THERMAL BEHAVIOR OF TEETH DURING RESTORATION PROCEDURE WITH

RESINS: EXPERIMENTAL TESTS AND NUMERICAL SIMULATION

M. Potenza1, G. Bovesecchi1, S. Corasaniti1, E. Armellin2, L. Cerroni2, P. Coppa1

1 Industrial Engineering Department - University of Rome “Tor Vergata” 2 Clinical Science and Translational Medicine Department - University of Rome “Tor Vergata”

Rome, 21-22 June 2017

PhD program of Industrial Engineering - Technological Advancement for Health

Conservative (or restoration) dentristry is carried out by dentists for different purposes:

Prevention and cure of carious and non carious lesions

Finding and stopping initial decalcification injuries

Conventional therapy of hard tissue injuries to maintain pulp vitality and prevent future damages of calcified tissues

Foreword

Two types of restoration are possible

Indirect Dental inlay

Dental filling Direct

Nowadays most restorations are carried out with composite resins, photopolymerized.

Foreword

MATERIALS USED: • Polymeric matrix: Bowen resin • Reinforcing filler: usually barium glass, borosilicate, alumina, zirconia. • Curing agent (e.g. silanes): owns two functional groups to bind different

components. Initial state of the resin matrix: monomers set BIS-GMA.

Photopolymerization: Establishment and cross linking of the polymeric chains. The composite acquires the wanted mechanical properties.

Composite resins and their polymerization

Fondamental of photopolymerization: • Photo-initiator (canforochinon). • Photopolymerizing lamp.

The tempeature increase is due to: • Irradiation of the lamp • Exothermic reaction due to photopolymerization

Composite resins and their polymerization

Activities 1. Temperature measurements of cylindric models of teeth, made of aluminum,

stainless steel, PVC, teflon, to simulate (physical models) the thermal tooth behaviour during recostruction.

2. Developement of a finite difference numerical model simulating the photopolymerization process.

Goals 1. To evaluate the thermal distribution inside the cylinders. 2. To compare the temperature distributions of different materials, with the ones of

real teeth (according to previous researches the tooth thermal trend is mostly influenced by tooth hydratation, and hence by the tooth thermal conductivity).

3. To evaluate the composite resin thermophysical properties (thermal conductivity and thermal diffusivity) and thermochemical (Arrhenius coefficient).

Carried out actions and goals

DURING TOOTH RECONSTRUCTION : • Heat propagates through conduction;

• Heat is transferred to the environment through convection;

• Radiation heat supplied to the resin and tooth by the led lamp;

• Heat generated by the exothermic reaction of the resin

photopolymerization. This heat depend on the reaction kinetics.

Thermal phenomena in teeth

Arrhenius equation:

Arrhenius coefficient k:

k depends on the luminous intensity of the radiation from the lamp, and changes with the sample depth due to the resin transparency, evaluated through Bougert-Beer law and its extinction coefficient. During polymerization tranparency, and hence this coefficient, changes

Only one substance , so x= 1:

( )1 xd kdtα α= ⋅ −

( )1 exp ktα = − −

Photopolymerization kinetics (k)

1 13

2 2

b bk b Ib I b

= + −+

Experimental setup • Cylindrical samples in stainless steel, aluminum,

teflon and PVC with a coaxial cavity on one surface; • Photopolymerizing lamp • Dental restoration composite • Thermometers (type J thermocouples, TC) and DAS

(data acquisition system)

Material Thermal Conductivity (Wm-1K-1)

Thermal diffusivity

(m2s-1) Aluminum 202 8.58 10-5

Stainless Steel (AISI 304) 12 3.89 10-6

Teflon 0.24 1.24 10-7 PVC 0.19 1.10 10-7

Experimental

1. 3 TCs are located on sample axis, one on the irradiated surface, one in the meddle, the third on the rear surface;

2. Filling: composite resin is inserted into the cavity;

3. Sample and TCs are positioned in front of the polymerizing lamp;

4. Irradiation by the lamp and contemporary TC data acquisition;

Experimental procedure

COMMENTS ON RESULTS • Reaction heat is variable according to

the different resin quantity in the cavity;

• Variation among post process tests is due to non constant distance between lamp and surface;

Sample temperatures were recorded both during polymerization and in a second test after polymerization (the same experiment with no reaction heat and no material change).

Experiment results

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Test #1 Test #2 Test #3 Test #4 Test #5

∆T ,

°C

τ , s

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Test #1 Test #2 Test #3 Test #4 Test #5

∆T ,

°C

τ , s

Stainless Steel

Stainless Steel

When data during polymerization and after polymerization are plotted together, the heat generated during the reaction is put into evidence (their difference).

Experimental results

0 10 20 30 40 50 600

2

4

6 Post Poly Poly Difference

T , °

C

τ , s

Stability and convergence criteria:

Basic conduction heat transfer equation:

Finite Difference Geometry

Numerical model

2 2

2 2

1 1r irrq qz r r rθ θ θ θ

λ α τ+∂ ∂ ∂ ∂

+ + + =∂ ∂ ∂ ∂

2 20.5 0.5Fo Foz r

α τ α τ∆ ∆= ≤ = ≤∆ ∆

z

r

H

R

Reaction heat:

Irradiated heat (Bouguer-Beer Law)

During polymerization, the extinction coefficient changes. The following empirical equation describing this change is assumed:

Finite difference equations:

Numerical model

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2

2 2

2

2 2

, , 1 , ,

1 1, , 1, ,2

1 , 1, , 1,2

1 1, , 1, , 2 , ,

1 , 1, , 1, 2 , ,

i j k i j k

i j k i j kr r

i j k i j kz z

i j k i j k i j kr r

i j k i j k i j kz z

θ θθτ τ

θ θ θ

θ θ θ

θ θ θ θ

θ θ θ θ

+ −∂=

∂ ∆

∂ = + − − ∂ ∆

∂ = + − − ∂ ∆

∂ = + − − − ∂ ∆

∂ = + − − − ∂ ∆

( )1 1 1 k kr r r r r r r

Q dq m q m q ke q keV V d V

τ τα ρτ τ

− −∆ = = = = ∆

( ) ( ) ( )0 expI z I z a zλ λ λ= = −

( ) ( ) ( )0 expirr irrq z q z a zλ= = −

( ) ( )( )0 exp 1 expa a c a cτ τ∞= − + − −

Finite difference final equation

Finite difference forward propagation and boundary conditions

Numerical model

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2

2

0

, , 1 , , 1, , 1, , 2 , ,

, 1, , 1, 2 , ,

1, , 1, ,2

a zk irrr r

i j k i j k i j k i j k i j kr

i j k i j k i j kz

i j k i j k q ke q er r

λτ

α τθ θ θ θ θ

α τ θ θ θ

α τ α τθ θ ρτ

−−

∆ + = + + + − − + ∆

∆ + + + − − + ∆

∆ ∆ + + − − + + ∆

4 spatial boundary conditions and 1 time condition

Numerical Model

)

[ ]

[ ]

)

[ ]

[ ]

,0,,0,

,0,,0,

, ,, ,

, ,, ,

1 For 0 with 0 and 0

2 For with 0 and 0

rr

rr

r Hr H

r Hr H

z r R t

A Q hAz

q hz

z H r R t

A Q hAz

q hz

ττ

ττ

ττ

ττ

θλ θ

θλ θ

θλ θ

θλ θ

= ≤ ≤ ≥

∂ − = − ∂

∂ − = − ∂

= ≤ ≤ ≥

∂ − = − ∂

∂ − = − ∂

)

)

[ ]

[ ]

0,z,

, ,, ,

, ,,z,

3 For 0 with 0 and 0

0

4 For with 0 and 0

For 0 with 0 and 00

R zR z

R zR

r z H

z

r R z H

A hAr

hz

z H r R

τ

ττ

ττ

τ

θ

τ

θλ θ

θλ θ

τθ

= ≤ ≤ ≥

∂ = ∂

= ≤ ≤ ≥

∂ − = ∂

∂ − = ∂

= ≤ ≤ ≤ ≤=

Edge boundary conditions: • Resin properties: thermal diffusivity,

thermal conductivity, density, reaction heat, start and final extinction coefficients a0, a ͚ ;

• Basic material properties (tooth or Al, SS, Teflon, PVC), for PVC and Teflon also extinction coefficient must be supplied;

• Lamp properties: Intensity I0; power for unit volume; time length of lighting;

• Convection heat transfer coefficient (constant over all external surfaces)

Numerical Model

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( )

( ) ( ) ( ) ( )

2

2

0 0 so 1 1

1,1, 1 , , 2,1, , , 2 1,1,

1,2, 1,0, 2 1,1,

2,1, 0, , 1,1, 1,1,2

with0,1, 2,1,

1,0, 1,2, 2 1,1, 1,1,

r irr

irr

r z i j

k i j k k i j k kr

k k kz

k j k q k q kr r

k kzk k q k h k

α τθ θ θ θ θ

α τ θ θ θ

α τ α τθ θτ

θ θ

θ θ θλ

= = = =

∆ + = + + − + ∆

∆ + + − + ∆

∆ ∆ + − + + ∆

=

∆= + −

Model parameters:

Experimental evaluation of the extinction coefficient for teflon and PVC (light intensity, as measured by a photodiode, versus material thickness)

Numerical model

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

PVC Fitting Experimental

Sgn

, V

x , mm

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

9

10

Teflon Fitting Experimental

Sgn

, V

x , mm

( )3.83exp 0.923y x= −

( )5.25exp 0.477y x= −

Model results

• The calculated temperature trend involves all the recognized and described phenomena. This means that the physical nature of the phenomena and the assumed hypotheses are acceptable.

• The presented results use plausible input data, but likely not the best. A non linear least square regression could supply a better set of the unknown parameters.

• Unfortunately the high number of the unknowns could make it hard to find a good set of them.

• The best compromise is to reduce the number of unknowns, measuring them as more as possible (e.g. Lamp characteristics, resin density, basic materials properties).

• The most difficult parameter to measure could then be evaluated from the experimental data (e.g. The thermal properties of the resins).

• Thermal conductivity and diffusivity of tooth materials (dentine, pulp, enamel) can thus be determined from data reduction of experiments.

Conclusions