Resonance frequencies of lipid-shelled microbubbles in the regime of nonlinear oscillations

Abstract

Knowledge of resonant frequencies of contrast microbubbles is important for the optimization of ultrasound contrast imaging and therapeutic techniques. To date, however, there are estimates of resonance frequencies of contrast microbubbles only for the regime of linear oscillation. The present paper proposes an approach for evaluating resonance frequencies of contrast agent microbubbles in the regime of nonlinear oscillation. The approach is based on the calculation of the time-averaged oscillation power of the radial bubble oscillation. The proposed procedure was verified for free bubbles in the frequency range 1-4 MHz and then applied to lipid-shelled microbubbles insonified with a single 20-cycle acoustic pulse at two values of the acoustic pressure amplitude, 100 kPa and 200 kPa, and at four frequencies: 1.5, 2.0, 2.5, and 3.0 MHz. It is shown that, as the acoustic pressure amplitude is increased, the resonance frequency of a lipid-shelled microbubble tends to decrease in comparison with its linear resonance frequency. Analysis of existing shell models reveals that models that treat the lipid shell as a linear viscoelastic solid appear may be challenged to provide the observed tendency in the behavior of the resonance frequency at increasing acoustic pressure. The conclusion is drawn that the further development of shell models could be improved by the consideration of nonlinear rheological laws.

Fig. 1

Experimentally determined resonance frequency versus equilibrium radius for lipid-shelled bubbles in the regime of linear oscillations. Circles indicate experimental estimates obtained by Sun et al. [8] for Definity^®. Asterisks indicate experimental estimates obtained by van der Meer et al. [9] for BR-14. The dashed line shows the linear resonance frequency for a free bubble, calculated by (2). The solid line shows the linear resonance frequency for an encapsulated bubble, calculated by (8) – (10).

Fig. 2

Oscillation power as a function of equilibrium radius for free bubbles. The excitation is a 20-cycle, 2.5 MHz, 100 kPa acoustic pulse.

Fig. 3

Resonance frequency versus equilibrium radius for free bubbles at increasing values of the acoustic pressure amplitude. The dotted line corresponds to the linear damped resonance frequency given by (2). The excitation is a 20-cycle acoustic pulse. The dashed lines represent results obtained in the case that the excitation is a continuous sinusoidal wave of the same amplitude.

Fig. 4

Oscillation power versus equilibrium radius for lipid-shelled bubbles insonified with a 20-cycle, 100 kPa acoustic pulse at four frequencies. Circles indicate results obtained from experimental radius-time curves. The solid lines show a polynomial interpolation for the envelopes of the experimental points.

Fig. 5

Oscillation power versus equilibrium radius for lipid-shelled bubbles insonified with a 20-cycle, 100 kPa acoustic pulse at four frequencies. Circles indicate results obtained from experimental radius-time curves. The solid lines show a polynomial interpolation for the envelopes of the experimental points.

Fig. 6

Resonance frequency versus equilibrium radius for lipid-shelled bubbles. Triangles and diamonds indicate experimental estimates following from Fig. 4 (100 kPa) and Fig. 5 (200 kPa), respectively. The F100 and F200 curves correspond to free bubbles under the same acoustic conditions. Circles indicate experimental estimates obtained by Sun et al. [8] for the linear regime.

Fig. 7

Comparison of the de Jong and the Maxwell shell models. The dJ and dJ200 curves are given by the de Jong model in the linear regime and at 200 kPa, respectively. The M and M200 curves are given by the Maxwell model in the linear regime and at 200 kPa, respectively.