A numerical model for large-amplitude spherical bubble dynamics in tissue

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Dinah Carr
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1 A numerical model for large-amplitude spherical 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 which focused shocks ablate tissue Primary damage mechanism: cavitation Soft tissue is

2 Motivation: ultrasound therapy & cavitation in polymers Histotripsy: therapeutic ultrasound procedure in which focused shocks ablate tissue Primary damage mechanism: cavitation Soft tissue is heterogeneous and viscoelastic Polymers: reduced cavitation activity Objective: to better understand bubble oscillations in viscoelastic media Luminescence in gelatin due to a passing bullet Aimed Research Shocks and cavitation in histotripsy, Maxwell et al. (submitted)

3 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. (JFM 2012) Missing elements: more sophisticated constitutive relations, thermal effects, finite-strain elasticity of the surroundings

4 Theoretical model: general approach Spherical bubble Uniform bubble pressure Zero mass transfer Compressibility of the surroundings

5 Governing equations: bubble Bubble equation (Keller-Miksis, 1980): ( 1 Ṙ ) R R+ 3 ( 1 Ṙ ) = 1 ( 1+ Ṙ + R )( d p p p 2S a(t) c 2 3c ρ c c dt R ) τ rr τ θθ +3 dr R r Bubble pressure equation: ṗ = 3 [ (κ 1)K T ] κpṙ R r R Energy equation (inside/outside bubble Stricker et al., 2012): [ κ 1 p T κ T t + 1 ( (κ 1)K T κp r rṗ ) ] T ṗ = (K T) 3 r T M t + R2 Ṙ T M r 2 r = D M 2 T M + 2 ρc v R 2 Ṙ r 3 (τrr τ θθ)

6 Governing equations: viscoelastic constitutive model τ = 2Gγ +2µ γ λ 1 τ +τ = 2µ γ λ 1 τ +τ = 2µ γ +2λ 2 µ γ λ 1 τ +τ = 2Gγ +2µ γ Kelvin-Voigt Maxwell Jeffreys Zener

7 Governing equations: viscoelastic constitutive model τ = 2Gγ +2µ γ λ 1 τ +τ = 2µ γ λ 1 τ +τ = 2µ γ +2λ 2 µ γ λ 1 τ +τ = 2Gγ +2µ γ Kelvin-Voigt Maxwell Jeffreys Zener ( ( ǫλ1 τ exp )+λ µ tr(τ) 1 τ +α τ τ ) ) = 2 (Gγ +µ γ +µλ 2 γ µ Nonlinear viscoelastic fluids: Upper-Convected Maxwell, Oldroyd-B, Giesekus, Phan-Tien-Tanner Nonlinear viscoelastic solids: hyperelasticity for any strain-energy function (Neo-Hookean, Mooney-Rivlin,...) ǫ τrr exp 1 λ 1 ( ) τrr R 2Ṙ τrr R 2Ṙ τrr + 2τθθ +λ 1 + ǫ 2 µ t r 2 + 4ǫ 2 r r 3 τrr + ǫ 3 τ rr 2 = 4Φ µ r 3 4ǫ 2 µλ R 4Ṙ2 2 r 6 ǫ τ θθ exp 1 λ 1 ( ) τ τθθ + 2τ θθ R 2Ṙ τ θθ R 2Ṙ θθ +λ 1 + ǫ 2 µ t r 2 2ǫ 2 r r 3 τ θθ + ǫ 3 τθθ 2 = 2Φ µ r 3 10ǫ 2 µλ R 4Ṙ2 2 r 6 where

8 Numerical method Spectral collocation method (Chebyshev + Gauss-Lobatto) Coordinate transformations for interior/exterior Gaussian pulse for verification Can transform PDEs for the stresses to ODEs for most models (except Giesekus and PTT)

9 Bubble response in linear viscoelastic media p A = 2MPa, 4MHz, R 0 = 1µm, µ = 30cP, λ 1 = 20ns, G = 1MPa Significant differences between different linear models

10 Bubble response in nonlinear viscoelastic fluids p A = 2MPa, 2MHz, R 0 = 1µm, µ = 30cP, λ 1 = 20ns Nonlinearities affect collapse properties

11 Bubble response in nonlinear (visco)elastic fluids p A = 2MPa, 2MHz, R 0 = 1µm, µ = 30cP, G = 1MPa Significant differences after first cycle

12 Inertial cavitation: past metrics do not hold Presence of microbubbles in US may lead to bleeding Bleeding depends on elastic + pulse properties (Patterson et al., JASA 2012) Inertial cavitation threshold = bioeffects threshold? Past metrics: R max /R o (Flynn, JASA 1975), T max > 5000K (Apfel & Holland, UMB 1991)

13 Viscoelastic media exhibit higher deviatoric stresses

14 Viscoelastic media exhibit higher deviatoric stresses

15 Mechanisms for higher stresses: geometry and properties

16 Mechanisms for higher stresses: geometry and properties

17 Conclusions Model development for bubble dynamics in (nonlinear) viscoelastic media Viscoelastic properties affect the bubble response Deviatoric stresses may cause damage Larger coefficients + geometry Need a new inertial cavitation metric in viscoelastic media Future work: Validation of model via experiments with Steve Ceccio, Zhen Xu (U. Michigan) Cloud initiation in histotripsy Cavitation in the brain Cloud initiation in histotripsy (Vlaisavljevich et al., in press) Acknowledgements: National Science Foundation (CAREER program), Rackham Graduate School Cavitation in the brain (hit to the head)

Models of Ultrasound Contrast Agents. Duncan Alexander Sutherland

Models of Ultrasound Contrast Agents Duncan Alexander Sutherland An essay submitted in partial fulfillment of the requirements for the degree of B.Sc. Honours) Applied Mathematics University of Sydney