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In vitro investigations of repulsion during laser lithotripsy using a pendulum set-up

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Abstract

Ureteroscopic laser lithotripsy is a commonly used technique to treat ureteral calculi.The type of energy source used is one of the main influences of retrograd calculi propulsion. Using a momentum pendulum under-water set-up the induced momentum and the initial velocity were investigated. Pulsed laser light from three different clinically available laser systems, including a Ho:YAG laser, a frequency-doubled double-pulse (second harmonic generation, SHG) Nd:YAG laser and a flash-lamp pumped dye (FLPD) laser, were transmitted via flexible fibres of different core diameter to the front of the pendulum sinker. Single pulses at variable pulse energy, according to the clinical laser parameter settings, were applied to the target sinker, thus causing a repulsion-induced deflection which was documented by video recording. The maximum deflection was determined. Solving the differential equation of a pendulum gives the initial velocity, the laser-induced momentum and the efficiency of momentum transfer. The induced deflection as well as the starting velocity of the two short-duration pulsed laser systems (SHG Nd:YAG, FLPD) were similar (s max = 2–3.6 cm and v 0 = 150–200 mm/s, respectively), whereas both values were lower using the Ho:YAG laser with a long pulse duration (s max = 0.9-–1.6 cm and v 0 = 60–105 mm/s, respectively). The momentum I induced by the Ho:YAG laser was only 50% and its transfer efficacy η Repuls was reduced to less than 5% of the values of the two short-pulsed laser systems. This investigation clearly showed the variable parts and amounts of repulsion using different pulsed lasers in an objective and reproducible manner. The momentum transfer efficiency could be determined without any physical friction problems. Further investigations are needed to compare stone fragmentation techniques with respect to laser repulsion and its clinical impact.

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Correspondence to Ronald Sroka.

Appendix: Theoretical basis for the determination of v 0

Appendix: Theoretical basis for the determination of v 0

The laser-induced momentum to the pendulum can be derived from differential equations of the pendulum movement and the amplitude of the maximum deflection.

As shown in Fig. 1, there are different forces affecting the movement of the compound pendulum. The Stokes friction F Stokes and the restore force F Restore are acting against the acceleration according to:

$$ F = F_{{{\text{Restore}}}} + F_{{{\text{Stokes}}}} $$
(A1)

After insertion of the parameters of the forces this equation changes to:

$$ m \cdot \ddot{s} = - m \cdot g \cdot \sin \theta - 6 \cdot \pi \cdot {\eta_{{{H_2}O}}} \cdot r \cdot \dot{s} $$
(A2)

Further approximating

$$ \begin{array}{*{20}{c}} {\sin \theta = \sin \frac{s}{L} \approx \frac{s}{L}} & {{\text{for}}\;s\; < < \;L} \\ \end{array} $$

and substitution of the Eigen angular frequency:

$$ \begin{array}{*{20}{c}} {{\omega_0} = \sqrt {{\frac{g}{L}}} } & {\text{and}} & {2 \cdot \delta = \frac{{6 \cdot \pi \cdot {\eta_{{{H_2}O}}} \cdot r}}{m}} \\ \end{array} $$

the differential equation according to Eq. A3 can be derived:

$$ \ddot{s} + 2 \cdot \delta \cdot \dot{s} + \omega_0^2 \cdot s = 0 $$
(A3)

By introducing of the dimensionless term \( \tau = {\omega_0} \cdot t \), Eq. A3 becomes Eq. A4:

$$ s\prime \prime + \frac{{2\cdot \delta }}{{{\omega _{0}}}}\cdot s\prime + s = 0 $$
(A4)

Further introduction of the Lehr damping module D results in Eq. A5:

$$ s\prime \prime + 2\cdot D\cdot s\prime + s = 0 $$
(A5)

with \( D = \frac{\delta }{{{\omega_0}}} = \frac{{3 \cdot \pi \cdot {\eta_{{{H_2}O}}} \cdot r}}{{m \cdot \sqrt {{\frac{g}{L}}} }} \)

D characterizes the method of damping. Weak damping is characterized by D values in the range 0–1.

The solution of to Eq. A5 is given by Eq. A6:

$$ s\left( \tau \right) = A \cdot {e^{{\lambda \cdot \tau }}} $$
(A6)

And the overall solution for weakly damped compound pendulum movement by Eq. A7:

$$ s\left( \tau \right) = {e^{{ - D \cdot \tau }}}\left( {{A_1} \cdot {e^{{i \cdot \sqrt {{1 - {D^2}}} \cdot \tau }}} + {A_2} \cdot {e^{{ - i \cdot \sqrt {{1 - {D^2}}} \cdot \tau }}}} \right) $$
(A7)

This can be changed using trigonometric expressions to:

$$ s\left( \tau \right) = {e^{{ - D \cdot \tau }}}\left( {A \cdot \cos \left( {\sqrt {{1 - {D^2}}} \cdot \tau } \right) + B \cdot \sin \left( {\sqrt {{1 - {D^2}}} \cdot \tau } \right)} \right) $$
(A8)

And finally substitution of \( \tau = {\omega_0} \cdot t \):

$$ s(t) = {e^{{ - D \cdot {\omega_0} \cdot t}}}\left( {A \cdot \cos \left( {\sqrt {{1 - {D^2}}} \cdot {\omega_0} \cdot t} \right) + B \cdot \sin \left( {\sqrt {{1 - {D^2}}} \cdot {\omega_0} \cdot t} \right)} \right) $$
(A9)

The constant A and B can be determined by the deflection under the starting conditions s 0 and velocity v 0 at time t = 0:

$$ \begin{array}{*{20}{c}} {A = {s_0}} & {\text{and}} & { B = \frac{{{v_0} + D \cdot {\omega_0} \cdot {s_0}}}{{{\omega_0} \cdot \sqrt {{1 - {D^2}}} }}} \\ \end{array} $$
(A10)

According to the experiments the starting conditions are:

  1. a

    deflection at t = 0: s 0 = 0

  2. b

    velocity at maximum deflection: \( \dot{s}\left( {{t_{{\max }}}} \right) = 0 \)

Using the starting conditions and theoretical evaluation of t max:

$$ t = {t_{{\max }}} = \frac{1}{{\sqrt {{1 - {D^2}}} \cdot {\omega_0}}}\arctan \frac{{\sqrt {{1 - {D^2}}} }}{D} $$

The overall solution looks like

$$ s(t) = {e^{{ - D \cdot {\omega_0} \cdot t}}}\left( {\frac{{{v_0}}}{{{\omega_0} \cdot \sqrt {{1 - {D^2}}} }} \cdot \sin \left( {\sqrt {{1 - {D^2}}} \cdot {\omega_0} \cdot t} \right)} \right) $$
(A11)

Using Eq. A11, the starting velocity v 0 can be derived as Eq. 3:

$$ {v_0} = \frac{{{s_{{\max }}} \cdot {\omega_0} \cdot \sqrt {{1 - {D^2}}} }}{{{e^{{ - D \cdot {\omega_0} \cdot {t_{{\max }}}}}} \cdot \sin \left( {\left. {\sqrt {{1 - {D^2}}} \cdot {\omega_0} \cdot {t_{{\max }}}} \right)} \right)}} $$
(3)

with

$$ {\omega_0} = \sqrt {{\frac{g}{L}}} $$
(3-1)

and

$$ {t_{{\max }}} = \frac{1}{{\sqrt {{1 - {D^2}}} \cdot {\omega_0}}}\arctan \frac{{\sqrt {{1 - {D^2}}} }}{D} $$
(3-2)

and

$$ D = \frac{\delta }{{{\omega_0}}} = \frac{{3 \cdot \pi \cdot {\eta_{{{H_2}O}}} \cdot r}}{{m \cdot \sqrt {{\frac{g}{l}}} }} $$
(3-3)

Abbreviations:

ω 0 :

Eigen angular frequency

s max :

maximum deflection

t max :

time to reach maximum deflection

m :

mass of the lead ball

r :

radius of the ball

s :

deflection

L :

length of pendulum

g :

gravity constant

Θ :

deflection angle

η H2O :

viscosity of water

D :

Lehr damping module

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Sroka, R., Haseke, N., Pongratz, T. et al. In vitro investigations of repulsion during laser lithotripsy using a pendulum set-up. Lasers Med Sci 27, 637–643 (2012). https://doi.org/10.1007/s10103-011-0992-0

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