The Physics of
Diagnostic Ultrasound
FRCR Physics Lectures
Session 1 & 2
Mark Wilson
Clinical Scientist (Radiotherapy)
mark.wilson@hey.nhs.uk
Hull and East Yorkshire Hospitals
NHS Trust
Session 1 Overview
Session Aims:
• Basic physics of sound waves
• Basic principles of image formation
• Interactions of ultrasound waves with matter
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Basic Physics
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Basic Physics
Wave Motion
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Basic Physics
Sound Waves
• Sounds waves are mechanical pressure waves which propagate
through a medium causing the particles of the medium to oscillate
backward and forward
•The term Ultrasound refers to sound waves of such a high frequency
that they are inaudible to humans
• Ultrasound is defined as sound waves with a frequency above 20 kHz
• Ultrasound frequencies used for imaging are in the range 2-15 MHz
• The velocity and attenuation of the ultrasound wave is strongly
dependent on the properties of the medium through which it is travelling
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Basic Physics
Wave Propagation
• Imagine a material as an array of molecules linked by springs
• As an ultrasound pressure wave propagates through the medium, molecules
in regions of high pressure will be pushed together (compression) whereas
molecules in regions of low pressure will be pulled apart (rarefaction)
• As the sound wave propagates through the medium, molecules will oscillate
around their equilibrium position
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Basic Physics
Power and Intensity
• A sound wave transports Energy through a medium from a source. Energy is
measured in joules (J)
• The Power, P, produce by a source of sound is the rate at which it produces
energy. Power is measured in watts (W) where 1 W = 1 J/s
• The Intensity, I, associated with a sound wave is the power per unit area.
Intensity is measured in W/m2
• The power and intensity associated with a wave increase with the pressure
amplitude, p
Power, P  p
Intensity, I  p2
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Basic Physics
Frequency (f):
Number of cycles per second
Unit: Hertz (Hz)
Speed (c):
Speed at which a sound wave
travels is determined by the
medium
Unit: Metres per second (m/s)
Air – 330 m/s
Water – 1480 m/s
Av. Tissue – 1540 m/s
Bone – 3190 m/s
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Basic Physics
Wavelength ():
Distance between consecutive
crests or other similar points on
the wave
Unit: Metre (m)
A wave from a source of
frequency f, travelling through a
medium whose speed of sound is
c, has a wavelength 
=c/f
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Basic Principles of
Image Formation
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Basic Principles of Image Formation
Pulse-Echo Principle
D
Source of sound
)
)
)
)
)
)
)
)
)
)
) )
) )
)
Sound reflected at boundary
Distance = Speed x Time
2D = c x t
Reduced signal amplitude
No signal returns
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Basic Principles of Image Formation
Pulse-Echo in Tissue
Tissue 1
Tissue 2
Tissue 3
Transducer
Can transmit
and receive US
• Ultrasound pulse is launched into the first tissue
• At tissue interface a portion of ultrasound signal is transmitted into the second
tissue and a portion is reflected within the first tissue (termed an echo)
• Echo signal is detected by the transducer
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Basic Principles of Image Formation
B-Mode Image
• A B-mode image is a cross-sectional image representing tissues and organ
boundaries within the body
• Constructed from echoes which are generated by reflection of US waves at
tissue boundaries, and scattering from small irregularities within tissues
• Each echo is displayed at a point in the image which corresponds to the
relative position of its origin within the body
• The brightness of the image at each point is related to the strength
(amplitude) of the echo
• B-mode = Brightness mode
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Basic Principles of Image Formation
B-Mode Image Formation
A 2D B-mode image is formed from a large number of B-mode lines, where each
line in the image is produced by a pulse echo sequence
Transducer
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Basic Principles of Image Formation
Arrays
Linear
Phased
Curvilinear
Rectangular FOV
Trapezoidal FOV
Useful in applications
where there is a need
to image superficial
areas at the same time
as organs at a deeper
level
Wide FOV near
transducer and even
wider FOV at deeper
levels
Sector FOV useful for
imaging heart where
access is normally
through a narrow
acoustic window
between ribs
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Basic Principles of Image Formation
B-Mode Image – How Long Does it Take?
1. Minimum time for one line = (2 x depth) / speed of sound = 2D / c seconds
2. Each frame of image contains N lines
3. Time for one frame = 2ND / c seconds
E.g. D = 12 cm, c = 1540 m/s, Frame rate = 20 frames per second
Frame rate = c / 2ND
N = c / 2D x Frame rate = 320 lines (poor - approx half of standard TV)
Additional interpolated lines are inserted between image lines to boost image
quality to the human eye
4. Time is very important!!!
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Basic Principles of Image Formation
Time Gain Compensation (TGC)
• Deeper the source of echo  Smaller signal intensity
• Due signal attenuation in tissue and reduction in initial US beam intensity by
reflections
• Operator can TGC use to artificially ‘boost’ the signals from deeper tissues
(like a graphic equaliser)
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Basic Principles of Image Formation
M-Mode Image
• Can be used to observe the motion of tissues (e.g. Echocardiography)
• One direction of display is used to represent time rather than space
Transducer at fixed point
Time
Depth
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Basic Principles of Image Formation
M-Mode Image of Mitral Valve
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Ultrasound
Interactions in Matter
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Ultrasound Interactions
• Reflection
• Scatter
• Refraction
• Attenuation and Absorption
• Diffraction
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Ultrasound Interactions
Speed of Sound, c
• The speed of propagation of a sound wave is determined by the medium it is
travelling in
• The material properties which determine speed of sound are density,  (mass
per unit volume) and elasticity, k (stiffness)
Atom / Molecule
Bond
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Ultrasound Interactions
Speed of Sound, c
• Consider a row of masses (molecules) linked by springs (bonds)
• Sound wave can be propagated along the row of masses by giving the first
mass a momentary ‘push’ to the right
• This movement is coupled to the second mass by the spring
m
K
m
K
m
K
m
Sound wave
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Ultrasound Interactions
Small masses (m) model a material of low density linked by springs of high
stiffness K
m
K
m
K
m
K
m
• Stiff spring will cause the second mass to accelerate quickly to the right and
pass on the movement to the third mass
• Smaller masses are more easily accelerated by spring
• Hence, low density and high stiffness lead to high speed of sound
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Ultrasound Interactions
Large masses (M) model a material of high density linked by springs of low
stiffness k
M
k
M
k
M
k
M
• Weak spring will cause the second mass to accelerate relatively slowly
• Larger masses are more difficult to accelerate
• Hence, high density and low stiffness lead to low speed of sound
Speed of Sound c =  k / 
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Ultrasound Interactions
Material
C (m/s)
Liver
1578
Kidney
1560
Fat
1430
Average Tissue
1540
Water
1480
Bone
3190
Air
330
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Ultrasound Interactions - Reflection
Reflection of Ultrasound Waves
When an ultrasound wave travelling through one type of tissue encounters an
interface with a tissue with different acoustic impedance, z, some of its energy
is reflected back towards the source of the wave, while the remainder is
transmitted into the second tissue
- Reflections occur at tissue boundaries where there is a change in acoustic
impedance
z1
z2
Transducer
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Ultrasound Interactions - Reflection
Acoustic Impedance (z)
• The acoustic impedance of a medium is a measure of the response of the
particles of the medium to a wave of a given pressure
• The acoustic impedance of a medium is again determined by its density, ,
and elasticity, k (stiffness)
• As with speed of sound, consider a row of masses (molecules) linked by
springs
m
K
m
K
m
K
m
Sound wave
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Ultrasound Interactions - Reflection
Small masses (m) model a material of low density linked by weak springs of low
stiffness k
m
k
m
k
m
k
m
• A given pressure is applied momentarily to the first small mass m
• The mass is easily accelerated to the right and its movement encounters little
opposing force from the weak spring k
• This material has low acoustic impedance, as particle movements are
relatively large in response to a given applied pressure
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Ultrasound Interactions - Reflection
Large masses (M) model a material of high density linked by springs of high
stiffness K
M
K
M
K
M
K
M
• In this case, the larger masses M accelerate less in response to the applied
pressure
• Their movements are further resisted by the stiff springs
• This material has high acoustic impedance, as particle movements are
relatively small in response to a given applied pressure
Can also be shown
Acoustic Impedance z =  k
Acoustic Impedance z = c
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Ultrasound Interactions - Reflection
z1
z2
pi , Ii
pt , It
pr , Ir
Amplitude Reflection Coefficient (r)
r=
pr
pi
=
Z2 – Z1
Z1 + Z2
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Ultrasound Interactions - Reflection
Intensity Reflection Coefficient (R)
R=
Ir
Ii
=
(
Z2 – Z1
Z1 + Z2
2
)
• Strength of reflection depends on the difference between the Z values of the
two materials
• Ultrasound only possible when wave propagates through materials with
similar acoustic impedances – only a small amount reflected and the rest
transmitted
• Therefore, ultrasound not possible where air or bone interfaces are present
Intensity Transmission Coefficient (T)
T=1-R
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Ultrasound Interactions - Reflection
Interface
R
T
Soft Tissue-Soft Tissue
0.01-0.02
0.98-0.99
Soft Tissue-Bone
0.40
0.60
Soft Tissue-Air
0.999
0.001
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Ultrasound Interactions - Reflection
Reflection at an Angle
• For a flat, smooth surface the angle of
reflection, r = the angle of incidence, i
• In the body surfaces are not usually
smooth and flat, then r  i
z1
z2
i
r
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Ultrasound Interactions - Scatter
Scatter
• Reflection occurs at large interfaces such
as those between organs where there is a
change in acoustic impedance
• Within most organs there are many small
scale variations in acoustic properties
which constitute small scale reflecting
targets
• Reflection from such small targets does
not follow the laws of reflection for large
interfaces and is termed scattering
• Scattering redirects energy in all
directions, but is a weak interaction
compared to reflection at large interfaces
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Ultrasound Interactions - Refraction
Refraction
When an ultrasound wave crosses a tissue boundary at an angle (non-normal
incidence), where there is a change in the speed of sound c, the path of the
wave is deflected as it crosses the boundary
c1
c2 (>c1)
Snell’s Law
i
t
sin (i)
c1
= c
sin (t)
2
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Ultrasound Interactions - Attenuation
Attenuation
• As an ultrasound wave propagates
through a medium, the intensity
reduces with distance travelled
Intensity, I
• Attenuation describes the reduction in
intensity with distance and includes
scattering, diffraction, and absorption
• Attenuation increases linearly with
frequency
Low freq.
High freq.
• Limits frequency used – trade off
between penetration depth and
resolution
I = Ioe- d
Distance, d
Where  is the attenuation coefficient
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Ultrasound Interactions - Attenuation
Absorption
• In soft tissue most energy loss (attenuation) is due to absorption
• Absorption is the process by which ultrasound energy is converted to heat in
the medium
• Absorption is responsible for tissue heating
Decibel Notation
Attenuation and absorption is often expressed in terms of decibels
Decibel, dB = 10 log10 (I2 / I1)
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Ultrasound Interactions - Diffraction
Diffraction
• Diffraction is the process by which the ultrasound wave diverges (spreads out)
as it moves away from the source
• Divergence is determined by the relationship between the width of the source
(aperture) and the wavelength of the wave
Low Divergence
Aperture small compared to 
High Divergence
Aperture large compared to 
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Break
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Session 2 Overview
Session Aims:
• Construction and operation of the ultrasound transducer
• Ultrasound instrumentation
• Ultrasound safety
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Ultrasound
Transducer
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Ultrasound Transducer
Transducer
• The transducer is the device that converts electrical transmission pulses into
ultrasonic pulses, and ultrasonic echo pulses into electrical signals
• A transducer produces ultrasound pulses and detects echo signals using the
piezoelectric effect
• The piezoelectric effect describes the interconversion of electrical and
mechanical energy in certain materials
• If a voltage pulse is applied to a piezoelectric material, the material will
expand or contract (depending on the polarity of the voltage)
• If a force is applied to a piezoelectric material which causes it to expand or
contract (e.g. pressure wave), a voltage will be induced in the material
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Ultrasound Transducer
Transducer
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Ultrasound Transducer
Transducer
• A transducer only generates a useful ultrasound beam at one given frequency
• This frequency corresponds to a wavelength in the transducer equal to twice
the thickness of the piezoelectric disk – This is due to a process known as
Resonance!
• Choice of frequency is important – remember that attenuation increases with
increasing frequency
• Image resolution increases with frequency
• Therefore, there is a trade-off between scan depth and resolution for any
particular application
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Ultrasound Transducer
Beam Shape – Diffraction
a
NEAR FIELD
FAR FIELD
NFL
Near Field Length, NFL = a2 / 
a = radius of transducer
 = Wavelength
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Ultrasound Transducer
Beam Shape - Diffraction
• In the near field region the beam energy is largely confined to the dimensions
of the transducer
• Need to select a long near field length to achieve good resolution over the
depth you wish to scan too
• Near field length increases with increasing transducer radius, a, and
decreasing wavelength, 
• Short wavelength means high frequency – not very penetrating
• Large transducer radius – Wide beam (poor lateral resolution)
• Trade-off between useful penetration depth and resolution!!
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Ultrasound Transducer
Beam Focusing
• An improvement to the overall beam width can be obtained by focusing
• Here the source is designed so that the waves converge towards a point in
the beam, the focus, where the beam achieves its minimum width
• Beyond the focus, the beam diverges again but more rapidly that for an
unfocused beam with the same aperture and frequency
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Ultrasound Transducer
Beam Focusing
F
a
W
Beam width at focus, W = F / a
At focal point:
• Maximum ultrasound intensity
• Maximum resolution
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Ultrasound Transducer
Beam Focusing
For a single element source, focusing can be achieved in one of two ways:
1)
A curved source
A curved source is manufactured with a radius of curvature of F and
hence produces curved wave fronts which converge at a focus F cm from
the source
Source
Focus
F
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Ultrasound Transducer
Beam Focusing
For a single element source, focusing can be achieved in one of two ways:
2)
An acoustic lens
An acoustic lens is attached to the face of a flat source and produces
curved wave fronts by refraction at its outer surface (like an optical lens).
A convex lens is made from a material with the lower speed of sound
than tissue.
Lens
Source
Focus
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Ultrasound Transducer
Beam Shape - Overlapping Groups of Elements
Fire elements
1-5 together
And then…
Fire elements
2-6 together
And so on…
Near field length increases as (N)2
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Ultrasound Transducer
Array Focusing
Waves from outer elements 1 and 5 have
greater path lengths than those from other
elements
Therefore signals do not arrive simultaneously
at the target and reflections do not arrive at all
elements at the same time
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Ultrasound Transducer
Array Focussing
Time delays
Introduce time delays to compensate for extra
path length on both transit and receive
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Ultrasound Transducer
Multiple Zone Focussing
• Fire transducer several times with different focus to compile better image
• However, more focus points decreases frame rate
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Ultrasound Transducer
Resolution
Resolution in three planes
Lateral
Axial
Slice Thickness
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Ultrasound Transducer
Resolution
Resolution
Depends on
Typical Value (mm)
Axial
Pulse length
0.2 - 0.5
Lateral
Beam width
2–5
Slice Thickness
Beam height
3-8
• Higher frequency improves resolution in all three planes
• Slice thickness is a hot topic for improvement – 2D arrays
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Instrumentation
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Instrumentation
Clock
Transmitter
TGC Generator
Beam Controller
Transducer
x, y
AD Converter
z
Signal Processor
Archive
Image Store
Display
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Instrumentation
Clock
• Command and control centre
• Sends synchronising pulses around the system
• Each pulse corresponds to a command to send a new pulse from the
transducer
• Determines the pulse repetition frequency (PRF)
PRF = 1 / time per line = c / 2D
Where c is speed of sound and D is maximum scan depth
If there are N lines, then Frame Rate = c / 2ND
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Instrumentation
Transmitter
• Responds to clock commands by generating high voltage pulses to
excite transducer
Transducer
• Sends out short ultrasound pulses when excited
• Detects returning echoes and presents them as small electrical
signals
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Instrumentation
AD Converter
• Converts analogue echo signals into digital signals for further
processing
Needs to:
• Be fast enough to cope with highest frequencies
• Have sufficient levels to create adequate grey scales (e.g. 256 or 512)
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Instrumentation
Signal Processor
Grey level
Liver
Heart
Linear
Carries out:
• TGC application
• Overall gain
Input Amp
• Signal compression – fits very large dynamic range ultrasound signal
on to limited greyscale display dynamic range
• Demodulation – removal of the carrier (ultrasound) frequency
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Instrumentation
Image Store
• Takes z (brightness) signal from processor
• Positions it in image memory using x (depth) and y (element position)
information from beam controller
• Assembles image for each frame
• Presents assembled image to display
• Typically have capacity to store 100-200 frames to allow cine-loop
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Ultrasound Safety
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Ultrasound Safety
Hazard and Risk
• Hazard describes the nature of the danger or threat (e.g. burning,
falling, etc)
• Risk takes into account the severity of the potential consequences
(e.g. death, injury) and the probability of occurrence
• There are two main hazards associated with ultrasound:
- Tissue heating
- Cavitation
• But is there any risk???
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Ultrasound Safety
Tissue Heating
• During a scan some of the ultrasound energy is absorbed by the exposed
tissue and converted to heat causing temperature elevation
• Elevated temperature affects normal cell function
• The risk associated with this hazard depends on the:
- Degree of temperature elevation
- Duration of the elevation
- Nature of the exposed tissue
Rate of energy absorption per unit volume
q = 2I
Where  = absorption coefficient,  = frequency, I = intensity
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Ultrasound Safety
Tissue Heating
• Thermal effects in patient are complex
• Temperature increase will be fastest at the focus resulting in a temperature
gradient
• Heat will be lost from focus by thermal conduction
• The transducer itself will heat up and this heat will conduct into tissue
enhancing the temperature rise near the transducer
• The presence of bone in the field will increase the temperature rise
• Blood flow will carry heat away from the exposed tissues
• It is impossible to accurately predict the temperature increase occurring in the
body and a simple approach to estimate the temperature increase is used to
provide some guidance - Thermal Index (TI)
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Ultrasound Safety
Thermal Index (TI)
TI = W / Wdeg
W = Transducer power exposing the tissue
Wdeg = The power required to cause a maximum temperature rise of 1oC
anywhere in the beam
• TI is a rough estimate of the increase in temperature that occurs in the region
of the ultrasound scan
• A TI of 2.0 means that you can expect at temperature rise of about 2oC
• The difficulty with calculating the TI lies mostly in the estimation of Wdeg
• To simplify this problem there are three TIs
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Ultrasound Safety
Soft-Tissue Thermal Index (TIS)
Maximum temperature
Soft tissue
Bone-at-Focus Thermal Index (TIB)
Maximum temperature
Bone
Soft tissue
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Ultrasound Safety
Cranial (or Bone-at-Surface) Thermal Index (TIC)
Maximum temperature
Bone
Soft tissue
All three TI values depend linearly on the acoustic power emitted by the
transducer
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Ultrasound Safety
Does Temperature Rise Matter?
• Normal core temperature is 36-38oC and a temperature of 42oC is “largely
incompatible with life”
• During an ultrasound examination only a small volume of tissue is exposed and
the human body is quite capable of recovering from such an event
• Some regions are more sensitive such as reproductive cells, unborn fetus, and
the CNS
• Temperature rises of between 3 and 8oC are considered possible under certain
conditions
• There has been no confirmed evidence of damage from diagnostic ultrasound
exposure
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Ultrasound Safety
Cavitation
• Refers to the response of gas bubbles in a
liquid under the influence of an ultrasonic wave
• Process of considerable complexity
• High peak pressure changes can cause microbubbles in a liquid or near liquid medium to
expand – resonance effect
• A bubble may undergo very large size variations
and violently collapse
• Very high localised pressures and temperature
are predicted that have potential to cause cellular
damage and free radical generation
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Ultrasound Safety
Cavitation
Micro-bubbles grow by resonance processes
Bubbles have a resonant frequency, fr, depending on their radius, R.
frR  3 Hz m
This suggests that typical diagnostic frequencies (3 MHz and above) cause
resonance in bubbles with radii of the order of 1 micrometer
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Ultrasound Safety
Mechanical Index (MI)
• The onset of cavitation only occurs above a threshold for acoustic pressure
• This has resulted in the formulation of a mechanical index (MI)
• Mechanical index is intended to quantify the likelihood of onset of cavitation
MI = pr / f
where pr is the peak rarefaction pressure and f is the ultrasound frequency
• For MI  0.7 the physical conditions probably cannot exist to support bubble
growth and collapse
• Exceeding this threshold does not mean there will be automatically be
cavitation
• Cavitation is more likely in the presence of contrast agents and in the
presence of gas bodies such as in the lung and intestine
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The End
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