Sound

Sound is a wave, i.e. traveling variations of some quantity (pressure). It involves mechanical motion in the medium through which it travels. Pressure variations cause particles of the medium to vibrate due to increase and decrease of density.

Sound : Frequency and Period

Sound is described by terms that describe waves:

Frequency: How many cycles occur in one second, measured in Hertz. (1 Hz = 1/s). Human hearing: 20 Hz to 20 000 Hz, or 20 KHz. Ultrasound: beyond 20 KHz. Frequency is important in ultrasound because of its impact on resolution and penetration of sonographic images.

Period: Time that takes for one cycle to occur. Inverse of frequency ? if frequency increases period decreases. For example, the period for 5 MHz (5 million Hertz) ultrasound is 1 / 5, 000, 000 = 0.0000002 = 0.2 µs. 1 µs is 1 millionth of a second (0.000001 s). Period is an important concept for pulsed ultrasound, as we'll see later.

Sound : Wavelength and Speed of Propagation

Wavelength: Length of space over which one cycle occurs. It is usually expressed in millimeters. One millimeter, 1 mm, is one thousandth of a meter (0.001 m). Wavelength is important when considering resolution of images.

Propagation Speed: Speed at which a wave moves through a medium. Measured in meters per second, or millimeters / microsecond.

Wavelength depends on the frequency and propagation speed:

Wavelength (mm) = Propagation Speed (mm/microsecond) / Frequency(MHz)

Previous relationship says that if frequency increases wavelength decreases.

Propagation speed depends on the medium. In soft tissue it averages 1540 m/s, or 1.54 mm/µs.

Density and stiffness determine propagation speed. Density is the concentration of matter (mass per unit volume: kg/m³ ). Hardness is the resistance of a material to compression (inverse of compressibility). Hardness is usually dominant factor on propagation speed:

Gas --> low propagation speed

Liquid -->higher propagation speed

Solid -->highest propagation speed.

Propagation speed is what imaging instruments use to correctly locate echoes on the display.

A-line

Positional information is determined by knowing the direction of the pulse entering the patient and measuring the time it takes the echo to return to the transducer. Range equation:

V = 2d / t

Sound : Harmonics

These are sinusoidal waves. Each curve is characterized by a single frequency (number of cycles per second).

Original frequency (black line) is the fundamental frequency, the even (red) and odd (green) multiples are the harmonics.

Strong pressure waves suffer deformation --> generation of harmonics --> non-linear propagation .

Harmonic frequency echoes improve the quality of sonographic images.

Pulsed Ultrasound

Frequency, period, wavelength and propagation speed are sufficient to describe continuous-wave (cw) ultrasound. Cycles repeat indefinitely.

Sonography uses pulsed ultrasound, i.e. a few cycles of ultrasound separated in time with gaps of no signal.

We need to define new parameters: pulse-repetition frequency, pulse-repetition period, pulse duration, duty factor, spatial pulse length.

Pulse repetition frequency (PRF): Number of pulses occurring in 1 s. Usually expressed in kHz.

Pulse repetition period (PRP): Time from the beginning of one pulse to the beginning of the next. Usually expressed in microseconds (µs).

PRP decreases as PRF increases. More pulses occur in a second, less time from one to the next.

PRF is controlled automatically in sonographic instruments, but operator may control it in Doppler instruments (more on this later).

Pulse duration: Time it takes for one pulse to occur = period times the number of cycles in the pulse. Expressed in ms.

Sonographic pulses ~ 2-3 cycles long, Doppler pulses ~ 5-20 cycles long. Pulse duration decreases if number of cycles in a pulse is decreased or if frequency is increased.

Operator chooses frequency.

Note: Frequency increases, period decreases, reducing pulse duration increases

Number of cycles in pulse decreases, pulse duration decreases.

Shorter pulses improve quality of images.

Duty factor: Fraction of time that pulsed US is on. Longer pulses increase the duty factor because the sound is on more of the time.

Higher PRF increase duty factor because there is less "dead" time between pulses.

Duty Factor = Pulse Duration (microseconds) / PRP (microseconds)

Dimensionless because it's a fraction. It is expressed as a decimal or as a percentage if multiplied by 100.

Example: Pulse duration is 4 µs, PRP is 160 µs:

Duty Factor = 4 / 160 = 0.025 = 2.5 %

Typical duty factors for sonography are ~0.1 to 1.0 %. For Doppler ~ 0.5 to 5.0 %.

Note: PRF increases, PRP decreases, duty factor increases

Spatial pulse length: Length of a pulse from front to back = length of each cycle times the number of cycles in the pulse. Shorter pulse length improves resolution.

Amplitude, intensity

Indicators of how strong or intense the ultrasound is.

Amplitude is the maximum variation occurring in an acoustic variable, i.e. how far the variable gets away from its normal, undisturbed value.

Amplitude is measured in units of pressure: MPa (Mega Pascals)

Intensity is the rate at which energy passes through unit area.

Average intensity of a sound beam is the total power in the beam divided by the cross-sectional area of the beam.

Note: Beam power increases, intensity increases. Beam area decreases (focusing), intensity increases

Power is the rate at which energy is transferred. Measured in watts.

Beam area is expressed in cm²

Therefore Intensity is measured in mW/cm²

Intensity is important when discussing bioeffects and safety. Intensity is proportional to the square of the amplitude. So if amplitude is squared, intensity is quadrupled.

Intensity varies in diagnostic ultrasound because it's highest at the center of the beam and falls off near the periphery.

It also varies along direction of travel due to focusing and attenuation.

In pulsed ultrasound, intensity varies with time: It's zero between pulses and not equal to 0 during each pulse.

Temporal peak (TP) is the greatest intensity found in a pulse.

Temporal average (TA) includes the "dead" time between pulses. It is the lowest value.

Pulse average (PA) is in between for a given pulse beam.

PA and TA are related by the duty factor:

TA intensity = PA intensity x Duty Factor.

Note: If duty factor increases, TA intensity increases.

If sound is continuous, duty factor is = 1 and PA and TA intensities are equal to each other.

In real-life equipment intensity is not constant within pulses.

Starts out high and decreases towards the end of the pulse. Damped pulses.

TA averaged over the pulse repetition period.

PA averaged over the pulse duration

TP no averaging.

Putting together spatial and temporal considerations we end up with 6 intensities: