Saudi Board of Radiology: Physics Refresher Course

Kostas Chantziantoniou, MSc2, DABRHead, Imaging Physics Section

King Faisal Specialist Hospital & Research CentreBiomedical Physics Department

Riyadh, Kingdom of Saudi Arabia

Ultrasound Physics

What is Ultrasound?

Ultrasound literally means “above or beyond sound”, that is, it is the sound above thehuman audible (hearing) range.

ASIDE• the frequency of sound that is in the human audible range is between 15 Hz to 20 kHz (Note: 1 Hz = 1 Hertz = 1 cycle/second, 1 kHz = 1,000 Hz)• ultrasound frequencies are higher than that of audible sound and thus comprises of sounds with frequencies greater than 20 kHz (similarly, sound under 20 kHz is called infrasound “below sound”)• the range of sound that is commonly used in Radiology (Diagnostic Ultrasound) is between 1 MHz and 20 MHz (Note: 1 MHz = 1,000 kHz = 1,000,000 Hz)

Although ultrasound was first used on a large scale practical basis in World War I inorder to detect the position and depth of submarines and was referred to as Sonar, itwas not until 20 years later that it was first applied to medicine (sonography).

Characteristics of Sound

A sound beam is similar to an x-ray ray beam in that both are waves transmittingenergy. A more important difference is that x-rays pass readily through a vacuum whilesound requires a medium for its transmission. The velocity of sound depends on the nature of the medium.

How does sound travel through a medium?

• each sphere represents an atom or a molecule• springs between spheres represent atomic interactions or molecular bonds• when the first particle is pushed, it moves and compresses the attached spring, thus exerting a force on the adjacent particle (sphere)• this sets up a chain reaction, but each subsequent particle moves a little less than its neighbor• the tension or pressure (piston device above) applied to the spring is greatest between the first two particles and less between any two down the line• if the driving force reverses its direction, the particles also reverse their direction• if the force vibrates to and fro like a cymbal that has been struck, the particles respond by oscillating back and forth• the particles in a sound beam behave in the same manner: that is, they oscillate back and forth, but over a short distance of only a few microns in liquids and even less in solids

• although the individual particles move only a few microns, we can see that the effect of their motion is transmitted through their neighbors over a much longer distance• during the same time that the first particle moves through a distance “a”, the effect of the motion is transmitted over a distance “b”• the velocity of sound is determined by the rate at which the force is transmitted from one molecule to another

NOTEIn summary then, sound waves are a mechanical disturbance that propagate through a medium.

Longitudinal Waves

Ultrasound waves are transmitted through tissue as longitudinal waves of alternatingcompression and rarefaction regions.

The term “longitudinal wave” means that the motion of the particles in the medium isparallel to the direction of wave propagation. The molecules of the conducting liquidmove back and forth, producing bands of compression and rarefaction.

• the wave front starts at time 1 when the vibrating piston compresses the adjacent material• a band of rarefaction is produced at time 2, when the piston reverses its direction• each repetition of this back-and-forth motion is called a cycle, and each cycle produces a new wave

The length of the wave (or wavelength) is the distance between two bands of compression or rarefaction and is represented by the symbol (has unit of millimeters).

Wavelength

ASIDEThe wavelength of a wave can also be represented using the following illustration

Compression region Rarefaction region

Frequency

The motion of the vibrating piston, plotted against time, forms the sinusoidal curveshown along the left side of the diagram below

The frequency is the number of compressions (or rarefactions) bands that pass anygiven point in space per unit time and is measured in Hertz which is defined as:

1 Hertz = 1 cycle per second

• frequency has the units of Hz (or 1/second or sec-1 or s-1)

Period

The period is the time between compression or rarefaction (oscillations) bands and hasthe units of time (seconds). In other words, the period is the time that it takes for onecycle to occur.

ASIDEThe period of a wave can be represented using the following illustration

The relationship between period and frequency is:

frequency (Hz) = 1 period (s) frequency period

Propagation Speed of Sound

Propagation speed is the speed with which a sound wave moves (travels) through a medium and for sound waves the velocity is given by:

velocity (m/sec) = wavelength (m) • frequency (Hz) = wavelength (m) period (sec)

For body tissues in the diagnostic ultrasound range, the speed of sound is(approximately) independent of frequency and depends primarily on the physicalmakeup of the material through which the sound is being transmitted. Because of theabove observation, that means:

frequency wavelength

NOTE• the average speed of sound in soft tissue (excluding in bone) is 1,540 m/s• all ultrasound scanners use this value for the speed of sound to compute tissue depth

and the speed of sound in soft tissue is a constant.

NOTE 1 s = 1 x 10-6 s

The physical properties of a material that directly affects the speed of sound are:

(1) Physical Density (concentration of mater, mass per unit volume) (2) Stiffness

Physical Density

Dense material tend to be composed of massive molecules, and these molecules have a great deal of inertia. They are difficult to move or to stop once they are moving.Because the propagation of sound involves the rhythmic starting and stopping ofparticulate motion, we would not expect a material made up of large molecules (i.e.:large in mass) such as mercury to transmit sound at as great a speed as a materialcomposed of smaller molecules, such as water.

• the speed of sound in a material decreases if the density is increased, assuming constant stiffness

Stiffness

Stiffness is the resistance of a material to compression. It is the opposite ofcompressibility, and in fact the less compressible a material is the more rapidly it will transmit sound. We have,

speed of sound 1 stiffness compressibility

• the speed of sound in a material increases if the stiffness is increased• sound travels slower in gases because the molecules are far apart [i.e.: they are held by loose “springs” (bonds) and a particle must move relatively a long distance before it can affect a nearest neighbor] • liquids and solids are less compressible because their molecules are closer together

Combined Effects of Density and Stiffness

It is generally true that media with higher densities also have higher stiffness.Because stiffness differences between materials generally dominate the effect ofdensity differences, higher-density materials usually have higher sound speeds than lower-density materials.

• in general speed of sounds are lower through gases, higher through liquids and highest through solids• this increasing sequence is due to the fact that the stiffness differences between gases, liquids and solids are greater than the density differences• mercury is a special case: mercury has a density 13.9 times greater than water and water has a compressibility that is 13.4 times greater than that of mercury (both effects balance out)

Acoustic Impedance

Acoustic impedance (Z) of a material is given by:

impedance (Rayl) = speed of sound (m/s) • density of material (kg/m3) in material

• the acoustic impedance unit is called the Rayl (kg/m2/s)• acoustic impedance can be considered to be a measure of a material’s ability to transmit acoustic energy (air and lung media have low values, and bone and metal have high values)• acoustic impedance is determined by the density and stiffness of a medium• since the speed of sound is independent of frequency in the diagnostic ultrasound range, acoustic impedance is also independent of frequency• acoustic impedance determines the amount of energy reflected at an interface• since the speed of sound in tissue is relatively constant in the diagnostic ultrasound range, then the acoustic impedance of most tissues is also a constant, they typically have values around 1.6 x 106 kg/m2/s (Rayls)

impedance density impedance speed of sound

unit conversion: g/cm2/sec = kg/m2/s (Rayls) 10

Intensity

The intensity (or loudness) of sound is determined by the length of oscillation of theparticles (vibration amplitude) conducting the waves.

• the harder the piston is struck, the more energy it receives and the wider its vibration amplitude• these wider excursions are transmitted to the adjacent conducting media and produce a more intense beam• in time the vibrations diminish in intensity although not in frequency, and the sound intensity decreases, producing a lower intensity beam• ultrasound intensities are expressed in power per unit area, where power (mW) is the rate at which acoustic energy is transferred and the area (cm2) is the area of the the ultrasound beam at some distance from the transducer surface, thus:

intensity (mW/cm2) = power (mW) area (cm2)

The amplitude of a wave is the size of the wave displacement or pressure.

• larger amplitudes of vibration produce denser compression bands and, hence, higher intensities of sound (i.e.: the greater the amplitude of oscillation the more intense the sound).• ultrasound beam intensity is a measure of the energy associated with the beam and is proportional to the square of the amplitude, that is:

intensity (amplitude)2

Relative Sound Intensity and Pressure (Decibels)

In diagnostic ultrasound it is not uncommon to receive ultrasound intensities (I) fromthe patient that are 1,000,000 to 1,000,000,000 times smaller than the originalintensity (I0). In fact depending on the depth of the I intensity source, we can have awide variation of the ratio of these two intensities. In order to reduce the ‘dynamic’ range of these ratio’s we introduce the following numerical scale:

relative intensity (B) = log10( I/I0)

relative intensity (dB) = 10 • log10( I/I0)

or

where I0 is the original intensity and I is the measured (received) intensity.

ASIDEThe logarithmic scale is used in mathematics to remap a very large number scale, like the ratio (I/I0), with a new much smaller number scale

1 B = 10 dB

where B is calledthe ‘bel’ and dB the‘decibel’

Since intensity is proportional to the square of the pressure amplitude (i.e.: I P2) then:

relative pressure (dB) = 20 • log10( P/P0)

• decibels may have either positive or negative values• a positive value corresponds to signal amplification (I > I0)• a negative value corresponds to signal attenuation (I < I0)

-

ExampleAn intensity reduction of 50% corresponds to - 3 dB (that is I = 0.5 I0). On the same note, a + 3dB change corresponds to a twofold increase in intensity (that is I = 2 I0).

Interaction of Ultrasound with Matter

Attenuation

Detector

Tissue Refracted(Scattered)

Reflected (Scattered)

Absorbed

With an unfocused beam in any medium, such as tissue, amplitude and intensity willdecrease as the sound travels through the medium. This reduction in amplitude (andthus intensity) is called attenuation. It encompasses absorption, reflection and scattering.

NOTE• the absorbed sound wave energy is converted into heat• absorption is normally the dominate contribution to attenuation in soft tissue

attenuation coefficient = attenuation (dB) path length (cm)

attenuation = - 10 log intensity at second point (I2) intensity at first point (I1)

(in dB)

Note The minus sign is often ignored in most textbooks

Note Path length is the distance between the first and second point

I1 I2

path length

attenuation coefficient attenuation attenuation path length

(in dB/cm)

• the attenuation of ultrasound in a homogeneous medium is exponential with penetration depth

1 dB/cm3 dB/cm

10 dB/cm

• the attenuation coefficient increases with frequency

• for soft tissue, there is a linear relationship between the frequency and attenuation

average attenuation coefficient = 0.5 dB/cm (for 1 MHz) = 2.5 dB/cm (for 5 MHz)

• for water and bone, attenuation increases approximately as frequency2

• since the attenuation for soft tissue is on the average 0.5 dB per centimeter for each MHz of frequency (i.e.: 0.5 dB/cm/MHz), then

attenuation (dB) = 0.5 • frequency (MHz) • path length (cm)

• attenuation is higher in lung than in other soft tissues, and it is higher in bone than in soft tissues (explains for poor US imaging of this tissue) • the practical consequence of attenuation is that it limits the depth at which images can be obtained• the imaging depth decreases as frequency increases

attenuation coefficient frequency attenuation frequency

frequency image depth

Absorption

The term “absorption” refers to the conversion of ultrasonic energy to thermal energy, and is the dominate contribution to attenuation in soft tissue.

The mechanisms involved in absorption are rather complex (and thus will not bediscussed in detail); the three primary factors that determine the amount of absorptionare:

(1) the viscosity of the conducting medium(2) the “relaxation time” of the medium; and(3) the frequency of the sound wave

Viscosity

• viscosity is an internal friction (or a frictional force) that opposes the motion of the particles in the medium• particle freedom decreases and internal friction increases with increasing viscosity• this internal friction absorbs the sound, or decreases its intensity, by converting sound into heat • in liquids, which have low viscosity, very little absorption takes place, in soft tissue viscosity is higher and a medium amount of absorption occurs, while bone shows high absorption of ultrasound

Relaxation Time

• the relaxation time is the time that it takes for a molecule to return to its original position after it has been displaced• when a molecule with short relaxation time is pushed by a longitudinal compression wave, the molecule has time to return to its resting state before the next compression wave arrives• a molecule with a longer relaxation time may not be able to return completely before a second compression wave arrives, when this happens, the compression wave is moving in one direction and the molecule in the opposite direction. Since more energy is required to reverse the direction of the molecule (in phase with compression wave) in order to transmit the sound wave further, more acoustic energy is used used up and is converted to heat

Frequency of Sound Wave

• since the frequency of sounds effects both of the above processes it also affects the amount of absorption produced by a medium• the higher the frequency (ie: the more often a particle moves back and forth in a given time), the more the particle motion is affected by the drag of a viscous material and its relaxation time

Reflection

A portion of the ultrasound beam is reflected at tissue interfaces as shown below. Thesound reflected back towards the source is called a echo and is used to generate the ultrasound image.

Like in the optical properties of light, in ultrasound we also have:

Angle of incidence (i) = Angle of reflection (r)

i r

• the percentage of ultrasound intensity reflected depends, in part, on the angle of incidence• as the angle of incidence increases, reflected sound is less likely to reach the transducer and thus no acoustic signal is received to image• no reflection is generally detected by the transducer, if the angle of incidence is greater than 3°• specular (smooth) reflection occurs from large, smooth surfaces (major contributor to ultrasound images)

• non-specular (diffuse) reflections is scatter from “rough” surfaces where the irregular contours are bigger than the ultrasound wavelength

specular reflections

Intensity Coefficients

The percentage of ultrasound reflected at a tissue interface is also dependent on the acoustic impedance of the two tissues.

intensity reflection coefficient (IRC) = reflected intensity (mW/cm2) incident intensity (mW/cm2) = (Z2 - Z1)2

(Z2 + Z1)2

where Z1 and Z2 is the impedance of tissue 1 and tissue 2

impedance difference IRC

For normal incidence (incident angle = reflected angle = 90°)

intensity transmission coefficient (ITC) = transmitted intensity (mW/cm2) incident intensity (mW/cm2) = 1 - IRC

= 4Z1Z2

(Z2 + Z1)2

IRC ITC

NOTEIRC is the fraction of the incident ultrasound intensity that is reflected, and ITC is thefraction transmitted at a interface of two different material (or tissues)

Common Interface Reflection Factors

• air/tissue interfaces reflect virtually all (99.9%) of the incident ultrasound beam, thus imaging through lungs (air) is generally not possible• gel is applied between the transducer (PZT) and the skin to displace the air and minimize large reflections (80%) that would interfere with ultrasound transmission into the patient• bone/tissue interfaces also reflect substantial fractions (30%) of the incident intensity• the lack of transmissions beyond these interfaces results in an area void of echoes called shadowing• in imaging the abdomen, the strongest echoes are likely to arise from gas bubbles• organs such as the kidney, pancreas, spleen and liver are composed of sub-regions that contain many scattering sites, which results in a speckled texture on ultrasound images• organs that contain fluids such as the bladder, cysts, and blood vessels have no internal structure and almost no echoes (i.e.: show black on images)

Refraction

Refraction is the change in direction of an ultrasound beam when passing through onemedium to another.

When ultrasound passes from one medium to another, the frequency remains the same but the wavelength changes to accommodate the new velocity of sound in thesecond medium (for diagnostic ultrasound, the speed of sound is independent of frequency).

i

t

Media 1

Media 2

Snell’s Law: sin(i) = velocity of sound in media 1 sin(t) velocity of sound in media 2

NOTEIf the speed of sound inmedia 2 is less than that ofmedia 1, then the wavelength of the sound beam in media 2 must be shortened in order to maintain a constant beam frequency.

• the transmitted angle (t) is determined by the speed of sound in both media’s and the incident angle (i) of the beam