Dynamics of a Contrast Agent Microbubble Attached to an Elastic Wall

A modified Rayleigh–Plesset equation is derived to model the oscillation of a contrast agent microbubble attached to an elastic wall. The obtained equation shows that contact with the wall affects the bubble oscillation as if the bubble oscillated in a liquid with a changed (effective) density. Depending on the wall properties, the effective density can be either higher or lower than the real liquid density and hence the natural frequency of the attached bubble can be either lower or higher than the natural frequency of the same bubble in an unbounded liquid. Numerical simulations are made for a contrast bubble with shell properties similar to those used in the Marmottant shell model. The cases of a rigid wall and a plastic wall are compared. The properties of the plastic wall are set to correspond to walls of OptiCell chambers commonly used in experiments. It is shown that contact with the rigid wall decreases the natural frequency of the bubble as compared to its natural frequency in an unbounded liquid, whereas contact with the OptiCell wall increases the natural frequency of the bubble. Bubble resonance curves for three cases are compared: the bubble in an unbounded liquid; the bubble at a distance from an OptiCell wall; the bubble in contact with an OptiCell wall. Results obtained for a 2-
$\mu{\rm m}$
-radius bubble insonified with a 10-cycle, 40 kPa, 2.1 MHz Gaussian pulse show that contact with the OptiCell wall leads to the following effects. The amplitude of the radial oscillation of the attached bubble is decreased by about 70% as compared to that of the same bubble in an unbounded liquid. The fundamental component in the spectrum of the scattered pressure of the attached bubble is decreased by 12 dB. A strong second harmonic appears in the spectrum of the scattered pressure of the attached bubble; its magnitude is about 11.5 dB higher than the level corresponding to the case of an unbounded liquid.