3.2 Bubble clouds in lithotripsy
3.2.3 Energy focused by bubbles
The energy stored by a cavitating bubble as it initially interacts with the pressure field is ultimately released during its violent collapse. Part of this energy will be released in the form of an expanding spherical pressure wave (which can be measured by a passive cavitation detector (PCD)). In the case where the collapse is asymmetric, part of the stored energy will be released in the form of a micro-jet which, if close to a solid
object such as a kidney stone, can cause pitting of the surface. The details of the amount of energy release in jetting versus pressure wave or heat generation are complex and have not yet been modeled in this work.
However, it is clear that the stone comminution capability of a cavitating bubble is a function of the work done on it by the mean pressure field.
In post-processing the simulation data, the energy released by a bubble was computed using a slightly different bubble model. We consider the Herring model (Herring 1941):
"
1−2 R˙
c
#
RR¨+3 2
"
1−4 3
R˙ c
#
R˙2= pB−p∞(t)
ρ +R
c d dt
pB−p∞(t) ρ
, (3.1)
which is nearly identical to the Gilmore model discussed earlier and has the advantage that certain terms can be integrated analytically. Rewriting the above equation:
1 R2
1 2
d dR
R3R˙2
+ 1 R2
3 2c
d dR
R3R˙3
=pB−p∞(t)
ρ +R
c d dt
pB−p∞(t) ρ
. (3.2)
Integrating the equation once:
1 2 1 + 3
R˙ c
!
R3R˙2= Z pB
ρ R2dR+
Z R3R˙ ρc
dpB
dRdR−1 ρ
Z
p∞+R c
dp∞
dt
R2dR+ constant. (3.3)
The ˙R/cdependence on the left hand side of the above equation was found to be negligible in cases relevant to this work. Similarly, the second term on the right hand side was also found to be negligible. The above equation can therefore be simplified down to
1
2R3R˙2≈ Z R
Ro
pB
ρ R2dR−1 ρ
Z R Ro
p∞+R c
dp∞ dt
R2dR, (3.4)
Kinetic energy: 1 2ρR3R˙2, Bubble potential energy:
Z
pBR2dR
R
, Initial bubble energy:
Z
pBR2dR
Ro
,
Work done by liquid:
Z R Ro
p∞+R c
dp∞ dt
R2dR.
To analyze the energy absorbed by a bubble during collapse in the context of lithotripsy, we compared
a simulation with and without bubble cloud cavitation. The parameters for the two simulations were the same except for the bubble number density. In the ‘no bubble cloud’ case, the bubble number density was set to zero which decouples the pressure field from the bubble dynamics. The bubble number density for the
’bubble cloud’ simulation was set to 10 bubble/cm3. As a first comparison, Figure 3.21 shows the pressure at the focus for both cases. It is interesting to note that apart from the pressure rise in the tail of the pulse, the two waves are very similar.
Time (µs)
Pressure (MPa)
170 172 174 176 178 180 182 184
-15 -10 -5
0 5 10 15 20 25 30
No bubble cloud interaction Bubble cloud interaction
Figure 3.21: Impact of bubble cloud on pressure at focus. Data fromRun000andRun000v.
Given the similarity between the above pressure waves, it would be expected that the resulting bubble dynamics should be also similar. However, the computer model predicts significantly different bubble be- havior for these cases. Figure 3.22 compares the bubble history, small-scale pressure fluctuation and bubble energy at the focal point for coupled and uncoupled cases. As seen in the upper part of Figure 3.22, the bubble growth and collapse for the bubble cloud case is far from symmetric. Due to the presence of the bubble cloud, the bubble does not absorb as much energy from the passage of the shock wave and exhibits a slower growth rate than in the case without the bubble cloud because of the trailing positive peak in the pressure (Figure 3.21). Furthermore, the expansion of the bubbles within the cloud displaces liquid outward and decreases the mixture pressure inside. Consequently, bubbles can grow for a longer duration and the maximum radius is achieved much later than in the noninteracting case. It is also important to note that based on the history of the bubble radius, bubble collapse inside the cloud appears more violent.
From the bottom plot of Figure 3.22, it is interesting to note that the energy at the peak bubble radius Rmax, is approximately the same, which leads to the conclusion that the exact details of the lithotripter
waveform are not as important as the maximum bubble size. Without the pressure rise due to interactions within the cloud, all waveforms producing similar peak bubble growth would be equivalent (assuming similar size bubble cloud). A second, perhaps even more important conclusion from these results, is that the collapsing bubble cloud provides nearly half the work done by the fluid on a bubble. This is particularly surprising since the pressure amplitudes related to the cloud collapse are two orders of magnitude smaller than that of the lithotripter shock wave. However, the work done (P dV work) by the fluid is very large because the bubble radius changes several orders of magnitude. In addition, since cloud interactions are a direct function of the void fraction, the energy released increases steadily with the maximum void fraction (see Figure 3.23),
As seen in Figures 3.22 and 3.23, a bubble at the focal point can absorb and release up to 50 µJ.
Considering that a bubble cloud can contain up to several hundred large bubbles, the amount of energy contained within the cloud can be estimated to be of the order of 1-10 mJ. This value appears 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, only a small portion of the wave energy is expected to be retained by the cloud.
Time (µs)
Pressure (MPa)
200 300 40 500
-0.4 -0.2 0 0.2
0.4 Time (µs)
Bubble radius (mm)
200 300 40 500
0 0.2 0.4 0.6 0.8
No= 0 bubbles/cm3 No=20 bubbles/cm3 Ro=20 µm
Time (µs)
Bubble energy (µJ)
200 300 40 500
0 5 10 15 20 25 30 35 40
Figure 3.22: Impact of bubble cloud on bubble radius, small-scale pressure and bubble energy at focus. Data fromRun000andRun000v.
Maximum void fraction in bubble cloud (%)
Energy released from bubble collapse (µJ)
0 2 4 6 8 10
0 10 20 30 40 50 60
Figure 3.23: Relationship between the energy released by bubble collapse and maximum void fraction. Data shown was taken from Table A.3 in Appendix A.