Sarah McBurnieProf Jon Chapman

OCIAM, University of Oxford

Medical Imaging – Injecting

Mathematics into the

Problem of Bubbly Blood

03/02/2009 2

Diagnostic Ultrasound

http://www.polhemus.com/

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

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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

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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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Close up of an attractive pattern

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…zooming out…

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…and again…

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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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Result

� and satisfies:

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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

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Any Questions?