Effects of Stone Size on the Comminution Process and Efficiency in Shock Wave Lithotripsy

Abstract

The effects of stone size on the process and comminution efficiency in shock wave lithotripsy (SWL) are investigated by experiments, numerical simulations, and scale analysis. Cylindrical BegoStone phantoms with approximately equal height and diameter of either 4-, or 7- or 10-mm, in a total aggregated mass of about 1.5 g, were treated in an electromagnetic shock wave lithotripter field. The resultant stone comminution (SC) was found to correlate closely with the average peak pressure, P+(avg), incident on the stones. The P+(avg) threshold to initiate stone fragmentation in water increased from 7.9 to 8.8 to 12.7 MPa, respectively, when the stone size decreased from 10 to 7 to 4 mm. Similar changes in the P+(avg) threshold were observed for the 7- and 10-mm stones treated in 1,3-butanediol where cavitation is suppressed, suggesting that the observed size dependency is due to changes in stress distribution within different size stones. Moreover, the slope of the correlation curve between SC and ln(Pfalse‒+(avg)) in water increased with decreasing stone size, while the opposite trend was observed in 1,3-butanediol. The progression of stone comminution in SWL showed a size-dependency with the 7- and 10-mm stones fragmented into progressively smaller pieces while a significant portion (> 30%) of the 4-mm stones were stalemated within the size range of 2.8 ~ 4 mm even after 1,000 shocks. Analytical scaling considerations suggest size-dependent fragmentation behaviour, a hypothesis further supported by numerical model calculations that exhibit changing patterns of constructive and destructive wave interference, and thus variations in the maximum tensile stress or stress integral produced in cylindrical and spherical stone of different sizes.

Shock-induced damage and dynamic fracture in cylindrical bodies submerged in liquid

“…Specifically, for specimen S6 ( L / L SW = 0.18), the value of D avg is less than 10% of that in S4. This trend is consistent with the finding of Zhang et al [23] that smaller specimens require more shock doses to break, except that they tested specimens in clusters instead of individual ones. We have found that when the specimen becomes too small compared to the length of the incident shock wave, the trailing tensile phase can no longer work jointly with the leading compressive phase — through wave superpositions described in Section 3.4 — to increase damage.…”

“…Previous studies have shown that fabricated BegoStone models can be considered as isotropic, and undergo brittle fracture under shock loading [22, 23]. Therefore, it is modeled here as a linear elastic and isotropic solid; and the constitutive equation is given by εij=1+vEσij−vEσkkδij, i,j=1,2,3, where εij=12∂ui∂xj+∂uj∂xi, i,j=1,2,3, is the infinitesimal strain tensor.…”

“…Its acoustic and mechanical properties can be tuned easily by varying the powder-to-water ratio [22]. It has been used as a model material for studying shock-induced damages and failures, mainly in the context of shock wave lithotripsy [23].…”

An experimentally-calibrated damage mechanics model for stone fracture in shock wave lithotripsy

“…In particular in all cases considered here, a secondary fracture area appears ahead of the main fracture. The main fracture has been argued to occur at approximately 2/3 of the stone length based upon ray tracing arguments [35]. Within this interpretation the appearance of a secondary fracture area can be interpreted as arising from reflection of incoming shocks at the interface formed by the first fracture.…”

“…If uniformly distributed, the energy within the newly created fracture volume would be W fr = fW Δ V = 0.06 mJ. Focalization of P , S -waves within the stone [35] lead to increases of stress by a factor of c ≅ 3 (Fig. 8) leading to an estimate of the energy release in the fracture volume of c 2 W fr ≅ 0.5 mJ, of the same order of magnitude as Δ U tot = 0.3 mJ.…”