Modeling and experimental analysis of acoustic cavitation bubbles for Burst Wave Lithotripsy

Abstract

A combined modeling and experimental study of acoustic cavitation bubbles that are initiated by focused ultrasound waves is reported. Focused ultrasound waves of frequency 335 kHz and peak negative pressure 8 MPa are generated in a water tank by a piezoelectric transducer to initiate cavitation. The resulting pressure field is obtained by direct numerical simulation (DNS) and used to simulate single bubble oscillation. The characteristics of cavitation bubbles observed by high-speed photography qualitatively agree withs the simulation result. Finally, bubble clouds are captured using acoustic B-mode imaging that works in synchronization with high-speed photography.

Burst Wave Lithotripsy (BWL) is a newly proposed non-invasive medical procedure to fragment kidney stones by focused ultrasound pulses of amplitude O(1) MPa[1]. BWL is a potential alternative to Shock Wave Lithotripsy (SWL), a widely used procedure in clinical treatments that typically utilizes shock waves of O(10) MPa for stone fragmentation; BWL enables comminution of stones with lower peak pressure of the sound field in a more controllable fashion. In the procedures of both BWL and SWL, acoustic cavitation can be initiated during the passage of the tensile parts of the pressure wave of amplitude O(1) MPa inside human body. Cavitation activity is important in such treatments because bubbles strongly interact with the incident sound field, potentially causing tissue injury and/or shielding the stone from the intended treatment. However, the fundamental physics of cavitation bubbles in strong interaction with the focused ultrasound field has not been fully studied in detail.

Here, we report a combined modeling and experimental study of cavitation bubbles that are initiated by the BWL type of ultrasound pulses in an open water tank. We generate waves using a piezoelectric transducer that has been used to break stones in both in vitro and in vivo environments. In the modeling, we analytically emulate the transducer and simulate the sound field generated by the transducer in detail using direct numerical simulation (DNS). In the experiment, we observe the evolution of the cavitation bubble cloud using high-speed photography combined with B-mode acoustic imaging. The captured bubble clouds are in a qualitative agreement with the radial evolution of single spherical bubble calculated using Keller-Miksis equation combined with the pressure field simulated by the DNS. Finally, through synchronization with the high-speed photography, we confirmed the potential of B-mode ultrasound imaging to effectively characterize bubble activity. The agreement among the modeling and the experimental measurements would lead to further understanding of the cavitation bubbles for BWL.

Figure 1 shows the experimental setup used for the generation and observation of cavitation bubbles. The piezoelectric transducer shown in figure 1b has 120 mm focal length and generates waves of frequency 335 kHz. An imaging probe (ATL HDI P4-2) is mounted in the center of the transducer. The high-speed camera (Model APX-RS camera, Photron USA, Inc., San Diego, CA) is triggered to captures images around the focal region of the transducer from the side of the tank (figure 1c) at 20,000 fps, when the waves arrive at the region. Tap water which was filtered and degassed to 65% O₂ level was used. The BWL pulses are characterized by the pulse repetition frequency (PRF), wave cycles per pulse (N_c), the total number of pulses (N_p) and maximum focal pressure P_f. In the present study, we use sinusoidal form of waves characterized by PRF= 200 Hz, N_c = 10, N_p = 100 and P_f = 8 MPa.

Figure 1.

The images and the schematic of the experimental setup.

Figure 2 shows the time evolution of the flooded pressure contour during the passage of the 10 cycles of left-going wave sent from the model transducer. The transducer is modeled as a spherical cup of 100 mm with its center located at the origin. The water is modeled as an inviscid fluid that follows stiffened gas equation of state. The acoustic sources are distributed on the model transducer such that the waves of designated amplitude are sent in one-way. The simulation algorithm uses a high-order accurate finite-volume weighted essentially non-oscillatory (WENO) reconstruction, to obtain the primitive flow variables at the cell-boundaries, and a Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver, to compute the resulting fluxes. Time marching is handled by a total-variation-diminishing Runge-Kutta (TVD-RK) time-stepper[2].

Figure 2.

The flooded pressure contour of the wave sent from the model transducer along the upper half of a plane of symmetry of the 2D cylindrical domain. The red and blue color denote high and low pressure regions, respectively. The number of grid points used in the simulation is [N_x; N_y] = 1250 × 300. The unit of the axis labels is mm.

Figure 3 shows the focal pressure evolutions obtained from a linear transducer model including ring-up/ring-down effects and that of the DNS shown in figure 2. Though the wave form of the DNS presents slight non-linear distortion and does not include the ring-up/ring-down effects, the values of peak maximum pressure as well as the qualitative wave forms from both agree.

Figure 3.

The focal pressure evolution of the waves sent from the transducer.

Figure 4 shows the representative images of cavitation bubbles captured by the high-speed camera at t = 80 µsec and 130 µsec during the 3rd pulse, where t = 0 is the moment when the transducer was triggered to generate the pulse. The location of the center of the images correspond to (−98, 0) in the images of figure 2, thus 98 mm distant from the center of the transducer. At t = 80 µsec, we see two bubble clouds as large as 3 mm. In the magnified inset, the presence of individual bubble can be confirmed in the leftmost cloud. At t = 130 µsec, the clouds disappear and a few bubbles are discretely observed. We note that this tendency of the shrinkage of the cloud was confirmed in the images from different pulses, though the locations and the sizes of the bubble clouds were not consistent.

Figure 4.

The high-speed images taken at t = 80 µsec and 130 µsec during the 3rd pulse. The scale bars denote 5 mm.

Figure 5 shows the pressure evolution at (−98, 0) obtained using the DNS and the corresponding radial evolution of pre-existing bubble nuclei with typical size, that are derived using Keller-Miksis equation[3] combined with the obtained pressure as the forcing term. Around 80 µsec, bubbles of all the radius largely oscillate to reach the maximum radius over 100 µ, while at t = 130 µsec, the waves are passed and only bubbles with large initial radius (10 µm and 20 µm) keep rebounding with peak radius less than 50 µm. Bubbles with initial radius 1 and 5 µm are immediately damped after the passage of the wave, due to strong acoustic radiations. Thus at t = 130 µm we would assume to see fewer number of bubbles of smaller size. This assumption corresponds to the observation in figure 4. Finally, figure 6 shows the representative images of cavitation bubbles obtained using synchronized B-mode imaging and high-speed photography. B-mode images were captured at 400 frames per the BWL, using the imaging probe with a Verasonics ultrasound engine (Kirkland, WA, USA). Because the open water tank is devoid of scatterers other than bubbles, the intense regions in reconstructed images should correspond to bubbles. Though the fps of the imaging pulse is much lower than that of high-speed photography, the size and location of bubble clouds in the B-mode image agrees well with those of the high-speed photography, suggesting its potential for quantifying bubble activities.

Figure 5.

The simulated pressure evolution at (x, y) = (−98, 0) and the corresponding radial evolution of single bubble of initial radius R₀ = 1, 5, 10 and 20 µm.

Figure 6.

The images of cavitation bubbles taken by synchronized B-mode acoustic imaging (the left and middle images) and high-speed photography (the right image) at t = 100 µsec. The waves propagate from the top to the bottom of the images. The scale bar in the photographic image denotes 5 mm.

In conclusion, we have confirmed the qualitative agreement between modeling and experiments in terms of the radial evolution of the bubbles based on a theory for that of single isolated bubble under pressure field provided by the DNS, as well as successfully synchronized the B-mode acoustic imaging and high-speed photography to capture cavitation bubble clouds. For further quantitative analysis of the acoustic cavitation, we plan to extend the present study by modeling bubble clouds taking inter-bubble interactions[4] into account and using high-speed photography with cameras of higher frame rate to capture the dynamics of the cloud in detail.