High order harmonic balance formulation of free and encapsulated microbubbles

The radial responses of free or encapsulated micro-bubbles excited by a plane wave of large wavelength are governed by NonLinear Ordinary Differential Equations (NL-ODEs). The nonlinear frequency response details the harmonic content of the time response and constitutes the expected outcome of a high order harmonic analysis. In this paper, a high order harmonic balance analysis of the ”RNNP” (bubble), Hoff and Marmottant (contrast agents) models are performed with the open-source Manlab software extensively described in a previous paper. With this purpose, the original NL-ODEs are recast into nonlinear systems for which the non-linearities are at most quadratic. In the spectral domain, this recast provides close form and aliasing-free solutions of arbitrary large number of harmonics. Relevant quantities such as primary and secondary resonances or the nonlinear amplitude threshold of the excitation wave are evaluated. The frequency curves drawn characterize the bending and quantify the jump frequencies and amplitudes of each harmonic component. The results obtained with this predictive methodology confirm the significant aid it may provide in the derivation of bubble nonlinear detection and sizing techniques or in the design of contrast agents and their optimal ultrasound activation.