17.7 Ultrasound

Ultrasound Fundamentals

Ultrasound uses sound waves with frequencies above the range of human hearing (typically above 20 kHz) to create images and measure motion inside the body. When these waves encounter boundaries between different tissues, some of the energy bounces back. By analyzing those reflections, we can build detailed pictures of internal structures without any incisions or radiation exposure.

Three core concepts drive how ultrasound works: acoustic impedance, the pulse-echo technique, and the Doppler effect. Together, they explain how images are formed and how motion (like blood flow) gets measured.

Acoustic Impedance in Ultrasound

Acoustic impedance ($Z$) describes how much a material resists the passage of sound waves. It's calculated as the product of the material's density ($\rho$) and the speed of sound in that material ($v$):

$Z = \rho v$

Materials with high acoustic impedance, like bone, reflect a large fraction of incoming sound energy. Materials with low acoustic impedance, like air, transmit sound more easily but create a huge mismatch with body tissues. That mismatch is actually the key to imaging.

When a sound wave hits the boundary between two materials with different acoustic impedances, part of the wave reflects back. The bigger the impedance difference, the stronger the reflection. For example, the soft tissue-to-bone interface produces a strong echo because bone's impedance is much higher than soft tissue's.

This is also why ultrasound gel is applied to the skin before scanning. Air between the transducer and skin would create a massive impedance mismatch, reflecting almost all the sound before it enters the body. The gel acts as an impedance matching layer, minimizing that surface reflection so more energy passes into the tissue.

Ultrasound Wave Properties and Interactions

A few wave properties matter most for understanding ultrasound:

• Frequency: The number of wave cycles per second, measured in Hertz (Hz). Medical ultrasound typically operates between 1 and 20 MHz. Higher frequencies give better image resolution but don't penetrate as deeply.
• Wavelength: The distance between consecutive wave peaks. Wavelength is inversely related to frequency, so higher-frequency ultrasound has shorter wavelengths and can resolve finer details.
• Intensity: The amount of energy the wave carries per unit area per unit time.

When ultrasound waves travel through the body, several interactions occur:

• Reflection: Waves bounce off interfaces between tissues with different acoustic impedances. This is the primary mechanism behind image formation.
• Refraction: Waves change direction when passing between materials where sound travels at different speeds, similar to how light bends through a lens.
• Scattering: Small structures or irregularities (like red blood cells) disperse sound waves in multiple directions. Scattering is what makes Doppler blood flow measurements possible.

Pulse-Echo Technique

The pulse-echo technique is the fundamental method behind ultrasound imaging. Here's how it works:

1. A transducer emits a short burst (pulse) of ultrasound waves into the body.
2. The waves travel through tissue until they hit a boundary between materials with different acoustic impedances.
3. Some of the wave energy reflects back toward the transducer as an echo.
4. The transducer detects the returning echo and records two things: the time delay (how long the echo took to return) and the amplitude (how strong the echo is).
5. The time delay tells you the distance to the reflecting surface, and the amplitude tells you how large the impedance mismatch is at that boundary.

By sending many pulses in slightly different directions and collecting all the echoes, the system assembles a 2D image of internal structures.

Ultrasound Calculations and Applications

Calculation of Acoustic Impedance

The formula $Z = \rho v$ is straightforward to use once you know the material's density and sound speed.

• Density ($\rho$): mass per unit volume, in kg/m³
• Sound speed ($v$): depends on the material's stiffness and compressibility, in m/s

Example: Calculate the acoustic impedance of water, given a density of 1000 kg/m³ and a sound speed of 1480 m/s.

$Z = \rho v = (1000 \text{ kg/m}^3)(1480 \text{ m/s}) = 1.48 \times 10^6 \text{ kg/(m}^2\text{·s)}$

The unit kg/(m²·s) is sometimes called the rayl. For comparison, soft tissue has an impedance of about $1.63 \times 10^6$ rayl, while bone is around $7.8 \times 10^6$ rayl. That large difference is why bone boundaries show up so clearly on ultrasound.

Doppler Effect for Velocity Measurement

The Doppler effect is the shift in frequency that occurs when a sound source and a receiver are moving relative to each other. You've experienced this with a passing ambulance siren: the pitch sounds higher as it approaches and lower as it moves away.

In ultrasound, the transducer sends a wave at a known frequency toward moving structures (most commonly red blood cells). The wave reflects off those moving cells and returns at a slightly different frequency. That frequency difference is the Doppler shift:

$\Delta f = f_r - f_t$

where $f_r$ is the received frequency and $f_t$ is the transmitted frequency. A positive $\Delta f$ means the blood is flowing toward the transducer; a negative value means it's flowing away.

The velocity of the moving object is then:

$v = \frac{c \, \Delta f}{2 f_t \cos \theta}$

• $c$ = speed of sound in the medium (about 1540 m/s in soft tissue)
• $\theta$ = angle between the ultrasound beam and the direction of motion
• The factor of 2 appears because the sound makes a round trip: it travels to the moving object and then back

Notice that when $\theta = 90°$, $\cos 90° = 0$, which makes the equation blow up. Physically, this means if the ultrasound beam is perpendicular to the blood flow, there's no Doppler shift at all. Sonographers aim for angles below 60° to get reliable velocity readings.

Example: Find the blood cell velocity given a transmitted frequency of 5 MHz, a received frequency of 5.1 MHz, sound speed in tissue of 1540 m/s, and a beam-to-flow angle of 45°.

1. Calculate the Doppler shift: $\Delta f = 5.1 \text{ MHz} - 5.0 \text{ MHz} = 0.1 \text{ MHz} = 100{,}000 \text{ Hz}$

2. Plug into the velocity equation: $v = \frac{(1540 \text{ m/s})(100{,}000 \text{ Hz})}{2(5{,}000{,}000 \text{ Hz})(\cos 45°)}$

3. Evaluate: $v = \frac{1.54 \times 10^8}{2(5 \times 10^6)(0.707)} = \frac{1.54 \times 10^8}{7.07 \times 10^6} \approx 21.8 \text{ m/s}$

Medical and Industrial Applications

Medical applications span both diagnosis and treatment:

• Diagnostic imaging visualizes internal structures in real time. In obstetrics, it tracks fetal development and screens for abnormalities. In cardiology, it evaluates heart valve function and measures cardiac output. Musculoskeletal ultrasound identifies tears, inflammation, and other soft tissue injuries.
• Therapeutic ultrasound uses sound energy to produce physical effects in tissue. Physiotherapy applications promote healing in conditions like tendinitis and muscle strains by gently warming deep tissue. High-intensity focused ultrasound (HIFU) concentrates enough energy at a focal point to destroy targeted tissue, and it's used to treat tumors in organs like the prostate and uterus without surgery.

Industrial applications rely on the same pulse-echo and Doppler principles:

• Non-destructive testing (NDT) inspects materials for internal defects like cracks, voids, or inclusions in welds and castings, all without damaging the part. It's also used for thickness measurements and corrosion monitoring in pipelines and storage tanks.
• Ultrasonic cleaning uses high-frequency vibrations to dislodge contaminants from surfaces, commonly used for jewelry and surgical instruments.
• Ultrasonic welding joins plastics and some metals using localized high-frequency vibrations, widely used in automotive manufacturing and packaging.
• Flow metering measures fluid flow rates in pipes using Doppler or transit-time methods, applied to water, oil, and gas systems.