Finite Element Modelling and Experimental Validation of the Enamel Demineralisation Process at the Rod Level

Abstract

In the past years, a significant amount of effort has been directed at the observation and characterisation of caries using experimental techniques. Nevertheless, relatively little progress has been made in numerical modelling of the underlying demineralisation process. The present study is the first attempt to provide a simplified calculation framework for the numerical simulation of the demineralisation process at the length scale of enamel rods and its validation by comparing the data with statistical analysis of experimental results. FEM model was employed to simulate a time-dependent reaction-diffusion equation process in which H ions diffuse and cause demineralisation of the enamel. The local orientation of the hydroxyapatite crystals was taken into account. Experimental analysis of the demineralising front was performed using advanced high-resolution synchrotron X-ray micro-Computed Tomography. Further experimental investigations were conducted by means of SEM and STEM imaging techniques. Besides establishing and validating the new modelling framework, insights into the role of the etchant solution pH level were obtained. Additionally, some light was shed on the origin of different types of etching patterns by simulating the demineralisation process at different etching angles of attack. The implications of this study pave the way for simulations of enamel demineralisation within different complex scenarios and across the range of length scales. Indeed, the framework proposed can incorporate the presence of chemical species other than H ions and their diffusion and reaction leading to dissolution and re-precipitation of hydroxyapatite. It is the authors' hope and aspiration that ultimately this work will help identify new ways of controlling and preventing caries.

Keywords: Demineralisation simulation; Dental demineralisation; Enamel; FEM; Reaction-diffusion; Synchrotron CT.

Graphical abstract

Fig. 1

Schematisation and evidence of the enamel structure at the micro-scale – a collection of images from the literature. (a) The left side of the diagram shows orientation of crystals in the forming rod head and tail, while the right part shows how forming crystals pack in the rod from the cell complex . (b) Image showing the different HAp crystals in the rod head, interrod and sheath domains . (c) Diagram of the six-sided ameloblasts overlying keyhole-shaped enamel rods .

Fig. 2

3D reconstructed volume of a demineralised enamel (colour online). (a) The region shown in blue is pristine enamel. Orange channels represent the demineralised enamel rods and the complete material removal (cavity formation). The region shown in green represents an empty volume (after thresholding) that correlates with an enamel fracture. (b) Visualisation excluding the surrounding pristine enamel.

Fig. 3

SEM and STEM imaging of etched enamel. (a) SEM etched enamel rods overview. (b) SEM close-up to the rod sheath and interrod enamel for the area indicated by red dashed box in (a). (c) STEM image of the HAp crystallites, in yellow a crystal highlighted. The STEM lamella was extracted from the region highlighted by the rectangular box in (b).

Fig. 4

Representative Volume Element characteristics. a) Periodicity representation and extraction of the RVE. b) Geometrical characteristics and relevant domains, along with a cartesian coordinate system. b) Hydroxyapatite crystals orientation indicated by arrows.

Fig. 5

HAp schematisation. Crystal axes and rotation angles. Reproduction from

Fig. 6

Two angles defining the positive orientation at the element centroids, around directions x and y. The dashed lines indicate the fitted polynomial function which has been implemented in the FEM modelling.

Fig. 7

Contour plots of the diagonal elements of the rotated diffusivity matrix $\stackrel{Ì¿}{D_G}$.

Fig. 8

Discretised RVE (a) and an example of the enamel erosion front extracted after 21 days of acid exposure (b).

Fig. 9

Lesion front change as a function of the time, in days. Modelling vs. experimental data (red solid line).

Fig. 10

Numerical simulation of the lesion advance. (a) Comparison of exposure to solutions at several levels of pH, after three weeks of continuous exposure. (b) Lesion front as a function of the pH, exposure time three weeks.

Fig. 11

Mixed mode etching type simulation. (a) Geometrical model and exposed surface in blue. (b) Demineralised front at several time frames. The colours represent the y-coordinates for each point of the domain (in millimetre).