Excitation threshold for subharmonic generation from contrast microbubbles

Six models of contrast microbubbles are investigated to determine the excitation threshold for subharmonic generation. The models are applied to a commercial contrast agent; its characteristic parameters according to each model are determined using experimentally measured ultrasound attenuation. In contrast to the classical perturbative result, the minimum threshold for subharmonic generation is not always predicted at excitation with twice the resonance frequency; instead it occurs over a range of frequencies from resonance to twice the resonance frequency. The quantitative variation of the threshold with frequency depends on the model and the bubble radius. All models are transformed into a common interfacial rheological form, where the encapsulation is represented by two radius dependent surface properties-effective surface tension and surface dilatational viscosity. Variation of the effective surface tension with radius, specifically having an upper limit (resulting from strain softening or rupture of the encapsulation during expansion), plays a critical role. Without the upper limit, the predicted threshold is extremely large, especially near the resonance frequency. Having a lower limit on surface tension (e.g., zero surface tension in the buckled state) increases the threshold value at twice the resonance frequency, in some cases shifting the minimum threshold toward resonance.