The effect of visco-elasticity on the stability of inertial cavitation

D. Sinden, E. Stride & N. Saffari

Department of Mechanical Engineering,

University College London

d.sinden@ucl.ac.uk

Effective Cavitation

Cavitation is more than just an indicator of the thermal and acoustical fields generated: cavitation can potentially enhance or inhibit effective treatment:

Inertial collapse of bubbles can localise heat deposition

Through shielding cavitation can defocus the ultrasonic beam and lead to the pre-focal destruction of healthy tissue.

Modelling Cavitation

For a unified analysis of cavitation in vivo two phenomena need to be modelled consistently:

The far-field effect of non-linear wave propagation through tissue

The near-field effect of the tissue on the bubble

Far-field Effects: Burgers' Equation

For nonlinear constitutive relations (Lardner 76):

the wave propagation model is equivalent to Burgers' equation

This has closed-form solutions:

Nonlinear Wave Propagation

Near-field Effects: Kelvin-Voigt Model

The effect of the nonlinear constitutive relations on the dynamics of a oscillating spherical bubble can be modelled as (Yang & Church 05):

where the Kelvin-Voigt model yields

Stability Thresholds Near Critical Blake Radius

Value of critical Blake radius increases:

Natural frequency of free oscillations also increases:

Stability Thresholds Near Critical Blake Radius

Using techniques from dynamical systems theory, (Harkin 1999) a revised stability threshold is found: indicating that, typically, unpredictable oscillations occur at higher driving pressures and larger smaller initial radii, furthermore:

Inertial collapse happens earlier within each cycle

Magnitude of collapse is less violent

Competing Influences in Unified Model

Nonlinear wave

propagation

Kelvin Voigt model

Fourier SpectraFor stable (repetitive) cavitation higher order harmonic components are observed in computations

Note: Received signal from cavitation activity needs to be an initial solution for Burgers' eq.

Primary Bjerknes Forcing

When nonlinear wave propagation is considered the effective force on the bubble is modified:

However, the effect of visco-elasticity inhibits translational motion (Magnaudet & Legendre 98)

Shape Instability

Possible additional resonances of higher order modes of oscillation excited higher order frequency components due to nonlinear wave propagation

Energy Transfer between modes

In the full modelling of shape oscillations, energy can be transfered between modes:

Main source of energy transfer is between radial motion (zeroth mode) and translational motion (first mode) however due to visco-elastic effects this is damped

Preliminary analysis suggests unstable, smaller, higher order modes are less damped in comparison with larger lower order modes

Summary

Unified analysis of effects of visco-elasticity in a uniform nonlinear medium as an example of the artefacts of associated with visco-elasticity

Thermal effects (in both tissue and bubbles) neglected

Revised stability thresholds

Effects of shape instability investigated

Effects on treatment monitoring highlighted

Implications

• Higher order modes of oscillations persist,leading to increased likelihood of instability.

• High Frequency components may not be detected

• Consequently thermal fields may be under estimated

16

Acknowledgements

The THIFU consortium: ‘Transcostal High-Intensity Focussed Ultrasound for the Treatment of Cancer’

UCL: Nader Saffari, Eleanor Stride, David Hawkes, Dean Barratt,Grigroy Vilenskiy, David Sinden, Erik-Jan Reijkhorst, Daniel Heanes, Pierre Gelat (NPL)

ICR: Gail ter Haar, Ian Rivens, Simon Woodford, Lise Retat, Richard Symonds-Tayler

University of Oxford BUBL: Constantin Coussios, Stephane Labouret, Miklos Gyongy, Ian Webb.

Oxford HIFU Unit: David Cranston, Tom Leslie

Funding & support: •EP/F025750/1•EP/F02617X/1•EP/F029217/1

References

X. Yang & C. C Church, “A model for the dynamics of gas bubbles in soft tissue”, J. Acoust. Am. Soc. 118 (2005)

R. W. Lardner “The development of plane shock waves in nonlinear viscoelastic media”, Proc R. Soc. Lond A 348 (1976)

A. Harkin, A. Nadim, T. J. Kuper “On acoustic cavitation of slightly subcritical bubbles” Phys. Fluids 11 (1999)

J. Maguadet & D. Legendre “The viscous drag force on a spherical bubble with a time-dependent radius” Phys. Fluids 10 (1998)