound) and electromagnetic; ultrasound is a high-frequency mechanical wave.

• Fundamental relationships: $c = f\lambda$ and $f = \frac{1}{T}.$

• Reflection and transmission at interfaces depend on impedance differences; high impedance mismatch yields stronger reflections (and potentially near-total reflection when one medium is air).

• Specular reflectors produce clear, directional reflections; scattering and interference contribute to image texture and signal characteristics.

• Absorption and scattering cause attenuation; the overall attenuation follows $A(x) = A<em>0 e^{-\alpha x}$ with $\alpha = \alpha</em>s + \alpha_a.$

• Attenuation increases with frequency; imaging depth vs. resolution is a trade-off guiding transducer choice.

• Time-of-flight (echo ranging) uses the relation $t = \frac{2d}{c}$ to determine depth, with typical tissue speeds around $c \approx 1540\ \mathrm{m/s}.$