Acta of Bioengineering and Biomechanics Original paperVol. 18, No. 3, 2016 DOI: 10.5277/ABB-00244-2014-03

Simulations of thermal processes in tooth proceedingduring cold pulp vitality testing

MARIUSZ CIESIELSKI1*, BOHDAN MOCHNACKI2, JAROSŁAW SIEDLECKI3

1 Institute of Computer and Information Sciences, Częstochowa University of Technology, Częstochowa, Poland.2 Higher School of Labour Safety Management, Katowice, Poland.

3 Institute of Mathematics, Częstochowa University of Technology, Częstochowa, Poland.

Purpose: This paper deals with the mathematical modeling of the thermal processes occurring in the tooth, during a very brief con-tact (a few seconds) with a very cold liquid on a part of the tooth crown. In this way one can simulate a heat transfer in tooth proceedingduring a dental diagnostic test – pulp vitality testing. The impact of rapid ambient thermal changes acting on the tooth can cause tooth-ache. Methods: The mathematical model: a system of partial differential equations with initial-boundary conditions (the axially-symmetrical problem) and their numerical solutions using the control volume method is discussed. Results: Simulation results of thekinetics of the temperature changes inside the tooth are presented. The example of the control volume mesh (using the Voronoi poly-gons) well describing the shape of a molar tooth is given. Conclusions: The simulation results (the temperature distribution in the tooth atany moment of the simulation time and the kinetics of temperature variation at the points of the tooth domain considered) can help den-tists in the selection of an appropriate method of treatment.

Key words: bio-heat transfer, mathematical modeling, thermal processes, tooth tissue, dental pulp testing

1. Introduction

Dental pulp testing (also known as a vitality test ora sensibility testing) [1], [2], [7], [8] is an investiga-tion that provides important diagnostic information tothe dental clinician. The tooth is composed of threelayers: enamel, dentil and the pulp. The outer layer,the enamel, is made of hard crystal and is the mostinorganic one. The dentin lies just under the enameland is the main structure of the tooth having proper-ties as a bone-like substance. The dentine consists ofmicroscopic fluid-filled channels called dentine tu-bules. In the middle of the tooth is the pulp. The“healthy” pulp is the vital tissue consisting of numer-ous blood vessels, nerves and cells, but the pulp in thetraumatized tooth is not necessarily innervated.

There are two general types of pulp testing [1],[7]: a thermal test (cold and heat) and an electrical

one. In this paper, only cold thermal test is considered.In this test, a refrigerant (i.e., dichlorodifluoro-methane, ethyl chloride at –50 °C) is sprayed ona small cotton pellet and applied to the tooth crown.This test causes contraction of the dentinal fluidwithin the dentinal tubules. The rapid flow of fluid inthese tubules results from the hydrodynamic forcesacting on the nerve fibers. This can cause a sharp sen-sation (pain) in the healthy, innervated pulp of thetooth, which takes a few seconds after the removalof cold stimulus [2].

The research presented in this paper is related tothe computations of temperature distribution in themolar tooth. The aim of this paper is the mathematicalmodeling of the thermal processes occurring in thetooth tissues (enamel, dentin, pulp) being in contactwith cold liquid (a moistened cotton pellet). Differentcontact times with the cold liquid are analyzed. Thegeometry of the tooth is treated here as an axially-

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* Corresponding author: Mariusz Ciesielski, Institute of Computer and Information Sciences, Częstochowa University of Technology,ul. Dąbrowskiego 73, 42-200 Częstochowa, Poland. Tel: +48 34 3250589, e-mail: mariusz.ciesielski@icis.pcz.pl

Received: November 28th, 2014Accepted for publication: September 21st, 2016

M. CIESIELSKI et al.34

symmetrical domain. The detailed knowledge of ther-mal processes occurring in the tooth domain will allowoptimization of diagnostics and treatment strategiesfor clinical applications. There is a need to study thethermal behavior of tooth, and it is the main aim ofthis research.

The heat transfer in the tooth domain was consid-ered, e.g., in work [18], but the mathematical model isbased on the Fourier equation. In works [5], [19],simulations were considered that related to differenttypes of dental fillings in a tooth which was in contactwith a cold liquid. The research in paper [10] dealswith the thermal stimulation (the 1D task) of dentinecorrelated with fluid flow in the dentinal tubule. Itshould be pointed out that the pulp is a vital tissue,and the models based on the Pennes equation con-taining terms with the blood perfusion and metabo-lism should be applied. Details concerning the bio-heat transfer models can be found, among others, in[4], [10], [12]–[14]. It is evident that the analyticalsolution of the problem considered is impossible andthe numerical methods should be used. In this paper,the control volume method (CVM) [3], [6] using theVoronoi tessellation [16] in order to construct thegeometrical mesh covering the tooth domain is ap-plied.

The choice of the numerical method is not acci-dental. The CVM (in particular using the Voronoitessellation) constitutes a very effective tool for anapproximate solution of boundary-initial problemsconnected with the mathematical models of heat trans-fer processes. The different shapes of control volumesallow us to reconstruct the real shape of the 2D objectboth in the case of the homogeneous and heterogene-ous domain. The domain discretization can be locallyconcentrated, for example, close to the externalboundary. The shape of Voronoi polygons offers usthe possibility of correct and exact recording energybalances formulation. They constitute the base forconstruction of the final system of linear equationscorresponding to the transition from time t to timet + t.

2. Materials and methods

2.1. Mathematical model

The tooth domain (treated as the axially symmetri-cal object) is shown in Fig. 1. The domain consists ofthe following sub-domains: the enamel (1), the den-

tin (2) and the pulp (3). The outer surface limitingthe domain (boundaries b1 and b2) is in thermal con-tact with environment (air). Additionally, in a shortperiod of time, the boundary b2 is subjected to thevery cold liquid. The boundary 0 represents contactwith the gum.

Fig. 1. Structure of a typical molar tooth [source:http://www.polskistomatolog.pl]

and the tooth domain considered (longitudinal section)

The temperature field in the domain considered isdescribed by the following system of equations

,3,2,1),()(

)],,()([),,()(

mTQTQ

tzrTTt

tzrTTc

mmetmper

mmm

m

(1)

where index m identifies the particular sub-domains(1 – the enamel, 2 – the dentin, 3 – the pulp), T [°C] isthe temperature, r, z [m], t [s] denote spatial coordi-nates and time, c [J/(m3 °C)], [W/(m °C)] are thevolumetric specific heat and the thermal conductivity,respectively. The Laplace operator in the cylindricalaxisymmetrical coordinates system is given as

.),,()(

),,()(1

)],,()([

ztzrTT

z

rtzrTTr

rr

tzrTT

mm

mm

mm

(2)

In equation (1), the terms Qper and Qmet [W/m3] arethe capacities of volumetric internal heat sources con-nected with the blood perfusion and metabolism, re-spectively. Assuming that the pulp is fed by a largenumber of uniformly spaced capillary blood vesselsand the blood vessels are not present in the dentil andenamel, one has

Simulations of thermal processes in tooth proceeding during cold pulp vitality testing 35

,3if)],,,()[(

,2,1if,0)(

33 mtzrTTTGcm

TQbbb

mper (3)

where Gb3 is the blood perfusion rate in the pulp[m3

(blood)/(s m3(tissue))], cb is the blood volumetric spe-

cific heat, and Tb is the blood temperature. The meta-bolic heat source Qmet 3 in the pulp sub-domain can betreated as a constant value or a temperature-dependentfunction [10], and simultaneously Qmet 1 = Qmet 2 = 0for the enamel and dentil sub-domains have been as-sumed.

Equation (1) is supplemented by the boundary-initial conditions. For t = 0 the initial condition isknown

.3,2,1,|),,( 0 mTtzrT inittm (4)

On the contact surfaces between tooth sub-domains, the continuity conditions are assumed

)},3,2(),2,1{(),(

),,(),,(

),,(),,(:),(

lk

tzrTtzrTn

tzrTn

tzrTzr

lk

ll

kk

lk

(5)

where /n is a normal derivative. On the externalsurfaces of the sub-domains, the Dirichlet (on thesurfaces – 0 and z) and the Robin (on the surface ofthe tooth crown – b1 and b2) boundary conditionsare given

3,2,1,),,(:},{),( 0 mTtzrTzr tissuemz , (6)

)],(),,([

),,(:),(

1

111

tTtzrT

ntzrTzr

airambair

b

(7)

],(for)](),,([

],(],0(for)](),,([

),,(:),(

11

1

211

1

112

contact

liqambliq

contact

airambair

b

tttt

tTtzrT

ttttt

tTtzrTn

tzrTzr

(8)

where air, liq [W/(m2 °C)] are the convective heattransfer coefficients and Tamb air, Tamb liq [°C] are thetemperatures of the air or fluid, respectively. In themathematical model the following simplifications areassumed: the temperature of the fluid grows according

to a given function and the heat transfer coefficientsare treated as the constant values (dependent on sev-eral factors (fluid velocity, surface geometry, natureof motion, etc.). Time t1 is a moment of simulationtime at which the contact of the moistened cottonpellet with cold liquid takes place. Time tcontact is thecontact time of the pellet with the tooth crown andt2 is the final time of simulation. In the time intervalt (0, t1], the patient has an open mouth and breath-ing – this causes cooling of the tooth with respect tothe initial temperature before starting a diagnostic test.On the boundary r, the non-flux boundary conditionis given

.3,2,1,0),,(:),(0

mr

tzrTzrr

mr (9)

2.2. Control volume method

At the stage of numerical modeling the controlvolume method (CVM) using the Voronoi tessellationhas been used. A similar version of CVM for the 2Dtask was discussed in detail by Ciesielski andMochnacki in [3], [6]. In this paper, the control vol-umes are in the shape of rings. So, the domain ana-lyzed (the longitudinal section) of the tooth is dividedinto N volumes (the section of the ring-shaped ele-ment corresponds to the shape of the Voronoi poly-gon). In Fig. 2, the example of the control volumemesh (N = 1835) and the selected control volume arepresented.

The CVM algorithm allows one to find the tran-sient temperature field at the set of nodes corre-sponding to the central points of the control volumes,while the nodal temperatures are found on the basis ofenergy balances for the successive CV.

In Fig. 3, the cross-section of control volume CViwith the central node pi = (ri, zi) is presented. Thiscross-section is a non-regular ni-sided polygon, at thesame time ni is the number of adjacent control vol-umes CVi(j), for j = 1, ..., ni, containing the nodes pi(j).Subscript i(j) indicates the index number of the adja-cent CV. The distance between nodes pi and pi(j) isdenoted by hi(j), whereas the area of contact surface(here, the surface obtained by rotation of the polygonside around the z axis) between two adjacent CVi andCVi(j) is equal to Ai(j) and the volume of ring-shaped CVi

is denoted by Vi. If the polygon surface Ai(j) is cov-ered by the outside boundary of sub-domains then the“virtual” neighbouring node pi(j) lies outside the do-main considered and in the computational algorithm,

M. CIESIELSKI et al.36

the index i(j) represents the index (tag) of the bound-ary (here, 0, b1, b2, r or z).

Fig. 2. The control volume mesh in the section of the toothand selected ring-shaped control volume

Fig. 3. Control volume CVi

It is assumed for each control volume CVi thatthe thermal capacities and the capacities of internalheat sources are concentrated at the nodes repre-senting elements, while the thermal resistances areconcentrated on the sectors joining the nodes. Theenergy balances corresponding to the heat exchangebetween the analyzed control volume CVi and ad-joining control volumes result from the integrationof energy equation (1) with respect to time t andvolume CVi. Let us consider the interval of timet = t f t f. Then,

.dd)]()([

dd)],,()([

dd),,()(

1

1

1

tVTQTQ

tVtzrTT

tVt

tzrTTc

mmet

t

tmper

CV

mm

t

t CV

mt

t CVm

f

fi

f

fi

f

fi

(10)

Applying the divergence theorem for the volumeCVi bounded by the surface Ai = in

j 1 Ai(j) one ob-tains

.dd)]()([

dd)],,()([

dd),,()(

1

1

1

tVTQTQ

tAtzrTT

tVt

tzrTTc

mmet

t

tmper

CV

mm

t

t A

mt

t CVm

f

fi

f

fi

f

fi

n

(11)

The numerical approximation of the left-hand sideof equation (11) can be accepted in the form

if

if

if

i

mt

t CVm

VTTc

tVt

tzrTTcf

fi

)(

dd),,()(

1

1

(12)

where fic is an integral mean of thermal capacity

and this value is approximated by the volumetricspecific heat corresponding to the temperature Tf

(explicit scheme). The source term in equation (11)for the pulp sub-domain 3 is treated in a similarway

tVTQTQ mmet

t

t CVmper

f

fi

dd))()((1

.])()()([

])()[(

3 tVQTTGc

tVQQ

ifimet

fib

fibb

ifimet

fiper

(13)

The term determining heat conduction betweenCVi and its neighbourhoods CVi(j) can be written (forthe explicit scheme) in the form

Simulations of thermal processes in tooth proceeding during cold pulp vitality testing 37

.)],,()([

d)],,()([

dd)],,()([

dd)],,()([

1 1)()()()()(

1)()()(

1)()()(

1

)(

1

1

i i

if

f

i

ji

f

f

f

fi

n

j

n

jji

fjiji

fjiji

n

jjijiji

t

t

n

jjiji

Aji

t

t

mm

t

t A

AtAtzrTTt

tAtzrTT

tAtzrTT

tAtzrTT

n

n

n

n

(14)

In the case when Ai(j) is placed between CVi andinternal CVi(j) then ( )

fi j is approximated as follows

fji

fi

fji

ji

fi

fjif

ij

fjimji

fji

RTT

hTT

tzrTT

)(

)(

)(

)(

)()()( )],,()([

n

(15)

where ij is the harmonic mean thermal conductivitybetween nodes pi and pi(j) defined as

fji

fi

fji

fif

ij)(

)(2

(16)

and fijji

fji hR /)()( is the thermal resistance. If Ai(j) is

a part of the boundary b1 or b2 then one of theboundary conditions (7) or (8) is used and in this case,the following formula is applied

1

2

)()(

)(

fi

ji

fi

fambf

ji hTtT (17)

where Tamb(tf) {Tamb air(tf), Tamb liq(tf)} and {air, liq} should be selected, respectively. One cansee that the denominator in the above formula cor-responds to the thermal resistance related to theRobin boundary condition. If Ai(j) is a part of theboundary 0 or z then the Dirichlet boundary con-dition (6) is used and the formula determining f

ji )(

is of the form

.

2)(

)(

fi

ji

fitissuef

ji hTT

(18)

In the case when Ai(j) coincides with the axis r = 0then

.i.e.,,equivalentor0 )()(

f

itissuefji

fji

TT (19)

At the stage of numerical computations, in theplace of a large value, i.e., 1010 can be assigned.

In order to ensure the unification of notations of theboundary formulas (17), (18) and (19) with the generalformula (15), one can use the following values

,2

,12

,12

and},),(),({

)()()()(

)(

fi

ji

liqfi

ji

airfi

jifji

tissuetissuef

liqambf

airambfji

hhhR

TTtTtTT

(20)

dependent on the appropriate boundary conditions.The energy balance (11) written in the explicit

form leads to the equation

tVQTTGc

AtVTTc

ifimet

fib

fibb

n

jji

fjii

fi

fi

fi

i

])()()([

)(

3

1)()(

1

(21)

or

.)()()(

1

3

1)(

)(

)(1

fimet

fib

fibb

n

jjif

ji

fi

fji

if

i

fi

fi

QTTGc

AR

TTVc

tTTi

(22)

The initial condition (4) is implemented as 0iT = Tinit,

i = 1, ..., N.In order to ensure the stability condition of explicit

scheme (22) the coefficient related with fiT must be

positive

0)(111

3)(

)(

in

j

fibbf

ji

ji

if

iGc

RA

Vct (23)

for all control volumes CVi, i = 1, …, N. Hence, itallows one to determine the critical time step t

in

j

fibbf

ji

ji

i

fi

GcRA

V

ct

13

)(

)( )(1. (24)

M. CIESIELSKI et al.38

3. Results

Two numerical simulations of thermal processesproceeding in the tooth domain have been executed.In Fig. 2, the shape and dimensions of the domainconsidered and the control volume mesh are pre-sented.

The following thermophysical parameters of thetooth layers have been assumed [9]: c1 = 750, c2 = 1170,c3 = 4200 J/(kg K), 1 = 2900, 2 = 2100, 3 =1000 kg/m3, 1 = 0.92, 2 = 0.63, 3 = 0.59 W/(m K).The initial temperature is Tinit = 36.6 C, the tem-perature of the ambient fluid is Tliq amb(t) =50 + 5·(t t1) C and the heat transfer coefficientis liq = 1000 W/(m2 °C), while parameters for theair are the following Tair amb(t) = 25 C and air =25 W/(m2 °C). In simulations two contact timestcontact = 2 s and tcontact = 4 s, the time limits t1 = 60 s,t2 = 90 s and the tissue temperature Ttissue = 36.6 Chave been assumed.

Fig. 4. Temperature distributionin the tooth sub-domains at time 60 s

In Figs. 4, 5 and 6, the isotherms in the tooth sub-domains for times 60 s, 60 s + tcontact, and 75 s areshown, respectively. The kinetics of temperaturevariation at selected points A, B and C located nearthe boundaries of tooth sub-domains (see Fig. 2) arepresented in Fig. 7. Figure 8 shows the time deriva-tives of temperature at points B and C. The controlvolumes represented by the central nodes B and C areplaced in the dentin and the pulp sub-domains, re-spectively.

Fig. 5. Temperature distributionin the tooth sub-domains at time 60 + tcontact [s]

Fig. 6. Temperature distribution in the tooth sub-domains at time 75 s

Fig. 7. The kinetics of temperature variation at selected points(see Fig. 2)

Simulations of thermal processes in tooth proceeding during cold pulp vitality testing 39

Fig. 8. The time derivatives at points B and C (see Fig. 2)

4. Discussion

First of all, the authors assumed that the tempera-ture of the tooth at the initial moment of time corre-sponds to the average body temperature (36.6 °C).Next, for 1 minute the patient has his mouth open,while the dentist prepares to perform a diagnostic test.At this time, the tooth is cooled by the process ofbreathing (periodic breathing out of cool air and warmair exhalation). At the stage of numerical modelingthe certain simplification is introduced, namely a con-stant average temperature in the oral cavity for a pe-riod of 1 minute is assumed. The amount of heat dis-sipated by the tooth at this time depends on, amongothers, the frequency and intensity of breathing, thegeometry of the mouth, the ambient temperature andambient humidity, etc. It is a complex process, and theheat exchange at the stage of calculations was essen-tially simplified by the assumption of the constantheat transfer coefficient occurring in the Robinboundary condition. On the basis of the temperaturefield shown in Fig. 4 one can see that, as a result ofcooling, the maximum temperature drop on the crownsurface equals about 3 K. The temperature field inFig. 4 corresponds to the state where a rapid coolingof the tooth (under the influence of an external factor)begins. The problem of rapid cooling in the modeledtask has also been simplified by the use of the thirdkind of boundary condition (the Robin condition) and

the adoption of the surrounding medium temperaturein the form of functional dependence, while the heattransfer coefficient is assumed to be a constant value.The course of the function Tliq amb(t) depends on manyfactors, but it should be an increasing function be-cause the soaked swab (the moistened cotton pellet) isheated by the warmer tooth in the mouth.

Figures 5 and 6 show the temperature fields ob-tained for different contact times between the coolantand tooth, while in Fig. 7 the temperature histories atselected points are presented. It can be seen that thetemperature differences on the crown surface in theperiod of two analyzed contact times during rapidcooling are small. In contrast, the differences are morenoticeable in the inner layers of the tooth – especiallyin the areas close to the boundary between the enameland dentin (point B). Analyzing the kinetics of tem-perature changes, one can conclude that the reductionof temperature inside the tooth takes place even afterthe tooth contact with a cotton swab. The temperaturedrop in the dentin layer at point B (see Fig. 8) is no-ticeable even a few seconds after the direct liquidcooling and the maximum value reaches almost 3 K/s.In the pulp layer at point C, the temperature drop issmaller (up to 0.5 K/s) and takes a long period oftime. The rapid drop in temperature in the dentin layercauses a change in pressure of the dentinal fluidwithin the channels and its contraction (according toone theory of pain sensation), so that movement of thefluid can cause irritation of the nerve fibers. In thedamaged (dead) tooth there are no active nerve fibersand the pain should not be felt.

So far, we did not find any similar results (fromboth experimental and mathematical models) in theliterature that can be used to analyse and comparewith our numerical simulation results. From an ex-perimental point of view, the use of the vital tooth (inthe in vivo investigation) is rather impossible for ethi-cal reasons. The widely used method to study thethermal behaviour of human tooth in vitro is based onthermocouples, but it can be technically challenging tothe study due to the small size and complex geometri-cal structure of the biological tooth. These experi-mental results may contain measurement errors andtheir cause is, e.g., low spatial resolution and contactmeasurement. For a vital tooth, the subgingival part ofthe tooth is surrounded by the environmental tissueswith blood vessels.

Several works were related to the experimentswith teeth. In work [20], the replica of an axisymmet-ric model of tooth for experimental purposes has beenused in order to simulate the thermal processes occur-ring while drinking hot liquids. Jakubinek et al. [9]

M. CIESIELSKI et al.40

have been modelled and simulated the process ofphotopolymerization and related to it the changes intemperature during light-curing of dental restorations.In paper [11], the authors presented the experimentalresults (also in vitro) as the field temperature distribu-tion on the surface of a cross-section of a human mo-lar tooth, sliced longitudinally into two halves. Thetooth was heated by circulating hot water and cooleddown by air, and at the same time the infrared cameraregistered the tooth surface temperature. One can no-tice that such experimental approaches (the researchcarried out in the laboratories) do not fully correspondwith the biological reality. The main aim of experi-mental studies, first of all, was to determine the ther-mophysical parameters of particular layers of a toothand the parameters of conditions acting on the tooth,which can be used in simulations performed on thebasis of mathematical models. It should be mentionedthat the way of modelling of the heat transfer problemproceeding during pulp vitality testing so far has notappeared in the known works.

5. Conclusions

A very common problem discussed in academicworks is tooth sensitivity to various external stimula-tions. The model presented can provide informationrelated to the temperature field in the tooth and thekinetics of the temperature changes in the varioustooth sub-domains and the simulation results can as-sist dentists in the selection of an appropriate methodof diagnostics and treatment. For example, directlyafter the completion of freezing the temperature atpoint A increases, while at points B and C it continuesto decrease. This results from the reduced temperatureof the enamel sub-domain. So, the cooling effect lastslonger than the thermal contact with the coolant. Inother words, a sudden drop of temperature at pointsplacing in pulp and dentin sub-domains can causesharp pain in the tooth, lasting even a few secondsafter completion of freezing.

The results of numerical simulations discussedhere concern the selected tooth geometry, but the al-gorithm presented can be used for the different pa-rameters occurring in the mathematical model and anygeometrical shape of the tooth.

In this paper, the possibilities of the CVM applica-tion for a numerical solution of the bioheat transfer areshown. The control volume meshes (using the Voronoipolygons) accurately reproduce the geometry of thetooth (if the assumption that 2D axially-symmetrical

approximation is acceptable) – it is an essential ad-vantage of the method proposed.

In the future, research is planned connected withthe elaboration of the numerical algorithm based onthe control volume method in which the thermophysi-cal parameters of tooth sub-domains will be treated asthe interval numbers [15], [17]. This results from thefact that the biological tissue properties are dependenton the individual characteristics such as gender, ageetc. Additionally, the experimental research using thethermal imaging techniques will be realized. Sucha study will allow one (at least) to observe the courseof transient temperature field on the surface of thetooth crown.

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