Based on in vitro experiments, it has been determined that the cloud of cavitating bubbles that forms in the wake of the shock wave is a crucial element in the rock crushing process. A diagram of the lithotripter used in this study with relevant dimensions is shown in Figure 1.2.
The pressure field produced by a lithotripter
In the work of Hamilton (1993), the spherical wave propagated to the wall of the reflector using linear geometric acoustics and the reflection (and associated diffraction) was calculated analytically using the Kirchoff integral method (see the work of Born & Wolf (1980). )). Another approach, based on geometric shock dynamics, was used in the work of Cates & Sturtevant (1997).
Bubble dynamic response to the pressure field
Numerical models were also used to better understand the effect of the shock wave on the bubble field. Mass transfer across the bubble interface is an important issue in the influence of pulse frequency.
Bubble cloud effects in SWL
Mixture compressibility
As depicted in Figure 1.12, the dual-pulse lithotripter consists of two conventional coaxial electrohydraulic lithotripters firing at the same target. In their study, Sokolovet al. 2001) collected high-speed images of the cavitation cloud in the focal region of a dual-pulse lithotripter (see Figure 1.13).
Cloud dynamics
An interesting feature observed in Figure 1.13 is the appearance of bands where hardly any bubbles can be observed. In Figure 1.14, this result is compared with Rayleigh-Plesset and Gilmore bubble models forced by shock waves based on Church's model for a pressure wave at the focus of a lithotripter (see Church 1989 and Figure 1.15).
The role of pulse rate frequency in lithotripsy
Overview of the thesis
Equations for continuum-based two-phase flow model
Flow field in the vicinity a bubble
Since the bubble is assumed to be spherical, it is limited to radial and translational motion. If the fluid surrounding the bubble is compressible, the finite speed at which waves propagate in the fluid must be taken into account.
Phase averaged equations
In addition, the momentum convection expression can be rewritten in terms of the primitive variables ρM and uM and the Reynolds stress expression:. 2.21). It is interesting to note that with a different approach to the averaging process, identical equations can be obtained for the same order of accuracy.
Final equations
This mean volume must be large enough to contain a large number of bubbles.
Initial conditions
Bubble field
Pressure field at early times
The added source term must meet two conditions: the velocity field must be zero before the shock wave. The strength of the distributed mass source is constant, which ensures a constantly decreasing velocity field between it and the spherical shock wave.
Grid and boundary conditions
Computational grid
However, in the opposite region of the domain where the shock wave focuses on the target area, a cylindrical grating is better suited for this task. On the plane between the two foci, the spheroidal lattice fits with first-order accuracy into the cylindrical lattice in the axial direction (and exactly in the radial direction).
Boundary conditions
This boundary condition is designed to absorb a plane wave impinging on the edge of a Cartesian grid. Hence, given a sufficiently large domain, the Thomson boundary condition can be applied with minimal reflection.
Mesh stretching
Using the same treatment as the symmetric boundary, the no-flow boundary condition can be approximately satisfied. This approach is based on a one-dimensional decomposition of the equation into (approximately) outgoing and incoming waves.
Shock capturing algorithm
For the case presented in the above figure, it is clear that the polynomial in the upper left corner is the best choice to evaluate the value of φi+1/2(t). For the purpose of this work, the measure of smoothness for the polynomial template was taken from the work of Jiang & Shu (1996) and is given by .
Time marching algorithm
Therefore, the waves caused by the rapid fluctuations of the bubble radius do not have time to travel outside the control volume and are averaged out. The timescale associated with bubble dynamics can be much shorter than that of the pressure field. This approach is useful in two ways: an appropriate time-marching algorithm is used for each stage, and the adaptive shrinking/expansion of the time step for a bubble at a given location is independent of other locations.
In terms of performance, it is therefore more advantageous to use a split march time with a small time step in order to compensate for the lower accuracy of the method.
Spatial and temporal resolution study
Due to the separate time integration, the choice of time step can affect the bubble dynamics. As mentioned in the previous section, a larger time step can cause a loss of precision in the coupling between the two phases. Based on the results in Table 2.2, we can conclude that the time steps used in this work (∆t = 0.020 µs and 0.025 µs) are small enough to ensure that the error caused by the discrete time integration is much smaller than that , which was introduced with other modeling aspects.
Shock wave focusing in lithotripters
Closed ellipsoidal bowl
The point F1 indicates the first focus of the reflector and the location of the spherically expanding wave. Due to the dependence of the speed of sound on pressure, the wave propagates slightly faster in the nonlinear Euler case. The amplitude and position of the reflected wave in the nonlinear model are slightly larger and propagate faster than linear cases, as expected (see A).
If the pulse thickness is reduced to a value much smaller than the length scale associated with the reflector, then the agreement with the geometric acoustic method would be better.
Edge wave and diffraction effects
Pressure field in a lithotripter
As for the discrepancy in the pressure rise of the shock wave, the sharp pressure jump is prevented by the dissipative properties of the WENO method used in the current numerical model. A grid with a significantly higher resolution (10-100 times) would reduce this damping and allow a sharp increase in pressure at the shock wave. The size of the focal area is often used as a means to compare different lithotripter models.
This region, often defined as a 6 dB drop from peak pressure, can provide some insight into the spatial distribution of shock wave energy.
Bubble clouds in lithotripsy
Cloud cavitation in a free-field lithotripter
This would also result in increased bubble activity in the wake of the reflected wave. The measured time to collapse for the Caltech-EHL has been reported to be in the region of 320 µs.1 This value provided guidance for the calibration of some of the simulation parameters. CasesRun105a (tc = 333 µs) andRun105d (tc = 315 µs) compared well with experimental images, not only in terms of lifetime but also in terms of bubble cloud dimensions (see Figure 3.14).
In describing the bubble field, Sokolovet al. 2001) reported that a peak bubble radius in the focal zone was approximately 0.5 mm.
Pressure waves in collapsing cloud
Energy focused by bubbles
It is interesting to note that apart from the increase in pressure at the tail of the pulse, the two waves are very similar. It is also important to note that based on the bubble radius history, the collapse of the bubble within the cloud appears more violent. This value seems relatively small compared to the energy contained in the initial spherical pulse, which is of the order of 100 mJ.
However, since the shock wave propagates relatively unhindered past the bubble cloud, it is expected that only a small portion of the wave energy is retained by the cloud.
Dual-pulse and partially obstructed lithotripter
- Bubbles and pressure superposition in dual-pulse lithotripter
- Structure in the dual-pulse bubble cloud
- Impact of an artificial stone in the lithotripter field
- Impact of a large cylindrical reflector in the field of a lithotripter
If the assumption of zero void fraction is relaxed, differences in the bubble cloud shape in the focal region of the dual-pulse lithotripter begin to appear. The void fraction in the focal region of a two-pulse lithotripter can grow up to 15%. As shown in Figure 3.29, the present numerical model predicted similar shapes and sizes for the cloud of bubbles in front of the stone.
An important finding that can be discerned from Figure 3.30 is the presence of a gap in the bubble field in front of the reflector.
Pulse-repetition frequency
- Single bubble analysis
- PRF in free-field lithotripter
- Impact of lithotripter intensity
- PRF and artificial stone in lithotripter field
- Concentration of gas in solution
In Figure 3.34 the increase in bubble radius between two pulses is plotted as a function of the initial bubble radius. As shown in Figure 3.37, an increase in the initial void fraction also leads to reduced shock wave intensity at the focal point (shielding). As a comparison, Figure 3.39 presents the values of peak pressure at the same location using our numerical model as a function of the energy of the spherical shock wave in the initial conditions.
As seen in Figure 3.40, the increased amplitude of the shock wave translates into an increase in the collapse energy of a bubble at that location.
Final conclusions
The banded structure of the bubble cloud initially observed in experiments was evident in the simulations performed with the present model. During the bubble growth phase, the pressure in the center of the bubble cloud decreased slightly below the ambient pressure for cases with non-zero void fraction. Because of this interaction between the bubble cloud and the pressure field, bubbles in the center of the cloud can show a much longer time to collapse than predicted given their maximum size and observed in experiments.
Furthermore, for the optimal case, the energy released by bubbles increased with number density due to the increase in interaction within the bubble cloud.
Practical implications for lithotripsy
During the collapse of the bubble cloud, the numerical model predicts the reverse interaction, where a rapid decrease in the void fraction causes the liquid to rush towards the center of the bubble cloud and cause a pressure rise. This pressure increase, although two orders of magnitude smaller than the initial shock wave, can result in up to a sixfold increase in bubble collapse energy release in cases where little or no cloud interactions are present. Based on the values obtained for the bubble collapse energy released for cases covering a range of simulation parameters, optimal damage conditions were determined.
Although the maximum bubble size tends to grow with core size, a high value of the initial void fraction results in shielding and a smaller peak radius.
Current limitations and future work
In Table A.3 all relevant data for the simulation of the free-field lithotripter based on the Caltech-EHL are collected. It should be noted that increasing the value of the nominal compression three times caused significant spurious oscillations. It should be noted that the impact of the correction term is significantly smaller than the error introduced by the WENO compression or the domain truncation.
For the present work, the choice of the compression parameter is arbitrary, so the data shown in Table B.1 provide a measure of the range of previous equally valid results.
Ensemble phase averaging
- Preliminaries
- Ensemble averaged Euler equations
- Equations for the disperse phase
- Bubble interior
- Modeling the probability distribution
- Relative motion of bubbles
Roi( ˙RoiP) = 0 (C.5) For negligible gas diffusion into the bubble (see section C.1.4), the last term in the above equation can be neglected. For the purpose of this work, the flow field near the bubble was assumed to be incompressible and irrotational (see Section C.1.3 for more details). Since the flow field is incompressible and irrotational near the bubble, we can write the potential as (note that the subscriptC for the continuous phase is dropped for convenience).
All equations are expressed as a function of the fluid pressure at the bubble wall pB.
Final equations
As mentioned earlier in this work, the numerical model requires some characterization of the bubble cloud. As illustrated in the above figure, an estimate can be made of the number of bubbles present within an area of known size. Directed diffusion during nonlinear pulsations of cavitation bubbles. Journal of the Acoustical Society of America.
Effect of overpressure and pulse repetition frequency on cavitation in shock wave lithotripsy.Journal of the Acoustical Society of America, submitted.
