Ultrasound Scattering
by Finite Objects and
Inhomogeneous Media
Ali Baghani
PhD Candidate
Robotics & Control Lab
University of British Columbia
Part I: Ultrasound Imaging*
*The lecture notes of Part I are partly based on lecture notes of the graduate course
EECE 546: “Medical Imaging” by Dr. Robert Rohling.
Introduction


Ultrasound is the world’s most frequently used
medical imaging modality, in terms of number
of images created annually.
As a comparison with x-ray radiography:
X-ray:
Ultrasound:
Advantages:
Advantages:
1.
Excellent Resolution
1.
Very safe at low powers
2.
Distinguishes bone
boundaries very well
2.
Differentiates soft tissue
Disadvantages:
1.
2.
Ionization hazard (esp. to
dividing cells)
Won’t differentiate soft tissue
well
Disadvantages:
1.
Resolution not as good as
ultrasound
2.
Won’t penetrate air (lung) or
bone areas
Ultrasound Imaging Machine






Ultrasound machine
Probe
Crystals
Boundaries
& scatterers
RF signal
Sequencing
Time to Depth Conversion

The speed of ultrasound is fairly constant in soft tissue.





Liver: ~1569 m/s
Strained muscle: ~1566 m/s
Water: ~1500 m/s
The speed of ultrasound is assumed constant at 1540
m/s (as an average) inside soft tissue.
This enables the processing unit to interpret time delay
between transmission and reception on the RF signal as
depth inside tissue.
t=0
pulse transmission
t
time delay
d=0
d
depth
Beam Forming


To get a high resolution, it is of interest to have a
narrow beam of ultrasound scanning a line of
tissue.
Unlike electromagnetic waves (laser),
mechanical waves such as acoustic or
ultrasound waves cannot be produced in a
narrow beam.
Beam Pattern of a Single Crystal

The wave equation could be solved for a
rectangular source of vibrations:
y
x
r
Φy
Φx
z
Beam Pattern of a Single Crystal
y
x
r
Φy
Φx
z
Source: D Christensen, Ultrasonic Bioinstrumentation , John Wiley, 1988.
Focusing: Transmission


Acoustic lenses can be used for
focusing the beam at a certain
depth.
Electronic focusing has more
flexibility.
Focusing: Reception


Dynamic electronic focusing is used.
Delayed version of signals received by successive
crystals are added together to


Blur out data from out of focus structures
Sharpen data from structures on the focal point.
Diagnostic Imaging Configurations



A-mode
B-mode
Doppler


Continuous Doppler
Pulsed Doppler






Spectral Doppler
Color Doppler
Power Doppler
3D and 4D (3D + time) imaging
Strain imaging
…
B-Mode imaging


The amplitude of the RF signal at each
point is used as the intensity (Brightness)
of the corresponding point on the image.
RF lines are acquired at successive spatial
locations to form a whole cross sectional
image of the tissue.
Photos: Courtesy of Koninklijke Philips Electronics (EnVisor™ machine)
Color Doppler Imaging



Moving scatterers cause
the frequency (phase) of
the RF signal to be
shifted.
The shift in frequency is
proportional to the speed
of the moving scatterers.
The phase shifts could be
used to obtain a velocity
profile of the scatterers.
Photo: Courtesy of Koninklijke Philips Electronics (EnVisor™ machine)
Part II: Ultrasound Scattering from
a Random Distribution of Finite
Objects
Motivation


Ultrasound contrast agents are used to increase
the scattering of ultrasound from blood.
They cause stronger reflections from blood, and
hence improve the contrast of the ultrasound
image.
Contrast Agents

Contrast agents are particularly important
for study of blood flow in small vessels
such as
 Observing
capillary circulation
And study of microcirculations such as
 Assessment of intra-cardiac shunts
 Detection of myocardial perfusion
Contrast Agents


A SuperSoft™ Plastic phantom is molded to test different
contrast agents.
The phantom has two holes which will are filled by the
sample suspension to be tested and water for
comparison
water only
water + contrast agent
Contrast Agents
water only
water + air bubbles
water only
water + cellulose
Encapsulation





Micro-bubbles create very good contrast.
They are clinically safe to inject.
Micro-bubbles do not last very long and pop.
Ultrasound radiation causes the bubbles to pop
faster.
The idea is to encapsulate the bubble inside a
shell.
Sci-Fi or Reality?

Commercial Products:
 Albunex®
Mallinckrodt, Inc., St. Louis, MO.
 Optison® Molecular Biosystems, San Diego,
CA.
 Sonazoid® Daiichi Sankyo, Tokyo, Japan.
…
Encapsulated Microbubble
Materials

Shell Material
 Flexible
 Lipid shell
 Phospho-lipid shell
 Rigid
 Albumin shell
 Gold or Silver?

Gas
 Air
 Octafluoropropane
(lower water solubility)
Shell Sizes, Resonant Frequencies

Radius
– 10 μm
 10 – 100 nm?
1

Thickness
– 200 nm
 Sub nm?
 10

Resonant Frequency
bubbles 3 – 10 μm: 320kHz – 1MHz
 Albunex® microbubbles : 2MHz – 12MHz
 Free
Another Application!


Scattering of acoustic
waves by a finite
spherical object is of
significance elsewhere!
This area was
investigated intensively
in the 1980s and 1990s
for detection of
undersea objects using
under-water acoustics.
Ultrasound Microbubble Interaction
Modeling



Because of linearity,
scattering from an
ensemble could be
modeled by
superimposing the results
from individual elements.
Repeated scattering also
takes place.
A statistical approach
could then be used for a
complete model (more on
this much later on in this
lecture)
Ultrasound Microbubble Interaction
Modeling


The goal is to study the
interaction of a longitudinal
plane wave propagating in
an infinite medium, with a
spherical object.
Since the microbubble can
be a multi-layered elastic
shell encapsulating a gas,
wave equations in elastic
media should be used.
z
y
x
Waves in Elastic Continuum

Two main types of waves can exist in an
infinite elastic continuum:
 Longitudinal
waves
 Shear waves
More on Shear Waves


There is only one direction for
particle motion in a plane
longitudinal wave.
The particles are free to move in a
plane (two directions) for a plane
shear wave.
Continuum Mechanics

Continuum mechanics is the modeling tool for
studying the deformations of a continuum of
matter which is subject to internal and/or
external [forces].

We will shortly see that force
is a secondary quantity, and
the fundamental quantity
from which it is derived, is
stress.
Continuum Mechanics


Consider a
continuum of
matter which has
deformed.
A small volume of
matter located at a
point x has been
displaced by u(x).
u(x)
x
Continuum Mechanics


To study how an
infinitesimal volume
has deformed, the
behavior of u(x) as
a function of x
should be studied.
If u(x) is constant as
a function of x, the
whole body has
displaced and no
deformation has
occurred!
Continuum Mechanics

Deformation occurs when u(x)
changes when x changes.

ux  0
ux  0

dy 0 
0 dz  
x+[dx 0 0 ]T
u x  dx 0 0
T
T
x
x+[0 0 dz]T
T
u
u(x  dx)  u(x)  dx
x
u(x)
Definition of Strain Tensor
 ux
 x
 u
u(x  dx)   y
 x
 uz

 x
ux
y
u y
y
uz
y

ux
2

x

1 u
u
u(x  dx)   y  x
2  x
y
 u u
 z x
z
 x
x+[dx 0 0 ]T
ux 
z 
u y 
dx
z 
uz 

z 
u x u y

y
x
u
2 y
y
uz u y

y
z
x
x+[0 0 dz]T
u(x)

u x uz 

0


z
x


u y uz 
1  u y u x

dx 

z
y 
2  x
y
 u u
uz 

 z x
2
z 
z
 x
Strain Tensor
 ij
u x u y

y
x
0
uz u y

y
z
ux uz 

z
x 

u y uz 

dx
z
y 


0

Rotation Tensor
Definition of Stress Tensor



There is no such thing as a force
acting on a point inside the
body!
Stress is itself a fundamental
concept which relates to the way
a small volume inside the
continuum experiences pushes
and pulls on its surfaces.
Force can be defined in terms of
the stress tensor.
τzz
τzx
τzy
τxz
τxx
τyz
τxy
τyx
τyy
Definition of Stress Tensor


The diagonal components
of the stress tensor create
forces which tend to
change the volume of the
small volume.
The off-diagonal
components of the stress
tensor create forces that
tend to change the shape
of the small volume and
give it a deviated
appearance.
τzz
τzx
τzy
τxz
τxx
τyz
τxy
τyx
τyy
Fundamental Conservation
Equations
The conservation laws of physics impose
conditions on the stress and strain
tensors.
 Conservation equations:

 Mass
 Linear
momentum
 Angular momentum
 Energy




  u x  u y  u z  0
t
x
y
z



 ix   iy   iz  ui
i  x, y , z
x
y
z
 ij   ji
i, j  x, y , z
e 
 
i x , y ,z j  x , y ,z

ji ij
Constitutive Equations




The conservation equations hold in each and
every continuum, regardless of the type of
material the continuum is made up of.
But each material deforms differently, when
subjected to stress.
The constitutive equations, describe the stressstrain relation of the continuum.
The constitutive equations depend on the type of
continuum.
Constitutive Equations
We are interested in elastic continua.
 For these continua, the stress-strain
behavior can be modeled as,


 ii   ( xx   yy   zz )  2 ii
i , j  x, y , z
 ij  2 ij
i , j  x, y , z
λ and μ are called Lamè constants.
Derivation of the Wave Equation

The wave equation in derived by substituting the
constitutive equations in conservation of linear
momentum equations.



 ix   iy   iz  ui
x
y
z

i  x, y , z
 ii   ( xx   yy   zz )  2 ii
i , j  x, y , z
 ij  2 ij
i , j  x, y , z
For example for the x coordinate:






( xx   yy   zz )  2    xx   xy   xz   ux
x
y
z 
 x
Derivation of the Wave Equation

The definition of strain tensor can now be
used to write the latter equation in terms of
displacements.





 ( xx   yy   zz )  2    xx   xy   xz   ux
x
y
z 
 x
 ij 


uj  uj
i
j
i , j  x, y , z
 2

 2
2
2
2
2 
(   ) 2 ux 
uy 
uz     2 ux  2 u y  2 uz   ux
xy
xz 
y
z
 x
 x

The Wave Equation

Using vector derivative notations, we can
write the wave equation in a more
compact form.
 2

 2
2
2
2
2 
(   ) 2 ux 
uy 
uz     2 ux  2 u y  2 uz   ux

x

x

y

x

z
y
z


 x


(   ) (  u)   u   u
2
The Two Components



The wave equation, although being a single equation,
actually models both longitudinal and shear waves.
These waves propagate independently except at the
boundaries.
It is possible to divide the wave equation to separate the
longitudinal and shear waves equations.
u      H,

(   2  )    

2H   H
2
Shear wave:
not present in
acoustics!
H  0
Longitudinal wave:
acoustic waves in
solids, etc.

c    
2
2
Mode Conversion

Longitudinal and shear waves can coexist
peacefully in an infinite medium.
 The
waves would propagate independently.
 The waves would not interact.

As soon as the waves meet a boundary (there
are no boundaries in a infinite medium),
 The
wave equations become coupled.
 The waves would interact.
Mode Conversion
A longitudinal wave, when hitting a
boundary can cause both reflected
longitudinal and shear waves.
 The same statement is true for a shear
wave incident on a boundary.
 The conversion of energy from shear wave
to longitudinal wave and vice-versa is
called mode conversion.

Back to the Future!


The goal still is to study
the interaction of a
longitudinal plane wave
propagating in an infinite
medium, with a spherical
object.
We are now equipped with
the wave equations in
elastic media!
z
y
x
Mode Conversion
z

As the longitudinal
wave interacts with the
spherical microbubble,
part of its energy is
converted into shear
waves!
y
x
Wave Fields

Three wave fields will be present:
 Incident field
 Exists outside the sphere.
 Is a longitudinal wave by assumption.
z
 Scattered field
 Exists outside the sphere.
 May have both longitudinal and shear
components.
 Refracted field
 Exists inside the sphere.
 May have both longitudinal and shear
components.
y
x
Spherical Coordinates
z

Because of the shape
of the boundary
conditions in this
problem, which are
spherical shells, the
most appropriate
coordinate system to
use, is the spherical
coordinate system.
r
θ
φ
x
y
Overview of the Solving Procedure




Axial symmetry of the problem around the z-axis
is used to simplify the form of u.
The wave equations (Helmholtz equations) are
written in the spherical coordinate system.
Separation of variables is used to solve the
equations for basis functions.
Any solution of the problem can then be written
as an appropriately weighted sum of the basis
functions.
What are We Looking for?

We are interested in the scattered wave field:
 How much of the wave energy is scattered?
 What is the directionality of scattering?
 How does the scattering pattern change, when
frequency of wave changes?
the
What are We Looking for?

We are particularly interested in the far-field
behavior of the waves:
 We
are not interested in the scattered wave field
close to the microbubble.
 We are interested in the scattered wave field far away
from it.
 The far-field behavior can be studied by studying the
asymptotic behavior of the solution for large r (much
larger than wavelength).
Scattering Modeling

Consider the scattered wave field part of
the solution to the wave equation,
i ( r, , )  0 exp  i(k1z  t ) 
   ( r,  )
c2 2  
s

We define the scattering function as,
 s ( r, )
f ( )  lim
r 
exp( i (k1r  t ))
0
r
Scattering Modeling

The scattering function for a spherical object
would always be of the form,

f ( )  k1  (2n  1) an Pn (cos  )
n 0


Pn are Legendre polynomials.
The coefficients an are determined by the
material composition of the spherical object (in
conjunction with the rest of the wave fields).
Scattering Modeling



an are complex numbers,
and so is the scattering
function.
This can cause
constructive or destructive
interference with the
incident wave.
Sound reflectivity R is
defined as,
2 f ( )
R
a
z
a
x
y
Published Simulation Results
sphere
g
 medium
h
csphere
cmedium
Source: Zhen Ye, “On sound scattering and attenuation of Albunex® bubbles,” J. Acoust. Soc. Am. 100
(4), Pt. 1, October 1996.
Published Simulation Results
Results for a spherical shell of relative thickness: 2.5%
Surrounding medium: water
Shell type: steel
Gas type: air
Source: John S. Allen, et. al, “Shell Waves and Acoustic Scattering from Ultrasound Contrast Agents,”
IEEE transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 48, No. 2, March 2001.
Published Simulation Results
Fig. 3. The reflectivity versus frequency for a 3.5-μm
radius albumin contrast agent in the limit of an
infinitesimally thin shell. The monopole peak dominates
the response, and a single dipole resonance response
occurs near 33MHz.
Fig. 4. Shown is the reflectivity of the 3.5-μm radius
albumin contrast agent when the shell thickness has been
increased to 100 nm. The monopole response has
diminished in size and has shifted to a higher frequency.
The dipole response has split into two resonance peaks
near 34MHz. The inset shows a close-up of the peak
centered about 33.08MHz.
Source: John S. Allen, et. al, “Shell Waves and Acoustic Scattering from Ultrasound Contrast Agents,”
IEEE transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 48, No. 2, March 2001.
Published Simulation Results
Fig. 5. Shown is the resonance frequency versus shell
thickness of the dipole peaks for the 3.5-μm radius agent
with a shear modulus of 88.8 MPa and a 100-nm thick
shell. Both of the closely spaced dipole peaks are plotted:
upper frequency peak (dashed line) and lower frequency
(solid line). The resonance frequency increases linearly
with shell thickness.
Fig. 6. The reflectivity versus shell thickness is shown for
the lower frequency peak (solid line) and upper frequency
peak (dashed line) for the 3.5-μm radius albumin contrast
agent (100 nm; 88.8 MPa).
Source: John S. Allen, et. al, “Shell Waves and Acoustic Scattering from Ultrasound Contrast Agents,”
IEEE transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 48, No. 2, March 2001.
Published Simulation Results
Fig. 8. Angular scattering calculations
reveal the monopole and dipole
patterns for the 3.5-μm radius
albumin contrast agent (100nm;
88.8MPa). The dotted line
corresponds to the monopole peak
(8.3MHz), and the solid line
corresponds to the dipole peak (34
MHz). A classic Figure 8 pattern is
found for the dipole, but the
monopole peak demonstrates
isotropic scattering.
Source: John S. Allen, et. al, “Shell Waves and Acoustic Scattering from Ultrasound Contrast Agents,”
IEEE transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 48, No. 2, March 2001.
Published Simulation Results
(a)
(b)
Fig. 3. The reflectivity versus the normalized frequency for scattering from an air-filled, encapsulating shell:
(a) double-layered with outer and inner radii: a = 3.5 μm, b = 3.4 μm, and interlayer radius: c = 3.45 μm; (b)
single-layered with outer and inner radii: a = 3.45 μm and b = 3.4 μm, and made of the same material as the
inner layer of the double-layered shell.
Source: Yuantai Hu, et. al, “Characteristics of Acoustic Scattering from a Double-Layered Micro Shell
for Encapsulated Drug Delivery,” IEEE transactions on Ultrasonics, Ferroelectrics, and Frequency
Control, Vol. 51, No. 7, July 2004.
Scattering from Ensembles



A statistical approach could be used to derive averages
of scattering quantities from ensembles of randomly
distributed scattteres.*
As the ultrasound travels through a medium of
scatterers, it loses its energy in the direction of
propagation due to scattering and absorption (mode
conversion, viscosity, etc).
The quantity used to describe the loss of energy of the
wave, is the extinction cross section ,which is the sum of
scattering cross section and absorption cross section:
e  s  a
*A Ishimaru, Wave Propagation in Random Media , Academic Press, Toronto, 1978.
Scattering from Ensembles


In the field of ultrasound, attenuation coefficients
are usually used to characterize the loss of
energy.
Attenuation coefficient is related to extinction
cross section by,
  4.34n e
dB/m
where n is the number density of the scatterers.
Forward Scattering Theorem

The forward scattering theorem* states the
relationship between the extinction cross
section σe with the previously derived
scattering function f(θ):
4
e  
Im f (  0)
k1
*A Ishimaru, Wave Propagation in Random Media , Academic Press, Toronto, 1978.
Scattering Cross Section for
Microbubbles
The reduced scattering
cross section due to
radial pulsation as a
function of the inner
radius of an Albunex®
bubble for various
driving frequencies
represented by the
numbers in the brackets
(unit MHz)
Source: Zhen Ye, “On sound scattering and attenuation of Albunex® bubbles,” J. Acoust. Soc. Am. 100
(4), Pt. 1, October 1996.
THE END