Transient FEM Calculation of the Spatial Heat Distribution in Hard Dental Tissue During and After IR Laser Ablation Günter Uhrig, Dirk Meyer, and Hans-Jochen Foth Dept. of Physics, University of Kaíserslautern, Germany

Contents Motivation Basics of model calculations Results –single Pulse –low number of pulses –large number of pulses –influence of repetition rate Conclusion

cw versus pulsed mode operation Dentin, CO 2 laser, 10.6  m 2 Watt, Super Pulse 20 Watt cw

CO 2 Laser 20 W, cw, no cooling

Laser System CO 2 laser, Sharplan 40C Pulse width in super pulse mode Correlation: Repetition rate to selected mean power

Thermal damage Important: Combination of temperature rise and time Temperature [°C] Time [s] Tissue damage No tissue damage

Experimental problems to measure the temperature T(x,y,z,t) at a point (x,y,z) inside the tissue for various times t Artefacts due to heat capacity and absorption of the thermocouples Only the surface is recorded

Experimental Set-Up for the Determination of Laser Induced Heat

Motivation for Model Calculation Laser induced heat deposition on surface or bottom of a crater Three-dimensional, transient calculation Surface temperature T S (x,y,z,t) Measurement of T S by IR Camera Inside temperature T inside (x,y,z,t) Good agreement ensures that calculation of T inside is correct

Principles of FEM Calculation FEM = Finite Element Method Generate Grid Points Equation for heat conduction with  = density c = heat capacity T = temperature t = time = heat conductivity Q = heat source  = Laplace operator Finite Elements With K = matrix of constant heat conduction coefficients C = matrix of constant heat capacity coefficients P = vector of time dependent heat flow

Gauß profil and Beer‘s law

Geometric Shape

Analytical Model Calculation

Solution

Results: 1Laser induced heat during the laser pulse interaction We can ignore heat conduction during the laser pulse

2Temperature distribution after one pulse

Temperature and temperature gradient along the symmetry axis z

Temperature gradient in the z-x-plane

What does these numbers mean ? Values were calculated using the thermodynamical values of dentin Density  2.03 g/cm 3 Specific Heat c1.17 J/(g·K) Heat Conduction W/(mm·K) Thermal Extension a /°C Elasticity Module E12,900 N/mm 2 Energy flow through the surface was 0.4 MW/cm 2 at a spot of 0.1mm radius Maximum of temperature slope dT/dz = - 16,400 °C/mm in a depth 60  m beneath the surface Mechanical stress up to ~ 1000 N/cm 2 = 10 MPa Maximum stress in dentin up to 20 MPa* * Private communication R. Hibst

3Low number of pulses Temperature evolution between two pulses 7 ms 12 ms 19 ms

Temperature after various pulses After 1st pulseAfter 2nd After 3rd After 4th

Temperature development at crater center

Temperature rise in the center of the crater Absolute value is not gauged

4Large number of pulses

Result of the movie After 10 Pulses: Temperature evolution between pulses is repeated Temperature distribution is moved into the tissue We reached dynamical confinement Computer program is o.k.

5Influence of repetition rate Results of Finite Element Calculation Compared to Analytical Approximation Temperatures at the points p 1 to p 3 Tissue is removed by laser pulses;  z = 40  m Point p 1

Results of Finite Element Calculation Compared to Analytical Approximation Point p 3 Point p 2 FEM: Three dimensional24 hours Analytical:one spatial point2 minutes

Which amount of heat is removed by the proceeding pulse?

Propagation of isotherms

Ablation depth versus repetition rate time between pulses [ms]

First laser pulse tissue ablated volume heat front Energy loss High ablation efficiency due to preheated tissue Next laser pulse

Speciality in Plexiglas Propagation of the isotherm of 160 °C (melting point)

CO 2 laser on Plexiglas, the influence of heat is visible by the thickness of the melting zone

Superposition of Crater 1 and 2

Conclusion cw laser mode gives deep thermal damage In pulse mode, low repetition rates are not automatically the best version, since high repetition rates giveless thermal stress higher efficiency for ablation This model was worked out by FEM and analytical model calculations and checked by experiments