ORIGINAL ARTICLE

Simulation of temperature and thermally induced stressof human tooth under CO2 pulsed laser beams using finiteelement method

Mohammad Sabaeian & Mohammadreza Shahzadeh

Received: 28 January 2013 /Accepted: 8 July 2013# Springer-Verlag London 2013

Abstract The authors report the simulation of temperaturedistribution and thermally induced stresses of human toothunder CO2 pulsed laser beam. A detailed tooth structurecomprising enamel, dentin, and pulp with realistic shapesand thicknesses were considered, and a numerical method offinite element was adopted to solve time-dependent bio-heatand stress equations. The realistic boundary conditions ofconstant temperature for those parts embedded in the gingivaand heat flux condition for those parts out of the gingivawere applied. The results which were achieved as a functionof energy density (J/cm2) showed when laser beam is irradi-ated downward (from the top of the tooth), the temperatureand thermal stresses decrease quickly as a function of depththat is a result of strong absorption of CO2 beams by enamel.This effect is so influential that one can use CO2 beams toremove micrometer layers while underlying tissues, espe-cially the pulp, are safe from thermal effects.

Keywords Laser dentistry . Bio-heat equation . Thermaleffects . Thermally induced stresses

Introduction

After invention of ruby laser by Maiman in 1960 [1], it wasGoldman et al. who investigated the possibility of using rubylaser for dental caries removal [2]. Later, several groups usedor investigated the feasibility of various types of lasers suchas CO2 [3–11], Er:YAG [10, 12], and Nd:YAG [10, 13] indental procedures.

Nowadays, lasers are being widely employed in dentaltreatments such as dental hypersensitivity [14] tooth whitening[15] esthetics [16], pulpotomy [7], etching [17], caries removaland prevention [5, 6, 18, 19], and drilling [20]. These vastapplications of laser beams in dentistry are due to noise,vibration, and pain reduction, automatically sterilization ofunder-operating tissues [9, 16] and fine removing of tissuesleading to a high degree of control [9].

The first laser approved by FDA for using in soft tissuesurgeries was carbon dioxide (CO2) laser with an efficientoutput at 10.6 μm [18, 21]. As the 10.6-μm beams areabsorbed strongly by the tooth tissues, it was recognized as apotential tool for dental treatments [6, 22, 23]. Among topmostadvantages of CO2 laser beams, one can name the capability ofimplanting fixed appliances just after the surgical procedure[16], increasing the uptake of fluoride and reduction of acidsolubility of enamel surface [24]. Q-switched 9.6 μm beamsalso gained success in dental hard tissue preparation. A9.6-μm wavelength of CO2 laser has been demonstrated suc-cessfully to alter the surface structure of tooth enamel toincrease its resistance against the caries [25]. Goodis et al.studied pulpal safety of tooth irradiated by 9.6 μm laser with arepetition rate of 10 Hz of microsecond pulses and reportedthis pulsed laser beam as a reliable and safe tool for cariesprevention in human tooth without any dangerous and perma-nent damages [25]. Also, Rosa et al. analyzed the thermaldamage zones in tooth after irradiated by microsecond andsub-microsecond laser pulses using polarized light microsco-py and synchrotron-radiation Fourier transform infraredspectro-microscopy and found out the thermal damage zonewas minimal for 9.6 μm CO2 laser pulses [26, 27].

Human tooth consists of threemain parts, namely the enamel,dentin, and pulp. Enamel which is the hardest tissue in thehuman body [28] has a pronounced absorption at around10.6 μm [6, 16]. Robust absorption of light in enamel resultsin a more temperature increase in enamel relative to other toothparts. Since the major portion of the laser beam energy is

M. Sabaeian (*) :M. ShahzadehPhysics Department, Faculty of Science, Shahid ChamranUniversity of Ahvaz, Ahvaz, Irane-mail: m_sabaian@yahoo.com

M. Sabaeiane-mail: sabaeian@scu.ac.ir

Lasers Med SciDOI 10.1007/s10103-013-1390-6

absorbed by the upper laying tissue, i.e., the enamel, the under-lying tissue say the pulp will be provided with ammunition [13,20, 24, 29]. The temperature intervals causing specific effectshave determined as follows [11, 30]: from 100 to 650°C tem-perature increase, the water content decreases, carbonate lossand carbonate converting to phosphate, and protein decomposi-tion occurs; from 650 to 1,100°C temperature increase, thethermal recrystallization and crystal size growth takes placeand the remaining of water and carbonate is lost; for temperatureincrease higher than 1,100°C, destructive phenomena such ashydroxyapatite melting and disproportionate take place.

In laser dentistry, modeling and experimentation of the toothtemperature and thermally induced stresses matter because itprovides us with a good insight to the laser parameters appro-priate for surgery. So, one can adjust them to reach betterresults. In this regard, Chiang et al. used 10.6 μm beams ofCO2 laser with 0.8mmbeam spot size to see themicrostructuralchanges in enamel and dentin during laser radiation [3]. Sagiet al. considering a homogeneous cube for the tooth calculatedthe temperature distribution of the tooth under laser beams [31].Malmstrom et al. inspected experimentally the effects of repe-tition rate of CO2 laser pulses on the temperature of the pulpchamber [32]. Prech et al. investigated the temperature effectson a 200-μm thick enamel slide and made a model for itstemperature changes under 100 ns pulsed CO2 beams [33].

In all of theoretical models, to the best of our knowledge,cubic geometries and homogenous constitutions were con-sidered for tooth. Furthermore, a comprehensive model forthermal stress is missing in the literature. Therefore, in thiswork by taking a realistic geometrical structure into accountalong with all tooth details, the temperature distribution andthermally induced stresses as a function of laser energydensity are simulated. To be precise, all tooth parts includingenamel, dentin, and pulp with realistic thicknesses andshapes are considered. The proper temperature boundaryconditions for tooth parts embedded in the gingiva and thoseout of the gingiva are employed.

Method and materials

The energy density of each pulse was calculated by Ed=2-E/πω2 where E is the pulse energy and w=0.8 mm is the laserbeam spot size [3, 11]. Time duration of each pulse was takento be τ=1 μs [34]. All calculations have been performed interms of energy density (J/cm2) which is a sensible quantityin these types of work.

Thermal modeling

The bio-heat equation governing the temperature distributionwithin the tissues is given by [35]:

ρC∂T∂t

þ ∇!: −K ∇

!T

� �¼ Qext ð1Þ

Where ρ is the mass density of tissue, C is the specificheat, K is the thermal conductivity, and Qext is the externalheat source. In our model, the optical absorptions of allsuccessive layers are taken into account with introducingthe following expressions as heat sources:

Qext;1 ¼2α1E

πω2τexp −2 x2 þ y2

� �=ω2−α1y−ln 2ð Þ t

2

τ2

� �ð2aÞ

Qext;2 ¼2α2E

πω2τexp −2 x2 þ y2

� �=ω2−α1y1

�

� exp −α2y−ln 2ð Þ t2

τ2

� �ð2bÞ

Qext;3 ¼2α3E

πω2τexp −2 x2 þ y2

� �.ω2−α1y1−α2y2

h i

� exp −α3y−ln 2ð Þ t2

τ2

� � ð2cÞ

Where 1, 2, and 3 refer to enamel, dentin, and pulp, respec-tively, with α1, α2, and α3 as their absorption coefficients. y1and y2 stand for average thicknesses of enamel and dentin justbelow the laser beam, respectively. As Eqs. (2a)–(2c) areshowing, Gaussian functions for spatial and temporal distribu-tion of laser beam have been considered.

For temperature, the general boundary condition of Robinis used which is given by [35, 36]:

−Kn:∇!T surfacej þ h T−T0ð Þsurface ¼ 0 ð3Þ

Where n is an outward unit vector perpendicular to thetooth boundaries, h is the heat convection coefficient, and T0is the ambient temperature. For those parts of tooth embeddedin the gingiva, the perfect cooling condition (h→∞ [35] ) isassumed, so Tsurface=T0=310.15K that is the human bodytemperature. For those parts of tooth above the gingiva, weassume a cooling via the heat flux with a heat convectioncoefficient of h=10W/m/K2 [35] and T0=300.15K as free-airtemperature. For inner interfaces, the temperature continuity isused. Figure 1a shows the scheme of tooth with its details.

The thermally induced stresses (or strains) are calculatedusing equilibrium equation given by [37]:

1−ν1þ ν

∇!

∇!: u!

� �−1−2ν1þ ν

∇!� ∇

!� u*

� �¼ αT ∇

!T ð4Þ

Where ν is the Poisson's ratio, and αT is the thermalexpansion coefficient. The solution to the above equationgives the displacement vector components (ux, uy, uz). The

Lasers Med Sci

first derivatives of displacement vector components definethe strain tensor components as:

εkl ¼ 1

2

∂uk∂xl

þ ∂ul∂xk

�ð5Þ

The generalized Hook's law can be used to calculate thestress components [37]:

σij ¼ Cijklεkl ð6Þ

Where σijs are the stress tensor components and Cijkls arethe stiffness tensor components, given in terms of Young'smodulus and Poisson's ratio [37].

The complex geometrical structure of tooth illustrated inFig. 1a along with its variety of boundary conditions do notallow to explore an analytical solution. So, in this work, weadopted a numerical method of finite element for solving thebio-heat and equilibrium equations with heat sources given

in Eqs. (2a)–(2c). The mechanical and thermo-optical prop-erties of human tooth at 10.6 μm are listed in Table 1.Approximate sizes for human tooth thicknesses have beentaken from ref. [43].

Results

To carry out the numerical calculations, the whole structureof tooth was divided into 26,007 triangular meshes shown inFig. 1b. For enamel, dentin, and pulp, 14,234, 8,965, and2,808 meshes were used, respectively.

Figure 2 shows the variations of temperature of the toothsurface (y=0) with time for two energy densities of 1.59 and2.19 J/cm2. As the curves are showing, the temperature isquickly raising in a small time interval that is of the order ofpulse duration (1 μs). In this period of time, the pulse energyis transferred into the tooth. Once the pulse vanishes, thetemperature drops more slowly proportional to the tooththermal relaxation time.

Figure 3 shows the temperature distribution along the ydirection from the tooth surface (y=0) towards the depth of25 μm. The calculations have been done for several energydensities of 3.38, 2.78, 2.19, and 1.59 J/cm2 shown in insetcorrespond to pulse energies of 34, 28, 22, and 16 mJ,respectively. The curves have been drawn at a time in whichthe temperature reaches to its maximum value. As the curvesis showing, for energy density of 3.38 J/cm2, the maximumtemperature reaches to ~1,450°C just at the top of the tooth.According to aforementioned temperature intervals, thistemperature lays in damaging range, so the enamel meltingor carbonization will take place. We notice a quick drop oftemperature with distance. For only a 25-μm distance awayfrom the tooth surface, the temperature falls into 200°C (for3.38 J/cm2) that is relatively a safe temperature. At thistemperature, the water lost and protein decomposition oc-curred are not considered dangerous. The quick temperaturedecay in enamel, when getting away from the irradiation point,is a result of strong absorption of 10.6μm beam by enamel andits low thermal conductivity. This quick temperature decaying

Fig. 1 a Scheme of tooth with its various parts, boundaries, anddirection of laser radiation and b meshed tooth

Table 1 Mechanical and ther-mo-optical properties of humantooth. The absorption coeffi-cients are given in 10.6 μm

Enamel Dentin Pulp

Average thickness (m) 1.45×10−3 2.32×10−3 6.45×10−4

Absorption coefficient(cm−1) 819 [22] 813 [22] 800 [38]

Thermal conductivity (W/m/K) 0.93 [3] 0.58 [3] 0.63 [39]

Specific heat capacity (J/Kg/K) 753.624 [3] 1,172.304 [3] 4,200 [39]

Density(Kg/m3) 2,950 [40] 2,140 [40] 1,000 [39]

Young's modulus (Pa) 8.41×1010 [41] 1.86×1010 [41] 2×103 [42]

Poisson ratio 0.30 [41] 0.31 [41] 0.45 [42]

Thermal expansion coefficient (K−1) 16.96×10−6 [23] 10.59×10−6 [23] 10−5 [23]

Lasers Med Sci

in enamel approves that CO2 laser beam can be considered as asuitable and reliable wavelength for enamel surgery becausethe lower layers would be immune from thermal effects.

In order to provide a more quantitative insight into thetooth temperature, we calculated the maximum temperatureat three separate planes perpendicular to beam direction,namely, the tooth surface and two distances of 10 and20 μm away from the tooth surface. The results are illustrat-ed in Fig. 3 which demonstrate the temperature versus theenergy density.

A range of energy density from 0.75 to 3 J/cm2 has beenchosen to cover the practical values. Figure 4 also reveals alinear function for temperature versus the energy density.This result is in good agreement with experimental resultsreported by Prech et al. [33] that achieved a similar linerfunction. Our achievements that have been explored forvarious depths indicate smaller steeps for lower layers. Most

notably, the results predict a very small steep for temperatureversus the energy density for a distance of 20 μm away fromthe tooth surface. This insures us that the temperature differ-ence between the surface and the underlying layers is so highthat the inner tissues experience a safe temperature. In par-ticular, for pulp that is highly sensitive to temperature riseand is located at approximately 3.78 mm below the toothsurface; no noticeable temperature changes are experienced.This result has also been reported in some experimentalworks approving an interaction range of 10 to 40 μm forCO2 laser beam in the tooth [4, 7, 44, 45]. For the pulp of upto 3.3°C temperature rise, the reversible histological effectsoccur. Loss of pulpal vitality will take place if the tempera-ture changes reach to 5.6°C [15]. Our model predicts evenlower values for temperature in the pulp for energy densitiesat which enamel starts melting.

In practical situation, the tooth is irradiated with a train ofsuccessive pulses. As one may expect in this situation, theaccumulation of heat due to the temporally slow spread ofthe energy would increase the tooth temperature. However,this event strongly depends on pulse repetition rate. Ourresults for repetition rate of v=20 Hz have been shown inFig. 5 in which the curves illustrate the temporal changes oftemperature for ten successive pulses. As the curves areshowing, for a repetition rate of v=20 Hz, the heat accumu-lation is very slow such that after a few number of pulses, thetemperature peaks reach to a steady state. However, when thepulse repetition rate is increased, a noticeable heat accumu-lation will take place and accordingly the temperature peaksgrow continuously. Calculations for pulse repetition rate ofv=500 Hz for three energy densities have been done and aredepicted in Fig. 6. We notice that the increasing in temper-ature could not persist if the shooting is cut or if the positionof laser spot is changed.

Fig. 2 Temporal variations of tooth temperature at y=0 for a single shot

Fig. 3 Temperature versus the distance from the top of the tooth atseveral energy densities

Fig. 4 Maximum temperature as a function of energy density (J/cm2) atthe surface (black square), 10 μm (black circle), and 20 μm (black up-pointing triangle) away from the tooth surface

Lasers Med Sci

The numerical procedure adopted for repetitively pulsedpumping calculations is as follows: For each pulse, thespatial temperature distribution of whole tooth at the finaltime step is considered as the initial time step of the nextpulse. This trend was repeated for all next pulses.

The thermal stresses induced by laser heating have alsobeen calculated and are illustrated in Fig. 7, where thestresses have been drawn versus distance from the toothsurface (y=0) to y=25 μm. According to the curves, theenamel stresses are compressive (negative in sign) and theirmaximum occurs at the tooth surface. For energy densities of1.59 to 3.38 J/cm2, the thermal stress ranges from ~1,250 to~2,800 MPa, respectively. It then decreases quickly with get-ting away from the surface and reaches to 400 to 475 MPa,respectively, at a typical depth of 25 μm in the enamel. Com-pressive stresses can be deduced from the natural tendency of

warmer internal parts of tooth to expand. The compressivestresses for mentioned energy densities exceed safe range thatis from 250 to 550 MPa for enamel [22] indicating seriouscracks or even fracture for upper layers [8–12]. As the stressdecreases noticeably in a few tens of micrometers, the proce-dure of removing thin layers from enamel surface can be donefinely. We notice the crack, and melting cannot happen simul-taneously. A closed inspection shows tooth cracking happenedbefore enameling for enamel.

We complete our investigation with inspecting the ther-mally induced stresses at the top and two depths of enamellocated at 10 and 20 μm below the surface. Figure 8 shows alinear relation between the stress and the energy density. Thesimilar effect was also observed for temperature (Fig. 4). Thevariations of thermal stress versus the energy density aresmoother considerably for deeper layers.

Fig. 5 Temporal variations of temperature at the surface of tooth forrepetitively pulse shooting with repletion rate of v=20 Hz

Fig. 6 Temporal variations of temperature at the surface of tooth forrepetitively pulse shooting with repletion rate of v=500 Hz

Fig. 7 Thermally induced stress as a function of distance (y) for variousenergy densities

Fig. 8 Thermally induced stress at three planes of tooth as a function ofenergy density (J/cm2)

Lasers Med Sci

Conclusion

In this work the temperature and thermally induced stressesof human tooth under CO2 laser pulses were simulated byfinite element method. To achieve applicable results, weconsidered detailed structure for the tooth with real dimen-sions. The aim of this work was to present quantitativevalues for temperature and thermal stress as a function ofenergy density.

The results predict a quick decay for temperature as wellas thermal stress with increasing the depth, approving CO2

laser as a very good tool for fine layer removing. This can betreated while underlying tissues are provided with hugeammunition from damaging thermal effects.

Acknowledgments The authors would like to thank Shahid ChamranUniversity of Ahvaz, Iran, for the financial support andMs. F. Khodarahmifrom the Health Department of York University, Toronto, Canada, for theuseful information about tooth.

References

1. Maiman TH (1960) Stimulated optical radiation in ruby. Nature187:493–494

2. Goldman L, Hornby P, Meyer R, Goldman B (1964) Impact of thelaser on dental caries. Nature 203:417

3. Chiang Y, Lee B, Wang Y, Cheng Y, Chen Y, Shiau J, Wang D, LinC (2008) Microstructural changes of enamel, dentin-enamel junc-tion, and dentin induced by irradiating outer enamel surfaces withCO2 laser. Lasers Med Sci 23:41–48

4. Wu C, Roan R, Chen J (2002) Sintering mechanism of the CaF2 onhydroxyapatite by a 10.6-μmCO2 Laser. Lasers Surg Med 31:333–338

5. Klocke A, Mihailova B, Zhang S, Gasharova B, Stosch R, Güttler B,Kahl-Nieke B, Henriot P, Ritschel B, Bismayer U (2007) CO2 laser-induced zonation in dental enamel: a Raman and IRmicrospectroscopicstudy. J Biomed Mater Res 81B:499–507. doi:10.1002/jbm.b.30690

6. Zuerlein MJ, Fried D, Featherstone JD, Seka W (1999) Opticalproperties of dental enamel in the mid-IR determined by pulsedphotothermal radiometry. IEEE J Sel Top Quant Electron 5:1083–1089. doi:10.1109/2944.796333

7. Arrastia AMA, Wilder-Smith P, Berns MW (1995) Thermal effectsof CO2 laser on the pulpal chamber and enamel of human primaryteeth: an in vitro investigation. Lasers Surg Med 16:343–350

8. McCormack SM, Fried D, Featherstone JDB, Glena RE, Seka W(1995) Scanning electron microscope observations of CO2 lasereffects on dental enamel. J Dent Res 74:1702–1708

9. Zuerlein MJ, Fried D, Seka W, Featherstone JDB (1998) Modelingthermal emission in dental enamel induced by 9–11 μm laser light.Appl Surf Sci 127–129:863–868

10. Yamada MK, UoM, Ohkawa S, Akasaka T, Watari F (2004) Three-dimensional topographic scanning electron microscope and Ramanspectroscopic analyses of the irradiation effect on teeth by Nd:YAG,Er: YAG, and CO2 lasers. J Biomed Mater Res 71B:7–15. doi:10.1002/jbm.b.30063

11. Lin CP, Lee BS, Kok SH, Lan WH, Tseng YC, Lin FH (2000)Treatment of tooth fracture by medium energy CO2 laser and DP-bioactive glass paste: thermal behavior and phase transformation ofhuman tooth enamel and dentin after irradiation by CO2 laser. JMater Sci Mater Med 11:373–381

12. Sasaki KM, Aoki A, Ichinose S, Ishikawa I (2002) Morphologicalanalysis of cementum and root dentin after Er:YAG laser irradia-tion. Lasers Surg Med 31:79–85

13. Zhao FY, Zhang KH, Wu MJ, Zhou H (1989) Threshold irradiationdose for damage to dental pulp irradiated by a Nd:YAG laser.Lasers Med Sci 4:187–191

14. Zhang C, Matsumoto K, Kimura Y, Harashima T, Takeda FH, ZhouH (1998) Effects of CO2 laser in treatment of cervical dentinalhypersensitivity. J Endod 24:595–597

15. Coutinho DS, Silveira L, Nicolau RA, Zanin F, Brugnera A (2009)Comparison of temperature increase in vitro human tooth pulp bydifferent light sources in the dental whitening process. Lasers MedSci 24:179–185

16. Kívia Correia Gama S, Martins de Araújo T, Luiz Barbosa PinheiroA (2008) Benefits of the use of the CO2 laser in Orthodontics.Lasers Med Sci 23:459–465

17. Jamjoum H, Pearson GJ, McDonald AV (1995) A comparativestudy of etching enamel by acid and laser. Lasers Med Sci 10:37–42

18. DunnWJ, Davis JT, Bush AC (2005) Shear bond strength and SEMevaluation of composite bonded to Er:YAG laser-prepared dentinand enamel. Dent Mater 21:616–624

19. Franklin SR, Chauhan P, Mitra A, Thareja RK (2005) Laser abla-tion of human tooth. J Appl Phys 97:094919

20. Nair PNR, Baltensperger M, Luder HU, Eyrich GKH (2005)Observations on pulpal response to carbon dioxide laser drilling ofdentine in healthy human third molars. Lasers Med Sci 19:240–247

21. Oliveira V, Sivakumar M, Vilar R (2006) A mathematical descrip-tion of surface texture development in laser-ablated dentin. J ApplPhys 100:104701

22. Vo-Dinh T (Ed) (2003) Biomedical photonics handbook. CRC.ISBN: 978-0-8493-1116-1

23. Xu HC, Lui WY, Wang T (1989) Measurement of thermal expan-sion coefficient of human teeth. Aust Dent J 34:530–535

24. González-Rodríguez A, López-González JD, Castillo JDL, Villalba-Moreno J (2011) Comparison of the effects of diode laser and CO2

laser on human teeth and their usefulness in topical fluoridation.Lasers Med Sci 26:317–324

25. Goodis HE, Fried D, Gansky S, Rechmann P, Featherstone JDB(2004) Pulpal safety of 9.6 μm TEA CO2 laser used for cariesprevention. Lasers Surg Med 35:104–110

26. Rosa AD, Sarma AV, Le CQ, Jones RS, Fried D (2004) Analysis ofperipheral thermal damage after laser irradiation of dentin usingpolarized light microscopy and synchrotron radiation infraredspectromicroscopy. Biomedical Optics 5313:31–40

27. Rosa AD, Sarma AV, Le CQ, Jones RS, Fried D (2004) Peripheralthermal and mechanical damage to dentin with microsecond andsub–microsecond 9.6, 2.79, and 0.355 μm laser pulses. Lasers SurgMed 35:214–228

28. Avery JK (2001) Oral development and histology. Thieme, New York29. Yazami H, Zeinoun T, Saba SB, Lamard L, Peremans A, LimmeM,

Geerts S, Lamy M, Nammour S (2010) Pulp temperature increaseduring photo-activated disinfection (PAD) of periodontal pockets:an in vitro study. Lasers Med Sci 25:655–659

30. Fried D, Glena RE, Featherstone JDB, Seka W (1997) Permanentand transient changes in the reflectance of CO2 laser-irradiateddental hard tissues at λ=9.3, 9.6, 10.3 and 10.6 μ and at Fluencesof 1-20 J/cm2. Lasers Surg Med 20:22–31

31. Sagi A, Segal T, Dagan J (1984) A numerical model for temperaturedistribution and thermal damage calculations in teeth exposed to aCO2 laser. Math Biosci 71:1–17

32. Malmstrom HS, Mc Cormack SM, Fried D, Featherstone JDB(2001) Effect of CO2 laser on pulpal temperature and surfacemorphology: an in vitro study. J Dent 29:521–529

33. Prech P, Jancarek A, Gavrilov P, Key PH (1998) Temperaturechanges induced by low-energy CO2 laser irradiation in enamel.Laser Phys 8:336–339

Lasers Med Sci

34. Fried D, Zuerlein MJ, Le CQ, Featherstone JDB (2002) Thermaland chemical modification of dentin by 9–11 μm CO2 laser pulsesof 5–100 μs duration. Lasers Surg Med 31:275–282

35. Sabaeian M (2012) Analytical solutions for anisotropic time-dependent heat equation with Robin boundary condition for cubic-shaped solid state laser crystals. Appl Opt 51:7150–7159

36. Sabaeian M, Nadgaran H, Mousave L (2008) Analytical solution ofthe heat equation in a longitudinally pumped cubic solid-state laser.App Opt 47:2317–2325

37. Mousavi L, Sabaeian M, Nadgaran H (2013) Thermally-inducedbirefringence in solid-core photonic crystal fiber lasers. OptCommun 300:69–76

38. Clayman L, Kuo P (1997) Lasers in Maxillofacial surgery andDentistry. Thieme

39. Lin M, Xu F, Lu TJ, Bai BF (2010) A review of heat transfer inhuman tooth—experimental characterization and mathematicalmodeling. Dent Mater 26:501–513

40. Preiskorn M, Zmuda S, Trykowski J, Panas A, Preiskorn M (2003)In vitro investigations of the heat transfer phenomena in humantooth. Acta Bioeng Biomech 5(2):23–36

41. Magne P (2007) Efficient 3D finite element analysis of dental restor-ative procedures using micro-CT data. Dent Mater 23:539–548

42. Romeed SA, Malik R, Dunne SM (2012) Stress Analysis ofOcclusal Forces in Canine Teeth and Their Role in theDevelopment of Non-Carious Cervical Lesions: Abfraction.International Journal of Dentistry Article ID 234845

43. Smith TM, Olejniczak AJ, Reid DJ, Ferrell RJ, Hublin JJ (2006)Modern human molar enamel thickness and enamel-dentine junc-tion shape. Arch Oral Biol 51:974–995

44. Wigdor HA,Walsh JT Jr, John DB F, Visuri SR, Daniel F,WaldvogelJL (1995) Lasers in dentistry. Lasers Surg Med 16:103–133

45. Marcon G, Vidal B (1992) Characterization of zones resulting fromthe action of carbon dioxide laser impacts on dentinal hydroxyap-atite. J Alloys Compd 188:147–151

Lasers Med Sci