Transient FEM Calculation of the Spatial Heat Distribution
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Transcript Transient FEM Calculation of the Spatial Heat Distribution
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 mm
2 Watt, Super Pulse
20 Watt cw
CO2 Laser 20 W, cw, no cooling
Laser System
CO2 laser, Sharplan 40C
250
120
measured
fit
100
Repetitionsrate [Hz]
Power [W]
200
150
FWHM
80µs
100
80
60
40
50
20
0
0
100
200
300
400
500
time [µs]
Pulse width in super pulse
mode
600
0,5
1,0
1,5
2,0
mittlere Leistung [W]
Correlation: Repetition rate to
selected mean power
2,5
Thermal damage
Important: Combination of temperature rise and time
Tissue damage
No tissue damage
Time [s]
Experimental problems
to measure the temperature T(x,y,z,t)
at a point (x,y,z) inside the tissue for various times t
Thermocouples
IR Camera
Laser
Laser
Tissue
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
Laser Beam
Infrarot Camera
Camera Processor
Video Recorder
PC + Videocart
1 Watt, cw, defokussiert
22,8
Temperatur [°C]
22,6
22,4
22,2
Data: Data17_B
Model: Impulsantwort
Chi^2 = 0.0042
P1 0.14485
±0.00508
P2 0.14654
±0.00812
P3 -0.48422
±0.16468
P4 21.45601
±0.04062
22,0
21,8
21,6
21,4
21,2
0
5
10
15
Zeit[s]
Time [s]
20
25
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)
Inside temperature
Tinside(x,y,z,t)
Measurement of TS
Good agreement ensures that
by IR Camera
calculation of Tinside is correct
Principles of FEM Calculation
FEM = Finite Element Method
Generate Grid Points
Equation for heat conduction
c
Node
dT ( x , t )
T ( x , t )
dt
with
= density
c = heat capacity
T = temperature
Finite Elements
Element
Q
i
t = time
= heat conductivity
Q = heat source
= Laplace operator
K T C T P
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
Equation for heat conduction
with
= density
c = heat capacity
T = temperature
c
dT ( x , t )
T ( x , t )
dt
t = time
= heat conductivity
Q = heat source
= Laplace operator
Analytical solution under boundary condition: T(x, 0) = f(x)
t
0
T ( x, t ) G x, s, t f ( s) ds G x, s, t Q s, t ds d
The Green´s function is given
G x , s, t
1
2 t
e
( x s)2
4t
Q
i
Solution
t d
2
T ( x , t ) G x , s, t Q s, t ds d M k (t ) e
d k 1
0 0
with
G x , s, t
2
kx
ks
sin
sin
e
d
d k 1 d
k
kx
M k (t ) f ( x ) sin
e
dx
d
d
0
0
d
t
( 1) k
k
d
t
e
0
k 2 2 t
d2
k 2 2 t
d2
k 2 2 t
d2
k 2 2 t
d2
h(t ) dt
g (t ) dt
kx
sin
d
Results:
1
Laser induced heat during the laser pulse interaction
1250
Temperature [°C]
1000
Temperature
750
Laser
Pulse (a.u.)
500
250
0
0
50
100
150
200
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
Temperature T [°C]
1600
1200
- dT/dz
800
T(z)
400
0
0
20
40
60
80
Depth z [µm]
100
120
140
Temperature gradient in the z-x-plane
What does these numbers mean ?
• Values were calculated using the
thermodynamical values of dentin
Density
Specific Heat
Heat Conduction
Thermal Extension
Elasticity Module
c
a
E
2.03 g/cm3
1.17 J/(g·K)
0.4 10-3 W/(mm·K)
11.9 10-6 1/°C
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 mm 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
7 ms
12 ms
20 ms
Time
19 ms
Temperature after various pulses
After 1st pulse
After 2nd
After 3rd
After 4th
20 ms
Time
Temperature development at crater center
550
temperature [°C]
500
450
400
350
300
250
200
150
100
0,00
0,02
0,04
time [s]
0,06
0,08
Temperature rise in the center of the crater
2600
temlperature [°C]
2400
2200
2000
1800
1600
0
2
4
6
pulses
Absolute value is not gauged
8
10
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
Thin Slice of Dentin
6000
FEM, Abstand 0,18 mm
WAERMEKO, 20 Hz
P1 P2
P3
Temperature [°C]
5000
4000
3000
2000
1000
Tissue is removed
by laser pulses;
z = 40 mm
0
0,00
0,05
0,10
Time [s]
Point p1
0,15
0,20
0,25
Results of Finite Element Calculation Compared to
Analytical Approximation
1800
400
1600
FEM, Abstand 0,22 mm
WAERMEKO, 20 Hz
FEM, Abstand 0,3 mm
WAERMEKO, 20 Hz
350
300
1200
Temperature [°C]
Temperature [°C]
1400
1000
800
600
400
200
250
200
150
100
50
0
0
0,00
0,05
0,10
0,15
0,20
0,25
0,00
Time [s]
0,05
0,10
0,15
Time [s]
Point p3
Point p2
FEM:
Three dimensional
24 hours
Analytical:
one spatial point
2 minutes
0,20
0,25
Which amount of heat is removed by the proceeding pulse?
100
Plexiglas
Pertinax
Bone
90
Removed Heat [%]
80
70
60
50
40
30
20
10
0
0
20
40
60
80
100
120
Repetition Rate [Hz]
140
160
180
200
Propagation of isotherms
95 Hz, 0.05 s, dT = 3.38 °C
q = 0.028, to = 0.18 s, zo = 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
1,6
Penetration Depth z[mm]
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0,00
0,25
0,50
0,75
1,00
1,25
Time t [s]
1,50
1,75
2,00
2,25
Ablation depth versus repetition rate
350
crater depth (OMECA MicroView)
linear fit
calculated depth
thermal diffusion length
300
depth [µm]
250
200
150
100
50
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
10
8
6.7
mean power [Watt]
40
20
13.3
time between pulses [ms]
First laser pulse
ablated volume
Next laser pulse
tissue
heat front
Energy loss
High ablation
efficiency due to
preheated tissue
Speciality in Plexiglas
Propagation of the isotherm of 160 °C (melting point)
90
Isotherme 160 °C ( T0 = 20 °C )
80
70
Eindringtiefe [µm]
60
Q = 30 mJ
50
40
Q = 19 mJ
30
20
10
0
0,00
0,01
0,02
0,03
Zeit [s]
0,04
0,05
0,06
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