[Paper Review] Nonlinear dynamics of acoustic bubbles excited by their pressure dependent subharmonic resonance frequency: oversaturation and enhancement of the subharmonic signal

Amin Jafari Sojahrood, R. Earl+4 more|arXiv (Cornell University)|Sep 9, 2019

Ultrasound and Hyperthermia Applications72 references4 citations

TL;DR

This paper investigates the nonlinear dynamics of uncoated acoustic bubbles excited at pressure-dependent subharmonic (SH) resonance frequencies, demonstrating that sonication near 1.5–1.8 times the linear resonance frequency (fr) induces a saddle-node bifurcation, leading to enhanced SH and ultraharmonic (UH) amplitudes—up to 7 dB in UH and 9.4× in scattered pressure—thereby increasing the contrast-to-tissue ratio and signal-to-noise ratio in ultrasound applications.

ABSTRACT

The acoustic bubble is an example of a highly nonlinear system which is the building block of several applications and phenomena ranging from underwater acoustics to sonochemistry and medicine. Nonlinear behavior of bubbles, and most importantly 1/2 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 their SH resonance frequency (fsh=2fr where fr is the linear resonance frequency), SHs are generated at the lowest excitation pressure. SHs then increase rapidly with pressure increase 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 has not been investigated. In this paper nonlinear dynamics of bubbles excited by frequencies in the range of fr

Motivation & Objective

• To understand the nonlinear dynamics of uncoated acoustic bubbles when excited near their pressure-dependent subharmonic (SH) resonance frequency.
• To identify conditions that enhance the upper saturation limit of SH and ultraharmonic (UH) signals beyond conventional excitation at 2fr.
• To explore bifurcation mechanisms, particularly saddle-node bifurcations, that lead to signal enhancement without bubble destruction.
• To optimize ultrasound exposure parameters for maximizing non-destructive SH and UH amplitudes in diagnostic and therapeutic applications.
• To establish a foundation for studying coated bubbles by first characterizing the simpler uncoated bubble system.

Proposed method

• Numerical simulations of the Rayleigh-Plesset equation for free bubbles (400 nm – 8 µm) under varying excitation frequencies (fr < f < 2fr) and pressures.
• Analysis of bifurcation structures using frequency and pressure sweeps to identify transitions between oscillation regimes (P1, P2) and the onset of period-doubling (PD).
• Identification of saddle-node bifurcations as the mechanism triggering a shift from low-amplitude to high-amplitude SH oscillations.
• Quantification of scattered pressure, wall velocity, and harmonic amplitudes (SH and UH) across different excitation frequencies and pressures.
• Use of amplitude modulation (AM) imaging principles to evaluate signal enhancement potential in diagnostic ultrasound.
• Exclusion of thermal damping and bubble-bubble interactions to isolate intrinsic nonlinear dynamics of isolated uncoated bubbles.

Experimental results

Research questions

• RQ1 How does the subharmonic resonance frequency of a bubble vary with excitation pressure?
• RQ2 What excitation frequency and pressure regime maximize the amplitude of subharmonic and ultraharmonic emissions without bubble destruction?
• RQ3 What bifurcation mechanisms underlie the enhancement of SH and UH signals in nonlinearly driven bubbles?
• RQ4 How does the upper saturation limit of SH and UH signals compare when exciting at the pressure-dependent SH resonance frequency versus the conventional 2fr?
• RQ5 Can the signal-to-noise ratio and contrast-to-tissue ratio be improved by tuning excitation frequency to 1.5–1.8fr?

Key findings

• The subharmonic resonance frequency (fsh) is pressure-dependent and decreases as excitation pressure increases.
• Sonication at f ≈ 1.5–1.8fr induces a saddle-node bifurcation, transitioning the bubble from P1/P2 to a higher-amplitude P2 oscillation regime.
• This bifurcation is accompanied by a 7 dB enhancement in ultraharmonic (UH) amplitude and a 9.4-fold increase in scattered pressure for a 1 µm bubble.
• The upper saturation limit of SH and UH signals is significantly enhanced when excited at the pressure-dependent fsh, rather than at the fixed 2fr.
• The highest non-destructive SH amplitude occurs at f ≈ 1.5–1.8fr, offering improved contrast-to-tissue ratio and signal-to-noise ratio in ultrasound imaging.
• Amplitude modulation (AM) imaging benefits from this enhancement, as pressure changes above and below the saddle-node threshold amplify residual bubble signals while suppressing tissue response.