Mechanism of Fluoride Uptake by Hydroxyapatite from Acidic Fluoride Solutions: I

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Mechanism of Fluoride Uptake by Hydroxyapatite from Acidic Fluoride Solutions: I. Theoretical Considerations

KENNETH G. NELSON* and WILLIAM 1. HIGUCHI College of Pharmacy, University of Michigan, Ann Arbor 48104, USA

A mathematical model for the mechanism terize the system in terms of the pH and of calcium fluoride formation on hydroxy- the buffer. A model for the mechanism of apatite in buffered fluoride solutions is pro- enamel dissolution has been proposed by posed. It takes into account the physical Higuchi et a18 and has been successfully and chemical processes that occur during the tested experimentally by a powder method9 reaction. With the model it is possible to as well as by a solid disk method.'0 This evaluate a priori the fluoride uptake poten- model is based on a simultaneous diffusion tial of fluoride solutions. and equilibrium chemical reaction in con- trast to other models7"'1-'4 that are based on It is well established that fluoride plays an an empirical kinetic expression of the type: important role in the prevention of dental rate = (constant) (concentration). The pres- caries. Fluoridation of drinking water has ent study is a derivation of a model that been a successful public health measure in represents the physical and chemical pro- reducing the prevalence of tooth decay.' cesses by which calcium fluoride is formed Beneficial results have also ensued from from hydroxyapatite and fluoride ion. incorporation of stannous fluoride into a dentifrice.2 Wellock and Brudevold3 and Materials and Methods others4,5 have reported a reduction in cari- The method used to derive the model for ous surfaces, as compared with controls, the mechanism of the reaction was to de- after one or more topical applications of an scribe the physical and chemical processes, acidic fluoride and phosphate solution to reduce these processes to mathematical equa- teeth. The tooth mineral hydroxyapatite de- tions, combine and solve the equations, and composes and calcium fluoride adheres to obtain theoretical data by calculation. The the apatite and can serve as a source of flu- last point will be reserved for a subsequent oride for subsequent conversion of hydroxy- report in which theoretical data will be com- apatite to fluorapatite.6 pared with experimental data. Recently there has been interest in the application of physical chemistry principles Results to hydroxyapatite, the primary mineral of teeth. Gray7 has studied dissolution behavior STEADY STATE CONSIDERATIONS.-If a flat of enamel and applied kinetics to charac- surface of hydroxyapatite be exposed to a stirred solution of fluoride, only the direc- This investigation was supported by a US Public tion normal to the surface, defined as the Health Service Fellowship 5-F1-DE-19, 742 and grant x direction, need be considered. The phys- DE-01830 from the National Institute of Dental Re- search, National Institutes of Health, Bethesda, Md, ical model is depicted in Figures 1 and 2. USA. Initially the apatite is in contact with the Based on a thesis submitted by K. G. Nelson to the treating solution (Fig 1), but after a finite graduate faculty, University of Michigan, Ann Arbor, time calcium fluoride forms and the solid- Mich, USA in partial fulfillment of the requirements = for the PhD degree. solid boundary, x s, progresses (Fig 2). Presented at the 46th annual Meeting of the IADR, Fluoride and acid diffuse into the reaction San Francisco, Calif, USA, March 21-24, 1968. site and phosphate diffuses out as the reac- Received for publication May 2, 1969. Hill'5 has shown that if the * Present address: College of Pharmacy University tion proceeds. of Minnesota, Minneapolis, Minn 55455, USA. ratio of the concentration difference of the 1541 1542 NELSON AND HIGUCHI J Dent Res November-December 1970

d2(HF2 ) (5) DHF2 dx + 46 0 Buffered Solution Hydroxyapatite d2(H3PO4) of 3PO4 = (6) Fluoride d2(PO0 ) 4 D ~±~ 0 (7) 3 4 dx2 D 21PO~4 d2(HPO=) DHPo4 dX2 -5 + <4 -O (8) FIG 1.-Physical model for formation of calcium..~~~~~fluoride on hydroxyapatite shows ini- d2(HPOf) tial situation. P04 dX2 P5 (9)

d2(H+) diffusing species to twice the amount bound D per unit volume in the new layer is suffi- H dx2 ciently small, steady state diffusion can be +(P4+ 65=O. (10) assumed to exist behind the moving bound- The parentheses indicate the concentra- ary. It can be shown that the present con- tions of the enclosed species and HB and ditions satisfy this criterion. Thus, the steady B - are two forms of a general buffer. The state diffusion equations for all of the dif- D's are effective diffusion coefficients in the fusing species in the region 0 c x c s are heterogeneous layer of the species indicated by the The co- d2(HB) subscripts. effective diffusion DHB -4hi° (1) efficient is related to the diffusion coefficient dx2 of the species in the liquid by, d2(B-) DB Deffective D liquid ±diXO (2) (11) dx2 lid-r

2P- (3 where 6 and T are the porosity and tortu- DHF dX2 +2 (6 ° (4) osity of the heterogeneous layer. The diffusion coefficient in the liquid is assumed to be that for the system of the d2 (F) binary single DF- +(P2-4)60 (4) species in the solvent. The terms ¢j through dx2 )6 are, respectively, the rates of reaction per unit volume of the following reactions that take place in the region 0 x s. HB-> H+ +B- Buffered HF-> H+ + F- Solution CaF2 Hydroxyapatite H3PO4 H+ + H2PO- Of 4 Fluoride H2PO_ H+ + HPO= 4 ~~~~~4 HPO=- H+ 4 +PO=4 HF + F- HF-. 2 0 5s These rates are indeterminate as discussed X-s by Olander.16 These terms can be eliminated FIG 2.-Physical model for formation of by appropriate combinations of equations 1 calcium fluoride on hydroxyapatite as reaction to 10, which yield the following four equa- progresses. tions: VOl 49 NO. 6 MECHANISM OF FLUORIDE UPTAKE 1543 DHB d2(HB) d2(B-) d(HF) D d(F-) + DHF ±D HB dX2 B dX2 HFdx dxA (12) + 2±2HF2'DH =(HF)d C4. (19) d2(H3PO4) d2(H2PO4) dX DH3PO4 dX2 + DH2P4 dX2 Fick's first law of diffusion for the flux, J, per cross-sectional area per unit time of a d2(HPOf=) d2(PO4) substance of concentration c, is + D1UP04 dX2 + DPO4 dX2 dC J=-D - (20) = 0 (13) dx d2(H+) d2(HB) Thus, physical significance can be ascribed DH dX2 + HB dX2 to the constants of integration in equations 16 to 19. d2(HF) d2(HF ) Cl= Jbuffer ± DHF dX2 + DHFO dX2 (21) C2 = Jtotal phosphate (22) d2(HPO4 ) DHpol dx2 C3= Jtotal acid (23) d2 (H2PO4 ) -C4= Jtotal fluoride- (24) + 2 D1H2P4 dX2 Since the buffer does not take part in the chemical reaction per se, the net flow of d2(H3PO4) 0 (14) ± 3 DH3PO4 dX2 buffer is zero: Buffer = (25) d2(HF) dF( ) 0. D + D Restrictions on equations 22 to 24 can be dx2 ~ ~ dX deduced from the stoichiometry of the net d2(HF2 reaction: + 2 DH 0. (15) Ca10(P04) 6(OH) 2 + 20 F- + 2 H+ 0 The first integration of equations 12 t 1OCaF2±6PO-+2H9O. yields the following equations, respect These restrictions are where the C's are constants of integra d(HB) 20 Total phosphate = +Dd(B-)±DB C 6 DHB dx dx Total fluoride (26) 20 Total acid = 2 Jtotal fluoride, (27) d(H3PO4) d(H2P( + Combining equations 16, 21, and 25 and DH3PO4 dx H2PO4 dx eliminating the dx term by multiplying both d(HPO47) d(PO. sides of the equation by it + dx + Dr4 dx DHr04 DEBI d(HB) + DB d(B-) = 0. (28) = C2 Combining equations 18, 19, 23, 24, and d(H±) d(HB) d(Jc] 27 and eliminating dx DH - + DHB D dx d(HB)+dx d± d D11d(H+) + DHBd(HB) + DHFd(HF) d(HF 2 ) d(HPO= ± DHFld(HF-) + DHPO4d(HPO=) + DH2 + DH dx A~04d + 2 D11PO4d(H2PO-) +2D d(H2PO4 ) + 2 DH2PO4 dx + 3 DH3PO4d(H3PO4) d(H3PO4) 2 2 + 3 (18) = - Dd(F-) ± -DHFd(HF) DH3PO4 dx = C3 20 20 1544 NELSON AND HIGUCHI J Dent Res November-December 1970 4 properties of fluorapatite.19 Because there + DH2d(HF-). (29) are finite concentrations of fluoride at the boundary x = s, the solid immediately on Combining equations 17, 19, 22, 24, and the apatite side of the boundary may be 26 and eliminating dx, considered fluorapatite (Fig 3). The stoichiometry of the net reaction, which has al- DPO4d(PO4 ) + DHP04d(HPO=) ready been considered, is still valid for the restrictions on the diffusion equations, but + DH2PO4d(H2PO-) + DH3PO4d(H3PO4) the reaction that governs the equilibrium 6 6 conditions at x = s is =- -DFd(F-) - DHFd(HF) 20 20 Calo(PO4)6F2 + 18 F- 12 -> 1OCaF2+ 6 PO0-. -2 D Fd(HF ) . ( 30) The equilibrium constant for this reaction is a combination of the solubility products The chemical reactions that occur in the of fluorapatite diffusion path 0 < x < s obey the following and calcium fluoride concentration ionization constant expres- sions: KFAP ,) = Keq - (37) KaF2 (F-) 18 K HRF -(H+)(F -) (31) -(HF) The s subscript refers to concentrations at K x = s. FAB refers to fluorapatite. The sub- (H+)(B-) (32) script o will refer to concentrations in the HB (HB) solution (presently, x 0). A combination of equations 34, 35, and 37 yields a new (H+) (H2PO4) constant K (33) (H3PO4) K1/6 (H2PO4 )8 X = =- (H+) (HPO=)4 K10./6 K3 pK2 P (F-)3 (H+)2 Kg P = (34) Call'2 S s (H2PO-) (38) The line integrals from concentrations at ) (H+)(PO4 (35) x = 0 to concentrations at x = s of equa- 3 P (HPO,-) tions 28, 29, and 30 are combined with equations 31 to 36 and 38 to yield two equations in two unknowns (HF2) (36) (HF)(F-)

An equation is now sought to describe the relationship between the participating Buffered entities of the chemical reaction that occurs Solution CaF2 Hydroxyapatite at x = s. If the reaction takes place rapidly of compared with the diffusion process, equi- Fluoride librium can be assumed to exist between the components in solution and in the solid phases,'7 ie, solubility products are valid. It luorapatite has been reported that the formation of calcium fluoride from calcium and fluoride ions 0 s occurs rapidly.'8 When hydroxyapatite is in contact with a FIG 3.-Physical model for formation of solution of fluoride, even at low fluoride calcium fluoride on hydroxyapatite with flu- concentration, it simulates the solubility orapatite as an intermediate phase. Vol 49 No. 6 MECHANISM OF FLUORIDE UPTAKE 1545 (QU- MV) + -/(MV - QU)2 - 4(LV - PU) (NV - RU) (39) 2(LV - PU)

(F-)3(H+)2V--P(F-)2 -Q(F-), - R K2P K3pK2P + -0 (40) DHP04 (H+) ±P04 (H+)2 where, INITIAL RATE EQUATION.-If the diffusion 16 KHF2 coefficients of the three species in equation L (H+)s 19 are assumed to be equal, then equations =--DHF220 KH-F 19 and 24 can be combined to give an 2 18 (H+)s equation for the flux of total fluoride per M =-DF- - DaF unit area, 20 20 KHF 2 18 Jtotalfluoride = - D d(TF) (41) N =-- DF(F-)O+ -DHF(HF)O 20 20 16 where TF is the sum of the hydrofluoric + acid, fluoride, and twice bifluoride. -DHFF)OD(H)20 HF2 (HF2 )0-D(H+) According to the liquid film theory, when + D11(H+)o the surface of apatite is exposed to the solution, a thin film of liquid exists adja- DHB(HB)O+ DB(B ) cent to the apatite across which the re- -DB DBKHB actants and products move. If the thickness DHB + of this film be defined as h, then the diffusion path at any time is from -h to s + DHB(HB)O + (Fig 4). For an initial rate, s = 0, so inte- DHPO4(HPO4 )° gration of equation 41 yields + 2 D 2O(H2PO4 ) ° (TF) - (TF)8 Jtotalfluoride =D + 3 DH3PO4(H3PO4)o h (42) 12 KHF2 In this case D is the diffusion coefficient in P= - DHF (+ ) 20 KII the homogeneous liquid and it need not be corrected for porosity and tortuosity. The 6 6 (H+)s subscript i refers to initial concentration in Q =- D DHF 20 20 KHF the solution. TIME DEPENDENT 6 6 EQUATION.-Integration of 41 from the total fluoride con- R =-DF(F-)o + DHF(HF)O equation 20 20 12

20 HF2(HF2)O + DH3PO4(H3PO4) O

+ D 2PO4(H2POj )o + DHPo4(HPO=)O 124PO4 4)4 + P04 (P°4=) 0 DHP04K2p U= (H) + 2 D (H+)( Iff2PO4 I m -hO s 3 DH3PO4 Kp X-_- DE3PO4(H+ )8 FIG 4.-Physical model for formation of V + DH2PO4 calcium fluoride on hydroxyapatite including a liquid diffusion layer. 1546 NELSON AND HIGUCHI J Dent Res November-December 1970 centration (TF) at x = -h to that at x - s (TF) - (TF)s gives + (TF), 1 n (TF)-(TF) - (TF)s- (TF) - AD Jtotalfluoride =-A - L l9(TF)&l2 vhh ] S +- LA2A(1- )D TAD- (43) -(TF) 1n(TF) S=t (48) where A is the area of apatite exposed to (TF)i- (TF)s the solution. The h had to be multiplied by Thus TF is implicitly a function of time. the quantity C/T as, recalling equation 11, be cal- the D in equation 43 is the effective dif- For any specific TF the time can heterogeneous culated. The fluoride uptake for this time fusion coefficient for the then can be calculated from equation 44. layer 0 . x I s and the -h I x I 0 region is the homogeneous liquid layer. The Discussion amount of fluoride lost from solution after an exposure of time t is To calculate an initial rate or the amount of fluoride uptake with time, it is first nec- v{(TF),- (TF)} (44) essary to solve the steady state, equations where v is the volume of the fluoride solu- 39 and 40, for (H+)8 and (F-)8. Since tion. Also, the amount of fluoride lost is dx was factored out and x does not appear the time integral of the flux of fluoride, in the steady state equations, (H+), and (F-)s are independent of the length of the t t ~~~~(TF)8 (TF) diffusion path. The values of (H+), and S Jtotalfluoride dt =-AD f dt. (F-), calculated from initial conditions of 0 0 he S +- the fluoride solution [(F-)o, (HB)O, etc] remain essentially constant during the (45) course of a reaction. This is true for two Combining equation 44 with the right side reasons. Diffusion coefficients for inorganic of 45 and taking the derivative with respect ions have numerical values that are about to time equal. Thus, the contribution by the diffusion coefficients in the steady state equations effectively vanishes. Secondly, the dd(TF) AD (TF) 8-(TF) dt he subscript o terms always appear in a group S +- within which they are related by stoichiometry, eg, R. The numerical value of any one (46) of these groups, then, remains constant be- The term s can be related to (TF) by cause as one term diminishes, another term equating the amount lost from solution (or terms) increases proportionately. Be- with the amount deposited in the calcium cause of the lengthy and repetitious calcu- fluoride layer. lations required to solve equations 39 and 40 simultaneously and to compute equation AsA(1 -, 48, it is necessary to use a computer. - =- ( v{(TF) (TF) I 19 Certain soluble species were not included (47) in the model. They are the complexes that calcium forms with phosphates.20 In acidic A is the density of fluoride in pure calcium solutions they typically account for less fluoride. The amount of fluoride in solution than 1% of the total dissolved calcium. in the pores of the layer is much smaller The numerical values needed to solve the and is omitted. Solving equation 47 for s, equations are obtained from various sources. substituting this into equation 46, and inte- The concentrations of the solutes, the vol- grating from the initial concentration to ume of solution, and the apatite area are that at time t particular experimental conditions. The 19v2 _ ionization constants, solubility products, and activity coefficients can be found in L A2A(1 - E)D i (TF) TF)i chemistry handbooks. Diffusion coefficients Vol 49 No. 6 MECHANISM OF FLUORIDE UPTAKE 1547 can also be found in handbooks, however, librium chemical reaction at a moving these numbers represent accurately deter- boundary. With the aid of a computer, it is mined values in two component systems. possible to calculate initial rates of reaction It would be expected that they would vary that can be used to evaluate a priori the somewhat in a multicomponent system. It fluoride uptake potential of various fluoride is a good practical assumption to allow all solutions. The variables that can be exam- diffusing species to have the same diffusion ined are the fluoride concentration, phos- coefficient. The effective diffusion layer phate concentration, pH, buffer type (pKa), thickness can be obtained by calibrating the and buffer concentration. A forthcoming experimental apparatus by a dissolution study will consider the comparison of ex- process with a compound of known solu- perimental studies with the model. bility and diffusivity. The porosity and tortuosity can be estimated by an experiment References involving the release of solute from one 1. BLAYNEY, J.R., and HILL, I.N.: Fluorine planar surface of a solution-saturated and Dental Caries, JADA 74:233-302, matrix.21 1967. The usefulness of the proposed model is 2. PEFFLEY, G.E., and MUHLER, J.C.: The threefold. First, it serves to illustrate the Effect of a Commercial Stannous Fluoride processes that apparently occur during ap- Dentifrice under Controlled Brushing Hab- plication of a fluoride to the teeth. its on Dental Caries Incidence in Children: It formalizes the relationships between the Preliminary Report, J Dent Res 39:871- diffusion, the ionic equilibriums, and the 874, 1960. chemical reaction at the 3. WELLOCK, W.D., and BRUDEVOLD, F.: A moving boundary. Study of Acidulated Fluoride Solutions II, Second, it enables the estimation of fluo- Arch Oral Biol 8:179-182, 1963. ride uptake with time for a given topical 4. PAMEIJER, J.H.N.; BRUDEVOLD, F.; and fluoride preparation. Third, it offers a means HUNT, E.E.: A Study of Acidulated Fluo- to calculate initial rates of fluoride uptake ride Solutions III, Arch Oral Biol 8:183- for a given fluoride solution. 185, 1963. In the case of fluoride uptake with time, 5. WELLOCK, W.D.; MAITLAND, A.; and absolute accuracy may be difficult to BRUDEVOLD, F.: Caries Increments, Tooth achieve because of the assumption of equal- Discoloration, and State of Oral Hygiene ity of all of the diffusion coefficients and in Children Given Single Annual Appli- of the and tor- cations of Acid Phosphate-Fluoride and the dependence porosity Stannous Fluoride, Arch Oral Biol 10:453- tuosity on the state of the hydroxyapatite. 460, 1965. In particular, it would be expected that the 6. BRUDEVOLD, F.: Chemical Composition of porosity and tortuosity of the calcium fluo- the Teeth in Relation to Caries, in SOGN- ride layer formed on a compressed disk of NAES, R.F. (ed): Chemistry and Preven- synthetic hydroxyapatite would be different tion of Dental Caries, Springfield, Ill.: from those of calcium fluoride formed on Charles C Thomas, 1962, pp 32-88. enamel. The use of initial rates to evaluate 7. GRAY, J.A.: Kinetics of the Dissolution of fluoride solutions circumvents the necessity Human Dental Enamel in Acid, J Dent of determining the porosity and tortuosity Res 41:633-645, 1962. of the calcium fluoride layer. The use of 8. HIGUCHI, W.I.; GRAY, J.A.; HEFFERREN, one initial rate relative to another precludes J.J.; and PATEL, P.R.: Mechanisms of Enamel Dissolution in Acid Buffers, J the necessity of determining the effective Dent Res 44:330-341, 1965. diffusion layer thickness. Thus, it is possible 9. PATEL, P.R.: Studies on Acid Demineral- to use initial rates to evaluate a priori the ization Kinetics of Enamel, PhD thesis, fluoride uptake potential of two or more University of Michigan, 1965. fluoride solutions without doing any experi- 10. BECKER, J.W.: Studies of a Proposed mental work. Model for Acid Demineralization of Hy- droxy apatite, PhD thesis, University of Conclusions Michigan, 1967. A physical chemical model for the mech- 11. ZIMMERMAN, S.O.: A Mathematical The- anism of fluoride uptake by hydroxyapatite ory of Enamel Solubility and the Onset was proposed. It was derived on the bases of Dental Caries: I. The Kinetics of Dis- of diffusion, ionic equilibriums, and equi- solution of Powdered Enamel in Acid 1548 NELSON AND HIGUCHI J Dent Res November-December 1970 Buffer, Bull Math Biophys 28:417-432, fer and Equilibrium Chemical Reaction, 1966. A I Ch E J 6:233-239, 1960. 12. ZIMMERMAN, S.O.: A Mathematical The- 17. CRANK, J.: The Mathematics of Diffusion, ory of Enamel Solubility and the Onset Oxford: Clarendon Press, 1956, p 121. of Dental Caries: II. Some Solubility 18. FARR, T.C.; TARBUTTON, G.; and LEWIS, Equilibrium Considerations of Hydroxy- H.T.: System CaO-P205-HF-H2O: Equi- apatite, Bull Math Biophys 28:433-442, 1966. librium at 25 and 500, J Phys Chem 66: 13. ZIMMERMAN, S.O.: A Mathematical The- 318-321, 1962. ory of Enamel Solubility and the Onset of 19. MIR, N.A.: The Mechanism of Action of Dental Caries: III. Development and Com- Solution Fluoride upon the Demineraliza- puter Simulation of a Model of Caries tion Rate of Enamel, PhD thesis, Univer- Formation, Bull Math Biophys 28:443- sity of Michigan, 1967. 464, 1966. 20. CHUGHTAI, A.; MARSHALL, R.; and NAN- 14. NAPPER, D.H., and SMYTHE, B.M.: The COLLAS, G.H.: Complexes in Calcium Dissolution Kinetics of Hydroxyapatite in Phosphate Solutions, J Phys Chem 72: the Presence of Kink Poisons, J Dent Res 208-211, 1968. 45:1775-1783, 1966. 21. DESAI, S.J.; SINGH, P.; SIMONELLI, A.P., 15. HILL, A.V.: The Diffusion of Oxygen and and HIGUCHI, W.I.: Investigation of Fac- Lactic Acid through Tissues, Proc Roy tors Influencing Release of Solid Drug Soc B 104:39-96, 1928. Dispersed In Inert Matrices II, J Pharm 16. OLANDER, D.R.: Simultaneous Mass Trans- Sci 55:1224-1229, 1966.