Shape stability of a gas cavity surrounded by linear and nonlinear elastic media

Abstract

A number of recent medical procedures such as histotripsy rely on inertially-dominated oscillating gas cavities (i.e. bubbles) inside soft tissue. As a first approximation, soft tissue can be modeled by linear or nonlinear elasticity models and the equations describing bubble dynamics in such media have been derived in previous works. However, these models assume that the bubble remains perfectly spherical at all times for all initial conditions, which is in contradiction with any practical setting or experiments. Ignoring non-spherical behavior could for instance lead to inaccurate predictions for the extent of tissue damage generated during these procedures. The use of such models in practice thus requires one to predict departures from spherical behavior. In this article, departures from sphericity are expressed by non-spherical perturbations. Two sets of equations describing the dynamics of all non-spherical modes are derived for a bubble surrounded by a medium described using linear elasticity and Neo-Hookean hyperelasticity. For both elasticity models and for given initial conditions, bubble shape stability is shown to be controlled by five dimensionless parameters: the Weber number We, the Cauchy number Ca, the dimensionless vapor pressure inside the bubble, the dimensionless initial non-condensible gas pressure inside the bubble and the dimensionless far-field pressure. A growth criterion indicating whether the amplitude of a given non-spherical mode increases exponentially with time is also derived for both models. Bubble shape stability is then compared for both elasticity models during a Rayleigh collapse. Overall, it is found that shape stability is promoted when the shear modulus of the surrounding medium is increased and when the initial step increase in the external pressure is reduced. It is also established that the bubble shape during a Rayleigh collapse is stable over a much wider range of parameters for a surrounding medium described using Neo-Hookean hyperelasticity as opposed to linear elasticity with a similar shear modulus. This could lead to the overprediction of the occurrence of bubble shape instabilities if the surrounding medium is described using linear elasticity, which is particularly problematic during violent bubble collapse.