Rectified growth of histotripsy bubbles

Abstract

Histotripsy treatments use high-amplitude shock waves to fractionate tissue. Such treatments have been demonstrated using both cavitation bubbles excited with microsecond-long pulses and boiling bubbles excited for milliseconds. A common feature of both approaches is the need for bubble growth, where at 1 MHz cavitation bubbles reach maximum radii on the order of 100 microns and boiling bubbles grow to about 1 mm. To explore how histotripsy bubbles grow, a model of a single, spherical bubble that accounts for heat and mass transport was used to simulate the bubble dynamics. Results suggest that the asymmetry inherent in nonlinearly distorted waveforms can lead to rectified bubble growth, which is enhanced at elevated temperatures. Moreover, the rate of this growth is sensitive to the waveform shape, in particular the transition from the peak negative pressure to the shock front. Current efforts are focused on elucidating this behavior by obtaining an improved calibration of measured histotripsy waveforms with a fiber-optic hydrophone, using a nonlinear propagation model to assess the impact on the focal waveform of higher harmonics present at the source’s surface, and photographically observing bubble growth rates.

Introduction

Histotripsy is a type of high intensity focused ultrasound (HIFU) that can be used to mechanically fractionate tissue without producing thermal effects like the denaturation of proteins [1, 2]. This technique is actively being explored for treating various tissues, including prostate, liver, and kidney. To produce these kinds of lesions, short acoustic pulses at very high intensities and relatively low duty cycles are used. However, a relatively large range of acoustic parameters can be used, and this parameter space can be broadly split into two categories. Shorter pulses at higher intensities can be used to directly excite bubble clouds to produce “cavitation histotripsy” [3, 4]; alternately, longer pulses at lower intensities can thermally generate bubble activity to produce “boiling histotripsy” [5, 6].

A feature common to both types of histotripsy is the central role played by bubbles. More specifically, both methods appear to rely on the development of relatively large bubbles: millimeter-sized bubbles are generated by boiling, while effective cavitation clouds comprise bubbles with maximum radii on the order of a hundred microns. The presence of large bubbles to provide a reflective acoustic interface has even been proposed as a mechanism for fractionating tissue into submicron-sized pieces through the phenomenon of ultrasonic atomization [7]. Accordingly, it is of interest to consider how endogenous bubble nuclei that are probably smaller than a micron [8] evolve into larger bubbles upon exposure to the intense ulrasound pulses characteristic of histotripsy.

A model of a single, spherical bubble was previously used to explore the bubble dynamics excited by nonlinear acoustic waveforms [9]. In that work, it was shown that the asymmetry between compressive and tensile portions of shocked waveforms characteristic of histotripsy can lead to rectified bubble growth. While heat and mass transport were found to affect the bubble dynamics, this rectified growth occurred even when transport was neglected in the model. As such, this phenomenon can be associated with liquid momentum and the asymmetry of the excitation waveform, and is not related to other behaviors such as rectified heat transfer [10] and rectified diffusion of non-condensable gases [11].

Methods and Results

In this effort, histotripsy bubble dynamics were investigated computationally to evaluate the impact different nonlinear waveforms. As in the previously cited work [9], an approach based on the Gilmore model for a single, spherical bubble in water [12] was used. Although heat and mass transfer were included in the calculations, transport behaviors did not substantially affect the results: Vapor dynamics were minimal because an ambient temperature of 25°C was used. Moreover, simulation times were too short for appreciable diffusion of non-condensable gases to occur. It is worthwhile to note here the limitations of a spherical model in studying bubbles excited by shocks. Though a basic assumption of Rayleigh-Plesset models is that the acoustic wavelength is much larger than the bubble radius, shocks can include length scales on the order of the bubble radius (or even smaller). Accordingly, the results presented here should be taken to represent a coarse estimate for how the fluid inertia associated with a large bubble responds to incident pressures.

Nonlinear acoustic waveforms used to excite the bubble were simulated using a KZK-type model [13]. For the same source at 2.158 MHz that was characterized in detail by Canney et al., the focal waveform was simulated using a uniform source pressure of p₀ = 0.57 MPa, where p₀ contains only the fundamental frequency ω₀. In addition, non-ideal source conditions were considered by adding a source-pressure component at the third harmonic. Third-harmonic components were taken to be 10% of the fundamental amplitude, either in phase or antiphase relative to the fundamental. Note that the total source power with the third harmonic component was normalized to match that of the fundamental case. The resulting excitation waveforms are plotted in Fig. 1, while the corresponding responses of a 0.5 μm bubble are plotted in Fig 2. Over the period simulated, the bubble undergoes rectified growth only for excitation by the waveform containing an in-phase, third-harmonic component. This case is notable in that the focal waveform has the largest negative pressure and the smallest positive pressure, though all waveforms are “balanced” and vanish when integrated. This result suggests that the bubble dynamics are sensitive to details of the waveform shape. Moreover, because ultrasonic transducers can be excited at their third harmonic, variations in waveform shape as depicted in Fig. 1 are plausible.

Figure 1.

Excitation waveforms calculated at the focus of the 2.158 MHz source described in detail by Canney et al. [13]. The pressure p₀ at the source was taken to comprise either the fundamental frequency ω₀ or the fundamental + 10% third harmonic (in phase or antiphase).

Figure 2.

Response of a 0.5 μm bubble upon excitation by each of the waveforms from Fig. 1.

Discussion

The rectification phenomenon described here can be explained by the asymmetry in nonlinear pressure waveforms, for which the duration of the tensile component exceeds that of the compressive component. As a bubble gets larger, its resonance frequency drops and its characteristic time scale for motion gets longer. Consequently, large bubbles in a nonlinear acoustic field preferentially respond to the tensile part of the wave. This behavior is clearly visible at the end of the red curve in Fig. 2, where the averaged radial growth accelerates as the bubble grows. Such behavior is characteristic of this rectification phenomenon, and implies threshold behavior: if a bubble gets big enough for long enough, rectified growth will ensue. However, this explanation appears overly simplistic as relatively subtle differences in nonlinear waveforms can significantly affect the potential for rectified bubble growth.

Ongoing efforts aim to characterize experimentally the waveforms and corresponding bubble dynamics generated by HIFU transducers used for histotripsy. Characterization includes focal pressure waveforms as measured by a fiber optic hydrophone, voltages used to drive piezoelectric transducers, and photographic observations of bubble behaviors.