The relation between structural, rugometric and fractal characteristics of hard dental tissues at micro and nano levels

Human tooth exhibits a structure of a mixture of inorganic hydroxyapatite nanocrystals and organic phases. The aim of this study is to investigate different tissues of human canine teeth surface along with the micro structure parameters of each tissue. X‐ray diffraction (XRD) is used to study the amorphous or crystalline nature of each tissue with different mineral compositions and crystalline structures where the highest crystalline quality is related to enamel. The surfaces are also examined by energy‐dispersive X‐ray spectrometry. Moreover, crystalline quality factor is carried out to estimate the crystallinity of the tissues. Also, based on the basic Scherrer equation, the Williamson–Hall equation is applied to extend the formula for the XRD. Enamel and cementum tissues of a typical human tooth, which look similar, are composed of a large variety of wide lines with different widths through Raman spectra analysis. In addition, the applied scanning electron microscopy extracts similar morphology for all tissues with round granular structures which are denser in the cementum. Atomic force microscopy is finally used for investigation of micro‐morphologies of the different tissues and the results are compared with the fractal analysis which ends to the bifractal and anisotropic nature of enamel and cementum along with monofractal and isotropic nature of dentin.


| INTRODUCTION
The teeth are intriguing parts of the human body due to their performance characteristics associated with material response (Gutiérrez-Salazara & Reyes-Gasga, 2003) and stability (Driessens, 1980). The teeth represent highly mineralized tissues and may be mainly regarded in terms of their constituent components and respective organization (Schroeder & Frank, 1985). In this respect hard dental tissues are found similar to bones, yet more mineralized (e.g., enamel) and thought of as living organs that form throughout life (e.g., dentin). The main obvious function of teeth is connected with their elastic properties-stiffness and hardness (Rey, Renugopalakrishnan, Shimizu, Collins, & Giimcher, 1991). This observation does not neglect other effects, for example biological interactions with the body and surrounding environment, however, for many applications such as identification of proper substituting materials, both for restorations (resin composites and glass-ionomer cements [gic]; Aroraa et al., 2004;Feilzer, De, & Davidson, 1988) and implants (titanium alloy for the screws replacing the roots and ceramics or metal/ceramics composites for the crowns; Barfeie, Wilson, & Rees, 2015), a simplified material-science point of view may be still successful.
Today's researches in dental science cover a broad spectrum of topics investigating complex relationships within structure-propertyperformance triangle for various materials on different length and time scales. The importance of these works is directly related to successful clinical applications of obtained results; however, even high-quality studies might suffer from disadvantages of traditional methods of data acquisition and analysis. Only few examples are mentioned below. Poggio, Ceci, Beltrami, Lombardini, and Colombo (2014) reported their study on remineralization processes within enamel and dentin based on measurements of the surface roughness. However, ordinary statistical measurements do not provide unique characteristics of asperities in terms of their distribution, high-order regularities etc. Marshall et al. (Marshall, Marshall, Balooch, & Kinney, 2004) performed similar experiments on demineralization of dentine by means of nanoindentation, which is thought of as a destructive method.
Another work by Sanches, Otani, Damião, and Miyakawa (2009) demonstrated the effect of making micro-porosities in dentine and enamel surfaces prior to dental restoration with adhesive composites, in which standard statistics were insufficient to reveal any geometric irregularities that are key factors determining dynamic interaction between contacting surfaces (Ţ alu et al., 2016). Domke et al. (2000) tested the biocompatibility of implant materials by examining adhesion behavior of osteoblast cells on grooved surfaces and pointed that the cells do not follow the grooves because bonding reactions have a fairly large random component. In such a case, biocompatibility might be achieved through duplication the entire surface texture on a wide range of length scales, but to this end, appropriate scale-invariant characteristics are necessary.
Apart from the above considerations, open questions of fundamental origin still arise in related fields such as physics and materials engineering. Fracture surfaces are known to undergo non-linear scaling behavior in correspondence with their mechanical performance . On the other hand, many processing operations produce rough-surface morphologies with fractal characteristics related to the crystalline structure of the bulk material. In order to get an insight into fundamental mechanisms behind fracture phenomena, superior approach for material characterization is required.
The presented work aims to demonstrate reliability and effectiveness of a joint technique of structural and topographical investigations of biological tissues (hard dental tissues as an example) to establish structure-property relationship of these materials on various length scales. Due to high inhomogeneous morphology of the tissues under study, obtained results might be of potential importance for understanding the mechanisms of adhesion of osteoblast cells to implants, better models of fracture mechanisms and crack propagation dynamics within hard dental tissues etc.

| EXPERIMENTAL DETAILS
A series of 10 permanent freshly extracted canine teeth of a group of 30-year old men were applied. Their similar results motivated us to choose only one of them as the main specimen in this study. Freshly extracted, healthy tooth was immersed in saline, transferred to the laboratory, and brushed with an ordinary tooth brush and standard dentifrice. Afterwards, it was ultrasonically cleaned in acetone and alcohol baths to remove remaining impurities and finally dried in the air. The measurements were focused on three hard dental tissues: enamel, dentin, and cementum. To get access to the dentine, the tooth was cross-sectioned longitudinally under profuse water irrigation using an Exakt sawing and grinding machine. Analysis of elemental composition was carried out using the energy-dispersive Xray spectroscopy (EDX), while scanning electron microscopy (SEM; KYKY-EM3200) and atomic force microscopy (AFM; Veeco, Santa Barbara, California) images were source data for further numerical analyzes concerning various aspects of surface roughness and morphology. Moreover, XRD (STOE-XRD) using a diffractometer with Cu Kα radiation (λ = 0.15406 nm) in the range of 2θ = 10 -70 together with the Raman spectroscopy were used to determine the crystalline structure of the specimen including, among others: stresses in the sample and mosaicity of coherently-diffracting domains.

| Characterization methods
Spatial characteristics of investigated dental tissues were derived from both AFM and scanning SEM images making use of the allometric similarity laws brought by the theory of fractal scaling (Farina et al, 1999;Stach et al, 2015;Ţ alu et al, 2015). AFM yields maps of height samples probed by the tip in the near-field regime, whereas SEM images contain gray-scale pixels that need to be linearly converted into pseudo-height entries prior to any numerical processing. After appropriate preparation of source data, the following multistep numerical procedure is carried out that begins with calculation of the autocorrelation function R according to the formula: where (m, n) enumerates discrete shift between the image and its lagged copy. Autocorrelation becomes zero for infinite lags, and the ratio of the extreme decay lengths τ defines the surface anisotropy ratio S tr : where a 1 , and a 2 are the directions of the fastest and the slowest autocorrelation decay, respectively.
In the next step, autocorrelation function is converted into the structure function according to: where S q is the surface roughness. The mean profile of the structure function averaged around origin exhibits specific scaling behavior: where D is the fractal dimension and K is pseudo-topothesy. In general, D and K correspond to the way, how the relative and absolute amplitudes of surface height variations vary with the wavelengths, respectively. On the other hand, the corner frequencies τcf are thresholds, at which the profile plot changes its slope.
Physical properties of the materials under investigation can be also studied making an insight into the arrangement of the crystal lattice. An effective way towards atomic structure of the materials is via X-ray diffraction (XRD) techniques. However, the drawback of this method in terms of materials with apparent mosaic structure is that the diffraction peaks overlap and the proper crystal structure cannot be easily determined using Braggs' law solely. In order to reconstruct accurate crystal lattice and determine the contributions of chemical compounds that form given structure, the Rietveld method was used in this work. Apart from that, the relative deformations of crystal lattices are determined from the Williamson-Hall plot.

| Chemical characterization
Representative EDX spectra of the enamel and cementum are shown in Figure 1. Peaks corresponding to Calcium (Ca), Phosphorus (P), Oxygen (O), Carbon (C), and Chlorine (Cl) can be seen for enamel and cementum (Povolo & Hamida, 2000).
Clearly, different relative compositions of the various elements are observed in Table 1. In enamel, only~3% (wt) can be ascribed to C, as expected, due to the higher mineralization of this tissue. It is high degree of mineral part, along with the microstructural organization in prisms, that imparts enamel its high stiffness and hardness, which is unfortunately associated also with some brittleness, similar to ceramics (Arsecularatne & Hoffman, 2012;Wu et al., 2013). The C contents reaches up to~49% in the cementum (Ghose, Viswanadhan, & Wendoloski 1999;Zhang, Arsecularatne, & Hoffman 2015): the latter value is obviously associated with imperfect removal of the periodontum from around the hard root surface. According to Table 1, the Ca/P ratios in enamel and cementum are almost the same nearly equal to 1.7-1.8, and both elements can be assigned to apatite crystals.
However, obtained results point at slightly non-stoichiometric hydroxyapatite (HAP) crystals embedded in the tissues, either Ca-abundant or P-deficient. Note also that HAP content within enamel is almost seven times higher than in cementum, while other studies report that this ratio is actually close to 2.
XRD was performed using STOE XRD in order to identify the crystalline phases within the tissues that form the tooth. Although the main component of all tissues was HAP (Cao, Mei, Li, Lo, & Chu, 2015;Selvig, 1970), different zones of hard dental tissues from enamel to dentin to cementum have different mineral compositions and hence crystal structures. Figure 2 represents normalized XRD spectra taken from various parts of the sample under study compared with the simulated spectrum of a perfect HAP structure. In general, each spectrum presented in Figure 2, exhibits different combinations of XRD peaks considering their visibility, positions, intensities and widths. Such a result points at the texture of the material, from which a given tissue is made of, that is, preferred orientations of ordered substructures within the sample. Detailed crystalline microstructure of the tissue depends strongly on its function.
Enamel is known to be the hardest material under investigation and its XRD spectrum in Figure 2 appears to be the sharpest among those under study. A bunch of XRD peaks is visible, each of which exhibits very high intensity to full-width at half maximum (FWHM) ratio which confirms good quality of crystals (long-range order). Compared to perfect HAP, the highest peak is (002), but not (211), which is caused by highly textured structure of that material.
To estimate the degree of crystallinity, crystalline quality factor (CQF) is proposed as a ratio of a difference between total and baseline spectrum areas to the total area, normalized with respect to the perfect HAP structure: where A is the total area below the XRD curve, B is the area under the background curve (baseline), and CQF HAP is the CQF equal to around 0.812 estimated for the XRD spectrum of a reference HAP structure.
In case of the enamel, CQF is found nearly 37%, the highest among the tissues under investigation. Table 2 summarizes the obtained results using the Rietveld refinement method that allows us to determine additional microstructural parameters of this material.
Both lattice constants, a 0 and c 0 , of enamel were found the largest compared to other tissue and the stress-free HAP crystal (9.526 and FIGURE 1 EDX spectra of the enamel and cementum. EDX, energydispersive X-ray spectroscopy [Color figure can be viewed at wileyonlinelibrary.com] The measured XRD spectra of enamel, dentin, and cementum compared with that of a model of HAP structure. All spectra were normalized with respect to their maximum intensities. HAP, hydroxyapatite; XRD, X-ray diffraction [Color figure can be viewed at wileyonlinelibrary.com] 6.903 for a 0 and c 0 , respectively). On the other hand, lattice parameters can be compared with the level of structural microstrains, ε, obtained using the Williamson-Hall equation (Figure 3). Based on the basic Scherrer equation, Williamson and Hall extended the formula for the XRD line broadening so that it included two terms associated with the finite crystallite size (mosaic domains) <D> and uniform microstrains ε (Williamson & Hall, 1953): Using the numerical least-square fit, strain and particle size can be determined from the slope and y-intercept of the fitted line, respectively, as in example plot shown in Figure 3. Considering the enamel, the microstrains approach the smallest value equal to 0.213%. In addition, the highest crystalline quality of enamel corresponds to the largest size of the mosaic domains responsible for coherent diffraction of the X-rays, equal to 33 nm.
In turn, much softer dentin yields lower but wider, hence less pronounced XRD peaks observed as a consequence of its poorer crystallinity. In this case, CQF is found lower at around 30% mostly due to increased intensity of a baseline, especially in a range of small 2θ angles, rather than decreased integral intensity of the XRD peaks.
Poorer crystallinity is accompanied by larger relative microstrains (0.335%, higher than in enamel) and mosaic domain size reduced to half of that in enamel (17 nm). Even though, the lattice parameters a 0 and c 0 are found less strained than in the enamel. However, the microstrains in the Williamson-Hall method are derived assuming isotropic lattice distortions, which might not be held in case of textured structure of the dentin (Carlén, Börjesson, Nikdel, & Olsson 1998).
Apart from that, the structure of this tissue appears to be composed of smaller grains of HAP crystals sorrounded by larger volume of amorphous material forming the grain boundaries and contributing to low-angle structural defects.
According to Figure 2, the XRD spectrum of the cementum is the weakest throughout the study, with sligthly marked thick peak at 2θ close to 32 . Other parameters of a crystalline purity also confirms the poorest quality of this material. The quality factor has dropped to the bottom and approached 2%, while the microstrains reached the highest level among the tissues under study at 0.335%. Simultaneously, the mosaic domains became even smaller than those in the dentin (11 nm), but the lattice parameters are found lower than in the stressfree HAP. Poor crystallinity strictly corresponds to extremely low HAP content seen in EDX spectra as trace amounts of Ca and P elements.
Two Raman spectra taken from different parts of the same human tooth are compared in Figure 4. The upper spectrum comes from the enamel, which is a tissue exposed to mechanical wear and the lower one from cementum, which is hidden in the gums. Both spectra look similar and are composed of a large variety of wide lines with different widths extending up to hundreds of reciprocal centimeters, which is the characterization of highly disordered solids with a complex a 0 , c 0 = lattice constants; γ 0 = orietation of the crystal lattice with respect to the incident beam; <D> = size of coherently-diffracting domains; ε = Relative structural microstrains; CQF = crystalline quality factor; XRD = X-ray diffraction. structure. On the other hand, XRD data showed that the crystalline structure of the enamel substantially differs from that of the cementum. Enamel is the most mineralized and the hardest tissue in the human body, composed mainly of the hydroxyapatite in which the stoichiometric Ca/P ratio is very close to 5/3. EDX analyses demonstrated that in the enamel, this ratio approaches 2.2, that is, more than 30% higher than in natural minerals that suggest the deficiency of phosphorus in the material. Similar ratio was found in the cementum (2.1) with fewer percentages of the elements. It means that both tissues are made of similar material, but they differ in the overall content. From this perspective, the Raman spectra bands, connected with specific groups of the hydroxyapatite lattice, should look similar with more intensity in enamel.
According to previous works (e.g., Kirchner, Edwards, Lucy, & Pollard, 1997), the main spectral feature which represents HAP, has a Another group of peaks in the Raman spectrum are those of amides, that is, the secondary structure of proteins. Previous works demonstrated the presence of significant peaks of 1,667, 1,450, and 1,245 cm −1 attributed to organic material. They are associated with, among others, peptide backbone vibrations of a double C-O bonds corresponding to a strong 1,600-1,700 cm −1 band, and side-chains of amino acids vibrations within N-H groups corresponding to a strong band centered at 1240 cm −1 . In the presented spectra, though, there is a very wide double band in the range from 1,100 to 1,600 cm −1 with dominant 1,280 and 1,390 cm −1 components assigned to strongly distorted amide bonds within NH and CH 2 groups, respectively. Similar discrimination can be made considering the results of fractal analysis shown in Table 3. As a rule (Ţ alu et al, 2016), the most outer tissues, that is, the enamel and cementum, are found bifractal, whereas the dentin-monofractal. Bifractal characteristics correspond to structures with interpenetrating spatial arrangements on different length scales, for example regular clusters, while monofractal ones correspond to structures exhibiting spatial patterns of the first order

| CONCLUSIONS
The paper reports the results of a joint study on the relationships between structure, material property and possible performance of hard dental tissues. The related preffered orientations of ordered structures were compared with the simulated spectrum of a perfect HAP structure via XRD. Enamel had the sharpest spectrum with very high intensity to FWHM and the the highest crystallinity through CQF estimatation while dentin was much softer with poorer crystallinity and cementum had the weakest spectrum. Additional microstructural parameters obtained using the Rietveld refinement method and the numerical least-square fit to determine the strain and particle size.
The highest crystalline quality of enamel is also related to the smallest value of its microstrains and the largest size of the mosaic domains which was reduced in dentin and caused poorer crystallinity. Studying Raman spectra of enamel and cementum led to similar spectra with a large variety of lines with different widths.
Moreover, AFM images presented different height variations of regular surface of each tissue which was significant in the inter enamel and inter dentin. Cementum revealed the largest contribution of short-wavelength height variation components. These results were in a good agreement with that of fractal analysis which resulted in the bifractal and anisotropic nature of enamel and cementum and the monofractal and isotropic nature of dentin. Finally, SEM images revealed similar morphology for all tissues with round granular structures which were denser and larger in the cementum. In the larger scan area of SEM images, the fractal analysis were in a good agreement with the results obtained from AFM analysis.
Although SEM gives deeper insight into topographical features of the samples under study, it is AFM that reveals more detailed surface structure at the nanometric level, which can be linked to various mechanical properties of the material.