Response of constrained and unconstrained bubbles to lithotripter shock wave pulses

Abstract

The Gilmore formulation for spherical bubble dynamics [F. R. Gilmore, The Growth or Collapse of a Spherical Bubble in a Viscous Compressible Liquid (California Institute of Technology, Pasadena, CA, 1952), Rep. No. 26-4] is used to investigate the response of air bubbles to a variety of lithotripter shock waveforms. A modification of the Gilmore model is proposed to account for partial constraint of the bubble expansion that can be caused by bubble coatings (such as in echo contrast agents) and by tissues or vessels surrounding bubbles in vivo. In the modified formulation, a viscoelastic membrane is assumed to exist at the bubble interface to include the possible effects of the nonlinear elasticity and strain rate dependent viscosity on the bubble response. The stress induced in the membrane is assumed to be an exponential function of the bubble radius, which tends to restrict the bubble expansion. The viscosity is assumed to increase with the strain rate. In the absence of the membrane, the maximum bubble wall pressure induced by a negative (tensile) pulse is much larger than that induced by a positive (compressive) pulse of the same pressure waveform and amplitude. This difference increases with decreasing initial bubble radius. The addition of the viscoelastic membrane significantly decreases the predicted maximum bubble pressure and the difference in response between the positive and negative pulses. The effect of the time delay between double pulses (positive followed by negative or negative followed by positive) is also investigated for unconstrained bubbles.

Bioeffects of positive and negative acoustic pressures in vivo

“…From this model, the magnitude of the negative as opposed to the positive pressure in the acoustic wave has come to be regarded as a predictor of the probability of inertial cavitation ͑AIUM/NEMA, 1992͒. However, if the expansion of a bubble during rarefaction is constrained, the violence of collapse is reduced ͑Ding and Gracewski, 1994;Church, 1995͒. Since bubbles are likely to be constrained in one way or another in the body, cavitation events may be less violent in vivo than they would be in open water. But, more important for the present investigation, if bubble expansion is constrained, the relative importance of negative as opposed to the positive pressure in bubble-related effects should be reduced ͑Ding and Gracewski, 1994͒.…”

Numerical analysis of bubble dynamics in electrohydraulic and electromagnetic shock wave lithotripsy

“…Coleman and colleagues also used the Gilmore model to study the cavitation produced by the 1 st and 2 nd shock wave in an HM-3 lithotripter, and confirmed the theoretical results by measurement of acoustic emission signals (Coleman et al, 1992). Ding and Gracewski proposed a modified Gilmore model, in which a viscoelastic membrane was included in the original Gilmore model, to model the dynamics of bubble with a viscoelastic wall (Ding and Gracewski, 1994). Zhu and Zhong used the Gilmore model coupled with zero-order gas diffusion to simulate the dynamics of bubble produced by different shock wave sequences of a modified XL-1lithotripter (Zhu and Zhong, 1999).…”

Time-lapse nondestructive assessment of shock wave damage to kidney stones in vitro using micro-computed tomography

“…This may not have eliminated cavitation altogether, but it is likely that gauze would interfere with bubble expansion, therefore, reducing the opportunity for effective bubble collapse. 10 This stone showed essentially no damage at its proximal face, rather it suffered the loss of a large chip from its distal side. This is consistent with failure by spall.…”

“…These include: Spall: the compressive component of the shock wave reflects off the distal surface and the stone fails in tension. 1,9,10 Cavitation: the tensile component of the shock wave makes small bubbles grow in the fluid surrounding the stone; the violent collapse of the bubbles acts principally on the proximal surface of the stone. 9,11,12 Squeezing: as the shock wave propagates through the stone a differential stress between the stone and the fluid develops which leads to a bulging and splitting along the SW axis.…”