Models of Ultrasound Contrast Agents · L. Ho , Acoustic Characterization of Contrast Agents for Medical Ultrasound Imaging, Springer 2001. C.E. Brennen, Cavitation and Bubble Dynamics,

Ultrasound Contrast agents Response of microbubble Models Results Limitations of models Summary

Models of Ultrasound Contrast AgentsApplied Mathematics Honours

Duncan Alexander Sutherland

Supervisor: Dr. R. S. Thompson

18th September, 2008

Duncan Alexander Sutherland Models of Ultrasound Contrast Agents


Ultrasound contrast agents: outline

Medical Ultrasonography

An acoustic signal in the 2− 15 MHz range is generated by atransducer

The signal returned to the transducer by tissue is detectedand interpreted to form an image

The reflected signal from blood is 30-60 dB less than thesignal from tissue

Microbubble contrast agents

Intravenous injection of gas filled microbubbles with lipid,polymer or albumin shells

Enhanced backscatter and frequency dependent oscillatoryresponse



Ultrasound contrast agents: enhanced image

Image showing EchoGen contrast agent in use from

http://people.maths.ox.ac.uk/ mcburnie/research.html



Response of the microbubble to driving ultrasound

Low power, low amplitudedriving pressure

The bubble has a sizedependent linear resonancefrequencyContrast agent size is1− 7µm ⇒ resonantfrequencies in the 2− 15 MHzrange

Increased driving pressureamplitude

Nonlinear bubble response.Generation of higherharmonics and sub harmonics



Models: Assumptions

Require a model for the gas, the shell if present, and the liquid

The bubble persists indefinitely. No mass transport across thebubble wall

Spherical symmetry at all times. Reasonable for smalloscillations far from boundaries

Fluid disturbances are irrotational thus ∇× u = 0. Velocitypotential u = ∇Φ, u is the velocity field

No body forces upon the bubble. Gravitational forces arenegligible due to the low mass of the contrast agent

The viscous effects of the gas are negligibly small. Thermalprocesses in the gas are assumed to be polytropicpV γ = constant



Models: Schematic diagram

Obtain an ODE that models the bubble radius R(t).



A first model: the Rayleigh Plesset equation

Additional assumptions: inviscid and incompressible fluid

Velocity by continuity equation u(r , t) =∂Φ

∂r(r , t) =

(R

r

)2

R

Momentum equation1

ρ

∂p(r , t)

∂r+∂u

∂t+ u

∂u

∂r= 0

Bernoulli equation∂Φ

∂t+

1

2

(∂Φ

∂r

)2

+

∫ p(r)

p(∞)

dp

ρ= 0

Boundary conditions

{p(R, t) = pL(t)

p(∞, t) = p∞ = p0 + P(t)

Rayleigh Plesset equation RR +3

2R2 =

pL(t)− p∞ρ

(1)



Improved model: Changed conditions at wall

Forces balance at the bubble pL(R, t) = pG (R, t)− 2σ

R− 4µR

R

Polytropic Gas pG (R, t) = pG (R0, 0)

(R0

R

)3γ



Improved model: RPNNP equation

Include viscous (Newtonian fluid) and surface tension terms:

pL(R, t) = pG

(R0

R

)3γ

− 2σ

R− 4µR

RModified BC

RR +3

2R2 =

1

ρ

((p0 +

2σ

R0)

(R0

R

)3γ

− p0 − P(t)− 4µ

RR − 2σ

R

)(2)

RPNNP (Rayleigh Plesset Noltingk Neppiras Poritsky) equation.



Improved model: Modified Herring

Finite, constant, sound speed c∞ in the liquid. Assume that thebubble oscillation creates a spherically diverging wave in the liquid.

Acoustic approximation, small amplitude M =R

c∞� 1

Velocity potential in the liquid

(∂

∂t+ c∞

∂

∂r

)(rΦ) = 0

Bernoulli equation∂Φ

∂t+

1

2

(∂Φ

∂r

)2

+

∫ p(r)

p(∞)

dp

ρ= 0

Modified Herring Model RR +3

2R2 − RpL

ρc∞=

pL − p∞ρ

(3)



Improved model: Modified Herring, comments

RR +3

2R2 − RpL

ρc∞=

pL − p∞ρ

Radiation damping correction term added to the RP equation (1)

Derivation proceeds differently to the Rayleigh Plesset, due to extracondition on φ

Radiation damping arises from calculating the time dependentpressure at the bubble wall

For an incompressible fluid c∞ →∞, gives the RP equation

The full Herring model contains further correction terms:

RR

(1− 2R

c∞

)+

3R2

2

(1− 4R

3c∞

)=

R

c∞ρ

dpL

dt

(1− R

c∞

)+

pL − p∞ρ



Shell model: diagram

(a) Diagram of shelled contrast agent (b) Schematic of shell

(a) is from MJK Blomley et al British Medical Journal (2001)



Shell model: Church’s model

For a bubble with an encapsulating shell, the momentum equationfor the liquid and shell is:

ρ(r)

(∂u

∂t+ u

∂u

∂r

)= −∂p

∂r+∂τrr (r)

∂r+

3τrr (r)

r

The density becomes a function of radius. τrr is the radialcomponent of stress (force per unit area). Proceeding as for theRP derivation yields:

R1R1

[1 +

ρL − ρS

ρS

R1

R2

]+ R1

2[

3

2+

(ρL − ρS

ρS

)4R3

2R1 − R41

2R42

]=

1

ρS

(pG (R1, t)− P(t)− p0 −

2σ1

R1− 2σ2

R2+

∫ R2

R1

3τSrr

rdr +

∫ ∞R2

3τLrr

rdr

)(4)



Shell model: continued

Expressions for the radial stresses in the shell and the liquid arerequired. The liquid is assumed Newtonian (a linear relationbetween stress and rate of strain) which yields:∫ ∞

R2

3τLrr

rdr = −4µLR

21

R32

R1 (5)

And the shell may be modelled as a linear viscoelastic materialwith viscosity µS and elastic modulus GS .∫ R2

R1

3τSrr

rdr = −[4GS(R1 − R1E

) + µS R1]

(R3

2 − R31

R32R1

)(6)

Equations (5) and (6) completely specify equation (4). Severalmodels exist for the shell, eg: Newtonian fluid, and exponentialmodel.



Results: Linearisation

Assume small perturbations in radiusR(t) = R0(1 + ε(t)) with ε(t)� 1

RP, RPNNP, mod. Herring andChurch’s equations (1), (2), (3), (4)reduce to linear harmonic oscillatorequations

ω0 = 1R0

√3γp∞

ρ + (other)

ρL = 1000Kg/m3, ρS = 1100Kg/m3,p0 = 0.1MPa, γ = 1.4,σ1 = 7Pa,σ2 = 0.5Pa, Gs = 88.8MPa,R2 − R1 = 15nm



Results: Response of RP bubble

(c) Amplitude: 0.1MPa (d) Amplitude: 2MPa

Bubblesim, MATLAB program using ode15s, Gear’s method solver.Hanning Pulse, centre frequency: 5MHz, air bubble with initialradius: 4µm, ω0 = 5.2MHz.



Limitations

Continuum models of shells maybe unrealistic

Shell is a few molecules thick

Spherically symmetricoscillations and a stationarybubble are unrealistic

Presence of boundaries (bloodvessels) and a non-stationaryfluid (blood)

Microbubble may rupture, theboundary or shell is permeableto gas

http://www.ntnu.no/ustech/ab



Limitations

Continuum models of shells maybe unrealistic

Shell is a few molecules thick

Spherically symmetricoscillations and a stationarybubble are unrealistic

Presence of boundaries (bloodvessels) and a non-stationaryfluid (blood)

Microbubble may rupture, theboundary or shell is permeableto gas

IEEE Ultrasonics, March 2002, cover.



Summary

Rayleigh Plesset models

Modified to include viscosity and radiation dampingLinearisation, predicts resonant frequencySimilar nonlinearities in R(t) ODEs for all models

Shell models

Rayleigh Plesset like modelModification of resonant frequenciesShell is difficult to model



Summary: oscillating contrast agent

Ultrasound contrast agent bubble oscillating at 0.5 MHz imaged at5 Mfps.

Looping is artificial. http://www.brandaris128.nl/



Summary: original papers and other references

L. Rayleigh, Philosophical Magazine, 1917

L. Trilling, Journal Applied Physics, 1951

C. Herring, OSRD Report, 1941.

C. Church, Journal of the Acoustical Society of America,1995.

L. Hoff, Acoustic Characterization of Contrast Agents forMedical Ultrasound Imaging, Springer 2001.

C.E. Brennen, Cavitation and Bubble Dynamics, OUP 1995.


Documents

Models of Ultrasound Contrast Agents · L. Ho , Acoustic Characterization of Contrast Agents for Medical Ultrasound Imaging, Springer 2001. C.E. Brennen, Cavitation and Bubble Dynamics,