The modelling of carious lesion progress

A carious lesion in the tooth enamel is a result of a chemical reaction between mobile organic acids and static hydroxyapatite. Hydroxyapatite is the main component of human tooth enamel, whereas, organic acids are produced in dental plaque by oral microorganisms which metabolize simple sugars from the diet. The organic acids can be transported into the enamel in a dissociated or undissociated form, which depends on the pH of the dental plaque. The transport of ions is diffusive and is motivated by the small diameter of the hydrogen ions compared to the size of the enamel tubules, whereas, when large acid molecules move into the enamel, subdiffusion occurs. Diffusion is a natural process involving the spontaneous spreading of a substance. Diffusion is characterized by the time dependence of the mean–square displacement of a random walker 〈 (∆x)(t) 〉 = Dαt /Γ(1+α), where α is a diffusion parameter, Dα is a diffusion coefficient measured in the units of m/s and Γ(x) denotes the Gamma function. For 0 < α < 1 we are dealing with subdiffusion; for α = 1, we have a situation of normal diffusion, and for α > 1, we encounter superdiffusion. Subdiffusion occurs in media in which the movement of the random walker is strongly hindered due to the complex structure of a medium and subdiffusion occurs—among other things—in porous media or gels. Equations describing the subdiffusive transport of acid molecules and their reaction with static hydroxyapatite are nonlinear partial differential equations with the Riemann-Liouville fractional time derivative ∂CA(x, t) ∂t = Dα ∂ ∂t1−α ∂CA(x, t) ∂x −Rα(x, t) , (1) ∂CB(x, t) ∂t = −Rα(x, t) , (2) where CA denotes the concentration of a mobile substance A, CB—the static substance B and the reaction term takes the form Rα(x, t) = ∂ ∂t1−α kCA(x, t)CB(x, t) , (3) and k is the reaction rate constant. For α = 1 we obtain equations describing a normal diffusion–reaction system. As far as we know, the general solutions, i.e. for arbitrary parameter values, to (sub)diffusion–reaction equations have not been found yet. Thus, in order to simplify the calculations, various approximations, such as the quasistationary approximation, the scaling method, or the perturbation method, are used. Employing these methods, characteristic functions of the system can be derived which include, among others things, the time evolution of the reaction front which can be identified with a lesion depth. Using the perturbation method we find approximate solutions to the normal diffusion–reaction equations which are quite satisfactory in comparison with experimental data. We will also find that the time evolution of the lesion depth in the cases of acid transport, normal diffusion and subdiffusion, is a power function of time xf ∼ t. This result is also in accordance with the experimental data.