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