Sarah McBurnieProf Jon Chapman
OCIAM, University of Oxford
Medical Imaging – Injecting
Mathematics into the
Problem of Bubbly Blood
03/02/2009 3
Ultrasound Contrast Agents
� Encapsulated gas bubbles
� Injected into the body to
enhance ultrasound imaging
of blood flow
� Characteristic nonlinear echo
http://www.amershamhealth-us.com
A micrograph of
Optison™ with red
blood cells
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And it really does make a difference…
M Main and P Grayburn, American Heart Journal. 137(1):144-153, January 1999
03/02/2009 5
Bubble, bubble, toil and trouble
� UCAs improve the image obtained but…
� Imaging artefacts can occur
� Can we improve the model describing the nonlinear passage of ultrasound through the bubbly blood, to get a better image from UCA-aided scanning?
Courtesy of Dr H. Becher & A. Ehlgen
03/02/2009 6
What is the acoustic response of a
bubbly liquid?
� Bubble interactions?
� Bubble centres drift?
� Asymmetric oscillations?
� Resonance?
� Bubble size and separation
� Incoming wave’s frequency
and speed
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Parameter Sizes
� Microbubble size: ~ 3 µm
� Bubble separation: ~ 0.1 mm
� Wavelength of 3 MHz diagnostic ultrasound in
blood: ~ 0.5 mm
� Image length: ~ 3 cm – 10 cm for cardiac
imaging
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Seek effective equations
� Want to know where regions of “bubbly
blood” are, not the location of every
individual bubble
� …an introduction to
mathematical homogenization…
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I. Homogenization via M.A.E.s
)( 3/1νO
We are seeking a continuum equation for the effective
medium on the outer region scale
)1(O
β = O(ν)≪ 1
ν≪ 1
)(νO
outer region intermediate region
inner region
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One Bubble Model
� Our building block is the response of one bubble:
),( tpp x∞→R(t)
0
pB(t)
Compressible fluid of density ρf ,
constant kinematic viscosity µf ,
speed of sound cf. as ∞→r
Encapsulative shell modelled as
a membrane (no bending
stiffness and isotropic)
Bubble assumed spherically
symmetric initially
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System to Solve
Conservation of mass:
Conservation of momentum:
subject to b.c. at infinity
kinematic condition at
bubble’s surface
dynamic condition at
bubble’s surface
where S is the surface tension and the bubble pressure is given by
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Near Field Solution
The (generalised)
Rayleigh Plesset
Equation
� In the near field we have:
� One bubble,
� which is effectively in incompressible fluid
(since length scale << wavelength),
� And sees only a uniform incoming pressure
(again since length scale << wavelength)
� This is a well known problem, which leads to:
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Far Field Solution
� By expanding (p=p0+ ν p1 + L) and matching we can show that, in the far field, the pressure perturbation p1 satisfies:
� Equivalently:
� So, in the multibubble problem, each bubble acts as a point source to all the other bubbles
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Multibubble ModelFor the bubbly liquid on the outer scale we have:
Homogenizing, by matching between the 3 different regions, we obtain:
where
bubble number density
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Comments
� Linearised solution recovers Foldy’s scattering result
� Nonlinear case returns Van Wijngaarden’sphenomenological equations
� Could (and will) analyse the acoustic response near resonance
� Thus far only looked at small volume fraction cases
� For large volume fraction only have two scales (as bubble size and separation are the same) – need a different technique to handle this…
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II. Homogenization via multiple scales
� Toy problem:
, are assumed 1-periodic in their second argument
� Define a microscale variable
� Seek as an expansion:
in which the are 1-periodic in
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0 0.2 0.4 0.6 0.8 10
5
10
a(x/ε)
x
0 0.2 0.4 0.6 0.8 1−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
x
y
exact solution for ε =0.5
exact solution for ε =0.1
exact solution for ε =0.05
homogenized solution
where
An example
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Extending this idea for our purposes
� 1D � 3D
� a and f discontinuous
� system of equations (connecting bubble
pressure, liquid pressure, bubble radii)
� aperiodic media
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0 0.2 0.4 0.6 0.8 10
5
10a(φ(x)/ε)
x
0 0.2 0.4 0.6 0.8 1−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
x
y
exact solution for ε =0.1
exact solution for ε =0.05
exact solution for ε =0.01
exact solution for ε =0.005
homogenized solution
where
An example
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State of play
Small volume
fraction
Large
volume
fraction
Linear Nonlinear
M.A.E.s
Multiple
Scales
Multiple
Scales
or
M.A.E.s