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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, CO2 laser, 10.6 m
2 Watt, Super Pulse 20 Watt cw
CO2 Laser 20 W, cw, no cooling
Laser SystemCO2 laser, Sharplan 40C
0 100 200 300 400 500 6000
50
100
150
200
250
FWHM
80µs
Po
we
r [W
]
time [µs]
measured fit
0,5 1,0 1,5 2,0 2,5
20
40
60
80
100
120
Rep
etiti
onsr
ate
[Hz]
mittlere Leistung [W]
Pulse width in super pulse mode
Correlation: Repetition rate to selected mean power
Thermal damage Important: Combination of temperature rise and time
Tem
pera
ture
[°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
Laser
IR Camera
Laser
Tissue
Thermocouples
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
0 5 10 15 20 2521,2
21,4
21,6
21,8
22,0
22,2
22,4
22,6
22,8
Data: Data17_BModel: ImpulsantwortChi 2 = 0.0042P10.14485 ±0.00508P20.14654 ±0.00812P3-0.48422 ±0.16468P421.45601 ±0.04062
1 Watt, cw, defokussiert
Tem
pera
tur [°
C]
Zeit[s]
Laser Beam Infrarot Camera
Camera Processor
Video RecorderPC + Videocart
Time [s]
Motivation for Model Calculation
Laser induced heat deposition on surface or bottom of a crater
Three-dimensional, transient calculation
Surface temperature
TS(x,y,z,t)
Measurement of TS
by IR Camera
Inside temperature
Tinside(x,y,z,t)
Good agreement ensures that
calculation of Tinside is correct
Principles of FEM Calculation
cdT x t
dtT x t Qi
( , )( , )
FEM = Finite Element Method
Node
Element
Generate Grid Points Equation for heat conduction
with = densityc = heat capacityT = temperature
t = time = heat conductivityQ = heat source = Laplace operator
Finite Elements K T C T P
WithK = matrix of constant heat conduction coefficientsC = matrix of constant heat capacity coefficientsP = vector of time dependent heat flow
Gauß profil and Beer‘s law
Geometric Shape
Analytical Model Calculation
cdT x t
dtT x t Qi
( , )( , ) Equation for heat conduction
with = densityc = heat capacityT = temperature
Analytical solution under boundary condition: T(x, 0) = f(x)
T x t G x s t f s ds G x s t Q s t dsdt
( , ) , , ( ) , , ,
0
The Green´s function is given G x s tte
x s
t, ,( )
1
2
2
4
t = time = heat conductivityQ = heat source = Laplace operator
Solution
T x t G x s t Q s t dsdd
M t ek x
d
dt
k
k t
d
k
( , ) , , , ( ) sin
00 1
22 2
2
with
G x s td
k x
d
k s
de
k
k t
d, , sin sin
2
1
2 2
2
M t f xk x
ddx
k
de g t dtk
d k t
dt
( ) ( ) sin ( )
0 0
2 2
2
( ) ( )1
2 2
2
0
k
k t
dtk
de h t dt
Results:1 Laser induced heat during the laser pulse interaction
0
250
500
750
1000
1250
200150100500
Temperature
Laser Pulse (a.u.)
Tem
pera
ture
[°C
]
Time [µs]
We can ignore heat conduction during the laser pulse
2 Temperature distribution after one pulse
Temperature and temperature gradient along the symmetry axis z
0 20 40 60 80 100 120 1400
400
800
1200
1600
- dT/dz
T(z)
Tem
pera
ture
T [
°C]
Depth z [µm]
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/cm3
Specific Heat c 1.17 J/(g·K)
Heat Conduction 0.4 10-3 W/(mm·K)
Thermal Extension a 11.9 10-6 1/°C
Elasticity Module E 12,900 N/mm2
• Energy flow through the surface was 0.4 MW/cm2 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/cm2 = 10 MPa
• Maximum stress in dentin up to
20 MPa*
* Private communication R. Hibst
3 Low number of pulses
Temperature evolution between two pulses
20 ms
Time
7 ms 12 ms 19 ms
Temperature after various pulses
20 ms
Time
After 1st pulse After 2nd After 3rd
After 4th
Temperature development at crater center
0,00 0,02 0,04 0,06 0,08100
150
200
250
300
350
400
450
500
550
tem
pera
ture
[°C
]
time [s]
Temperature rise in the center of the crater
0 2 4 6 8 10
1600
1800
2000
2200
2400
2600
tem
lpe
ratu
re [
°C]
pulses
Absolute value is not gauged
4 Large 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.
5 Influence of repetition rate
Results of Finite Element Calculation Compared to Analytical Approximation
• Temperatures at the points p1 to p3
P1 P2 P3
Thin Slice of Dentin
Tissue is removed by laser pulses;
z = 40 m
0,00 0,05 0,10 0,15 0,20 0,25
0
1000
2000
3000
4000
5000
6000
FEM, Abstand 0,18 mm WAERMEKO, 20 Hz
Te
mp
era
ture
[°C
]
Time [s]
Point p1
Results of Finite Element Calculation Compared to Analytical Approximation
Point p3Point p2
0,00 0,05 0,10 0,15 0,20 0,250
50
100
150
200
250
300
350
400
FEM, Abstand 0,3 mm WAERMEKO, 20 Hz
Te
mp
era
ture
[°C
]
Time [s]
0,00 0,05 0,10 0,15 0,20 0,25
0
200
400
600
800
1000
1200
1400
1600
1800
FEM, Abstand 0,22 mm WAERMEKO, 20 Hz
Te
mp
era
ture
[°C
]
Time [s]
FEM: Three dimensional 24 hours
Analytical: one spatial point 2 minutes
Which amount of heat is removed by the proceeding pulse?
0 20 40 60 80 100 120 140 160 180 2000
10
20
30
40
50
60
70
80
90
100
Plexiglas Pertinax Bone
Rem
oved
Hea
t [%
]
Repetition Rate [Hz]
Propagation of isotherms
0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,250,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
95 Hz, 0.05 s, dT = 3.38 °C q = 0.028, t
o = 0.18 s, z
o = 0.36 mm
95 Hz, 0.05 s, dT = 5.44 °C q = 0.028, to = 0.15 s, zo = 0.15 mm ISOKORR, dT = 3.38 °C ISOKORR, dT = 5.44 °C
Pen
etra
tion
Dep
th z
[mm
]
Time t [s]
Ablation depth versus repetition rate
0,0 0,5 1,0 1,5 2,0 2,5 3,00
50
100
150
200
250
300
350 crater depth (OMECA MicroView) linear fit calculated depth thermal diffusion length
dept
h [µ
m]
mean power [Watt]
10 8 6.713.32040
time between pulses [ms]
First laser pulse
tissue
ablated volume
heat front
Energy lossHigh ablation efficiency due to preheated tissue
Next laser pulse
Speciality in PlexiglasPropagation of the isotherm of 160 °C (melting point)
0,00 0,01 0,02 0,03 0,04 0,05 0,060
10
20
30
40
50
60
70
80
90
Isotherme 160 °C ( T0 = 20 °C )
Q = 19 mJ
Q = 30 mJ
Zeit [s]
Ein
drin
gtie
fe [µ
m]
CO2 laser on Plexiglas, the influence of heat is visible
by the thickness of the melting zone
Crater 1: 10 Pulses, 22 Hz Crater 2: 10 Pulses, 72 Hz
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 give less thermal stress
higher efficiency for ablation
• This model was worked out by FEM and analytical model calculations and checked by experiments