A numerical model for large-amplitude spherical bubble dynamics …mwarnez/ASA2013.pdf ·...

A numerical model for large-amplitudespherical bubble dynamics in tissue

Matthew Warnez, Renaud Gaudron & Eric Johnsen

ASA meeting, San Francisco, Dec. 2-6, 2013

Motivation: ultrasound therapy & cavitation in polymers

Histotripsy: therapeutic ultrasound procedure in whichfocused shocks ablate tissue

Primary damage mechanism: cavitationSoft tissue is heterogeneous and viscoelastic

Polymers: reduced cavitation activity

Objective: to better understand bubble oscillations inviscoelastic media

Shocks and cavitation in histotripsy, Maxwell et al. (submitted)

Luminescence in

gelatin due to a

passing bullet

Aimed Research

M. Warnez, R. Gaudron & E. Johnsen, U. Michigan Spherical bubble dynamics in tissue

Past theoretical work on spherical bubble dynamics in viscoelastic media

Bubble dynamics: Rayleigh-Plesset/Keller-Miksis

Constitutive relations:

Maxwell models: Fogler & Goddard (PoF 1970), Allen & Roy(JASA 2000), Jimenez-Fernandez & Crespo (US 2006), Brujan(2010), ...Kelvin-Voigt models: Yang & Church (JASA 2005), Hua &Johnsen (PoF 2013), ...Finite-strain elasticity (for viscoelastic shell): Liu et al. (JFM2012)

Missing elements: more sophisticated constitutive relations,thermal effects, finite-strain elasticity of the surroundings


Theoretical model: general approach

Spherical bubble

Uniform bubble pressure

Zero mass transfer

Compressibility of thesurroundings


Governing equations: bubble

Bubble equation (Keller-Miksis, 1980):(

1−R

c∞

)

RR+3

2

(

1−R

3c∞

)

=1

ρ∞

(

1 +R

c∞+

R

c∞

d

dt

)(

p− p∞ − pa(t)−2S

R

+3

∫

∞

R

τrr − τθθ

rdr

)

Bubble pressure equation:

p =3

R

[

(κ− 1)K∂T

∂r

∣

∣

∣

R

− κpR

]

Energy equation (inside/outside bubble – Stricker et al., 2012):

κ− 1

κ

p

T

[

∂T

∂t+

1

κp

(

(κ− 1)K∂T

∂r−

rp

3

)

∂T

∂r

]

− p = ∇ · (K∇T )

∂TM

∂t+

R2R

r2∂TM

∂r= DM∇

2TM +

2

ρcv

R2R

r3(τrr − τθθ)


Governing equations: viscoelastic constitutive model

τ = 2Gγ + 2µγ

λ1τ + τ = 2µγ

λ1τ + τ = 2µγ + 2λ2µγ

λ1τ + τ = 2Gγ + 2µγ

Kelvin-Voigt

Maxwell

Jeffreys

Zener


Governing equations: viscoelastic constitutive model

τ = 2Gγ + 2µγ

λ1τ + τ = 2µγ

λ1τ + τ = 2µγ + 2λ2µγ

λ1τ + τ = 2Gγ + 2µγ

Kelvin-Voigt

Maxwell

Jeffreys

Zener

τ exp

(

ǫλ1

µtr (τ )

)

+ λ1

(

▽

τ +ατ · τ

µ

)

= 2

(

Gγ + µγ + µλ2

▽

γ

)

Nonlinear viscoelastic fluids: Upper-Convected Maxwell, Oldroyd-B,Giesekus, Phan-Tien-Tanner

Nonlinear viscoelastic solids: hyperelasticity for any strain-energyfunction (Neo-Hookean, Mooney-Rivlin, ...)

τrr exp

ǫ1λ1

µ

(

τrr + 2τθθ

)

+λ1

∂τrr

∂t+ ǫ2

R2R

r2

∂τrr

∂r+ 4ǫ2

R2R

r3τrr +

ǫ3

µτ2rr

= −

4Φ

r3−4ǫ2µλ2

R4R2

r6

τθθ exp

ǫ1λ1

µ

(

τθθ + 2τθθ

)

+λ1

∂τθθ

∂t+ ǫ2

R2R

r2

∂τθθ

∂r− 2ǫ2

R2R

r3τθθ +

ǫ3

µτ2θθ

=2Φ

r3−10ǫ2µλ2

R4R2

r6

whereG (

3 3)

2(

2 2)


Numerical method

Spectral collocation method (Chebyshev + Gauss-Lobatto)

Coordinate transformations for interior/exteriorGaussian pulse for verification

Can transform PDEs for the stresses to ODEs for most models(except Giesekus and PTT)


Bubble response in linear viscoelastic media

pA = 2MPa, 4MHz, R0 = 1µm, µ = 30 cP, λ1 = 20ns,G = 1MPaSignificant differences between different linear models


Bubble response in nonlinear viscoelastic fluids

pA = 2MPa, 2MHz, R0 = 1µm, µ = 30 cP, λ1 = 20ns

Nonlinearities affect collapse properties


Bubble response in nonlinear (visco)elastic fluids

pA = 2MPa, 2MHz, R0 = 1µm, µ = 30 cP, G = 1MPa

Significant differences after first cycle


Inertial cavitation: past metrics do not hold

Presence of microbubbles in US may lead to bleeding

Bleeding depends on elastic + pulse properties (Patterson etal., JASA 2012)

Inertial cavitation threshold = bioeffects threshold?

Past metrics: Rmax/Ro (Flynn, JASA 1975), Tmax > 5000K(Apfel & Holland, UMB 1991)


Viscoelastic media exhibit higher deviatoric stresses


Mechanisms for higher stresses: geometry and properties


Conclusions

Model development for bubble dynamics in(nonlinear) viscoelastic media

Viscoelastic properties affect the bubbleresponse

Deviatoric stresses may cause damage

Larger coefficients + geometryNeed a new inertial cavitation metric inviscoelastic media

Future work:

Validation of model via experiments withSteve Ceccio, Zhen Xu (U. Michigan)Cloud initiation in histotripsyCavitation in the brain

Acknowledgements: National Science Foundation(CAREER program), Rackham Graduate School

Cloud initiation in histotripsy

(Vlaisavljevich et al., in press)

Cavitation in the brain

(hit to the head)


A numerical model for large-amplitude spherical bubble dynamics …mwarnez/ASA2013.pdf ·...

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