Introduction

The tooth represents one of the most interesting tissues in the human body, as to the performances in terms of material response [1]. This is also the case thanks to the stability of the adult teeth [2], which present a large mineralized portion in terms of their constituent materials and organization [3]. In this respect, the teeth are similar to bones, yet their degree of mineralization is higher, and, even if they are also live organs—which is especially clear when considering their interfaces at the periodontum ligament. The main teeth function is connected to the mechanical properties, mainly the elastic ones—stiffness and hardness [4]. This observation does not intend to minimize the effect of the biological interactions of the teeth with the body as well as with the environment. Nevertheless, for many applications still a simplified field of material-science may successfully apply to the investigation of their properties, which is also functional to the identification of appropriate substituting materials i.e., implants (metal/ceramics for the crowns and titanium alloy for the screws replacing the roots) and restorations (glass-monomer cements [gic] and resin composites) [5,6,7].

Several studies have already been carried out on the materials constituting the teeth, over the past decades, with the most disparate techniques, yet based mainly on a combination of microscopy and elastic characterization [8, 9]. Enamel, the universally recognized hardest and stiffest material in our body is composed of both organic and inorganic phase [10]. Dentin is obviously less stiff and plays a support role in buffering biting forces [2] that comprises a higher amount of organic phase [11]. For dentin, the situation is more difficult, since the material is a porous composite, and therefore a correct characterization should account for the porosity. The dentin microstructure consists of dentinal tubules that radiate outward through its volume, from the pulpal chamber to the exterior enamel or cementum [2]. Cementum is the last hard tissue of the tooth, which covers the surface in the sub-gingival region, with the compact structure like enamel (i.e., non-porous). However, it has the highest percentage of organic materials compared to the other regions [12].

On the other hand, different experimental and theoretical studies have reported the materials constituting human teeth, but there still remain many questions at basic levels about the structure–property relationship of these 3-D bio-structures across the length scales of the human teeth structural architecture [13,14,15,16,17].

Several recent nano-technological research lines emerge in dentistry that can provide new innovative ideas and concepts for solving pathological mechanisms and lead to novel therapeutic orientations [18,19,20]. In this context the 3-D shape models can be used to design, model, simulate, and visualize the teeth 3-D geometry and to help in computer-aided diagnosis [21,22,23].

Although the 3-D surface morphology of the human teeth is an extremely irregular and complex bio-structure with fine anatomic detail and surface texture, it can be analyzed quantitatively by applying the stereometric [24,25,26], fractal [27,28,29] or multifractal [17, 20] analyses of AFM micrograph samples.

The aim of this research was to examine the 3-D micromorphology of human canine teeth materials using AFM images and multifractal analysis.

Experimental details

Ten extracted permanent canine teeth of 40 year old men were collected for the experiment. Because of their similarity in their results, we represent the results of one of these specimens.

As the first step, all samples were placed in saline and brought to the laboratory. They were then brushed with standard dentifrice and tooth brush and kept in acetone and alcohol baths to remove their impurities ultrasonically. After drying them in air, all samples were cross-sectioned so that we were able to access to the inner parts.

AFM (Veeco, Santa Barbara, CA) images were applied to provide quantitative data about surface roughness and morphology for studying the structural topography of enamel, dentin, and cementum in the healthy tooth. The room temperature and ambient pressure was the same for all steps of experiments as 297 ± 1 K and 50 ± 1%, respectively.

Multifractal analysis of the 3-D surface micro-texture

The analysis of surface topography with the fractal method attracts a lot of interest in the field of biomaterials surface [18,19,20,21,22] since it is considered as a useful tool for analysing irregular and complex structures of various types of biomaterials [26,27,28].

For a 3-D fractal surface, the statistical index D is defined as fractal dimension with 2 ≤ D ≤ 3 which extracts the complexity of structure [17]. It should be noted that higher D values refer to the irregular surface with more complexity since it gives information about non-differentiability and self-similarity mathematically [17, 22].

However, when fractal analysis cannot describe surface topography due to its complexity, multifractal analysis is the most appropriate tool for characterization of surface morphology [30,31,32,33,34].

By considering q as the order moment with real values from − ∞ (less dense areas) to + ∞ (dense areas), Dq and α(q) represent information about fractal/multifractal geometry and are measured as the generalized dimension of Hölder exponents of q while f(α) refer to singularity spectrum. Two multifractal parameters are related by the Legendre transformation [32,33,34].

The generalized fractal dimensions Dq can be computed using the following formula [17]:

$$D_{q} = \tau (q)/\left( {q{-} 1} \right)$$
(1)

where τ(q) is the mass (or correlation) exponent of the qth order; and q is the order moment.

The multifractal spectrum function can be expressed as [17]:

$$f\left( {\alpha \left( q \right)} \right) \, = q\alpha (q){-}\tau \left( q \right)$$
(2)

where α(q) = dτ(q)/dq. αq represents Hölder exponents of the qth order moment.

On the other hand, Dq for q = 0, 1, 2 is known as the capacity, information entropy, and correlation dimensions, respectively. The relation between these parameters is D0 > D1 > D2. In addition, αmax = D− ∞ and αmin = D+ ∞ for q = ± ∞ with the width spectrum of Δα = αmax − αmin, and heights difference of Δf = f(αmin)− f(αmax) [32,33,34].

When Δf < 0 and Δf > 0, the fragments are explained by the low and high probability value predominate, respectively. The left arm (positive q) and right arm (negative q) of multifractal spectrum are related to irregular and flat areas, respectively [35]. By decreasing the value Dq affected by increase of q, multifractal structure is created which demonstrated the dependency of f(α) to α and Dq to q by box counting method [17].

Statistical analysis

The GraphPad InStat version 3.20 computer software package (GraphPad, San Diego, CA, USA) was applied here to perform statistical investigations. Meanwhile, Kolmogorov–Smirnov test was applied to extract normal distributions. Moreover, T test and Mann–Whitney U test were used to compare various areas in typical sample and reveal statistical differences, respectively.

Results and discussion

Scanning electron microscopy (SEM) images shown in Fig. 1 reveal the presence of granular structures on different hard dental tissues. The sizes of these grains are about 70–150 nm.

Fig. 1
figure 1

SEM images of: a enamel, b inter enamel, c inter dentin, and d cementum

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The representative 3-D AFM images for 1 µm × 1 μm scanning square areas of hard dental tissue of the human canine teeth are shown in Fig. 2.

Fig. 2
figure 2

AFM micrographs of the human canine teeth materials: a enamel, b inter enamel, c inter dentin, and d cementum

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The functional dependences R2-error versus q are shown in Fig. 3. After small values of R2-error in low values of q, it is rapidly raised. For q > − 1.2, fractal properties exist for all samples.

Fig. 3
figure 3

The graphs of coefficient R2-error versus q for all dental tissue

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Figure 4 shows multifractal analysis based on the box-counting method which determines functional dependency of f(α) to α and q to Dq on each region of interest (ROI) (where − 20 ≤ q ≤ 20 for successive 1.0 steps).

Fig. 4
figure 4

Generalized a fractal and b multifractal spectrum for each tissue

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AFM images of each tissue exhibit multifractal nature of dental surface along with non-homogeneous nature.

For the surfaces of the human canine teeth, different fractal dimensions follows D0 > D1 > D2 (Fig. 4a) which indicates the non-uniform features of 3-D surfaces. Figure 4b illustrates the asymmetric non-linear curve of f(q,r) −α(q,r) with two arms and Δα width.

All the values of Δα > 0 and Δf > 0 (Tables 1, 2, 3 and 4).

Table 1 The values of Dq, R2-error, f(α) for the surfaces of enamel on square areas of 1 μm × 1 μm
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Table 2 The values of Dq, R2-error, f(α) for the surfaces of inter dentin on square areas of 1 μm × 1 μm
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Table 3 The values of Dq, R2-error, f(α) for the surfaces of inter enamel on square areas of 1 μm × 1 μm
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Table 4 The values of Dq, R2-error, f(α) for the surfaces of cementum on square areas of 1 μm × 1 μm
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Compared to other tissue, longer right shoulder of f(α) along with the most irregular topography (Δα = 2.8361) are illustrated in inter dentin. On the other hand, the most regular topography was observed in cementum (Δα = 2.6804) (according Table 5).

Table 5 The multifractal width spectrum, Δα = αmax − αmin and the spectrum arms’ height Δf = f(αmin)− f(αmax)
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A similar situation is observed for values of Δf: the maximum value Δf = 0.7859 is illustrated in inter dentin and the minimum value of Δf = 0.3028 is observed in cementum (according Table 5).

Due to increasing surface roughness, the f(q,r) −α(q,r) curve deviation and hence, the inhomogeneity are increased. Larger values of Δα end to more differences between the lowest and highest probability along with wider probability distributions [35].

The Δf = f(αmin) − f(αmax) is considered as the ratio of maximum-to-minimum probability. Here, the negative values of Δf indicate the fragments explained by low probability value predominate. Moreover, significant correlation (P < 0.05) confirmed to accuracy of multifractal theory in estimating surface morphology which reveal non-uniform surfaces depend to human teeth materials. In addition, statistical parameters in Tables 1, 2, 3 and 4 are used to represent a powerful complement for 3-D surface morphology investigations more explicitly.

Conclusions

The micromorphology of human canine teeth materials can be evaluated by multifractal analysis and revealed the multifractal geometry of samples at nanometer scale. The results represented here are new insights for the understanding of the structure–properties relationships (local composition, crystalline structure, local mechanical information) and to study of various physical phenomena at the nanoscale scale, for both experimental and theoretical perspectives.

These models provide a basis for further research and analysis with 3-D modern mathematical methods for the simulation of mechanical and physical phenomena at different length and time scales, such as fracture and cracks evolution of nanocrystalline human canine teeth materials for fatigue loading using computer graphics.