Modelling Lipid-Coated Microbubbles in Focused Ultrasound Applications at Subresonance Frequencies

We present a computational study of the behaviour of a lipid-coated SonoVue microbubble with initial radius 1 µm ≤ R0 ≤ 2 µm, excited at frequencies (200-1500 kHz) significantly below the linear resonance frequency and pressure amplitudes of up to 1500 kPa-an excitation regime used in many applications of focused ultrasound. The bubble dynamics are simulated using the Rayleigh-Plesset equation and the Gilmore equation, in conjunction with the Marmottant model for the lipid monolayer coating. Also, a new continuously differentiable variant of the Marmottant model is introduced. Below the onset of inertial cavitation, a linear regime is identified in which the maximum pressure at the bubble wall is linearly proportional to the excitation pressure amplitude and the mechanical index. This linear regime is bounded by the Blake pressure, and, in line with recent in vitro experiments, the onset of inertial cavitation is found to occur at an excitation pressure amplitude of approximately 130-190 kPa, depending on the initial bubble size. In the nonlinear regime the maximum pressure at the bubble wall is found to be readily predicted by the maximum bubble radius, and both the Rayleigh-Plesset and Gilmore equations are shown to predict the onset of sub- and ultraharmonic frequencies of the acoustic emissions compared with in vitro experiments. Neither the surface dilational viscosity of the lipid monolayer nor the compressibility of the liquid has a discernible influence on the quantities studied, but accounting for the lipid coating is critical for accurate prediction of the bubble behaviour. The Gilmore equation is shown to be valid for the bubbles and excitation regime considered, and the Rayleigh-Plesset equation also provides accurate qualitative predictions, even though it is outside its range of validity for many of the cases considered.