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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 $1 \, μ\text{m} \leq R_0 \leq 2 \, μ\text{m}$, 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 focused ultrasound applications. 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, likewise, 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 approximately at an excitation pressure amplitude of 130-190 kPa, dependent 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 to in vitro experiments. Neither the surface dilatational viscosity of the lipid monolayer nor the compressibility of the liquid have a discernible influence on the studied quantities, yet accounting for the lipid coating is critical for the accurate prediction of the bubble behaviour. The Gilmore equation is shown to be valid for the considered bubbles and excitation regime, and the Rayleigh-Plesset equation also provides accurate qualitative predictions, even though it is outside its range of validity for many of the considered cases.