Suppression of shocked-bubble expansion due to tissue confinement with application to shock-wave lithotripsy

Estimates are made of the effect of tissue confinement on the response of small bubbles subjected to lithotriptor shock pressures. To do this the Rayleigh-Plesset equation, which governs the dynamics of spherical bubbles, is generalized to treat a bubble in a liquid region (blood), which is in turn encased within an elastic membrane (like a vessel's basement membrane), beyond which a Voigt viscoelastic material models the exterior tissue. Material properties are estimated from a range of measurements available for kidneys and similar soft tissues. Special attention is given to the constitutive modeling of the basement membranes because of their expected importance due to their proximity to the bubble and their toughness. It is found that the highest expected values for the elasticity of the membrane and surrounding tissue are insufficient to suppress bubble growth. The reduced confinement of a cylindrical vessel should not alter this conclusion. Tissue viscosities taken from ultrasound measurements suppress bubble growth somewhat, though not to a degree expected to resist injury. However, the higher reported viscosities measured by other means, which are arguably more relevant to the deformations caused by growing bubbles, do indeed significantly suppress bubble expansion.