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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 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 diffu- sion 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 stoichi- ometry 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.
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