Numerical Studies of Dental Plaque and Caries Formation

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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

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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

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List of symbols

Name Symbol Dimensions

Concentration of a microbial group i, constant in time

(i = STA, STN, ACT, VEL)

CX,i (kg dry biomass) (m–3 plaque) 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–3 Diffusive flux at the plaque-saliva interface Np,j mol s–1 m–2 Diffusive flux at the plaque-tooth interface Npt,j mol s–1 m–2 Net reaction rate of each component j in the saliva film Rf,j mol s–1 m–3 Net reaction rate of each component j in the dental plaque Rp,j mol s–1 m–3

Electrical potential ɸ V

Diffusion coefficient of each component j Dj m2 s–1 Substrate specific uptake rate qm mol s–1 g–1 Monod substrate saturation constant KS mol L–1

Inhibition constant KI mol L–1

Degree of saturation with respect to hydroxyapatite DSHAP (mol m–3)4/9 Degree of saturation with respect to fluorohydroxyapatite DSHAP(xf) (mol m–3)4/9 Ionic product with respect to hydroxyapatite IPHAP mol4 m–12 Ionic product with respect to fluorohydroxyapatite IPHAP(xf) mol4 m–12 Halving time in the saliva film th,f s Halving time in the saliva bulk th,s s

Volume of saliva film Vf m3

Volume of saliva bulk Vs m3

Area of saliva-plaque interface Af m2 Volumetric flowrate of saliva film Qf m3 s–1 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 Minimum glucose concentration (during resting time) Cs,Glu,min mol m–3 Maximum glucose concentration (during feeding time) Cs,Glu,max mol m–3

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

,

f i

G

' kJ mol–1 Gibbs energy changes of the anabolic reaction ΔGana kJ Gibbs energy changes of the catabolic reaction ΔGcat kJ Concentration of bacterial group j (varying in space and time) Xp,j mol m–3

Advective velocity uB m s–1

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Name Symbol Dimensions

Concentration of polyglucose stored within bacteria Cp,sto mol m–3 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–1 Reaction rate constant of acid-base equilibria kj m3 mol–1 s–1 Equilibrium constant for the dissociation of species i Ke,i mol L–1 Equilibrium constant for water dissociation KH2O mol2 L-2 Volumetric demineralisation rate of hydroxyapatite rd,HAP molHAP m–3 s–1 Reaction rate constant of hydroxyapatite demineralisation ken s–1

Enamel solubility constant KS,HAP(en) mol9 L–9 Hydroxyapatite solubility constant KS,HAP mol9 L–9 Hydroxyapatite molar weight MHAP g mol–1

Hydroxyapatite density ρHAP kg m–3

Radius of hydroxyapatite rod rrod m

Demineralisation rate constant kd mol0.7 m–1.1 s–1 Remineralisation rate constant kr mol0.25 m0.75 s–1 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

Specific HAP crystal surface available for demineralisation ad m2 m–3 Specific HAP crystal surface available for remineralisation ar m2 m–3 Surface-based demineralisation rate *

,

d Ca

r molCa m–2 s–1

Surface-based remineralisation rate * ,

r Ca

r molHAP m–2 s–1 Volumetric remineralisation rate rr,HAP molHAP m–3 s–1 Molar fraction of fluoride in fluorohydroxyapatite (FHAP) xF mol mol–1 Mixed FHAP/HAP solubility constant Kx mol9 L–9 Two-dimensional Cartesian coordinate system m–1 AHL production rate after induction rAHL Pmol L–1s–1 Basal AHL production rate kAHL ni μmol L–1 s–1 AHL production rate after induction kAHL in μmol L–1 s–1 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–3 Affinity coefficient KHill μmol L–1

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 Molecular flux of AHL at the boundary JAHL mol s–1 m–2

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 -

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Con$nuous Model, 1D

Dental Plaque: One Species; No Growth

Tooth: Represented as 1-‐d domain

tooth

plaque

saliva

Chapter 4

Numerical modelling of tooth enamel subsurface lesion forma8on induced by

dental plaque

Con$nuous Model, 1D

Dental Plaque: Mul$species; No Growth

Tooth: Represented as boundary

tooth

plaque

saliva

Chapter 2

Mathema8cal modelling of tooth demineralisa8on and pH proﬁles in dental plaque

Con$nuous Model, 1D

Dental Plaque: Mul$species; Growth

Tooth: Represented as boundary

tooth

plaque

saliva

plaque grown

Chapter 3

Mathema8cal modelling of microbial dynamics in dental plaque

Individual Based Model, 3D

Bioﬁlm: One Species; Growth

Chapter 6

A three-‐dimensional numerical study on the privacy of cell-‐cell

communica8on

Con$nuous Model, 2D

Dental Plaque: One Species; No Growth

Tooth: Represented as 1-‐d domain

tooth

plaque

Chapter 5

Two-‐ dimensional mathema8cal modelling of tooth enamel subsurface

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1. Introduction

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Introduction

1.1. 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

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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)

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“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.”

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13 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.

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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).

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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).

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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.

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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

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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.

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19 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

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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

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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

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2. Mathematical modelling

of tooth demineralisation

and pH profiles in dental

plaque

2

Mathematical

modelling of tooth

demineralisation

and pH profiles in

dental plaque

Dental caries is currently one of the most widespread conditions associated with oral hygiene. The term dental caries is used to describe the results – signs and symptoms – of a localized chemical dissolution of the tooth surface caused 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 Chapter published in Journal of Theoretical Biology 309:159-175 (2012)

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25 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-one-dimensional pH profiles in the depth of

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26

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

Streptococcus, Actinomyces, Veillonella)

One generic species

(presumably aciduric Streptococcus) Polysaccharide (polyglucose) storage and consumption Only polysaccharide storage Five acids:

lactic, acetic, propionic, formic, succinic

Two acids: lactic, to account for strong organic 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

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27 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

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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).

Name Symbol _{[(kg dry biomass) (m}Concentration –3_plaque)] Concentration _{[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

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Table 2. 3 M obil e che m ical species and their asso ciated p

roperties in the curren

t model Na m e Sy m b o l Form ula Cha rge Diffusion coefficient Initial concentrati o n in saliva film a n d plaque [10 -9 m 2 s –1] Reference [m m o l L –1] Reference/Calcula tion Acetate Ace – C2 H3 O2 – –1 1.38 Cussler, 2009 Va nýse k , 2 001 0 equilibr ium, CAce ,to t = 0 Acetic acid AceH C2 H4 O2 0 1.64 Cussler, 2009 Va nýse k , 2 001 0 equilibr ium Fo rmate Fo r– CHO 2 – –1 1.84 Va nýse k , 2 001 0 equilibr ium, CFor,tot = 0 Fo rmic acid Fo rH CH 2 O2 0 1.79 Cussler, 1984 0 equilibr ium Lactate Lac – C3 H5 O3 – –1 1.31 Va nýse k , 2 001 0 equilibr ium, CLac,tot = 0 Lactic acid LacH C3 H6 O3 0 1.41 Assumed 0 equilibr ium Pro p ionate Pro – C3 H5 O2 – –1 1.20 Cussler, 2009 Va nýse k , 2 001 0 equilibr ium, CPro,tot = 0 Pro p ionic acid Pro H C3 H6 O2 0 1.34 Cussler, 1984 0 equilibr ium Su ccinate Su c2 – C4 H4 O4 2 – –2 0.99 Va nýse k , 2 001 0 equilibr ium, CSuc,tot = 0 H ydrogen succinate Su c– C4 H5 O4 – –1 1.10 Assumed 0 equilibr ium Su ccinic acid Su cH C4 H6 O4 0 1.19 Cussler, 1984 0 equilibr ium Bicarbonate HCO3 – CHO 3 – –1 1.50 Va nýse k , 2 001 4.17 equilibr ium, CCO2,tot = 5.1, Mars h et al, 200 9 Carbon dioxide CO2 CO 2 0 2.43 Cussler, 2009 0.93 equilibr ium Hy drogen P hosp h ate Ph o2 – HPO 4 2 – –2 0.96 Va nýse k , 2 001 2.06 equilibr ium, CPho,tot = 5 .4 , F ej ersk o v a n d Kid d , 20 08 Dihy drogen Phosphate Ph o – H2 PO 4 – –1 1.22 Va nýse k , 2 001 3.34 equilibr ium Hy droxy l HO – HO – –1 6.7 Cussler, 2009 Va nýse k , 2 001 10 –4 equilibr ium Pro ton H+ H + +1 11. 81 Cussler, 2009 Va nýse k , 2 001 10 –4 Fejerskov an d Kidd, 200 8 Calcium Ca2+ Ca 2+ +2 1 .00 Cussler, 2009 Va nýse k , 2 001 0.04 equilibr ium, CCa2+,tot = 1.32, Fejersk ov and K idd, 2008 Anion (C hlori d e) Cl – Cl – –1 2.57 Cussler, 2009 Va nýse k , 2 001 40 Fejerskov an d Kidd, 200 Cation (Potass ium) K+ K + +1 2.49 Cussler, 2009 Va nýse k , 2 001 Cs,K+ = 49 Cp,K+ = 80. 8 6

charge balance in saliva charge balance in

plaque Oxy g en O2 O2 0 2.66 Cussler, 2009 0.15 Assumed Glucose Glu C6 H12 O6 0 0.85 Va nýse k , 2 001 0.07 Van d er Hoeven et al., 1 990 Eth anol Eth C2 H6 O 0 1.57 Va nýse k , 2 001 0 Assumed

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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

Name Symbol Formula Charge Initial concentration

[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

Fixed charge cationic sites SH2+ {SH2+ +1 3.59 × 10–5 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.

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(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).

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Table 2.

5

Stoichi

o

metr

y (molar) and rates of microb

ial p

roc

esses c

onsidered in the model

d ental pla que Glu O2 Suc 2 La c Pr o Ace Fo r Et h HCO3 – H+ H2O Pg st a Pg st n P gac t Rat es Refe ren ce 1. A n aer o b ic h igh c o n cen tr at ion g lu cos e f erm en ta ti on STA –1 2 2 qm,STA ,G lu,H C STA M( CGl u ) I (C H+ ) Van B ee len e t a l., 19 86 STN –1 2 2 qm,STN ,G lu,H C STN M( CGl u ) I(C H+ ) Van B ee len e t a l., 19 86 ACT –1 2 2 qm,ACT ,Gl u, H C ACT M( CGl u ) I( CH+ ) Van B ee len e t a l., 19 86 2. A n aer o b ic l ow con cen tr ati o n glu cos e fe rmen ta ti on STA –1 1 2 1 3 –1 qm,STA ,G lu,L C STA M( CGl u ) I (C H+ ) I (CO2 ) I( CGl u ) Van B ee len e t a l., 19 86 STN –1 1 2 1 3 –1 qm,STN ,G lu,L C STN M( CGl u ) I (C H+ ) I (C O2 ) I( CGl u ) Van B ee len e t a l., 19 86 ACT –1 1 1 1 –1 3 1 qm,ACT ,Gl u, L C ACT M( CGl u ) I (C H+ ) I (CO2 ) I( CGl u ) De Jo ng et al ., 1 9 88 3 . Aero bic g lucose co nv er si o n STA –1 –1 2 2 4 qm,STA ,G lu,L,O 2 C STA M( CGl u ) M (CO2 ) I( CH+ ) I( CGl u ) Van B ee len e t a l., 19 86 STN –1 –3/2 2 1 1 4 qm,STN ,G lu,L,O 2 C STN M( CGl u ) M (C O2 ) I( CH+ ) I( CGl u ) Van B ee len e t a l., 19 86 ACT –1 –2 2 2 4 qm,A C T ,Gl u, L ,O2 C ACT M( CGl u ) M( CO2 ) I( CH+ ) I( CGl u ) De Jo ng et al ., 1 9 88 4. P o lyglu cos e st or ag e STA –1 1 qm,STA ,s to C STA M( CGl u ) I (CH+ ) I( CPg st a ) Ass u m ed STN –1 1 qm,STN ,s to C STN M( CGl u ) I (C H+ ) I( CPg st n ) Ass u m ed ACT –1 1 qm,ACT ,s to C ACT M( CGl u ) I(C H+ ) I (C P gact ) Ass u m ed 5. A n aer o b ic p o ly gl u cos e co n v er si on STA 1 2 1 3 –1 –1 qm,STA ,P g st a C STA M( CPg st a ) I (C H+ ) I (C O2 ) I( CGl u ) S imila r to p ro ce ss ( 2 ) STN 1 2 1 3 –1 –1 qm,STN ,P g st n C STN M( CPg st n ) I (C H+ ) I (CO2 ) I( CGl u ) S imila r to p ro ce ss ( 2 ) ACT 1 1 1 –1 3 1 –1 qm,AC T,P gact C ACT M( CP gact ) I (C H+ ) I (C O2 ) I (CGl u ) S imila r to p ro ce ss ( 2 ) 6. A er o b ic p o ly gl u cos e con v er si on STA –1 2 2 4 –1 qm,STA ,P g st a C STA M( CPg st a ) M(C O2 ) I(C H+ ) I(C Gl u ) S imila r to p ro ce ss ( 3 ) STN –3/2 2 1 1 4 –1 qm,STN ,P g st n C STN M( CPg st n ) M(C O2 ) I(C H+ ) I(C Gl u ) S imila r to p ro ce ss ( 3 ) ACT –2 2 2 4 –1 qm,AC T,P gact C ACT M( CP gact ) M(C O2 ) I(C H+ ) I(C Gl u ) S imila r to p ro ce ss ( 3 ) 7. L a ct at e ferm en ta tion VEL –1 2/3 1/3 1/3 1/3 qm,V E L ,La c– C VE L M( CLac – ) I (C H+ ) S eelig er et al., 20 02 ; , , j j X S j j C K C C M j j X I j X I j C K K C I , , , , . S ubs cri p ts s tan d f or: S su bs tr at e, I inhibition, X bac teri al sp ec ie s ( i.e ., S T A, S T N, VE L o r ACT) an d j c h em ical sp eci es ( i.e., G lu, O 2, L ac, H+, Pg sta , P gstn , P gact)

Numerical Studies of Dental Plaque and Caries Formation