Numerical Studies of Dental Plaque and Caries Formation

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft; op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben; voorzitter van het College voor Promoties

in het openbaar te verdedigen op 28 april 2014 om 12:30 uur

door

Olga ILIE

Engineer Industrial Biochemistry, “Politehnica” University of Bucharest geboren te Boekarest, Roemenië

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. M.C.M. van Loosdrecht

Copromotor: Dr. ir. C. Picioreanu

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter Prof.dr.ir. M.C.M. van Loosdrecht ………………..., Technische Universiteit Delft, promotor Dr.ir. C. Picioreanu ………………………………...., Technische Universiteit Delft, copromotor Prof. dr. J.M. Ten Cate …………………………….., University of Amsterdam Prof. dr. H.J. Busscher …………………………….., University of Groningen Prof. dr. J.S. Vrouwenvelder …....…...…………...., Technische Universiteit Delft Prof. dr. ir. J.F.M. van Impe …………………….…., Katholieke Universiteit Leuven Prof. dr. G.J. Witkamp …………....……………….., Technische Universiteit Delft Prof. dr. ir. L.A.M. van der Wielen ……..………....., Technische Universiteit Delft, reservelid

This study was funded by the Netherlands Organization for Scientific Research (NWO VIDI grant No. 864-06-003).

Cover design: Andrei Iancu Inside cover and art-work: Raluca Iosifescu Caries ilustration copyright Raluca Iosifescu Picture of tooth worm sculpture copyright W.O. Funk

Table of contents

List of symbols...... 8 1. Introduction ...... 11 1.1. Caries and the dental plaque ...... 11 1.2. The tooth demineralisation profile ...... 14 1.3. Numerical modelling ...... 17 1.4. Objectives and thesis layout ...... 20 1.5. References ...... 21 2. Mathematical modelling of tooth demineralisation and pH profiles in dental plaque ...... 24 2.1. Model description ...... 26 2.1.1. Components ...... 26 2.1.2. Processes ...... 30 2.1.3. Model domains ...... 36 2.1.4. Model solution ...... 40 2.2. Results and discussion ...... 40 2.2.1. Standard case ...... 40 2.2.2. Plaque thickness and area ...... 46 2.2.3. Drinking habits ...... 47 2.2.4. Comparison with other models...... 50 2.2.5. Outlook and future model development ...... 52 2.3. Conclusions ...... 53 2.4. References ...... 54 3. Mathematical modelling of microbial dynamics in dental plaque ...... 58 3.1. Model description ...... 60 3.1.1. Components ...... 61 3.1.2. Processes ...... 62 3.1.3. Component balances ...... 64 3.1.4. Model solution ...... 71 3.2. Results and discussion ...... 71 3.2.1. Standard Case ...... 72 3.2.2. Effect of storage polymers...... 76 3.2.3. Drinking habits ...... 78

3.2.4. Tooth brushing ...... 79 3.3. Conclusions ...... 82 3.4. References ...... 82 4. Numerical modelling of tooth enamel subsurface lesion formation induced by dental plaque ...... 85 4.1. Model description ...... 87 4.1.1. Processes ...... 87 4.1.2. Balance equations ...... 94 4.1.3. Model solution ...... 96 4.2. Results ...... 97 4.2.1. Case 1. Tooth enamel consisting of hydroxyapatite (HAP) ...... 97 4.2.2. Case 2. Tooth enamel includes fluorohydroxyapatite (FHAP) ...... 102 4.2.3. Comparison of the lesions ...... 103 4.3. Discussion ...... 105 4.4. References ...... 106 5. Two-dimensional mathematical modelling of tooth enamel subsurface lesion formation induced by dental plaque ...... 110 5.1. Model description ...... 112 5.1.1. Geometries ...... 112 5.1.2. Processes ...... 113 5.1.3. Balance equations ...... 118 5.1.4. Model solution ...... 120 5.2. Results and discussion ...... 121 5.2.1. HAP demineralisation-remineralisation (Case 1) ...... 121 5.2.2. FHAP demineralisation (Case 2) ...... 125 5.2.3. Comparing the lesion formation mechanisms ...... 127 5.2.4. Impact of model assumptions ...... 128 5.3. Conclusions ...... 131 5.4. References ...... 132 6. A three-dimensional numerical study on the privacy of cell-cell communication ...... 134 6.1. Model Description ...... 137 6.1.1. Signal production and spreading ...... 137 6.1.2. Geometry cases ...... 140 6.1.3. Model solution ...... 141

6.2. Results and Discussion ...... 142 6.2.1. Colony size at induction ...... 143 6.2.2. Privacy of communication ...... 148 6.3. Conclusions ...... 150 6.4. References ...... 150 7. Outlook ...... 152 7.1. Main results ...... 153 7.2. Possible developments ...... 154 7.3. References ...... 156 Summary ...... 157 Samenvatting ...... 160 Acknowledgements ...... 163 Curriculum vitae ...... 165

List of symbols Name Symbol Dimensions –3 Concentration of a microbial group i, constant in time CX,i (kg dry biomass) (m plaque) (i = STA, STN, ACT, VEL) –3 Concentration of solute species j in the saliva film Cf,j mol m –3 Concentration of solute species j in the saliva bulk Cs,j mol m –3 Concentration of species j in the dental plaque Cp,j mol m –3 Concentration of species j in the tooth Ct,j mol m –1 –2 Diffusive flux at the plaque-saliva interface Np,j mol s m –1 –2 Diffusive flux at the plaque-tooth interface Npt,j mol s m –1 –3 Net reaction rate of each component j in the saliva film Rf,j mol s m –1 –3 Net reaction rate of each component j in the dental plaque Rp,j mol s m Electrical potential ɸ V 2 –1 Diffusion coefficient of each component j Dj m s –1 –1 Substrate specific uptake rate qm mol s g –1 Monod substrate saturation constant KS mol L –1 Inhibition constant KI mol L –3 4/9 Degree of saturation with respect to hydroxyapatite DSHAP (mol m ) –3 4/9 Degree of saturation with respect to fluorohydroxyapatite DSHAP(xf) (mol m ) 4 –12 Ionic product with respect to hydroxyapatite IPHAP mol m 4 –12 Ionic product with respect to fluorohydroxyapatite IPHAP(xf) mol m

Halving time in the saliva film th,f s

Halving time in the saliva bulk th,s s 3 Volume of saliva film Vf m 3 Volume of saliva bulk Vs m 2 Area of saliva-plaque interface Af m 3 –1 Volumetric flowrate of saliva film Qf m s Volumetric flowrate of saliva bulk Q m3 s–1

Dental plaque thickness Lp m

Maximum dental plaque thickness Lp, max m

Length of the tooth domain Lt m

Charge number zj -

Feeding period during a feeding/clearance/resting cycle tfeed s

Transition time from Cs,Glu,min to Cs,Glu,max tstep s

Length feeding/clearance/resting cycle tcycle s –3 Minimum glucose concentration (during resting time) Cs,Glu,min mol m –3 Maximum glucose concentration (during feeding time) Cs,Glu,max mol m Faraday’s constant F C mol–1 Universal gas constant R J mol–1K–1 Temperature T K

Stoichiometric coefficient of the anabolic reaction νana, i mol Standard Gibbs energy of formation 01 kJ mol–1 'Gfi,

Gibbs energy changes of the anabolic reaction ΔGana kJ

Gibbs energy changes of the catabolic reaction ΔGcat kJ –3 Concentration of bacterial group j (varying in space and time) Xp,j mol m –1 Advective velocity uB m s –3 Biomass density in the plaque ρB g m

Name Symbol Dimensions –3 Concentration of polyglucose stored within bacteria Cp,sto mol m –1 Dental plaque growth rate up,gro m s –1 Dental plaque detachment rate up,det m s –1 Reaction rate of acid-base equilibria re,i s 3 –1 –1 Reaction rate constant of acid-base equilibria kj m mol s –1 Equilibrium constant for the dissociation of species i Ke,i mol L 2 -2 Equilibrium constant for water dissociation KH2O mol L –3 –1 Volumetric demineralisation rate of hydroxyapatite rd,HAP molHAP m s –1 Reaction rate constant of hydroxyapatite demineralisation ken s 9 –9 Enamel solubility constant KS,HAP(en) mol L 9 –9 Hydroxyapatite solubility constant KS,HAP mol L –1 Hydroxyapatite molar weight MHAP g mol –3 Hydroxyapatite density ρHAP kg m

Radius of hydroxyapatite rod rrod m 0.7 –1.1 –1 Demineralisation rate constant kd mol m s 0.25 0.75 –1 Remineralisation rate constant kr mol m s

Demineralisation reaction order nd -

Demineralisation reaction order md -

Remineralisation reaction order nr -

Reduction factor for diffusion coefficients in the plaque fp - Tooth enamel porosity E m3 m–3 2 –3 Specific HAP crystal surface available for demineralisation ad m m 2 –3 Specific HAP crystal surface available for remineralisation ar m m Surface-based demineralisation rate * molCa m–2 s–1 rdCa, Surface-based remineralisation rate * molHAP m–2 s–1 rrCa, –3 –1 Volumetric remineralisation rate rr,HAP molHAP m s –1 Molar fraction of fluoride in fluorohydroxyapatite (FHAP) xF mol mol 9 –9 Mixed FHAP/HAP solubility constant Kx mol L –1 Two-dimensional Cartesian coordinate system m –1 –1 AHL production rate after induction rAHL Pmol L s –1 –1 Basal AHL production rate kAHL ni μmol L s –1 –1 AHL production rate after induction kAHL in μmol L s –3 Total AHL concentration cAHL mol m –3 Concentration of AHL produced by the reference colony cAHL,R mol m –3 Concentration of AHL produced by the background colonies cAHL,B mol m –1 Affinity coefficient KHill μmol L Hill coefficient n - Cell crowdedness over the computational domain C cell cell–1

Local cell crowdedness Clocal cells Number of cells N cells

Length of the computational domain in x, y and z directions Lx , Ly , Lz μm –1 –2 Molecular flux of AHL at the boundary JAHL mol s m

Basic induction number N0 cells

Number of cells in the reference colony at themoment of induction Nref col cells Reduction quotient Q cell cell–1

Local privacy PL - Privacy factor P -

Chapter 2 Mathemacal modelling of tooth demineralisaon and pH proﬁles in dental plaque

Connuous Model, 1D Dental Plaque: Mulspecies; No Growth Tooth: Represented as boundary tooth plaque saliva

Chapter 3 Mathemacal modelling of microbial dynamics in dental plaque

Connuous Model, 1D Dental Plaque: Mulspecies; Growth tooth plaque saliva plaque grown Tooth: Represented as boundary

Chapter 4 Numerical modelling of tooth enamel subsurface lesion formaon induced by dental plaque

Connuous Model, 1D Dental Plaque: One Species; No Growth tooth plaque saliva Tooth: Represented as 1-d domain

Chapter 5 Two- dimensional mathemacal modelling of tooth enamel subsurface plaque tooth lesion formaon induced by the dental plaque

Connuous Model, 2D Dental Plaque: One Species; No Growth Tooth: Represented as 1-d domain

Chapter 6 A three-dimensional numerical study on the privacy of cell-cell communicaon Individual Based Model, 3D Bioﬁlm: One Species; Growth

Introduction

1. Introduction1.1. Caries and the dental plaque

Dental caries disease has preoccupied humans since

the beginning of time due to the impact it has on the inflicted

persons in the advanced stages (e.g., severe pain, discomfort

in mastication, tooth loss etc.). In the antiquity and medieval

times it was believed that a worm, called the tooth worm, was

responsible for the formation of the tooth holes and for the

tooth pain (Figure 1.1a). The first mention of such a belief is

in a Sumerian text from 5000 BC called ‘The Legend of

Worms’ that was discovered on a clay tablet in the Euphrates

Valley. The Greek poet Homer also blamed toothache on the

worms, while in 1800 BC Mesopotamia the tooth worm even

has its own creation myth.

It was thought that the tooth worm caused a toothache

by wriggling around, and the pain subsided once the worm

rested. The description of the worm varied from culture to

culture: British folklore had the tooth worm resembling an eel,

while the Germans believed the maggot-like worm was red,

blue and gray in colour. In ancient Rome the doctors mistook

tooth nerves for tooth worms and extracted both tooth and

(a) (b)

Figure 1.1. (a) Ivory sculpture of a tooth worms devouring people in the left and in the right, people suffering in hell as a metaphor of toothache. Anonymous French artist, 18th century. Collection of the Deutsches Medizinhistorisches Museum, Ingolstadt, Germany. Photo copyright: W.O. Funk, Bergisch Gladbach, Germany. (b) Fragment from Antoni van Leeuwenhoek’s letter to Francois Aston (van Leeuwenhoek, 1683; IRef 2) nerve in an extremely painful (sometimes opium was used as anaesthetic) predecessor to the modern day root canal (IRef 1). Ironically, although the idea behind such an intervention was wrong, it might have helped in relieving some of the patients' pain, since a toothache usually occurs when the nerve is affected by the tooth decay. Around the XVIIIth century, the idea that worms caused tooth decay started to be rejected by physicians and scientists. Pierre Fauchard, known as the father of modern dentistry, was the first to note that sugar was detrimental to the teeth and gums, while Antoni van Leeuwenhoek, in Delft, observed that

“Mijn gewoonte is des mergens myn tanden te vryven met zout, en dan myn mont te spoelen met water, en wanneer ik gegeten heb, veeltijts myn kiezen met een tandstoker te reinigen;.... Dat in de gezeide materie waren, veele zeer kleine dierkens, die haar zeer aardig beweegden. De grootste soort,was van Fig.A. dezelfve hadden een zeer starke beweginge, en schoten door het water,of speeksel, als een snoek door het water doet;deze waren meest doorgaans weinig in getal” 1 (van Leeuwenhoek, 1683; IRef 2)

1 “In the morning I used to rub my teeth with salt and rinse my mouth with water and after eating to clean my molars with a toothpick.... I then most always saw, with great wonder, that in the said matter there were many very little living animalcules, very prettily a-moving. The biggest sort had a very strong and swift motion, and shot through the water like a pike does through the water; mostly these were of small numbers.” 12

This may be the first scientific observation of the dental plaque, although the connection between these “little bugs” and dental caries was not made yet. Antoni van Leeuwenhoek was also the first to observe under his microscope the canals present inside the teeth (Figure 1.1b). At the end of the XIXth century Willoughby D. Miller formulated the so called chemo-parasitic caries theory (Miller, 1890) on which is based the current explanation of caries etiology. He found that the mouth was populated by bacteria and the mixed bacteria contained in saliva were able in the presence of fermentable carbohydrates to produced acids that decalcified tooth structures. The work of Rodriguez Vargas (1922) and Clarke (1924) further added to Miller’s theory, leading to the specific plaque hypothesis. This theory stated that dental caries can be attributed to individually identifiable bacterial species, (e.g., Lactobacillus strains and Streptococcus mutans) and it received a lot of attention in the dentistry literature of the last decades. The goal was for all the cariogenic bacterial groups to be successfully identified and dental caries avoided by targeting the treatment on those specific groups. However, recent studies have shown that dental caries occur also in the absence of the notorious cariogenic strains (Jenkinson, 2011). The reverse also holds true: that is, no caries were present at sites where S. mutans and/or Lactobacillus were identified (Tahmourespour, 2013). Therefore, while it is still widely acknowledged that the presence of some bacterial groups (the aciduric and acidogenic ones) is usually harmful for the tooth, it is also accepted that the problem is far more complex than this and the dental caries disease cannot be explained only from the perspective of dental plaque composition. A relatively new ecological plaque hypothesis (Marsh, 1989; Loesche, 1986; Kleinberg 2002; Takahashi and Nyvad 2011) considers the dental caries disease to be a consequence of an imbalance in the oral ecosystem, caused by a modification in the diet, diseases, modification of hygiene habits etc. This imbalanced ecosystem could have consequences on the dental plaque composition and on a shift of the chemical processes occurring inside the tooth enamel towards the tooth dissolution. In 1940, Robert Stephan showed how the sugar consumption leads to a sudden drop in the pH of the dental plaque due to the microbial metabolic activity (bacteria present in the dental plaque, consume the sugar and produce organic acids) followed by a slow recovery to steady state values as a consequence of oral clearance (Stephan, 1940; Stephan and Miller, 1943). This type of pH progressions in time are now commonly called Stephan curves and they clearly show the impact of eating and drinking on the tooth health. There is a pH “danger zone” for tooth dissolution (i.e., caries formation) under the value of 5.5. Therefore, the longer the period spent under the critical pH, the bigger the negative impact on the tooth.

1.2. The tooth demineralisation profile Dental enamel consists of approximately 99% (dry weight) microscopic calcium phosphate crystals (rod-shaped) resembling the mineral hydroxyapatite (HAP),

Ca5(PO4)3(OH), together with impurities such as carbonate, sodium, fluoride and other ions (Figure 1.2a) (Fejerskow and Kidd, 2008). The inter-rod space in the enamel is filled with water and organic matter, allowing diffusion of various small ions and molecules through the enamel. The pH decline (acidification) following sugar consumption triggers a variation in the degree of saturation (DSHAP) of plaque liquid with respect to HAP. When the pH drops below a critical value, the plaque fluid becomes undersaturated with respect to HAP (DSHAP < 1) which has as a direct consequence the dissolution (demineralisarion) of the mineral. As the pH values restore above 5.5 during oral clearance, the plaque fluid becomes once again oversaturated with respect to HAP (DSHAP > 1). In these conditions, the remineralisation process may be activated. One important point of the ecological plaque hypothesis is that in a healthy individual, these two processes of demineralisation and remineralisation balance each other and, finally, there is no net loss of HAP after a meal followed by a resting time. In a person with active caries however, this balance is shifted towards demineralisation. As the research methods developed (e.g., better microscopes, more sensitive and refined laboratory techniques) our understanding of dental caries formation became more complex. In spite of this increasing comprehension, there are still aspects eluding our understanding. One such aspect is the typical mineral profile of incipient carious lesions, which shows a region of roughly 100 μm at the tooth surface seemingly unaffected by demineralisation (Fejerskov and Kidd, 2008), with the main body of the lesion present under this surface layer (Figure 1.2b). The differences in mineral content between the two zones can become considerable, especially in the more advanced stages of caries development: up to 99% mineral content in the surface layer compared to just 50 - 75% in the body of the lesion (Robinson et al., 2000; Fejerskov and Kidd, 2008). This particular profile has been observed for the first time by Hollander and Saper (1935) who mistook it for a photographic artefact. Over the years many theories have been proposed to explain this “signature profile” of the dental caries: 1. Unbalanced demineralisation and remineralisation. Dental caries are the consequence of an imbalance between the two concurrent processes that occur naturally in vivo: demineralisation and remineralisation of tooth enamel, and which have the same thermodynamic driving force (Loeshe, 1986; Fejerskov and Kidd, 2008).

(a) (b)

Figure 1.2 (a) Dental enamel structure. (R) HAP rod and (IR) inter-rod space. (b) Dental caries mineral profile. Images from (Fejerskov and Kidd, 2008).

There is a debate in the dentistry field regarding the importance of the remineralisation phenomena for the subsurface layer in vivo. One point of view is that what appears to be a restored surface can be partly explained in terms of wear and polishing (Fejerskov et al., 2008). Another argument is that inhibitor molecules present in saliva (e.g, statherin) prevent in vivo precipitation at crystal surface by blocking crystallization nuclei (Santos et al. 2008). It was also argued that the very fast uptake of calcium and phosphate by the HAP makes the pore liquid in the deep layers of lesion only marginally saturated (Larsen and Fejerskov, 1989) thus remineralisation can only be very slow. 2. Less soluble mineral crystals at the tooth enamel surface. This might be due to: i. Fluorohydroxyapatite. a fluoride distribution over the enamel depth with higher fluoride content at the surface. The fluoride is incorporated into hydroxyapatite (HAP) structure forming a more stable mineral, fluorohydroxyapatite (FHAP), which makes the enamel less soluble during acid attacks (Koutsoukos et al, 1980; Robinson et al., 2000; Fejerskov and Kidd, 2008); ii. Ripening. crystals with higher purity (hence, lower solubility) may develop at the enamel surface through a process called “Ostwald ripening” (Fejerskov and Kidd, 2008). Following repeated cycles of demineralisation and remineralisation, HAP loses the impurity inclusions (e.g., carbonate, magnesium, sodium) and crystals of higher purity emerge (Robinson et al., 2000).

15 iii. Brushite. During an acid attack, the crystal at the surface demineralises and crystallizes again in a more stable form at a low pH, i.e., dicalcium phosphate dehydrate (DCPD or brushite) at pH 4.3 (Margolis et al, 1999; Fejerskov and Kidd, 2008). Phase transformations of HAP to DCPD at the enamel surface during an acid attack have not been verified experimentally. According to Arends and Cristoffersen (1986) the experimental observation that a surface layer once formed has a nearly constant thickness, supports the idea that the surface layer forms because of a solubility gradient along the enamel depth. 3. Diffusion barrier for remineralisation. The remineralisation of the surface layer occurs relatively sooner than in the subsurface layers which creates a region with lower porosity at the enamel surface. As a result, a diffusion barrier is created. The enhanced retardation of most ions, compared to that of protons delays the remineralisation of the subsurface region, while it stimulates demineralisation (Silverstone, 1977; White et al, 1988). Although the surface layer acts as a diffusion delaying barrier due to its lower porosity, the delay for all the species will be the same. That is, the protons will diffuse slower together with all the other species (i.e., calcium, phosphate, organic acid anions etc.) present inside the tooth. 4. Remineralisation inhibition. The remineralisation process might be less efficient in the subsurface layer due to the presence of inhibiting species uniformly distributed into the enamel matrix and released during extended demineralisation. Once released, these species would diffuse towards the enamel surface and part of them is presumably absorbed onto crystals surface (White et al, 1988). However, it is not experimentally determined what compound within the enamel would serve as an inhibitor. 5. Demineralisation inhibition. Salivary proteins present in the acquired enamel pellicle adsorb onto HAP crystals at the tooth surface hence inhibiting demineralisation and remineralisation only in the surface layer (White et al, 1988). It is not clear if the polar bounds formed between the salivary protein and HAP are strong enough to resist acidic pH during meal time and low protein concentration in the biofilm. One may also argue that the protein would serve even as a feeding substrate for some bacterial species. This theory speculates on components present in the acquired enamel pellicle. However, demineralisation occurs at tooth surfaces covered with biofilm where there is no pellicle anymore. Also, it is not experimentally determined if these salivary proteins have inhibitory concentration in the biofilm. 6. Different acid speciation. Organic acids produced by dental plaque would diffuse in an undissociated form inside the tooth enamel and dissociate once they are deeper inside. This would generate acidity and trigger demineralisation in deeper layers of the enamel, while the surface remains intact (Loeshe, 1986). It is not clear however, what conditions at the tooth surface would make an acid to remain in its undissociated form.

1.3. Numerical modelling Most of the studies on caries formation are either dealing with the processes occurring in the dental plaque alone (Dibdin and Reece, 1984; Dawes and Dibdin, 1986; Dibdin, 1990; Dibdin, 1997; Dibdin and Dawes, 1998) or with the chemical processes of dissolution / remineralisation of the tooth enamel (Holly and Gray, 1968; Zimmerman, 1966 a,b,c; Fox et al., 1978; Van Dijk et al., 1979; Ten Cate, 1983). It must be considered however, that one precondition for the development of dental caries is the presence of dental plaque on the affected tooth enamel. This essential biological factor makes the dental caries formation more complex than simple mineral demineralisation and remineralisation. Thus, for a correct understanding of the mechanisms governing this disease, the processes occurring inside the tooth enamel must be studied in relation to the (micro)biological and chemical processes occurring in the dental plaque. It is very difficult to handle experimentally all these chemical, physical and biological factors in an integrated manner, especially when some aspects regarding dental caries are still unclear (such as the subsurface lesion formation). Moreover, given the complexity of this disease where factors like genetics and individual behaviour (sugar consumption frequency, oral hygiene) combine with the long time span required for cavities to develop, it is rather impossible to make any prediction regarding the outcome of an experiment testing a hypothesis by using only simple calculations and intuition. For this reason, mathematical modelling (sometimes called in silicio experimentation) offers a different research perspective by surmounting some of the problems of the in vivo and in vitro studies. Using numerical tools it is possible to create a perfectly controlled environment (e.g., saliva composition and flow, or the active biochemical reactions and transport processes occurring inside the plaque) in which reality is simplified to the relevant aspects for the studied problem. This is also offering the possibility of easily adding or removing different physical, chemical or biological processes in order to study in separation their influence on the system which, from an experimental point of view, can be very difficult to achieve. Further advantages are shorter times to obtain results compared to the in vivo situation, the possibility to study a wider variety of conditions and also the elimination of the ethical concerns present in the in vivo studies, regarding the influence of the experiment on the health of the patient. However, mathematical models must be based on experiments. Input parameters (such as solute transport properties, chemical and microbial reaction rate coefficients, etc.) have to be determined experimentally, or, at least theoretically estimated based on knowledge acquired in an empirical manner. It is evident that

17 the model outcome depends on the quality of input parameters. Furthermore, model results have to be compared with observational data, so that the numerical model can be “validated”. Only when the model is refined and tested enough against the reality, useful predictions can be made and confidence in the model outcome can be gained. Given the complexity of dental caries reflected by the multitude of factors involved in its development, the research of dental plaque and caries formation is a multi-disciplinary activity. Ideally, for a deep understanding of this disease, dentists, microbiologists, biologists, chemists, physicists and modellers all have to exchange data and integrate information. This can be complicated for a number of reasons: it is not easy to gather a team of scientists with such different backgrounds even now when long distance communication and travelling possibilities are easier than ever in the human history. Also, once gathered such a team the communication can be at times difficult due to the different goals of each group but also the differences that exist in the language between the scientific fields.

Tooth demineralisation/remineralisation models A number of numerical models for tooth de- and remineralisation only (thus, without dental plaque being included) was developed starting with the middle of the 1960’s. The main purpose of those models was to derive the kinetic expression of HAP demineralisation during caries formation and to reproduce the typical subsurface lesion observed in all the initial caries. The models developed by Zimmerman, (1960 a, b, c) described a mathematical approach to determine theoretically an expression for the kinetics of enamel dissolution. The model involves diffusion of calcium and phosphate ions as well as organic acids in both dissociated and un-dissociated form and the demineralisation reaction is assumed to be fast in comparison with the diffusion processes. No subsurface lesion was obtained with this model. A one-dimensional time dependent numerical model was developed by Holly and Gray, 1968 to represent an incipient carious lesion (also called a “white spot”). The model consisted of two membranes on top of each other: first membrane with a constant thickness acted for the surface layer and the second membrane with an increasing thickness approximated the subsurface lesion. When this model was developed, no kinetic expression for tooth enamel demineralisation had been determined experimentally; therefore, the purpose of the model was to predict the behaviour of the subsurface demineralisation process. Consequently, two mathematical expressions have been established that related the rate of advancement of the subsurface lesion in time with the concentration of undissociated acids.

Another diffusion-reaction mathematical model of tooth demineralisation was proposed by Fox et al., 1978. The goal was to establish a correlation between three proposed HAP demineralisation kinetics and the different levels of undersaturation of dental plaque fluid (partially saturated vs. nearly completely unsaturated), as well as examining the potential for remineralisation in each of the studied cases. A more sophisticated model (Van Dijk et al., 1979) takes into account factors such as the ionic product of the solution to which the tooth is exposed and using Nernst-Plack equation to describe the transport of ions. Van Dijk et al. obtained a surface layer with their model, but only after imposing at least one gradient in: enamel solubility, or demineralisation rate constant, or enamel porosity. Ten Cate, 1983 proposed a first model for studying the remineralisation of tooth enamel. The model takes into account diffusion through the lesion pores, diffusion in the close vicinity of the pore walls, adsorption-desorption occurring at the pore surface and mineralisation in the pores. This model was a valuable contribution to understanding the subtleties of enamel remineralisation. However, enamel demineralisation kinetics and metabolic processes of microorganisms in the dental plaque were beyond the scope of the model.

Dental plaque models The first mathematical model accounting for dental plaque, diffusion of glucose, glucose conversion to lactate and enamel demineralisation was developed by Higuchi et al. (1970). Among the main limitations of this model are the steady state assumption, the constant glucose concentration in saliva and the lactic acid production not inhibited at low pH. Following Higuchi’s model, Dibdin and Reece (1984) developed the first model in a series of numerical descriptions of dental plaque activity (Dawes and Dibdin, 1986; Dibdin, 1990; Dibdin et al., 1995; Dibdin, 1997; Dibdin and Dawes, 1998). These models aimed at calculating the time-dependent one- and two-dimensional pH profiles in the depth of dental biofilms, as a consequence of the metabolic processes taking place in the dental plaque when sucrose is provided. Moreover, the change of the Stephan curve in different conditions was also explained by these models (Dawes and Dibdin, 1986; Fejerskov and Kidd, 2008). For this purpose, the models developed by Dibdin’s group considered: multiple solutes transport by diffusion and ion migration; charge balancing using a novel charge-coupling algorithm; a single generic microbial group (presumably aciduric Streptococcus); two acids, lactic, to account for strong organic acids, and acetic, to account for weak acids and only anaerobic metabolism; polysaccharide storage by microorganisms but not its consumption;. The enamel

19 was assumed to be inert in most of these models. The work of Dibdin was a starting point for the building of the dental plaque numerical model described in the first chapter of this thesis.

1.4. Objectives and thesis layout The dental plaque models presented in this thesis are the first to integrate existing knowledge on biofilm processes (i.e., mass transfer, microbial composition, microbial conversions and substrate availability) with tooth demineralisation and remineralisation kinetics. These models simulate the pH variation (i.e., the Stephan curve) under the influence of microbial metabolism occurring in dental plaque and successfully reproduce the formation of a subsurface lesion even in the absence of any pre-imposed gradients. The research topics and study aims in this thesis follow a progressive approach. First, a one-dimensional (1-d) model of dental plaque metabolism (Chapter 2) was developed. The 1-d plaque model was then extended to represent the dynamics of microbial populations (Chapter 3) and of sub-surface lesion formation in the enamel (Chapter 4). Furthermore, the sub-surface lesions were studied in a more complex two-dimensional (2-d) setup (Chapter 5). Finally, in a three-dimensional model all the current theories regarding bacterial communication were integrated. This was to study how and why bacteria would maintain the inter-colonial communication private. The time-dependent 1-d and 2-d numerical models for caries formation are continuous (volume averaged) and built conveniently in commercial software dedicated to solution of partial differential equations (PDE), COMSOL Multiphysics (COMSOL releases 4.1 and 3.5a, Comsol Inc, Burlington, MA, www.comsol.com). The 3-d individual based model to study the privacy of cell-cell communication has a more sophisticated construction, being built in MATLAB (MATLAB 2008b, The MathWorks, Natick, MA, www.mathworks.com) and integrated with COMSOL (for PDE solution) and with self-made Java routines (for individual-based microbial colony formation). The questions to be addressed in the current work are:

1. How do behavioural factors influence the development of initial dental caries? The factors considered are sugar consumption patterns and oral hygiene by tooth brushing.

2. Is there any influence of the bacterial storage compounds (e.g., polyglucose, glycerol) on the evolution of caries?

3. What are the microbial shifts occurring in dental plaque in different oral environmental conditions (i.e., different regimes of sugar consumption, often or rare tooth brushing, etc.)?

4. Which of the mechanisms proposed in the literature are responsible for the typical profile of a dental caries (i.e., a healthy enamel layer covering an area of high porosity)?

5. What is the influence of the geometry of the tooth site at which the caries develops (e.g., occlusal area, smooth surface) on the profile of the lesion?

6. How can we integrate the existing theories on bacterial communication (that is, quorum sensing, diffusion sensing and efficiency sensing) in order to find out how bacteria keep their communication private? The first two research questions were addressed in Chapter 2 and then re-evaluated in Chapter 3. Question 3 was extensively research in Chapter 3, while Chapter 4 focused only on question 4. Chapter 5 re-evaluated question 4 and addressed as well question 5. Chapter 6 is dealing exclusively with question 6.

1.5. References Antonie van Leewenhoek, Letter of Antonie van Leewenhoek to Francois Aston, Delft, The Netherlands, 12 September 1683, Pag.11 Arends J, Christoffersen J: The nature of early caries lesions in enamel. J Dent Res 1986; 65(1):2-11. Clarke JK, On the Bacterial Factor in the Ætiology of Dental Caries, British Journal of Experimental Pathology (1924); 5: 141–7 Dawes C, Dibdin GH: A theoretical analysis of the effects of plaque thickness and initial salivary succrose concentration on diffusion of succrose into dental plaque and its conversion to acid during salivary clearance. J Dent Res 1986; 65(2):89-94. Dibdin GH, Dawes C, A mathematical model of the influence of salivary urea on the pH of fasted dental plaque and on the changes occurring during a cariogenic challenge, Caries Res 1998; 32:70-74. Dibdin GH, Dawes C, Macpherson LMD, Computer modelling of the effects of chewing sugar-free and sucrose-containing gums on the pH changes in dental plaque associated with a cariogenic challenge at different intra-oral sites, J Dent Res 1995; 74(8):1482-1488. Dibdin GH, Mathematical modelling of biofilms, Adv Dent Res 1997; 11(1):127-132. Dibdin GH, Reece GL, Computer simulation of diffusion with reaction in dental plaque, Caries Res 1984; 18(2):191-192. Dibdin GH: Effect on a cariogenic challenge of saliva/plaque exchange via a thin salivary film studied by mathematical modelling. Caries Res 1990; 24(4):231-238. Fejerskov O, Kidd E: Dental caries: The disease and its clinical management, 2nd ed, Chicester, United Kingdom, Blackwell Munksgaard, 2008.

Fox JL, Higuchi WI, Fawzi MB, Wu M-S, A new two-site model for hydroxyapatite dissolution in acidic media, J Colloid Interf Sci 1978; 67(2):312-330. Higuchi WI, Young F, Lastra JL, Koulourides T, Physical model for plaque action in the tooth-plaque-saliva system, J Dent Res 1970; 49:47–60. Hollander F, Saper E: The apparent radiopaque surface layer of the enamel, Dent Cosmos 1935; 77:1187-1197 Holly FJ, Gray JA: Mechanism for incipient carious lesion growth utilizing a physical model based on diffusion concepts, Archs Oral Biol 1968; 13:319-334. Internet reference 1: http://www.medelita.com/blog/beware-evil-tooth-worm/, accessed on 2nd of October 2013 Internet reference 2: http://www.faculty.umb.edu/gary_zabel/Courses/Spinoza/Texts/WAW%20Moll%20Antonie %20van%20Leeuwenhoek.htm, accessed on 2nd of October 2013 Jenkinson H, Beyon the oral microbiome, Environ Microbiol 2011; 13(12):3077-3087 Kleinberg I, A mixed-bacteria ecological approach to understanding the role of the oral bacteria in dental caries causation: an alternative to Streptococcus mutans and the specific- plaque hypothesis, Crit Rev Oral Biol Med (2002). 13(2): 108–25. Koutsoukos P, Amjad Z, Tomson MB, Nancollas GH: Crystalization of calcium phosphates. A constant composition study, J Am Chem Soc 1980; 102(5):1553-1557. Larsen MJ, Fejerskov O: Chemical and structural challenges in remineralisation of dental enamel lesions, Scand J Dent Res 1989; 97:285-96. Loesche WJ, Role of Streptococcus mutans in human dental decay, Microbiol Rev (1986); 50:353-380. Margolis HC, Zhang YP, Lee CY, Kent RL, Moreno JR, Moreno EC: Kinetics of enamel demineralisation in vitro, J Dent Res 1999; 78(7):1326-1335. Marsh PD, Host defenses and microbial homeostasis. Role of microbial interactions. J Dent Res (1989); 68(Spec Iss):1567-1575. Miller WD, The microorganisms of the human mouth. Philadelphia (1890), PA: SS White and Co. Reprinted, 1973. Basel: Karger. Robinson C, Shore RC, Brookes SJ, Strafford S, Wood SR, Kirkham J: The chemistry of enamel caries, Crit Rev Oral Biol M 2000; 11(4):481-495. Rodriguez Vargas FE, The specific study of the bacteriology of dental cavities, Military Dental Journal, (1922), December Santos O, Kosoric J, Hector MP, Anderson P, Lindh L: Adsorption behavior of statherin and a statherin peptide onto hydroxyapatite and silica surfaces by in situ ellipsometry, J Colloid Interf Sci 2008; 318(2):175-82. Silverstone LM: Remineralization phenomena. Caries Res 1977; Suppl. 1:59-84.

Stephan RM, Changes in hydrogen-ion concentration on tooth surfaces and in carious lesions. J Am Dent Assoc, 1940, 27:718-723. Stephan RM, Miller BF, A quantitative method for evaluating physical and chemical agents which modify production of acids in bacterial plaques on human teeth, Journal of Dental Research, 1943, 22(1):45-51. Tahmourespour A, Nabinejad A, Shirian H, Ghasemipero N, The comparison of proteins elaborated by Streptococcus mutans strains isolated from caries free and susceptible subjects, Iranian Journal of Basic Medical Sciences 2013; 16 (4):648-652. Takahashi N, Nyvad B, The role of bacteria in the caries process: ecological perspectives, J Dent Res (2011); 90(3):294-303 Ten Cate JM: A model for enamel lesion remineralisation, in: S.A. Leach, W.M. Edgar (Eds.), Demineralization and Remineralization of the Teeth, Irl. Pr., 1983, pp. 129-144. Van Dijk JWE, Borggreven JMPM, Driessens FCM: Chemical and mathematical simulations of caries, Caries Res 1979; 13:169-180. White DJ, Chen WC, Nancollas GH: Kinetic and physical aspects of enamel remineralisation - a constant composition study. Caries Res 1988; 22:11-19. Zimmerman SO, A mathematical theory of enamel solubility and the onset of dental caries: I. The kinetics of dissolution of powdered enamel in acid buffer, B Math Biophys 1966a; 28:417-432. Zimmerman SO, A mathematical theory of enamel solubility and the onset of dental caries: II. Some solubility equilibrium considerations of hydroxyapatite, B Math Biophys 1966b; 28: 433-441. Zimmerman SO, A mathematical theory of enamel solubility and the onset of dental caries: III. Development and computer simulation of a model of caries formation, B Math Biophys 1966c; 28:443-464.

Mathematical

modelling of tooth 2 demineralisation

and pH profiles in

dental plaque

2. Mathematical modelling of tooth demineralisationDental caries is currently one of the most widespread conditions associated with oral hygiene. The term dental and pH profilescaries is used to describe in the dentalresults – signs and symptoms – of a localized chemical dissolution of the tooth surface caused plaque by metabolic events taking place in the biofilm (dental plaque) covering the affected area (Fejerskov and Kidd, 2008). Dental enamel consists of approximately 99% (dry weight) microscopic calcium phosphate crystals (called rods)

resembling the mineral hydroxyapatite (HAP), Ca5(PO4)3(OH), together with impurities such as carbonate, sodium, fluoride and other ions. The inter-rod space in the enamel is filled with water and organic matter, allowing diffusion of protons and other ions through the enamel. Carbohydrate consumption by microorganisms present in the dental plaque leads to organic acid production, which causes an acidic pH in the plaque. This triggers a variation in

24 Chapter published in Journal of Theoretical Biology 309:159-175 (2012) the degree of saturation of plaque liquid with respect to HAP. Dental caries is the direct consequence of the resulting dissolution of HAP due to the decreased pH. The caries formation process is extended over a long time span, which correlated with the conditions present in the mouth, makes it very difficult for in vivo studies. Currently, the in vitro studies are limited mainly by two factors: (1) the difficulty of creating HAP crystals with a similar structure to the natural enamel, for the studies using artificial teeth. This problem can be surpassed using extracted human or animal teeth (Larsen and Pearce, 199; Featherstone et al., 1979; Lippert et al., 2004); (2) the long time periods required to develop caries reproducing the conditions present in the mouth: acid attacks during eating and drinking (resulting in demineralisation) interposed with fasting periods (with the potential of remineralisation). Therefore, continuous acid attack is often used in experimental studies (Larsen and Pearce, 1992; Lippert et al., 2004, Margolis and Moreno, 1992). Mathematical modelling offers a different research perspective by surmounting some of the problems the experimental studies are facing, such as the long time period required for the development of the caries from the subclinical level until the clinically visible caries. Using numerical tools it is possible to create a controlled environment in which reality is simplified to the relevant aspects for the problem to be studied. This is also offering the possibility of easily adding or removing different physical, chemical or biological processes in order to study in separation their influence on the system which, from an experimental point of view can be very difficult to achieve. Because of its high relevance, dental caries problem has been intensively studied and, as soon as the computational power allowed it, the first mathematical models for tooth demineralisation (Holly and Gray, 1968; Zimmerman, 1966a; Zimmerman, 1966b; Zimmerman, 1966c; Fox et al., 1978) have been developed. As the experimental studies over the years offered more insight into the problem, new and more sophisticated numerical models emerged (Van Dijk et al., 1979; Ten Cate, 1983). These models are dealing exclusively with the processes taking place in the enamel, while the plaque itself is not considered. Higuchi et al. (1970) developed a first mathematical model accounting for dental plaque, diffusion of glucose, glucose conversion to lactate and enamel demineralisation. Major limitations of this model are the steady state assumption, the constant glucose concentration in saliva and the lactic acid production not inhibited at low pH. Dibdin and Reece (1984) developed the first model in a series of mathematical models of dental plaque (Dawes and Dibdin, 1986; Dibdin, 1990a; Dibdin, 1990b; Dibdin et al., 1995; Dibdin, 1997; Dibdin and Dawes, 1998). These models aimed at calculating the one- dimensional and, further (Dawes, 1989), the two-dimensional pH profiles in the depth of

25 dental biofilms, as a consequence of the metabolic processes taking place in the dental plaque when sucrose is provided. Moreover, the variation of plaque pH in time (the so-called Stephan curves; Dawes and Dibdin, 1986; Fejerskov and Kidd, 2008) in different conditions was also explained by these models. Surprisingly, development of these models has not been continued during the recent years. We decided to extend Dibdin's work by making advantage of both increased understanding of plaque processes and function and of computational resources. In this paper we describe the development of a model integrating complex metabolic, chemical and mass transfer processes occurring in the dental plaque with the rate of enamel demineralisation. The main improvements in the current model in respect with the previous work of Dibdin et al. are summarized in Table 2.1. Our aim was to (1) investigate the process of caries formation in a more quantitative and structured manner, (2) identify critical parameters playing a role in caries formation and (3) guide new research in respect with relevant and not understood aspects of caries formation.

Table 2.1 Comparison of current model with the model of Dawes and Dibdin, 1986.

Current model Dibdin’s models Ion migration rate with Nernst-Planck equation Charge balancing Multiple bacterial species (aciduric and nonaciduric One generic species Streptococcus, Actinomyces, Veillonella) (presumably aciduric Streptococcus) Polysaccharide (polyglucose) storage and consumption Only polysaccharide storage Five acids: Two acids: lactic, to account for strong organic lactic, acetic, propionic, formic, succinic acids, and acetic, to account for weak acids Aerobic and anaerobic metabolism included Only anaerobic metabolism

Reliable demineralisation kinetics for the tooth Enamel is assumed to be inert

2.1. Model description The mathematical model considers planar dental plaque geometry with only one-dimensional gradients of solute concentration from the saliva perpendicular to the tooth surface. Multiple microbial and chemical species are taken into account together with a selection of the most relevant metabolic and chemical processes that have the potential to influence significantly the plaque pH.

2.1.1. Components The model has to describe the chemical interactions between several types of microorganisms present in the dental plaque via different substrates and metabolic products. Therefore, in the

26 present model we first distinguish between microbial components and chemical components. The microbial components are immobile and they are the constituents of the dental plaque. Further, the chemical components can be divided into mobile and immobile. While we assume that the microbial and the immobile chemical components are present only in the dental plaque, the mobile chemical ones exist both in the saliva and dental plaque. In the following sections a detailed description of the model components and processes in which they are involved is provided.

(a) Microbial components One of the goals of this study was to find out whether it is relevant to include multiple microbial components performing a series of bio-transformations or it is sufficient just to lump all active microbial species into a single generic component (such as in the reference work of Dibdin, Dawes and Dibdin, 1986; Dibdin, 1990a; Dibdin, 1990b; Dibdin, 1997). There is vast amount of literature reporting studies on the microbial composition of dental plaque (e.g., Fejerskov and Kidd, 2008; Marsh et al., 2009; Ritz, 1967; Bowden, 2000; Takahashi and Nyvad, 2008; Filoche et al., 2010). Among the main microbial groups present and active in dental plaque are: Streptococcus, Actinomyces, Lactobacillus, Veillonella, Propionibacterium, Bifidobacterium and Capnocytophaga (Hojo et al., 2009). In order to keep the model within reasonable limits of complexity, while still considering the dominant microorganisms in the plaque, the following microbial groups have been selected as plaque constituents for the current study: ● Streptococcus is a facultative anaerobic bacterium that can represent up to 85% (Fejerskov and Kidd, 2008; Marsh et al., 2009) of the organisms present in dental plaque. Two Streptococcus categories must be differentiated because they have different niches in the dental plaque: • non-aciduric Streptococci (S. milleri, S. sanguis, S. salivarius etc.) living in the near neutral pH range, denoted by STN in this model, are the majority. • aciduric Streptococci (S. mutans, S. sobrinus) active until pH as low as 4.5 (Hamilton and Buckley, 1991; Bowden, 2000), symbolized by STA, appear as a minority group in dental plaque. Even though STA are usually accounting for a small percentage of the plaque (between 0-23%, Marsh et al., 2009), their influence seems to be important for the purpose of this study, since an increase in the concentration of aciduric Streptococci is usually associated

27 with the appearance of dental caries around pH 5.5. Therefore, both STA and STN have to be included in the dental plaque model. ● Actinomyces is the second most present microbial group in the dental plaque (up to 45%, Fejerskov and Kidd, 2008; Marsh et al., 2009) and it is also facultative anaerobe (Ritz, 1967). In the model we represented all Actinomyces as a generic group, ACT, taking into account the main glucose conversions occurring in this genus. ● Veillonella are strictly anaerobes and they are the third most abundant group of microorganisms present in mature oral biofilms (sometimes up to 40%, Fejerskov and Kidd, 2008; Marsh et al., 2009). The Veillonella are also represented in the model as a lumped metabolic group, VEL. We neglected several other microbial groups reported as being possibly important. Among these, Lactobacillus is an anaerobe usually present in low amounts in the plaque and especially in the advanced stages of caries formation process (Loesche, 1986).

Microbial components are described by their concentration in the plaque CX,i (g dry biomass / m3 plaque). For simplification, in the current study we did not include microbial growth neither a layered plaque structure. It was considered that during the limited period of time studied by the present simulations (~2-3 hours), the plaque thickness and the plaque microbial composition would remain relatively constant. The plaque composition was assumed (Table 2.2) based on the values reported in Marsh et al., 2009 and Fejerskov and Kidd, 2008. There is therefore no microbial state variable whose distribution in space and time must be computed by the model.

Table 2.2 Microbial components in the model and their constant concentrations in dental plaque (assumed cf. Fejerskov and Kidd, 2008). Concentration Concentration Name Symbol –3 [(kg dry biomass) (m plaque)] [mass %] Aciduric Streptococcus CX,STA 4 5 Non-aciduric Streptococcus CX,STN 36 45 Actinomyces CX,ACT 32 40 Veillonella CX,VEL 8 10

(b) Chemical components The chemical model components are chemical species relevant for describing the cariogenic effect of dental plaque. We distinguish between mobile and immobile (fixed) chemical species. Each mobile chemical species (Table 2.3) is characterized by a constant diffusion coefficient Di and charge zi, and it is described by a state variable, concentration Ci, changing in time in saliva and at different depths in the dental plaque. In the first place, substrates for

Table 2.3 Mobile chemical species and their associated properties in the current model

Diffusion coefficient Initial concentration in saliva film and plaque Name Symbol Formula Charge -9 2 –1 [10 m s ] Reference [mmol L–1] Reference/Calculation – Acetate Ace– C2H3O2 –1 1.38 Cussler, 2009 0 equilibrium, CAce,tot = 0 Vanýsek, 2001 Acetic acid AceH C2H4O2 0 1.64 Cussler, 2009 0 equilibrium Vanýsek, 2001 – Formate For– CHO2 –1 1.84 Vanýsek, 2001 0 equilibrium, CFor,tot = 0

Formic acid ForH CH2O2 0 1.79 Cussler, 1984 0 equilibrium – Lactate Lac– C3H5O3 –1 1.31 Vanýsek, 2001 0 equilibrium, CLac,tot = 0

Lactic acid LacH C3H6O3 0 1.41 Assumed 0 equilibrium – Propionate Pro– C3H5O2 –1 1.20 Cussler, 2009 0 equilibrium, CPro,tot = 0 Vanýsek, 2001 Propionic acid ProH C3H6O2 0 1.34 Cussler, 1984 0 equilibrium 2– Succinate Suc2– C4H4O4 –2 0.99 Vanýsek, 2001 0 equilibrium, CSuc,tot = 0 – Hydrogen succinate Suc– C4H5O4 –1 1.10 Assumed 0 equilibrium

Succinic acid SucH C4H6O4 0 1.19 Cussler, 1984 0 equilibrium – Bicarbonate HCO3– CHO3 –1 1.50 Vanýsek, 2001 4.17 equilibrium, CCO2,tot = 5.1, Marsh et al, 2009

Carbon dioxide CO2 CO2 0 2.43 Cussler, 2009 0.93 equilibrium 2– Hydrogen Phosphate Pho2– HPO4 –2 0.96 Vanýsek, 2001 2.06 equilibrium, CPho,tot = 5.4, Fejerskov and Kidd, 2008 – Dihydrogen Phosphate Pho– H2PO4 –1 1.22 Vanýsek, 2001 3.34 equilibrium Hydroxyl HO– HO– –1 6.7 Cussler, 2009 10–4 equilibrium Vanýsek, 2001 Proton H+ H+ +1 11.81 Cussler, 2009 10–4 Fejerskov and Kidd, 2008 Vanýsek, 2001 2+ Calcium Ca2+ Ca +2 1.00 Cussler, 2009 0.04 equilibrium, CCa2+,tot = 1.32, Fejerskov and Kidd, 2008 Vanýsek, 2001 Anion (Chloride) Cl– Cl– –1 2.57 Cussler, 2009 40 Fejerskov and Kidd, 200 Vanýsek, 2001 + Cation (Potassium) K+ K +1 2.49 Cussler, 2009 Cs,K+= 49 charge balance in saliva Vanýsek, 2001 Cp,K+= 80.86 charge balance in plaque Oxygen O2 O2 0 2.66 Cussler, 2009 0.15 Assumed

Glucose Glu C6H12O6 0 0.85 Vanýsek, 2001 0.07 Van der Hoeven et al., 1990

Ethanol Eth C2H6O 0 1.57 Vanýsek, 2001 0 Assumed the considered microbial components must be included, such as glucose (for aciduric and non- aciduric Streptococcus and Actinomyces) and lactate (for Veillonella), plus the dissolved oxygen. Secondly, the organic acids produced in anaerobic fermentations or aerobic conversions (e.g., lactic, acetic, propionic, succinic, formic) are taken into account because they are the main cause of caries formation. Thirdly, enamel ions such as calcium and phosphate are included to be able to calculate the enamel dissolution rates. In addition, other background electrolytes, cations (K+) and anions (Cl–) are needed for charge balancing. The final group of chemical components is needed for speciation in acid-base equilibria (the base of pH and charge balance calculations), namely the main dissociation/association species of each chemical component (e.g., lactic acid and lactate ion, proton and hydroxyl ions, carbon dioxide and bicarbonate ions, etc.). There are two types of immobile chemical components in the plaque. First, the "surface species" account for the buffering capacity of the dental plaque (Shellis and Dibdin, 1988; Hong and Brown, 2006). They are functional groups bound to the bacterial cell wall or extracellular polymers and they participate in the ion speciation and charge balancing. Different chemical compounds stored within the microbial cells, such as a generic polyglucose component, are also accounted for. The immobile chemical components (Table 2.4) are characterized by a concentration variable in time and at different depths in the plaque.

Table 2.4 Fixed chemical species and their properties Initial concentration Name Symbol Formula Charge [kg m–3] Reference/Calculation Polyglucose in ACT Pgact (C6H12O6)n 0 0 Assumed Polyglucose in STA Pgsta (C6H12O6)n 0 0 Assumed Polyglucose in STN Pgstn (C6H12O6)n 0 0 Assumed + –5 Fixed charge cationic sites SH2+ {SH2 +1 3.59 × 10 equilibrium CS,tot = 28.8, Hong and Brown, 2006 Fixed charge neutral sites SH {SH 0 0.13 equilibrium Fixed charge anionic sites S– {S– –1 30.59 equilibrium Complex of fixed charge and Ca2+ SCa+ {SCa+ +1 1.28 equilibrium

2.1.2. Processes Each of the microbial groups considered active in dental plaque model carries out several metabolic processes, according to different pathways (e.g., aerobic, anaerobic, low or high glucose). These pathways will be called here biological reactions. In addition, because the main goal of the current model is to calculate pH changes in the plaque leading to enamel demineralisation, several acid-base and complexation equilibria must be included together with the buffering effect of plaque by surface charge equilibria. Finally, a dissolution reaction occurs at the enamel surface due to the generated acid environment.

(a) Biological reactions Two general types of biological transformations performed by the microorganisms are included in the model: (i) the conversion of substrate and other nutrients and (ii) the production and consumption of internal storage compounds. All biological transformations considered in the model, together with their reaction stoichiometry and rate expressions are presented in Table 2.5. The biological rate parameters are listed in Table 2.6.

Glucose conversion Glucose was chosen as substrate because it is used (directly, or indirectly under the form of sucrose) in many clinical studies (Tanzer et al., 1969; Dong et al., 1999; Pearce et al., 1999), and it is also the most readily available carbon source during prandial periods. In the model glucose is consumed by Streptococcus and Actinomyces, while Veillonella take up only the lactate generated in the glucose fermentation. For simplification, sucrose and its metabolic reactions were not considered in the case studies presented here, although Streptococcus mutans metabolizes sucrose to form extracellular polymers that in part account for its enhanced cariogenicity versus non-mutans streptococci (Marsh et al., 2009). The anaerobic glucose fermentation is the process with the highest impact on the caries formation. Depending on the environmental conditions, it occurs differently for the various organisms considered. There are two mechanisms for glucose uptake, as a function of its available concentration. The high affinity pathway is active at low glucose concentrations (i.e., during inter-prandial periods), whereas the low affinity pathway occurs at high glucose concentrations (Van der Hoeven et al., 1985; Colby and Russell, 1997). Low concentration anaerobic glucose conversion occurs with the same stoichiometry for both Streptococcus types (producing acetate, formate and ethanol) but the rates differ in the degree of pH inhibition (Van Beelen et al., 1986, Table 2.5, Table 2.6). Actinomyces is consuming low concentration glucose anaerobically by using a different metabolic pathway with formation of acetate, formate and succinate (De Jong et al., 1988). At high glucose concentrations both Streptococcus and Actinomyces are using the same pathway with production of lactate (Van Beelen et al., 1986; De Jong et al., 1988). Since Streptococci and Actinomyces are facultative anaerobes, and oxygen may be present in a shallow superficial layer in the dental plaque exposed to saliva, the glucose can also be converted according to aerobic pathways (Table 2.5). The aerobic conversion of glucose is different for each of the above mentioned microorganisms, with different ratios of acetate and formate being produced (van Beelen et al., 1986; De Jong et al., 1988). 31

Table 2.5 Stoichiometry (molar) and rates of microbial processes considered in the model dental plaque Glu O2 Suc2 Lac Pro Ace For Eth HCO3– H+ H2O Pgsta Pgstn Pgact Rates Reference

1. Anaerobic high concentration glucose fermentation

STA –1 2 2 qm,STA,Glu,H CSTA M(CGlu) I(CH+) Van Beelen et al., 1986 STN –1 2 2 qm,STN,Glu,H CSTN M(CGlu) I(CH+) Van Beelen et al., 1986 ACT –1 2 2 qm,ACT,Glu,H CACT M(CGlu) I(CH+) Van Beelen et al., 1986 2. Anaerobic low concentration glucose fermentation STA –1 1 2 1 3 –1 qm,STA,Glu,L CSTA M(CGlu) I(CH+) I(CO2) I(CGlu) Van Beelen et al., 1986 STN –1 1 2 1 3 –1 qm,STN,Glu,L CSTN M(CGlu) I(CH+) I(CO2) I(CGlu) Van Beelen et al., 1986 ACT –1 1 1 1 –1 3 1 qm,ACT,Glu,L CACT M(CGlu) I(CH+) I(CO2) I(CGlu) De Jong et al., 1988 3. Aerobic glucose conversion STA –1 –1 2 2 4 qm,STA,Glu,L,O2 CSTA M(CGlu) M(CO2) I(CH+) I(CGlu) Van Beelen et al., 1986 STN –1 –3/2 2 1 1 4 qm,STN,Glu,L,O2 CSTN M(CGlu) M(CO2) I(CH+) I(CGlu) Van Beelen et al., 1986 ACT –1 –2 2 2 4 qm,ACT,Glu,L,O2 CACT M(CGlu) M(CO2) I(CH+) I(CGlu) De Jong et al., 1988 4. Polyglucose storage STA –1 1 qm,STA,sto CSTA M(CGlu) I(CH+) I(CPgsta) Assumed STN –1 1 qm,STN,sto CSTN M(CGlu) I(CH+) I(CPgstn) Assumed ACT –1 1 qm,ACT,sto CACT M(CGlu) I(CH+) I(CPgact) Assumed 5. Anaerobic polyglucose conversion STA 1 2 1 3 –1 –1 qm,STA,Pgsta CSTA M(CPgsta) I(CH+) I(CO2) I(CGlu) Similar to process (2) STN 1 2 1 3 –1 –1 qm,STN,Pgstn CSTN M(CPgstn) I(CH+) I(CO2) I(CGlu) Similar to process (2) ACT 1 1 1 –1 3 1 –1 qm,ACT,Pgact CACT M(CPgact) I(CH+) I(CO2) I(CGlu) Similar to process (2) 6. Aerobic polyglucose conversion STA –1 2 2 4 –1 qm,STA,Pgsta CSTA M(CPgsta) M(CO2) I(CH+) I(CGlu) Similar to process (3) STN –3/2 2 1 1 4 –1 qm,STN,Pgstn CSTN M(CPgstn) M(CO2) I(CH+) I(CGlu) Similar to process (3) ACT –2 2 2 4 –1 qm,ACT,Pgact CACT M(CPgact) M(CO2) I(CH+) I(CGlu) Similar to process (3) 7. Lactate fermentation VEL –1 2/3 1/3 1/3 1/3 qm,VEL,Lac– CVEL M(CLac–) I(CH+) Seeliger et al., 2002 C j K I ,X , j . Subscripts stand for: S substrate, I inhibition, X bacterial species (i.e., STA, STN, VEL or ACT) and j chemical species (i.e., Glu, O2, Lac, H+, Pgsta, Pgstn, Pgact) M C j ; I C j K S,X , j C j K I ,X , j C j

Table 2.6 Rate parameters for biological processes Parameter name Symbol Value Reference Substrate specific uptake rate qm,STA,Glu,H 96.70 Van der Hoeven et al., 1985 –7 –1 –1 (10 mol s g ) qm,STA,Glu,L 5.00 Assumed as qm,STN,Glu,L qm,STA,Glu,L,O2 5.00 Assumed as qm,STN,Glu,L qm,STA,Pgsta,sto 15.6 Assumed as qm,STN,Pgstn,sto (a) qm,STA,Pgsta 1 Assumed 1/5 qm,STA,Glu,L

qm,STN,Glu,H 138 Van der Hoeven et al., 1985 qm,STN,Glu,L 5.00 Van der Hoeven et al., 1985 qm,STN,Glu,L,O2 5.00 Assumed as qm,STN,Glu,L qm,STN,Pgstn,sto 15.6 Hamilton, 1968 (a) qm,STN,Pgstn 1 Assumed 1/5 qm,STN,Glu,L

qm,ACT,Glu,H 22.00 Van der Hoeven and Gottschal, 1989 (b) qm,ACT,Glu,L 0.88 Assumed 1/25 qm,ACT,Glu,H qm,ACT,Glu,L,O2 11.8 Van der Hoeven and Gottschal, 1989 qm,ACT,Pgact.sto 1.40 Assumed based on Komiyama et al., 1986 and Komiyama and Khandelwal, 1992 (a) qm,ACT,Pgact 0.176 Assumed 1/5 qm,ACT,Glu,L qm,VEL,Lac– 252 Assumed based on Gerritse et al., 1992 and Seeliger et al., 2002

Substrate saturation constant KS,Glu,H 1220 Assumed based on Hamilton and Martin, 1982; (10–6 mol L–1) Van der Hoeven et al., 1985 and Dawes and Dibdin, 1986 KS,Glu,L 8.04 Hamilton and Martin, 1982 KS,O2 6.00 Van der Hoeven and Gottschal, 1989 KS,Lac– 290 Seeliger et al., 2002 KS,Pgsta 0.2×KS,Glu,H Assumed KS,Pgstn 0.2×KS,Glu,H Assumed KS,Pgact 0.2×KS,Glu,H Assumed

(c) Inhibition constant KI,STA,H+ 15.8 Assumed (10–6 mol L–1) (d) KI,STN,H+ 1.58 Assumed KI,ACT,H+ 1.58 Assumed as KI,STN,H+ KI,VEL,H+ 15.8 Assumed as KI,STA,H+

KI,STA,O2 0.2 Van der Hoeven and Gottschal, 1989 KI,STN,O2 0.2 Assumed as KI,STA,O2 KI,ACT,O2 0.2 Van der Hoeven and Gottschal, 1989

KI,STA,Glu 2200 Assumed based on Hamilton, 1968 KI,STN,Glu 2200 Assumed based on Hamilton, 1968 KI,ACT,Glu 2200 Assumed based on Hamilton, 1968

(e) KI,Pgsta 0.05 Based on Hamilton, 1968 (e) KI,Pgstn 0.05 Based on Hamilton, 1968 (e) KI,Pgact 0.05 Based on Hamilton, 1968 (a) Rate value decreased to also account for slow hydrolysis of polyglucose (b) Assumption based on the fact that qm,STA,Glu,H/ qm,STA,Glu,L ≈ 20 and qm,STN,Glu,H/ qm,STN,Glu,L ≈ 27 (c) Optimum pH for glycolysis in STA is 6 (Dashper and Reynolds, 1992). The reaction is assumed to be half inhibited at a pH value smaller with 1.2 units than the optimum one. (d) Optimum pH for glycolysis in STN is 7 (Hamilton, 1968). The reaction is assumed to be half inhibited at a pH value smaller with 1.2 units than the optimum one. (e) Microorganisms cannot store more polyglucose than 50% their cell dry weight.

Besides the glucose taken up during meals, the model considers also a continuous low-rate production of glucose in the saliva. This process, mimics production of carbohydrate substrates (e.g., by mucin biodegradation) that feeds the resting plaque leading to a slightly acidic environment. Storage compounds During periods of glucose abundance Streptococcus and Actinomyces are storing substrate in the form of a glucose polymer, which we call here generically polyglucose. Polyglucose storage is a process inhibited at low pH (4.5) and the maximum concentration of polyglucose that can be stored equals half of the cells dry weight (Hamilton, 1968). When the glucose becomes depleted in the environment, bacteria consume the polyglucose reserves using the high affinity pathway. The model assumes the same stoichiometry for polyglucose conversion as for the equivalent low glucose concentration anaerobic and aerobic processes.

Lactate fermentation This process is performed by Veillonella, and the overall expected effect is a reduction in the local acidity values, due to the lower acidity of the reaction products (acetate, propionate and bicarbonate) than reactant acidity (lactate).

(b) Chemical reactions Diverse chemical reactions can take place in the saliva, in the plaque or at the tooth surface. The model accounts for acid-base and complexation equilibria involving both mobile chemical components and fixed surface species; these reactions are treated volume-based. In addition, the enamel dissolution takes place at a solid-liquid interface, being thus treated as a surface-based reaction.

Acid-base and complexation equilibria These equilibria are used to calculate the pH in any point of the computational domain, whether plaque or saliva. There are equilibria for the mobile dissociable chemical species, but also for the fixed charged groups on the bacterial surface. The equilibria included in the model are presented in Table 2.7 and Table 2.8. All the equilibrium processes are considered to be very fast, therefore their both forward and backward reaction rate constants are assigned very large values.

Dissolution reaction Enamel demineralisation occurs according to the stoichiometry and with the reaction rate proposed by Margolis and Moreno (1992). The dissolution rate takes place when the degree of saturation (DSHAP) of solution in contact with the tooth is less than 1, i.e., when the solution is undersaturated in respect to hydroxyapatite (HAP). Because the concentrations of phosphate 34

3 – ( PO4 ) and hydroxyl (HO ) ions are very low compared to those of other species, this would introduce numerical instability (scaling) in the model. These concentrations have been replaced by other species via chemical equilibria (CPO4–3 = Ke,Pho2– · CPho2– /CH+ and CHO- 5 3 –4 = Ke,H2O/ CH+), leading to the ionic product IPHAP = (CCa2+) (CPho2–) (CH+) and the degree of 1/9 saturation DSHAP = (IPHAP·KS,HAP(en)) . The complete rate is given in Table 2.7.

Table 2.7 Acid-base equilibria stoichiometry and rate expressions

Reaction Stoichiometry Reaction rate

Water dissociation H2O ' HO– + H+ re,H2O = kH2O [1 – (CH+ CHO–)/Ke,H2O] Lactic acid dissociation LacH ' Lac– + H+ re,LacH = kLacH [CLacH – (CH+ CLac–)/Ke,LacH] Formic acid dissociation ForH ' For– + H+ re,ForH = kForH [CForH – (CH+ CFor–)/Ke,ForH] Acetic acid dissociation AceH ' Ace– + H+ re,AceH = kAceH [CAceH – (CH+ CAce–)/Ke,AceH] Propionic acid dissociation ProH ' Pro– + H+ re,ProH = kProH [CProH – (CH+ CPro–)/Ke,ProH] Succinic acid dissociation SucH ' Suc– + H+ re,SucH = kSucH [CSucH – (CH+ CSuc–)/Ke,SucH] Hydrogen succinate dissociation Suc– ' Suc2– + H+ re,Suc– = kSuc– [CSuc– – (CH+ CSuc2–)/Ke,Suc–] Carbonic acid dissociation CO2 + H2O ' HCO3– + H+ re,CO2 = kCO2 [CCO2 – (CH+ CHCO(3)–)/Ke,CO2] Dihydrogen phosphate dissociation Pho– ' Pho2– + H+ re,Pho– = kPho– [CPho– – (CH+ CPho2–)/Ke,Pho–] Dissociation cationic sites SH2+ ' SH + H+ re,SH2+ = kSH2+ [CSH2+ – (CH+ CSH)/Ke,SH2+] Dissociation neutral sites SH ' S– + H+ re,SH = kSH [CSH – (CH+ CS–)/Ke,SH] Dissociation calcium complex SCa+ ' S– + Ca2+ re,SCa+ = kSCa+ [CSCa+ – (CCa2+ CS–)/Ke,SCa+] 2.8 0.3 Hydroxyapatite (HAP) dissolution Ca5(PO4)3OH + 4H+ rd,HAP = ken (1–DSHAP) (∑CA(i)H) , 5Ca2+ + 3Pho2– + H2O i – acids 1/9 DSHAP = (IPHAP · KS,HAP(en) ) 5 3 4 IPHAP = [(CCa2+) (CPho2–) ] / (CH+) 3 KS,HAP(en) = [(Ke,Pho2–) Ke,H2O] / KS,HAP ∑CA(i)H = CCO2 + CPho– + CLacH + CAceH + CForH + CProH + CSucH + CSuc– –3 –1(b) ken = 0.42 u 10 min –12.35 –1 Ke,Pho2– = 10 mol L –55 9 –9(c) KS,HAP = 5.5 u 10 mol L (b) (Margolis and Moreno, 1992) (c) (Moreno and Zahradnik, 1974)

Table 2.8 Rate parameters for acid-base equilibira Reaction Reaction rate Acidity Reference constant(a) constant(b) (c) 7 –14 Water dissociation kH2O = 10 Ke,H2O = 10 Atkins and de Paula, 2009 7 –3.86 Lactic acid dissociation kLacH = 10 Ke,LacH = 10 Atkins and de Paula, 2009 7 –3.75 Formic acid dissociation kForH = 10 Ke,ForH = 10 Atkins and de Paula, 2009 7 –4.76 Acetic acid dissociation kAceH = 10 Ke,AceH = 10 Atkins and de Paula, 2009 7 –4.87 Propionic acid dissociation kProH = 10 Ke,ProH = 10 Atkins and de Paula, 2009 7 –4.20 Succinic acid dissociation kSucH = 10 Ke,SucH = 10 Martell and Smith, 1976 7 –5.63 Hydrogen succinate dissociation kSuc– = 10 Ke,Suc– = 10 Martell and Smith, 1976 7 –6.35 Carbonic acid dissociation kCO2 = 10 Ke,CO2 = 10 Atkins and de Paula, 2009 7 –7.21 Dihydrogen phosphate dissociation kPho– = 10 Ke,Pho– = 10 Atkins and de Paula, 2009 7 –3.45 Dissociation cationic sites kSH2+ = 10 Ke,SH2+ = 10 Hong and Brown, 2006 7 –4.62 Dissociation neutral sites kSH = 10 Ke,SH = 10 Hong and Brown, 2006 7 –3 Dissociation calcium complex kSCa+ = 10 Ke,SCa+ = 10 Assumed based on Rose et al., 1993 (a) Assumed an arbitrarily high value for very fast equilibria. (b) The values of acidity constants are given at 25 °C. The variation with temperature from 25 °C to 37 °C was neglected. (c) Units for rate constants are [s-1] and for acidity constants are [mol/L], except for the water dissociation which is measured in [mol2/L2]. 35

Cf(t) Qf Qf Vf Cs(t) Cf Saliva film

NpA ● Cf,i(t)

Lp● ● ● ● ● ● x ● C (t, x) ● p,i ● Φ(t, x) ● ● 0 ● ■ Tooth MODEL COMPARTMENTS MODEL AND LINKS VARIABLES

Figure 2.1. The model computational domains. Saliva is represented by two ideally mixed compartments (bulk and film on top of the dental plaque) between which there is an exchange flow rate Qf. The saliva film is perfectly mixed, having the same concentration Cf of the components over the entire volume Vf. The saliva film is exchanging components with the dental plaque, with the flux Np·A. If in the entire saliva domain (bulk and film) the components concentration (Cs, respectively Cf) were varying only in time, in the plaque domain the component’s concentration Cp is varying in time as well as in space (over the plaque length Lp). On the right side of the figure is shown schematically the one-dimensional representation of the model: the saliva and tooth domains are represented as a boundary conditions and the plaque domain is represented as a line perpendicular to the tooth surface. The state variables for dental plaque domain (concentrations of chemical components, Cp(t,x) and the electric potential, Φ(t,x)) are calculated in each point of the domain (depending on the domain discretization).

2.1.3. Model domains We are interested in the rate of tooth demineralisation, which depends on the activity (here simplified as concentration) of several chemical species at the tooth surface. These concentrations, however, can be very different from those in saliva, being the result of (bio)chemical reactions and transport processes. Because both the presence of certain reactions and transport mechanisms are different in the dental plaque and saliva, we separate these two domains (see Figure 2.1). For each domain, a set of chemical and/or microbial species is defined, as well as a set of associated processes. Further, solving the mass balances for each chemical species in each of the domains will give the time-dependent concentrations of all chemical species. In the following sections detailed descriptions of every compartment will be provided.

(a) Saliva The plaque is in direct contact with a thin film of saliva present on the teeth, communicating with saliva that has not come yet into contact with the plaque. This volume of saliva not yet in contact with the plaque is named in this model “bulk saliva”. The fact that saliva film and bulk 36 saliva are into permanent contact, is leading to different concentrations in these two volumes. We calculate in this model only the change in concentrations of chemical species in the saliva film, Cf,j (t). The saliva film represents only a small volume of the total saliva present in the mouth. By assuming complete mixing in the saliva film, the mass balances for each mobile chemical species j (from Table 2.3) in the saliva film f take the general form (2.1):

dCfj, Q f A f CCsj,,jfj fj, Npj , R fj , (2.1) dtVV V V ffnetne reaction accumulation rate rate exchange with exchange the saliva bulk with plaque –3 allowing for calculation of Cf,j, the concentration of species j in the saliva film (mol m ) at each time t. The mass balance (2.1) is based on the species exchange with the saliva bulk represented by the input (Qf Cs,j / Vf) and output (Qf Cf,j / Vf) terms. There is also exchange of chemical species with the dental plaque (Np,j Af/Vf), with the flux Np,j calculated as the diffusive flux at the plaque-saliva interface (x = Lp), Dj ∂Cp,j / ∂x. Qf is the volumetric flowrate of saliva film, Vf is the volume of saliva film, Af is the area of saliva-plaque interface and Cs,j is the concentration of species j in the saliva bulk. The saliva film compartment is characterized by a halving time th,f = 0.5 min (Dibdin, 1990b) giving the residence time 4 –1 Qf / Vf = ln(2) / th,f. The plaque area per saliva volume ratio was Af / Vf = 10 (m ) implying that the salivary film thickness is Vf / Af = 100 Pm (Fejerskov and Kidd, 2008). A series of reactions lead to the net reaction rate of each component, Rf,j. First, there is a constant production of glucose at very low rate in the saliva, used here to represent small amounts of carbohydrates (e.g., fucose and sialic acid) that originate from glycoproteins such as mucin (Marsh et al., 2009). Even if specifically glucose is not produced from glycoproteins, these carbohydrates are metabolizable by certain Streptococci and Actinomyces, which could explain the slightly acidic pH (~ 6.5) of resting plaque. The value of glucose production rate in saliva used in this model, 0.02 (mol m−3 s−1), was calculated so that the resting pH will have the value reported in the literature (Marsh et al., 2009; Fejerskov and Kidd, 2008). Second, all the mobile components considered in the model are present in saliva, therefore acid-base equilibria are included (Table 2.7) with kinetic parameters presented in Table 2.8.

The input concentrations of mobile chemical species Cs,j are set in several ways. For species involved in acid-base equilibria (lactate, acetate, formate, propionate, succinate, carbonate, phosphate and hydroxyl), a total concentration is given, Cs,j,tot, from which the equilibrium concentrations of each form are calculated (Table 2.3). The pH of secreted saliva is also set, thus Cs,H+ is known. Chemical species existing in only one form (oxygen, ethanol, chloride anion and calcium) have a fixed concentration (Table 2.3). The charge balancing 37 cation (e.g., potassium) concentration is calculated from an electroneutrality condition, which must be fulfilled at any moment both in the input and in the film of saliva. (2.2) ¦ zCjsj, 0 j In order to achieve electroneutrality, two extra species are introduced in the system: cations

(K+) and anions (Cl−). While the concentration of anions Cf,Cl- is fixed, the concentration

Cf,K+ is calculated from equation (2.2) based on the concentrations of all other ions present in the saliva film. For glucose, there are different types of input concentrations. In the feeding regime, Cs,Glu was imposed to increase from zero to a maximum value of Cs,Glu,max = 500 mM within tstep = 10 s using a step function. This maximum concentration was maintained during the entire feeding period (tfeed = 2 min). At the end of the feeding period the clearance begins, where the glucose concentration starts to decrease exponentially: ªºQ CtCtts,, Glu() s Glu ( step feed )exp«» ttt ( step feed ) (2.3) V ¬¼s

Given the halving time in the saliva bulk compartment th,s = 2 min (Dibdin, 1990b), the residence time can be expressed as Q/Vs = ln(2)/th,s. The glucose concentration in the bulk saliva for one feeding-clearance cycle is presented in Figure 2.2a. The initial concentrations for equations (2.1) are equal with the input concentrations

Cf,j(0) = Cs,j. The initial glucose concentration is set to a low value (Table 2.3).

(b) Dental plaque In the plaque domain are present all microbial groups (Table 2.2), all mobile chemical species (Table 2.3) and the fixed (immobile) chemical species (Table 2.4). Because microbial growth is not considered, the concentrations of microorganisms remain constant in time and have the same value throughout the plaque. The concentrations of chemical species, j, however, will change in time and also along the plaque depth, Cp,j(t,x) (see Figure 2.1).

Domain equations Within the one-dimensional plaque domain two types of material balances were defined: for the mobile (solute) species and for the fixed chemical species. The nj balances for mobile species (2.4) are Nernst-Planck equations for transport of dilute species by molecular diffusion and ion electro-migration, including also a reaction term (Newman, 1991):

2 wwCCzFpj,, pj j ww)§· DDCRj jjpj ¨¸pj ,,, pj (2.4) wwwwttRTxxww©¹x2 RTww xxx©¹x netn reaction accumulationcumulation diffusion ion migration rate rate rate rate 38 where Dj is the diffusion coefficient of solute j in the plaque (at 37 ºC and constant in space –1 and time), zj charge number, F Faraday’s constant (96485 (C mol )), R universal gas constant (8.314 (J mol–1K–1)) and T is the temperature (constant 310 K = 37 ºC). Because immobile chemical species are either fixed on the surface of bacteria or inside them (and bacteria are also considered immobile within the plaque) their ni material balances do not contain any transport terms: dC pi, R (2.5) dt pi, accumulation net reaction rate rate

With Cp,i(t,x) being the concentration of immobile chemical species i at a certain point x in the plaque domain (mol m−3). The additional state variable )(t, x) in equation (2.4) is the potential field developed due to the different ion transport rates. Calculation of potential necessitates the introduction of an electroneutrality condition including both mobile and fixed charges:

¦ zCjpj, 0 (2.6) j

Coupling equations (2.4), (2.5) and (2.6) will result in a system of (nj+ni+1) equations with nj+ni unknown concentrations and one unknown potential.

Initial conditions The initial values of each mobile component concentrations in any point of the plaque domain are equal with those in the initial saliva composition, Cp,j(0) = Cf,j(0). The initial values for storage compounds are all zero. For the surface charges, the initial concentrations result from the acid-base equilibria, given the total surface charge concentration (Table 2.4). Like in saliva, the balancing mobile cations in the plaque result from a charge balance including not only the mobile but also the fixed charged species.

Boundary conditions To the set of equations (2.4) two boundary conditions are attached. For the saliva-plaque interface

(at x = Lp) the concentrations of all mobile chemical components are equal to those in the saliva film. At the plaque-tooth interface, flux conditions are implemented for all mobile species, where the molar flux due to diffusion and migration is:

wCzFw) −2 −1 ND pj, DC j (mol m s ) (2.7) pt,, jjjpjjjj wwxRTxjjpj p, j x diffusion ion migration flux flux The molar flux of species involved in the dissolution reaction (j = H+, Ca2+, Pho2–) equals their net formation/consumption rate due to the HAP dissolution reaction (rate rd,HAP and 39 stoichiometric coefficient Qj according to Table 2.7) calculated with concentrations at the tooth surface (Cpt,j): NrCCC Q ,, (2.8) pt,,,,2,2 j j d HAP pt H pt Ca pt Pho

For the rest of chemical species Npt,j = 0, i.e., these do not penetrate into the tooth. In addition, the electric potential is set to a reference value at the saliva-plaque interface (Φ = 0), while an electrical insulation condition is applied at the tooth surface (∂Φ / ∂x = 0).

2.1.4. Model solution The model was implemented in COMSOL Multiphysics software (COMSOL 4.1, Comsol Inc, Burlington, MA, www.comsol.com), which allows a very flexible and well-structured model construction and solves the resulting systems of partial differential equations by finite elements method. The plaque domain has been discretized using a uniform mesh with 5 Pm size. The following solution strategy was used. First, from the initial condition the time- dependent equations were solved until no more changes occurred in concentrations of mobile species in saliva and plaque – a steady state representing a situation of resting plaque, typically encountered after maximum 2 × 104 s. Second, the simulations continued from the steady state (initial time set here to 0) for another period of 2 × 104 s. Output from the time dependent solver was taken at intervals of 10 s in the first 2000 s (i.e., during the glucose pulse disturbance and until the system roughly recovered from it) followed by intervals of 100 s until the end of the simulation. The integration time step was automatically adjusted by COMSOL so that the relative tolerance of 10–4 and absolute tolerance of 10–5 were satisfied.

2.2. Results and discussion Results of the 1D plaque model were first analysed for a chosen standard case. Thereafter, the influence of different factors such as oral hygiene (e.g., plaque thickness) and eating habits on the pH change in time and tooth demineralisation were studied.

2.2.1. Standard case The standard case was chosen to correspond with common situations created in clinical studies (Tanzer et al., 1969; Dong et al., 1999; Pearce et al., 1999). The dental plaque had a thickness of 500 Pm. A high glucose concentration (0.5 M) is maintained for 2 min in saliva (glucose pulse), followed by oral clearance with a halving time of 2 min. 40

(a) (b) 14 total acetic 500 bulk saliva film saliva 12 total lactic tooth total formic 10 total propionic 400 total succinic 8 300 6 200 4

[mM] concentrations acid 100 2 [mM] concentration glucose 0 0 0 50 100 150 0 50 100 150 time [min] time [min] 7 (c) (d) 20 total acetic 6 pH 18 Dental plaque lacking in Veillonella total lactic 16 total formic total succinic 5 14

12 DS 4 10 8 m Demin.rate x 106 -2 -1 3 [mg Ca mm min ] 6

[mM] concentrations acid 4 2 2 C [mM] Ca 0 0 50 100 150 1 time [min] C [mM] HPO4-2

0 0 50 100 150 time [min] Figure 2.2 (a) Glucose concentration profile in the saliva bulk, saliva film and at the tooth surface (i.e., tooth- plaque interface). (b) Total (anionic and protonated) acetate, lactate, formate, propionate and succinate concentrations in time at the tooth-plaque interface. (c) Total (anionic and protonated) acetate, lactate, formate, propionate and succinate concentrations in time at the tooth-plaque interface for a dental plaque composition lacking in Veillonella. (d) Variation in time, at the tooth-plaque interface of the main factors influencing tooth demineralisation: pH, degree of saturation (DSHAP), calcium and hydrogen phosphate concentrations and demineralisation rate.

Glucose profiles The imposed glucose pulse in the saliva bulk, together with the calculated glucose concentrations in the saliva film and in the plaque at the tooth surface are represented in Figure 2.2a. The glucose in the bulk saliva is cleared first (~15 min), while in the saliva film the glucose remains for a longer time (~25 min). At tooth level glucose concentrations start to decrease only after 10 min. Even if the plaque was exposed for only 2 min to high glucose concentration, glucose is present in the dental plaque for 25 min. This underlines the importance of mass transport processes (i.e., diffusion) in dental plaque when analysing caries formation.

Acid production The effect of glucose presence in the microbial film is the formation of organic acids. The change in concentration of acids included in the model (acetic, lactic, formic, propionic and 41 succinic) over time at the tooth surface is illustrated in Figure 2.2b. When considering the total concentrations of acids (sum of anionic and protonated forms) in fermenting plaque, the propionic acid (resulting from lactate consumption by Veillonella) is dominant, followed by acetic acid (resulting in this case mainly from lactate consumption) and lactic acid (from glucose/polyglucose fermentations). However, when only the anionic forms are considered, lactate is by far present in the highest concentration. The acid dominance in the resting plaque agrees with the results from Borgström et al. (2000) where acetate is the main component, followed by propionate and very low amounts of lactate (see Figure 2.2b, time greater than 100 min). The total concentration of lactate/lactic acid present in the system follows the glucose pulse, reaching its maximum soon after the glucose concentration is at its highest value (around 6 min). Once the glucose concentration starts to decrease the lactate production is decreasing as well, ceasing completely once the glucose was cleared from the saliva (after ~35 min). Without production, lactate will only diffuse out of the dental plaque or be consumed by the Veillonella. Noticeably, even after glucose depletion lactate is still present in the plaque due to the diffusion time needed, thereby maintaining low pH values. For a dental plaque composition without Veillonella (Figure 2.2c), there is much more lactic acid present in the fermenting plaque (i.e., maximum 18 mM compared to 8 mM) and also for more extended period of time (i.e., ~50 min compared to ~35 min). In addition, the lactic acid is dominant in the resting plaque without Veillonella. In contrast, when Veillonella was consuming lactate (Figure 2.2b), the lactate concentration was already negligible when glucose had been depleted at the tooth surface. Experimental observations (Bowden, 2000) also show that Veillonella’s presence in a plaque has a protective effect due to the lactic acid consumption. Formic and succinic acids (Figure 2.2b) are produced at the beginning and at the end of the feeding pulse when the glucose concentrations are low. Once the glucose is cleared, the microorganisms start to consume the stored polyglucose. Products of polyglucose degradation include mainly acetic and formic acids, whose concentrations increase after the glucose depletion. The succinate production (Figure 2.2b) is very low and it only reaches negligible concentrations. pH curves The current mathematical model can reproduce the experimental Stephan curves (pH change in time) reported in the literature (Aamdal-Scheie et al., 1996; Zaura and ten Cate, 2004; Deng and ten Cate, 2004). The model describes the three main areas of the pH curve (Figure 2.2d), 42

7 7 no storage p 6.5 6.5

m standard 6 6 standard o 5.5 5.5 pH pH

5 m no Veillonella 5

4.5 4.5

4 4 0 50 100 150 0 50 100 150 time [min] time [min] Figure 2.3 Influence over the pH profile at the tooth- Figure 2.4 Influence of storage compounds over the plaque interface of a plaque where Veillonella is not pH profile at tooth surface present, compared to the standard case as observed experimentally in caries active subjects. First, the steep pH decrease at the beginning of the pulse is due to the conversion of glucose into acids. Second, a plateau area corresponding to minimum pH between 4.5 and 5 arises because of the pH inhibition on microorganisms. Finally, once most of the glucose is consumed, the pH restores to the steady state level (pH 6.5) of resting plaque. Noticeably, during the restoration period there is a short pH plateau. This second plateau corresponds to acids production from the stored polyglucose. Veillonella’s presence in dental plaque increases the minimum pH (Figure 2.3) due its ability to consume lactic acid. Subsequently, this also reduces the time spent under critical pH. The role of storage polymers can be shown by comparing pH profiles in a plaque with and without glucose storage (Figure 2.4). The consumption rate of storage compounds is very low when pH < 5.5 and reaches the maximum only when pH > 5.5. In consequence, storage compounds appear to have little impact on the amount of demineralised calcium because the resulted acids are mostly produced at pH > 5.5.

Enamel demineralisation

The degree of saturation (DSHAP) of plaque solution (at the tooth surface) in respect to HAP is the main factor affecting tooth demineralisation rate. Furthermore the DSHAP is determined by pH and concentrations of calcium and hydrogen phosphate ions, all represented in

Figure 2.2d. The sharp decrease of pH is immediately correlated with the DSHAP fall and with a sudden increase in the demineralisation rate triggering the increase in the calcium ion concentration. Model simulations show that the calcium concentration can restore to its initial level soon after the pH increases above 5.5. This is the pH value recognized to be critical for tooth enamel demineralisation (Schmidt-Nielsen, 1946). Consequently, when pH is higher than 5.5 the demineralisation stops. Importantly, the pH remains in the critical range for

7 6.5 6 5.5 5 pH 4.5

4 standard no demin 3.5 no pH inhib

3 0 50 100 150 200 250 300 time [min] Figure 2.5 pH profiles at the tooth surface for the standard case, and two other systems similar to the one described in the standard case, but where there is no demineralisation, respectively no pH inhibition demineralisation long after the glucose pulse has been cleared. While glucose is cleared in about 20 min, it takes 40 min for the pH to increase above the critical level for demineralisation and 140 min to restore to the steady state (resting) value of 6.5. Overall, the amount of demineralised calcium after one acid attack obtained with this model (10-4 (mgCa mm–2)) is in the range experimentally observed by Margolis and Moreno (1992).

Effect of phosphate release on pH A common assumption is that the plateau area of minimum pH is due to the buffering effect of the phosphate resulting from the demineralised HAP (Loesche, 1986). In order to evaluate this hypothesis, two situations have been tested, with and without demineralisation. In addition, a case without pH inhibition on microbial activity is also represented in Figure 2.5. The model indicates that the buffering effect due to the demineralisation is minor, i.e., it does not change significantly the shape of Stephan curve or the minimum pH. Conversely, without acid inhibition of microbial activity the pH is decreasing abruptly not to 4.5 but further down to about 3. The pH curve has also a different shape in this case, with sharp changes in slope when the low glucose mechanism is active. Most importantly, there is no plateau area at the minimum pH. The slower pH restoration corresponding to storage compounds consumption is more extended because the polyglucose storage was not inhibited by pH either. We believe that the minimum pH plateau is better explained in the current model by pH inhibition of microbial activity and not by the buffering effect of tooth demineralisation products (e.g., phosphates).

(a) (b) 250 m enamel saliva o 9 8 200 5 3 7 5 3 6 1 150 9 12 5 9

12 4 100 20 3 1 0.5 50 0.5 2 glucose concentration [mM] concentration glucose

20 total lactate concentration [mM]1 0 0 0 100 200 300 400 500 0 100 200 300 400 500 plaque length [Pm] plaque length [Pm] (c) (d) 6 7

C (20 min) 6 PO4 total 5.5 0.5 C (0 min) 5 PO4 total

5 1 4 pH

3 3 5 C (20 min) 4.5 9 [mM] concentration Ca 12 20 2 C (0 min) Ca 4 1 0 100 200 300 400 500 0 100 200 300 400 500 plaque length [Pm] plaque length [Pm]

Figure 2.6 Concentrations of solutes and pH along the dental plaque. Plaque length represents here the distance from enamel (x=0) to the plaque surface (x=Lp=500 Pm). (a) Glucose concentration profiles at different time steps: at the beginning of glucose pulse (0.5 min), at its maximum (1 min) and during the oral clearance (3 min, 5 min, 9 min, 12 min, 20 min) (b) Total lactate concentration profiles at different time steps (0.5 min, 1 min, 3 min, 5 min, 9 min, 12 min, 20 min) (c) pH profiles at different time steps (0.5 min, 1 min, 3 min, 5 min, 9 min, 12 min, 20 min) (d) Concentration profiles of species directly influenced by the demineralisation process: calcium and the total phosphate present in the system. The profiles are represented at the initial time of the simulation, 0 min (i.e., steady state) and at a time step (20 min) during the minimum pH period when the rate of demineralisation is at its maximum.

Solute gradients in plaque Unlike for the other species present in the system, the glucose concentration gradients can be very strong in the plaque especially at the beginning of glucose pulse (Figure 2.6a). In the first minutes of the pulse, glucose is diffusing from the saliva towards the tooth surface while being consumed by the microorganisms. After 5 min from the beginning of the pulse the glucose concentration in the plaque becomes almost constant and it reached the maximum value at tooth surface. Then, because of salivary clearance creating low concentrations outside plaque, glucose starts to diffuse out the plaque while still being consumed by microorganisms. The total lactate and glucose concentrations in the plaque are, as expected, correlated (see Figure 2.6a and Figure 2.6b). Almost all the glucose is converted into lactate through the 45

-4 (a) (b) x 10

7 ] 2 100 Pm 6.5 2.5 1000 Pm 250 Pm 6 2 500 Pm

5.5 1.5 pH 1000 Pm 500 Pm 5 1 250 Pm 0.5 4.5 100 Pm

[mg/mm Ca demineralized of mass 4 0 0 50 100 150 0 50 100 150 time, [min] time [min] (c) 7

0.1*a Figure 2.7 (a) pH profiles at the tooth surface for 6.5 different plaque thicknesses. a (b) Total amount of calcium demineralised in time 6 measured at the tooth surface (i.e., tooth-plaque 5.5 interface) for the plaque thicknesses studied pH (c) pH profiles at the tooth surface for different 5 plaque areas (a=Af/Vf standard conditions, 0.1*a = 10% of the tooth surface covered with plaque). 4.5

4 0 50 100 150 time, [min] high glucose fermentation pathway, as glucose penetrates in the plaque. Lactate will still be present at the tooth surface even after the glucose has been consumed (Figure 2.6b). The pH value remains almost constant over the entire length of the dental plaque, except for the time at the beginning of the glucose pulse (0.5 min) when the pH value is decreasing drastically (Figure 2.6c). It is relevant to compare the levels of calcium and total phosphate concentration in the resting plaque (i.e., in steady state at 0 min) and at a time during the minimum pH period (20 min), shown in Figure 2.6d. In steady state (i.e., resting plaque), the concentrations are constant over the entire plaque depth. However, when demineralisation rate is strong the calcium and phosphate concentrations are increasing at the tooth-plaque boundary and form a gradient from the tooth surface to the saliva.

2.2.2. Plaque thickness and area This case study is relevant from the perspective of oral hygiene, because it could be related to tooth brushing habits. The impact of plaque thickness on the cariogenic potential is not fully understood since a variety of other factors (e.g., amount and frequency of sugars consumption, microbial composition of the plaque) are also involved. The plaque model was used to evaluate the effect of plaque thickness on the sugar induced acidification and subsequent tooth demineralisation. The pH curves were simulated with plaque thicknesses of 100 Pm, 250 Pm, 500 Pm, and 1000 Pm.

In general, the thicker the plaque the larger the negative effect of acidification. The same glucose pulse (Figure 2.2a) resulted in very different pH profiles in plaque with different thicknesses (Figure 2.7a). Thin plaque produced less acidification, while the pH at the tooth surface with thicker plaque reached lower values. Most importantly, not only the pH is lower but the period of critical pH is more extended for the thicker plaque. Thicker plaque progressively results in increased tooth dissolution (Figure 2.7b). The calculated amount of demineralised calcium increases with the time when the plaque pH remains under the critical value of 5.5. In conclusion, this model shows a quasi-linear dependency between the mass of demineralised calcium and dental plaque thickness. For thick plaque (i.e., 500 Pm in this case) the minimum pH at the beginning of the acid attack period (e.g., 1 min, Figure 2.6c) is not reached at the tooth surface, but rather closer to saliva. This suggests that for even thicker plaque (1000 μm) a secondary effect could diminish the cariogenic potential, that is, glucose consumption in shallow plaque layers. Similar model results were reported by Dibdin (Dawes and Dibdin, 1986). However, the overall impact is minor, because during most of the acid attack the pH at tooth surface is < 5.5, therefore leading to demineralisation. The tooth area covered by plaque can also have a strong impact on the Stephan curve.

If only 10% of the tooth surface develops plaque of 500 Pm (a = 0.1Af/Vf ), then the amount of acids produced is lower. Although the pH minimum is only slightly higher than in the standard case, the period of acid attack is significantly shorter (Figure 2.7c). At the same time, the resting plaque pH is also higher for smaller plaque coverage area: 6.7 instead of 6.5 in the standard case.

2.2.3. Drinking habits This case study shows the importance of sugar intake patterns on the pH profiles, when the same amount of sugar is ingested. For this purpose three hypothetical regimes have been evaluated, all based on the consumption of a high (0.5 M glucose) sugar containing soft drink. The 0.5 M glucose concentration used is approximately the sugar content in most of the commercialized soft drinks, also referred to as “liquid candies” by nutritionists. The regimes were defined as follows: (1) Thirsty: a glass of sugar containing drink (250 mL) consumed in 15 s drinking continuously, (2) Short-Sipping: the same sugar containing drink consumed every minute, i.e., in 5 equal portions for 3 s (5×50 mL) at intervals of 57 s, and (3) Long- Sipping: the same sugar containing drink consumed every 10 min, i.e., in 5 portions for 3 s (5×50 mL) at longer intervals each (597 s).

(a) (b)7

500 Long-Sipping Short-Sipping 6.5 Thirsty Thirsty 400 6 Long-Sipping 300 5.5 Short-Sipping pH

200 5

100 glucose concentration [mM] concentration glucose 4.5

0 0 50 100 150 4 time [min] 0 50 100 150 -4 (c) x 10 time [min]

] 2 Figure 2.8 (a) Variation of glucose concentration 2.5 Long-Sipping from bulk saliva in time corresponding to the three drinking patterns studied: Thirsty, Short-Sipping 2 and Long-Sipping

1.5 (b) pH profiles in time for the studied drinking Short-Sipping patterns: Thirsty, Short-Sipping and Long-Sipping 1 (c) Total amount of calcium demineralised in time Thirsty for the studied drinking patterns: Thirsty, Short- 0.5 Sipping and Long-Sipping

mass of demineralized Ca [mg/mm Ca demineralized of mass 0 0 50 100 150 time [min] The input glucose profiles in saliva are shown in Figure 2.8a. In both Short-Sipping and Long-Sipping cases the maximum value of glucose concentration is the same for each sip because the diluting effect of the saliva already present in the mouth was neglected. This approximation is justified because the volume of saliva present in the mouth, ca. 1 mL (Lagerlöf and Dawes, 1984), is very low compared to the 50 mL of one sip of drink. Overall, the consumption of the sugar-containing drink over extended periods of time leads to longer times of tooth exposure to low pH. The calculated pH variations at the tooth surface in different drinking patterns for the 500 μm thick plaque are shown in Figure 2.8b. Interestingly, irrespective of drinking regimes the pH minima were all found to be around 4.5, due to the pH inhibition on microbial conversions (i.e., below pH 4.5 there is no more production of acids). However, clear differences were observed when looking at the time the pH remained under the critical value for demineralisation. In the case of drink consumption every 10 min (Long-Sipping), the pH remained at its minimum value during the entire consumption time. Sufficient acids were generated and remained at the tooth surface in between the consumption events, even though the glucose concentrations varied depending on the sip intervals.

(a) 25 (b)20 c x 0.1 c x 0.1 glucose glucose c c 20 polyglucose polyglucose 3 rx x 103 15 rx x 10 polyglucose storage polyglucose storage

15 10

5 5

0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 time [min] time [min] Figure 2.9 Glucose concentration, total polyglucose concentration stored by all bacterial groups, and the rate of polyglucose storage by all bacterial groups at tooth surface for two studied drinking patterns (a) Short-Sipping (b) Long-Sipping

By correlating the observations regarding the pH profiles with the calculated amount of calcium demineralised (Figure 2.8c) it appears that the highest amount of calcium is lost in the extended drink consumption (Long-Sipping), when the time with pH<5.5 at tooth surface is the longest (~75 min). These model calculations indicate that drinking a glass of sugar- containing liquid in small sips over an extended period of time has a higher cariogenic effect than drinking it at once. Furthermore, the longer the break between sips, the greater is the damage inflicted on the tooth. Noticeably, the amount of demineralised calcium during extended drinking time (Long-Sipping case) is the same with the amount lost for thick dental plaque (1000 μm), suggesting that a person with a good oral hygiene but bad sugar consumption habits is subjected to the same risks as a person with poor oral hygiene. Polyglucose storage is limited by the length of the period with glucose present in the plaque, rather than by glucose concentration. In both Short-Sipping (Figure 2.9a) and Long- Sipping (Figure 2.9b) cases, at the time when glucose is depleted in the plaque (~30 min, respectively ~1 h) polyglucose storage also ceases. In fact, considering the assumption that microorganisms can store polyglucose up to 50% of their weight (Hamilton, 1968), and considering that in the current model the total concentration of microorganisms is 80 (kg m–3), it means the theoretical maximum concentration of glycogen of 40 (kg m–3) was not reached not even for the social drinking case Long-Sipping. The difference in the amount of stored polyglucose is reflected in the extent of the second plateau of the pH curve corresponding to the polyglucose consumption (Figure 2.8b). The plateau is longer in the Long-Sipping case compared to Short-Sipping because of higher amounts of polyglucose available for consumption. 49

7 7 6.5 6.5 6 6 5.5 5.5 pH pH 5 5 100 O standard 2 4.5 100 monosp STN 4.5 500 O 4 monosp STA 2 monosp ACT 500 4 3.5 0 50 100 150 0 50 100 150 200 250 300 time [min] time [min] Figure 2.10 pH profiles of a strictly anaerobic plaque Figure 2.11 Comparison between the standard (similar to standard case but without aerobic glucose case and three equivalent monospecies systems of conversion) for two plaque thicknesses (100 Pm and aciduric Streptococcus (STA), non-aciduric 500 Pm) compared to the pH profiles obtained in the Streptococcus (STN) and Actinomyces, standard case and a case with a 100 Pm plaque thickness. respectively.

Johansson et al. (2004) studied experimentally the influence of different drinking patterns on the pH at tooth surface by using an acidic drink. Their work refers to tooth erosion (i.e., dental plaque was not present) and the conditions are not identical with those used in the present model, therefore we cannot make a rigorous comparison. However, their findings could confirm that the pH profile at the tooth surface depends on the drinking pattern. Cases resembling the three drinking patterns simulated with the model led to similar pH profiles: faster recovery for Sucking case (similar to Thirsty case), followed by Short-Sipping and Long-Sipping.

2.2.4. Comparison with other models Compared to Dibdin’s work (Dawes and Dibdin, 1986; Dibdin and Wimpenny, 1999; Dibdin et al., 1995, Dibdin and Dawes, 1998), the present model introduces new aspects, such as degradation of storage compounds, mixed microbial species plaque, multiple acid production, aerobic/anaerobic metabolism and tooth demineralisation rate. The differences between the current model and the model proposed by Dawes and Dibdin (1986) are listed in Table 2.1.

Storage compounds In the model of Dawes and Dibdin (1986) the substrate (sucrose) can be stored, but there is no further conversion of the storage compounds by microorganisms. Therefore, the process of storage is merely acting as a sink of substrate, a competitive process to glucose fermentation. This approach does not allow to study the effect of acids produced by storage compounds

50 degradation resulting in prolonging the period of low pH under critical values for demineralisation (Zero et al., 1986). However, in the context of current model, the polyglucose degradation does not have an important impact on tooth demineralisation because it occurs at higher pH values, when demineralisation does not take place. In this situation, we can even infer that storage of glucose may have a protective effect on the tooth: the more glucose is stored, the less is available for acid formation under critical conditions (Figure 2.4). Although on the short term polyglucose storage may reduce the damage, over long periods of time part of the stored polymer can be used for growth, which will result in more biomass and higher cariogenic potential (equivalent with an increase of plaque thickness).

Strictly anaerobic metabolism vs. aerobic and anaerobic metabolism Previous plaque models (e.g., Dawes and Dibdin, 1986) considered that only anaerobic processes take place in dental plaque. However, because the mouth is also an aerobic environment, it seems more realistic to also include facultative aerobic conversion of glucose in the model. To test the effect of aerobic processes on the pH profiles, we compared the pH curves obtained with a thin plaque (100 μm) and thicker plaque (500 μm), in aerobic and anaerobic conditions (Figure 2.10). In the thin plaque the oxygen can diffuse throughout the whole plaque depth, while the thick plaque can have both aerobic and anaerobic regions. However, model simulations indicate that the presence of oxygen has little effect on acid production and pH, because aerobic processes are producing mostly weak acids and they are inhibited at low pH. Anaerobic lactic fermentation is still the dominant generator of acidity.

Monospecies plaque vs. multispecies plaque pH changes in plaque are mainly due to production of various acids during microbial fermentations. Dawes and Dibdin (1986) considered only two acids: lactic to account for strong acids and acetic as a representative of the weak acids. We proposed here more rigorous microbial process stoichiometry, including therefore other acids such as succinic, formic and propionic, together with several microbial types. Depending on the percentage of Veillonella present the amount of lactic acid/lactate can vary significantly (Figure 2.2b and Figure 2.2c). Irrespective of Veillonella’s presence, at the time when formic acid is the dominant acid in the plaque (during storage compounds hydrolysis) the pH is already above the critical value for demineralisation. So, even though formic acid maintains the acidic pH for longer time in plaque, this has no influence on tooth demineralisation. Depending on the model purpose

(e.g., whether a strict distribution of acids is required or not, or whether detailed microbial activity is not essential) it seems reasonable to consider only lactic and acetic acids products of microbial metabolism.

Reducing the complex ecology of the dental plaque (there are over 500 bacterial species identified so far in the dental plaque – Rosan and Lamont, 2000) to a single microbial group (Dawes and Dibdin, 1986) could be seen as a too strong model simplification. We compared the Stephan curves obtained with the standard model (multiple microbial groups) with the pH resulted when the plaque contains exclusively each of the main groups, while keeping the same total biomass concentration in the plaque (80 (kg m–3)) (Figure 2.11). If only Actinomyces (ACT) were in the plaque, the acid attack would be shorter and the pH would decrease to 4.7 only, compared to 4.5 in standard case. With only non-aciduric Streptococci (STN), the pH drop would be slightly more than in standard case, due to the absence of further lactic conversion by Velionella. However, when the plaque was containing only aciduric Streptococci (STA) then a minimum pH of 3.5 could be obtained and even the resting pH was below the critical value of 5.5. As expected, this would result in the strongest acid attack on the tooth enamel. Species composition could therefore indeed strongly influence the potential of caries formation.

2.2.5. Outlook and future model development The model can be developed by adding different levels of complexity: new chemical and microbial components, new chemical / biological / transport processes, various other compartments or extension to two- or three-dimensional geometry. Plaque domain. To study the influence of microbial composition on the Stephan curve, other potentially important microbial groups (e.g., Lactobacillus) could be added. In addition, the bacterial composition should develop in time by including growth of several microbial groups in the biofilm (as in other biofilm models, e.g., Wanner and Gujer, 1985). Together with growth, microbial decay or inactivation could be included. If acidity decreasing compounds (e.g., urea, arginine, ammonium) and the processes in which these are involved are taken into account, then the minimum pH should be at least half a pH unit higher as shown by Dibdin and Dawes (1998). Saliva domain. In order to account for different salivary composition function of the tooth’s position in the mouth, saliva flow over the tooth surface should be included. This would help

52 evaluating if the increased saliva flow during meals does not only help the digestion, but has also a protective effect on the tooth during a cariogenic challenge. Tooth. For a more realistic representation of the total amount of HAP lost during feeding/resting cycles, tooth remineralisation should also be included. In order to study the evolution of caries profile in time, i.e., variation of the HAP content in depth of the tooth, a separate computational domain representing the tooth has to be added. An important addition to the current model (suggested also by Dibdin and Wimpenny, 1999), would be the two-dimensional geometry in order to account for the pH gradients along the tooth – plaque interface and for a better representation of microbial competition for space within the plaque. A two- or three-dimensional approach would also allow considering the localized effect of dental plaque on the tooth and simulation of caries formation on different places on the tooth.

2.3. Conclusions In the current paper we describe a one-dimensional numerical model of dental plaque. The model simulates the pH variation (i.e., the so-called Stephan curve) under the influence of microbial metabolism occurring in dental plaque, followed by the tooth demineralisation at pH < 5.5. Using the current model several hypothesis have been tested: (1) Poor oral hygiene (having as consequence a thick dental plaque) leads to caries development. Model results confirm that for thicker plaque the amount of demineralised HAP is higher than for thinner plaque. (2) Consumption of small amounts of sweets extended over long periods of time leads to tooth decay. When studying different drinking behaviours it appears that slow social drinking (the so called Long-Sipping in the current model) is the most harmful from all the tested patterns, especially when compared with the Thirsty case. The amount of demineralised calcium in the Long-Sipping case is similar to the case of poor oral hygiene (i.e., 1000 μm dental plaque thickness) when the same amount of glucose is consumed at once. (3) The presence of bacterial storage compounds may not have a direct harmful effect on the teeth by extending the period of acid attack because these compounds are metabolized to acids only when the pH had been restored above 5.5. (4) The presence of Veillonella in dental plaque has a protective effect due to its ability to consume lactic acid and convert it in weaker acids and the effect is stronger at higher percentage of Veillonella in the dental plaque. 53

This model is a good base for future developments, such as: including the changes of microbial composition in time, plaque growth and decay, tooth remineralisation, extensions to two- or three-dimensional geometries, or saliva flow over the tooth and plaque.

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Takahashi N, Nyvad B, Caries ecology revisited: microbial dynamics and the caries process, Caries Res 2008; 42:409-418. Tanzer JM, Krichevsky MI, Keyes PH, The metabolic fate of glucose catabolized by a washed stationary phase caries-conductive Streptococcus, Caries Res 1969; 3:167-177. Ten Cate JM, A model for enamel lesion remineralisation, in: S.A. Leach, W.M. Edgar (Eds.), Demineralization and Remineralization of the Teeth, Irl. Pr., 1983, pp. 129-144. Van Beelen P, Van der Hoeven JS, De Jong MH, Hoogendoorn H, The effect of oxygen on the growth and acid production of Streptococcus mutans and Streptococcus sanguis, FEMS Microbiol Ecol 1986; 38:25-30. Van der Hoeven JS, De Jong MH, Camp PJM, Van den Kieboom CWA, Competition between oral Streptococcus species in the chemostat under alternating conditions of glucose limitation and excess, FEMS Microbiol Ecol 1985; 31:373-379. Van der Hoeven JS, Gottschal JC, Growth of mixed cultures of Actinomyces viscosus and Streptococcus mutans under dual limitation of glucose and oxygen, FEMS Microbiol Ecol 1989; 62:275-284. Van der Hoeven JS, van den Kieboom CWA, Camp PJM, Utilization of mucin by oral Streptococcus species, Anton Leeuw Int J G 1990; 57(3):165-172. Van Dijk JWE, Borggreven JMPM, Driessens FCM, Chemical and mathematical simulations of caries, Caries Res 1979; 13:169-180. Vanýsek P, Handbook of chemistry and physics, 82nd ed., Boca Raton: CRC Press LLC, 2001, pp. 5-95 and pp. 6-194. Wanner O, Gujer W, A multispecies biofilm model, Biotechnol Bioeng 1986; 28:314-328. Zaura E, Ten Cate JM, Dental Plaque as a biofilm: a pilot study of the effects of nutrients on plaque pH and dentin demineralization, Caries Res 2004; 38(suppl 1):9-15. Zero DT, van Houte J, Russo J, Enamel demineralization by acid produced from endogenous substrate in oral Streptococci, Arch Oral Biol 1986; 31(4):229-234. Zimmerman SO (a), A mathematical theory of enamel solubility and the onset of dental caries: I. The kinetics of dissolution of powdered enamel in acid buffer, B Math Biophys 1966; 28:417-432. Zimmerman SO (b), A mathematical theory of enamel solubility and the onset of dental caries: II. Some solubility equilibrium considerations of hydroxyapatite. B Math Biophys 1966; 28:433-441. Zimmerman SO (c), A mathematical theory of enamel solubility and the onset of dental caries: III. Development and computer simulation of a model of caries formation. B Math Biophys 1966; 28:443-464.

Mathematical

modelling of 3 microbial

dynamics in

dental plaque

3. Mathematical modelling Dental caries is primarily a biofilm induced disease of microbialsince a pre- shiftscondition for the developmin entdental of a carious lesion is the presence of a dental plaque on top of the affected area. plaque This means that dental caries formation is more than just a

chemical dissolution and the processes occurring in the plaque

are critical for the initiation of tooth enamel demineralisation.

For decades there is a debate in the dentistry literature regarding

the influence of dental plaque microbial composition on the

process of caries formation (Ritz, 1967; Loesche, 1986;

Marsh, 2003; Takahashi and Nyvad, 2011).Commonly,

Streptococcus mutans species have been isolated from dental

plaque collected at carious active sites, which led to the

widespread idea that these species are associated with the

development of enamel lesions (Loesche, 1986; Kleinberg

2002). Supporting this idea is the high acidogenicity and

aciduricity of S. mutans which gives this species a significant

58 advantage over non-aciduric bacterial species (such as Actinomyces, Veillonella, non-mutans Streptococcus, Neisseria etc.). However, it was observed that dental caries can form also at sites where there are no mutans Streptoccoci or other aciduric bacteria (e.g., Lactobacillus) present (Jenkinson, 2011).The reverse was noticed as well: aciduric bacterial groups were identified in the dental plaque of caries-free individuals (Tahmourespour, 2013). Adding to the complexity is the influence of sugar consumption habits and oral hygiene over the microbial community in dental plaque and therefore over tooth enamel dissolution. It is believed and supported by several experimental studies (Ten Cate, 2006) that often consumption of sugars, has a double negative effect: an immediate result by maintaining the low pH that favours the tooth demineralisation (Fejerskov and Kidd, 2008) and a long term effect (i.e., weeks or months) by selecting for the aciduric bacterial species and causing a shift towards a plaque composition favourable to caries formation (Loesche, 1986). The view on the dental caries disease as an ecological imbalance is commonly called the “ecological plaque hypothesis” (Marsh, 2003). Numerous experimental studies have been performed in order to understand the nature and the role of microbial shifts in the plaque on the process of caries formation (Bradshaw et al. 1989). While many of these studies led to better insight in the caries problem, some of them gave results that at the current level of understanding appear to be contradicting. An example in this sense is the controversial role of mutans Streptococci in caries formation. In vivo studies are difficult to perform due to the large time span in which microbial shifts occur, but also due to high individual variations in plaque composition of each patient, depending on personal factors difficult to control (e.g., genetic variations, immune system, etc.). In this situation, a new approach by means of mathematical modelling could provide new insights in the role of plaque composition on dental caries formation. The advantages of using numerical modelling include: full control over the conditions of the study (e.g., saliva composition and flow, active processes inside the plaque), shorter times to obtain results, the possibility to study a wide variety of conditions and also the elimination of the ethical concerns intrinsic to in vivo studies (e.g., what is the influence of the experiment on the health of the patient?). Nevertheless, numerical (computational) models must be complemented by experimental studies, not only for obtaining reliable model parameters but also for model validation. This numerical study presents a one-dimensional mathematical model of dental plaque including microbial growth and population shifts in time. The model couples the metabolic and chemical processes occurring in the plaque with transport processes and tooth enamel

59 demineralisation. This computational approach has been used to answer several questions regarding caries formation and microbial shifts in the dental plaque: 1) Is there any long term effect of the bacterial storage polymers on the growth of dental plaque and tooth demineralisation? This question addresses a hypothesis present in the literature (Zero et al., 1986) stating that the ability of bacteria to store the excess glucose and consume it at times of nutrient depletion increases its cariogenic potential. In a previous study (Ilie et al., 2012) we argued that there is no immediate negative effect since the stored polymers are consumed only after the sugar have been depleted, at pH already higher than the critical values for demineralisation, and their slow consumption does not cause a large pH drop anymore. However, since the storage polymers can be used for microbial growth, there might be still an overall negative impact on long term. 2) What is the effect of drinking patterns (i.e., sugar consumption habits) on plaque community and demineralisation? One of the main identified causes for dental caries development is the often consumption of sugars. The Vipeholm clinical study (Gustafsson et al., 1954) showed that when patients consumed small amounts of sugar between the main meals the negative effect on the teeth was much higher than for those who consumed a large desert right after the meal. This was confirmed also by the results of a previous numerical model (Ilie et al., 2012) showing that when some sweet drink is consumed over a long time, the amount of demineralised tooth is much greater than when the same amount is drank at once. The same drinking habits (called here Thirsty, Short-Sipping, Long-Sipping) will be tested using the current model to evaluate if they could lead to different microbial compositions of the dental plaque. Specifically, we are also interested to see if often sugar consumption is selecting for aciduric bacteria. 3) What is the effect of dental hygiene (i.e., tooth brushing) on plaque community? This effect was tested by periodically resetting the length of the dental plaque with a frequency corresponding to good hygiene (Often brushing, 3 times a day) and poor hygiene (Rare brushing, once a day). The effect on plaque community and dental demineralisation was evaluated.

3.1. Model description The goal of the mathematical model described in this study is to analyse the shifts occurring in the microbial population of a growing dental plaque in different oral environment conditions. For this purpose, a previous numerical model for calculation of pH and solute concentrations in an active multispecies dental plaque (Ilie et al., 2012) was extended to include microbial 60 growth. At the same time the complexity of metabolic processes in the plaque was slightly decreased by not considering some less important factors (i.e., no aerobic processes, and no buffering effect of the plaque by fixed charges on the bacterial surface). The following paragraphs describe the main extensions made to the basic model from Ilie et al. (2012).

3.1.1. Components Both microbial and chemical components are considered in the model, with concentrations that vary in space and time due to different production or consumption reactions coupled with transport processes.

(a) Microbial components Four generic bacterial groups were considered, the same as in the previous study (Ilie et al., 2012): aciduric Streptococcus (STA), non-aciduric Streptococcus (STN), Actynomices (ACT) and Veillonella (VEL). The Streptococcus species are usually the most abundant in a mature dental plaque (up to 85%) followed by ACT (up to 45%) and VEL (rarely reaching 40%) (Fejerskov and Kidd, 2008; Marsh et al., 2009). In the model, these percentages are allowed change in time, starting from a very thin microbial layer (2 μm) containing the four groups in equal concentrations. This choice would give equal chances to each microbial group to, depending on conditions, become dominant in the mature plaque. Microbial components are 3 described by their concentration in the plaque CX,i (C-mol dry biomass / m plaque) which changes in space and time depending on the solutes concentrations and microbial growth. The total microbial concentration in plaque is fixed to 3250 (C-mol dry biomass / m3 plaque).

(b) Chemical components The model takes into account solutes (Table 3.1) and the bacterial storage compounds (particulate matter). The solutes are present in all model compartments and consist of substrates for bacterial growth (glucose for STA, STN and ACT and lactate for VEL, plus ammonium as N-source), products of bacterial metabolism (bicarbonate, ethanol, organic acids their anions), ions resulting from tooth dissolution (phosphates and calcium) and ions used for charge balancing (potassium and chloride). Each solute i (Table 3.1) is characterized by a constant diffusion coefficient Di (Chapter 2, Table 2.3), charge number zi, and a concentration Ci changing in time in saliva and in time and along the plaque thickness in the plaque.

In the presence of glucose bacteria store internally a part of the excess glucose in the form of polymers (e.g., glycogen, polyglucose etc.). In the current model these storage polymers have been lumped into one generic component named polyglucose. A differentiation was made among the polyglucose stored and consumed by each microbial group: STA will store and consume Pgsta, while Pgstn and Pgact are associated to STN and ACT, respectively. VEL does not consume glucose therefore no polyglucose is stored. Being associated to the biomass, the polyglucose is present only in the plaque compartment and it is transported together with the biomass over the dental plaque thickness.

Table 3.1 Chemical species considered in the model Name Symbol Standard Gibbs energy(a) (kJ mol–1) Acetate Ace– –369.41 Acetic acid AceH Formate For– –335 Formic acid ForH Lactate Lac– –517.18 Lactic acid LacH Propionate Pro– –361.08 Propionic acid ProH Succinate Suc2– –690.23 Hydrogen succinate Suc– Succinic acid SucH Bicarbonate HCO3– –586.85 Carbon dioxide CO2 Hydrogen Phosphate Pho2– Dihydrogen Phosphate Pho– Hydroxyl HO– Proton H+ –81.41 Calcium Ca2+ Anion (Chloride) Cl– Cation (Potassium) K+ Ammonium NH4+ –79.37 Ammonia NH3 Glucose Glu –917.22 Ethanol Eth –181.75 Biomass Biomass –67 Water H2O –237.18 (a) Gibbs free energies of formation from the elements for the compounds involved in the microbial growth reactions, under biochemical standard conditions (pH = 7; 1 atm; 1 M; 25 °C), from Heijnen (1999).

3.1.2. Processes Each of the considered microbial groups performs a series of biological processes (conversions). There are also chemical processes such as acid-base equilibria and the dissolution of tooth enamel that occurs at low pH values. 62

(a) Biological processes The biological (microbial) reactions that take place in the plaque are of special interest because: (i) metabolites released by the bacteria can attack the tooth enamel (e.g., organic acids), and (ii) these reactions represent also the microbial growth, the basic process leading to microbial shifts in multispecies dental plaque. Stoichiometry of microbial reactions can be derived from the energetics of separate catabolic and anabolic reactions (Heijnen and Kleerebezem, 2010). Through a catabolic reaction the microorganisms produce energy by oxidizing an electron- donor. The catabolic processes considered in the current model have the same stoichiometry and rate expressions as in the previous plaque model (Ilie et al., 2012). Glucose at high concentrations is either stored as polyglucose, or consumed by STA, STN and ACT in a lactic fermentation (the low affinity glucose uptake pathway, Table 3.2, processes 1 and 3). At low glucose concentrations, the STA, STN and ACT produce formate, acetate, succinate and ethanol (the high affinity glucose uptake pathway, Table 3.2, process 2). The same pathway is used for the conversion of polyglucose when glucose is present at low concentrations in the plaque (Table 3.2, process 4). VEL converts the lactate into propionate and acetate. The energy supplied by the catabolism must be sufficient to allow the anabolic reaction leading to biomass growth to occur (eq. 3.1). The five stoichiometric coefficients of the anabolic reaction, νana,i, were easily determined from the element (C, H, N, O) and charge balances. Q C-SourceQ NH ana,C-Source ana,NH4+ 4 (3.1) + 1BiomassQQana,H+ H ana,HCO3 HCO 3 Q ana,H2O H 2 O 0 STA, STN and ACT use glucose as C-source and electron donor, while VEL uses the lactate. Further, each of the considered catabolic processes (specific to the microbial group considered) has been energetically coupled with the anabolic process (the same for all microbial groups) using the method from (Heijnen and Kleerebezem, 2010).The Gibbs energy changes of the anabolic, ΔGana, and catabolic, ΔGcat, reactions were calculated from standard Gibbs energy 01 of formation, 'G fi, (Table 3.1). The Gibbs energy required for biomass growth, ΔGgro, was also estimated using the correlation from Heijnen and Kleerebezem (2010) for heterotrophic growth (function of the number of carbons and degree of reduction of the electron donor). Then, the factor k = (ΔGgro – ΔGana)/ ΔGcat represents how many times the catabolic reaction needs to be performed in order to produce enough energy for the formation of one C-mol of biomass.

Therefore, the stoichiometry of the whole biological reaction (i.e., coefficients νi) was obtained by multiplying the stoichiometry of the catabolic reaction (νcat,i) with k and adding this to the coefficients (νana,i) of the anabolic reaction: νi = k νcat,i + νana,i. The coefficients of the complete biological reaction have been normalised to the C-source. The resulting stoichiometry matrix and the kinetic expressions of each microbial process are presented in Table 3.2 and Table 3.3 respectively, while the corresponding kinetic parameters are in Table 3.4. 63

Although experimentally proven, inactivation of plaque in its depth was ignored in the current model (i.e., no metabolic processes occurring in a thick plaque close to the tooth surface, and bacteria being in a dormant state).

(b) Chemical processes The anions resulted from the bacterial metabolism are in equilibrium with their corresponding organic acids via acid-base equilibria (Table 3.5). Protonation equilibria are also included for phosphate (product of tooth enamel dissolution), bicarbonate and ammonium (involved in biomass growth reaction). The remaining two phosphate equilibria (dissociation of hydrogen phosphate and phosphoric acid) were not considered since they become important at very alkaline pH values which are not reached in the current model. The acid-base equilibria are considered to be very fast, therefore their associated reaction rate constants have been assigned arbitrary very large values. The surface-based reaction of tooth enamel dissolution takes place when the degree of saturation (DSHAP) of the solution in contact with the tooth surface is below 1 (i.e., undersaturation) (Table 3.5). The demineralisation process was included in order to compare the damage inflicted on the teeth in different situations studied with the current model. Although experimentally proven, the enamel remineralisation process is not essential for the purpose of the current work.

3.1.3. Component balances Similar to Ilie et al. (2012), the numerical model presented in the current study considers chemical components into saliva, dental plaque and tooth enamel, while the microbial components are only in the plaque. The novelty of the current model consists in introducing separate mass balances for the microbial species, allowing the calculation of shifts in microbial populations along the plaque thickness.

(a) Saliva The mass balances for the chemical components are exactly as in Ilie et al. (2012) and Chapter 2. The saliva is considered and ideally mixed volume containing all the solutes (microbial conversions are negligible when compared with those in plaque) performing their associated chemical reactions (Table 3.5). The saliva is divided into two volumes: (1) Vf, a thin film of saliva with a thickness of 100 μm (Fejerskov and Kidd, 2008) found in direct contact with the dental plaque, and (2) Vs, bulk saliva which communicates permanently with the saliva film. The change in concentrations of chemical species present in the saliva film,

Cf,j (t), is (eq. 3.2): 64

Table 3.2 Stoichiometry of microbial conversions considered in the model Reference catabolic Glu Suc2– Lac Pro Ace For Eth HCO3– H+ NH4+ H2O Pgsta Pgstn Pgact XSta XStn XAct XVel reaction 1. Anaerobic high concentration glucose fermentation STA –1 1.607 0.056 1.888 –0.224 0.45 1.122 Van Beelen et al., 1986 STN –1 1.607 0.056 1.888 –0.224 0.45 1.122 Van Beelen et al., 1986 ACT –1 1.607 0.056 1.888 –0.224 0.45 1.122 Van Beelen et al., 1986 2. Anaerobic low concentration glucose fermentation STA –1 0.787 1.573 0.787 0.061 2.665 –0.224 –0.298 1.219 Van Beelen et al., 1986 STN –1 0.787 1.573 0.787 0.061 2.665 –0.224 –0.298 1.219 Van Beelen et al., 1986 ACT –1 0.755 0.755 0.755 –0.685 2.615 –0.280 1.316 1.4 De Jong et al., 1988 3. Polyglucose storage STA –1 1 Assumed STN –1 1 Assumed ACT –1 1 Assumed 4. Anaerobic polyglucose conversion STA 0.787 1.573 0.787 0.061 2.665 –0.224 –0.298 –1 1.219 Assumed as Process (2) STN 0.787 1.573 0.787 0.061 2.665 –0.224 –0.298 –1 1.219 Assumed as Process (2) ACT 0.755 0.755 0.755 –0.685 2.615 –0.280 1.316 –1 1.4 Assumed as Process (2) 5. Lactate fermentation VEL –1 0.625 0.312 0.321 0.294 –0.036 0.072 0.180 Seeliger et al., 2002

Table 3.3 Rates of microbial conversions in the dental plaque Rates 1. Anaerobic high concentration glucose fermentation

STA qm,STA,Glu,H CSTA M(CGlu) I(CH+) STN qm,STN,Glu,H CSTN M(CGlu) I(CH+) ACT qm,ACT,Glu,H CACT M(CGlu) I(CH+) 2. Anaerobic low concentration glucose fermentation

STA q,STA,Glu,L CSTA M(CGlu) I(CH+) I(CGlu) STN qm,STN,Glu,L CSTN M(CGlu) I(CH+) I(CGlu) ACT qm,ACT,Glu,L CACT M(CGlu) I(CH+) I(CGlu) 3. Polyglucose storage

STA qm,STA,sto CSTA M(CGlu) I(CH+) I(CPgsta) STN qm,STN,sto CSTN M(CGlu) I(CH+) I(CPgstn) ACT qm,ACT,sto CACT M(CGlu) I(CH+) I(CPgact) 4. Anaerobic polyglucose conversion

STA qm,STA,Pgsta CSTA M(CPgsta) I(CH+) I(CGlu) STN qm,STN,Pgstn CSTN M(CPgstn) I(CH+) I(CGlu) ACT qm,ACT,Pgact CACT M(CPgact) I(CH+) I(CGlu) 5. Lactate fermentation

VEL qm,VEL,Lac– CVEL M(CLac–) I(CH+) C j Monod substrate limitation factor for species j MC j KCSX,, j j K I ,X , j Inhibition factor for species j. I C j K I ,X , j C j Subscript X is the microbial species (i.e., STA, STN, VEL or ACT).

Table 3.4 Rate parameters for microbial conversions Parameter name Symbol Value Reference Substrate specific uptake rate qm,STA,Glu,H 96.70 Van der Hoeven et al., 1985 (10–7 mol s–1 g–1) qm,STA,Glu,L 5.00 Assumed as qm,STN,Glu,L qm,STA,Pgsta,sto 15.6 Assumed as qm,STN,Pgstn,sto (a) qm,STA,Pgsta 1 Assumed 1/5 qm,STA,Glu,L qm,STN,Glu,H 138 Van der Hoeven et al., 1985 qm,STN,Glu,L 5.00 Van der Hoeven et al., 1985 qm,STN,Pgstn,sto 15.6 Hamilton, 1968 (a) qm,STN,Pgstn 1 Assumed 1/5 qm,STN,Glu,L qm,ACT,Glu,H 22.00 Van der Hoeven and Gottschal, 1989 (b) qm,ACT,Glu,L 0.88 Assumed 1/25 qm,ACT,Glu,H qm,ACT,Pgact.sto 1.40 Assumed based on Komiyama et al., 1986 and Komiyama and Khandelwal, 1992 (a) qm,ACT,Pgact 0.176 Assumed 1/5 qm,ACT,Glu,L qm,VEL,Lac– 0.252 Seeliger et al., 2002

Substrate saturation constant KS,Glu,H 1220 Assumed based on Hamilton and Martin, 1982; (10–6 mol L–1) Van der Hoeven et al., 1985 and Dawes and Dibdin, 1986 KS,Glu,L 8.04 Hamilton and Martin, 1982 KS,Lac– 290 Seeliger et al., 2002 KS,Pgsta 0.2×KS,Glu,H Assumed KS,Pgstn 0.2×KS,Glu,H Assumed KS,Pgact 0.2×KS,Glu,H Assumed

Parameter name Symbol Value Reference (c) Inhibition constant KI,STA,H+ 15.8 Assumed (10–6 mol L–1) (d) KI,STN,H+ 1.58 Assumed KI,ACT,H+ 1.58 Assumed as KI,STN,H+ KI,VEL,H+ 15.8 Assumed as KI,STA,H+ KI,STA,Glu 2200 Assumed based on Hamilton, 1968 KI,STN,Glu 2200 Assumed based on Hamilton, 1968 KI,ACT,Glu 2200 Assumed based on Hamilton, 1968 (e) KI,Pgsta 0.05 Based on Hamilton, 1968 (e) KI,Pgstn 0.05 Based on Hamilton, 1968 (e) KI,Pgact 0.05 Based on Hamilton, 1968 (a) Rate value decreased to also account for slow hydrolysis of polyglucose (b) Assumption based on the fact that qm,STA,Glu,H/ qm,STA,Glu,L ≈ 20 and qm,STN,Glu,H/ qm,STN,Glu,L ≈ 27 (c) Optimum pH for glycolysis in STA is 6 (Dashper and Reynolds, 1992). The reaction is assumed to be half inhibited at a pH value smaller with 1.2 units than the optimum one. (d) Optimum pH for glycolysis in STN is 7 (Hamilton, 1968). The reaction is assumed to be half inhibited at a pH value smaller with 1.2 units than the optimum one. (e) Microorganisms cannot store more polyglucose than 50% their cell dry weight.

Table 3.5 Acid-base equilibria and rate expressions

Reaction Stoichiometry Equilibrium Reaction rate, re,i constant(a), Ke,i -14 2 2 Water dissociation H2O ' HO– + H+ 10 mol /L kH2O [1 – (CH+ CHO–)/Ke,H2O] –3.86 Lactic acid dissociation LacH ' Lac– + H+ 10 mol/L kLacH [CLacH – (CH+ CLac–)/Ke,LacH] –3.75 Formic acid dissociation ForH ' For– + H+ 10 mol/L kForH [CForH – (CH+ CFor–)/Ke,ForH] –4.76 Acetic acid dissociation AceH ' Ace– + H+ 10 mol/L kAceH [CAceH – (CH+ CAce–)/Ke,AceH] –4.87 Propionic acid ProH ' Pro– + H+ 10 mol/L kProH [CProH – (CH+ CPro–)/Ke,ProH] dissociation –4.20 Succinic acid SucH ' Suc– + H+ 10 mol/L kSucH [CSucH – (CH+ CSuc–)/Ke,SucH] dissociation –5.63 Hydrogen succinate Suc– ' Suc2– + H+ 10 mol/L kSuc– [CSuc– – (CH+ CSuc2–)/Ke,Suc–] dissociation –6.35 Carbonic acid CO2 + H2O ' 10 mol/L kCO2 [CCO2 – (CH+ CHCO(3)–)/Ke,CO2] dissociation ' HCO3– + H+ –7.21 Dihydrogen phosphate Pho– ' Pho2– + H+ 10 mol/L kPho– [CPho– – (CH+ CPho2–)/Ke,Pho–] dissociation –9.2 Ammonium dissociation NH4+ ' NH3 + H+ 10 mol/L kNH(4)+ [CNH(4)+ – (CH+ CNH3)/Ke,NH(4)+] 2.8 0.3 Hydroxyapatite (HAP) Ca5(PO4)3OH + 4H+ rd,HAP = ken (1–DSHAP) (∑CA(i)H) , i – dissolution 5Ca2+ + 3Pho2– acids 1/9 + H2O DSHAP = (IPHAP·KS,HAPen) 5 3 4 IPHAP = [(CCa2+) (CPho2–) ] / (CH+) 3 KS,HAP(en) = [(Ke,Pho2–) Ke,H2O] / KS,HAP ∑CA(i)H = CCO2 + CPho– + CLacH + CAceH + CForH + CProH + CSucH + CSuc– –3 –1(b) ken = 0.42 u 10 min –12.35 –1 Ke,Pho2– = 10 mol L –55 9 –9(c) KS,HAP = 5.5 u 10 mol L (a) The values of all the acidity constants are taken from Atkins and de Paula, 2009 except for Ke,SucH and Ke,Suc– which are taken from Martell and Smith, 1976 (b) (Margolis and Moreno, 1992) (c) (Moreno and Zahradnik, 1974)

dCfj, Q f A f CCsj,, fj N pj ,, R fj (3.2) dt Vff V

The mass balance (3.2) is based on the exchange with the saliva bulk with a flow rate Qf, exchange with the dental plaque of area Af with the flux Np,j (diffusive flux at the plaque-saliva interface) and the net reaction rate of each component, Rf,j (Table 3.6). The saliva inlet concentrations for chemical species Cs,j are set as in a previous study (Ilie et al., 2012), based on chemical speciation (i.e., mass-action laws and charge balance). The initial concentrations for equations (3.2) are equal with the input concentrations Cf,j(0) = Cs,j. The total input concentrations for the master species (i.e., CO2, Pho, Ca2+, H+, Ace, For, Pro, Lac, Suc, Glu) have the same values as in Ilie et al. (2012) and Chapter 2, Table 2.3, except for chloride which has an input concentration of 50 (mol m–3). The input values for the remaining solutes are resulting from chemical equilibira.

Table 3.6 Parameters associated to processes in the saliva domain Parameter Symbol Value Reference Time for the transition from CGlu,min to CGlu,max tstep 10 s Assumed Feeding period tfeed 2 min Assumed Length feeding/resting cycle tcycle 90 min Assumed Halving time in the saliva film th,f 0.5 min Dibdin, 1990 Halving time in the saliva bulk th,s 2.17 min Dibdin, 1990 Residence time in the saliva film Qf / Vf ln(2) / th,f min Calculated Residence time in the saliva bulk Q / Vs ln(2) / th,s min Calculated 3 –1 (a) Area per volume ratio in saliva film Af / Vf 10 m Calculated Minimum glucose concentration (between meals) Cs,Glu,min 0.07 mM Van der Hoeven et al., 1990 Maximum glucose concentration (during pulse) Cs,Glu,max 560 mM Dirksen et al., 1962 (a) Assumes that only 10% of the tooth is covered with plaque.

The inlet glucose concentration was set through repeated feeding/clearance/resting cycles called feeding cycles and having the total time of 2 h per cycle (eq. 3.3, similar to eq 2.3). The feeding phase was represented by a glucose pulse given at the beginning of the cycle in the bulk saliva. Cs,Glu quickly increased within tstep from Cs,Glu,min to a maximum concentration Cs,Glu,max, which was maintained for the entire feeding time, tfeed. After the feeding phase, clearance begins and glucose concentration decreases exponentially (with the residence time Q/Vs ) until the end of the resting period. ªºQ CtCs,,, Glu s Glu max tt step feedexp «» ttt step feed C s ,, Glu min (3.3) ¬¼Vs

(b) Dental plaque The dental plaque is represented as a 1-d domain perpendicular to the tooth surface and in direct contact with the saliva film at one boundary (x=Lp) and with the tooth surface at the other (x=0). 68

All microbial groups (i.e., STA, STN, ACT and VEL) and chemical species (Table 3.1 and the polyglucose stored insde the bacteria: pgsta, pgstn and pgact) together with all their associated processes (Table 3.2, Table 3.3, Table 3.4 and Table 3.5) are present in the plaque domain.

Mass balances for the chemical components The change in the concentration of each solute j in time and along the plaque thickness,

Cp,j(t,x), are calculated based on Nernst-Planck equations (eq. 3.4) coupled with an electroneutrality condition (eq. 3.5), the same as in Ilie et al., (2012):

2 wwCCDpj,, pj j ww)§· (3.4) (3.5) DzFCRj2 j¨¸ pj,, pj ¦ zCjpj, 0 wwtxRTxx ww©¹ j where Dj is the diffusion coefficient of solute j in the plaque (constant in space and time) –1 (Table 2.3), F Faraday’s constant (96485 C mol ), T is the temperature (310 K) and Rp,j is the net reaction rate of each solute, accounting for both chemical and microbial processes. The additional state variable )(x,t) is an electrical potential field developed due to ions being transported with different diffusion rates. The initial values for all concentrations were considered equal to those in the saliva film, while the potential )(x,0) = 0. For the mass balances of solutes (eqs. 3.4) the concentrations of all the species at the saliva-plaque interface (x=Lp) are equal to those in the saliva film. At the plaque-tooth interface the molar flux due to diffusion and migration, Npt,j, is set to zero for all the chemical species except for those involved in the dissolution reaction (j = H+, Ca2+, Pho2–). For these species, the molar flux equals their net formation/consumption rate due to the HAP dissolution reaction (rate rd,HAP and stoichiometric coefficient Qj according to Table 3.5) calculated with concentrations at the tooth surface (Cpt,j): Npt,j = νj rd,HAP (Cpt,H+, Cpt,Ca+, Cpt,Pho2–). In addition, the electric potential is set to a reference value at the saliva-plaque interface (Φ = 0), while an electrical insulation condition is applied at the tooth surface (∂Φ / ∂x = 0). Mass balances for the microbial components (biomass) The most important development of the current model is the inclusion of microbial growth within the dental plaque, according to the model of Wanner and Gujer (1986). The concentration of each bacterial species (j = STA, STN, ACT and VEL), Xp,j(t,x), changes in time along the plaque length according to eq. (3.6). R ww ¦ pj, XuXpj,,() B pj wwuuBBj Rpj, (3.6) D (3.7) wwtx wwtx UB

The mass balance (3.6) for each bacterial group includes the net growth rate, Rp,j, accounting for the contribution to growth of every couple of catabolic-anabolic reaction (biological reactions in Table 3.2), and a transport term corresponding to a very slow advective velocity, uB. The advective velocity (same for all the microbial groups) is the result of biomass generation within the dental plaque and is variable in space and time cf. eq. (3.7). The continuity equation (3.7) results after summation of all equations (3.6) and imposing the condition that the total biomass density in the biofilm remains constant (here, 3 –1 ρB = 3250 C-mol dry biomass / m plaque = 80 g L and the damping coefficient D = 1 s/m).

The initial velocity uB(0,x) = 0 and the initial microbial concentrations are all equal, Xp,j(0,x) =

ρB /4. The boundary condition associated to the mass balances (3.6) is a zero biomass flux at the tooth-plaque interface (i.e., bacteria do not penetrate into the tooth: ww()/0uXBpj, x at x=0 and any t). For eq. (3.7) there is zero advective velocity (i.e., no movement possible) at the tooth surface, uB(t,0)=0. Associated to the biomass are the storage compounds that is, polyglucose stored by STA, STN and ACT inside the bacterial cells. These components move together with the microbial cells and have similar mass balances with the biomass components: a time- dependent advection-reaction equation (eq. 3.8). wwCuC() psto,, B psto R (3.8) wwtxpsto,

In the above mass balance Cp,sto is the concentration of polyglucose stored by each bacterial species (sto = pgsta, pgstn and pgact) and Rp,sto is the rate of polyglucose storage (Table 3.2). The boundary conditions are the same as for biomass, and the initial concentration of storage compounds is zero.

Change in biofilm thickness

A direct consequence of bacterial growth is the increase of the plaque thickness (Lp). The change of plaque thickness is determined by the balance between the plaque growth rate up,gro and the biomass detachment rate up,det: dL p uu (3.9) dt pgro,, pdet The growth rate is actually the biofilm advective velocity taken at the plaque-saliva surface,

uutLpgro, B(, p ). The detachment rate was set so that it strongly increases when the plaque thickness Lp approaches the maximum plaque length Lp,max = 500 μm:

4 uu ªº1/ LL . In this way, a steady plaque thickness can be obtained when p,det p,, gro¬¼«» p p max

uupgro,, pdet. An initial value Lp(0)=2 μm was set for the plaque thickness.

3.1.4. Model solution The model equations were implemented in COMSOL Multiphysics software (COMSOL 4.3, Comsol Inc, Burlington, MA, www.comsol.com), which allows a very flexible and well- structured model construction. The system of differential equations was solved by finite element methods. The plaque domain was discretized on a uniform mesh with the maximum element size of 0.04 μm. As the plaque thickness grows, the mesh is automatically refined. First, the equations for solute transport and reaction in plaque and saliva were solved for a time interval of 30 minutes, sufficient to let the pH and solute concentrations stabilise in the saliva and plaque. This situation was assumed to correspond to a steady-state plaque in resting state. Second, the state of resting plaque was used as initial condition for the next set of time-dependent simulations performed during 60 sequential feeding cycles of 2 h each. The simulations in time included the complete model equations in saliva (eq. (3.2)) and plaque (eqs.(3.4)-(3.9)) with the associated boundary conditions.

3.2. Results and discussion The multispecies plaque model was first applied to a designated standard case. For this situation the immediate changes that occur during a feeding/resting cycle were analysed (e.g., Stephan curves at the tooth surface and the amount of demineralised calcium per cycle), then their added effect over a larger time span was evaluated. Since the main goal of the current work is to study the slow microbial shifts in the dental plaque, these cumulative long term effects of the feeding/resting cycles are of particular interest (e.g., variation of plaque thickness and of microbial community distribution along the plaque). Thereafter, the influence of storage polymers (i.e., polyglucose) on microbial growth and tooth demineralisation was evaluated. Last, two cases are studied: (1) the impact of sugar consumption habits (drinking habits) and (2) oral hygiene (tooth brushing) on plaque community and tooth demineralisation.

-3 x 10 Ca2+ 1.7 7 demin 350 305

) c

-2 1.68 glucose 6.5 300 1.66 pH 1.64 6 250 Plaque length 300 m) P 1.62 5.5 200 1.6 pH 5 150 1.58 295

1.56 4.5 100 Plaque ( length 1.54 4 50 Glucose concentration (mM) 1.52 Mass of demineralized Ca (mg mm 3.5 0 290 0 20 40 60 80 100 120 Time (min) Figure 3.1 Glucose concentration, pH and amount of demineralised calcium at the tooth surface (plaque-tooth boundary) together with the change in the dental plaque thickness during the 30th feeding/resting cycle.

3.2.1. Standard Case The standard case was designed to correspond with common situations created in clinical studies (Tanzer et al., 1969; Dong et al., 1999; Pearce et al., 1999). The dental plaque (2 μm initial thickness) was periodically exposed to high concentrations of nutrients (i.e. glucose and lactate) until maximum plaque thickness of 500 μm was reached. A high glucose concentration (560 mM, Dirksen et al., 1962) was maintained for 2 min in saliva (glucose pulse), followed by oral clearance with a halving time of 2 min (Dibdin, 1990) and a resting phase lasting until the end of the 2 h feeding/resting cycle.

(a) Solute concentration profiles during a feeding/resting cycle The glucose from saliva diffuses in the dental plaque and it is metabolized by the microorganisms to organic acids (i.e. lactic, formic, acetic, succinic and propionic) and other metabolic products (i.e., ethanol and bicarbonate, see Table 3.2 for the detailed stoichiometry). The formation of acids explains the sudden drop in the pH that follows a glucose pulse (Figure 3.1). In time, due to oral clearance and consumption by plaque bacteria, the glucose concentration returns to the almost zero steady state value (after ~20 min from the beginning of the pulse). All this time the pH is maintained at its minim value of 3.9 by the lactic acid produced in high concentration by bacteria. When glucose is depleted, the pH quickly increases up to a second plateau region starting at pH 6 and corresponding to the consumption of the storage compounds (i.e., polyglucose). Only after all the polyglucose has been consumed, the pH will restore to its resting value of 6.6.

-3 x 10 500 6 ) -2 400

m) 4 P 300

Plaque length o

200 m Ca2+ demin 2 Plaque length ( Plaque ( length Critical thickness 100 for demin p mm demineralized Ca (mg of Mass 0 0 0 20 40 60 80 100 120 Time (h)

Figure 3.2 Dental plaque thickness and the amount of demineralised calcium during 60 feeding/resting cycles of 2 h each. The red dot represents the critical plaque thickness (15 μm) above which, according to the current model, tooth demineralisation occurs.

The most important aspect regarding the caries formation, is that once the pH is below 5.5 the plaque fluid at the tooth surface becomes unsaturated with respect to hydroxyapatite (HAP) leading to tooth demineralisation until the pH is restored to values higher than 5.5 (~25 min). The time when tooth dissolution is active and its extent are visualised in Figure 3.1 by representing the total amount of calcium ions lost from the tooth surface. Therefore, after an exposure to high glucose concentration of only two minutes in the saliva, it takes about 20 minutes for the glucose at the tooth surface to be consumed and even longer (~30 min) for the tooth enamel demineralisation to stop. The nutrients provided (sugars) or generated (acids) during the glucose pulse will support the microbial growth. While the concentrations of different groups of bacteria in the plaque are changing, the thickness of the dental plaque increases. The variation of dental plaque thickness during one feeding/resting cycle (Figure 3.1) can be divided into three regions. In the first 25 min the growth is the fastest because the glucose is abundant (the lactic fermentation occurs in this period). When the extracellular glucose concentration is restored to a very small value, bacteria begin to consume from the internally stored polyglucose using the high affinity glucose uptake pathway. This significantly slows down the microbial growth which is reflected in a profile with a much less steep slope in this region. Finally, once the polyglucose is also finished, the growth slows down once again as the microorganisms are thriving on the very low glucose concentration present in the resting plaque. The most significant biofilm growth occurs during the feeding phase when glucose is present in high concentration. 73

80 saliva 70 p ) -1 60 STA n tooth 50

30 STN

20 Microbial concentration (g L (g Microbial concentration 10 ACT

0 VEL 0 20 40 60 80 100 120 Time (h)

Figure 3.3 Microbial shifts at the plaque surface (continous line) and at the tooth surface (dotted line) for the four components of the dental plaque during 60 feeding/resting cycles. Aciduric Streptococcus (STA) become the dominant microbial group, while Actinomyces (ACT) and Veillonella (VEL) are quickly outcompeted.

(b) Microbial shifts and other long term effects Although the variations in plaque thickness and amount of demineralised calcium per feeding/resting cycle are not large, when cummulated over a long period of time these changes can become significant. When analysing the plaque thickness variation over 60 feeding/resting cycles and the amount of demineralised calcium during the same time period (Figure 3.2) it becomes clear that there is a critical dental plaque thickness (~15 μm) under which no tooth dissolution can occur because the pH never falls under 5.5 in such a thin plaque. This observation qualitatively confirms the results obtained using a previous model (Ilie et al., 2014) and reinforces the conclusion that there is no need to totally remove the dental plaque in order to prevent caries formation. Instead, it is essential to maintain a good oral hygiene with regular tooth brushing in order to keep the plaque under the critical length for enamel dissolution. The shifts in the composition of the plaque community were analised at the plaque surface (plaque-saliva boundary) and at the tooth surface (plaque-tooth boundary) during 60 sequential feeding/resting cycles (Figure 3.3). The model indicates that in the studied conditions the dominant species on the long term at both the tooth and the plaque surface is the aciduric Streptococcus (STA) folowed by much lower concentrations of non-aciduric Streptococcus (STN) and by negligible amounts of Actinomyces (ACT) and Veillonella (VEL). The selection towards a STA-dominated plaque is made relatively soon, after the first ~8 cycles, showing that a diet rich in “sugar snacks” selects efficiently for the most cariogenic 74 type present in the plaque. This result is in accordance with the theory supported by Loesche (1986) that a high incidence of dental caries is associated with high concentrations of Streptococcus mutans (belonging to the STA cathegory) in the dental plaque. There are however recent studies showing that caries can be developed in the absence of STA (Jenkinson, 2011) and, conversely, that STA is present in high concentrations in dental plaque but without significant cariogenic effect (Tahmourespour, 2013). When analysing the pattern of demineralised calcium from Figure 3.2, it appears that in the later stages (e.g., after 30 h) the mass of lost calcium per cycle is larger than at the beginning. This could be caused both by an increased amount of STA in the plaque and by the thicker plaque compared to previous cycles. It was noticed (Figure 3.1) that most of the damage on the tooth occurs during and soon after the glucose pulse. Therefore, the mere presence of STA in a dental plaque if not combined with often sugar uptake could be, in theory, harmless for the tooth. By selectively removing STA from the plaque (e.g., with targeted antibiotics) as suggested in literature (Losche, 1986), while maintaining bad sugar consumption habbits will still result in dental caries formation even in the absence of the aciduric bacteria. On the other hand, it is also true (Figure 3.3) that if STA is present in the plaque even in low amounts, through repeated acid attacks STA will become the dominant species and in this situation it is an indicator of a rich cariogenic activity. Bradshaw et al. (1989) studied the dynamics of a population of nine bacterial groups, aciduric and non-aciduric, present in a chemostat that received glucose pulses at regular time intervals. In the absence of any pH control, the concentration of Streptococcus mutans (belonging to STA group) increased greatly, but when the pH was maintained at 7 almost no variation was registered. This result confirms qualitatively the trend observed using the current model: that STA concentration increases very much in the absence of a pH control. It also prooves once more that STA has such a strong competitive advantage over other species due to its acidogenicity and aciduricity. In the study of Bradshaw however, the domiant species was in both studied cases VEL, followed in the low pH case by STA, then STN, and finally by ACT. The non-aciduric Streptococci (STN) proliferate faster initially , but they are dominated by the steadyly increasing concentration of STA. In the absence of STA, STN is the dominant species, while ACT and VEL are outcompeted since the constant microbial reaction rate of STN is much higher than those of ACT and VEL.

80 STA 60 h 70 ) -1 STA 30 h 0.6 60

(g L ACT 30 h x

50 ) 0.4 STN 6 h -1

40 (g L x STA 6 h c 0.2 VEL 30 h ACT 60 h 30

0 VEL 60 h 20 STN 30 h 0 250 500 x ( Pm)

Microbial concentration, c Microbial concentration, STN 60 h 10 6 h ACT VEL 0 0 100 200 300 400 500 Plaque length, x ( Pm) Figure 3.4 Microbial distributions along the dental plaque after: 6 h (dotted line), 30 h (dashed line) and 60 h (continous line).

For a better understanding of the evolution of dental plaque community it is important to study also the microbial distributions along the plaque depth at few moments in time (Figure 3.4). At the end of the thrid feeding/resting cycle (i.e., after 6 h) the dominant species appears to be STN, followed closely by STA, whereas ACT and VEL are mostly eliminated. Not much differences in microbial composition along the plaque depth can be observed. After 15 cycles (i.e., 30 h) the STA has become the dominant bacterial group along the entire plaque, with a higher concentration at the plaque surface. In the current model STA has an advantage towards the other three bacterial groups because its metabolism can withstand lower pH values than the other groups (the glycolisis becomes half inhibited in STA at pHSTA = 4.8 and in the other groups at pH = 5.8). Therefore, as the pH drops under 5.8, STN, ACT and VEL become pH inhibited and cannot consume the abundency of glucose at their maximum capacity, while STA can take advantage of this period of nutrient abundency and low competition. As stated eralier, STN dominates over ACT and VEL due to its higher microbial reaction rate constant (higher even than STA).

3.2.2. Effect of storage polymers In order to evaluate the effect of storage compounds on microbial growth and enamel demineralisation, in a second simulated case the glucose storage in microorganisms was disabled. The model without polyglucose storage and consumption was run in the same conditions as the standard case and only small differences were observed in plaque growth (Figure 3.5a) and in the amount of released calcium ions (Figure 3.5b). The only marginally 76

(a) (b) -3 x 10 500 6

) polyglucose o -2 5 400 polyglucose o

m) m no polyglucose 4 P m no polyglucose 300

3 200 2 Plaque ( length 100 1

mm demineralized Ca (mg of Mass

0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (h) Time (h)

Figure 3.5 Dental plaque thickness (a) and amount of demineralised calcium (b) for the standard case compared with a case without polyglucose storage. thicker plaque obtained when considering storage compounds can be explained by the much slower microbial growth on stored polyglucose (Figure 3.1) than on glucose, during the feeding phase. This small difference in plaque thickness leads to only slightly higher amount of demineralised calcium when polyglucose can be formed and consumed. The concentration profiles of the four microbial groups with and without storage were very similar (Figure 3.6). Therefore, within the current model conditions, the storage and consumption of polyglucose did not affect the microbial shifts and no long term effect of polyglucose storage was observed, as proposed in a previous study (Ilie et al., 2012). 80 polyglucose p 70

) n

-1 STA 60 no polyglucose

30 STN

20 Microbial concentration (g L (g Microbial concentration 10 ACT

0 VEL 0 20 40 60 80 100 120 Time (h)

Figure 3.6 Microbial shifts at the the plaque surface for the standard case (continous line) and a similar case whithout polyglucose storage (dotted line) during 60 feeding/resting cycles. Polyglucose storage leads only to a minor advantage of the aciduric Streptococcus (STA).

-3 (a) (b) x10 500 18

Short-Sipping o ) 16 -2 400 14 Long-Sipping o

m) Long-Sipping o m Thirsty 12 P 300 10

8 200 6

Plaque length ( Plaque ( length Short-Sipping p 4 100 n Thirsty mm demineralized Ca (mg of Mass 2 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (h) Time (h)

Figure 3.7 (a) Dental plaque thickness and (b) mass of demineralised calcium, in three drinking patterns of sugar-containing drinks.

3.2.3. Drinking habits One of the most acknowledged reasons for caries formation is the often consumption of sugars. A previous model (Ilie et al., 2012) evaluated the effect of three drinking patterns on the amount of calcium demineralised. The same amount of glucose was ingested using three different drinking patterns: 5 times one sip of 3 s with 10 min break in between the sips (Long-Sipping case), 5 times one sip of 3 s with 1 min break (Short-Sipping case) and one sip of 15 s (the Thirsty case). The same model drinking cases were used in the current study to evaluate the effect of drinking patterns on plaque growth and tooth dissolution. The Long-Sipping case had the highest impact on the dental plaque growth (Figure 3.7a) while the Short-Sipping and Thirsty cases provided similar results. The plaque thickness in the Long-Sipping case already reached the allowed maximum of 500 μm after 70 h (~35 feeding cycles), while for the other two cases the 500 μm thick plaque was obtained only after more than 120 h. Although the amount of glucose consumed was equal for the three studied cases, there were large differences between the amount of glucose present in the plaque for the Long-Sipping case and the amount of glucose present for any of the remaining two cases (Short-Sipping or Thirsty) This situation is similar to the one described in Chapter 2, Figure 2.8 where the glucose concentration at the tooth surface in Long-Sipping case is significantly higher and maintained high for a longer time than the one for Short- Sipping case. The consequence is that the dental plaque in the Long-Sipping has more glucose available for growth, therefore a thicker plaque will form in this case. The amount of demineralised calcium followed the same trend (Figure 3.7b), with three times more

Long-Sipping 80 p

70 STA Short-Sipping

) p n -1 60 Thirsty

20 STN

L (g Microbial concentration 10 ACT VEL 0 0 20 40 60 80 100 120 Time (h)

Figure 3.8 Microbial shifts at the the plaque surface for the Long-Sipping (dashed line), Short-Sipping (dotted line) and Thirsty (continous line) cases, during 60 feeding/resting cycles. demineralisation in the Long-Sipping case than in the Short-Sipping and Thirsty cases. The explanation for this situation is that the consumption of the glucose-containing drink over extended periods of time leads to longer times of tooth exposure to critical pH for demineralisation. The lohger the time between two sips, the longer the demineralisation period. The fastest shift towards the dominance of aciduric streptococci (STA) occurred during Long-Sipping case, while the Short-Sipping and Thirsty had very similar profiles (Figure 3.8). Again, Long-Sipping is the most aggressive case because the plaque growth is the most accelerated in this case for the reasons exposed earlier. Therefore, the population dynamics are the fastest and the shift towards STA dominance occurs sooner. Clearly, sugar consumption habits play a key role in the caries formation, and this role becomes even more important on longer time periods.

3.2.4. Tooth brushing Another aspect generally accepted in dentistry (Fejerskov and Kidd, 2008) is the necessity of tooth brushing as an aspect of a good oral hygiene, aimed at keeping the plaque thickness low, and therefore limiting the damage on the tooth during acid attacks. In order to test the efficacy of tooth brushing on tooth demineralisation and microbial shifts, two cases have been studied: (1) Often Brushing corresponds to a good oral hygiene in accordance with the ideal recommendations of 3 times tooth brushing per day (i.e., brushing every 8 h, therefore after 79

(a) (b) 120 -3 x 10 2.5 100 Rare brushing o ) -2 2

m) 80

P Rare brushing o 60 1.5

Often 40 1 Plaque ( length brushing p 20 0.5 Often brushing

Mass of demineralized Ca (mg mm demineralized Ca of (mg Mass p 0 0 20 40 60 80 100 120 0 Time (h) 0 20 40 60 80 100 120 Time (h)

Figure 3.9 Dental plaque thickness (a) and amount of demineralised calcium (b) for the Often brushing (three times a day) and Rare brushing (once a day). every 4 feeding/resting cycles); (2) Rare brushing, corresponds to a poorer dental hygiene of brushing once a day (every 24 h, that is every 12 feeding/resting cycles). In both cases, at the moment when the tooth brushing was active, the scheduled glucose pulse was skipped. During the tooth brushing, the length of the plaque was reset to a minimum value of 5 μm (i.e., under the critical value for demineralisation, but not completely eliminating the plaque). The dental plaque is maintained thin with both proposed tooth brushing regimes (Figure 3.9a). Obviously, more frequent brushing leads to a thiner plaque: for the Often brushing case is reached a maximum plaque thicknes of ~30 μm, whereas for the Rare brushing the maximum plaque thickness is ~110 μm. In both cases the tooth brushing corresponds to the sudden drop in plaque thickness (Figure 3.9a). The effect of brushing on tooth dissolution is also straightforward: more often brushing prevents damage on the tooth during the acid attack (Figure 3.9b). In the Rare brushing case, the plateau areas correspond to the periods after brushing, when the plaque has been reset to non-dangerous levels for demineralisation. Simulations indicate a large difference (one order of magnitude) between the amounts of calcium lost through demineralisation if tooth brushing is done 3 times compared to only one time per day. The current model shows why tooth brushing is a very efficient way of limiting the damage on the tooth during the acid attacks caused by dental plaque. An analysis of microbial shifts in the plaque for the two brushing cases (Figure 3.10) reveals that the switch to the STA dominance occurs much later in the case of Often brushing (after ~25 h) than in the case of Rare brushing (~15 h). Considering the increased cariogenic potential of STA, this delay in the proliferation of an aciduric microbial community 80

80 Rare brushing p 70

) STA -1 60 Often brushing

50 p

Microbial concentration (g L (g concentration Microbial STN 10 ACT VEL 0 0 20 40 60 80 100 120 Time (h) Figure 3.10 Microbial shifts at the the plaque surface for the Often brushing (continuous line) and Rare brushing (dotted line) cases , during 60 feeding/resting cycles. contributes to the tooth protection from demineralisation. All the results obtained with the current model for the two brushing cases emphasize how important this aspect of preventive dentistry is in limiting the cariogenic potential of dental plaque.

In all the studied situations (e.g., standard case, drinking habits, tooth brushing habits) the repeated exposure to glucose pulses selected for a microbial population dominated by aciduric streptococci (STA), with a smaller amount of non-aciduric streptococci (STN) always present. This is not surprising since STA are more adapted for the acidic environment (less pH inhibition) maintained with often glucose pulses, and both STA and STN have higher growth rates compared to ACT and VEL (see Table 3.4 for the values). However, for a more realistic study of microbial population dynamics in the dental plaque it might be useful to include also several survival strategies used by ACT and STN, which are not accessible to STA (Takahashi and Nyvad, 2011): (1) degradation of salivary glycoproteins (e.g., mucin) to sugars and aminoacids and their consumption in times of glucose scarcity (between meals); (2) utilization of arginine and arginine-containing peptides available in saliva for growth and local increase of the pH around STN and ACT; (3) utilization by ACT of urea (Kleinberg, 2002) and lactic acid (Takahashi and Yamada, 1996) for growth. It is also important to include in a future model the plaque microbial inactivation in depth (Ten Cate, 2006).

3.3. Conclusions The dental plaque model proposed in this study indicates that the microbial storage compounds (e.g., polyglucose) do not have a marked effect on tooth demineralisation. Model simulations show that drinking sugar-containing beverages over extended periods of time is more tooth damaging case than drinking at once. This frequent plaque exposure to glucose leads to the highest percentage and fastest shift towards the dominance of aciduric streptococci (STA) from all analysed cases. Through its ability to withstand the acidic environment the STA is clearly advantaged during an acid attack. Clearly, the most efficient way of limiting the damage of tooth enamel caused by microbial plaque is often tooth brushing.

3.4. References Atkins P, De Paula J, Physical Chemistry, 9th ed., W. H. Freeman, 2009. Bradshaw DJ, McKee AS, Marsh PD, Effects of carbohydrate pulses and pH on population shifts within oral microbial communities in vitro, J Dent Res 1989; 68(9):1298-1302. Dashper SG, Reynolds EC, pH regulation by Streptococcus mutans, J Dent Res 1992; 71:1159-1165. De Jong MH, Van der Hoeven JS, Van den Kieboom CWA, Camp PJM, Effects of oxygen on the growth and metabolism of Actinomyces viscous, FEMS Microbiol Ecol 1988; 53:45-52. Dibdin GH, Effect on a cariogenic challenge of saliva/plaque exchange via a thin salivary film studied by mathematical modelling, Caries Res 1990; 24:231-238. Dirksen TR, Little MF, Bibby BG, Crump SL: The pH of carious cavities. I - The effect of glucose and phosphate buffer on cavity pH. Arch Oral Biol 1962; 7:49-57. Dong Y-M, Pearce EIP, Yue L, Larsen MJ, Gao X-J, Wang J-D, Plaque pH and associated parameters in relation to caries, Caries Res 1999; 33:428-436. Fejerskov O, Kidd E: Dental caries: The disease and its clinical management, 2nd ed, Chicester, United Kingdom, Blackwell Munksgaard, 2008. Gustafsson BE, Quensel CE, Lanke LS, Lunquist C, Grahnen H, Bonow BE, Krasse B, The Vipeholm dental caries study. The effect of different levels of carbohydrate on caries activity in 436 individuals observed for 5 years. Acta Odontol Scand 1954; 11:232-364. Hamilton IR, St. Martin EJ, Evidence for the involvement of proton motive force in the transport of glucose by a mutant of Streptococcus mutans Strain DR0001 defective in glucose- phosphoenolpyruvate phosphotransferase activity, Infect Immun 1982; 36(2):567-575. Hamilton IR, Synthesis and degradation of intracellular polyglucose in Streptococcus salivarius, Can J Microbiol 1968; 14(1):65-77.

Heijnen JJ, Bioenergetics of microbial growth, Encyclopedia of Bioprocess Technology: Fermentation, Biocatalysis, and Bioseparation, M. C. Flickinger and S. W. Drew (Eds.), New York, Wiley-Interscience, 1999 Heijnen JJ, Kleerebezem R, Bioenergetics of microbial growth, Encyclopedia of Industrial Biotechnology: Bioprocess, Bioseparation, and Cell technology, M. C. Flickinger (Ed.), New York, John Wiley & Sons, 2010. Ilie O, Van Loosdrecht MCM, Picioreanu C: Mathematical modelling of tooth demineralisation and pH profiles in dental plaque. J Theor Biol 2012; 309:159–175. Jenkinson HF, Beyond the oral microbiom, Environ Microbiol 2011; 13(12): 3077-3087 Kleinberg I., A mixed-bacteria ecological approach to understanding the role of the oral bacteria in dental caries causation: an alternative to streptococcus mutans and the specific- plaque hypothesis, Crit Rev Oral Biol M 2002; 13(2):108-125. Komiyama K, Khandelwal RL, Acid production by Actinomyces viscosus of root surface caries and non-caries origin during glycogen synthesis and degradation at different pH levels, J Oral Pathol Med 1992; 21(8):343-347. Komiyama K, Khandelwal RL, Duncan DE, Glycogen synthetic abilities of Actinomyces viscosus and Actinomyces naeslundii freshly isolated from dental plaque over root surface caries lesions and non-carious sites, J Dent Res 1986; 65(6):899-902. Loesche WJ: Role of Streptococcus mutans in human dental decay. Microbiol Rev 1986; 50(4):353-380. Margolis HC, Moreno EC, Kinetics of hydroxyapatite dissolution in acetic, lactic and phosphoric acid solutions, Calcified Tissue Int 1992; 50:137-143. Marsh PD, Are dental diseases examples of ecological catastrophes?, Microbiology 2003; 149:279-294. Marsh PD, Martin MV, Lewis MAO, Williams DW: Oral Microbiology, 5th ed., Churchill Livingstone Elsevier, 2009. Martell AE, Smith RM, Critical stability constants, vol. 1–4, Plenum Press: New York, 1976. Moreno EC, Zahradnik RT, Chemistry of enamel subsurface demineralization in vitro, J Dent Res 1974; 53(2):226-235. Pearce EIF, Margolis HC, Kent Jr. RL, Effect of in situ plaque mineral supplementation on the state of saturation of plaque fluid during sugar-induced acidogenesis, Eur J Oral Sci 1999; 107:251-259. Ritz HL, Microbial population shifts in developing human dental plaque, Archs Oral Biol 1967; 12:1561-1568. Seeliger S, Janssen PH, Schink B, Energetics and kinetics of lactate fermentation to acetate and propionate via methylmalonyl-CoA or acrylyl-CoA, FEMS Microbiol Lett 2002; 211:65-70. Tahmourespour A, Nabinejad A, Shirian H, Ghasemipero N, The comparison of proteins elaborated by Streptococcus mutans strains isolated from caries free and susceptible subjects, Iranian Journal of Basic Medical Sciences 2013; 16 (4):648-652.

Takahashi N, Yamada T, Catabolic pathway for aerobic degradation of lacate by Actinomyces naeslundii, Oral Microbiol Immunol 1996; 11:193-198. Takahashi N., Nyvad B., The role of bacteria in the caries process: ecological perspectives, J Dent Res 2011; 90(3):294-303. Tanzer JM, Krichevsky MI, Keyes PH, The metabolic fate of glucose catabolized by a washed stationary phase caries-conductive Streptococcus, Caries Res 1969; 3:167-177 Ten Cate JM, Biofilms, a new approach to the microbiology of dental plaque, Odontology 2006; 94:1-9. Van Beelen P, Van der Hoeven JS, De Jong MH, Hoogendoorn H, The effect of oxygen on the growth and acid production of Streptococcus mutans and Streptococcus sanguis, FEMS Microbiol. Ecol 1986, 38:25-30 Van der Hoeven JS, De Jong MH, Camp PJM, Van den Kieboom CWA, Competition between oral Streptococcus species in the chemostat under alternating conditions of glucose limitation and excess, FEMS Microbiol. Ecol, 1985; 31:373-379. Van der Hoeven JS, Gottschal JC, Growth of mixed cultures of Actinomyces viscosus and Streptococcus mutans under dual limitation of glucose and oxygen, FEMS Microbiol Ecol 1989; 62:275-284. Van der Hoeven JS, van den Kieboom CWA, Camp PJM, Utilization of mucin by oral Streptococcus species, Anton Leeuw Int J G 1990; 57(3):165-172. Wanner O, Gujer W, A multispecies biofilm model, Biotechnol Bioeng 1986; 28:314-328. Zero DT, Van Houte J, Russo J, Enamel demineralization by acid produced from endogenous substrate in oral Streptococci, Arch Oral Biol 1986; 31(4):229-234.

Numerical

modelling of tooth enamel 4 subsurface lesion

formation induced

by dental plaque

4. Numerical modelling of tooth enamelDental caries subsurfaceusually start with a lesion in the superficial enamel layer (i.e., initial caries) developed under the influence lesion formationof dental plaque metabolism. induced If the process is not stopped,by such a lesion would advance until the dentino-enamel junction dental plaqueand even beyond, into the dentin and pulp. The study of the initial stages of caries development is important in understanding the disease as a whole but also in coming up with restorative strategies since in the early stages the process is reversible (Fejerskov and Kidd, 2008). One aspect that currently is not well understood concerns the aspect of initial lesions. The typical mineral profile of initial dental caries shows a region of roughly 100 μm at the tooth surface seemingly unaffected by demineralisation (Fejerskov and Kidd, 2008), with the main body of the lesion present under this surface layer. The differences in mineral content between the two zones can become considerable, especially in the

85 Chapter published in Caries Research 48:73-89 (2014) The movies refered to in this chapter can be downloaded from: http://biofilms.bt.tudelft.nl/ more advanced stages of caries development: up to 99% mineral content in the surface layer compared to just 50 - 75% in the body of the lesion (Robinson et al., 2000; Fejerskov and Kidd, 2008). This particular profile has been observed for the first time by Hollander and Saper (1935) who mistook it for a photographic artefact. Since then, much research – both experimental and theoretical – has been carried in order to confirm and explain the phenomena responsible for subsurface lesion formation (Gray and Francis, 1963; Van Dijk et al., 1979; Langdon et al., 1980; Ten Cate, 1983; Margolis and Moreno, 1985). From the hypotheses made to explain the mechanisms behind the surface layer formation, two have received special attention. These hypotheses have been tested in the current study. 1) Initial dental caries is mainly the consequence of an imbalance between two concurrent processes, demineralisation and remineralisation of tooth enamel (Loeshe, 1986; Hicks et al., 2004; Fejerskov and Kidd, 2008). These processes have the same thermodynamic driving force and both can occur in vivo. Therefore, the formation of a surface layer covering a subsurface lesion is seen as the result of an imbalance following repeated cycles of demineralisation (during meals) and remineralisation (between meals). 2) The mineral crystals are less soluble at the enamel surface. This might be due to: (i) a fluoride distribution in the enamel depth with the highest content at the surface (Arends and Christoffersen, 1986; Fejerskov and Kidd, 2008). The fluoride is incorporated into hydroxyapatite (HAP) structure forming a more stable mineral, fluorohydroxyapatite (FHAP), which makes the enamel less soluble during acid attacks (Koutsoukos et al., 1980; Robinson et al., 2000); (ii) HAP crystals with higher purity (hence, lower solubility) may develop at the enamel surface through a process called “Ostwald ripening” (Robinson et al., 2000; Fejerskov and Kidd, 2008). Other hypotheses in the dentistry literature have been proposed to explain the surface layer: faster remineralisation and delayed diffusion at the tooth surface (Silverstone, 1977; White et al, 1988); presence of chemical species that inhibit subsurface remineralisation (White et al, 1988); presence of salivary proteins that inhibit surface demineralisation and remineralisation (White et al., 1988); phase transformation at tooth surface resulting in the formation of dicalcium phosphate dehydrate (Margolis and Moreno, 1985; Fejerskov and Kidd, 2008). Although plausible, some of these premises are too speculative or cannot explain alone the formation of the surface layer. The interaction between physical, chemical and (micro)biological aspects involved in the subsurface lesion formation is highly complex. Intuition and simple calculations may not

86 be sufficient to allow a thorough theoretical evaluation of a conceptual hypothesis. Therefore, a series of numerical models of caries formation have been developed (Zimmerman, 1966a, b,c; Holly and Gray, 1968; Van Dijk, 1978; Ten Cate, 1983, Arends and Christoffersen, 1986). These models could test with minimum effort hypotheses for which laborious experiments (some even impossible) would have been required. A disadvantage though is that all these models are simplified to deal exclusively with the processes occurring in the superficial layer of the tooth (demineralisation or remineralisation), without considering the active influence of the dental plaque. Moreover, demineralisation and remineralisation were not studied together, most of the models focusing on demineralisation only, with only few (Ten Cate, 1983; Ten Cate, 2008) investigating tooth remineralisation. The current work presents a one-dimensional (1-d), time dependent mathematical model that integrates the kinetics of tooth demineralisation and remineralisation with mass transfer and with microbial conversions occurring in the dental plaque. The study is based on a previously developed numerical model for tooth demineralisation under the influence of an oral biofilm (Ilie et al., 2012), extended to evaluate the two main hypotheses for subsurface lesion formation. Our purpose was to establish which of the two theories can best explain the formation of a surface layer with the simplest assumptions. Case 1 assumes the tooth enamel is formed by HAP only and both demineralisation and remineralisation processes can occur, depending on the degree of saturation (DSHAP). Case 2 assumes that the tooth enamel includes FHAP crystals and studies the surface layer formation in the presence of a solubility gradient when only tooth demineralisation occurs.

4.1. Model description The one-dimensional (1-d) numerical model for calculation of pH and solute concentrations in active dental plaque (Ilie et al., 2012) was extended to describe the subsurface lesion formation in the tooth enamel. In the present model, the dental plaque composition was simplified to only one generic microbial group with one fermentative pathway, in order to allow an increased complexity of tooth chemistry (while maintaining a reasonable computational effort).

4.1.1. Processes (a) Microbial reactions From the multitude of microbial processes occurring in dental plaque, only lactic fermentation was considered since it has the highest impact on caries formation (Dawes and Dibdin, 1986; 87

Ilie et al., 2012). This simplification still offers a good description of acidity production due to microbial metabolism, while allowing to focus the study on the porosity profile developed in the tooth. Consequently, the dental plaque was reduced to one microbial group, the aciduric

Streptococci, with a constant concentration CX in the plaque (i.e., no microbial growth was considered). This microbial group is commonly associated with caries development due to its high acidogenicity and aciduricity (Loesche, 1986; Marsh et al., 2009). The anaerobic fermentation at high glucose concentration (eq. (4.1)) is the main source of lactic acid during a cariogenic attack. Moreover, lactic acid is the strongest and most abundant acid present in fermenting dental plaque (Borgström et al., 2000).

CGlu KIH, CH6126 Oo 2CHO 353 2H rqCGlu S,max X (4.1) CKCKGlu S,, Glu H I H

–1 –1 For the glucose uptake rate rGlu (mol L s ), Monod kinetics with glucose limitation

(concentration CGlu) and acid inhibition (proton concentration CH+) was assumed, with rate parameters given in Table 4.1 (Van Beelen et al., 1986).

(b) Acid-base equilibria The simplified model includes very fast protonation equilibria for lactate (microbial metabolic product) and hydrogen phosphate (tooth demineralisation product) in all model compartments (saliva, plaque and tooth) (eqs. (4.2) and (4.3)), with parameters in Table 4.1):

CHO ' CHO H rkCKCC (4.2) 363 353 LacH LacH LacH LacH H Lac HPO ' HPO2 H rkCKCC (4.3) 24 4 Pho Pho Pho Pho H Pho2 The other two phosphate equilibria were not considered because they only become important at pH values not reached in the current conditions. Although present in the previous dental plaque - model, other possible inorganic buffers (e.g., CO2/HCO3 ) as well as complexation equilibria with Ca2+ and microbial surface charges were neglected here to simplify the analysis of results.

(c) Tooth demineralisation and remineralisation Following an analysis of mechanisms suggested in the literature for the formation of subsurface enamel lesions, two theories have been studied: (Case 1) an alternation of two competing processes, enamel demineralisation and remineralisation, assuming only hydroxyapatite (HAP) crystals; (Case 2) demineralisation of the tooth enamel composed of fluorhydroxyapatite (FHAP) with higher concentration of fluoride at the enamel surface slowing down the dissolution rate. The model was adapted to each case, by considering different chemical reactions to occur in the tooth domain in addition to the acid-base equilibria. 88

Table 4.1 Model parameters

Parameter Symbol Value Reference –1 Biomass concentration in plaque CX 100 g L Morgenroth et al, 2000 –6 –1 –1 Maximum specific uptake rate qS,max 9.67×10 mol g s van der Hoeven et al, 1985 –4 –1 Monod half-saturation coefficient for glucose KS,Glu 1.22×10 mol L Hamilton and Martin, 1982 van der Hoeven et al, 1985 –6 –1 Proton inhibition constant KI,H+ 10 mol L Assumed for inhibition at pH = 4.5 6 –1 (a) Rate coefficients for dissociation equilibria kLacH 10 s Assumed kPho– –3.86 –1 Acidity constant for lactic acid KLacH 10 mol L Atkins and de Paula, 2009 –7.21 –1 Acidity constant for dihydrogen phosphate KPho– 10 mol L Atkins and de Paula, 2009 dissociation –12.32 -1 Acidity constant for hydrogen phosphate KPho2– 10 mol L Atkins and de Paula, 2009 dissociation –14 2 -2 Acidity constant for water dissociation KH2O 10 mol L Atkins and de Paula, 2009 –60 9 –9 Enamel solubility constant KS,HAP(en) 7.41×10 mol L Fejerskov and Kidd, 2008 –55 9 –9 HAP solubility constant KS,HAP 5.50×10 mol L Fejerskov and Kidd, 2008 –1 HAP molar weight MHAP 502 g mol Fejerskov and Kidd, 2008 –3 HAP density ρHAP 3160 kg m Fejerskov and Kidd, 2008 –8 Radius of HAP rod rrod 2.5×10 m Fejerskov and Kidd, 2008 –6 0.7 –1.1 –1 Demineralisation rate constant kd 7×10 mol m s Margolis and Moreno, 1999 –3 0.25 0.75 –1 Remineralisation rate constant kr 7.44×10 mol m s Nancollas, 1983 Demineralisation reaction order nd 0.3 Margolis and Moreno, 1992 Demineralisation reaction order md 2.8 Margolis and Moreno, 1992 Remineralisation reaction order nr 1.25 Nancollas, 1983 Length of tooth domain Lt 100 μm Assumed (b) Length of dental plaque domain Lp 250 μm Assumed Reduction factor for diffusion coefficients in the fp 0.25 Dibdin, 1981 plaque Time for the transition from CGlu,min to CGlu,max tstep 10 s Assumed Feeding period tfeed 2 min Assumed Length feeding/resting cycle tcycle 90 min Assumed Halving time in the saliva film th,f 0.5 min Dibdin, 1990 Halving time in the saliva bulk th,s 2.17 min Dibdin, 1990 Residence time in the saliva film Qf / Vf ln(2) / th,f min Calculated Residence time in the saliva bulk Q / Vs ln(2) / th,s min Calculated 3 –1 (c) Area per volume ratio in saliva film Af/ Vf 10 m Calculated Minimum glucose concentration (between meals) CGlu,min 0.07 mM Van der Hoeven et al., 1990 Maximum glucose concentration (during pulse) CS,Glu,max 560 mM Dirksen et al., 1962 Universal gas constant R 8.314 J mol–1 K–1 - Temperature T 310 K Chosen, 37 qC Faraday’s constant F 96485 C mol–1 - (a) Assumed an arbitrarily high value for very fast equilibria. (b) Based on the measured thickness of a 96 h dental plaque, grown in vivo (Wood et al., 2000). (c) Assumes that the saliva film thickness is 100 μm (Fejerskov and Kidd, 2008) and only 10% of the tooth is covered with plaque.

It was assumed that tooth enamel consists of parallel rods (represented as cylindrical crystals) of hydroxyapatite (HAP), the spaces between them being filled with aqueous solution (Ten Cate, 1979). Enamel porosity is the total volume of inter-rod spaces divided by total volume of the enamel (Figure 4.1a). Enamel porosity E is further used to characterize a lesion by the mineral content profile (i.e., the variation of (1 – E) over the first micrometers of enamel) and the lesion severity (i.e., the maximum porosity E reached within the lesion).

7 (a) (b) x 10 8 E > 0.215 0.01 ≤ E < 0.215

a = a = f(cylinder specific surface) ar = f(parallelepiped specific surface) m total specific surface (a ) r d 7 d

rrod 6 ) -3 5 m 2 (m r 4

3 and a

HAP d a m specific growth surface (a ) rod 2 r

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Porosity, E (m3 m-3) Figure 4.1. (a) Schematic representation of the volume inside tooth enamel. The hydroxyapatite (HAP) rods are assumed cylinders with radius rrod, while the inter-rod space is a parallelepiped. Calculation of the specific surface for remineralisation (ar) at: (Left) high enamel porosity function of area of the HAP cylinders, and (Right) low enamel porosity function of inter-rod surface area. (b) Calculated total specific surface, ad, and specific growth surface, ar, function of enamel porosity E.

Case 1. Tooth enamel consisting of hydroxyapatite (HAP) The first hypothesis is that the subsurface lesion is formed as a consequence of an imbalance between two concurrent processes, demineralisation and remineralisation. It was assumed that the tooth enamel consists of homogeneous HAP, therefore removing the bias of a solubility gradient due to, for example, a fluoride distribution over the enamel depth. All the chemical species listed in Table 4.2, with the exception of fluoride, have been included in this case.

Table 4.2 Chemical components in the model

Diffusion coefficient in water(c) Saliva concentrations Initial(a) Name Symbol Inlet(b) (10–9 m2s–1) Reference C (0) Reference f,j (mol m–3) (mol m–3) Lactate ion Lac– 1.31 Vanýsek, 2001 0 0 Assumed Lactic acid LacH 1.31 Vanýsek, 2001 0 0 Assumed –4 –4 Proton H+ 11.81 Vanýsek, 2001 1.29×10 8.71×10 [Fejerskov and Kidd, 2008] Equilibrium, –3 CPho,tot initial= 13.2 mol m Hydrogen phosphate Pho2– 0.96 Vanýsek, 2001 4.27 2.24 –3 CPho,tot inlet= 5.4 mol m [Fejerskov and Kidd, 2008] Dihydrogen phosphate Pho– 1.22 Vanýsek, 2001 8.93 3.16 Equilibrium Calcium ion Ca2+ 1.00 Vanýsek, 2001 3.5 1.32 [Fejerskov and Kidd, 2008] Cation (Potassium ion) K+ 2.49 Vanýsek, 2001 51.5 19.4 [Fejerskov and Kidd, 2008] Anion (Chloride ion) Cl– 2.57 Vanýsek, 2001 41.03 14.4 Charge balance Fluoride ion F– 1.48 Vanýsek, 2001 6 1 [Fejerskov and Kidd, 2008] Van der Hoeven et al., 1989; Glucose Glu 0.85 Vanýsek, 2001 0.07 C (t) s,Glu Calculated for inlet (a) Values corresponding those measured in human resting plaque (assumed to be in a steady state) (b) Values corresponding those measured in unstimulated mixed (from all salivary glands) human saliva (c) Diffusion coefficients for all solutes in water are reduced in plaque by a factor of 0.25 (Dibdin, 1981) and in the enamel reduced by the porosity E. 90

For the HAP demineralisation reaction (eq. (4.4)) Ca PO OH ' 5Ca23 3PO HO (4.4) 543 4 * –2 –1 the empirical equation for the surface-based demineralisation rate, rdCa, (mol Ca m s ), was proposed in Margolis and Moreno (1992) (eqs. (4.5) and (4.6), Table 4.1).

n * md d rkDSCdCa,() d 1 HAP ¦ AiH for DSHAP < 1 (4.5)

* rdCa, 0 for DSHAP ≥ 1 (4.6)

The demineralisation rate is a function of the concentration of all protonated acids, CA(i)H, and the degree of saturation of the solution with respect to HAP, DSHAP (eq. (4.7)):

19 §·IP CC53 CC 53 DS ¨¸HAP and IP Ca23 Pho Ca 22 Pho K3 K (4.7) HAP ¨¸K HAPCC4 Pho2 H2O ©¹SHAPen,() HO H

DSHAP depends on the ionic product IPHAP, and the solubility constant, KS,HAP(en). Due to the impurities present in the tooth, the enamel solubility KS,HAP(en) has a lower value than that of pure HAP (Table 4.1). The demineralisation occurs only when the solution is undersaturated in respect to HAP, DSHAP < 1. –3 –1 The volumetric demineralisation rate of HAP (eq. (4.8)), rd,HAP (mol HAP m s ), was 2 –3 derived by multiplying the surface-based rate with the specific crystal surface, ad (m m ), and considering the reaction stoichiometry (eq. (4.4)):

* rradHAP,, 0.2 dCa d (4.8)

The specific crystal surface ad was calculated as the area/volume ratio of a rod with a constant radius, rrod, multiplied with the volume fraction of crystal in the enamel (the mineral content, 1–E) (Ten Cate, 1979) (eq. (4.9) and Figure 4.1b): 2 aEd 1 (4.9) rrod

* –2 –1 The surface-based relation for HAP remineralisation rate, rrHAP, (mol HAP m s ) (eqs. (4.10) and (4.11)) was empirically determined by Nancollas (1983) from in vitro experiments on crystallisation of bovine enamel blocks.

* nr rkDSr, HAP r HAP 1 for DSHAP ≥ 1 (4.10)

* rrHAP, 0 for DSHAP < 1 (4.11)

The lumped remineralization rate constant, kr, and the reaction order, nr, were calculated from the results of Nancollas (1983) (values in Table 4.1).

–3 –1 The volumetric remineralisation rate, rr,HAP (mol HAP m s ), was obtained from the surface-based rate multiplied with the specific growth surface, ar (eq. (4.12)):

* rrarHAP,, rHAP r (4.12)

The specific growth surface, ar, is the fraction of the total specific surface, ad , that is available for crystal growth (Figure 4.1a). Depending on the enamel porosity values (E), the specific growth surface is calculated in two ranges: (i) At porosity higher than 0.215 (corresponding to perfectly cylindrical HAP rods Ten Cate, 1979) the entire crystal surface is available for remineralisation (eq. (4.13)). 2 aard 1 E, when E > 0.215 (4.13) rrod (ii) At porosity values lower than 0.215, as the inter-rod space gets filled, it was assumed that only a part of the total specific crystal surface ad remains available for crystal growth. The ar area was assumed to become zero when porosity reaches a minimal value Emin

(i.e., remineralisation stops when Emin = 0.01). No pre-existing gradients in porosity were assumed and a uniform initial porosity Emin for healthy enamel was considered (Fejerskov and Kidd, 2008). The specific growth surface was in this case calculated from a sigmoid function making the growth specific area zero at Emin, mimicking a shrinking parallelepiped inter-rod space (Figure 4.1b) (eq. (4.14)).

§·1 aa ¨¸1 , when 0.01 < E < 0.215 (4.14) rd¨¸ ©¹1 exp 45 E 0.15 Finally, the change in enamel porosity in time (i.e., the lesion formation) is the difference between the volumetric demineralisation and remineralisation rates as shown in eq. (4.15) (adapted from van Dijk, 1978):

dE M HAP rrd,, HAP r HAP with EE 0 min (4.15) dt UHAP

Case 2. Tooth enamel includes fluorohydroxyapatite (FHAP) This case tests the hypothesis that subsurface lesion is the result of an enamel solubility gradient caused by the presence of less soluble fluoride inclusions (fluorohydroxyapatite, FHAP) near the enamel surface. To represent this situation, fluoride ions (F–) were introduced in the model (Table 4.2). Fluoride was considered non-reactive in saliva and dental plaque and exchangeable between all the model compartments. Only FHAP demineralisation (eq. (4.16)) is allowed in this case, without any remineralisation. 92

-27 10 0.5 K S, HAP (x ) F 0.4

-28 10 )

F 0.3 F x S, HAP (x S,

K 0.2 -29 10

0.1 x F

-30 10 0 -15 -10 -5 0 Enamel depth (Pm)

Figure 4.2. Imposed distribution of fluoride fraction xF and the mixed FHAP/HAP crystal solubility constant KS,FHAP(xF) within 15 μm from the enamel surface.

The solubility of a mixed enamel crystal containing FHAP is lower than that of HAP. It was experimentally observed that fluoride diffuses and partly replaces the hydroxyl within the first 100 μm of enamel (Moreno et al, 1974; Fejerskov and Kidd, 2008) according to the reaction (eq. (4.16))

23 Ca54 PO OH Fx ' 5Ca 3PO4 1xxFF HO F (4.16) 31xF F

The molar fraction of fluoride, xnnnFF /( FHO ) decreased exponentially within 5 μm from the enamel surface from its maximum value xF,max = 0.5 to a minimum xF,min = 0.05 (Moreno et al, 1974) according to eq. (4.17) (x in Pm) (Figure 4.2):

xxFF ,min x F ,max x F ,min exp x 100 (4.17)

The dependency of the mixed FHAP/HAP solubility constant, Kx, on the fraction of fluoride

2 was measured by Moreno et al., 1974 as KxxxFF exp 8.7923 10.2691 31.9308 . These results were adjusted proportionally for the replacement of hydroxyl in enamel instead of pure HAP, KKKK / . The fraction of fluoride and the FHAP solubility SHAPx,()F x SHAPen ,() SHAP , constant within enamel in the first 15 μm from the tooth surface are displayed in Figure 4.2. We considered the same type of rate expression for the dissolution of mixed -3 -1 FHAP/HAP crystals (called here HAP(xF)) as for HAP, so that rd,HAP(xF) (mol HAP(xF) m s ) becomes:

m n rkDSCa 0.2 1 d d (4.18) dHAPx,()FF d HAPx () ¦ AiH () d

19 §·IP 53 xF HAP() xF CCCa22 Pho C F 3 DSHAP() x ¨¸ and IPHAP() x K Pho 2 K H2O (4.19) F ¨¸K F C4xF ©¹SHAPx,()F H

Considering that a steady fluoride content in enamel builds up over years and that the time periods simulated were relative short (hours), the contribution of FHAP remineralisation was assumed negligible and the gradient of fluoride content is stable in time. The variation of FHAP enamel porosity in time was calculated similarly to eq. (4.15) without remineralisation: dE M r HAP EE0 (4.20) dHAPx,()F min dt UHAP

4.1.2. Balance equations The change of tooth porosity in time depends on the spatial distribution of the concentrations of several solute species (Table 4.2). We assumed one-dimensional (1-d) concentration gradients in tooth enamel and in the dental plaque on a direction perpendicular to the tooth surface. The entire saliva volume is considered well-mixed and the concentrations are varying only in time. The 1-d tooth enamel domain extends from x=0 (deep enamel) to x=Lt (enamel- plaque interface) and the plaque is between x=Lt and x=Lt+Lp (the plaque-saliva interface). Chemical species (solutes) were exchangeable between these model compartments. To calculate the solute concentrations and the amount of demineralised tooth enamel, mole and charge balances were defined on enamel, plaque and saliva domains.

(a) Saliva

The saliva film in direct contact with the plaque has a volume Vf, and represents only a small fraction of the total saliva present in the mouth. The change in concentration of each chemical species j (Table 4.2), Cf,j (t), is calculated from eq. (4.21), similar to the approach used in (Ilie et al., 2012):

dCfj, Q f A f CCsj,, f j N pj , R j (4.21) dt Vff V

Eq. (4.21) is based on the species exchange with the saliva bulk with a flow rate Qf, exchange with the dental plaque of area Af with the flux Np,j (diffusive flux at the plaque-saliva interface) and the net reaction rate of each component, Rj (Table 4.1). The saliva inlet concentrations for chemical species Cs,j are set as in a previous study (Ilie et al., 2012) (Table 4.2), based on chemical speciation (mass-action laws) and charge balance (solution electroneutrality, ¦ zCjsj, 0, where zj is the charge number). j For inlet concentration of glucose, repeated feeding/clearance/resting cycles having the total time of 1 hour were imposed. In the feeding regime, Cs,Glu quickly increased within tstep 94 from CGlu,min to a maximum concentration Cs,Glu,max, which was maintained for the entire feeding time, tfeed. After feeding, clearance begins and glucose concentration decreases exponentially (eq. (4.22)) until the end of the resting period:

ªºQ CtCtts,, Glu() s Glu ( step feed )exp«» ttt ( step feed ) C Glu , min (4.22) ¬¼Vs

The residence time (Q/Vs = ln(2)/th,s) was expressed based on the halving time in the saliva bulk compartment th,s (Dibdin, 1990).

(b) Dental Plaque

In the dental plaque the solute concentrations Cp,j change from the plaque surface to the tooth surface. Therefore, 1-d mole balances are set for all solutes j, as in (Ilie et al., 2012) and based on Nernst-Planck equations (4.23) coupled with the electroneutrality condition (4.24).

Equations include net reaction rates Rp,j and the transport by molecular diffusion (Dp,j, diffusion coefficient of a solute in the plaque at 37 ºC, calculated as 25% from the diffusion coefficients in water, Dj, according to Dibdin, 1981) and ion electro-migration (zj charge number, F Faraday’s constant, R universal gas constant and T temperature):

2 wwCCzFpj,, pj j ww)§· DDCR (4.23) zC 0 (4.24) pj,,,,2 pj¨¸ pj pj ¦ jpj, wwwtRTxxw©¹x j

Rp,j is the net reaction rate of each compound, accounting for both chemical and microbial processes. The additional state variable )(x,t) is an electrical potential field developed due to ion transport with different diffusion rates. The initial values for all concentrations were considered equal to those in the saliva film, while the potential )(x,0) = 0.

For the saliva-plaque interface (at x = Lp+Lt) the concentrations of all chemical components are equal to those in the saliva film and Φ = 0 as reference value. For the plaque- tooth interface, total flux continuity is set for all the components except glucose, for which the tooth is impermeable (zero-flux condition) (Fremlin and Mathieson, 1961).

(c) Tooth The most important development brought by the current model compared to Ilie et al. (2012) is the explicit calculation of mineral content variation in the tooth depth. This required the representation of the tooth as another 1-d domain adjacent to the plaque domain. Because the focus of this study was on the initial development of subsurface lesions, only the first 100 μm in depth from the tooth surface were considered.

As in the plaque domain, the variation in time of solute concentrations (all species but glucose) is calculated by the same Nernst-Plank equations coupled with charge balance:

2 wwCCzFtj,, tj j ww)§· DDCR (4.25) zC 0 (4.26) tj,,,,2 tj¨¸ tj tj ¦ jtj, wwwtRTxxw©¹x j

The net reaction rate, Rt,j, accounts only for chemical processes (acid-base equilibria and tooth (de)remineralisation) since there are no microbial species present inside the tooth. Lower enamel porosity leads to slower diffusion of any chemical species in the tooth domain

DDEtj, j in respect to diffusivities in water, Dj, from Table 4.2 (Ten Cate, 1979). The enamel porosity is variable in time and over the tooth depth, E(x,t), and it is calculated differently, function of the tested hypothesis, with equations (4.15) (Case 1) or (4.20) (Case 2). The boundary conditions were: flux continuity at the plaque-tooth interface and zero-flux / electrical insulation at the deep enamel boundary (x = 0). Initial concentrations of chemical components in the tooth are equal to those in the plaque and the initial porosity is Emin (i.e., healthy enamel).

4.1.3. Model solution The model equations were implemented in COMSOL Multiphysics software (COMSOL 4.1, Comsol Inc, Burlington, MA, www.comsol.com), which allows a very flexible and well- structured model construction. COMSOL solves the resulting system of ordinary differential, partial differential and algebraic equations by finite element methods. The plaque domain was discretized on a mesh of 100 elements with finer mesh size next to both the saliva and enamel boundaries. The mesh in the tooth domain contained also 100 elements of size and finer meshing at the enamel surface, where the concentration gradients can become very steep. First, the mole and charge balances in plaque and enamel eqs. (4.23)- (4.26) were solved towards the steady state solution, which represents a situation of resting plaque in contact with constant composition saliva. Second, the steady state solution was used as initial condition for the time-dependent simulations performed during 80 sequential feeding /clearance/resting cycles of 1.5 hours each. The time-dependent simulations included the complete model equations in saliva (eq. (4.21)), plaque (eq. (4.23), (4.24)) and tooth (eq. (4.25), (4.26) and (4.15) or (4.20)), with the associated boundary conditions and constitutive (rate) equations.

4.2. Results The numerical model described in this study is aiming to offer a better understanding of the mechanisms leading to a tooth subsurface lesion under the influence of acids produced in dental plaque. Two cases were chosen in order to distinguish if the lesion could be caused solely by an alternation of demineralisation/remineralisation processes (i.e., originating from the same thermodynamic force), or by a fluoride gradient (and implicitly solubility gradient). In Case 1, tooth enamel consists of HAP with demineralisation and remineralisation, and in Case 2 tooth enamel contains a mixture of HAP and FHAP with only demineralisation occurring. For each case we analysed (1) the influence of glucose pulses on the pH and the total amount of demineralised enamel at tooth surface in time, (2) the pH profile, concentration of calcium ions and total phosphate species over the plaque depth as well as demineralisation and remineralisation rates within the enamel at a certain time, (3) the evolution of porosity profiles inside the enamel.

4.2.1. Case 1. Tooth enamel consisting of hydroxyapatite (HAP) (a) Concentration profiles Feeding/resting cycle. The glucose concentration in the bulk saliva and its influence on the pH profile at the tooth surface (Stephan curve) and subsequently on the amount of HAP lost in time are shown in Figure 4.3a. Moments after the concentration of glucose in saliva increases, the pH at the tooth surface is sharply decreasing below the critical value for demineralisation

(pHcritical = 5.5). In these conditions, the plaque fluid at the tooth surface becomes undersaturated with respect to HAP (DSHAP < 1) and the tooth enamel starts to demineralise. As long as there is glucose in the plaque, the pH stays acidic and an increasing amount of HAP is lost. After approximately 20 min from the beginning of the cycle, when the glucose was cleared from the saliva and dental plaque, the pH starts to restore towards the steady state value. As soon as the pH rises above 5.5, the amount of HAP lost decreases, meaning that remineralisation is now active. While remineralisation is active (i.e., E > 0.01 and DSHAP > 1) the pH cannot restore to the steady state value of 7 due to the continuous consumption of hydroxyl ions during this process (eq. (4.4)). Restoration to steady state values in tooth and plaque can occur only when the tooth is completely remineralised (i.e., E = Emin = 0.01). In the current case, the tooth is not fully remineralised at the end of the 90 min cycle, and there is a net loss of approx. 5 × 10– 4 (mol HAP m–2) per cycle (Figure 4.3a). This corresponds to 10– 4 (mg Ca mm–2) lost per cycle, an amount in the same order of magnitude with the experimental 97

observations of Margolis and Moreno (1992) and with the results of our previous model (Ilie et al., 2012).The net amount of lost HAP per feeding/resting cycle depends thus on the length of the remineralisation stage.

Demineralisation phase. An example of concentration profiles along the dental plaque and

tooth domains for the ions influencing DSHAP (i.e., calcium, total phosphates and protons/pH), after 1 min in the feeding period (i.e., during the acid attack and maximum demineralisation rate) is presented in Figure 4.3b. At this time, the maxima of total phosphate and calcium concentrations are close to the enamel surface, where these components are released by HAP demineralisation, and diffuse towards the plaque surface (therefore the linear concentration profile, showing also that a quasi-steady state was reached in the plaque). These species diffuse also inside the tooth, but much slower, therefore the concentration profiles are curved and far from steady state. The HAP demineralisation in which these species are produced

occurs only in a very narrow region in the tooth where pH < pHcritical. In the plaque depth, the pH decreases from nearly neutral in saliva to almost 5 at the tooth surface. This decrease is caused by the microbial production of organic acids. Within the tooth enamel the pH increases to neutral values far from the tooth surface due to the slower diffusion of protons inside the tooth compared to the plaque. The proton consumption in demineralisation leads also to a small pH increase in the plaque depth. (a) -3 x 10 (b) tooth plaque 1.4 7.5 500 5 25 7 ) -1

) 7 -2 HAP 4.5 s 1.2 demin -3 6.5 7 (mM) c 400 4 6 20 glucose 1 pH 3.5 pH 6 6.5 5 300 3 15 0.8 5.5 6 4 pH 2.5 pH c 0.6 Pho total 5 200 2 3 10 5.5 0.4 1.5 2 4.5 100 c Glucose concentration (mM) 1 Ca2+ 5 0.2 5 Calcium concentration (mM) m r 1 demin HAP 4 Total demineralised HAP (mol m m (mol HAP demineralised Total

0.5 concentration phosphate Total

m (mol rate Demineralisation HAP 0 4.5 0 0 0 0 3.5 0 20 40 60 80 -100 -50 0 50 100 150 200 250 t (min) Length (Pm) (c) tooth plaque 5 25 7 Figure 4.3. Calculated concentration profiles for Case ) 7 -1 1, when the tooth enamel contains only HAP.

4.5 s pH -3 6.5 4 6 20 (mM) (a) pH, glucose concentration and total amount of demineralised HAP at the tooth surface during one 3.5 6 5 feeding/resting cycle; (b) Concentrations of calcium 3 15 5.5 4 and total phosphate, pH and rate of HAP 2.5 pH c demineralisation along the dental plaque and in the Pho total 5 2 3 10 tooth enamel after 1 min from the beginning of the 1.5 2 4.5 first feeding/resting cycle. The grey dotted line c 1 Ca2+ 5 represents the plaque-tooth boundary. Concentrations Calcium concentration (mM) 1 4 of calcium and total phosphate, pH and rate of HAP

0.5 concentration phosphate Total r remin HAP m m (mol rate Remineralisation HAP 0 0 0 3.5 remineralisation profile along the dental plaque and in -100 -50 0 50 100 150 200 250 the tooth enamel after 30 min from the beginning of Length (Pm) the first feeding/resting cycle. 98

Remineralisation phase. After 30 min from the beginning of the feeding/resting cycle, during the remineralisation stage, the calcium and phosphate are consumed in the tooth, therefore their fluxes are reversed compared to the demineralisation stage (Figure 4.3c). At this time, the maximum concentrations are at the saliva-plaque boundary and the minimum at the tooth surface where the ions are consumed. Although the pH is above the critical value along the entire tooth domain, the remineralisation reaction takes place only in a narrow region at the very surface of the tooth. This means that during the previous demineralisation stage, only a narrow region of HAP enamel was demineralised leading to porosity values higher than Emin. The maximum calculated rate of remineralisation is approximately ten times smaller than the maximum rate of demineralisation and is in the same range as the rates experimentally observed by de Rooij and Nancolas (1984) and White et al. (1988).

Mineral profiles during one cycle. The mineral content along the first 5 μm from the tooth surface at different stages after the beginning of the feeding/resting cycle is represented in Figure 4.4 for the 80th cycle. The model clearly demonstrates how a subsurface lesion is formed after many feeding/resting cycles only by alternating phases of demineralisation and remineralisation. The most chemically active part of the tooth appears to be near its surface: the HAP content during demineralisation (22 min) is 15% lower than the content of the same region at the end of the cycle (90 min). The current model suggests that a pre-existing gradient (such as in the enamel solubility constant (van Dijk et al., 1979; Arends and Cristoffersen, 1986), or demineralisation/ /remineralisation reaction rate constants (ten Cate, 1979), or fluoride distribution (Wang et al.,

plaque-tooth boundary 100 p 90 min 90 45 min 22 min 80 70

(%) content Mineral 50

30 -5 -4 -3 -2 -1 0 Enamel depth (Pm) Figure 4.4. Calculated mineral content profiles in the tooth enamel at different stages during the 80th feeding/resting cycle: 22 min - end of demineralisation stage, 45 min - during remineralisation and 90 min - at the end of the remineralisation (resting) period. 99

1996)) is not required in order to develop a subsurface lesion when remineralisation is present. The remineralisation rate is highest at the tooth surface. If the remineralisation period is long enough, the entire lesion can remineralise. However, for short recovery times (as it is the case for frequent sugar consumption) the lesion can remineralise only superficially, while higher porosity builds up in the lesion depth. Therefore, the porosity of the surface layer will remain within certain limits (1-15% in the current model) while the porosity in the subsurface area will slowly increase with each demineralisation-remineralisation cycle (for example, up to 75% as shown in Movie 1 in the Supplementary Material).

(b) Sensitivity analysis Important factors to be considered when discussing the development of a mineral surface layer via remineralisation are: (1) the diffusion coefficient inside the tooth; (2) the HAP remineralisation rate; (3) length of the remineralisation period (resting time). To study their influence, a sensitivity analysis was performed. For these parameters, simulations were run for 25 feeding/resting cycles, with the base value and with increased/decreased parameter values (Table 4.3).

Table 4.3. Sensitivity analysis results

Parameter Diffusion inside the tooth (Dt,j) Remineralisation rate constant (km) Length of feeding/resting cycle (tcycle) name (fmin = 0.2) (fmin = 0.2; fmax = 5) (fmin = 0.5; fmax = 2)

(a) (a) (a) Lost HAP Min. HAP Lost HAP Min. HAP Lost HAP Min. HAP (mmol HAP m – 2) content(b) (%) (mmol HAP m – 2) content(b) (%) (mmol HAP m – 2) content(b) (%) Parameter value f × Standard min 12.6 40 14.9 69 29 36 value Standard value 14.6 64 14.6 64 14.6 64 f × Standard max - - 14.4 60 1.5 96 value (a) Amount of HAP lost at the end of all the 25 feeding / resting cycles. (b) Minimum content of HAP at the end of the 25th feeding / resting cycle.

Diffusion inside the tooth. A subsurface lesion was obtained for each tested value of the diffusion coefficients in the tooth (Figure 4.5a). However, it appears that the faster the diffusion, the less pronounced (higher mineral content) but wider and deeper within the enamel the lesion forms. Indeed, faster diffusion creates a more evenly distributed

concentration profile of ions (and consequently DSHAP) within the enamel, and the demineralisation reaction is active over an extended depth. Although the lesion appears to be less severe for faster diffusion, the total amount of mineral lost is still the highest, with ~10% more HAP lost compared to the standard case (Table 4.3). Therefore, the effective diffusion of chemical species within the tooth has a significant influence on the shape of the lesion and on the time when the porosity becomes large enough to make the lesion clinically observable. 100

plaque-tooth boundary p

100 100 90 90 (a) (b) 80 80

70 70

60 60 50 50 40 40

30 0.2 x D (%) content Mineral 30

Mineral content (%) content Mineral 0.2 x k t,j m 20 D 20 k t,j m 10 5 x D 10 5 x k t,j m 0 0 -30 -25 -20 -15 -10 -5 0 -30 -25 -20 -15 -10 -5 0 Enamel depth (Pm) Enamel depth (Pm) 50 100 150 100 100

90 200 (c) (d) 80 80 250 70 100 60 60 80 50 300 40 40 60

Mineral content % content Mineral

Mineral content (%) content Mineral 30 0.5 x t 400 Pm cycle 40

% mineral Minimum 20 t 20 cycle 20 10 2 x t 0 200 400 cycle Plaque thickness (Pm) 0 0 -30 -25 -20 -15 -10 -5 0 -30 -25 -20 -15 -10 -5 0 Enamel depth (Pm) Enamel depth (Pm) Figure 4.5. Mineral content in the enamel at the end of the 25th feeding/resting cycle, with various parameter values for the demineralisation-remineralisation model (Case 1): (a) Effective diffusion coefficient in the enamel; (b) Remineralisation rate constant; (c) Length of the feeding/resting cycle; (d) Plaque thickness between 50 and 400 μm. The figure insert shows the minimum mineral content reached after 25 cycles as a function of the plaque thickness.

Remineralisation rate. Kinetics of remineralisation may influence the mineral profile,

therefore the remineralisation rate constant (km) was also varied (Figure 4.5b). The simulated

lesions had very similar mineral profiles, with only 1% more HAP lost for the highest km value compared with the standard case (Table 4.3). For slower remineralisation rate, the surface layer remains porous for extended periods of time, thus with increased the effective diffusivity. This allows ions to penetrate deeper inside the tooth and to better restore the demineralised enamel. In conclusion, small variations in remineralisation rate may not influence significantly the formation of a subsurface lesion, but only the “quality” of the surface layer formed. 101

Length of the remineralisation phase. The highest impact on the lesion formation proved to be the length of the remineralisation period. This is a behavioural factor, determined by the eating habits of the individual because caries formation is mainly associated with frequent sugar intake, therefore short remineralisation time. For longer remineralisation times th (tcycle = 180 min) there is no lesion formed at the end of the 25 feeding/resting cycle (Figure 4.5c). This shows that given enough time the lesion formed after an acid attack could be fully remineralised. However, for a shorter remineralisation time (tcycle = 30 min), the lesion is formed directly at the tooth surface and no surface layer is present, giving the highest amount of HAP lost from all the cases studied (Table 4.3). The restoration period is therefore very important and the numerical simulations support the widespread observation that often consumption of sugars is a key factor in caries formation.

Plaque thickness. Interestingly, the model of enamel demineralisation and remineralisation allows the identification of a minimum plaque thickness that can induce a lesion. With the current model parameters (Case 1), for plaque thickness Lp smaller than 150 μm there was no significant mineral loss (Figure 4.5d). Above 150 μm, the thicker the plaque the more extensive enamel lesions developed. It should be noted that the model only indicates the existence of a critical plaque thickness needed for the development of an incipient caries lesion. The value of this critical thickness is subjected to many parameters, such as the length of remineralisation phase, plaque composition and activity, salivary flow and composition, etc. Showing that caries onset can be prevented by reducing, though not completely eliminating, the plaque would be relevant to understanding appropriate oral hygiene protocols.

4.2.2. Case 2. Tooth enamel includes fluorohydroxyapatite (FHAP) This case tested in what extent the subsurface lesion is the result of an increased enamel resistance to acid attacks due to the presence of less soluble fluoride inclusions near the enamel surface. The glucose concentration in the saliva bulk, the pH at the tooth surface and the total amount of demineralised FHAP during one cycle in this case are displayed in Figure 4.6a. The same correlation is noticed between the glucose pulse, the sudden pH drop and the increasing amount of demineralised FHAP. Because there is no remineralisation process in this case, the amount of FHAP remains constant once the demineralisation stopped. The lower solubility of FHAP compared to HAP is reflected in the smaller amount of demineralised FHAP compared with Case 1. The minimum pH value is lower than the one obtained for the HAP case, due to the lower buffer capacity at lower concentrations of phosphate released during demineralisation. 102

(a) -3 x 10 (b) tooth plaque

1.4 7.5 500 5 25 ) 7 -1

) 7 s

-2 pH 4.5 -3 1.2 7 6.5 400 4 6 20 c glucose pH 1 FHAP 3.5 6 6.5 demin 5 300 3 15 0.8 5.5 4 6 2.5 pH pH c 0.6 Pho total 5 200 2 3 10 5.5 1.5 0.4 2 4.5 c 1 Ca2+ 5 100 Calcium concentration (mM) 5 Glucose concentration (mM) 0.2 1 4

0.5 Total phosphate concentration (mM) Total demineralised FHAP (mol m r m demin FHAP FHAP Demineralisation rate (mol m 0 4.5 0 0 0 0 3.5 0 20 40 60 80 -100 -50 0 50 100 150 200 250 t (min) Length (Pm) Figure 4.6. Calculated concentration profiles for Case 2, when the tooth enamel contains a fluoride distribution. (a) pH, glucose concentration and total amount of demineralised FHAP at the tooth surface during one feeding/resting cycle; (b) Concentrations of calcium and total phosphate, pH and rate of FHAP demineralisation along the dental plaque and in the tooth enamel after 1 min from the beginning of the first feeding/resting cycle. The grey dotted line represents the plaque-tooth boundary.

Qualitatively, the profiles of the total phosphate species and calcium ion concentrations in this case (Figure 4.6b) resemble those for Case 1 (Figure 4.3b), one minute after the beginning of the feeding/resting cycle. The maximum rate of FHAP demineralisation was in this case 6 times lower than the HAP demineralisation rate and it led to lower concentrations of calcium and total phosphate species in the plaque. As expected, the first few micrometers of enamel do not dissolve due to the fluoride presence, but there is still mineral loss under this surface layer. This was also assumed in the model of Wang et al (1996) on fluoride adsorption in enamel and experimentally observed by Chu et al, 1989. It remains questionable whether the surface layer is completely inert in vivo during lesion development. Experimental data is available on the development of subsurface lesions in environments both with and without fluoride (Theuns et al, 1984; Margolis et al, 1999; Nancollas, 1983). Although the fluoride presence may not be the sole reason for the development of a subsurface lesion, the model shows how fluoride distribution alone can lead to subsurface lesions comparable with the experimental observations.

4.2.3. Comparison of the lesions Formation of surface layers in time was compared for the studied cases: (1) HAP without remineralisation (Figure 4.7a); (2) HAP by alternating demineralisation/remineralisation periods (Figure 4.7b); (3) HAP/FHAP by imposing a fluoride distribution over the tooth domain (Figure 4.7c). As expected, there was no surface layer present and the carious lesion formed at the tooth surface without remineralisation because there was no restoring process to balance the demineralisation damage (Figure 4.7a). After each demineralisation cycle, the mineral

103

(a) Tooth enamel consisting of HAP plaque-tooth (b) Case 1: Tooth enamel consisting of HAP No remineralisation boundary 100 100 p 80 80

Time 60 60 Time 40 40

Mineral content (%) content Mineral Mineral content (%) content Mineral 20 20

0 0 -35 -30 -25 -20 -15 -10 -5 0 -35 -30 -25 -20 -15 -10 -5 0 Enamel depth (Pm) Enamel depth (Pm) (c) Case 2: Tooth enamel consisting of FHAP Figure 4.7 Calculated mineral content profiles 100 (progression of caries lesion) inside the tooth enamel at the end of each 8th feeding/resting cycle. The 80 plaque-tooth boundary is represented at depth zero. (a) Only demineralisation occurs without any 60 remineralisation; (b) Demineralisation and Time remineralisation of HAP (Case 1); (c) Demineralisation of FHAP/HAP (Case 2). 40 Animations of calculated mineral content evolution

Mineral content (%) content Mineral are presented in Supplementary Material, Movie 1 and 20 Movie 2.

0 -35 -30 -25 -20 -15 -10 -5 0 Enamel depth (Pm) content near the surface would decrease, and the simulated lesion extended in the enamel depth at a quasi-constant rate reaching approximately 30 μm after 80 feeding/resting cycles. When remineralisation was active (Case 1), at the end of the 80th cycle the minimum mineral content reached 30% within the first 5 μm of tooth enamel, with a distinct surface layer formed at the tooth-plaque interface (Figure 4.7b). The decrease in mineral content near the surface layer to the minimum value in the body of the lesion was very abrupt, while the damage caused deeper within the tooth was more gradual. This can be explained by the faster mineral regeneration at the tooth surface than in the deeper layers, leading to a fully regenerated surface. As the number of cycles (i.e., acid attacks) increases, a more pronounced subsurface lesion developed inside the tooth (see Movie 1 in Supplementary Material). The obtained profile is very similar to the mineral content profiles determined experimentally by Margolis et al., 1999. In case the tooth enamel contained FHAP crystals but no remineralisation occurred (Case 2), the obtained profiles resembled qualitatively those obtained for HAP with demineralisation/remineralisation (Figure 4.7c). The most notable differences were however

104 the almost complete mineral dissolution in the subsurface lesion after 80 cycles and a faster progression of the lesion in the tooth depth (see Movie 2 in Supplementary Material). However, these late stages of lesion development are less probable to occur in vivo. There is a limit to the mineral content in the subsurface lesion after which the surface layer brakes during mastication and cavitation would expose the body of the lesion.

4.3. Discussion This study developed a numerical model that couples tooth demineralisation and remineralisation with metabolic processes in the dental plaque. It was found that a subsurface lesion can be achieved using the same thermodynamic driving force (degree of saturation leading to demineralisation or remineralisation) and without any pre-imposed gradients. There is a debate in dentistry regarding the importance of the remineralisation phenomena for the formation of a surface layer. The model results are discussed here in the context of early stage carious lesions, the so called “white spots”. One point of view is that what appears to be a restored surface, is partly explained in terms of wear and polishing (Fejerskov et al., 2008). Although it is possible that polishing plays a role in the aspect of a restored tooth surface this does not exclude the existence of remineralisation. Furthermore, there is experimental evidence that the surface layer has more than just the aspect of a restored surface; it also has a mineral content close to that of healthy enamel (Fejerskov et al., 2008; Silverstone, 1977). Another argument is that inhibitor molecules present in saliva (e.g, statherin) prevent in vivo precipitation at crystal surface by blocking crystallization nuclei (Santos et al. 2008). However, the presence of these inhibitors does not imply that the process of remineralisation is entirely blocked, but only indicates that remineralisation in vivo would occur slower (Figure 4.5b). It was also argued that the very fast uptake of calcium and phosphate by the HAP makes the pore liquid in the deep layers of lesion only marginally saturated (Larsen and Fejerskov, 1989) thus remineralisation can only be very slow. A widespread theory is also that since the surface enamel remineralises faster, an area of lower porosity at the tooth surface is created. This surface layer acts as a diffusion barrier that restricts further access of HAP constituent ions in the deeper layers of the lesion and stops the remineralisation (Larsen and Fejerskov, 1989; Silverstone, 1977; Robinson et al., 2000). Although it is true that such a barrier is formed, this can only delay the diffusion of the ions

105 inside the lesion (Figure 4.5a). If the remineralisation time is long enough, the deep layers of the lesion can still be restored (Figure 4.5c). According to Arends and Cristoffersen (1986) the experimental observation that a surface layer once formed has a nearly constant thickness, supports the idea that the surface layer forms because of a solubility gradient along the enamel depth. With the model described in this study, a surface layer of nearly constant thickness was obtained in both cases. The surface layer can develop only by alternating periods of demineralisation (corresponding to sugar consumption) and remineralisation (corresponding to resting between meals) in tooth enamel with uniform mineral composition. Sensitivity analysis showed that the surface layer formation is strongly dependent on the length of remineralisation and demineralisation cycles, and on the plaque thickness. This confirms once more that the behavioural aspect of caries formation (i.e., length of resting phase, frequency of sugar consumption, tooth brushing) is very important. It was also shown how the presence of a fluoride gradient leads to subsurface lesions. The calculated profiles of mineral content in enamel are similar to those observed experimentally in vitro and in vivo (Margolis et al., 1999). Most probably, both mechanisms studied in the present paper are interacting in vivo in the process of caries development.

4.4. References Arends J, Christoffersen J: The nature of early caries lesions in enamel. J Dent Res 1986; 65(1):2-11. Borgström MK, Edwardsson S, Sullivan Å, Svensäter G: Dental plaque mass and acid production activity of the microbiota on teeth. Eur J Oral Sci 2000; 108:412-417. Chu JS, Fox JL, Higuchi WI: Quantative study of fluoride transport during subsurface dissolution of dental enamel. J Dent Res 1989; 68:32-41. Dawes C, Dibdin GH: A theoretical analysis of the effects of plaque thickness and initial salivary succrose concentration on diffusion of succrose into dental plaque and its conversion to acid during salivary clearance. J Dent Res 1986; 65(2):89-94. De Rooij JF, Nancollas GH: The formation and remineralization of artificial white spot lesions - a constant composition approach. J Dent Res 1984; 63:864-867. Dibdin GH: Diffusion of sugars and carboxylic acids through human dental plaque in vitro. Arch Oral Biol 1981; 26:515-523. Dibdin GH: Effect on a cariogenic challenge of saliva/plaque exchange via a thin salivary film studied by mathematical modelling. Caries Res 1990; 24(4):231-238.

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Dirksen TR, Little MF, Bibby BG, Crump SL: The pH of carious cavities. I - The effect of glucose and phosphate buffer on cavity pH. Arch Oral Biol 1962; 7:49-57. Fejerskov O, Kidd E: Dental caries: The disease and its clinical management, 2nd ed, Blackwell Munksgaard, 2008. Fremlin JH, Mathieson J: A microchromatic study of the penetration of enamel by C14- labelled glucose. Archs Oral Biol 1961; 4(C):92-96. Grey JA, Francis MD: Physical chemistry of enamel dissolution; in Destruction of hard tissues, RF Sognnaes (ed) Washington, 1963: Publication No 75 of the American Association for the Advancement of Science, pp 213-260. Hamilton IR, St. Martin EJ: Evidence for the involvement of proton motive force in the transport of glucose by a mutant of Streptococcus mutans Strain DR0001 defective in glucose- phosphoenolpyruvate phosphotransferase activity. Infect Immun 1982; 36(2):567-575. Hicks J, Garcia-Godoy F, Flaitz C: Biological factors in dental caries enamel structure and the caries process in the dynamic process of demineralization and remineralization (part 2). J Clin Pediatr Dent 2004; 28(2):119-124. Hollander F, Saper E: The apparent radiopaque surface layer of the enamel. Dent Cosmos 1935; 77:1187-1197. Holly FJ, Gray JA: Mechanism for incipient carious lesion growth utilizing a physical model based on diffusion concepts. Archs Oral Biol 1968; 13:319-334. Ilie O, Van Loosdrecht MCM, Picioreanu C: Mathematical modelling of tooth demineralisation and pH profiles in dental plaque. J Theor Biol 2012; (309):159–175 Koutsoukos P, Amjad Z, Tomson MB, Nancollas GH: Crystalization of calcium phosphates. A constant composition study. J Am Chem Soc 1980; 102(5):1553-1557. Langdon DJ, Elliott JC, Fearnhead RW: Microradiografic observation of acidic surface decalcification in synthetic apatite aggregates. Caries Res 1980; 14:359-366. Larsen MJ, Fejerskov O: Chemical and structural challenges in remineralisation of dental enamel lesions. Scand J Dent Res 1989; 97:285-96. Loesche WJ: Role of Streptococcus mutans in human dental decay. Microbiol Rev 1986; 50(4):353-380. Margolis HC, Moreno EC: Kinetic and thermodynamic aspects of enamel demineralization. Caries Res 1985; 19:22-35. Margolis HC, Moreno EC: Kinetics of hydroxyapatite dissolution in acetic, lactic and phosphoric acid solutions, Calcified Tissue Int 1992; 50:137-143. Margolis HC, Zhang YP, Lee CY, Kent RL, Moreno JR, Moreno EC: Kinetics of enamel demineralisation in vitro. J Dent Res 1999; 78(7):1326-1335. Marsh PD, Martin MV, Lewis MAO, Williams DW: Oral Microbiology, 5th ed., Churchill Livingstone Elsevier, 2009. Moreno EC, Kresak M, Zahradnik RT: Fluoridated hydroxyapatite solubility and caries formation. Nature 1974; 247: 64 – 65. 107

Morgenroth E, Eberl H, van Loosdrecht MCM: Evaluating 3-D and 1-D mathematical models for mass transport in heterogeneous biofilms. Water Sci Technol 2000; 41(4-5):347-356. Nancollas GH: Kinetics of demineralisation and remineralisation; in: S.A. Leach, W.M. Edgar (Eds.), Demineralization and Remineralization of the Teeth, Irl. Pr., 1983, pp. 113–128. P. Atkins, J. De Paula, Physical Chemistry, 9th ed., W. H. Freeman, 2009. Robinson C, Shore RC, Brookes SJ, Strafford S, Wood SR, Kirkham J: The chemistry of enamel caries. Crit Rev Oral Biol M 2000; 11(4):481-495. Santos O, Kosoric J, Hector MP, Anderson P, Lindh L: Adsorption behavior of statherin and a statherin peptide onto hydroxyapatite and silica surfaces by in situ ellipsometry. J Colloid Interf Sci 2008; 318(2):175-82. Silverstone LM: Remineralization phenomena. Caries Res 1977; Suppl. 1:59-84. Ten Cate JM: A model for enamel lesion remineralisation, in: S.A. Leach, W.M. Edgar (Eds.), Demineralization and Remineralization of the Teeth, Irl. Pr., 1983, pp. 129-144. Ten Cate JM: Remineralization of deep enamel dentine caries lesions. Aust Dent J 2008; 53: 281–285. Ten Cate JM: Remineralization of enamel lesion. A study of the physico-chemical mechanism. (Dissertation), University of Groningen, The Netherlands, 1979. Theuns HM, van Dijk JWE, Driessens FCM, Groeneveld A: The Surface layer during artificial carious lesion formation. Caries Res 1984; 18:97-102. Van Beelen P, Van der Hoeven JS, De Jong MH, Hoogendoorn H: The effect of oxygen on the growth and acid production of Streptococcus mutans and Streptococcus sanguis. FEMS Microbiol Ecol 1986; 38:ss25-30. Van der Hoeven JS, De Jong MH, Camp PJM, Van den Kieboom CWA: Competition between oral Streptococcus species in the chemostat under alternating conditions of glucose limitation and excess. FEMS Microbiol Ecol 1985; 31:373-379. Van der Hoeven JS, Gottschal JC: Growth of mixed cultures of Actinomyces viscous and Streptococcus mutans under dual limitation of glucose and oxygen. FEMS Microbiol Ecol 1989; 62:275-284. Van der Hoeven JS, van den Kieboom CW, Camp P J: Utilization of mucin by oral Streptococcus species. Anton van Lee 1990; 57:165-172. Van Dijk JWE, Borggreven JMPM, Driessens FCM: Chemical and mathematical simulations of caries. Caries Res 1979; 13:169-180. Van Dijk JWE: The Electrochemistry of Dental Enamel and Caries. (Dissertation), The Netherlands, 1978. Vanýsek P: Handbook of chemistry and physics, 82nd ed., Boca Raton: CRC Press LLC, 2001, pp. 5-95 and pp. 6-194. Wang Z, Fox JL, Baig AA, Otsuka M, Higuchi WI: Calculation of intercrystalline solution composition during in vitro subsurface lesion formation in dental minerals. J Pharm Sci 1996; 85(1):117-28. 108

White DJ, Chen WC, Nancollas GH: Kinetic and physical aspects of enamel remineralisation - a constant composition study. Caries Res 1988; 22:11-19. Wood SR, Kirkham J, Marsh PD, Shore RC, Nattress B, Robinson C: Architecture of Intact Natural Human Plaque Biofilms Studied by Confocal Laser Scanning Microscopy. J Dent Res 2000; 79(1):21-7. Zimmerman SO, A mathematical theory of enamel solubility and the onset of dental caries: I. The kinetics of dissolution of powdered enamel in acid buffer. B Math Biophys 1966a; 28:417-432. Zimmerman SO, A mathematical theory of enamel solubility and the onset of dental caries: II. Some solubility equilibrium considerations of hydroxyapatite. B Math Biophys 1966b; 28: 433-441. Zimmerman SO, A mathematical theory of enamel solubility and the onset of dental caries: III. Development and computer simulation of a model of caries formation. B Math Biophys 1966c; 28:443-464.

109

Two -dimensional

mathematical

5 modelling of tooth

enamel subsurface

lesion formation

induced by dental plaque

Dental caries formation has been an intensively studied process due to the impact it can have on the inflicted persons if left untreated (e.g., severe pain, discomfort in mastication, tooth loss etc.). One precondition for the development of dental caries is the presence of dental plaque (oral biofilm) on Two-5. dimensionalthe affected tooth enamel. This essential biological factor makes the dental caries formation a more complex matter than mathematicalsimple mineral modellingdissolution and mineralisation. Forof a correct understanding of the mechanisms governing this disease, the tooth enamelprocesses occurring subsurface inside the tooth enamel must be studied in lesion formationrelation to the (micro)biological induced and chemical processesby occurring in the dental plaque. Due to the high complexity of the dentalthe plaque-tooth interaction to which behavioural aspects are added (e.g., frequency of sugar consumption, tooth brushing habits etc.) it is useful for the experimental studies to benefit from a thorough theoretical understanding of the mechanisms underlying dental caries formation.

110 This chapter will be submitted in the current form to the Journal of Theoretical Biology. A common issue when studying dental caries is how the typical profile of an incipient enamel lesion is formed. This profile involves a layer of enamel at the very surface of the tooth that is apparently inert to any dissolution. Under this layer the so called “body of the lesion” develops – an area of reduced mineral content that further advances in the enamel depth. Many studies have focused on identifying the mechanisms leading to such a profile (e.g., Robinson et al., 2000; White et al., 1988; Gray and Francis, 1963; Van Dijk et al., 1979; Langdon et al., 1980). From numerous hypotheses present in the literature, two have received special attention. First, the subsurface lesion is the consequence of an imbalance between two opposite processes originating from the same thermodynamic driving force – demineralisation and remineralisation of tooth enamel, mainly composed of hydroxyapatite, HAP (Fejerskov and Kidd, 2008). Second, the subsurface lesion results from a solubility gradient caused by a fluoride distribution inside the tooth (fluorohydroxyapatite, FHAP), which makes the tooth surface insoluble at the pH values reached in the plaque fluid during an acid attack (Moreno et al., 1974). In a previous numerical modelling study (Ilie et al., 2014) these two theories have been critically tested using a one-dimensional (1-d) mathematical model for the demineralisation and remineralisation of tooth enamel under the influence of the dental plaque. The main conclusion of that work was that both proposed mechanisms can lead to the formation of a subsurface lesion and most probably they collaborate in vivo in the process of caries development. Although the HAP demineralisation-remineralisation mechanism alone can explain the formation of a subsurface lesion (i.e., the entire dynamics of caries formation relies on the degree of saturation at the tooth surface), the second theory cannot be discarded either, since the presence of a fluoride gradient inside the tooth is well-established experimentally (Moreno et al., 1974). To better distinguish between these two mechanisms and gain further insight into the way they would act in a more realistic tooth geometry, the 1-d model has been extended in this study to a two-dimensional (2-d) setup. This improvement allows considering the local concentration gradients present along the plaque-tooth interface, thereby leading to a more accurate calculation of the subsurface lesion profile. Moreover, the calculated 2-d mineral distribution could be directly compared with experimentally observed profiles in thin enamel slices (Fejerskov and Kidd, 2008). The 2-d calculations are expected to reveal the areas of preferential enamel dissolution along the plaque-tooth interface, as pointed by experimental studies (Fejerskov and Kidd, 2008). An important application of the 2-d model is in the study of the influence of tooth site geometry at which the dental caries develop on the localized lesion profile and surface layer formation. From this point of view, the carious lesion formed 111 as a consequence of plaque gathering inside a fossa on the occlusal surface of a molar or in a micro-fissure inside the enamel, might be structurally different from a lesion developed on the smooth surface of a tooth or molar. Since these two geometrical situations (plaque inside a pit and on top of smooth enamel surface) can be present even on the same molar, a comparative study could show that different types of lesions can form in rather similar conditions (salivary flow, plaque composition etc.) only as a consequence of the different tooth surface geometry.

5.1. Model description In order to study the effect of the tooth geometry and dental plaque location on the initiation of enamel lesions, a two-dimensional (2-d) model was set up based on the one-dimensional (1-d) numerical model presented in (Ilie et al., 2014). While the dental plaque composition, with all associated biological and chemical reactions inside the plaque and the tooth were the same as in the 1-d model, the two mechanisms for lesion formation were re-evaluated in two typical plaque/tooth 2-d geometries.

5.1.1. Geometries To determine the spatial distribution of enamel demineralisation both in the tooth depth and along the tooth surface, as a localized effect of dental plaque, two simplified geometries were studied (Figure 5.1). Both geometries were constructed in a manner that allows comparison with the previous 1-d models (Ilie et al., 2012; Ilie et al., 2014), while still describing realistic situations. The Occlusal geometry represents a pit filled with dental plaque inaccessible to brushing on the occlusal area of a molar (Fejerskov and Kidd, 2008). The Smooth geometry represents dental plaque accumulated on a smooth surface (buccal or lingual) of the tooth. In order to increase the calculation speed, the size of the computational domain can be reduced by considering a symmetry axis along the pit depth (Figure 5.1). For example, the abstracted 3-d geometry from Figure 5.1 could be obtained by revolving the simplified 2-d geometry around the symmetry axis. Furthermore, solute concentration and enamel porosity profiles along the vertical symmetry axis in both situations are comparable (though not identical) with those obtained in the 1-d geometry considered in Ilie et al. (2014), with a plaque depth of 250 μm (Figure 5.1). Since the focus of this study was the initial development of subsurface lesions, only the first 100 μm in depth from the tooth surface were considered. In both geometries, the plaque and tooth domains do not change in time. The Moving Boundary is a particular case of the Occlusal geometry where the tooth dissolution/re-mineralization was not represented in the volume, but as a set of processes occurring on a moving plaque-tooth 112

3-d Occlusal Geometry

3-d Smooth Geometry

plaque plaque

enamel 2-d 2-d Occlusal enamel Occlusal Moving Boundary

2-d 100 μm saliva saliva Smooth

plaque plaque

μ saliva m m μ μ plaque 250 enamel 250 moving boundary m 250 μ

m no flux enamel μ symmetry 100

100 axis symmetry symmetry saliva axis no flux axis

plaque

m μ 1-d along the 1-d along the symmetry axis 250 symmetry axis

μ enamel

100 Figure 5.1. Evaluated model geometries, from the real 3-d tooth representation to 2-d axisymmetric and 1-d profile along the symmetry axis.

boundary due to enamel dissolution (i.e., similar to the development of a surface lesion). The purpose of this case was to assess the importance of considering the tooth mineral as a separate domain (at the cost of slower computations) or as a boundary condition (faster computationally, but with a possible qualitative impact on the results). In this case, also the plaque geometry changes in time due to the moving boundary.

5.1.2. Processes The 2-d model includes one generic glucose performed by the bacteria present in the dental plaque, acid-base equilibria and demineralisation / mineralisation reactions of the tooth 113 enamel (Table 5.1) involving a series of solute chemical species (Table 5.2). The dental plaque was reduced to one general microbial group, the aciduric Streptococci, having constant concentration CX in the plaque and performing only lactic fermentation. The reaction stoichiometry and glucose uptake rate with glucose limitation and acid inhibition are given in Table 5.1 (process 1). Very fast protonation equilibria for lactate (microbial metabolic product) and hydrogen phosphate (tooth demineralisation product) occur in all model compartments (saliva, plaque and tooth) (processes 2 in Table 5.1). The less important phosphate equilibria and other possible inorganic buffers, complexation equilibria with Ca2+ and microbial surface charges were here neglected to simplify the analysis of results. The two theories for subsurface lesion formation on which the previous study (Ilie et al. 2014) was focused, have been re-evaluated using the 2-d model. Case 1 is an alternation of two competing processes, enamel demineralisation and remineralisation, assuming the entire tooth composition solely of hydroxyapatite (HAP) rods. Both HAP demineralisation and HAP remineralisation reactions can occur in the superficial enamel layer, depending on the degree of saturation of the plaque fluid with respect to HAP, DSHAP (processes 3 and 4 in Table 5.1). The difference between the volumetric demineralisation and remineralisation rates determines the change in enamel porosity E in time. A special situation is the Modified Case 1, in which only demineralisation with HAP takes place (process 3 in Table 5.1). In Case 2, only demineralisation occurs for a tooth with higher concentration of fluoride at the enamel surface slowing down the dissolution rate (process 5 in Table 5.1). This case tested the hypothesis that a subsurface lesion can form through the demineralisation only if the enamel has a heterogeneous solubility, consequence of a fluoride distribution (Fejerskov and Kidd, 2008) assumed constant in time. It was experimentally observed that fluoride diffuses and partly replaces the hydroxyl from the structure of HAP with the stoichiometry from Table 5.1 (Moreno et al, 1974). The resulting HAP crystal with FHAP inclusions is less soluble than pure HAP. The dependency of mixed FHAP/HAP solubility constant, Kx, on the fraction of – fluoride xF was described in Ilie et al (2014). To represent this situation, fluoride ions (F ) were introduced in the model (Table 5.2). Fluoride was considered non-reactive in saliva and dental plaque and exchangeable between all the model compartments.

114

Table 5.1. Processes considered in the model and their associated parameters (Ilie et al., 2014).

Constants and variables Processes Name and symbol Value and units Reference –6 –1 –1 1. Lactic fermentation in plaque Maximum specific uptake rate, qS,max 9.67×10 (mol g s ) van der Hoeven et al, 1985 –4 –1 C K Monod half-saturation coefficient for 1.22×10 (mol L ) Hamilton and Martin, 1982 o Glu IH, (5.1) CH6126 O 2CHO 353 2H rqCGlu S,max X glucose, KS,Glu van der Hoeven et al, 1985 CKCKGlu S,, Glu H I H –6 –1 Proton inhibition constant, KI,H+ 10 (mol L ) Assumed inhibition at pH = 4.5 6 –1 2. Acid base equilibria Rate coefficients, kLacH, kPho– 10 (s ) Assumed arbitrarily large –7.21 –1 Acidity constant lactic acid, KLacH 10 (mol L ) Atkins and de Paula, 2009 CHO363 CHO35 353 H rkCCCKLacH LacH LacH H Lac/ LacH (5.2a) Acidity constant dihydrogen- 10–12.16 (mol L-1) Atkins and de Paula, 2009 HPO HPO2 H rkCCCK / (5.2b) 244 Pho Pho Pho H Pho2 Pho phosphate, KPho–

–8 3. Demineralisation of hydroxyapatite (HAP) Radius of HAP rod, rrod 2.5×10 (m) Fejerskov and Kidd, 2008 23 –6 0.7 –1.1 –1 Ca PO OHo 5Ca 3PO HO * (5.3a) Demineralisation rate constant, kd 7×10 (mol m s ) Margolis and Moreno, 1999 54 3 4 rradHAP,,0.2 dCa d Demineralisation reaction order, md 2.8 Margolis and Moreno, 1992 arE 21 demineralisation area (5.3b) drod Demineralisation reaction order, nd 0.3 Margolis and Moreno, 1992 n –60 9 –9 md d forDS 1 Enamel solubility constant, KS,HAP(en) 7.41×10 (mol L ) Fejerskov and Kidd, 2008 * °kDSCdHAPAiH 1 ¦ () HAP –12.32 -1 rdCa, ® (5.3c) Acidity constant for hydrogen 10 (mol L ) Atkins and de Paula, 2009 forDSHAP t 1 ¯° 0 phosphate dissociation, KPho2– –14 2 -2 19 53 53 Water ionic product, K 10 (mol L ) Atkins and de Paula, 2009 §·IP CC CC H2O DS ¨¸HAP , IP Ca23 Pho Ca 22 Pho K3 K (5.3d) HAP ¨¸HAP4 Pho2 H2O ©¹KSHAPen,() CCHO H

–3 0.75 0.25 –1 4. Remineralisation of HAP Remineralisation rate constant, kr 7.44×10 (m mol s ) Nancollas, 1983 23 * 5Cao 3PO HO Ca PO OH rra (5.4a) Remineralisation reaction order, nr 1.25 Nancollas, 1983 4543 rHAP,, rHAP r 2 -3 Specific surface demineralisation, ad (m m ) a 2 -3 ° d forE ! 0.215 Specific surface remineralisation, a (m m ) a ® (5.4b) r r aa1 expªº 45 E 0.15 for 0.01 dE 0.215 ¯° dd^`¬¼ nr t * °kDSrHAP 1 for DSHAP 1 r ® (5.4c) rHAP, for DS 1 ¯° 0 HAP

5. Demineralisation of fluorohydroxyapatite (FHAP) Minimum fraction of fluoride, xF, min 0.05 Moreno et al., 1974 23 Maximum fraction of fluoride, x 0.5 Moreno et al., 1974 Ca54 PO OH FxFFo 5Ca 3PO 4 1xx HO F F,max 31xF F –55 9 –9 HAP solubility constant, KS,HAP 5.50×10 (mol L ) Fejerskov and Kidd, 2008 md nd rkDSCadHAPx,() 0.2 d 1 HAPx () AiH () d for DSHAP() x 1 (5.5a) Depth in tooth, x (μm) FF¦ F Fluoride fraction in tooth, xF (-) xxFF ,min x F ,max x F ,min exp x 100 (5.5b) KKK / exp 8.7923 x2 10.2691 x31.9308 (5.5c) S,() HAP xF S ,() HAP en S , HAP F F 19 §·IP 53 xF HAP() xF CCCa22 Pho C F 3 DSHAP() x ¨¸, IPHAP() x K Pho 2 K H2O (5.5d) F ¨¸K F C4xF ©¹SHAPx,()F H

Table 5.2 Chemical components in the model. Diffusion coefficient (a) D Saliva concentrations Charge p,j Initial (b) Name Symbol number Inlet ( c) (10–9 m2s–1) Reference C (0) Reference (z ) f,j (mol m–3) j (mol m–3) Lactate ion CLac– –1 1.31 Vanýsek, 2001 0 0 Assumed Lactic acid CLacH 0 1.31 Vanýsek, 2001 0 0 Assumed –4 –4 Proton CH+ +1 11.81 Vanýsek, 2001 1.29×10 8.71×10 [Fejerskov and Kidd, 2008] Equilibrium, –3 CPho,tot initial= 13.2 mol m Hydrogen phosphate CPho2– –2 0.96 Vanýsek, 2001 4.27 2.24 –3 CPho,tot inlet= 5.4 mol m [Fejerskov and Kidd, 2008] Dihydrogen phosphate CPho– –1 1.22 Vanýsek, 2001 8.93 3.16 Equilibrium Calcium ion CCa2+ +2 1.00 Vanýsek, 2001 3.5 1.32 [Fejerskov and Kidd, 2008] Cation (Potassium ion) CK+ +1 2.49 Vanýsek, 2001 51.5 19.4 [Fejerskov and Kidd, 2008] Anion (Chloride ion) CCl– –1 2.57 Vanýsek, 2001 41.03 14.4 Charge balance Fluoride ion CF– –1 1.48 Vanýsek, 2001 6 1 [Fejerskov and Kidd, 2008] Van der Hoeven et al., 1989; Glucose C 0 0.85 Vanýsek, 2001 0.07 C (t) Glu s,Glu Calculated for inlet (a) Aqueous diffusion coefficient (b) Values corresponding those measured in human resting plaque (assumed to be in a steady state) (c) Values corresponding those measured in unstimulated mixed (from all salivary glands) human saliva

116

Table 5.3. Mass balances for each model compartment and their associated parameters.

Associated parameters Balance equations Name and symbol Value and units Reference

Tooth domain (2-d) Diffusion coefficient of solute j in the tooth, E·Dj Ten Cate, 1979 Dt,j and in water, Dj wCzFtj, 2 j DCDtj,,, tj tj C tj , ) R tj , (5.6a) Volume based demineralisation rate of Table 5.1 – wtRT HAP, r zC 0 (5.6b) d,HAP ¦ jtj, Volume based remineralisation rate of Table 5.1 – j HAP, rr,HAP dE M HAP –1 HAP: rr (5.7a) HAP molar weight, MHAP 502 (g mol ) Fejerskov and Kidd, 2008 dHAP,, rHAP –3 dt UHAP HAP density, ρHAP 3160 (kg m ) Fejerskov and Kidd, 2008 dE M Volume based demineralisation rate of Table 5.1 – FHAP: r HAP (5.7b) dHAPx,()F FHAP, rd,HAP(xF) dt UHAP Ionic charge, zj Table 5.2 Dental plaque domain (2-d) Diffusion coefficient of solute j in the Table 5.2 – wCzF plaque, Dp,j pj, 2 j –1 –1 DCDpj,,, pj pj C pj , ) R pj , (5.8a) Universal gas constant, R 8.314 (J mol K ) - wtRT Temperature, T 310 (K) Assumed, 37 C zC 0 (5.8b) –1 ¦ jpj, Faraday’s constant, F 96485 (C mol ) - j Electrical potential field, Ф(x,t) (V) Saliva (0-d) Residence time in the saliva film, Qf / Vf ln(2) / th,f (min) Calculated dC Q A Halving time in the saliva film, th,f 0.5 (min) Dibdin, 1990 fj, f f (5.9) 3 –1 (†) CCsj,, fj N pj , R fj , Area per volume ratio in saliva film, Af/ Vf 10 (m ) Calculated dt Vff V Residence time in the saliva bulk, Q / Vs ln(2) / th,s (min) Calculated Halving time in the saliva bulk, th,s 2.17 (min) Dibdin, 1990 ªºQ Time for the transition from minimum to 10(s) Assumed CtCtt() ( )exp ttt ( ) C (5.10) s,, Glu s Glu step feed«» step feed Glu , min maximum glucose concentration, t ¬¼Vs step Feeding period, tfeed 2 (min) Assumed Length feeding/resting cycle, tcycle 60 (min) Assumed Minimum glucose concentration, CGlu min 0.07 (mM) Van der Hoeven et al., 1990 Glucose concentration during pulse, CGlu max 560 (mM) Dirksen et al., 1962

The mathematical operator is defined for a two-dimensional Cartesian coordinate system: w wxy w w . The subscripts s, f, p and t used in the above equations stand for saliva bulk, saliva film, plaque, and tooth respectively. The subscript j denotes each of the chemical species for which a mass balance is calculated (see Table 5.1 for the complete list of all the chemical species considered in the current model) (†)Assumes that the saliva film thickness is 100 μm (Fejerskov and Kidd, 2008) and only 10% of the tooth is covered with plaque.

117

5.1.3. Balance equations The model assumes solute exchange between three compartments: saliva, dental plaque and tooth enamel. In plaque and tooth enamel a 2-d concentration distribution is calculated, while the entire saliva volume is assumed well-mixed with concentrations changing only in time. To calculate the solute concentrations and the amount of demineralised tooth enamel, mole and charge balances were applied on the three compartments.

(a) Tooth The change of tooth mineral content in time (i.e., the carious lesion development) depends on the spatial distribution of several solute species (all but glucose in Table 5.2). The concentrations of each solute species j in the tooth mineral (Ct, j) was calculated from Nernst- Planck equations coupled with charge balance (eqs. (5.6a) and (5.6b), Table 5.3). Molecules diffuse and migrate in the tooth and participate in heterogeneous (mineral dissolution/precipitation) and homogeneous (acid-base) reactions (Table 5.1) resulting in the net reaction rate, Rt,j. The additional state variable ) is a minute electrical potential field developed due to ion transport with different diffusion rates, contributing to charge balancing by transporting ions by migration. Low enamel porosity leads to slower diffusion in the tooth domain (Dt,j = DjE) than in plaque (Ten Cate, 1979). The boundary conditions included flux continuity at the plaque/tooth interface; no flux and electrical insulation at the deep enamel boundary and no flux on the symmetry axis (see Figure 5.1). The local enamel porosity E was also variable in time, and calculated differently in each of the tested hypothesis: as the difference between demineralisation and remineralisation rates (eq. 5.7a, Table 5.3) in Case 1 (HAP) or from demineralisation rate only (eq. 5.7b,

Table 5.3) in Case 2 (FHAP) (van Dijk, 1978). An initial enamel porosity Emin = 0.01 was taken as corresponding to healthy enamel (Fejerskov and Kidd, 2008).

(b) Dental plaque

In the plaque (biofilm), the 2-d distributions of solute concentrations Cp,j depend on the variation of glucose concentration in saliva, the metabolism of present bacteria, chemical species transport (diffusion and ion-migration), dissolution/mineralisation reactions occurring inside the tooth, etc. These factors are accounted for in mole balances set for all solutes j based on Nernst-Planck equations coupled with the electroneutrality condition (eq. (5.8a) and

(5.8b), respectively, in Table5. 3). The net reaction rates Rp,j include chemical and microbial processes (Table 5.1). According to Dibdin (1981) the diffusion coefficients in plaque (Dp,j) may be about four times smaller than those in water (Daq,j). However, the previous 1-d model

(Ilie et al., 2014) revealed qualitatively similar results for Dp,j = Daq,j and Dp,j = Daq,j/4. Since the focus of the current study is not towards absolute values but rather to indicate qualitative trends, it was assumed that Dp,j = Daq,j, in order maintain simplify the results interpretation. The initial values for all concentrations in plaque were equal to those in initial saliva and )(t=0) = 0. For the saliva/plaque interface the concentrations of all chemical components were equal to those in the saliva and Φ = 0 as reference value. The symmetry axis was set as no flux boundary. At the plaque/tooth interface, total flux continuity was set for all components except glucose, for which the tooth is impermeable (no flux condition) (Fremlin and Mathieson, 1961) (Figure 5.1). In a special case (Moving Boundary), we investigated the possibility of considering the enamel totally impermeable to solutes and the plaque/tooth interface as a reactive and moving boundary. In this case, the plaque geometry changed in time and the solute flux was linked to the enamel dissolution rate (similarly to the approach used in Ilie et al., 2012). HAP demineralisation was implemented by setting the molar flux Npt,j of the species involved in the enamel dissolution ( i.e., H+, Ca2+, phosphates) equal with their net formation / consumption

* rate NrCCCpt,,,,2,2 j Q j d HAP(, pt H pt Ca , pt Pho ). The molar flux was calculated using the

* concentrations at the tooth surface, while the stoichiometric coefficients Qj and rate rdHAP, were according to eq. (5.3a)-( 5.3c) from Table 5.1. Zero-flux was set for the rest of chemical species. Finally, the displacement of the tooth boundary (i.e., surface lesion progression) was related to the HAP demineralisation by defining the boundary velocity, upt, function of the surface-based calcium dissolution rate, uNpt pt,2 Ca M HAPU HAP .

(c) Saliva Using the same approach as in previous models (Ilie et al., 2012; Ilie et al., 2014), saliva constituted the boundary of the dental plaque domain. It was assumed that the saliva film in direct contact with the plaque has a perfectly mixed volume Vf, which represents a small fraction of the total saliva present in the mouth and it exchanges solutes with the remaining saliva volume (Vs, referred to as “bulk saliva”) and with the dental plaque. The concentration of each chemical species in the saliva film, Cf,j, changes in time according to the equation (5.9) from Table 5.3, based on the exchange with the saliva bulk with a flow rate Qf, exchange with the dental plaque of area Af with the flux Np,j and the net reaction rate in saliva,

Rf,j (Table 5.3). The saliva inlet concentrations Cs,j were set based on chemical speciation equilibria and solution electroneutrality (Ilie et al., 2012 and Ilie et al., 2014, Table 5.2). 119

For the inlet concentration of glucose, Cs,Glu, successive feeding/clearance/resting cycles (further named simply “feeding cycles”) with a total duration of 4000 s were imposed.

In the feeding phase a pulse of glucose with maximum concentration Cs,Glu,max was maintained for two minutes (tfeed) in the saliva bulk. At the end of the feeding phase, oral clearance allowed the glucose concentration to decrease exponentially (eq. (5.10) from Table 5.3).

5.1.4. Model solution (a) Solvers The model equations were implemented in COMSOL Multiphysics software (COMSOL 4.3a, Comsol Inc, Burlington, MA, www.comsol.com), which allows a very flexible and well- structured model construction. COMSOL solves the resulting system of ordinary differential, partial differential and algebraic equations by finite element methods. First, the mole and charge balances in plaque and enamel, equations (5.6a,b) and (5.8a,b) were solved towards the steady state solution, which represents a situation of resting plaque in contact with constant composition saliva. Second, the steady state solution was used as initial condition for the time-dependent simulations performed during 80 sequential feeding cycles of 4000 s each. The time-dependent simulations included the complete model equations in saliva (eq. (5.9) and (5.10)), plaque (eq. (5.8a,b)) and tooth (eqs. (5.6a,b) and (5.7a) or (7b)), with their associated boundary conditions and constitutive (rate) equations.

(b) Meshing The 2-d dental plaque and tooth enamel domains must be discretised in a carefully chosen mesh of points. The large difference between the diffusion coefficients inside the plaque and tooth creates steep concentration gradients near the plaque/tooth interface. These large gradients can create numerical instabilities, unless this region is very finely meshed. Because in time the lesion progresses in the tooth enamel depth, the steep concentration gradients also advance deeper in the tooth domain. Therefore, in the absence of an automatic re-meshing procedure, the entire tooth enamel domain needed a fine mesh. Figure 5.2 represents the constructed mesh using quadrilateral elements for the Occlusal geometry (the same kind of meshing was also used for the Smooth geometry). Because steep gradients may develop near the plaque/saliva interface, a fine mesh was also needed near the plaque surface (Figure 5.2). The plaque domain was discretized on a triangular mesh with a maximum element size of 5 μm. The mesh had to be refined around the saliva-plaque boundary (minimum element size of 2 μm) due to the steep gradients developed during the feeding period. 120

Figure 5.2. Finite element mesh used in the 2-d Occlusal geometry (left) and a close up of the refined mesh near the plaque-saliva and plaque-tooth boundaries (right).

5.2. Results and discussion For each of the two proposed mechanisms for subsurface lesion formation (HAP demineralisation-remineralisation and FHAP demineralisation, corresponding to Case 1 and Case 2, respectively) the calculated mineral content inside the tooth enamel at the end of 80 feeding cycles of 4000 s each was analysed. The impact of the tooth site at which the carious lesion develops was evaluated by comparing the two mineral content profiles for the Occlusal and Smooth geometry for each studied case. First, the two proposed mechanisms have been evaluated by comparing the mineral profile obtained for a given geometry. Second, the calculated mineral profiles were qualitatively compared with experimental data available in the literature. Finally, the impact of some modelling decisions on the model outcome is discussed.

5.2.1. HAP demineralisation-remineralisation (Case 1) This case evaluates the hypothesis proposing that the tooth enamel has a homogenous composition of hydroxyapatite crystals and the initial caries lesion is the result of repeated alternations of demineralisation and remineralisation periods.

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Figure 5.3. Mineral profiles inside the tooth enamel for Occlusal (top panels) and Smooth (bottom panels) geometries for the different mechanisms of lesion formation studied. (a) HAP demineralization/remineralisation (Case 1) ; (b) HAP demineralization (Modified Case 1); (c) FHAP demineralization (Case 2). Mineral content in the enamel is shown on a colour scale from white (100%) to black (0 % mineral).

(a) Occlusal geometry At the end of 80 feeding cycles a subsurface lesion developed at the bottom of the pit (fossa) from the Occlusal geometry (Figure 5.3 – upper panels). To better understand the interaction between the demineralisation and remineralisation processes during incipient stages of caries formation, it is useful to analyse the mineral profile calculated in a special situation of Case 1 when the remineralisation is absent (i.e., Modified Case 1). With HAP demineralisation only, no surface layer developed and the area where demineralisation was the most extensive was close to the middle of the cavity (~130 μm in depth of the pit, Figure 5.3b1). Conversely, with remineralisation the maximum porosity was reached at the bottom of the pit (Figure 5.3a1) and an incipient surface layer can be observed close to the pit bottom. Such a mineral porosity pattern agrees with the literature reports stating that the maximum porosity in an incipient carious lesion formed in a similar geometry (e.g., enamel fissure, occlusal fossa) is at the sides of the fissure and at its bottom (Silverstone et al, 1981; Loesche, 1986; Fejerskov and Kidd, 2008). By comparing these two profiles, it appears that although the demineralisation is more 122

(a) HAP demineralisation-remineralisation saliva 1 5 15 20 25 30 45 60 65 min

plaque

pit bottom

tooth enamel (b) 1 5 15 20 25 30 45 min

FHAP demineralisation

Figure 5.4. Fluxes of Ca2+ at the tooth surface for the Occlusal geometry at different times during the first feeding cycle. (a) HAP demineralization/remineralisation (Case 1) ; (b) FHAP demineralization (Case 2). The arrows are proportional with the flux value and the same vector scale was used for both cases. intense at the sides of the pit, the sides are also efficiently remineralised, which leads to net demineralisation only at the bottom of the fossa. The more aggressive demineralisation in the middle of the pit can be visualised by plotting the vectors of Ca2+ flux released from the enamel along the plaque-tooth boundary (Figure 5.4a). Only demineralisation takes place in the first 20 min of the feeding cycle, on most of the tooth surface in contact with plaque. At the end of the acid attack (25 min) the remineralisation process becomes active, first near the saliva boundary (top), then gradually extending towards the bottom of the pit. While the top surfaces remineralise, the pit bottom is still mildly affected by demineralisation – showing that the two opposing processes can coexist in the same lesion. After 45 min, remineralisation is the only process active and the thin enamel layer restores along the whole tooth surface. Because in this model occurrence of remineralisation is conditioned by both non-acidic pH

123

7 s 10 s 1 min

plaque tooth

12 min 23 min 30 min

Figure 5.5. Variation of pH distribution in plaque and tooth enamel at different moments during one feeding cycle of 4000 s.

(leading to supersaturated conditions) and the presence of mineral porosity above 1%, the remineralisation rate decreases in the middle of the fossa as the surface layer is restored. The upper part of the enamel (close to saliva) is not affected at all by demineralisation (Figure 5.3 – upper panels and Figure 5.4a). In this area, the plaque fluid does not become undersaturated with respect to HAP because the pH is never acidic enough (Figure 5.5). The quasi-neutral pH of the saliva combined with the acidity generated by the microbial metabolism results in a pH gradient in the dental plaque, which creates different local conditions for demineralisation along the plaque-tooth interface (Figure 5.5).

(b) Smooth geometry A surface lesion developed also when the biofilm grows on the smooth buccal or lingual surface of a tooth. A hemispherical biofilm colony was considered in this case (called Smooth 124 geometry) (Figure 5.3a – lower panels). Without remineralisation (Figure 5.3b2) it appears that the largest area of maximum enamel porosity (tooth damage) was at the symmetry axis, in the middle of the tooth area occupied by the colony, where the strongest acid concentrations formed. Like in the Occlusal case, the enamel region covered by plaque close to plaque-saliva boundary was not affected by demineralisation. As expected, also in this case no surface layer develops in the absence of remineralisation. Interestingly however, in the remineralisation case, after 80 feeding cycles no enamel lesion formed (Figure 5.3a2). Although there was significant demineralisation during the feeding and clearance phases (reaching 10% porosity), the remineralisation was very effective during the resting phase, so that the lesion completely healed by the end of the feeding cycle. The Smooth and Occlusal geometries differ not only in shape but also in biofilm volume, which makes the comparison of the results obtained in the two cases more difficult. However, it can be noted that the larger plaque volume in the Smooth case will produce more acids than the Occlusal biofilm, triggering more HAP demineralisation per cycle. On the other hand, for the Smooth case there is less limitation in the ions diffusion outside the plaque due to the higher plaque surface to plaque volume ratio. This could make the remineralisation more efficient, which could explain why no lesion was obtained in the Smooth case. Another straightforward influence of the tooth site geometry is on the place where demineralisation starts. For demineralisation only the lesion is deeper closer to the plaque- saliva boundary in the Occlusal geometry (Figure 5.3b1), whereas the deepest lesion is at the symmetry axis in the Smooth geometry (Figure 5.3b2). When remineralisation occurs, however, in the Occlusal geometry a lesion develops at the bottom of the fossa and a small

surface layer of 1-2 μm sound enamel remains.

5.2.2. FHAP demineralisation (Case 2) This case represents the hypothesis in which the characteristic profile of an initial carious lesion is the result of a solubility gradient in the enamel depth caused by a heterogeneous mineral composition, assumed to contain a higher fluoride content towards the tooth surface.

(a) Occlusal Geometry With FHAP demineralisation, a quasi-uniformly distributed surface layer developed along the tooth-plaque boundary starting from ~40 μm from the saliva boundary and continuing until the bottom of the pit. The influence of the fluoride distribution leading to a less soluble enamel is best observed by comparing the mineral profiles obtained for FHAP

125 demineralisaion (Case 2, Figure 5.3c1) with the corresponding ones calculated for HAP demineralisation (Modified Case 1, Figure 5.3b1). The induced solubility gradient in FHAP favours the development of a surface layer, which is not present at all in HAP. The highest enamel porosity in Case 2 developed about the middle of the pit. Although the effect is less pronounced, this resembles the profile calculated for HAP demineralisation. The more uniform lesion formation along the plaque-tooth boundary can be seen from the flux of Ca2+ displayed in Figure 5.4b. In time, after more acid attacks, the area of lower mineral content appears more clearly. Nevertheless, the FHAP containing enamel degrades much less than the HAP-only containing enamel. Similarly with the HAP demineralisation, in the FHAP demineralisation the start point of the lesion is at ~40 μm from the tooth surface.

(b) Smooth geometry At the end of the 80 feeding cycles of 4000 s each, a subsurface lesion developed close to the symmetry axis. By comparing this lesion (Figure 5.3c2) with the one obtained for the same geometry in the HAP demineralisation case (Figure 5.3b2), it clearly appears that less enamel demineralised and the resulting sub-surface lesion is less deep for the fluoride-containing enamel. The diffusion barrier represented by the surface layer in Case 2 also makes the caries progression slower. In both geometries a surface layer developed in the presence of an enamel solubility gradient. This gradient in fluoride makes the first few micrometers of the enamel along the plaque-tooth interface insoluble over the pH range reached in the plaque during acid attacks. Like in the case of HAP demineralisation/remineralisation (Case 1), the first 40 μm of enamel close to the biofilm surface were not demineralised at all. Although the development of a surface layer in the Case 2 (FHAP demineralisation) was expected, actual profiles obtained for the Occulsal geometry (more demineralisation on the sides of the pit compared to the bottom) could not be intuitively predicted. Interestingly, the maximum porosity (the severity of the lesion) in the Smooth geometry was higher than for Occlusal, for the same reasons exposed in the previous case (i.e., more dental plaque combined with faster diffusion of the species from the plaque in saliva).

(c) Lesion progression For both Case 1 and Case 2 there are two types of dynamics occurring at different time scales. First, short term variations occur during one feeding cycle (e.g., glucose concentration, pH, concentrations of acids produced by dental plaque metabolism, etc.). For example, the pH

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Figure 5.6. Progression of the enamel lesion with the FHAP demineralisation mechanism (Case 2). Each feeding/clearance/resting cycle lasted 4000 s. The tooth demineralisation is a long term process, visible on a longer timescale. Mineral content in the enamel is shown on a colour scale from white (100%) to black (0 % mineral). distribution in plaque and tooth at different moments during the cycle is presented in Figure 5.5. A few seconds after the beginning of the glucose pulse, the pH becomes acidic in the plaque and the low pH front quickly progresses towards the pit bottom. After 1 min the critical pH value (5.5) for enamel dissolution is reached over most of the pit and this is maintained for another 20 minutes, after which clearance leads to a gradual pH restoration to quasi-neutral values (30 min). Second, there are long term changes which become noticeable only over large time intervals, such as the evolution of tooth porosity, as presented in Figure 5.6. The short and long term effects are correlated: the pH variation during one cycle (Figure 5.5) cumulated over several cycles will generate the appearance and progression of the subsurface lesion, as shown in Figure 5.6.

5.2.3. Comparing the lesion formation mechanisms Although with both proposed mechanisms a subsurface lesion could be obtained in the enamel, the profiles and the origin of the lesions differ. In the Occlusal geometry, the alternation of demineralisation periods with remineralisation periods led to a lesion only at the bottom of the pit (with the maximum porosity close to the symmetry axis). However, the surface layer is much less defined than in the fluoride-containing enamel. In the FHAP case,

127 the lesion developed as a consequence of a solubility gradient inside the tooth enamel. The surface layer is well defined and uniform, while the lesion is more demineralised in the pit middle compared to the bottom. Interestingly, the combination of these two types of profiles (maximum demineralisation at the bottom of the pit and on its sides) corresponds to the experimentally measured mineral content of natural carious lesions occurring in enamel fissures (Silverstone et al, 1981; Loesche, 1986; Fejerskov and Kidd, 2008). For the Smooth geometry, the subsurface lesion developed only in the absence of remineralisation and mainly in the middle of the tooth enamel area covered by plaque.

5.2.4. Impact of model assumptions A mathematical model is an abstract representation of a real situation by making use of simplifying assumptions and mathematical equations. The assumptions are an essential aspect in the conceptual phase of a model: they must be purpose-oriented and preserve the essential features of the problem to be studied (i.e., keep the model realistic enough). Yet, the assumptions should also reduce the complexity in order to allow the model to be solved numerically, but also to reduce the number of measurable model parameters. Therefore the output of a model relies greatly on the simplifications made in the initial stages of the model development. Two important assumptions made in the current work are: (i) the two- dimensional geometry and (ii) the representation of the tooth compartment as an explicit 2-d domain. The impact of these two decisions will be analysed by first comparing the current 2-d results with those obtained in a simplified 1-d geometry (as in Ilie et al., 2014). Then, results are compared with those obtained with a similar model in which the tooth has been implemented as a boundary condition to the dental plaque.

(a) One-dimensional vs. two-dimensional geometry The number of spatial dimensions considered has an impact on the realism of the mathematical model (a 3-d model should be the closest to reality). On the other hand, increasing the number of space dimensions leads to a significant escalation of the number of degrees of freedom (points in which the state variables are calculated) of the problem to be solved. This increase can be of several orders of magnitude and it impacts the computational speed. For example, the 1-d plaque-tooth model (Ilie et al., 2014) runs 80 feeding cycles in 1 to 5 hours, whereas the current 2-d plaque-tooth model requires 5 days to compute the same 80 feeding cycles at comparable mesh size with the 1-d model. Another decision factor in

128

choosing of the model geometry is the purpose of the model. For example, if the purpose is to show whether a subsurface lesion can form at all inside the tooth then a 1-d geometry would suffice. However, if the aim is to show the influence of the pH gradients along the plaque- tooth interface on the profile of the carious lesion at different locations on the tooth, then 2-d or 3-d geometries are required. Solute concentration and enamel mineral content profiles obtained with both 2-d model geometries along their symmetry axis are comparable to those from previous 1-d models, in both mechanisms (Ilie et al., 2014, Ilie et al., 2012). The mineral profiles along the symmetry axis for the Occlusal and Smooth 2-d geometries together with the profile obtained for an equivalent 1-d situation are presented in Figure 5.7. For the HAP demineralisation / remineralisation case a carious lesion developed for the 2-d Occlusal geometry but not for the Smooth 2-d and 1-d geometries Figure 5.7a. The lesion obtained in the Occlusal case could be the result of the “closed” nature of the geometry which makes the diffusion of protons outside the plaque and tooth possible only in the upward direction and the transfer to the saliva on a smaller area (the saliva-plaque boundary) compared with Smooth case when the diffusion outside the plaque into saliva is possible on a much wider area. Also for the FHAP demineralisation case it is visible that the 2-d Smooth and the 1-d profiles are very similar, although not identical due to the added diffusion in the 2-d case. The 2-d Occlusal profile shows less demineralisation along the symmetry axis than in the other two geometries. It was noticed in Figure 5.3c1 that most of the demineralised FHAP is from the middle of the pit, not at its bottom. Therefore, although it appears that there is less demineralisation in this case, this line (the axis of symmetry) is not the most representative one to compare the 3 cases.

FHAP demineralisation HAP demineralisation / remineralisation tooth-plaque TITLE(Case 2) XXX(Case 1) boundary 100 100 p

80 80

60 60

40 40 Mineral content (%) Mineral content (%) content Mineral 20 1-d 20 1-d 2-d Smooth 2-d Smooth 2-d Occlusal 2-d Occlusal 0 0 -30 -25 -20 -15 -10 -5 0 -30 -25 -20 -15 -10 -5 0 Enamel depth (Pm) Enamel depth (Pm)

Figure 5.7. Mineral content after 80 feeding cycles for the FHAP case in the 2-d geometries along the symmetry axis compared with a 1-d geometry model. 129

(a) (b) (c) 15 min in cycle 80 15 min in cycle 400

time

Figure 5.8. Lesion formation when the tooth is represented as a boundary (case Moving boundary) compared with tooth represented as a 2-d domain (case HAP demineralisation only). (a) Ca2+ fluxes (arrows) on the tooth moving boundary after 80 (left) and 400 (right) feeding cycles; (b) Lesion formation by moving the tooth boundary (red lines: tooth surface position after 100, 200, 300 and 400 feeding cycles); (c) Ca2+ fluxes (arrows) at the tooth surface in the HAP demineralisation case and mineral content in the enamel (same color scale as in Figure 5.3, from white 100% to black 0 % mineral).

(b) Tooth as a domain or boundary condition In addition to the main demineralisation mechanisms, we also implemented a model with moving enamel surface, based on the simplifying assumption that no solutes diffuse in the enamel. As the cavity develops, the plaque-tooth boundary moves and the biofilm grows inside the enamel. This model simplification could be useful computationally because the whole enamel domain is here reduced to a boundary condition. Indeed, when the demineralisation is modelled only on the tooth surface (Moving boundary case), the needed computational times were a quarter of those for HAP demineralisation only with tooth as a domain (Modified Case 1). However, there are notable differences in the amount of demineralised HAP and lesion position among the two cases. For the 2-d moving boundary an almost uniform demineralisation along the tooth was found below 100 μm down the fossa (Figure 5.8a), whereas for the 2-d tooth domain the demineralisation is more active at a medium depth in the pit (Figure 5.8c). On the other hand, ~4 times less HAP was lost with the 2-d moving boundary than with the 2-d tooth domain (4×10-6 (mol HAP) compared with 16×10-6 (mol HAP) after 80 feeding cycles). This important difference is due to the acids penetration in the tooth enamel, which leads to more dissolution in the volume than on the

130 surface only. Only after 400 feeding cycles in the moving boundary case, the lesion will reach the same severity as after 80 cycles with tooth domain (Figure 5.8b). Moreover, the restoration of concentrations after the acid attack in the enamel is slower than in the biofilm, because of slower diffusion. Therefore the impact of small differences in HAP degree of saturation and demineralisation at the middle of the cavity are maintained longer when an enamel domain is present.

5.3. Conclusions A carious subsurface lesion can form using any of the two studied hypotheses: (1) HAP demineralisation/remineralisation, when the tooth is made only from HAP with homogenous solubility and both demineralisation and remineralisation processes are active and (2) FHAP demineralisation, when the tooth enamel is made of FHAP with a solubility gradient lower at the surface and only demineralisation occurring. The influence of tooth site geometry at which the caries is formed is important. The lesion obtained in the Occlusal geometry for both cases is different: (i) more at the enamel surface and only at the bottom of the pit (near the symmetry axis) in the HAP demineralisation / remineralisation case, and (ii) uniformly distributed along the plaque-tooth interface with a clearly defined surface layer and more FHAP demineralisation at the middle of the occlusal fossa in the FHAP demineralisation case. For the Smooth geometry a lesion formed only with the FHAP demineralisation mechanism (close to the symmetry axis), while using the other mechanism the remineralisation was so efficient that for the studied length of the feeding cycle all the HAP lost during the acid attack was remineralised by the end of each 4000 s cycle and no lesion was formed after 80 feeding cycles. By comparing the results obtained for the two geometries within the same tested mechanism, it appears that the most “damaged” by the acidity of the dental plaque are the occlusal sites compared to the smooth surfaces. This is in agreement with the general conception based on experimental observations that caries are formed with predilection at protected site on the tooth (e.g., occlusal surface, approximal areas). In both cases was noticed that the geometry influences the shape and location of the lesion as well as its severity. From a mathematical modelling point of view, greater insight can be obtained by using a 2-d numerical model that can account for local concentration gradients along the plaque- tooth boundary, than using a 1-d model which does not offer the possibility to study local effects along the interface. Also, if the tooth is represented as an explicit 2-d domain makes a difference in the profile of the calculated lesion by pointing at areas of preferential

131 demineralisation/remineralisation compared to the case when the tooth is represented as a boundary and the demineralisation appears to be uniformly distributed at the tooth surface. With a few additions to the current model, other problems related to dental caries formation can be studied: by considering plaque growth, mixed bacterial populations and microbial activity gradients it is possible to study microbial shifts in the dental plaque and their effect on the initiation of dental caries; adding protein adsorption on the enamel surface could give more insight into the mechanisms of initial colonisation of the tooth by bacteria interacting with the acquired enamel pellicle; a model accounting for inhomogeneities and defects in the crystal structure and composition would help to evaluate the theory that dental caries are initiated at sites with higher tension in the HAP crystal lattice (e.g., curved surfaces) or with defects in the structure (e.g., small fissures, inclusions of foreign ions in the HAP crystal).

5.4. References Atkins P, De Paula J, Physical Chemistry, 9th ed., W. H. Freeman, 2009. Dibdin GH, Diffusion of sugars and carboxylic acids through human dental plaque in vitro. Arch Oral Biol 1981; 26:515-523. Dibdin GH, Effect on a cariogenic challenge of saliva/plaque exchange via a thin salivary film studied by mathematical modelling, Caries Res 1990; 24:231-238. Dirksen TR, Little MF, Bibby BG, Crump SL: The pH of carious cavities. I - The effect of glucose and phosphate buffer on cavity pH. Arch Oral Biol 1962; 7:49-57. Fejerskov O, Kidd E, Dental caries: The disease and its clinical management, 2nd ed, Blackwell Munksgaard, 2008. Fremlin JH, Mathieson J, A microchromatic study of the penetration of enamel by C14- labelled glucose. Archs Oral Biol 1961; 4(C):92-96. Grey JA, Francis MD: Physical chemistry of enamel dissolution; in Destruction of hard tissues, RF Sognnaes (ed) Washington, 1963: Publication No 75 of the American Association for the Advancement of Science, pp 213-260. Hamilton IR, St. Martin EJ, Evidence for the involvement of proton motive force in the transport of glucose by a mutant of Streptococcus mutans Strain DR0001 defective in glucose- phosphoenolpyruvate phosphotransferase activity. Infect Immun 1982; 36(2):567-575. Ilie O, Van Loosdrecht MCM, Picioreanu C, Mathematical modelling of tooth demineralisation and pH profiles in dental plaque. J Theor Biol 2012; 309:159–175. Ilie O, Van Turnhout AG, Van Loosdrecht MCM, Picioreanu C, Numerical modelling of tooth enamel subsurface lesion formation induced by dental plaque. Caries Res 2014; 48:73-89. Langdon DJ, Elliott JC, Fearnhead RW: Microradiografic observation of acidic surface decalcification in synthetic apatite aggregates. Caries Res 1980; 14:359-366. 132

Loesche WJ, Role of Streptococcus mutans in human dental decay. Microbiol Rev 1986; 50(4):353-380. G.H. Margolis HC, Moreno EC, Kinetics of hydroxyapatite dissolution in acetic, lactic and phosphoric acid solutions. Calcified Tissue Int 1992; 50:137-143. Moreno EC, Kresak M, Zahradnik RT, Fluoridated hydroxyapatite solubility and caries formation. Nature 1974; 247: 64 – 65. Nancollas GH: Kinetics of demineralisation and remineralisation; in: S.A. Leach, W.M. Edgar (Eds.), Demineralization and Remineralization of the Teeth, Irl. Pr., 1983, pp. 113–128. Robinson C, Shore RC, Brookes SJ, Strafford S, Wood SR, Kirkham J: The chemistry of enamel caries. Crit Rev Oral Biol M 2000; 11(4):481-495. Silverstone LM, Wefel JS, Zimmerman BF, Clarkson BH, Featherstone MJ, Remineralization of natural and artificial lesions in human dental enamel in vitro. Effect of calcium concentration of the calcifying fluid, Caries Res. 1981; 15(2):138-57. Ten Cate JM, Remineralization of enamel lesion. A study of the physico-chemical mechanism. (Dissertation), University of Groningen, The Netherlands, 1979. Van der Hoeven JS, De Jong MH, Camp PJM, Van den Kieboom CWA, Competition between oral Streptococcus species in the chemostat under alternating conditions of glucose limitation and excess. FEMS Microbiol Ecol 1985; 31:373-379. Van der Hoeven JS, Gottschal JC: Growth of mixed cultures of Actinomyces viscous and Streptococcus mutans under dual limitation of glucose and oxygen. FEMS Microbiol Ecol 1989; 62:275-284. Van der Hoeven JS, van den Kieboom CW, Camp P J: Utilization of mucin by oral Streptococcus species. Anton van Lee 1990; 57:165-172. Van Dijk JWE, Borggreven JMPM, Driessens FCM: Chemical and mathematical simulations of caries. Caries Res 1979; 13:169-180. Van Dijk JWE, The Electrochemistry of Dental Enamel and Caries. (Dissertation), The Netherlands, 1978. Vanýsek P, Handbook of chemistry and physics, 82nd ed., Boca Raton: CRC Press LLC, 2001, pp. 5-95 and pp. 6-194. White DJ, Chen WC, Nancollas GH: Kinetic and physical aspects of enamel remineralisation - a constant composition study. Caries Res 1988; 22:11-19.

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A three- dimensional 6

numerical study on the privacy of

cell-cell

communication

6. A three-dimensional Different levels of understanding the cell-cell communication numericalhave study crystallized in three on main theories, the with focus on several types of information cells can obtain from the environment privacy ofvia cell-cellsignaling compounds. Chronologically, the first theory was the quorum sensing (QS). QS is defined as the ability of communicationcells to sense their own cell density in a population, in order to behave as a population instead of individually (Fuqua et al., 1994). Sensing how large the cell population is may be advantageous in initiating group actions such as virulence for pathogenic bacteria (e.g., Pseudomonas aeruginosa causing cystic fibrosis disease (Kirisits and Parsek, 2006; Girard and Bloemberg, 2008)) or luminescence for Vibrio fischeri bacteria living in symbiosis with certain fish species that use this bioluminescence to attract prey (Dunlap, 1999). In addition to the quorum sensing concept, Redfield (2002) proposed the diffusion sensing hypothesis. According

134 to this theory, group behavior is regulated not only by cell density, but also by the physical properties of the environment such as the intensity of transport processes: diffusion, mixing, presence of flow, etc. Using sensing molecules that are metabolically cheap to produce and are not naturally present in the environment, microorganisms can test if the surroundings are favorable for secreting other compounds requiring more energy to be synthesized. Hense et al. (2007) introduced the concept of efficiency sensing, as a unifying theory for bacterial communication. This hypothesis states that cells, using signaling molecules, are able not only to measure the combination of cell density and mass transfer limitations, but also their spatial distribution. Microbial cells can communicate via signaling molecules (Nealson et al., 1970). Cells secrete metabolically inexpensive molecules in the environment (acyl homoserine lactones – AHL – for gram negative bacteria and peptides for gram positive bacteria) until a threshold concentration is reached, then the cells switch in a different metabolic state called the induced state. Associated to the induction is also an increased production rate for the signaling molecules (Fekete et al., 2010). This increase in the production of signaling molecules is called positive feedback kinetics (or simply, feedback kinetics). Because signaling molecules trigger the cell induction, they are also called autoinducer molecules. Therefore, the bacterial group behavior can be orchestrated by the signaling molecules via “sensing” mechanisms. This may be an efficient system, but not perfect since it cannot prevent the appearance of so-called “cheaters”. In this context, cheaters are mutant or intruding bacterial cells enjoying the benefits of living in a community without investing the metabolic costs of producing public goods (here, signal molecules). Since the presence of cheaters cannot be eliminated, bacteria may have developed mechanisms of coping with their presence and minimize the negative effects of “free riders”. One such mechanism could be to keep the communication private between their kin i.e., per colony. Maintaining the privacy of communication within a colony makes perfect sense in such a context since a colony is mostly formed by clones of the same cell, therefore, the risk of having cheaters around is considerably lower (Hense et al., 2007). If this is the case, the efficiency sensing theory (i.e., sensing the spatial arrangement of bacterial cells) could obtain new support. In order to get more insight into this type of bacterial communication, the current work tries to answer three central questions: (1) What is the effect of cell clustering on signaling and up regulation for clustered cells and lone cells compared to the random distribution?

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(2) Who listens to whom? To what extent is signaling within clusters influenced by signaling in the neighborhood? (3) What are the effects of positive feedback? Mathematical modelling is one way to investigate these questions because of its capacity of simplifying the reality to the essential characteristics regarding the problem of interest. Dockery and Keener (2001) developed a metabolic model at intracellular level, including the regulatory pathways with focus on the quorum sensing theory. Their conclusion was that quorum sensing is based on a biochemical switch between two stable steady states, one with low levels of autoinducer and one with high levels of autoinducer. At the same time, Ward et al, (2001) conducted another study focused at population level and using a black box approach, reaching similar conclusions to Dockery and Keener. Horswill et al. (2007) built a diffusion-reaction one-dimensional (1-d) biofilm model where both quorum sensing and diffusion sensing aspects (regarding molecular diffusion and advection) were integrated. They concluded that the flow rate of the liquid washing the surface of a signal producing biofilm affects the degree of induction within the biofilm: for laminar flow, the lower the flow rate, the more induced the biofilm is. Vaughan et al. (2010) used a continuous two-dimensional (2- d) mathematical model and they also concluded that increasing the flowrate in a system has a negative effect on QS due to the signal molecules being washed away faster. The roughness of the biofilm surface however, has a positive effect on the induction because it creates protected area from the flow, where the signal can accumulate. A three-dimensional (3-d) model was included in the work of Hense et al (2007) to support the description of the newly introduced efficiency sensing showing that the same amount of cells, placed in identical environmental conditions will become induced if grouped together and will remain in a non-induced state if spread away from each other. In the present study, we develop the ideas from Hense et al. (2007) in a more elaborate model, based on measurable parameters and aiming at quantitative results. Therefore, a new 3-d model was built to integrate in a rigorous manner the available knowledge, using experimentally measured parameters (Fekete et al., 2010). We first assess the three theories on bacterial communication by studying the influence of cell density (QS), system geometries and spatial distribution on the necessary size of a given colony at induction. The importance of positive feedback in AHL production kinetics from a cell-cell communication point of view has been numerically evaluated. The second part of the study is dedicated to the privacy of communication and how this can be influenced by factors such as system geometry and signal production rates.

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6.1. Model Description 6.1.1. Signal production and spreading The cell-cell signaling model used in this work describes the effect of microbial distribution on communication between microbial colonies attached on a planar surface (i.e., biofilms). Soluble signal distributions were calculated in a three-dimensional space (3-d) for different cases with an equivalent in nature.

(a) Microbial cells Only one microbial species is considered in the model (e.g., Pseudomonas putida). The biological parameters (e.g., AHL production kinetics) are the same for all the individual cells and they are taken as for Pseudomonas putida cf. Fekete et al. (2010) (Table 6.1). However, in order to study the privacy of cell-cell communication, a distinction had to be made between two subpopulations: cells from a “reference” colony and the “background” cells. The reference colony contained cells on which the effect of foreign signal on the induction parameters was monitored. This colony was placed in the middle of the computational domain. The measure used to quantify the effect of different parameters on privacy of communication was the number of cells present in the reference colony at the moment of induction. The size of the reference colony was increased starting from a single cell until the colony reached a critical number of cells for which the induction took place. The mechanism for colony growth was adapted from the individual-based model of Kreft et al. (2001). Background cells are the producers of the “foreign” signal having different distributions on the planar surface in the neighborhood of the reference colony (see Figure 6.1). In order to have comparable contributions of background cells to the signal production, the distribution of background cells was kept the same while increasing the size of the reference colony. Both types of cells are represented as individual spheres, characterized by masses between 0.1 and 0.2 fg and radii between 0.6 and 0.8 Pm. Background cells repartition on the surface is described by three parameters: population density, crowdedness and randomness. Population density allows setting the intensity of the background concentration of signal perceived by the reference colony. The population density is defined as the number of background cells per unit area attachment surface. Furthermore, by varying the population density it is also possible to test the quorum sensing theory in the current model setting. Population crowdedness is a measure originally used in ecology for spatial statistics (Dale et al., 2002). In the current context, crowdedness was used to quantify and control the

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Reference colony

Background colonies

Figure 6.1. Representation of the two subpopulations of cells present in the computational domain. White – reference colony, Pink – background colonies degree of “agglomeration” of background cells. Crowdedness is the handle to set up the non- homogeneity of the background signal spatial distribution. Low values of crowdedness indicate a homogenous distribution of the background signal, while higher values of crowdedness means a non-homogenous signal distribution. The crowdedness, C (cell/cell), (eq. 6.1) was calculated as the arithmetic mean of local crowdeness, Clocal (cells), where the local crowdedness represents the number of neighbouring cells found in a circle surrounding the cell for which the crowdedness is calculated. The surface of this circle is equal with the area of the entire computational domain, divided by the total number of background cells, N (cells). C C ¦ local (6.1) N Distributions of background cells with the same population density and crowdedness are, however, not unique. Randomness of distribution is therefore another parameter influencing the spatial arrangement of background cells. Two types of spatial distributions were used: random and uniform. Ten random distributions were created to study the influence of parameters such as crowdedness, density, geometry and kinetics on general aspects of communication. Uniform distribution, when cells are positioned equidistantly on a grid, was used for the specific study of privacy of communication.

(b) Signal compound There is only one signal molecule considered, acyl homoserine lactone (3 oxo-C12- HSL), named AHL. 138

(a) (b) kAHL,in+kAHL,ni

kAHL,ni

Figure 6.2. Influence of rate parameters on the Hill kinetics (eq.6.2). (a) variable Hill exponent n at constant

KHill=0.07 PM; (b) variable affinity coefficient KHill at constant n=2.

Production. Both cell subpopulations produce AHL with the same kinetics. Previous studies Fekete et al., (2010)) have revealed that cells produce the signal molecule with a positive feedback kinetic based on AHL. A constant basal rate of AHL generation, kAHL,ni, was assumed regardless of cell state (non-induced or induced). In addition, cells become induced when the AHL concentration reaches a certain threshold (70 nM). Consequently, the AHL production significantly increases according to a Hill kinetic expression, kAHL,in ∙ fHill. The total AHL production therefore describes a positive feedback effect on the AHL rate:

n cAHL rkAHL AHL,ni k AHL,in nnn n (6.2) KcHill+ AHL

fHill In eq.(6.2), the characteristic kinetic parameters are the constant AHL production rate in the basal state, kAHL,ni, the maximum rate of AHL production in the induced state, kAHL,in, the Hill coefficient, n, and the affinity coefficient, KHill. KHill controls the threshold AHL concentration and n affects the steepness of the sigmoidal transition to induced AHL production (Figure 6.2). Values of the kinetic parameters used, as measured by Fekete et al., (2010), are listed in Table 6.1. Model simulations were performed in both kinetic hypotheses: (i) without feedback (zero order rate, rAHL=kAHL,ni) and (ii) with positive feedback rate (eq. 6.2).

Table 6.1. AHL production kinetic parameters. Values are considered according to Fekete et al., (2010). Parameter Symbol Value Units

Basal AHL production rate kAHL ni 0.064 Pmol/L/s

AHL production rate after induction kAHL in 0.64 Pmol/L /s Hill coefficient n 2 –

Affinity coefficient KHill 0.07 Pmol/L 139

(a) (b) (c)

Figure 6.3. System configurations. (a) Pit case: all boundaries impermeable to AHL (zero-flux, JAHL=0) except the upper one which is an infinite sink (CAHL=0); (b) Open case: all boundaries open (CAHL=0) except the impermeable attachment surface (JAHL=0); (c) Thin Layer: lateral boundaries are open (CAHL=0), while the lower and upper ones are closed (JAHL=0). The gray boxes contain the cells distributed in equal computational volumes, while the white boxes in (b) and (c) are "buffer" regions where the signal can diffuse but no cells are present.

Transport. Only diffusive transport of AHL was taken into account in the present study. This situation corresponds either to stagnant aqueous conditions or to a computational domain included in the diffusive boundary layers (if flow were present in the system). The 3-d diffusion-reaction mass balance of AHL is:

222 wcAHL§· www ccc AHL AHL AHL DrcAHL¨¸222 AHL AHL (6.3) wwwwtxyz©¹ –11 2 were DAHL is the diffusion coefficient (6 × 10 m /s, cf. Hense et al., 2007). Different types of boundary conditions associated to eq. (6.3) were set according to the particular case studied. Initially, there was no AHL present in the computational domain.

6.1.2. Geometry cases The effect of spatial microbial distribution on communication between microbial colonies was studied in three different cases, corresponding to possible real-life situations (see Figure 6.3). The three cases studied represent different degrees of "opening" for the system: Pit Case is the most insulated to signal dissipation in the environment, the Open Case allows the most signal loss, while the Thin Layer Case is an intermediate situation. These three systems have different geometries and boundary conditions.

(a) Pit Case This case is similar to a situation when cells are attached at the base of a cavity ("pit"), filled with a stagnant layer of liquid. It is assumed that above the pit a stream of liquid would wash the entire signal produced by the cells. The computational domain is rectangular, with size

Lx × Ly × Lz. Two sub-cases were studied: (1) Pit Base Case: 200 × 200 × 100 microns (2) Pit

Shallow: 200 × 200 × 25 microns. The upper limit at z = Lz constitutes an infinite sink for the

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AHL, thus cAHL = 0. The five pit walls are closed boundaries, with zero flux of AHL normal to the wall surface ( wwcnAHL /0 at x = 0, x = LX, y = 0, y = LY and z = 0).

(b) Open Case An equivalent for this model setup is represented, for example, by the biofilm growing on a rock on the bottom of a lake. Compared to the Pit case, the signal produced by the cells can leave the system also through the lateral boundaries. However, setting an infinite AHL sink on the lateral boundaries would not be a representative approximation because the AHL concentration would drop too steeply to zero (i.e., these boundaries are too close to the background cells, while the upper boundary is still distant enough). Realistically, it would be preferable to have the zero AHL concentration at infinite distance. A numerical compromise was therefore adopted: a "buffer volume" was introduced around the main computational domain (see Figure 6.3b). The main computational domain (internal box) is identical to the one from Pit Base Case (200 × 200 × 100 microns), but it is surrounded by a larger “box” with the dimensions 400 × 400 × 100 microns. The lateral boundaries of the internal domain imply flux and concentration continuity, while for the outer zone AHL sinks can be set

(CAHL=0).

(c) Thin Layer Case This case simulates, for example, dental plaque having a thin layer of stagnant liquid (e.g., saliva) above. The boundaries and the domain resemble the Open Case with the only difference that the upper limit is closed (JAHL = 0). For this geometry, as for the Pit Case, two sub-cases were discussed: (1) Thin Layer Base 400 × 400 × 100 microns; (2) Thin Layer Shallow 400 × 400 × 25 microns.

6.1.3. Model solution When solving the numerical model, the aim was to find the critical size of the reference colony for induction. The computer code implementing the model was written in MATLAB (MATLAB 2008b, The MathWorks, Natick, MA, www.mathworks.com), linked with embedded solution algorithms in Java and COMSOL (COMSOL 3.5a, Comsol, Burlington, MA, www.comsol.com). The solution algorithm steps are: (i) An initial distribution of background cells is generated on the attachment surface, with a defined population density, crowdedness and randomness. 141

(ii) The AHL concentration field, cAHL(x,y,z), is calculated for this cell distribution, according to eq. (6.3) and the boundary conditions corresponding to each studied case. The 3-d concentration field is found by finite element methods, using the solvers fromCOMSOL. The maximum mesh size in the whole domain was 50 Pm. To speed-up the calculations, the steady-state diffusion-reaction equation for AHL is solved when the colony sizes are much smaller than the induction size (i.e., the microbial colonies grow much slower than the change in solute concentration, so that the AHL spatial distribution can be considered at steady state at any time step of microbial growth). However, at times close to the colony induction, the steady state assumption is no longer valid due to the sudden changes involving the AHL kinetics with feedback. Therefore, the solution of the time-dependent diffusion-reaction eq. (6.3) was found by time stepping during the induction. For verification, time-dependent simulations were run also for the whole duration of colony growth and induction and the results were very similar to those obtained using steady state AHL distribution during colony growth and transient during the induction period. (iii) Check if induction has been achieved. A microbial population is defined here as induced when the maximum local concentration of AHL present in the computational domain exceeds the KHill threshold concentration: cAHL(x,y,z) > KHill. (iv) Colony size increases incrementally, by successive divisions and movement of cells based on the individual-based algorithm from Kreft et al (1998, 2001) and Picioreanu et al 2004. The individual-based colony growth model was implemented in Java and linked to the MATLAB main code. For accuracy, smaller time steps were used near the induction point.

6.2. Results and Discussion We evaluated the effects of cell population crowdedness, population density, AHL kinetics and system geometry on the cell-cell communication with emphasis on privacy of communication. In each geometry case, we investigated the communication behavior at population densities of 625, 5625, 15625, 30625 and 50625 background cells per mm2 of domain (i.e., 25, 225, 625, 1225 and 2025 cells per computational domain of 200 × 200 Pm2). In addition, for each population density, crowdedness took the values 1, 2, 5, 10, 20, 30, 40, 60 and 80. For each pair of density/crowdedness values, 10 simulations were performed, each with different random distribution of cells. All these sets of simulations were performed both with and without feedback in the AHL production rate.

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Results were evaluated from several perspectives. The reference colony size at induction is probably the most important indicator for these combined effects. Further, a privacy factor was defined to quantify how much a micro-colony (the reference colony) is influenced by its own signal and how much is by the signal produced by its neighbors (the background cells).

6.2.1. Colony size at induction The number of cells in the reference colony needed for induction when no background cells are present in the environment (i.e., the blank case) is here called basic induction number (N0).

Table 6.2 presents the basic induction numbers obtained in all simulated cases. N0 was used as a reference to evaluate the effect of background cells with different population densities and crowdedness in various geometries and kinetic regimes on the induction of the reference colony. When background cells are present, more AHL is released in the environment and the induction point can be reached sooner. Therefore the number of cells in the reference colony at induction Nref col will be less than N0.

Table 6.2 Basic induction numbers (N0) in the reference colony Hill Kinetics Zero Order Kinetics System Boundaries (feedback present) (feedback absent) Deep Pit 501 4164 Shallow Pit 592 10048 Thin Layer Base 529 3850 Thin Layer Shallow 260 1964 Open 608 4301

(a) Effect of AHL kinetics The most important effect of the feedback on AHL production is the decrease in the number of cells in the reference colony at induction, Nref col, compared with the kinetics without feedback. In all studied cases, the colony size at induction is about one order of magnitude smaller in the presence of positive feed back, as shown in Table 6.2 for all geometry cases studied. By comparing the sequence of images in Figure 6.4 and Figure 6.5 it can be observed that when using the basal AHL production rate (without feedback), the colony induction takes place in a very slow and gradual manner (Figure 6.5).

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(a)(a)

63 cells 433 cells Induction 461 cells 1616 cells (b)(b )

Figure 6.4. 6 4 Spreading of induction for signal production with feedback kinetics (background(background population density 15600 cells/mm2, population crowdedness 30, geometry case Deep Pit). (a) Spatial distribution of AHL concentration for different numbers of cells in the reference colony in the middle of the domain (numbers indicated below images), while the background cells remain unchanged. The two slices through the 3-d domain are: x-y at z=0 and x-z at y=100 Pm. Color scale indicates AHL concentration (blue = 0, red = 4 × 10-4 mol/m3). White – non-induced cells in the reference colony, magenta– non-induced background cells, black – induced cells. (b) 3-d distributions of cells. White – all non-induced cells, red – all induced cells. Intermediate pink shades show different levels of induction.

(a) x-zx-z

x-yx-y

3840 cells Induction 4015 cells 12386 cells 384 cells (b)

FigureFi 66.5. 5 SdiSpreading offid inductiondi for f sigilnal production di withoutith t ffeedbackdb k (background(b k d populationlti density d it 15600 cells/mm2, population crowdedness 30, geometry case Deep Pit). (a) Spatial distribution of AHL concentration for different numbers of cells in the reference colony (numbers indicated below images), in two slices: x-y at z=0 and x-z at y=100 Pm. (b) 3-d distributions of cells. Color scales as in Figure 6.4. 144

As a general tendency, model simulations demonstrate that with feedback kinetics

(Hill rate added to the basal rate) the crowdedness influences the induction parameter Nref col. For example, in the Deep Pit case presented in Figure 6.6 for the same population density the induction number decreases notably at higher crowdedness values (i.e., clustering the cells leads to sooner induction). Conversely, in the absence of feed back, crowdedness variation has no effect on the induction number Nref col (Figure 6.6).

(b) Effect of background population density The influence of background population density on decreasing the induction number of reference colony, Nref col, compared with the basic induction N0 can be quantified by a reduction quotient Q. A high value of the reduction quotient Q, means a strong influence of the background cell density on the reference colony. NN Q 0 ref col u100 (6.4) N0 The reduction quotients averaged over 10 simulations obtained at the highest population density and crowdedness 1 (i.e., when background cells are uniformly distributed) in the absence of feedback for each geometry case are presented in Table 6.3.

Table 6.3 Reduction quotients for different geometry cases at background cell density 50625 cells mm–2 and crowdedness 1. System geometry Q (feedback present) Deep Pit 36 % Shallow Pit 7 % Thin Layer Base 48 % Thin Layer Shallow 25 % Open 15 %

In accordance with the quorum sensing theory, the reference colony induces sooner for higher background population densities: more signal sources means a higher AHL concentration in the background and therefore faster reaching of the threshold concentration for induction. This was indeed observed in all performed simulations (Figure 6.6). However, this effect can be amplified or diminished by the system geometry and in some cases by the crowdedness. In the case of feedback kinetics of signal production, the Deep Layer and Deep Pit show the most pronounced effects of background cell density increase on the induction number Nref col (Figure 6.6a and Figure 6.6.c, respectively). In these cases, the signal has less possibilities of leaving the system: the pit has 5 impermeable walls, while the layer has two. Consequently, in these two cases the reduction quotient has the largest values, 36% for pit and 48% for the layer (Table 6.3). 145

(a) 600 (b) 600

500 500

400 400

300 300

200 200 (#cells at induction) (#cells at induction) (#cells # Cells in focal colony at induction # Cells in focal colony at induction 100 100 ref col ref col

N 25_cells 225_cells 625_cells 1225_cells 2025_cells N 625_cells 1225_cells 2025_cells 0 0 0 102030405060708090 0 102030405060708090 Crowdedness Crowdedness (c) (d) 600 600 625_cells 1225_cells 2025_cells

500 500

400 400

300 300

200 200 (#cells at induction) (#cells at induction) 100 # Cells in focal colony at induction 100 ref col # Cells in focal colony at induction ref col N 625_cells 1225_cells 2025_cells N 0 0 0 102030405060708090 0 102030405060708090 Crowdedness Crowdedness

(e) 600 Figure 6.6. Variation of the induction number

500 Nref col for different population densities and crowdedness values in the presence of feedback. 400 The different geometry cases are: (a) Deep Pit; (b) Shallow Pit; (c) Thin Layer Base; (d) Thin 300 Layer Shallow; and (e) Open. Error bars indicate deviations among the 10 simulations performed for 200 each pair of values with different random

(#cells at induction) background colony distributions. The orizontal red 100 line represents the basic induction number, N0, for # Cells in reference colony at induction (N)

ref col 625_cells 1225_cells 2025_cells 225_cells 25_cells

N 0 each case 0 102030405060708090 Crowdedness When the signal is produced with a feedback rate, the Thin Layer Base and Deep Pit show

the most pronounced effects of background cell density increase on the induction number Nref col (Figure 6.6a and Figure 6.6.c, respectively). In these cases, the signal has less possibilities of diffusing away in the environment: the pit has five impermeable walls, while the layer has two, therefore an increase in the number of background cells means an increase of the total AHL concentration and an earlier induction of the reference colony. Consequently, in these two cases the reduction quotient has the largest values, 36% for pit and 48% for the layer (Table 6.3). In contrast, for the Open system only the attachment surface is impermeable, while the signal can diffuse out in all other directions, therefore the threshold concentration is reached more difficultly. Hence, the effect of background population density is rather minor with Q=15%

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(Table 6.3, Figure 6.6). Similarly, the Shallow Pit system (Figure 6.6) shows little variation in the size of reference colony at induction with increasing cell density. This reveals the importance of the distance to the upper sink of signal: when positioned close, the signal can rapidly diffuse away. In the case of Thin Layer Shallow (Figure 6.6), the geometry effect overrides the cell density effect, suggesting that diffusion sensing is more important than quorum sensing. Closing the upper boundary therefore keeps the signal to accumulate in the system and accelerates the induction.

(c) Effect of background population crowdedness The influence of population crowdedness on the homogeneity of the background signal is shown in Figure 6.7. For lower values of crowdedness the background signal is more homogenously distributed in space than for high crowdedness. For all the analyzed cases, crowdedness has an effect on the induction number only in the presence of feedback. There is a faster induction for increasing crowdedness (Figure 6.6). At higher crowdedness, the background colonies are larger therefore the local AHL concentration is increased. This triggers more signal production overall due to feedback, which means higher AHL concentrations everywhere in the domain, thus faster induction also for the reference colony. As expected, the crowdedness effect is less pronounced for the Open and Shallow Cases, when the signal leaves the system at a higher rate (Figure 6.6). This is in agreement with the efficiency sensing theory (Hense et al., 2007), showing that for the same number of cells present in the system the induction can occur at different times, depending on the spatial distribution of cells.

(a)

(b)

Crowdedness = 10; Crowdedness = 20; Crowdedness = 60; Background cells = 625 Background cells = 625 Background cells = 625 Figure 6.7. (a) Two-dimensional AHL concentration field at the attachment surface (z=0), corresponding to the three distributions of background cells, calculated for Deep Pit case; (b) Examples of spatial distribution of background cells for different values of crowdedness. 147

(a) (b)

Figure 6.8 Background cells placed on a square grid to generate a regular field of background AHL concentration in the Deep Pit case: (a) signal concentration produced by reference colony (CAHL,R), (b) signal 3 concentration produced by background cells (CAHL,B). Color scale: AHL concentration in mol/m .

6.2.2 Privacy of communication In order to evaluate the degree of privacy in bacterial communication, an artificial distinction for the provenience of the signal was required. Two separate AHL concentrations (same diffusion coefficient) were used: cAHL,R produced by the reference colony cells and cAHL,B produced by the background cells. The AHL production rate of any cell in the system still depends on the total concentration cAHL = cAHL,R + cAHL,B ( Figure 6.8). We evaluated the effects of AHL kinetics and background cell density on the privacy of communication. In this case, a regular field of background signal was employed by placing the background colonies on a square grid of 3×3, 5×5, 7×7 and 9×9 colonies per domain. The intensity of the background signal was increased by increasing the number of cells per background colony: 8, 32, 64, 100, 242 and 512 cells per colony ( Figure 6.9).

(a) Privacy factor The question regarding privacy of communication is in which extent a microcolony (the reference colony) is influenced by its own signal and how much by the signal produced by its neighbors (the background colonies). To quantify the privacy of communication in the reference colony we introduced the privacy factor, P, calculated first at a cell level as local privacy, PL. The local privacy represents the fraction of signal produced by the reference colony related to the total detected signal, as perceived by each cell:

cAHL,R PL (6.5) ccAHL,R AHL,B 148

(a)

(b)

Figure 6.9. Colonies of background cells placed on a square grid (3×3, 5×5 and 7×7) to generate a regular field of background AHL concentration in the Thin Layer Base case. The intensity of the background signal field is increased by increasing the number of cells in the background colonies: (a) colonies of 8 cells per colony; (b) colonies of 64 cells per colony.

The arithmetic mean of all local privacies of cells in the reference colony represents the privacy factor, P. When the privacy factor in the reference colony is higher than 0.95, then the communication in the colony is considered to be private. The Thin Layer Base case was chosen to study the privacy of communication in the reference colony. This choice was based on the observation that the induction of the reference colony appeared to be the most influenced by an increase in the density of background cells in Thin Layer Base case. The level of privacy reached in the focal colony at the moment of induction for a uniform background signal is represented in Figure 6.10 as function of background colony size and background colony density. The lack of feedback in the production of AHL provides a wide transition zone from full privacy (P=1, blue) to total lack of privacy (P=0, red, Figure 6.10b). From this point of view, it seems that the effect of signaling feedback on privacy (Figure 6.10a) is a more extreme behaviour of the focal colony, which either totally lacks privacy or it has a very private communication (i.e., there is only a narrow transition area between the private/non-private states). Noticeably, the absence of feedback cancels the effect of colony density on the privacy and the only factor with an impact is the size of the background colonies. The implications of this behaviour could be that feedback kinetics makes the privacy more sensitive to spatial distribution of the background colonies (how far the nearest neighbour is). This combines the effects of efficiency sensing and quorum sensing on the privacy of communication. 149

(a) (b)

Figure 6.10 Privacy in the focal colony at the moment of induction. (a) Signal production rate with feedback; (b) Constant signal production rate (no feedback). Color scale from red for lack of privacy to blue for privacy. Results were obtained with background cells arranged on a square grids (3×3; 5×5; 7×7 and 9×9 cells) with all the background colonies having the same size per studied case (8, 32, 64, 100, 242, 512).

6.3. Conclusions (1) We built a numerical model, which integrates metabolic aspects with signaling compound kinetics and 3-d mass transfer of AHL signal at the population level. (2) Clustering of cells (crowdedness) has a minor effect on the colony induction in the presence of feedback rate of signal production, but it has no effect at constant signal production rate. (3) Other colonies “listen” to the induction of the largest colony (the reference colony in this case) in the presence of positive feedback (4) Positive feedback has an influence on the privacy of communication by making a sharp transition of the focal colony from no privacy to full privacy.

6.4. References Dale MRT, Dixon P, Fortin M-J, Legendre P, Myers DE, Rosenberg MS, Conceptual and mathematical relationships among methods for spatial analysis, Ecography 2002; 25:558-577. Dockery JD, Keener JP, A mathematical model for quorum sensing in Pseudomonas aeruginosa, B Math Biol 2001; 63:95-116.

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Dunlap PV, Quorum regulation of luminescence in Vibrio fischeri, J Mol Microbiol Biotechnol, 1999; 1(1):5-12. Fekete A, Kuttler C, Rothballer M, Hense BA, Fischer D, Buddrus-Schiemann K, Lucio M, Müller J, Schmitt-Kopplin P, Hartmann A, Dynamic regulation of N-acyl-homoserine lactone production and degradation in Pseudomonas putida IsoF, FEMS Microbiol Ecol. 2010, 72(1):22-34 Fuqua WC, Winans SC, Greenberg EP, Quorum sensing in bacteria: the LuxR-LuxI family of cell density-responsive transcriptional regulators, J Bacteriol 1994, 176:269-275. Girard G, Bloemberg GV, Central role of quorum sensing in regulating the production of pathogenicity factors in Pseudomonas aeruginosa, Future Microbiol, 2008; 3(1):97-106. Hense BA, Kuttler C, Müller J, Rothballer M, Hartman A, Kreft J-U, Does efficiency sensing unify diffusion and quorum sensing?, Nat Rev Microbiol 2007; 5:230:239 Horswill AR, Stoodley P, Stewart PS, Parsek MR, The effect of the chemical, biological,a nd physical environment on quorum sensing in structured microbial communities, Anal Bioanal Chem 2007; 387:371-380 Kirisits MJ, Parsek MR, Does Pseudomonas aeruginosa use intercellular signalling to build biofilm communities?, Cell Microbiol, 2006; 8(12), 1841–1849 Kreft J-U, Booth G, Wimpenny WT, BacSim, a simulator for individual-based modelling of bacterial colony growth, Microbiology 1998, 144:3275-3287. Kreft J-U, Picioreanu C, Wimpenny JWT, Van Loosdrecht MCM, Individual-based modelling of biofilms, Microbiology 2001, 147:2897-2912. Nealson KH, Platt T, Hastings JW, Cellular control of the synthesisand activity of the bacterial luminescent system, J Bacteriol 1970; 104:313-322. Redfield RJ, Is quorum sensing a side effect of diffusion sensing?, Trends Microbiol 2002; 10(8):365-370. Vaughan BL, Smith BG, Chopp DL, The influence of fluid flow on modelling quorum sensing in bacterial biofilms, B Math Biol 2010, 72(5):1143:1165. Ward JP, King JR, Koerber AJ, Williams P, Croft JM, Sockett RE, Mathematical modelling of quorum sensing in bacteria, IMA J Math Appl Med 2001, 18:263-292.

151

Outlook

7. Outlook

Dental caries is a widespread infectious disease that can have 1. Research questionsserious and implications approach for human health when left untreated (e.g., severe pain, teeth lose). In spite of its high occurrence

there are still essential aspects of the disease, especially

related to its incipient stages and to the processes occurring

inside the dental plaque, which are not fully understood. The

current thesis presents a series of numerical models that

couple for the first time the (micro)biological processes

occurring in the dental plaque, with transport processes,

chemical speciation and realistic kinetics for the chemical

dissolution and mineralisation of the tooth enamel under the

influence of sugar intake. These models where used to answer

questions related to (i) the influence of some behavioural

factors (tooth brushing and frequency of sugar intake) and

biological factors (storage of polyglucose) on the 152 development of dental caries and on the evolution of the microbial population dynamics, (ii) to identify the main mechanisms from those presented in the literature which are responsible for the typical profile of a dental caries, as well as the influence of the geometry of the tooth site at which the caries develops (occlusal area vs. smooth surface) and finally (iii) to study several aspects related to the bacterial communication in biofilm communities.

7.1. Main results The results obtained with the current models indicate that the behavioural factors have the highest impact on the caries formation from all the studied factors. Our results confirmed the clinical observations that often tooth brushing can significantly reduce the amount of demineralised enamel and that the consumption of the same amount of sugar at shorter time intervals compared to longer intervals can also dramatically influence the amount of demineralised enamel. However, it was not confirmed the idea present in the literature that the storage compounds (named here generically polyglucose) increase the cariogeneity of the dental plaque. According to our model, the storage compounds have no significant long term or short term effect on the mass of lost enamel because their consumption by the bacteria present in the plaque occurs at pH > 5.5 (the critical values for hydroxyapatite demineralisation) and because they are stored in too small amounts to make a significant contribution for bacterial growth. One of the most important outcomess of the current work is to show that theoretically it is possible for a carious lesion to develop from the interplay of two opposite processes: demineralisation and remineralisation. Although it was clinically proven for a long time that the presence of fluoride has a protective role for the tooth enamel (Fejerskov and Kidd, 2008) (and this was confirmed as well with the current models) it is important to be also aware of the significance of remineralisation in the formation and prevention of dental caries. By acknowledging the role of remineralisation and through a better understanding of the mechanisms in which it is involved, new preventive strategies for caries formation can be developed. According to the current research the simplest method for caries prevention is to extend the length of remineralisation periods between two acid attacks. This reinforces the advice present for years in the dentistry literature that rare consumption of sugars and snacks between meals is the best strategy to prevent caries formation. Although it is an important part of the bacterial lifestyle in a biofilm, EPS was not explicitly considered in any of the current models. The most important characteristic of EPS 153 would have been as a physical barrier to delay the diffusion in the dental plaque (this delayed diffusion was considered in Chapter 3). However, all the models presented here are qualitative models and following a sensitivity analysis it was concluded that by considering this property of EPS, the results would have change only quantitatively but not from a qualitative point of view. Therefore, for the purpose of the current studies, the presenceof EPS can be neglected. Regarding the cell-cell communication study it appeares that the spatial arrangement of colonies (quantified by crowdedness) becomes important only when AHL is produced according to feed-back kinetics (which corresponds to the kinetics observed in reality). The topic of cell-cell communication might have some future in conditions such as periodontal disease but for the incipient stages of caries formation, it shows little relevance since the fermentative processes carried by oral bacteria, don’t seem to beinfluenced by factors as quorum sening or efficiency sensing.

7.2. Possible developments The models described in the current thesis are a good base for future developments that would allow the study of other aspects related to dental caries formation. An example in this direction is the study of the development of dental plaque on a tooth surface starting from the initial phases of bacterial adhesion. Such a topic would be relevant for at least two reasons: first, to help in the identification of new, cheaper methods to disrupt the formation of the plaque and second, to understand if there is a clear connection between the types and concentrations of bacteria found in the saliva and the composition of mature dental plaque at different sites in the mouth. In the initial phases of plaque development bacteria belonging to the initial colonizers groups (S. oralis, S. mitis, S. gordonii, S. sanguis (Kolenbrander et al., 2002)) interact with the proteins from the acquired enamel pellicle and break the natural defence of the tooth against bacterial colonization. Interacting with these initial colonizers, the second colonizers would join the plaque community followed by other bacterial groups attaching to the latter. Eventually, depending on the conditions in the oral environment, a steady state composition corresponding to a mature plaque would be reached. For such a model to be developed several additions to the current models are required. Inclusion of proteins and protein adsorption on the tooth surface is necessary to simulate the acquired enamel pellicle. Also, weak physical rejection forces between these proteins and the other microbial groups that do not belong to the initial colonizers must be included. The acquired enamel pellicle can also be represented as a membrane at the tooth surface with weak 154 rejection forces. Unlike in all the models from the current thesis, the saliva should not be sterile but it should contain in planctonic form all the microbial groups that will be part of the plaque comunity. New levels of complexity can be added to the microbial shifts study presented in Chapter 2 by including microbial activity gradients in the dental plaque as it was experimentally observed that the plaque is biologically inactive at the tooth surface (Ten Cate, 2006). For more realistic results it is important to include survival strategies of non-mutans Streptococcus (STN) and Actynomices (ACT) by adding to the model a number of processes characteristic to these species but not to mutans Streptococcus (STA): degradation of glycoproteins widely available in saliva (e.g., mucin) to sugars and aminoacids and their consumption in times of glucose scarcity (between meals), utilization of arginine and arginine-containing peptides available in saliva. This last addition is important for STA and ACT also because it allows them to locally increase the pH around them in times of high acidity helping them to survive (Takahashi and Nyvad, 2011). In addition, ACT is also ureolytic (Kleinberg, 2002) and it can use lactic acid for growth (Takahashi and Yamada, 1996). Adding these processes would give a chance to the STN and ACT bacterial groups to also become dominant in the plaque, even in the cases when glucose is consumed very often (as studied in Chapter 2). It is known that during and following a meal, the salivary flow is higher and has a different composition than in the resting phase (between meals). This difference in flows would speed-up the oral clearance and could have positive effects on limiting the tooth dissolution. The change in composition (more “watery” during meals and more “viscous” and rich in mucins between meals) could also have an influence in the development of bacterial communities. These effects are difficult to intuitively predict due to complexity of the processes involved in caries formation. Another possibility opened by explicitly including saliva flow in the model would be to make possible the modelling of caries formation at different locations in the mouth. It is known (Fejerskow and Kidd, 2009) that different locations in the mouth are characterized by different flow rates for instance, the molars in the back of the mouth, being closer to several important salivary glands are washed by higher salivary flow rates than the upper incisives. Finally, a model accounting for inhomogeneities and defects in the crystal structure and composition would help to evaluate the theory that dental caries are initiated at sites with

155 higher tension in the HAP crystal lattice (e.g., curved surfaces) or with faults in the structure (e.g., small fissures, inclusions of foreign ions in the HAP crystal). The most important aspect when considering the future of mathematical modelling of dental caries however is to couple the modelling work with laboratory experiments and clinical studies. Such an approach would maximize the usefulness of any numerical model: the model would be used to help guiding the experiments, and the experiments could validate the mathematical model.

7.3. References Fejerskov O, Kidd E: Dental caries: The disease and its clinical management, 2nd ed, Chicester, United Kingdom, Blackwell Munksgaard, 2008. Kolenbrander PE, Andersen RN, Blehert DS, Egland PG, Foster JS, Palmer Jr RJ, Communication among oral bacteria, Microbiol Mol Biol R 2002, 66(3):486-505. Takahashi N, Nyvad B, The role of bacteria in the caries process: ecological perspectives, J Dent Res (2011); 90(3):294-303 Takahashi N, Yamada T, Catabolic pathway for aerobic degradation of lacate by Actinomyces naeslundii, Oral Microbiol Immunol 1996; 11:193-198. Kleinberg I, A mixed-bacteria ecological approach to understanding the role of the oral bacteria in dental caries causation: an alternative to Streptococcus mutans and the specific- plaque hypothesis, Crit Rev Oral Biol Med (2002). 13(2): 108–25. Ten Cate JM, Biofilms, a new approach to the microbiology of dental plaque, Odontology 2006; 94:1-9.

156

Summary

The process of dental caries formation is highly complex and dynamic, involving both chemical and (micro)biological aspects. The fact that dental caries cannot occur in the absence of the dental plaque shows that the development of carious lesions is far more complex than simple mineral dissolution and remineralisation. Thus, for a correct understanding of the mechanisms governing this disease, the processes occurring inside the tooth enamel must be studied in relation to those occurring in the dental plaque. The current thesis presents an integrated approach using mathematical modelling as a tool to research a number of aspects regarding caries formation. Several one-dimensional, two-dimensional and three-dimensional time dependent numerical models have been developed by coupling existing knowledge on biofilm processes (mass transfer, microbial compositions, microbial conversions and substrate availability) with tooth composition and the demineralisation and remineralisation kinetics. The main questions addressed in the current work are: 1. How do behavioural factors influence the development of initial dental caries? The factors considered are sugar consumption patterns and oral hygiene (tooth brushing). 2. Is there any influence of the bacterial storage compounds (e.g., polyglucose, glycerol) on the evolution of caries? 3. What are the microbial shifts occurring in a dental plaque in different oral environmental conditions (i.e., different regimes of sugar consumption, often or rare tooth brushing)? 4. Which of the mechanisms presented in the literature are responsible for the typical profile of a dental caries (i.e., a healthy enamel layer covering an area of high porosity)? 5. What is the influence of the geometry of the tooth site at which the caries develops (e.g., occlusal area, smooth surface) on the profile of the lesion? 6. How can we integrate the existing theories on bacterial communication (that is, quorum sensing, diffusion sensing and efficiency sensing) in order to find out how bacteria keep their communication private? Chapters 2 and 3 focus on the first two research questions. In Chapter 2, a one- dimensional (1-d) time dependent mathematical model of a multispecies dental plaque is presented. The model can successfully simulate the pH variation under the influence of the microbial metabolism following a pulse of glucose and the subsequent tooth demineralisation.

157

The conclusions of this first model are that the thicker the dental plaque is (this corresponds to poor oral hygiene), the more severe tooth demineralisation will be (i.e., higher amounts of calcium are being demineralised). Also, slow consumption of sugar containing drinks proved to be more harmful than drinking the same amount over a short period of time. Since microbial growth was not included in this first model, it was not possible to study the effect of bacterial storage compounds for longer period of time. However, for the limited time period studied (~2–3 h) the storage compounds appeared to have no effect on demineralisation. Although it was not possible to study the microbial shifts using the model presented in Chapter 2, it was shown that higher percentage of Veillonella present in a dental plaque with a composition stable in time has a protective effect on tooth demineralisation due to lactate consumption during acid attack period. In the third chapter, the previous model was extended to include microbial growth while several aspects that appeared to have less impact on the Stephan curves of the previous study were omitted (the buffering effect of the plaque through the presence of fixed charges on the bacterial walls, complexation with calcium and aerobic processes). Using this new multispecies model, the first two research questions were revisited, but the main focus pertains to the third research question, regarding the modifications in the dental plaque community. The previous conclusions regarding the damaging effect of the slow drinking and poor oral hygiene have been reconfirmed. In all the studied cases, the steady state composition of the plaque has been dominated by the aciduric Streptococcus (STA) bacterial group. The fastest switch towards STA domination occurred in the slowest social drinking case (Long- Sipping) while the slowest switch was observed in the case corresponding to a three times a day brushing (Often Brushing). The focus in Chapter 4 was changed from the processes occurring in the plaque to those occurring in the tooth enamel. For this purpose, a new 1-d model was developed from the previous models, but with a simplified biological compartment (only one bacterial species considered and one metabolic process) and a more complex dental enamel compartment. This time, the tooth was represented as a 1-d computational domain (similar to the dental plaque domain) unlike in the previous models where it was represented as a boundary condition at the plaque-tooth interface. The purpose of this model was to evaluate the main hypotheses present in the literature regarding the formation of enamel subsurface lesions. Two mechanisms have been tested: (1) the formation of the lesion as a consequence of an imbalance between tooth demineralisation and remineralisation and (2) the formation of the

158 lesion as a result of a preferential subsurface demineralisation due to an insoluble fluoride containing surface. The most important result of this study was to show for the first time that, in theory, it is possible to create a subsurface lesion by alternating periods of demineralisation (during prandial periods) with periods of remineralisation (during inter-prandial periods). Likewise, the second tested mechanism led to a subsurface lesion covered by a surface layer. It was concluded that most probably in vivo these two mechanisms interact in the process of caries development. The two mechanisms mentioned earlier have been re-evaluated using a two- dimensional numerical model that was also able to predict the effect of tooth geometry on the profile of the subsurface lesion. This model and its results were presented in Chapter 5 together with some considerations regarding the impact of modelling decisions such as building a 1-d or a 2-d model or the choice of geometry. When an occlusal geometry was considered, both studied mechanisms yielded a subsurface lesion, while for the smooth surface geometry a lesion developed only for the case considering fluoride. All these results offered a greater insight into the profile of the lesion along the plaque-tooth interface than the corresponding 1-d model, showing that although computationally slower the 2-d models are very useful for a deeper understanding of a phenomenon as complex as caries formation. The sixth chapter presents a three-dimensional time-dependent, individual based model used to study the aspect of privacy in bacterial communication among Pseudomonas putida cells. The model is used to evaluate how different factors can influence the nature of the signal “sensed” by a reference colony (self-produced or background signal). The factors considered are: the properties of the environment (different geometries and boundary conditions), the number of cells and their distributions and the kinetics of signal molecule production (feedback vs. zero-order kinetics). A conclusion is that feedback kinetics are important because they allow the bacteria to take advantage of favorable environmental conditions faster once detected, than they would in the absence of feedback. Also, the feedback kinetics and a “closed” geometry (cells are positioned in a pit) are factors that decrease the privacy of communication in the reference colony.

159

Samenvatting

De vorming van tandcariës is een zeer complex en dynamisch proces waarin zowel chemische als (micro)biologische aspecten een rol spelen. Het feit dat tandcariës niet kunnen voorkomen in de afwezigheid van tandplaque, toont aan dat de ontwikkeling van cariës veel complexer is dan het slechts oplossen en remineraliseren van mineralen. Dus voor een goed begrip van de mechanismen die aan de oorsprong deze aandoening staan, moeten de processen in het tandglazuur worden bestudeerd in samenhanging met de processen die in tandplaque voorkomen. In dit proefschrift wordt een dergelijke geïntegreerde aanpak bescheven; wiskundig modelleren wordt toegepast om een aantal aspecten met betrekking tot cariësvorming te onderzoeken. Eendimensionale, tweedimensionale en driedimensionale tijdsafhankelijke numerieke modellen zijn ontwikkeld door kennis van biofilm processen (massa overdracht, microbiële samenstelling, microbiële omzettingen en aanwezigheid van substraat) te combineren met tandsamenstelling, en de kinetiek van demineralisatie en remineralisatie. De belangrijkste vragen die in dit werk worden behandeld zijn: 1. Hoe beinvloeden gedragspatronen het onstaan van vroege cariës? De factoren die worden beschouwd zijn patronen van suiker consumptie alsmede mondhygiene (tandenpoetsen). 2. Is er enige invloed van de bacteriële reservestoffen (bijv. polyglucose, glycerol) op het verloop van cariës? 3. Welke veranderingen treden op in een microbiële samenstelling van tandplaque in verschillende orale milieus (bijv. verschillende diëten van suiker consumptie, vaak of zelden tandenpoetsen)? 4. Welke mechanismen, bekend uit de literatuur, zijn verantwoordelijk voor het typische profiel van cariës (dus: een gezonde glazuurlaag die een zeer poreus deel bedekt? 5. Wat is de invloed van de geometrie van de tand daar waar de cariës ontstaat (bijv. omsloten of glad oppervlak) op het profiel van de laesie? 6. Hoe kunnen bestaande theorieën met betrekking tot bacteriële communicatie (dus; quorum waarneming, diffusie waarneming en efficiëntie waarneming) worden geïntegreerd, om te onderzoeken hoe bacteriën strict binnen de eigen soort communiceren? In het tweede en derde hoofdstuk van het proefschrift ligt de nadruk op de eerste twee onderzoeksvragen. In hoofdstuk 2 wordt een een-dimensionaal (1-d) tijdsafhankelijk wiskundig model van een tandplaque bestaande uit meerdere soorten bacteriën beschreven. Het model beschrijft met succes de variatie in pH onder invloed van het microbiële metabolisme na een 160 glucose piek en de daaropvolgende tand-demineralisatie. De conclusies van dit eerste model zijn dat hoe dikker de tandplaque is (dit komt overeen met slechte orale hygiëne), hoe erger de tand-demineralisatie zal zijn. (bijv. grotere hoeveelheden calcium worden gedemineraliseerd). Verder blijkt langzame consumptie van suikerhoudende drankjes schadelijker dan eenzelfde hoeveelheid binnen korte tijd te drinken. Aangezien microbiële groei in het eerste model niet in acht was genomen, was het niet mogelijk het effect van bacteriële reservestoffen over langere tijd te bestuderen. Echter, in het bestudeerde gelimiteerde tijdsinterval (~2-3 u) bleek niet dat de reservestoffen een effect op demineralisatie hadden. Hoewel het niet mogelijk was veranderingen in de microbiële samenstelling te bestuderen met het model beschreven in hoofdstuk 2, is aangetoond dat een hoger percentage Veillonella aanwezig in een tandplaque, met een samenstelling die stabiel blijft in de tijd, een beschermende werking tegen tand demineralisatie heeft dankzij de lactaat consumptie tijdens een zuuraanval periode. In het derde hoofdstuk is het eerder beschreven model uitgebreid door toevoeging van microbiële groei, terwijl een aantal aspecten die minder invloed op de Stephan curves van de eerdere studie bleken te hebben zijn weggelaten (het buffer effect van de tandplaque door de aanwezigheid van ladingen op de bacteriële celwand, complexvorming met calcium en aerobe processen). In dit nieuwe model met meerdere soorten bacteriën, zijn de eerste twee onderzoeksvragen opnieuw bestudeerd, maar de aandacht ging vooral uit naar de derde onderzoeksvraag, betreffende de veranderingen in de microbiële samenstelling van de tandplaque. De eerdere conclusies over het schadelijke effect van langzaam drinken en slechte gebitsverzorging worden opnieuw bevestigd. In alle bestudeerde scenarios werd de steady state samenstelling van de plaque gedomineerd door bacteriële groep van de zuurtolerante Streptococcus (STA). De snelste omschakeling naar dominatie van STA vond plaats in het langzaamste sociale drinker scenario (kleine slokjes gedurende lange tijd) terwijl de langzaamste omschakeling werd gezien in een geval waar drie maal daags gepoetst wordt (vaak poetsen). In hoofdstuk 4 is de focus verlegd van de processen die plaatsvinden in de plaque naar de processen die plaatsvinden in het glazuur. Hiervoor is een nieuw 1-d model ontwikkeld, uitgaande van de eerdere modellen, maar met een vereenvoudigd biologisch gedeelte (slechts één bacterie soort en één metabolisch proces wordt beschreven) en een complexer tandglazuur gedeelte. Nu wordt de tand weergegeven als een 1-d rekenkundig domein (overeenkomstig met het tandplaque domein) in tegenstelling tot de eerdere modellen waar het werd weergegeven als grensconditie op de grens tussen plaque en tand. Het doel van dit model was de belangrijkste hypothese in de literatuur omtrent de vorming van laesies onder het glazuur te evalueren. Twee mechanismen zijn getest: (1) de vorming van de laesie ten gevolge van een

161 onbalans tussen demineralisatie en remineralisatie van de tanden en (2) de vorming van de laesie ten gevolge van de voorkeur voor demineralisatie onder het glazuuroppervlak vanwege een onoplosbaar fluoride bevattend oppervlak. Het belangrijkste resultaat van deze studie is dat voor de eerste keer is aangetoond dat het in theorie mogelijk is een laesie onder het oppervlak te creëren door alternerende periodes van demineralisatie (tijdens de prandiale periodes) en periodes van remineralisatie (tijdens de inter-prandiale periodes). Het als tweede geteste mechanisme leidde eveneens tot een afgedekte laesie onder het oppervlak. Waarschijnlijk hebben deze twee mechanismen in vivo een wisselwerking in het proces van cariësvorming. De twee eerdergenoemde mechanismen zijn opnieuw geëvalueerd met een twee- dimensionaal rekenkundig model waarbij ook het effect van de geometie van de tand op de laesie onder het oppervlak kon worden voorspeld. Dit model en zijn resultaten zijn beschreven in hoofdstuk 5 samen met enkele overwegingen betreffende de invloed van de modelopzet, zoals het maken van een 1-d of 2-d model of de keuze van geometrie. Een omsloten geometrie resulteerde bij beide bestudeerde mechanismen in een laesie onder het oppervlak, terwijl bij een gladde geometrie alleen in het geval waarbij fluoride van invloed is een laesie ontstond. Alle resultaten gaven, vergeleken met het 1-d model, een breder inzicht in het profiel van de laesie over het grensvlak tussen plaque en tand, wat aantoont dat hoewel het berekenen langer duurt, de 2-d modellen zeer nuttig zijn voor een ruimer begrip van een natuurverschijnsel met de complexiteit van cariës vorming. Het zesde hoofdstuk toont een driedimensionaal tijdsafhankelijk, individu gebaseerd model dat wordt gebruikt om het aspect van privacy in bacteriële communicatie onder Pseudomonas putida cellen te bestuderen. Het model wordt toegepast om te bepalen hoe verscheidene factoren van invloed zijn op het karakter van een signaal "waargenomen" door een referentie kolonie (zelf geproduceerd of achtergrondsignaal). De factoren die in overweging genomen worden zijn: de eigenschappen van de omgeving (verschillende geometriën en grenscondities), het aantal cellen en hoe deze verdeeld zijn en de kinetiek van de productie van signaal moleculen (feedback tegenover nulde orde kinetiek). Een conclusie is dat feedback kinetiek belangrijk is, omdat het de bacteriën in staat stelt sneller te profiteren van gunstige omstandigheden, waargenomen in de omgeving, in vergelijking met de afwezigheid van feedback. Verder zijn feedback kinetiek en een "gesloten" geometrie (cellen zijn geplaatst in een holte) factoren die communicatie privacy van de referentie kolonie verlagen.

162

Acknowledgements

When I think of all the people who made it possible for me to complete my PhD, I realize that there are so many! I would like to warmly thank them all for their constant support, understanding and for all the valuable knowledge that they shared with me during these years. I would have never managed to reach this stage without them! It is difficult to find the words to express my deep gratitude towards my two promoters, Prof. Dr. Ir. Mark van Loosdrecht and Dr. Ir. Cristian Picioreanu, who shaped my scientific thinking and catalyzed my development. I would like to thank Prof. Mark van Loosdrecht for his wise guidance and clear-sighted advice. With his broad interdisciplinary understanding and his practical sense, he always emphasized the important aspects of the research questions from the “rubbish”, non-essential ones. Furthermore, I want to express my endless gratitude for his kind and unconditional support, especially in these past months when he helped me deal with the bureaucracy both in the Netherlands and in Switzerland. I will never be able to thank Dr. Cristian Picioreanu enough for his essential role during my PhD training at TU Delft. Through his vast amount of knowledge, scientific rigor, and the passion he puts in his daily work, he was a continuous source of inspiration for me during the past years. A great deal of what I know today about mathematical modelling of biofilms and scientific research in general comes from Cristian’s advising. I also remember with great fondness our cultural dialogues and I would like to thank Cristian for encouraging me to discover new literature, for showing me outstanding classical music interprets, for helping me learn more about the music of Bach and, his all-time favorite, Mozart. I will always remember with a happy smile our incursion to the Copenhagen opera house for Le nozze di Figaro. I would also like to acknowledge Dr. Jan Kreft from the University of Birmingham for our discussions on efficiency sensing and the privacy of bacterial communication and for always making the time for me when I needed his help. I also sincerely appreciated the talks on bacterial cell-cell communication with our collaborators Dr. Burkhard Hense and Dr. Andreas Dötsch. I am grateful to André van Turnhout with whom I collaborated for the two studies on tooth enamel subsurface lesion formation as a part of his MSc graduation project. His 163 dedication and high quality work were key factors in making the project a successful one and we are now able to publish two articles based on that collaboration. I would also like to acknowledge the members of the PhD committee who evaluated my PhD dissertation and the Netherlands Organisation for Scientific Research (NWO) for providing the financial support required for my research. A special thank you goes to Andrea Radu who was much more than a great colleague. She helped me with numerous scientific discussions and observations, but she is also a wonderful friend. Andrea is an exceptional human being whose correctitude, altruism and kindness never cease to amaze and inspire me. I also want to show appreciation to my office colleagues Florence, Marco, Geert and Oezlem with whom I shared not only the office room but also numerous ideas and coffee breaks. A thank you also goes to all my other colleagues from the EBT group and especially Jelmer, Udo, Mario, Tommaso and Sheevita who always made coffee time a cheerful and interesting moment. I am forever grateful to my dear friends who are always there for me and helped me with their continuous support, love and even scientific advice. Andrei is an extraordinary counselor and was also the art director for all my articles and for this thesis (if I would be a president, you would be my prime minister!). Eugen was always there to advise me when I needed an “outside opinion” on my research with his sharp and clear thinking. Raluca (my sister of soul) was my graphic artist and art co-director for the third article and for this thesis (thank you for the great drawings!). I would also like to acknowledge my friends Bogdan, Johann, Diego, Monika, Barbara and Ada for making my time in The Netherlands sunny (inspite of the weather) and culturally intensive. I am grateful to the entire Crefcoeur family (my Dutch adoptive family) for always being there for me and caring for me as one of their own. Especially, I want to thank Paul for his precious observations on my models and articles from a dentist’s point of view. A very special thank you goes to Remco, because without his love, encouragements, support and understanding I would have never managed to finish my PhD studies. (I am fortunate to have you by my side!) Also, I would like to thank him for the Dutch translation of the thesis summary and the propostitions. Last but not least, I am grateful to my parents for their support, their incommensurable love and for all the sacrifices they made so I can have a good education and never lack anything. Thank you! (Romanian: Doresc sǎ le mulţumesc pǎrinţilor mei pentru suport, pentru dragostea lor neţǎrmuitǎ şi pentru toate sacrificiile pe care le-au fǎcut pentru ca eu sǎ pot beneficia de o educaţie bunǎ şi sǎ nu îmi lipseascǎ nimic vreodatǎ. Mulţumesc!)

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Curriculum Vitae

Olga Ilie, born 25th of June 1981, Bucharest, Romania 2000 – 2005 University “Politehnica” of Bucharest BSc in Chemical Engineering MSc in Biochemical Engineering. Graduation thesis: The modulator activity of some vegetal lectins on normal and tumor cell cultures (Prof Dr. D. Gabor) and The industrial process for ethanol production via fermentation starting from potatoes(Prof. Dr. Ir. O. Muntean)

2005 National Agency for Environment Protection – Junior Counselor

2005 – 2006 Biochemistry Institute of the Romanian Academy – Assistant Researcher

2006 – 2008 Delft University of Technology, Biotechnology Department PDEng stage. Professional Doctor in Engineering – Bioprocess Design Group design project: Design a Novel Concept in Sugar Cane Industry (project owner from DSM: Dr. Peter Nossin) Individual Project: Modelling of the Human Digestive Tract as a Chemical Plant (Advisers: Prof. Dr. Ir. Klaas van’ Riet and Dr. Rolf Bos)

2008 – 2014 Delft University of Technology, Biotechnology Department Doctoral stage. Numerical Studies of Dental Plaque and Caries Formation (promoters: Prof. Dr. Ir. M.C.M van Loosdrecht and Dr. Ir. C. Picioreanu)

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