Nonlinear dynamics of acoustic bubbles excited by their pressure-dependent subharmonic resonance frequency: influence of the pressure amplitude, frequency, encapsulation and multiple bubble interactions on oversaturation and enhancement of the subharmonic signal

Nonlinear behavior of bubbles, and most importantly 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document}-order subharmonics (SH), are used to increase the contrast-to-tissue ratio (CTR) in diagnostic ultrasound (US) and to monitor bubble-mediated therapeutic US. It is shown experimentally and numerically that when bubbles are sonicated with a frequency that is twice their resonance frequency (fr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{r}$$\end{document}), SHs at half the excitation frequency are generated at the minimum excitation pressure. Thus, f=2fr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=2f_\mathrm{r}$$\end{document} is defined as the SH resonance frequency. SHs increase rapidly with pressure and reach an upper limit of the achievable SH signal strength. Numerous studies have investigated the pressure threshold of SH oscillations; however, conditions to enhance the saturation level of SHs have not been investigated. In this paper, nonlinear dynamics of bubbles excited at frequencies in the range of fr<f<2fr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{r}<f<2f_\mathrm{r}$$\end{document} is studied for different sizes of bubbles (400 nm–8 μm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu \mathrm{m}$$\end{document}). In agreement with previous studies, we show that the SH resonance frequency is pressure dependent and decreases as pressure increases. When a bubble is sonicated with its pressure-dependent SH resonance frequency, oscillations undergo a saddle node bifurcation from a P1 or P2 regime to a P2 oscillation regime with higher amplitude. The saddle node bifurcation is concomitant with over-saturation of the SH and UH amplitude and eventual enhancement of the upper limit of SH and UH strength (e.g., ≈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx $$\end{document} 7 dB in UH amplitude). This can increase the CTR and signal-to-noise ratio in applications. Here, we show that the highest non-destructive SH amplitude occurs when f≊1.6-1.8fr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\approxeq 1.6-1.8f_\mathrm{r}$$\end{document}. In the case of interacting bubbles, fr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{r}$$\end{document} and pressure threshold of SH emissions decrease with increasing concentration, and supersaturation amplitude decreases above an upper limit of concentration.