Cavitation-induced pressure saturation: a mechanism governing bubble nucleation density in histotripsy

Abstract

Objective.Histotripsy is a noninvasive focused ultrasound therapy that mechanically disintegrates tissue by acoustic cavitation clouds. In this study, we investigate a mechanism limiting the density of bubbles that can nucleate during a histotripsy pulse. In this mechanism, the pressure generated by the initial bubble expansion effectively negates the incident pressure in the vicinity of the bubble. From this effect, the immediately adjacent tissue is prevented from experiencing the transient tension to nucleate bubbles. Approach.A Keller-Miksis-type single-bubble model was employed to evaluate the dependency of this effect on ultrasound pressure amplitude and frequency, viscoelastic medium properties, bubble nucleus size, and transducer geometric focusing. This model was further combined with a spatial propagation model to predict the peak negative pressure field as a function of position from a cavitating bubble.Main results. The single-bubble model showed the peak negative pressure near the bubble surface is limited to the inertial cavitation threshold. The predicted bubble density increased with increasing frequency, tissue viscosity, and transducer focusing angle. The simulated results were consistent with the trends observed experimentally in prior studies, including changes in density with ultrasound frequency and transducerF-number.Significance.The efficacy of the therapy is dependent on several factors, including the density of bubbles nucleated within the cavitation cloud formed at the focus. These results provide insight into controlling the density of nucleated bubbles during histotripsy and the therapeutic efficacy.

Keywords: acoustic cavitation; cavitation nucleation; focused ultrasound; histotripsy.

Figure 1.

Example images of cavitation clouds in 1% agarose hydrogel captured as shadowgraphs after application of an ultrasound pulse of the same pulse duration (1 cycle), amplitude (34.7 MPa peak negative pressure), and frequency (500 kHz) from transducers with f-number of 0.54 (top) and 0.89 (bottom). Smaller aperture angle (larger f-number) appears to reduce bubble nucleation density within the cloud.

Figure 2.

Example bubble radius vs. time plot showing the initial explosive growth of the bubble (top) in response to an incident pressure pulse (B), with the start of growth occurring at the time $t=0.71\mu \text{s}$ corresponding with the pulse pressure exceeding the inertial cavitation threshold $p\text{\_{-}}=-28.2\text{Mpa}$ for a bubble with initial radius $R_0=2.5\text{nm}$. This time also corresponds to a sudden increase in the bubble-induced pressure (3). The inset in the top pane shows the bubble radius in microns vs. time in microseconds over the entire period of growth and collapse in response to the initial pulse.

Figure 3.

Incident, bubble-generated, and total pressure fields as a function of time at points with different distance from the bubble wall. The figure shows the pressure in the immediate vicinity of the bubble is negated as the bubble reaches the threshold for inertial cavitation. At further distances, the radiated pressure declines and the local pressure field trends towards the incident pressure. To simplify comparisons, the peak of the incident wave is aligned to $t=1\mu \text{s}$ for each position $r_0$.

Figure 4.

Peak negative pressure generated by the combined bubble induced pressure $p_B$ and the incident wave $p_a$. The pressure initiates near the inertial cavitation threshold at the bubble surface and declines back towards the incident pressure (−40 MPa) over 1 mm, following $1/r$ trend for large $r$ (dashed line).

Figure 5.

(Top) Transient pressure signatures for bubbles with different incident pressures. (A) shows the pressure near the bubble surface ($r_0=1\mu m$) while (B) shows it at a distance of $r_0=200\mu m$. The colors differentiate pressure amplitude. (C,D) Peak negative pressure experienced in the medium surrounding the bubble as a function of the initial distance from the bubble $r_0$ for 3 incident pressure amplitudes. (C) shows absolute peak negative pressure while (D) shows the fraction of the peak negative pressure relative to the incident wave.

Figure 6.

(A,B) Transient pressure vs. time for bubbles with initial radius of $R_{\text{0}}=2.5nm$, 10 nm, and 1 μm. (A) shows the pressure near the bubble surface ($r_0=1\mu m$) while (B) shows it at a distance of $r_0=200\mu m$. The colors differentiate nucleus size. (C,D) Peak negative pressure experienced as a function of the initial distance from the bubble ($r_0$). (C) shows absolute peak negative pressure with corresponding inertial cavitation thresholds denoted by the labeled horizontal lines. (D) shows the fraction of the peak negative pressure relative to the incident wave.

Figure 7.

Peak negative pressure in the medium as a function of the initial distance from the bubble ($r_0$)for 3 pulse durations corresponding to frequencies of 0.5, 1, and 3 MHz.

Figure 8.

(A) shows transient pressure signatures for bubbles in a viscoelastic medium with varying shear modulus $G\text{=1}$, 100, and 510 kPa with fixed viscosity $\mu =0.001$, while (B) shows the bubble radius vs. time for each elasticity. The different colors show the curves for each elasticity. (C) shows the transient pressure signatures for bubbles in a viscoelastic medium with varying viscosity $\mu =0.001$ – 0.2 Pa-s with fixed $G\text{=1}\text{kPa}$, while (D) shows the bubble radius vs. time for each viscosity. The different colors show the curves for each viscosity.

Figure 9.

Peak negative pressure experienced as a function of the initial distance from the bubble ($r_0$) for 3 different viscoelastic conditions. The results demonstrate a markedly greater peak negative pressure with increased viscosity of the medium.

Figure 10.

(A) Diagram showing the distance of the bubble-induced pressure $p_b$ vs. the incident pressure $p_i$ arriving at an angle for two bubbles along a horizontal acoustic axis. The incident wave arrives at the second bubble sooner than the bubble-induced pressure wave with greater angle $\theta$ (B) The relation between maximum half-angle of incidence and transducer $F$, as well as the corresponding difference in time of arrival between the two waves. (C) Transient pressure signatures at a distance of $r_0=200\mu \text{m}$ along the acoustic axis, experiencing incident waves + bubble-induced pressure for a 2.5-nm radius bubble positioned at $r_0=0$, for a −40-MPa wave propagating parallel to the acoustic axis $(F\to \infty )$ and at an angle of approximately 41 degrees $\text{(}F\text{=0}.77)$, creating a shift of 100 ns. (D) Peak negative pressure at a distance $r_0=50$, 100, and 200 μm from a bubble as a function of incident wave angle.

Figure 11.

(Left) Peak negative pressure profiles from simulation for 3 different frequency waves with −34 MPa incident amplitude from Edsall 2021. The data points indicate the calculated distance between bubbles derived from experimental measurements. (Right) Simulated peak negative pressure vs distance $r$ for 3 different F-numbers of transducers used in Vlaisavljevich 2017. The data points indicate the calculated distance based on experimentally observed densities.

Figure 12.

(Top panel) A distribution of bubble nuclei ($n_b=50$, $R_0=2.5$) projected onto the X-Z plane. (Middle and bottom panels) Transient pressure plots showing the effect of the multi-bubble distribution on the plane pressure wave propagation. As the plane wave propagates through the cloud, a loss of negative pressure occurs locally around the center of the cloud, and a positive wave forms behind the incident wave. The color bar indicates pressure in MPa.

Figure 13.

(A) Peak negative pressure over the spatial area in the X-Z plane for varying numbers of bubbles in the field. The top image is for a single bubble positioned at the origin, while the lower 4 panels display the peak negative pressure distribution for a specific randomly positioned set of bubbles (see distribution in Figure 12) with different numbers $n_b$. The peak negative pressure distal to the cloud is reduced to ~−28 MPa as the number of bubbles increases. (B) Average peak negative pressure as a function of position along the acoustic axis within the 1.2-mm lateral area where bubbles are populated for a random distribution of bubbles between $n_b\text{=1–100}$. As $n_b$ increases, it is seen to asymptotically approach a value near the cavitation threshold in the distal focus. (C) The maximum peak negative pressure achieved distal to the cloud ($z=4\text{mm}$) as a function of the density of bubbles in the volume corresponding with $n_b=1$ to 100 bubbles.