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 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)
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
Theoretical model: general approach Spherical bubble Uniform bubble pressure Zero mass transfer Compressibility of the surroundings
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)
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
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
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)
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
Bubble response in nonlinear viscoelastic fluids p A = 2MPa, 2MHz, R 0 = 1µm, µ = 30cP, λ 1 = 20ns Nonlinearities affect collapse properties
Bubble response in nonlinear (visco)elastic fluids p A = 2MPa, 2MHz, R 0 = 1µm, µ = 30cP, 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 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)
Viscoelastic media exhibit higher deviatoric stresses
Viscoelastic media exhibit higher deviatoric stresses
Mechanisms for higher stresses: geometry and properties
Mechanisms for higher stresses: geometry and properties
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)