Numerical Modelling of Tooth Enamel Subsurface Lesion Formation Induced by Dental Plaque

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1 Originl Pper Cries Res 214;48:73 89 DOI: / Received: My 2, 213 Accepted fter revision: June 28, 213 Published online: November 14, 213 Numericl Modelling of Tooth Enmel Subsurfce Lesion Formtion Induced by Dentl Plque O. Ilie A.G. vn Turnhout b M.C.M. vn Loosdrecht C. Piciorenu Deprtments of Biotechnology nd b Geoscience nd Engineering, Delft University of Technology, Delft, The Netherlnds Key Words Deminerlistion Enmel Fluoride Mthemticl model Orl biofilms Subsurfce lesion Abstrct Using one-dimensionl mthemticl model tht couples tooth deminerlistion nd reminerlistion with metbolic processes occurring in the dentl plque, two mechnisms for subsurfce lesion formtion were evluted. It ws found tht subsurfce lesion cn develop only s the result of lternting periods of deminerlistion (cid ttck during sugr consumption) nd reminerlistion (resting period) in tooth enmel with uniform minerl composition. It ws lso shown tht minimum plque thickness tht cn induce n enmel lesion exists. The subsurfce lesion formtion cn lso be explined by ssuming the existence of fluoridecontining lyer t the tooth surfce tht decreses enmel solubility. A nerly constnt thickness of the surfce lyer ws obtined with both proposed mechnisms. Sensitivity nlysis showed tht surfce lyer formtion is strongly dependent on the length of reminerlistion nd deminerlistion cycles. The restortion period is very importnt nd the numericl simultions support the observtion tht often consumption of sugrs is key fctor in cries formtion. The clculted profiles of minerl content in enmel re similr E-Mil krger@krger.com S. Krger AG, Bsel /14/481 73$39.5/ to those observed experimentlly. Most probbly, both studied mechnisms interct in vivo in the process of cries development, but the simplest explntion for subsurfce lesion formtion remins the lterntion between deminerlistion nd reminerlistion cycles without ny pre-imposed grdients. 213 S. Krger AG, Bsel Dentl cries usully strts with lesion in the superficil enmel lyer (i.e., initil cries) developing under the influence of dentl plque metbolism. If the process is not stopped, such lesion will dvnce until the dentino-enmel junction nd even beyond, into the dentin nd pulp. The study of the initil stges of cries development is importnt in understnding the disese s whole, but lso in coming up with restortive strtegies since in the erly stges the process is reversible [Fejerskov nd Kidd, 28]. One spect tht is not well understood currently concerns the spect of initil lesions. The typicl minerl profile of initil dentl cries shows region of roughly 1 μm t the tooth surfce seemingly unffected by deminerlistion [Fejerskov nd Kidd, 28], with the min body of the lesion present under this surfce lyer. The differences in minerl content between the two zones cn become considerble, especilly in the more dvnced Olg Ilie Deprtment of Biotechnology, Delft University of Technology Julinln 67 NL 2628 BC Delft (The Netherlnds) E-Mil tudelft.nl /18/213 4:4:53 PM

2 stges of cries development: up to 99% minerl content in the surfce lyer compred to just 5 75% in the body of the lesion [Robinson et l., 2; Fejerskov nd Kidd, 28]. This prticulr profile hs been observed for the first time by Hollnder nd Sper [1935], who mistook it for photogrphic rtefct. Since then, much reserch both experimentl nd theoreticl hs been crried out in order to confirm nd explin the phenomen responsible for subsurfce lesion formtion [Gry nd Frncis, 1963; Vn Dijk et l., 1979; Lngdon et l., 198; Ten Cte, 1983; Mrgolis nd Moreno, 1985]. From the hypotheses mde to explin the mechnisms behind surfce lyer formtion, two hve received specil ttention. These hypotheses hve been tested in the current study. (1) Initil dentl cries is minly the consequence of n imblnce between two concurrent processes, deminerlistion nd reminerlistion of tooth enmel [Loesche, 1986; Hicks et l., 24; Fejerskov nd Kidd, 28]. These processes hve the sme thermodynmic driving force nd cn both occur in vivo. Therefore, the formtion of surfce lyer covering subsurfce lesion is seen s the result of n imblnce following repeted cycles of deminerlistion (during mels) nd reminerlistion (between mels). (2) The minerl crystls re less soluble t the enmel surfce. This might be due to the following: (i) A fluoride distribution in the enmel depth with the highest content t the surfce [Arends nd Christoffersen, 1986; Fejerskov nd Kidd, 28]. The fluoride is incorported into hydroxyptite (HAP) structure forming more stble minerl, fluorohydroxyptite (FHAP), which mkes the enmel less soluble during cid ttcks [Koutsoukos et l., 198; Robinson et l., 2]. (ii) HAP crystls with higher purity (hence, lower solubility) my develop t the enmel surfce through process clled Ostwld ripening [Robinson et l., 2; Fejerskov nd Kidd, 28]. Other hypotheses in the dentistry literture hve been proposed to explin the surfce lyer: fster reminerlistion nd delyed diffusion t the tooth surfce [Silverstone, 1977; White et l., 1988], presence of chemicl species tht inhibit subsurfce reminerlistion [White et l., 1988], presence of slivry proteins tht inhibit surfce deminerlistion nd reminerlistion [White et l., 1988], nd phse trnsformtion t the tooth surfce resulting in the formtion of diclcium phosphte dehydrte [Mrgolis nd Moreno, 1985; Fejerskov nd Kidd, 28]. Although plusible, some of these premises re too specultive or cnnot lone explin the formtion of the surfce lyer. 74 Cries Res 214;48:73 89 DOI: / The interction between physicl, chemicl nd (micro)biologicl spects involved in subsurfce lesion formtion is highly complex. Intuition nd simple clcultions my not be sufficient to llow thorough theoreticl evlution of conceptul hypothesis. Therefore, series of numericl models of cries formtion hve been developed [Zimmermn, 1966 c; Holly nd Gry, 1968; Vn Dijk, 1978; Ten Cte, 1983; Arends nd Christoffersen, 1986]. These models could test with minimum effort hypotheses for which lborious experiments (some even impossible) would hve been required. A disdvntge though is tht ll these models re simplified to del exclusively with the processes occurring in the superficil lyer of the tooth (deminerlistion or reminerlistion), without considering the ctive influence of the dentl plque. Moreover, deminerlistion nd reminerlistion were not studied together, most of the models focusing on deminerlistion only, with only few [Ten Cte, 1983, 28] investigting tooth reminerlistion. The current work presents one-dimensionl, timedependent mthemticl model tht integrtes the kinetics of tooth deminerlistion nd reminerlistion with mss trnsfer nd with microbil conversions occurring in the dentl plque. The study is bsed on previously developed numericl model for tooth deminerlistion under the influence of n orl biofilm [Ilie et l., 212], extended to evlute the two min hypotheses for subsurfce lesion formtion. Our purpose ws to estblish which of the two theories cn best explin the formtion of surfce lyer with the simplest ssumptions. Cse 1 ssumes the tooth enmel is formed by HAP only nd both deminerlistion nd reminerlistion processes cn occur, depending on the degree of sturtion. Cse 2 ssumes tht the tooth enmel includes FHAP crystls nd studies the surfce lyer formtion in the presence of solubility grdient when only tooth deminerlistion occurs. M o d e l D e s c r i p t i o n The one-dimensionl numericl model for clcultion of nd solute concentrtions in ctive dentl plque [Ilie et l., 212] ws extended to describe the subsurfce lesion formtion in the tooth enmel. In the present model, the dentl plque composition ws simplified to only one generic microbil group with one fermenttive pthwy, in order to llow n incresed complexity of tooth chemistry (while mintining resonble computtionl effort). Ilie /vn Turnhout /vn Loosdrecht / Piciorenu /18/213 4:4:53 PM

3 Tble 1. Model prmeters Prmeter Symbol Vlue Reference Biomss concentrtion in plque C X 1 g l 1 Morgenroth et l., 2 Mximum specific uptke rte q S,mx mol g 1 s 1 vn der Hoeven et l., 1985 Monod hlf-sturtion coefficient for glucose K S,Glu mol l 1 Hmilton nd St Mrtin, 1982 vn der Hoeven et l., 1985 Proton inhibition constnt K + I,H 1 6 mol l 1 ssumed for inhibition t 4.5 Rte coefficients for dissocition equilibri 1 6 s 1 ssumed k LcH k Pho Acidity constnt for lctic cid K LcH mol l 1 Atkins nd De Pul, 29 Acidity constnt for dihydrogen phosphte dissocition K Pho mol l 1 Atkins nd De Pul, 29 Acidity constnt for hydrogen phosphte dissocition K 2+ Pho mol l 1 Atkins nd De Pul, 29 Acidity constnt for wter dissocition K H2 O 1 14 mol 2 l 2 Atkins nd De Pul, 29 Enmel solubility constnt K S,HAP(en) mol 9 l 9 Fejerskov nd Kidd, 28 HAP solubility constnt K S,HAP mol 9 l 9 Fejerskov nd Kidd, 28 HAP molr weight M HAP 52 g mol 1 Fejerskov nd Kidd, 28 HAP density ρ HAP 3,16 kg m 3 Fejerskov nd Kidd, 28 Rdius of HAP rod r rod m Fejerskov nd Kidd, 28 Deminerlistion rte constnt k d mol.7 m 1.1 s 1 Mrgolis nd Moreno, 1992 Reminerlistion rte constnt k r mol.25 m.75 s 1 Nncolls, 1983 Deminerlistion rection order n d.3 Mrgolis nd Moreno, 1992 Deminerlistion rection order m d 2.8 Mrgolis nd Moreno, 1992 Reminerlistion rection order n r 1.25 Nncolls, 1983 Length of tooth domin L t 1 μm ssumed Length of dentl plque domin L p 25 μm ssumed b Reduction fctor for diffusion coefficients in the plque f p.25 Dibdin, 1981 Time for the trnsition from C Glu,min to C Glu,mx t step 1 s ssumed Feeding period t feed 2 min ssumed Length of feeding/resting cycle t cycle 9 min ssumed Hlving time in the sliv film t h,f.5 min Dibdin, 199 Hlving time in the sliv bulk t h,s 2.17 min Dibdin, 199 Residence time in the sliv film Q f /V f ln(2)/t h,f min clculted Residence time in the sliv bulk Q/V s ln(2)/t h,s min clculted Are per volume rtio in sliv film A f /V f 1 3 m 1 clculted c Minimum glucose concentrtion (between mels) C Glu,min.7 mm Vn der Hoeven et l., 199 Mximum glucose concentrtion (during pulse) C S,Glu,mx 56 mm Dirksen et l., 1962 Universl gs constnt R J mol 1 K 1 Temperture T 31 K chosen, 37 C Frdy s constnt F 96,485 C mol 1 Assumed n rbitrrily high vlue for very fst equilibri. b Bsed on the mesured thickness of 96-hour dentl plque, grown in vivo [Wood et l., 2]. c Assumes tht the sliv film thickness is 1 μm [Fejerskov nd Kidd, 28] nd tht only 1% of the tooth is covered with plque. P ro c e s se s Microbil Rections From the multitude of microbil processes occurring in dentl plque, only lctic fermenttion ws considered since it hs the highest impct on cries formtion [Dwes nd Dibdin, 1986; Ilie et l., 212]. This simplifiction still offers good description of cidity production due to microbil metbolism, while llowing to focus the study on Numericl Modelling of Subsurfce Lesion Formtion Induced by Dentl Plque the porosity profile developed in the tooth. Consequently, the dentl plque ws reduced to one microbil group, the ciduric Streptococci, with constnt concentrtion C X in the plque (i.e., no microbil growth ws considered). This microbil group is commonly ssocited with cries development due to its high cidogenicity nd ciduricity [Loesche, 1986; Mrsh et l., 29]. The nerobic fermenttion t high glucose concentrtion (eqution 1) is the min source of lctic cid dur- Cries Res 214;48:73 89 DOI: / /18/213 4:4:53 PM

4 E >.215 r = d = f(cylinder specific surfce) E <.215 r = f(prllelepiped specific surfce) r rod Color version vilble online Acid-Bse Equilibri The simplified model includes very fst protontion equilibri for lctte (microbil metbolic product) nd hydrogen phosphte (tooth deminerlistion product) in ll model comprtments (sliv, plque nd tooth) (equtions 2 nd 3), with prmeters in tble 1 : ing criogenic ttck. Moreover, lctic cid is the strongest nd most bundnt cid present in fermenting dentl plque [Borgström et l., 2]. 76 d nd r (m 2 m 3 ) b HAP rod R totl specific surfce ( d ) R specific growth surfce ( r ) Porosity (m 3 m 3 ) Fig. 1. Schemtic representtion of the volume inside the tooth enmel. The HAP rods re ssumed cylinders with rdius r rod, while the inter-rod spce is prllelepiped. Left: clcultion of the specific surfce for reminerlistion ( r ) t high enmel porosity function of re of the HAP cylinders. Right: low enmel porosity function of the inter-rod surfce re. b Clculted totl specific surfce, d, nd specific growth surfce, r, function of enmel porosity E. CH O 2CHO 2H C K Glu I, H rglu qs, mx CX C Glu K S, Glu C H K I, H For the glucose uptke rte r Glu (mol l 1 s 1 ), Monod kinetics with glucose limittion (concentrtion C Glu ) nd cid inhibition (proton concentrtion C H + ) ws ssumed, with rte prmeters given in tble 1 [Vn Beelen et l., 1986]. Cries Res 214;48:73 89 DOI: / (1) CHO CHO H rlch klch CLcH KLcH C H CLc 2 HPO HPO4 H r Pho k Pho C Pho KPho C H C Pho (3) The other two phosphte equilibri were not considered becuse they only become importnt t vlues not reched in the current conditions. Although present in the previous dentl plque model, other possible inorgnic buffers (e.g., CO 2 /HCO3 ) s well s complextion equilibri with C 2+ nd microbil surfce chrges were neglected here to simplify the nlysis of results. Tooth Deminerlistion nd Reminerlistion Following n nlysis of mechnisms suggested in the literture for the formtion of subsurfce enmel lesions, two theories hve been studied: cse 1 n lterntion of two competing processes, enmel deminerlistion nd reminerlistion, ssuming only HAP crystls; cse 2 deminerlistion of the tooth enmel composed of FHAP with higher concentrtion of fluoride t the enmel surfce slowing down the dissolution rte. The model ws dpted to ech cse, by considering different chemicl rections to occur in the tooth domin in ddition to the cid-bse equilibri. It ws ssumed tht the tooth enmel consists of prllel rods (represented s cylindricl crystls) of HAP, the spces between them being filled with queous solution [Ten Cte, 1979]. Enmel porosity is the totl volume of inter-rod spces divided by totl volume of the enmel ( fig. 1 ). Enmel porosity E is further used to chrcterize lesion by the minerl content profile (i.e., the vrition of (1 E ) over the first micrometres of enmel) nd the lesion severity (i.e., the mximum porosity E reched within the lesion). Cse 1: Tooth Enmel Consisting of HAP The first hypothesis is tht the subsurfce lesion is formed s consequence of n imblnce between two concurrent processes, deminerlistion nd reminerlistion. It ws ssumed tht the tooth enmel consists of homogeneous HAP, therefore removing the bis of solubility grdient due to, for exmple, fluoride distribu- Ilie /vn Turnhout /vn Loosdrecht / Piciorenu (2) /18/213 4:4:53 PM

5 Tble 2. Chemicl components in the model Nme Symbol Diffusion coefficient in wter c Sliv concentrtions 9 (1 m 2 s 1 ) reference initi l C f,j (), inlet b, mol m 3 mol m 3 reference Lctte ion Lc 1.31 Vnýsek, 21 ssumed Lctic cid LcH 1.31 Vnýsek, 21 ssumed Proton H Vnýsek, Fejerskov nd Kidd, 28 Hydrogen phosphte Pho 2.96 Vnýsek, equilibrium, C Pho,tot initil = 13.2 mol m 3 C Pho,tot inlet = 5.4 mol m 3 Fejerskov nd Kidd, 28 Dihydrogen phosphte Pho 1.22 Vnýsek, equilibrium Clcium ion C Vnýsek, Fejerskov nd Kidd, 28 Ction (potssium ion) K Vnýsek, Fejerskov nd Kidd, 28 Anion (chloride ion) Cl 2.57 Vnýsek, chrge blnce Fluoride ion F 1.48 Vnýsek, Fejerskov nd Kidd, 28 Glucose Glu.85 Vnýsek, 21.7 C s,glu (t) Vn der Hoeven nd Gottschl, 1989; clculted for inlet Vlues corresponding to those mesured in humn resting plque (ssumed to be in stedy stte). b Vlues corresponding to those mesured in unstimulted mixed (from ll slivry glnds) humn sliv. c Diffusion coefficients for ll solutes in wter re reduced in plque by fctor of.25 [Dibdin, 1981] nd in the enmel reduced by the porosity E. tion over the enmel depth. All the chemicl species listed in tble 2, with the exception of fluoride, hve been included in this cse. For the HAP deminerlistion rection (eqution 4) C 5 (PO4 ) 3 (OH) 5C2+ + 3PO HO (4) the empiricl eqution for the surfce-bsed deminerlistion rte, r * d,c (mol C m 2 s 1 ), ws proposed in Mrgolis nd Moreno [1992] (equtions 5 nd 6) ( tble 1 ). md d, C d 1 HAP AiH Numericl Modelling of Subsurfce Lesion Formtion Induced by Dentl Plque r k DS C for DS HAP <1 (5) r * r,c = for DS HAP 1 (6) The deminerlistion rte is function of the concentrtion of ll protonted cids, C A(i)H, nd the degree of sturtion of the solution with respect to HAP, DS HAP (eqution 7): DS HAP IP K HAP S, HAPen 1/9 nd nd C C C C IP K K C Pho C Pho 3 HAP 2 4 Pho H2O C HO C H DS HAP depends on the ionic product IP HAP, nd the solubility constnt, K S,HAP(en). Due to the impurities present in (7) the tooth, the enmel solubility K S,HAP(en) hs lower vlue thn tht of pure HAP ( tble 1 ). The deminerlistion occurs only when the solution is understurted in respect to HAP, DS HAP <1. The volumetric deminerlistion rte of HAP (eqution 8), r d,hap (mol HAP m 3 s 1 ), ws derived by multiplying the surfce-bsed rte with the specific crystl surfce, d (m2 m 3 ), nd considering the rection stoichiometry (eqution 4): r d,hap =.2 r * d,c d (8) The specific crystl surfce d ws clculted s the re/volume rtio of rod with constnt rdius, r rod, multiplied with the volume frction of crystl in the enmel (the minerl content, 1 E ) [Ten Cte, 1979] (eqution 9 nd fig. 1 b): 2 d 1 E (9) r rod The surfce-bsed reltion for HAP reminerlistion rte, r * r,hap (mol HAP m 2 s 1 ) (equtions 1 nd 11) ws empiriclly determined by Nncolls [1983] from in vitro experiments on crystllistion of bovine enmel blocks. r r,hap * = k r (DS HAP 1) n r for DS HAP 1 (1) r r,hap * = for DS HAP <1 (11) Cries Res 214;48:73 89 DOI: / /18/213 4:4:53 PM

6 The lumped reminerliztion rte constnt, k r, nd the rection order, n r, were clculted from the results of Nncolls [1983] (vlues in tble 1 ). The volumetric reminerlistion rte, r r,hap (mol HAP m 3 s 1 ), ws obtined from the surfce-bsed rte multiplied with the specific growth surfce, r (eqution 12): 78 K S,HAP(xF) r r,hap = r * r,hap r (12) The specific growth surfce, r, is the frction of the totl specific surfce, d, tht is vilble for crystl growth ( fig. 1 ). Depending on the enmel porosity vlues (E), the specific growth surfce is clculted in two rnges: (i) At porosity higher thn.215 (corresponding to perfectly cylindricl HAP rods) [Ten Cte, 1979], the entire crystl surfce is vilble for reminerlistion (eqution 13). 2 1, r d E when E >.215 (13) r rod K S,HAP(xF) x F Enmel depth (μm) Fig. 2. Imposed distribution of fluoride frction x F nd the mixed FHAP/HAP crystl solubility constnt K S,FHAP(xF) within 15 μm from the enmel surfce. Cries Res 214;48:73 89 DOI: / x F Color version vilble online (ii) At porosity vlues lower thn.215, s the interrod spce gets filled, it ws ssumed tht only prt of the totl specific crystl surfce d remins vilble for crystl growth. The r re ws ssumed to become zero when porosity reches miniml vlue E min (i.e., reminerlistion stops when E min =.1). No pre-existing grdients in porosity were ssumed nd uniform initil porosity E min for helthy enmel ws considered [Fejerskov nd Kidd, 28]. The specific growth surfce ws in this cse clculted from sigmoid function mking the growth-specific re zero t E min, mimicking shrinking prllelepiped inter-rod spce ( fig. 1 b) (eqution 14). r 1 1 d, when.1 < E <.215 (14) 1exp45E. 15 Finlly, the chnge in enmel porosity in time (i.e., the lesion formtion) is the difference between the volumetric deminerlistion nd reminerlistion rtes s shown in eqution 15 [dpted from Vn Dijk, 1978]: de MHAP rd, HAP r, HAP, with E() = E min (15) dt HAP Cse 2: Tooth Enmel Includes FHAP This cse tests the hypothesis tht subsurfce lesion is the result of n enmel solubility grdient cused by the presence of less soluble fluoride inclusions (FHAP) ner the enmel surfce. To represent this sitution, fluoride ions (F ) were introduced in the model ( tble 2 ). Fluoride ws considered non-rective in sliv nd dentl plque nd exchngeble between ll the model comprtments. Only FHAP deminerlistion (eqution 16) is llowed in this cse, without ny reminerlistion. The solubility of mixed enmel crystl contining FHAP is lower thn tht of HAP. It ws experimentlly observed tht fluoride diffuses nd prtly replces the hydroxyl within the first 1 μm of enmel [Moreno et l., 1974; Fejerskov nd Kidd, 28] ccording to the following rection: C 5 (PO 4 ) 3 (OH)1 x F F x F 5 C PO (1 x F )HO + x F F (16) The molr frction of fluoride, x F = n F / (n F + n HO ), decresed exponentilly within 5 μm from the enmel surfce from its mximum vlue x F,mx =.5 to minimum x F,min =.5 [Moreno et l., 1974] ccording to eqution 17 ( x in μm) ( fig. 2 ): x F = x F,min + ( x F,mx x F,min ) exp ( x 1) (17) The dependency of the mixed FHAP/HAP solubility constnt, K x, on the frction of fluoride ws mesured by Moreno et l. [1974] s K x = exp( x 2 F x F ). These results were djusted proportionlly for the replcement of hydroxyl in enmel insted of pure HAP, K S,HAP(x F ) = K x K S,HAP (en) / K S,HAP. The frction of fluoride nd the FHAP solubility constnt within enmel in the first 15 μm from the tooth surfce re displyed in figure 2. We considered the sme type of rte expression for the dissolution of mixed FHAP/HAP crystls (clled here Ilie /vn Turnhout /vn Loosdrecht / Piciorenu /18/213 4:4:53 PM

7 HAP(x F )) s for HAP, so tht r d,hap(xf) (mol HAP(x F ) m 3 s 1 ) becomes: md d r 2. k 1 DS C, d HAP xf d HAP xf A i H DS HAP xf HAP xf IP K HAP xf S, HAPxF 1/9 nd C C C IP K K 5 3 xf 2 C Pho2 F xf Pho C H H2O n d (18) (19) Considering tht stedy fluoride content in enmel builds up over yers nd tht the time periods simulted were reltive short (hours), the contribution of FHAP reminerlistion ws ssumed negligible nd the grdient of fluoride content is stble in time. The vrition of FHAP enmel porosity in time ws clculted similrly to eqution 15 without reminerlistion: de M r E() = E min (2) dt HAP,, d HAP xf HAP Blnce Equtions The chnge of tooth porosity in time depends on the sptil distribution of the concentrtions of severl solute species ( tble 2 ). We ssumed one-dimensionl concentrtion grdients in the tooth enmel nd in the dentl plque on direction perpendiculr to the tooth surfce. The entire sliv volume is considered well-mixed nd the concentrtions re vrying only in time. The one-dimensionl tooth enmel domin extends from x = (deep enmel) to x = L t (enmel-plque interfce) nd the plque is between x = L t nd x = L t + L p (the plque-sliv interfce). Chemicl species (solutes) were exchngeble between these model comprtments. To clculte the solute concentrtions nd the mount of deminerlised tooth enmel, mole nd chrge blnces were defined on enmel, plque nd sliv domins. S liv The sliv film in direct contct with the plque hs volume V f, nd represents only smll frction of the totl sliv present in the mouth. The chnge in concentrtion of ech chemicl species j (tble 2 ), C f,j (t), is clculted from eqution 21, similr to the pproch used in Ilie et l. [212]: dc f, j Qf Af Cs, jcf, j Np, j R (21) j dt V V f f Eqution 21 is bsed on the species exchnge with the sliv bulk with flow rte Q f, exchnge with the dentl plque of re A f with the flux N p,j (diffusive flux t the plque-sliv interfce) nd the net rection rte of ech component, R j ( tble 1 ). The sliv inlet concentrtions for chemicl species C s,j re set s in previous study [Ilie et l., 212] ( tble 2 ), bsed on chemicl specition (mssction lws) nd chrge blnce (solution electroneutrlity, j zc j s, j, where z j is the chrge number). For inlet concentrtion of glucose, repeted feeding/ clernce/resting cycles hving the totl time of 1.5 h were imposed. In the feeding regime, C s,glu quickly incresed within t step from C Glu,min to mximum concentrtion C s,glu,mx, which ws mintined for the entire feeding time, t feed. After feeding, clernce begins nd glucose concentrtion decreses exponentilly (eqution 22) until the end of the resting period: Cs, Glu t Q C t t exp tt t C V s s, Glu step feed step feed Glu, min (22) The residence time ( Q/V s = ln(2)/ t h,s ) ws expressed bsed on the hlving time in the sliv bulk comprtment t h,s [Dibdin, 199]. Dentl Plque In the dentl plque the solute concentrtions C p,j chnge from the plque surfce to the tooth surfce. Therefore, one-dimensionl mole blnces re set for ll solutes j, s in Ilie et l. [212] nd bsed on Nernst- Plnck eqution 23 coupled with the electroneutrlity condition (eqution 24). Equtions include net rection rtes R p,j nd the trnsport by moleculr diffusion ( D p,j, diffusion coefficient of solute in the plque t 37 C, clculted s 25% from the diffusion coefficients in wter, D j, ccording to Dibdin [1981]) nd ion electromigrtion ( z j = chrge number, F = Frdy s constnt, R = universl gs constnt nd T = temperture): C C zf t x RT x x 2 p, j p, j j Φ Dp, j D 2 p, j C p, j Rp, j (23) zc j p, j (24) j Numericl Modelling of Subsurfce Lesion Formtion Induced by Dentl Plque Cries Res 214;48:73 89 DOI: / /18/213 4:4:53 PM

8 R p,j is the net rection rte of ech compound, ccounting for both chemicl nd microbil processes. The dditionl stte vrible Φ (x,t) is n electricl potentil field developed due to ion trnsport with different diffusion rtes. The initil vlues for ll concentrtions were considered equl to those in the sliv film, while the potentil Φ( x,) =. For the sliv-plque interfce (t x = L p + L t ) the concentrtions of ll chemicl components re equl to those in the sliv film nd Φ = s reference vlue. For the plque-tooth interfce, totl flux continuity is set for ll the components except glucose, for which the tooth is impermeble (zero-flux condition) [Fremlin nd Mthieson, 1961]. To oth The most importnt development brought by the current model compred to Ilie et l. [212] is the explicit clcultion of minerl content vrition in the tooth depth. This required the representtion of the tooth s nother one-dimensionl domin djcent to the plque domin. Since the focus of this study ws on the initil development of subsurfce lesions, only the first 1 μm in depth from the tooth surfce were considered. As in the plque domin, the vrition in time of solute concentrtions (ll species but glucose) is clculted by the sme Nernst-Plnk equtions coupled with chrge blnce: 8 C C zf t x RT x x 2 t, j t, j j Φ Dt, j D 2 t, j C t, j Rt, j zc j t, j j Cries Res 214;48:73 89 DOI: / (25) (26) The net rection rte, R t,j, ccounts only for chemicl processes (cid-bse equilibri nd tooth (de)reminerlistion) since there re no microbil species present inside the tooth. Lower enmel porosity leds to slower diffusion of ny chemicl species in the tooth domin D t,j = D j E in respect to diffusivities in wter, D j, from tble 2 [Ten Cte, 1979]. The enmel porosity is vrible in time nd over the tooth depth, E(x,t), nd it is clculted differently, function of the tested hypothesis, with eqution 15 (cse 1) or eqution 2 (cse 2). The boundry conditions were: flux continuity t the plque-tooth interfce nd zero-flux/electricl insultion t the deep enmel boundry ( x = ). Initil concentrtions of chemicl components in the tooth re equl to those in the plque nd the initil porosity is E min (i.e., helthy enmel). Model Solution The model equtions were implemented in COMSOL Multiphysics softwre (COMSOL 4.1, COMSOL Inc., Burlington, Mss., USA, which llows very flexible nd well-structured model construction. COMSOL solves the resulting system of ordinry differentil, prtil differentil nd lgebric equtions by finite element methods. The plque domin ws discretized on mesh of 1 elements with finer mesh size next to both the sliv nd enmel boundries. The mesh in the tooth domin contined lso 1 elements of size nd finer meshing t the enmel surfce, where the concentrtion grdients cn become very steep. First, the mole nd chrge blnces in plque nd enmel (equtions 23 26) were solved towrds the stedy stte solution, which represents sitution of resting plque in contct with constnt composition sliv. Second, the stedy stte solution ws used s initil condition for the time-dependent simultions performed during 8 sequentil feeding/clernce/resting cycles of 1.5 h ech. The time-dependent simultions included the complete model equtions in sliv (eqution 21), plque (equtions 23, 24) nd tooth (equtions 25, 26 nd 15 or 2), with the ssocited boundry conditions nd constitutive (rte) equtions. R e s u l t s The numericl model described in this study is iming to offer better understnding of the mechnisms leding to tooth subsurfce lesion under the influence of cids produced in dentl plque. Two cses were chosen in order to distinguish whether the lesion could be cused solely by n lterntion of deminerlistion/reminerlistion processes (i.e., originting from the sme thermodynmic force) or by fluoride grdient (nd implicitly solubility grdient). In cse 1, tooth enmel consists of HAP with deminerlistion nd reminerlistion, nd in cse 2 tooth enmel contins mixture of HAP nd FHAP with only deminerlistion occurring. For ech cse we nlysed (1) the influence of glucose pulses on the nd the totl mount of deminerlised enmel t tooth surfce in time, (2) the profile, concentrtion of clcium ions nd totl phosphte species over the plque depth s well s deminerlistion nd reminerlistion rtes within the enmel t certin time, nd (3) the evolution of porosity profiles inside the enmel. Ilie /vn Turnhout /vn Loosdrecht / Piciorenu /18/213 4:4:53 PM

9 Totl deminerlised HAP (mol m 2 ) c glucose HAP demin Glucose concentrtion (mm) Color version vilble online t (min) Clcium concentrtion (mm) b Tooth R r demin HAP Plque c C Length (μm) c Pho totl HAP deminerlistion rte (mol m 3 s 1 ) Totl phosphte concentrtion (mm) Fig. 3. Clculted concentrtion profiles for cse 1, when the tooth enmel contins only HAP., glucose concentrtion nd totl mount of deminerlised HAP t the tooth surfce during one feeding/resting cycle. b Concentrtions of clcium nd totl phosphte, nd rte of HAP deminerlistion long the dentl plque nd in the tooth enmel fter 1 min from the beginning of the first feeding/resting cycle. The grey dotted line represents the plquetooth boundry. c Concentrtions of clcium nd totl phosphte, nd rte of HAP reminerlistion profile long the dentl plque nd in the tooth enmel fter 3 min from the beginning of the first feeding/resting cycle. Clcium concentrtion (mm) c Tooth R r remin HAP Plque c C Length (μm) c Pho totl HAP reminerlistion rte (mol m 3 s 1 ) Totl phosphte concentrtion (mm) Numericl Modelling of Subsurfce Lesion Formtion Induced by Dentl Plque Cries Res 214;48:73 89 DOI: / /18/213 4:4:53 PM

10 82 Cse 1: Tooth Enmel Consisting of HAP Concentrtion Profiles Feeding/Resting Cycle. The glucose concentrtion t the tooth surfce nd its influence on the profile t the tooth surfce (Stephn curve) nd subsequently on the mount of HAP lost in time re shown in figure 3. Moments fter the concentrtion of glucose in sliv hs incresed, the t the tooth surfce shrply decreses below the criticl vlue for deminerlistion ( criticl = 5.5). In these conditions, the plque fluid t the tooth surfce becomes understurted with respect to HAP ( DS HAP < 1) nd the tooth enmel strts to deminerlise. As long s there is glucose in the plque, the stys cidic nd n incresing mount of HAP is lost. After pproximtely 2 min from the beginning of the cycle, when the glucose is clered from the sliv nd dentl plque, the strts to restore towrds the stedy stte vlue. As soon s the rises bove 5.5, the mount of HAP lost decreses, mening tht reminerlistion is now ctive. While reminerlistion is ctive (i.e., E >.1 nd DS HAP > 1), the cnnot restore to the stedy stte vlue of 7 due to the continuous consumption of hydroxyl ions during this process (eqution 4). Restortion to stedy stte vlues in tooth nd plque cn occur only when the tooth is completely reminerlised (i.e., E = E min =.1). In the current cse, the tooth is not fully reminerlised t the end of the 9 min cycle, nd there is net loss of pproximtely mol HAP m 2 per cycle ( fig. 3 ). This corresponds to 1 4 mg C mm 2 lost per cycle, n mount in the sme order of mgnitude with the experimentl observtions of Mrgolis nd Moreno [1992] nd with the results of our previous model [Ilie et l., 212]. The net mount of lost HAP per feeding/resting cycle depends thus on the length of the reminerlistion stge. Deminerlistion Phse. An exmple of concentrtion profiles long the dentl plque nd tooth domins for the ions influencing DS HAP (i.e., clcium, totl phosphtes nd protons/), fter 1 min in the feeding period (i.e., during the cid ttck nd mximum deminerlistion rte) is presented in figure 3 b. At this time, the mxim of totl phosphte nd clcium concentrtions re close to the enmel surfce, where these components re relesed by HAP deminerlistion, nd diffuse towrds the plque surfce (therefore the liner concentrtion profile, showing lso tht qusi-stedy stte ws reched in the plque). These species diffuse lso inside the tooth, but much slower, therefore the concentrtion profiles re curved nd fr from stedy stte. The HAP deminerlistion in which these species re produced occurs only in Cries Res 214;48:73 89 DOI: / Minerl content (%) Plque-tooth boundry 9 min f 22 min Enmel depth (μm) 45 min Fig. 4. Clculted minerl content profiles in the tooth enmel t different stges during the 8th feeding/resting cycle. 22 min: end of deminerlistion stge; 45 min: during reminerlistion; 9 min: t the end of the reminerlistion (resting) period. very nrrow region in the tooth where < criticl. In the plque depth, the decreses from nerly neutrl in sliv to lmost 5 t the tooth surfce. This decrese is cused by the microbil production of orgnic cids. Within the tooth enmel the increses to neutrl vlues fr from the tooth surfce due to the slower diffusion of protons inside the tooth compred to the plque. The proton consumption in deminerlistion leds lso to smll increse in the plque depth. Reminerlistion Phse. After 3 min from the beginning of the feeding/resting cycle, during the reminerlistion stge, the clcium nd phosphte re consumed in the tooth, therefore their fluxes re reversed compred to the deminerlistion stge ( fig. 3 c). At this time, the mximum concentrtions re t the sliv-plque boundry nd the minimum t the tooth surfce where the ions re consumed. Although the is bove the criticl vlue long the entire tooth domin, the reminerlistion rection tkes plce only in nrrow region t the very surfce of the tooth. This mens tht during the previous deminerlistion stge, only nrrow region of HAP enmel ws deminerlised, leding to porosity vlues higher thn E min. The mximum clculted rte of reminerlistion is pproximtely ten times smller thn the mximum rte of deminerlistion nd is in the sme rnge s the rtes experimentlly observed by de Rooij nd Nncolls [1984] nd White et l. [1988]. Ilie /vn Turnhout /vn Loosdrecht / Piciorenu Color version vilble online /18/213 4:4:53 PM

11 Tble 3. Sensitivity nlysis results Prmeter nme/ Prmeter vlue Diffusion inside the tooth (D t,j ) (f min =.2) Reminerlistion rte constnt (k m ) (f min =.2; f mx = 5) Length of feeding/resting cycle (t cycle ) (f min =.5; f mx = 2) lost HAP, mmol HAP m 2 min. HAP content b, % lost HAP, mmol HAP m 2 min. HAP content b, % lost HAP, mmol HAP m 2 min. HAP content b, % f min stndrd vlue Stndrd vlue f mx stndrd vlue Amount of HAP lost t the end of ll the 25 feeding/resting cycles. b Minimum content of HAP t the end of the 25th feeding/resting cycle. Minerl Profiles during One Cycle. The minerl content long the first 5 μm from the tooth surfce t different stges fter the beginning of the feeding/resting cycle is represented in figure 4 for the 8th cycle. The model clerly demonstrtes how subsurfce lesion is formed fter mny feeding/resting cycles only by lternting phses of deminerlistion nd reminerlistion. The most chemiclly ctive prt of the tooth ppers to be ner its surfce: the HAP content during deminerlistion (22 min) is 15% lower thn the content of the sme region t the end of the cycle (9 min). The current model suggests tht pre-existing grdient (such s in the enmel solubility constnt [Vn Dijk et l., 1979; Arends nd Christoffersen, 1986], or deminerlistion/reminerlistion rection rte constnts [Ten Cte, 1979], or fluoride distribution [Wng et l., 1996]) is not required in order to develop subsurfce lesion when reminerlistion is present. The reminerlistion rte is highest t the tooth surfce. If the reminerlistion period is long enough, the entire lesion cn reminerlise. However, for short recovery times (s it is the cse for frequent sugr consumption) the lesion cn reminerlise only superficilly, while higher porosity builds up in the lesion depth. Therefore, the porosity of the surfce lyer will remin within certin limits (1 15% in the current model) while the porosity in the subsurfce re will slowly increse with ech deminerlistion-reminerlistion cycle (for exmple, up to 75% s shown in movie 1 in the online supplementry mteril; for ll online suppl. mteril, see www. krger.com/doi/1.1159/354123). Sensitivity Anlysis Importnt fctors to be considered when discussing the development of minerl surfce lyer vi reminerlistion re (1) the diffusion coefficient inside the tooth, Numericl Modelling of Subsurfce Lesion Formtion Induced by Dentl Plque (2) the HAP reminerlistion rte nd (3) length of the reminerlistion period (resting time). To study their influence, sensitivity nlysis ws performed. For these prmeters, simultions were run for 25 feeding/resting cycles, with the bse vlue nd with incresed/decresed prmeter vlues ( tble 3 ). Diffusion inside the Tooth. A subsurfce lesion ws obtined for ech tested vlue of the diffusion coefficients in the tooth ( fig. 5 ). However, it ppers tht the fster the diffusion, the less pronounced (higher minerl content) but wider nd deeper within the enmel the lesion forms. Indeed, fster diffusion cretes more evenly distributed concentrtion profile of ions (nd consequently DS HAP ) within the enmel, nd the deminerlistion rection is ctive over n extended depth. Although the lesion ppers to be less severe for fster diffusion, the totl mount of minerl lost is still the highest, with 1% more HAP lost compred to the slower diffusion cse ( tble 3 ). Therefore, the effective diffusion of chemicl species within the tooth hs significnt influence on the shpe of the lesion nd on the time when the porosity becomes lrge enough to mke the lesion cliniclly observble. Reminerlistion Rte. The kinetics of reminerlistion my influence the minerl profile, therefore the reminerlistion rte constnt (k m ) ws lso vried ( fig. 5 b). The simulted lesions hd very similr minerl profiles, with only 1% more HAP lost for the highest k m vlue compred with the stndrd cse ( tble 3 ). For slower reminerlistion rte, the surfce lyer remins porous for extended periods of time, thus with incresed effective diffusivity. This llows ions to penetrte deeper inside the tooth nd to better restore the deminerlised enmel. In conclusion, smll vritions in reminerlistion rte my not influence significntly the formtion of subsurfce lesion, but only the qulity of the surfce lyer formed. Cries Res 214;48:73 89 DOI: / /18/213 4:4:53 PM

12 Minerl content (%) D t,j D t,j Plque-tooth boundry f Enmel depth (μm) Minerl content (%) b k m k m 5 k m Enmel depth (μm) Color version vilble online Minerl content (%) c t cycle t cycle 2 t cycle Enmel depth (μm) Minerl content (%) Minimum minerl content (%) μm Plque thickness (μm) d Enmel depth (μm) Fig. 5. Minerl content in the enmel t the end of the 25th feeding/resting cycle, with vrious prmeter vlues for the deminerlistion-reminerlistion model (cse 1): effective diffusion coefficient in the enmel; b reminerlistion rte constnt; c length of the feeding/resting cycle; d plque thickness between 5 nd 4 μm. The insert shows the minimum minerl content reched fter 25 cycles s function of the plque thickness. Length of the Reminerlistion Phse. The highest impct on lesion formtion proved to be the length of the reminerlistion period. This is behviourl fctor, determined by the eting hbits of the individul becuse cries formtion is minly ssocited with frequent sugr intke nd therefore short reminerlistion time. For longer reminerlistion times ( t cycle = 18 min) there is no 84 Cries Res 214;48:73 89 DOI: / lesion formed t the end of the 25th feeding/resting cycle ( fig. 5 c). This shows tht given enough time, the lesion formed fter n cid ttck could be fully reminerlised. However, for shorter reminerlistion time ( t cycle = 3 min), the lesion is formed directly t the tooth surfce nd no surfce lyer is present, giving the highest mount of HAP lost from ll the cses studied ( tble 3 ). The res- Ilie /vn Turnhout /vn Loosdrecht / Piciorenu /18/213 4:4:53 PM

13 Totl deminerlised FHAP (mol m 2 ) c glucose FHAP demin Glucose concentrtion (mm) Color version vilble online t (min) Fig. 6. Clculted concentrtion profiles for cse 2, when the tooth enmel contins fluoride distribution., glucose concentrtion nd totl mount of deminerlised FHAP t the tooth surfce during one feeding/resting cycle. b Concentrtions of clcium nd totl phosphte, nd rte of FHAP deminerlistion long the dentl plque nd in the tooth enmel fter 1 min from the beginning of the first feeding/resting cycle. The grey dotted line represents the plque-tooth boundry. Clcium concentrtion (mm) b Tooth Plque c Pho totl 3 2 c C 2+ 1 R r demin FHAP Length (μm) FHAP deminerlistion rte (mol m 3 s 1 ) Totl phosphte concentrtion (mm) tortion period is therefore very importnt nd the numericl simultions support the widespred observtion tht often consumption of sugrs is key fctor in cries formtion. Plque Thickness. Interestingly, the model of enmel deminerlistion nd reminerlistion llows the identifiction of minimum plque thickness tht cn induce lesion. With the current model prmeters (cse 1), for plque thickness L p <15 μm there ws no significnt minerl loss ( fig. 5 d). Above 15 μm, the thicker the plque the more extensive enmel lesions developed. It should be noted tht the model only indictes the existence of criticl plque thickness needed for the development of n incipient cries lesion. The vlue of this criticl thickness is subjected to mny prmeters, such Numericl Modelling of Subsurfce Lesion Formtion Induced by Dentl Plque s the length of reminerlistion phse, plque composition nd ctivity, slivry flow nd composition, etc. Showing tht cries onset cn be prevented by reducing, though not completely eliminting, the plque would be relevnt to understnding pproprite orl hygiene protocols. Cse 2: Tooth Enmel Includes FHAP This cse tested in wht extent the subsurfce lesion is the result of n incresed enmel resistnce to cid ttcks due to the presence of less soluble fluoride inclusions ner the enmel surfce. The glucose concentrtion t the tooth surfce, the t the tooth surfce nd the totl Cries Res 214;48:73 89 DOI: / /18/213 4:4:53 PM

14 Minerl content (%) Tooth enmel consisting of HAP No reminerlistion time Plque-tooth boundry f Minerl content (%) Cse 1: tooth enmel consisting of HAP time Color version vilble online Enmel depth (μm) b Enmel depth (μm) 1 Cse 2: tooth enmel consisting of FHAP 8 Fig. 7. Clculted minerl content profiles (progression of cries lesion) inside the tooth enmel t the end of ech 8th feeding/resting cycle. The plque-tooth boundry is represented t depth zero. Only deminerlistion without ny reminerlistion. b Deminerlistion nd reminerlistion of HAP (cse 1). c Deminerlistion of FHAP/HAP (cse 2). Animtions of clculted minerl content evolution re presented in online supplementry movies 1 nd 2. Minerl content (%) c time Enmel depth (μm) mount of deminerlised FHAP during one cycle in this cse re displyed in figure 6. The sme correltion is noticed between the glucose pulse, the sudden drop nd the incresing mount of deminerlised FHAP. Since there is no reminerlistion process in this cse, the mount of FHAP remins constnt once the deminerlistion hs stopped. The lower solubility of FHAP compred to HAP is reflected in the smller mount of deminerlised FHAP compred with cse 1. The minimum vlue is lower thn the one obtined for the HAP cse, due to the lower buffer cpcity t lower concentrtions of phosphte relesed during deminerlistion. Qulittively, the profiles of the totl phosphte species nd clcium ion concentrtions in this cse ( fig. 6 b) resemble those for cse 1 ( fig. 3 b), one minute fter the beginning of the feeding/resting cycle. The mximum rte of FHAP deminerlistion ws in this cse 6 times 86 Cries Res 214;48:73 89 DOI: / lower thn the HAP deminerlistion rte nd it led to lower concentrtions of clcium nd totl phosphte species in the plque. As expected, the first few micrometres of enmel do not dissolve due to the fluoride presence, but there is still minerl loss under this surfce lyer. This ws lso ssumed in the model of Wng et l. [1996] on fluoride dsorption in enmel nd experimentlly observed by Chu et l. [1989]. It remins questionble whether the surfce lyer is completely inert in vivo during lesion development. Experimentl dt re vilble on the development of subsurfce lesions in environments both with nd without fluoride [Nncolls, 1983; Theuns et l., 1984; Mrgolis et l., 1999]. Although fluoride presence my not be the sole reson for the development of subsurfce lesion, the model shows how fluoride distribution lone cn led to subsurfce lesions comprble with the experimentl observtions. Ilie /vn Turnhout /vn Loosdrecht / Piciorenu /18/213 4:4:53 PM

15 Comprison of the Lesions Formtion of surfce lyers in time ws compred for the studied cses: (1) HAP without reminerliztion ( fig. 7 ); (2) HAP by lternting deminerlistion/reminerlistion periods ( fig. 7 b); (3) HAP/FHAP by imposing fluoride distribution over the tooth domin ( fig. 7 c). As expected, there ws no surfce lyer present nd the crious lesion formed t the tooth surfce without reminerlistion becuse there ws no restoring process to blnce the deminerlistion dmge ( fig. 7 ). After ech deminerlistion cycle, the minerl content ner the surfce would decrese, nd the simulted lesion extended in the enmel depth t qusi-constnt rte, reching pproximtely 3 μm fter 8 feeding/resting cycles. When reminerlistion ws ctive (cse 1), t the end of the 8th cycle the minimum minerl content reched 3% within the first 5 μm of tooth enmel, with distinct surfce lyer formed t the tooth-plque interfce ( fig. 7 b). The decrese in minerl content ner the surfce lyer to the minimum vlue in the body of the lesion ws very brupt, while the dmge cused deeper within the tooth ws more grdul. This cn be explined by the fster minerl regenertion t the tooth surfce thn in the deeper lyers, leding to fully regenerted surfce. As the number of cycles (i.e., cid ttcks) increses, more pronounced subsurfce lesion developed inside the tooth (see movie 1 in the online suppl. mteril). The obtined profile is very similr to the minerl content profiles determined experimentlly by Mrgolis et l. [1999]. In cse the tooth enmel contined FHAP crystls but no reminerlistion occurred (cse 2), the obtined profiles resembled qulittively those obtined for HAP with deminerlistion/reminerlistion ( fig. 7 c). The most notble differences were however the lmost complete minerl dissolution in the subsurfce lesion fter 8 cycles nd fster progression of the lesion in the tooth depth (see movie 2 in online suppl. mteril). However, these lte stges of lesion development re less probble to occur in vivo. There is limit to the minerl content in the subsurfce lesion fter which the surfce lyer brkes during mstiction nd cvittion exposes the body of the lesion. Discussion This study developed numericl model tht couples tooth deminerlistion nd reminerlistion with metbolic processes in the dentl plque. It ws found tht Numericl Modelling of Subsurfce Lesion Formtion Induced by Dentl Plque subsurfce lesion cn be chieved using the sme thermodynmic driving force (degree of sturtion leding to deminerlistion or reminerlistion) nd without ny preimposed grdients. There is debte in dentistry regrding the importnce of the reminerlistion phenomen for the formtion of surfce lyer. The model results re discussed here in the context of erly stge crious lesions, the so-clled white spots. One point of view is tht wht ppers to be restored surfce is prtly explined in terms of wer nd polishing [Fejerskov nd Kidd, 28]. Although it is possible tht polishing plys role in the spect of restored tooth surfce, this does not exclude the existence of reminerlistion. Furthermore, there is experimentl evidence tht the surfce lyer hs more thn just the spect of restored surfce; it lso hs minerl content close to tht of helthy enmel [Silverstone, 1977; Fejerskov nd Kidd, 28]. Another rgument is tht inhibitor molecules present in sliv (e.g., sttherin) prevent in vivo precipittion t crystl surfce by blocking crystlliztion nuclei [Sntos et l., 28]. However, the presence of these inhibitors does not imply tht the process of reminerlistion is entirely blocked, but only indictes tht reminerlistion in vivo would occur more slowly ( fig. 5 b). It ws lso rgued tht the very fst uptke of clcium nd phosphte by the HAP mkes the pore liquid in the deep lyers of lesion only mrginlly sturted [Lrsen nd Fejerskov, 1989], thus reminerlistion cn only be very slow. A widespred theory is lso tht since the surfce enmel reminerlises fster, n re of lower porosity t the tooth surfce is creted. This surfce lyer cts s diffusion brrier tht restricts further ccess of HAP constituent ions in the deeper lyers of the lesion nd stops the reminerlistion [Silverstone, 1977; Lrsen nd Fejerskov, 1989; Robinson et l., 2]. Although it is true tht such brrier is formed, this cn only dely the diffusion of the ions inside the lesion ( fig. 5 ). If the reminerlistion time is long enough, the deep lyers of the lesion cn still be restored ( fig. 5 c). According to Arends nd Christoffersen [1986], the experimentl observtion tht surfce lyer once formed hs nerly constnt thickness supports the ide tht the surfce lyer forms becuse of solubility grdient long the enmel depth. With the model described in this study, surfce lyer of nerly constnt thickness ws obtined in both cses. The surfce lyer cn develop only by lternting periods of deminerlistion (corresponding to sugr consumption) nd reminerlistion (corresponding to rest- Cries Res 214;48:73 89 DOI: / /18/213 4:4:53 PM