Ultrasonics 58 (2015) 53–59

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Combined short and long-delay tandem shock waves to improve shock wave lithotripsy according to the Gilmore–Akulichev theory Miguel de Icaza-Herrera a, Francisco Fernández a, Achim M. Loske a,b,⇑ a b

Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Boulevard Juriquilla 3001, Querétaro, Qro. 76230, Mexico División de Ciencias de la Salud, Universidad del Valle de México, Villas del Mesón 1000, Querétaro, Qro. 76230, Mexico

a r t i c l e

i n f o

Article history: Received 27 September 2014 Received in revised form 11 December 2014 Accepted 11 December 2014 Available online 19 December 2014 Keywords: Extracorporeal shock wave lithotripsy Gilmore–Akulichev formulation Tandem shock waves Mechanical stress Acoustic cavitation

a b s t r a c t Extracorporeal shock wave lithotripsy is a common non-invasive treatment for urinary stones whose fragmentation is achieved mainly by acoustic cavitation and mechanical stress. A few years ago, in vitro and in vivo experimentation demonstrated that such fragmentation can be improved, without increasing tissue damage, by sending a second shock wave hundreds of microseconds after the previous wave. Later, numerical simulations revealed that if the second pulse had a longer full width at half maximum than a standard shock wave, cavitation could be enhanced significantly. On the other side, a theoretical study showed that stress inside the stone can be increased if two lithotripter shock waves hit the stone with a delay of only 20 ls. We used the Gilmore–Akulichev formulation to show that, in principle, both effects can be combined, that is, stress and cavitation could be increased using a pressure pulse with long full width at half maximum, which reaches the stone within hundreds of microseconds after two 20 ls-delayed initial shock waves. Implementing the suggested pressure profile into clinical devices could be feasible, especially with piezoelectric shock wave sources. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Extracorporeal shock wave lithotripsy (SWL) continues to be a worldwide accepted therapy [1–3]; however, new-generation lithotripters are often less effective than the first-generation ones [4–6]. Because of this, research groups and manufacturers are still looking for novel techniques to improve clinical outcomes [1–3,7]. During the treatment’s early stage, urinary stones fragment as a consequence of longitudinal and transverse stress waves, as well as of surface waves and circumferential squeezing. Since urinary stones are often surrounded by urine and pooled blood with cavitation nuclei and micrometer-sized bubbles, as fissures get filled with fluid, cavitation erodes the fragments [8–12]. The positive pressure pulse of each incoming shock wave produces a fast compression of the microbubbles. Shortly after, the trailing tensile phase of the shock wave contributes to an explosive bubble growth, followed hundreds of microseconds later by a violent collapse, resulting in the development of high-speed microjets that lead to pits on the stone surface [13–20]. Stress wave-induced

⇑ Corresponding author at: Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Boulevard Juriquilla 3001, Querétaro, Qro. 76230, Mexico. Tel.: +52 442 238 11 64; fax: +52 442 238 11 65. E-mail address: [email protected] (A.M. Loske). http://dx.doi.org/10.1016/j.ultras.2014.12.002 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.

fracture is useful at the beginning of the treatment while cavitation is important during the remaining phase [12,21]. We implemented a numerical model to analyze the dynamics of a small spherical bubble immersed in water and subjected to different shock wave profiles. The purpose of this article was to propose the use of a novel pressure profile to enhance stressinduced fracture and acoustic cavitation simultaneously.

2. Background and hypothesis We will refer to conventional lithotripter shock waves as C pressure profiles. When running a simulation on an air bubble immersed in water, subjected to a C pressure profile, the bubble shows a fast forced collapse reducing its radius to a first minimum (Rmin1) and then expands and elastically bounces several times before reaching equilibrium (Fig. 1). In a real situation the life of the bubble would be shorter, since the second collapse would be asymmetric and the bubble would disintegrate after this collapse. The maximum radius Rmax1 has been used as a way of comparing the potential fragmentation efficiency of different C pressure profiles [22–24]. Another criterion to determine bubble-collapse intensity is the minimum bubble radius Rmin2 at the second bubble collapse [25]. The larger a cavitation bubble expands, the smaller Rmin2 will result and the more violent the collapse will be.

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It is known that the collapse energy of the microbubbles can be increased significantly if a second shock wave arrives shortly before the bubbles, generated by the previous shock wave, start to collapse [7,26–36]. By adjusting the delay (Dt) between both shock waves it is possible to enhance microjet emission. In this article dual-pulse pressure profiles are named long-delay tandem shock waves (C–C profile) if the time delay is between 100 and 900 ls (Fig. 2a). In a preceding article [25] we reported that the delay leading to the most violent collapse i.e., the delay to obtain the lowest Rmin2 value, for a spherical bubble (initial bubble radius R0 = 70 lm) in water exposed to a tandem shock wave profile composed of two C shock waves (P+ = 101 MPa, P = 16.4 MPa) was 287.0 ls. At this optimal delay, Rmin2 was equal to 1.48 lm. Because the full width at half maximum (FWHM) of the positive phase of a C pulse is short, compared to the bubble collapse time, the negative phase of the shock wave arrives during collapse, reducing the bubble collapse intensity. So-called modified tandem shock waves (C–S profiles) i.e., a conventional shock wave (C) followed by a slower pressure pulse (S) were proposed as a solution. The S pulse was designed so that its positive pressure acted on the bubble during the whole collapse. For the C–S pressure profile the optimal delay was 150.5 ls and Rmin2 resulted to be 0.72 lm. This is about half the value obtained with the C–C profile. Tham and colleagues [37] compared the stress field induced inside kidney stones by single-pulse shock waves with the stress produced by short-delay (Dt = 20 ls) tandem shock waves. The authors modeled the conventional lithotripter shock wave according to a simplified expression proposed by Church [38]:

 p PðtÞ ¼ 2Pþ ea1 t cos x1 t þ ; 3

ð1Þ

where P+ is the pressure amplitude of the shock wave (47 MPa), a1 = 9.1  105 s1 is the decay constant and x1 = 2pf1 is the angular frequency ( f1 = 83.3 kHz). They also analyzed other shock wave profiles, which we do not mention here, because they did not lead to improved results. Their simulation revealed that the short-delay tandem pulses enhanced peak stress in the front and back of the kidney stones; however, running the code with single pulses produced reduced stress only on the back side of the stone. According to their results, tandem shock waves with a 20 ls inter-pulse delay could be more effective than single-pulse SWL to break-up kidney stones during the initial part of a two-phase treatment to expose a larger total stone surface (more fragments) to cavitation erosion in the second phase of the treatment. Our study was motivated by the possibility of designing a shock wave profile that takes

advantage of both shear stress and cavitation, during the whole treatment, to enhance stress-induced fracture and acoustic cavitation simultaneously. The hypothesis was that a short-delay tandem pressure profile (referred to as CC profile) i.e., a C shock wave followed 20 ls later by another C pressure wave (Fig. 2b), enhances stone fragmentation by increasing stress, but does not affect bubble collapse. If this proved to be true, then a third shock wave could be used to enhance the collapse of the bubbles produced by the first shock wave and it could be possible to benefit from stress-inducing short-delay tandem pulses and bubble-collapse enhancing longdelay tandem shock waves at the same time. Stress and amplified cavitation could work synergistically from the beginning of the treatment.

3. Methods 3.1. The pressure profiles We defined the C pressure profile by Eq. (1), where 2P+ = 101 MPa, a1 = 9.1  105 s1 and f1 = 83.3 kHz. Under these conditions, the pressure profile includes a negative pressure (tension) tail as low as P = 16.4 MPa. For simplicity, the rise time (defined in this study as the time taken for the pressure to increase from 10 to 90% of P+) chosen for the C pulses was tr = 0. The pulse duration t+ (FWHM of P+) and the time (tsw) measured from the maximum positive peak amplitude to the minimum value of the negative pressure peak, were 0.338 ls and 1.998 ls, respectively. A graph of a long-delay tandem shock wave, referred to as C–C pressure profile, is shown in Fig. 2a. As can be seen in Fig. 2b, the CC profile is a short-delay tandem shock wave that consists of a C pulse followed 20 ls later by another C pulse. In the first part of the study, the 20 ls-delay in all CC pulses was fixed. The CC–C profile was defined as a CC pressure variation, followed hundreds of microseconds later by a third C pulse (Fig. 2c). Similarly, the CC–S profile (Fig. 2d) was made up from a CC profile followed hundreds of microseconds later by an S pulse which has the same P+ amplitude as a conventional lithotripter shock wave C; however, with a larger FWHM of P+. The S pulse was defined as

P S ðt Þ ¼

t þ P ; tr

ð2Þ

and a1 t 8

PS ðt Þ ¼ 2Pþ e

Fig. 1. Numerical simulation of the dynamics of a spherical air bubble (initial radius R0 = 70 lm) in water exposed to a standard lithotripter shock wave (P+ = 101 MPa, P = 16.4 MPa). The second minimum Rmin2 = 2.06 lm occurred at t = 290.5 ls.

for 0 6 t 6 tr

cos



x1 t 8

þ

p 3

 ;

for t > tr :

ð3Þ

Under these conditions, tr = 0.8 ls, t+ = 3.5 ls and tsw = 15 ls. The C pressure profile (Eq. (1)) was used in this study since it mimics a standard lithotripter shock wave. The C–C profile (Fig. 2a) is important, because it has shown to produce increased stone fragmentation in vitro and in vivo and can be considered as the best available pressure profile for SWL so far. C–S shock waves were included because previous numerical simulations revealed that they are superior to C–C shock waves. Simulations with the CC waveform (Fig. 2b) were crucial, because we wanted to prove that it does not significantly influence bubble collapse. Simulations were also run with a CC–C profile to compare its effects on bubble dynamics with those of previous results obtained with a C–C profile. Finally, a comparison between the bubble dynamics produced by a standard lithotripter shock wave and the CC–S pressure waveform was needed to demonstrate that the novel pressure waveform could be more efficient to pulverize urinary stones during SWL.

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Fig. 2. Graph of (a) a long-delay tandem shock wave C–C, i.e., a conventional C shock wave, followed several tenths of microseconds later by another C pulse, (b) a short-delay tandem shock wave CC, i.e., a conventional lithotripter shock wave C, followed 20 ls later by another C pulse, (c) a CC–C pressure profile, i.e., a short-delay tandem shock wave CC, followed several tenths of microseconds later by a third C pulse, and (d) a CC–S pressure profile, i.e., a short-delay tandem shock wave CC, followed several tenths of microseconds later by an S pulse.

3.2. The numerical simulation The dynamics of a single spherical air bubble in water, exposed to passage of the above-mentioned pressure variations was modeled by the Gilmore–Akulichev equation [38–40]. Gilmore’s theory [39] is particularly well suited for conditions of high pressure in which the compressibility of the liquid plays an important role [41]. It depends on Eq. (8), where it is assumed that the bubble, the surrounding liquid, and the flow have spherical symmetry. The flow is irrotational since the eulerian velocity is parallel to the radial direction. Next, the Navier–Stokes equation is simplified according to the spherical symmetry and assuming that all thermodynamical variables depend only on the pressure. This last assumption, satisfied by the adiabatical nature of the process, takes the form of an isentropical transformation. If / is the velocity potential, Gilmore [39] showed that

@/ 1 2 ¼ u þ @t 2

Z

p

p1

dp

q

;

ð4Þ

where u is the velocity of the fluid, q = q(p) is the density of the fluid, p1 is the pressure at an infinite distance from the bubble and the integral is the enthalpy change h(p). Gilmore assumed that / may be written as a spherical wave,

/¼

  1 r ; f t r c1

ð5Þ

where r is the distance measured from the center of the bubble and c1 is the speed of sound at an infinite distance from the bubble. Since r/ ¼ f ðt  r=c1 Þ; w ¼ @ðr/Þ satisfies @t

c1

@ @ ðrwÞ ¼ ðrwÞ: @r @t

ð6Þ

The letter c in these equations should not be confused with the capital letters C used to designate the different pressure profiles. Because Eq. (6) neglects the velocity of the fluid u and takes the speed of sound calculated using the initial conditions, Gilmore proposed a time evolution equation, in agreement with Kirkwood and Bethe [42]:

ðc þ uÞ

@ @ ðrwÞ ¼ ðrwÞ: @r @t

ð7Þ

Gilmore’s equation is Eq. (7) where o//ot is given by Eq. (4):

ðc þ uÞ

      @ u2 @ u2 ¼ : r hþ r hþ @r @t 2 2

ð8Þ

Let R stand for the bubble radius at the time t, and HðRÞ ¼ limr!Rþ hðrÞ for the limit of the fluid enthalpy at the bubble surface. From Eq. (8) Gilmore showed that:

dU 1 ¼ dt Rð1  UcÞ

     U 3 U U dH H U2 þ 1 1þ : c 2 3c c dR

ð9Þ

In this equation, all variables should depend on the time t: _ To calculate the speed of sound c in water at the R = R(t), U = RðtÞ. bubble wall and the enthalpy H in the liquid, we need an adiabatic connection between the pressure p and the liquid density q, which according to both Gilmore [39] and Church [38] is given by the Tait equation:



p  p0 m ; A 1

qðpÞ ¼ q0 1 þ 2

ð10Þ

where A ¼ c1mq0 and m = 7. H and c may be easily calculated from the equations shown above. The connection between the liquid pressure p, immediately at the bubble wall is given by [38,39]:

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p ¼ pi 

M. de Icaza-Herrera et al. / Ultrasonics 58 (2015) 53–59

2 4lU  ; R R

ð11Þ

where r is the surface tension constant, l the viscosity coefficient of the liquid and pi the gas pressure inside the bubble, expressed as [38]

  3g 2r R0 pi ¼ p0 þ ; R R

To evaluate the response of the bubble (R0 = 70 lm) to C–C tandem shock wave profiles, the delay Dt was varied until R2 reached its minimum (Rmin2 = 1.48 lm). As mentioned before, this occurred at a delay of 287.0 ls (Fig. 3) and is in agreement with our previous article [25].

ð12Þ 4.3. Short-delay tandem shock waves

where g is the polytropic exponent of the gas. Finally, the pressure p1 is given by

p1 ¼ p0 þ Ps ðtÞ;

4.2. Long-delay tandem shock waves

ð13Þ

where Ps(t) is the input pressure profile. The resulting equation is an ordinary second order differential equation. As Eq. (9) shows, the right side has a pole for U = c, that is, when the time rate of change of the bubble’s radius equals the speed of sound at the pressure p; a condition requiring higher shock wave pressures than used in this study. This equation may be transformed into a first order ordinary differential equation in two dimensions. The right side of Eq. (9) being bounded fulfills the Lipschitz condition for the existence and uniqueness of the solution. The numerical solution may be worked out by any numerical method. We have used the so-called Runge–Kutta method [43]. If Matlab (The Math Works Inc., MA, USA) software is used, the approximate running time of the program on a personal computer with a 2 Intel(R), Xeon(R) CPU w3503 @2.4 GHz is 18 s. The relative difference between the solution and the numerical approximation was less than 104. This model, used by several authors to describe the dynamics of a bubble exposed to lithotripter shock waves [38,40,44,45], assumes a non vanishing compressibility of the liquid, that the bubble, initially in equilibrium, is embedded in an infinite liquid and also that the flow is isentropic. Furthermore, it is supposed that the initial radius of the bubble (R0) is much smaller than the length of the driving pressure pulse (approximately 2500 lm). In this study, bubbles of three different sizes (R0 = 7, 70 and 700 lm) were exposed to the five above-mentioned pressure profiles (C, C–C, CC, CC–C and CC–S). The largest radius was chosen to be 700 lm, because bubbles of this size have been produced and exposed to long-delay tandem shock waves (C–C) in our laboratory. The initial radii of the other two bubbles were arbitrarily selected to be 10 and 100 times smaller. To reduce the extent of this article, most of the results reported in the next section correspond to an initial bubble radius R0 = 70 lm. The whole simulation was also repeated extending the tail of the lithotripter pulse using the decay constant and the angular frequency proposed by Matula and colleagues [45] i.e., a2 = 3.5  105 s1 and f2 = 50 kHz. Our code was also run reducing the P+ value of the S pulse in the CC–S pressure profile by 50%. The temperature was fixed at 20 °C for all simulations.

Fig. 4 shows the dynamics of the bubble after exposure to a short-delay tandem pressure profile (CC). In this case, initially only one numerical simulation was done, because the delay (Dt = 20 ls) was not varied. The graph reveals that the arrival of the second shock wave inhibits bubble growth for an instant; however, the inertia of the liquid surrounding the bubble pulls radially outwards, increasing the bubble’s size. The bubble collapses 283.8 ls after passage of the first shock wave to a final radius Rmin2 of 2.12 lm. Comparing this value to the Rmin2 radius for a bubble exposed to a single C shock wave (Rmin2 = 2.06 lm), it is evident that the influence of the second pulse of the CC pressure profile on Rmin2 was very low. 4.4. Combined short- and long-delay shock waves When exposing our ideal bubble to the CC–C pressure profile shown in Fig. 2c, the bubble was compressed and expanded by the first shock wave (Fig. 5). After 20 ls the outward motion of the bubble was slowed down by the second compressive pulse. However, as already observed, this second compressive wave cannot force the bubble to collapse before the negative pulse of the second wave contributes to its expansion. The radius R2 reached its minimum (Rmin2 = 1.59 lm) for Dt = 282.4 ls. As shown in Fig. 6, the numerical simulation revealed that the CC–S profile (Fig. 2d) enhanced bubble collapse even more. For R0 = 70 lm, the smallest R2 radius (Rmin2 = 0.73 lm) was obtained using a delay between the first and the third pressure pulses of 148.5 ls. Because the FWHM of the S pressure pulse in the CC–S profile is relatively long, the bubble collapse occurs before the arrival of the negative phase of the S pulse and is only affected

4. Results 4.1. Standard lithotripter shock waves As the positive pressure pulse of the standard lithotripter shock wave C reaches the bubble, its initial radius (R0 = 70 lm) is reduced to Rmin1 = 4.14 lm (Fig. 1). After shock wave passage, the bubble grows, reaching a large maximum radius (Rmax1 = 1.58 mm) and finally collapses, reducing its radius down to Rmin2 = 2.06 lm, exactly 290.5 ls after arrival of the shock wave. This agrees with previous publications [25,38].

Fig. 3. Numerical simulation of the dynamics of a spherical air bubble (initial radius R0 = 70 lm) in water exposed to a C–C pressure profile as shown in Fig. 2a. The smallest R2 radius (Rmin2 = 1.48 lm) was obtained at a delay between the first and the second pressure pulse of 287.0 ls and occurred at t = 288.7 ls. The arrow shows the instant when the second shock wave hits the bubble.

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Fig. 4. Numerical simulation of the dynamics of a spherical air bubble (initial radius R0 = 70 lm) in water exposed to a CC pressure profile as shown in Fig. 2b. The smallest R2 radius (Rmin2 = 2.12 lm) occurred at t = 283.8 ls. The arrow shows the instant when the second shock wave hits the bubble.

Fig. 5. Numerical simulation of the dynamics of a spherical air bubble (initial radius R0 = 70 lm) in water exposed to a CC–C pressure profile as shown in Fig. 2c. The smallest R2 radius (Rmin2 = 1.59 lm) was obtained at a delay between the first and the third pressure pulse of 282.4 ls and occurred at t = 283.3 ls. The arrows show the instant when the second and the third shock wave hit the bubble.

Fig. 7. Graph of the minimum bubble radius at the second collapse as a function of the delay between the first and third shock wave for (a) a CC–C pressure profile as the one shown in Fig. 2c and (b) a CC–S pressure profile as the one shown in Fig. 2d.

Table 1 Comparison between the minimum radii at second collapse of an air bubble subjected to a standard lithotripter shock wave and to a novel pressure profile.

a b c d

Fig. 6. Numerical simulation of the dynamics of a spherical air bubble (initial radius R0 = 70 lm) in water exposed to a CC–S pressure profile as shown in Fig. 2d. The smallest R2 radius (Rmin2 = 0.73 lm) was obtained at a delay between the first and the third pressure pulse of 148.5 ls and occurred at t = 154.7 ls. The arrows show the instant when the second and the third shock wave hit the bubble.

by the compression of the S pulse. We believe that the atypical behavior of the graph after the second bubble collapse is due to the influence of the negative phase of the S pulse. A graph of Rmin2 as a function of the delay between the first and the third shock wave is shown in Fig. 7a and b for the CC–C and the CC–S pressure profiles, respectively. For all delays, between the first and third shock waves, Rmin2 was much smaller with the CC–S profile.

R0 (lm)a

Pressure profile

Rmin2 (lm)d

700 700 70 70 7 7

Cb CC–Sc C CC–S C CC–S

525.270 28.947 2.061 0.728 0.069 0.003

Initial bubble radius. Standard lithotripter shock wave. Novel pressure profile (see Fig. 2d). Minimum bubble radius at the second collapse.

Results obtained by using a2 and x2 are not presented here, because they did not reveal new information. Reducing the amplitude of the positive pressure peak of the S pulse by 50% only changed the Rmin2 value for the CC–S profile (R0 = 70 lm) from 0.73 to 0.85 lm. This demonstrates that the S pressure pulse may have smaller amplitude than the two leading C waves and still be very effective. Table 1 shows the Rmin2 values obtained after running the numerical simulation with the standard lithotripter profile C and the new CC–S profile for the three bubble radii. The Rmin2 values corresponding to the novel profile are significantly smaller than those obtained with the standard lithotripter shock wave,

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indicating that, for the three studied bubble sizes, CC–S shock waves are expected to produce much more violent bubble collapses. 5. Discussion During SWL shear forces and cavitation contribute synergistically to stone pulverization [12]. We designed a novel shock wave profile that enhances both stone damage due to shear and pitting. The first two shock waves of the novel profile hit the stone with a fixed time delay of 20 ls, increasing shear forces as reported by Tham et al. [37]. The third pressure pulse, emitted hundreds of microseconds after the first, intensified bubble collapse. The main objective of our study was to analyze if, in principle, the novel pressure profile could improve SWL outcomes compared to standard shock waves. A main concern was if the second shock wave of the CC–S profile would stop and reverse the growth of the bubbles generated by the first pressure pulse. In that case, stress and cavitation could not have been increased simultaneously and a two-stage SWL treatment as suggested by Tham and colleagues [37] would be needed. Our numerical model showed that the second shock wave does not interfere significantly with inertial bubble growth, so that bubble collapse can indeed be enhanced by a third pulse i.e., short-delay and long-delay tandem shock waves could be used simultaneously, resulting in improved stone comminution. The results demonstrate that the CC–S pressure profile produces much smaller Rmin2 values than a standard lithotripter shock wave. By comparing Fig. 7a and b, it is evident that an advantage of using the S pulse is that intensified bubble collapses (small Rmin2 values) are achieved for a much larger range of delays. The CC–C shock waves achieve their best result only at a very narrow range of delays (Fig. 7a); however, the CC–S profile produces very small Rmin2 values almost over the whole range of delays (Fig. 7b). Table 1 reveals that the novel pressure waveform (CC–S) produces much smaller Rmin2 radii than standard lithotripter shock waves (C) for the three different bubble radii tested in this study. The temperature was fixed at 20 °C, because previous studies revealed that Rmin2 did not vary significantly running the code at 0 °C and 30 °C [46]. A concern about using the CC–S pressure profile could be the possibility of an increased tissue damage, because enhancing bubble cluster collapse accelerates stone fracture [12,47]; however, it could also produce tissue hemorrhage [48,49]. Fortunately, blood and most mammalian tissue have few cavitation nuclei [50]. Furthermore, in blood vessels bubble collapse is weaker than in a free field, since bubble expansion is constrained by the vessel wall. Because of this, the collapse time of in vivo bubbles was found to be significantly reduced from that in vitro [51,52]. Since long-delay tandem shock waves did not increase tissue damage in vivo [32] we believe that the combination of short- and long-delay tandem shock waves should not cause more tissue damage than standard single-pulse shock waves; however, this assertion must be confirmed by extensive in vivo experiments in the future. The proposed CC–S pressure profile is expected to improve fragmentation efficiency only if the urinary stone is surrounded by fluid i.e., only if cavitation can act. The importance of creating a so-called fluid filled expansion chamber to allow bubble growth and collapse during SWL has been demonstrated [34]. Stress waves play an important role when the size of a stone or stone fragment is larger than half the compressive wavelength inside it [12,53]. Since stone fragments get smaller as the treatment progresses, gradually reducing the 20-ls delay between first and second shock wave of the CC–S pressure profile could help to maintain a high stress level inside the fragments. Simulations exposing the 70 lm-radius bubble to CC pressure profiles with a delay between C shock waves of 5, 10 and 15 ls resulted in Rmin2

equal to 2.15, 2.12 and 2.12 lm, respectively, confirming that the bubble expansion produced by the first shock wave was not significantly reduced by the second 5, 10 or 15 ls-delayed pulse. Implementing CC–S shock wave generation on piezoelectric lithotripters should be feasible, because the pressure profile can be altered by modifying the excitation of the piezoelectric elements. Long-delay (C–C) tandem shock waves have already been tested in vitro and in vivo using Piezolith 2300 and Piezolith 2501 (Richard Wolf GmbH, Knittlingen, Germany) shock wave sources [34,35]. Smaller double-layer piezoelectric shock wave generators were developed by the same manufacturer and used in the mobile modular Piezolith 3000 lithotripter [54]. In this device, the front layer is excited at a fixed delay after the back layer, so that the waves generated by both layers superpose at the front surface. Arora et al. [55] provided this type of shock wave source with circuitry to trigger both layers independently at an arbitrary delay and measured the spatial and the temporal evolution of cavitation clusters. To generate combined short- and long-delay tandem shock waves of large enough peak pressure with this source by varying the delay between the excitation of the piezoelectric layers may not be easy [58]; however, series of three shock waves could be emitted if both layers are triggered at the required delays while maintaining their original time shift. The pressure profile proposed in this article may not be generated using electrohydraulic lithotripters, because single-spark gap systems cannot produce two consecutive shock waves at delays shorter than 10 ms [56]. However, shock wave generators based on underwater multichannel electrical discharges [56–58] could emit pressure profiles as proposed in this article. Electromagnetic lithotripters have also been used to generate long-delay tandem shock waves [33]. Whether this shock wave generation method is capable of producing two C shock waves at delays as short as required for the CC–S profile has still to be answered. We believe that the simplified model discussed in this article is valid, because it was useful to analyze the feasibility of enhancing simultaneously stress and cavitation during SWL. 6. Conclusions The Gilmore–Akulichev model is suitable to predict if a novel shock wave profile could enhance microbubble collapse. Implementing the CC–S shock wave profile into clinical devices may improve SWL outcomes, thus being a promising method for manufacturers, physicians and patients. Acknowledgements Paula Bernardino, Juan Carlos Álvarez and Guillermo Vázquez are acknowledged for assistance. References [1] J.J. Rassweiler, T. Knoll, K.U. Köhrmann, J.A. McAteer, J.E. Lingeman, R.O. Cleveland, M.R. Bailey, C. Chaussy, Shock wave technology and application: an update, Eur. Urol. 59 (2011) 784–796. [2] J.E. Lingeman, Lithotripsy systems, in: A.D. Smith, G.H. Badlani, D.H. Bagley, R.V. Clayman, S.G. Docimo, G.H. Jordan, L.R. Kavoussi, B.R. Lee, J.E. Lingeman, G.M. Preminger, J.W. Segura (Eds.), Smith’s Textbook on Endourology, B.C. Decker Inc., Hamilton, Ontario, Canada, 2007, pp. 333–342. [3] A.M. Loske, Shock Wave Physics for Urologists, CFATA-UNAM, Querétaro, Mexico, 2007, ISBN 978-970-32-4377-8. [4] J.E. Lingeman, Stone treatments: current trends and future possibilities, J. Urol. 172 (2004) 1774. [5] R. Gerber, U.E. Studer, H. Danuser, Is newer always better? A comparative study of 3 lithotripter generations, J. Urol. 173 (2005) 2013–2016. [6] A. Skolarikos, G. Alivizatos, J. de la Rosette, Extracorporeal shock wave lithotripsy 25 years later: complications and their prevention, Eur. Urol. 50 (2006) 981–990.

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