Physics

The physical principles behind neurovascular ultrasound — sound waves, transducers, the Doppler effect, hemodynamics, and the key equations you need at the machine.

Comprehensive neurosonology reference covering the physics of ultrasound, transducer design, Doppler principles, and the clinical application of carotid/vertebral duplex and transcranial Doppler (TCD), including velocity criteria for stenosis and vasospasm.

Physics of Sound

Sound Basics

• Audible sound is 20–20,000 Hz. Ultrasound is > 20 KHz.
• Acoustic variables: pressure – density – distance.
• Sound is a longitudinal wave; particles move in the same direction as the wave.

The 7 Parameters of Sound Waves

• Period: time it takes for one cycle. Determined by source, not affected by medium, cannot be changed by sonographer.
• Frequency: number of waves per second. Determined by source, not affected by medium, cannot be changed by sonographer. Typically ranges between 1–10 MHz. (Period × frequency = 1; period ∝ 1/frequency.)
• Wavelength: length of one wave in space. Determined by both source and medium; cannot be changed by sonographer. Typically ranges from 0.1–0.8 mm in soft tissue. Wavelength = propagation speed (mm/µs) / frequency (MHz); wavelength ∝ 1/frequency.
• Propagation speed: speed of sound propagation in a medium (units mm/µs or m/s). A pure property of the medium, not related to the sound source and not changeable by the sonographer. Determined by medium density and stiffness (propagation speed ∝ stiffness/density — "S&S → same direction").
  • Propagation speed in soft tissue is 1540 m/s = 1.5 mm/µs.
  • Propagation speed in air is 330 m/s.
  • Compressibility is the opposite of stiffness, hence opposite of speed.
  • Gases are not stiff → low speed; bone is stiff and not very dense → high speed.
• Amplitude, power and intensity: all are measures of the strength of sound waves, initially determined by the sound source, not affected by medium, decrease as sound propagates, and all can be changed by the sonographer.
• Amplitude: strength of sound wave; the difference between the average (not minimum) and maximum value of the acoustic variable (pressure or density). Interference of waves: in-phase waves augment (constructive interference) and out-of-phase waves reduce (destructive interference).
• Power: also strength of sound wave, units are watts. Power ∝ (amplitude)² — power increases 9 times when amplitude triples.
• Intensity: concentration of power in a sound beam, units watts/cm². Intensity = power / beam area (when power doubles, intensity doubles); intensity ∝ (amplitude)² (if amplitude is quartered, intensity & power are reduced to 1/16).

The 5 Parameters of Pulsed Ultrasound

• Pulse duration: time when the pulse is on; determined only by source, cannot be changed. Related to the number of cycles and periods of waves in the pulse. A short pulse has few cycles and short periods (higher frequency). Shorter pulses produce higher quality images. (Ultrasound pulses are 2–3 cycles; Doppler pulses are 5–30 cycles.)
• Pulse repetition period (PRP): time from the start of one pulse to the start of the next pulse. Can be changed by the sonographer (changing listening time) while adjusting depth. Typically ranges from 100s of µs to 1 ms. PRP = pulse duration + listening duration.
• Pulse repetition frequency (PRF): number of pulses in a second. Also can be changed by adjusting depth. (↑ depth → ↑ PRP and ↓ PRF.)
• Duty factor: percentage of time when the pulse is on; no units, ranges from 0 to 1 or 0% to 100%. Also changed by adjusting depth. Typically ranges from 0.001 to 0.01, and is 1 in continuous-wave ultrasound. Duty factor = (pulse duration / PRP) × 100.
• Spatial pulse length (SPL): length of the pulse in space (when the pulse is on). Related to source (cycles per pulse) and medium (wavelength). SPL = waves per pulse × wavelength.

Intensity

• Intensity is not the same throughout the wave; it varies by time and location in the wave.
  • Temporal peak: maximum intensity at any point in time.
  • Temporal average: average throughout all the time (transmitting and receiving).
• Spatial × temporal combinations: SPTP (spatial peak temporal peak), SATP (spatial average temporal peak), SPTA (spatial peak temporal average), SATA (spatial average temporal average).
• Pulse-wave ultrasound is about time — holding and beaming — so another variable was introduced, resulting in two more measurements:
  • Pulse average: average for only when the pulse is on.
  • SPPA (spatial peak pulse average) and SAPA (spatial average pulse average).
• Pulse average (PA) is always higher than temporal average (TA), because TA includes off time.
• SAPA value is between SATP and SATA. SPPA value is between SPTP and SPTA.
• Im is the average intensity during the most intense half cycle. Typically similar to SPTP (highest intensity).
• Beam Uniformity Coefficient (SP/SA factor): ratio that describes how uniform a wave is. Minimum is 1 (when both SP and SA are the same) with unlimited maximum. SP/SA factor = spatial peak intensity / spatial average intensity.
• Converting intensities:
  • To convert spatial intensity, use the SP/SA factor: SA = SP/BUC → SATA = SPTA / (SP/SA).
  • To convert temporal intensity, use the duty factor: TA = PA × duty factor → SPTA = SPPA × duty factor.

Amplitude (Decibels)

• When expressing the value for a variable with an extremely wide range, we use logarithms (10 = log 1; 1,000,000 = log 6; 1 billion = log 9).
• A decibel is not an actual value; rather it is used to express the change in value using logs. If above 10 dB, each 0 means a 10-times increase. If below 10 dB, each 3 dB means doubled one time:
  • 3 dB = double; 6 dB = double double (4 times); 9 dB = doubled 3 times (8 times).
  • 10 dB = 10-times increase; 20 dB = 100 times; 30 dB = 1000 times.

Attenuation

• Depends on distance traveled and frequency of sound.
• The greater the frequency, the more the attenuation.
• We may choose a lower frequency transducer to image deeper structures (same for TCD).
• Attenuation occurs through different mechanisms:
  • Absorption: directly related to frequency (increases with frequency).
  • Reflection: occurs at boundaries of media with different impedance.
    • Specular reflection: if the boundary is smooth.
    • Scattering: if the boundary is irregular, smaller than the wavelength. It increases with increased frequency.
    • Rayleigh scattering: waves scatter in all directions when the reflector is much smaller than the wavelength. Occurs with red blood cells.
• Attenuation coefficient: amount of attenuation per cm of tissue, units dB/cm.
  • The higher the frequency, the higher the attenuation coefficient.
  • Typically for soft tissue it is about half the frequency. If a 5 MHz transducer, the attenuation coefficient will be 2.5 dB/cm.
  • Attenuation coefficient is higher in bone (absorber) and lungs (scatterer).
• Total attenuation = attenuation coefficient × distance travelled.
• Penetration depth (half value layer thickness): the thickness at which half of intensity is lost (attenuation is −3 dB). Typically between 0.3–1 cm. Penetration depth = 3 / attenuation coefficient (dB/cm).

Impedance

• Resistance to sound traveling through the medium. A characteristic of the medium only, represented by the letter Z, units are Rayls. Typically 1.25–1.75 MRayls (million).
  • Impedance = density × propagation speed.
  • Impedance is high in bone (dense and high propagation speed).
• Used to determine reflection. Reflection occurs only between media of different impedance.

Reflection

• Types of reflection:
  • Specular reflection: occurs at smooth surfaces.
  • Non-specular reflection (scattering): when the surface is irregular with irregularities smaller than the wavelength.
  • Rayleigh scattering: when sound is scattered in all directions; occurs with very small particles, smaller than the wavelength, such as RBCs and intracellular particles.
• Types of incidence:
  • Normal incidence: sound strikes the surface at 90°.
  • Oblique incidence: sound strikes the surface at an acute or obtuse angle.
• Intensity reflection coefficient (IRC): percent of sound waves that is reflected.
• Intensity transmission coefficient (ITC): percent of sound waves that pass through. IRC + ITC always = 1.
• Reflection at normal incidence depends on impedance; strong reflection occurs between soft tissue and bone, near-total between soft tissue and air. IRC = ((Z2 − Z1) / (Z2 + Z1))².
• Reflection at oblique incidence is very complex; we cannot predict if reflection will occur.

Refraction

• Change of direction of sound waves when passing from one medium to another.
• Occurs only when there is oblique incidence and different propagation speeds.
• Snell’s law: sine of transmission angle / sine of incidence angle = propagation speed of medium 2 / propagation speed of medium 1.
• If medium 2 speed > medium 1, the transmission angle > incidence angle.

Range Equation

• Process of calculating depth based on time-of-flight.
• Sound travels 1 inch in 16 µs, 1 cm in 7.5 µs; goes and returns from 1 cm in 13 µs.
• Depth (mm) = velocity × (time-of-flight / 2).
• Depth for soft tissue = 0.77 × time-of-flight. (1 cm = 13 µs → 13 × 0.77 = 10 mm.)

Physics of Transducers

Anatomy

• Active element: a ferroelectric (piezoelectric) material that vibrates when an electric current passes through it (piezoelectric effect).
  • Curie temperature: the temperature at which ferroelectric materials lose their piezoelectric properties.
  • Types of active elements: natural (quartz, tourmaline); artificial (PZT – lead zirconate titanate, lead metaniobate, barium titanate).
• Damping material: material attached to the back of the active element to shorten its vibration. Not used in continuous-wave Dopplers (they don’t use pulses). Effects:
  • Shorter pulse duration.
  • Shorter pulse length → increased picture quality (improves resolution).
  • Increases bandwidth (adds a range of frequencies) → decreases quality factor.
  • Reduces amplitude, reduces efficiency and sensitivity to see weak echoes.
• Matching layer: a layer between the active element and skin; aims to make a gradual step-wise decrease in impedance while sound moves towards the skin to avoid reflection. It has an impedance between that of PZT and the skin. Gel has the same function, with an impedance between matching layer and skin.

Transducer Frequency

• Continuous wave: frequency is the same as the electrical frequency used.
• Pulsed ultrasound:
  • PRF equals the frequency of electrical spikes that reach the PZT.
  • Frequency is determined by the PZT itself — the thickness and propagation speed in the crystal: PZT frequency in pulsed US = propagation speed (mm/µs) / (thickness (mm) × 2). The thinner the crystal and the more speed, the higher the frequency.

Transducer Bandwidth

• PZT produces one frequency, then damping material adds a band of frequencies.
• Bandwidth is the difference between the lowest and highest frequencies. If a 4 MHz transducer has a range of 2–5 MHz, bandwidth is 3 MHz.
• If a 5 MHz transducer has a bandwidth of 2.5, it can’t work at 3 or 7 MHz (range is 3.75–6.25).
• Quality factor is used to express how pure the sound is: quality factor = resonant frequency / bandwidth.
• Diagnostic ultrasounds have a lower quality factor compared with therapeutic ultrasound.

Transducer Sound Beam

• One may think of sound spreading away in all directions once it is produced by the sound source (diffraction).
• In transducers this doesn’t happen; instead it makes an hour-glass pattern of sound beam, because the tiny particles of PZT produce wavelets that interact with each other (destructive and constructive interference) producing this shape.
• Focus: the location where the beam is narrowest.
• Focal length: distance between transducer and focus.
  • Depends on PZT diameter (or transducer aperture size) and frequency.
  • Larger crystal diameter and higher frequency (thin crystals) produce deeper focus.
  • For unfocused transducers, at 1 focal length the beam is half the transducer diameter; at 2 focal lengths it is the same as the transducer diameter.
• Near zone (Fresnel): area between transducer and focus.
• Far zone: area beyond the focus.
• Divergence: at deeper distances the beam will diverge; divergence is more pronounced with smaller diameter crystals.
• Crystal diameter: large diameter causes deeper focus and less divergence in the far field.
• Crystal thickness: the thinner, the higher the frequency and deeper the focus.

Transducer Resolution

• The shorter and narrower the pulse, the better the resolution.
• Longitudinal resolution (axial, radial, depth): ability to distinguish two structures that are in line with the sound wave direction (front and back) as separate. The shorter the pulse, the better (smaller) the longitudinal resolution. Longitudinal resolution = spatial pulse length / 2. Depends on sound source and medium.
• Lateral resolution (transverse, angular): ability to distinguish two structures that are side by side as separate. The smaller the beam, the better (smaller) the lateral resolution. Lateral resolution = beam diameter.
• Temporal resolution: ability to determine position of moving structures. Depends on frame rate. A higher frame rate is limited by image resolution (number of lines), whether multi-focus is used (for better resolution at different depths), and the depth of the image.
• Typically the longitudinal resolution is better than the lateral resolution.

Scan Modes

• A mode: plots depth against amplitude. Looks like Manhattan skylines.
• B mode: plots depth and uses a gray scale to show amplitude. The usual gray-scale picture.
• C mode: constant depth mode; uses only a specific gate of specific depth. Produces a slice of a particular depth.
• M mode: motion mode — the only mode that plots motion.

Types of Transducers

• Steering: angling the sound beam to make it at an angle; used for Doppler and for scanning larger sections than the aperture size.
• Focusing: adjusting the focus to adapt to a new depth.
• Sequential: no steering and fixed focus.
• Mechanical: the array can be tilted mechanically.
• Phased array: can do both (steering and focusing) electronically.

• Linear: sequential transducer, fixed focus, good for the near-field. Rectangular images.
• Curved: sequential transducer, fixed focus, good for the far-field (abdominal, transvaginal, rectal). Blunted sector-shaped images.
• Phased array: steering is made electronically, focus can be adjusted; has a small footprint, poor near-field resolution, best used for cardiac, abdominal and brain. Sector-shaped images.
• Vector array: combination of phased and linear sequential. Picture is trapezoid shaped.

Receiver Actions

• Amplification (receiver gain or overall gain): too little — some signals will be dropped; too much (saturated) — all signals are bright.
• Compensation (time gain compensation or swept gain compensation): the same reflectors at different depths may show different signals due to more attenuation with deeper reflectors. Compensation will boost deeper structures to have the same intensity (brightness). Problems with compensation affect deeper structures.
• Rejection (threshold): rejects very low-noise signals.
• Compression: squeezes the signals into a smaller range to be shown on the image.

Artifacts

• Acoustic speckle: tissue appears more granular. Caused by interaction between the ultrasound wavelet and scattering of ultrasound within the tissue.
• Slice thickness: causes small cysts to appear as filled-in or solid. The beam is thick and will include tissues above and below the cyst, causing it to appear gray, not black, from inside. To overcome, decrease beam thickness by activating tissue harmonic imaging.
• Refraction: structure appears at a different location.
• Reverberation: sound is trapped between two surfaces causing multiple reflections; multiple equally spaced reflections due to sound trapped between two reflectors. Only the first two reflectors are real.
• Comet tail: reverberation from two reflectors that are very close. Causes reverberations very close to each other, appearing as a comet tail.
• Mirror image.
• Multipath.
• Side lobes: caused by single-crystal transducers.
• Grating lobes: duplicate structures lateral to the true ones. Caused by phased array transducers producing off-axis waves. These off-axis waves may cause reflectors outside the imaging area to appear in improper locations.
• Shadowing (after a reflector) & enhancement (after a hypoechoic structure).
• Propagation speed errors: displace structures axially. Caused by fluid or fat tissues.
• Range ambiguity: structures appear closer to the surface. During deeper imaging, when pulses get delayed and are received with the next pulse. Prevented by reducing PRF (giving more time for returning echoes) in deeper scans.
• Aliasing in Doppler: the most common artifact with Doppler. Causes improper positioning of signals on the spectrum; positive waves appear as negative. Caused by exceeding the Nyquist limit (not enough sampling rate); with high velocities, the sampling rate has to be high. Corrected by increasing PRF, shifting the baseline, or using a transducer with lower frequency. The Nyquist limit is the minimum sampling needed to avoid aliasing — sampling rate has to be double the highest frequency.
• Aliasing in color Doppler: same principle, caused by exceeding the Nyquist limit. Will cause color to switch abruptly from the highest velocity in one direction to the highest velocity in the opposite direction.
• Doppler range ambiguity.

Ultrasound Safety

• Ultrasound causes:
  • Heat: SPTA < 100 mW/cm² has been shown to be safe; a thermal index < 2 is not harmful. Thermal index: the power needed to raise tissue temperature by 1 degree.
  • Mechanical: radiation force (force exerted by the sound beam on an absorber); streaming; cavitation (stable cavitation from bubbles changing their size with the sound wave; collapsing cavitation when it is a more severe change, causing bubbles to rupture and more injury).

Doppler

• Moving objects cause sound frequency to change.
• Doppler shift = (2 × reflector speed × incident frequency × cosine angle) / (propagation speed + reflector speed).
• Velocity = (77 × Doppler shift) / source frequency.
• Doppler-measured velocity is more accurate when in-line with the direction of movement.
• Actual velocity = measured velocity / cosine angle.
  • Cosine of 0° and 180° is 1.
  • Cosine of 45° is 0.7.
  • Cosine of 90° is 0 → no Doppler signal.
• Types of Doppler:
  • Continuous wave: duty factor is 1, can measure very high velocities, produces a sound that is an estimate of speed, can’t produce images, no gate.
  • Pulsed wave Doppler: can produce an image but limited max speed. We can identify the gate (depth) of interest by adjusting listening time. Most important feature is identifying the tested location (gate). Most limiting feature: high velocities appear negative, called aliasing.
• Aliasing:
  • May occur only with pulsed wave, never with continuous Doppler.
  • Positive waves appear negative on screen.
  • The Nyquist limit is the velocity beyond which velocities are considered in the negative direction: Nyquist limit = PRF / 2. Since PRF changes with depth, the shallower the depth the higher the Nyquist limit; the deeper, the smaller the limit and the more aliasing.
  • To overcome aliasing: use a transducer of low frequency; try to use shallower depth.
• Doppler packets:
  • Doppler sends packets of pulses to detect velocity, not just a single pulse.
  • The fewer the packets, the less accurate; the more packets, the lower the frame rate. Each packet produces a single velocity measurement.
  • Velocity mode: the average of velocities is shown (single number).
  • Variance mode: shows both average velocity and the variability. When flow is smooth and laminar, the variance in velocities is small, presented as yellow on a variance map. With turbulent flow, different velocities are picked up, appearing as green on a variance map.

Hemodynamics

Flow Rate

• Flow of fluid in a long tube. Flow depends on pressure difference, viscosity and diameter.
• Flow rate ∝ pressure difference / resistance.
• Resistance ∝ (viscosity × tube length) / (radius)⁴.

Types of Flow

• Plug flow: all parts move as one unit, same velocity for all layers. Seen in spectral narrowing, in locations with severe stenosis.
• Laminar flow: flow in parallel lines, maximum speed at the center.
• Disturbed flow: streamlines are not straight, as at a bifurcation or focal stenosis.
• Turbulent flow: non-laminar flow, chaotic speeds and directions.

Bernoulli Effect

• As flow increases, pressure decreases. Pressure is reduced at a stenosis compared with proximal and distal to it, which creates a pressure difference for the blood to accelerate at the stenosis.

Spectral Variations

• Spectrum size: the number of frequencies presented on the spectrum. If the spectrum is full, it means large variation in frequencies, variable speeds, turbulent flow, or a very large sample volume. Typically the sample volume should be about 3 mm, at the center of the vessel. If the volume is large enough to include the entire vessel diameter, it will bring all kinds of frequencies from slow laminar flow close to the vessel wall. With severe stenosis, all flow laminae will take high speed due to the high pressure difference across the stenosis, causing spectral narrowing.
• Distal resistance: with high distal resistance, the ED wave will decrease in size until it disappears.

Key Equations

• Wavelength (mm) = propagation speed (mm/µs) / frequency (MHz).
• Propagation speed ∝ stiffness / [compressibility (density)].
• Power ∝ (amplitude)² — power increases 9 times when amplitude triples; if amplitude is halved, power is quartered.
• Intensity = power / beam area — when power doubles, intensity doubles.
• Intensity ∝ (amplitude)² — if amplitude is quartered, intensity & power are reduced to 1/16.
• Duty factor = (pulse duration / PRP) × 100.
• BUC (SP/SA factor) = SP/SA — 1 is uniform, > 1 is less uniform.
• TA = PA × duty factor.
• Attenuation ∝ frequency — higher frequency attenuates more.
• Attenuation coefficient for soft tissue = frequency / 2.
• Penetration depth = 3 / attenuation coefficient (dB/cm).
• Impedance = density × propagation speed.
• Intensity reflection coefficient = ((Z2 − Z1) / (Z2 + Z1))².
• Snell’s law: sine transmission angle / sine incidence angle = propagation speed medium 2 / propagation speed medium 1.
• Depth (mm) = velocity × (time-of-flight / 2); depth for soft tissue = 0.77 × time-of-flight.
• PZT frequency in pulsed wave = propagation speed (mm/µs) / (thickness (mm) × 2).
• PZT frequency in continuous wave = electrical frequency.
• Longitudinal resolution = spatial pulse length / 2.
• Lateral resolution = beam diameter.
• Actual velocity = measured velocity / cosine angle.
• Doppler shift = (2 × reflector speed × incident frequency × cosine angle) / (propagation speed + reflector speed).
• Nyquist limit = PRF / 2.

Ahmed Koriesh, MD