Microstructure, micromorphology, and fractal geometry of hard dental tissues: Evaluation of atomic force microscopy images

Determining surface topography of different tissues of the molar tooth with novel analytical methods has opened new horizons in dental surface measurements which characterize tooth surface quality in dentistry. Studying surface topological measurements and comparing surface morphology of hard tissue of the molar tooth are the ultimate goals of the present study. Ten molar teeth have been chosen for investigating their surface characteristics through image processing techniques. The power spectral density (PSD) and fast Fourier transform algorithms of every molar tooth containing enamel, dentin, and cementum have determined that the characterization of surface profiles is possible. As can be seen, PSD along with fractal dimensions leads to good results for teeth surface topography. Moreover, PSD angular plot assures appropriate description of surface.


| INTRODUCTION
Recently, applications of nanotechnology in dentistry have been reported in order to improve tooth properties (Salerno & Diaspro, 2015;Ţ alu et al., 2016;Ţ alu, Yadav, Lainovi c, et al., 2018;Uskokovi c & Bertassoni, 2010). Enamel, dentin, and cementum as the hard tissue along with dental pulp are the main parts of every natural tooth.
Enamel with its "keyhole-like" structure is perpendicular to the junction of enamel and dentin with hydroxyapatite (HAP) crystals composition (Habelitz, Marshall, Marshall, & Balooch, 2001;Kerebel, Daculsi, & Kerebel, 1979;Solaymani et al., 2019). Dentin with collagen fibers composition and dentinal tubules microstructures is surrounded by root canals and placed beneath the enamel (Driessens, 1980;Garberoglio & Brannstrom, 1976). Moreover, cementum is composed of apatite such as calcium along with organic components. The hardness of cementum is less than dentin (Ghose, Viswanadhan, & Wendoloski, 1999). Identification of dental implant components along with tooth surface properties such as surface chemistry and surface topography attracts a lot of attention during the last decade (Le, Soueidan, Layrolle, & Amouriq, 2007;Rompen, Domken, Degidi, Pontes, & Piatelli, 2006), because surface morphology and microstructure of human teeth will strongly affect their physical and chemical properties (Wennerberg, Sawase, & Kultje, 1999). However, most of recent studies have reported average roughness parameters and deviations in two dimensions (Hosoya, Honda, Lino, & Arai, 2003;Whitehead, Lo, Watts, & Wilson, 1997). Atomic force microscope (AFM), with its high resolution both vertically and spatially, is applied to investigate microroughness characterization and surface information (Binnig, Quate, & Gerber, 1986;Stach et al., 2015;Ţ alu, Yadav, Šik, et al., 2018). Whitehead et al. applied a profilometric measuring system for studying surface texture parameters which were extracted by Ra parameter and using bearing ratio. Moreover, Edblad et al. used interferometry method for characterizing enamel and cementum at the cervical region of healthy teeth which is a noncontacting optical technique analyzing surface topography. Their experiment was carried out on nine premolar teeth. They also investigated the influence of a protein-dissolving enzyme on surface topography (Edblad et al., 2009). In addition, Kocher et al. examined different methods for studying topography and roughness of extracted teeth. For example, they used laser profilometry for 3D roughness analysis of teeth instrumented in vivo with teflon-coated sonic scaler insert and conventional ultrasonic device. They compared the values of Ra and Rz. They also studied roughness parameters of teeth which were instrumented by periotor and curettes instruments (Kocher, Rosin, Langenbeck, & Bernhardt, 2001). Besides, Mehl et al. found a basic mathematical description of the human first lower molars by introducing biogeneric tooth model by computer-aided design processes (Mehl, Blanz, & Hickel, 2005).
Among several methods including average roughness, root mean square roughness (R rms ), PSD, and a peak to valley roughness for surface roughness, PSD as the frequency distribution is related to the spectroscopic nature of the tooth surface and extracts more information by assuming fast Fourier transforms (FFTs) as the PSD background. We show that the correct determination of surface morphology using scanning force microscopy (SFM) imaging and power spectral density (PSD) analysis of the surface roughness is an extremely demanding task that is easily affected by experimental parameters such as scan speed and feedback parameters. We present examples where the measured topography data are significantly influenced by the feedback response of the SFM system and the PSD curves calculated from this experimental data do not correspond to that of the true topography. Instead, either features are "lost" due to low pass filtering or features are "created" due to oscillation of the feedback loop. In order to overcome these serious problems, we show that the interaction signal (error signal) can be used not only to quantitatively control but also to significantly improve the quality of the topography raw data used for the PSD analysis. In particular, the calibrated error signal image can be used in combination with the topography image in order to obtain a correct representation of surface morphology ("true" topographic image). From this "true" topographic image, a faithful determination of the PSD of surface morphology is possible. The corresponding PSD curve is not affected by the finetuning of feedback parameters, and allows for much faster image acquisition speeds without loss of information in the PSD curve. Moreover, Mandelbrot proposed the relation between fractal geometry which is a simple rule for complex objects, and surface morphology (Dalouji, Elahi, Ghaderi, & Solaymani, 2016) so that at each scale, a set of geometric parameters characterize surface morphology. In the present work, AFM analysis yields valuable information about tooth surface roughness, while PSD plots are used for investigating spatial frequencies of height distribution and ends to a correlation between the percentage of atomic elements and film morphology (Solaymani et al., 2013).
The main goal of the present work is to evaluate morphological topographical characterizations of enamel out, interenamel, dentin, and cementum of the healthy molar tooth through 3D AFM images for the first time.

| EXPERIMENTAL DETAILS
Among the analysis of 10 selected permanent molar teeth included in the present study, the results of just one molar tooth of a 29-year-old man was presented due to similar results. The healthy molar teeth which were extracted by ordinary routines for orthodontic purposes were collected in a dental clinic. The extracted tooth was inserted into saline and transferred to the laboratory. Afterward, it was cleaned by standard dentifrice and tooth brush. It was then cleaned ultrasonically in acetone and alcohol baths in order to remove impurities and then air-dried. The selected tooth was cross-sectioned longitudinally just to access to the interenamel.
Energy-dispersive X-ray spectroscopy (EDX) was applied for investigating the elemental percentage of tooth composition in every tissue. In addition, micromorphology and surface roughness investigation was carried out by using a Nanoscope Multimode AFM (Veeco, Santa Barbara, CA) in noncontact mode with scan rates of 10-20 μm/s to obtain 256 × 256 pixel images. Relative humidity and temperature were set 44±1% and 298±1 K, respectively, and all images were obtained over square areas of 1 μm × 1 μm by cantilevers with 25 μm width, 180 μm length, 10 nm tip radius, and 4 μm thickness along with quality factor (Q) of 100, Young's modulus (E) of 1.3 × 10 11 Pa, mass density (ρ) of 2,330 kg/m 3 , and Poisson ratio (ν) of 0.28 for force-distance curve measurements. Raman spectra analysis was used to provide information about the structural quality and stresses and variations of crystalline nature of enamel and cementum. Moreover, X-ray diffraction (XRD) (STOE-XRD) using a diffractometer with CuKα radiation (λ = 0.15406 nm) in the range of 2θ = 10 -70 was used to determine the crystal structure of each part of the specimen.

| RESULTS
EDX analysis of enamel and cementum confirms HAP composition of all dental tissues with calcium (Ca), oxygen (O), and phosphorus (P) with various percentages of each element in each tissue that can be seen in Table 1 so that about 3% (wt) of enamel and 49% of cementum are carbon which means higher mineralization of enamel.
The ratio of Ca/P in enamel and cementum is identical which is an important parameter and will be discussed later.
In order to investigate the crystalline structure of each dental hard tissue with HAP composition, XRD was applied with STOE XRD (Cao, Mei, Li, Lo, & Chu, 2015;Selvig, 1970). XRD spectra of all are presented in Figure 1 in which each tissue contains different peaks from intensity, position, and width aspect. Good crystallinity of enamel out is obvious due to its sharp peaks compared with other tissue along with high ratio of intensity to full-width at half-maximum with the in which hi is the ith pixel's height, h is the total mean height, and N is the number of image pixels. One of the problems of R rms evaluation method is that it does not give information about height scale. Therefore, the PSD method is replaced with it which estimates the variation of roughness with length scale. Moreover, FFT obtains frequency distribution over the whole range of frequency and ends to the PSD function of | F(x, y) | 2 where F(x, y) is the FFT coefficients for the digitized surface profile z(x, y). Angular average of | F(x, y) | 2 of a 2D isotropic surface results in PSD where the radius at reciprocal space is considered as f. PSD normalization ends to units of (length) 4 (Jiang, Hall, Ho, & Morin, 2005).
PSD versus spatial frequency or wavelength gives valuable information about fractals and microstructures which characterizes the surface better than RMS roughness (Sahoo, Thakur, Senthilkumar, Bhattacharyya, & Das, 2003) and can be derived via several types of method including surface profiles measurements by a mechanical or optical profiler or AFM data (Elson & Bennett, 1995). PSD function in this work follows the equation below (Ferre-Borrull, Duparre´, & Quesnel, 2001): where S 2 denotes 2D PSD, L 2 is the surface area, N is data points per line which is considered to be 181 in the present work, and Zmn is the surface height function at (m, n) position. fx and fy are the spatial frequency of x and y directions, respectively. ΔL = L/N defines sampling distance (Martínez, Carvajal, Abad, & Colchero, 2012). Going to polar coordinates in frequency space angular averaging ({φ}), we have: and the dependency of PSD to f is: The inverse slope of the log-log plot of PSD versus high spatial frequency is defined as the power γ so that power PSD is the spatial length K to γ. γ. Here, L is considered to be 3.0 μm with the sample rate of 3Mm −1 /2 = 30.12 μm −1 which specified the band- and frequency k is shown in the following equation and the slope of a Log S(k)-Log k plot lead to fractal limitations (Solaymani, Ghaderi, & Nezafat, 2012) S Moreover, the relation between fractal dimension and the slope of log-log plot which is presented by β is as follows: where D determines the irregulations of surface at different scales.  Table 2 and the PSD plots of each hard tissue are shown in Figure 3 calculated through the FFT algorithm. Typical features of all PSD plots include a plateau with low spatial frequency along with an inverse slope with high spatial frequency. k-correlation is applied to investigate plot features (Sahoo et al., 2003) which is defined as PSD ABC for f as the spatial frequency as below: in which "A" is the magnitude of low f defined by the surface height (Palasantzas, 1993), "B" is the "knee" position obtained from the slope of a line which connect two points on the surface of dental tissue, and "C" is the slope at high spatial frequency with the constant value which refer to the nature of roughness.
Region I in Figure 3 refers to the region with high spatial frequency. As can be seen, the PSD of enamel out and cementum are compared in Figure 3(a) and the comparison of PSD of interenamel and dentin is carried out in Figure 3 (b). Despite low-frequency and mid-frequency regions in which the shape of all curves are almost similar, there are some differences in the shape of these curves in Region I which reflects surface information. It should be noticed that the majority of energy is in the low-frequency and mid-frequency regions.
In addition, fine structures with complexity along with increasing fractal dimension (D) are observable. The curves of the log PSD have different shapes in the high-frequency range of films with different hard tissues of molar tooth. The decrease of slope at high-frequency region makes an increase in fractal dimension (Shaheen & Ruzic, 1993). D in Equation (6) is summarized in Table 2 for each part of dental tissue containing interenamel, enamel out, dentin, and cementum and are compared together. It is deduced that for frequency region I, by moving from enamel out to cementum and from interenamel to dentin, the fractal dimension is decreased.
The slope variation (D) of the log-log diagram of PSD to frequency for typical enamel out is presented in Figure 4 for three frequency regions, which confirms the dependency of fractal dimension to frequency, spectral density, and spectrum region. Fractal exponents are extracted from mentioned log-log plot in Figure 4 and are summarized F I G U R E 3 Comparing the spectral plots of the log power spectral density-log frequency of (a) enamel out and cementum, and (b) interenamel and dentin [Color figure can be viewed at wileyonlinelibrary.com] in Table 3. Moreover, lines I and II are the slopes of one and two log cycle of data, respectively, while the best fitted line is line III. Slope variation strongly depends on the data size, frequency, and spectral range (Solaymani et al., 2012). Therefore, in enamel out, increasing frequency lead to fractal dimension increase which are the same for interenamel, dentin, and cementum which are summarized in Tables 4-5, and 6, respectively.
In addition, similar Raman spectra of enamel and cementum in Figure 5 confirm their highly disordered and complex nature based on the variety of wide lines with more intensity in enamel. PO 4 functional group at 960 cm −1 is surrounded by peaks with weaker intensity (Kirchner, Edwards, Lucy, & Pollard, 1997). Here, there is a wide band from 900 to 1,100 cm −1 in cementum. However, in the Raman spectra of enamel, a double feature at 976 cm −1 is related to PO 4 vibrations, and 1,040 cm −1 assigned to out-of-phase vibrations of phosphate.
The similarities of phosphate bands in the Raman spectra are not in agreement with their obvious differences through XRD pattern which might be due to the anisotropy of scattering light during the experimental procedure.
Moreover, the Raman peaks of amides, organic materials such as peptides (C O bonds), and amino acids (N H bonds) can be seen at 1600-1700, 1,240, and 1,280-1,390 cm −1 , respectively.

| CONCLUSIONS
In the present work, morphology and structural properties of the hard tissue of the molar tooth containing enamel, dentin, and cementum were studied. Compared with enamel, as the hardest tissue with sharpest peaks in XRD, dentin was softer. Besides, cementum with thick XRD peaks has the poorest crystallinity. Raman spectra extract similar spectra for cementum and enamel which confirm similar types of composition with different percentage of its content and are in good agreement with EDX results. Surface characterization as significant criteria affected by external factors was also taken into consideration for different tissues based on PSD analysis and fractal dimension through AFM images. 3D AFM images and surface parameters of enamel out, interenamel, dentin, and cementum gave a precise picture of the surface and makes the precise evaluation of surface roughness parameters which led to direct relation between the slope of log PSD-log frequency and the fractal dimensions.

CONFLICT OF INTEREST
The authors report no conflict of interests. The authors alone are responsible for the content and writing of the paper.