Regulated fracture in tooth enamel: A nanotechnological strategy from nature

Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Regulated fracture in tooth enamel: A nanotechnological strategy from nature Elnaz Ghadimi a,1, Hazem Eimar a,1, Jun Song b, Benedetto Marelli b, Ovidiu Ciobanu a, Mohamed-Nur Abdallah a, Christoph Stähli b, Showan N. Nazhat b, Hojatollah Vali a, Faleh Tamimi a,n a b

Faculty of Dentistry, McGill University, Montreal, QC, Canada H3A 0C7 Department of Mining and Materials Engineering, McGill University, Montreal, QC, Canada H3A 2A7

art ic l e i nf o

a b s t r a c t

Article history: Accepted 7 April 2014

Tooth enamel is a very brittle material; however it has the ability to sustain cracks without suffering catastrophic failure throughout the lifetime of mechanical function. We propose that the nanostructure of enamel can play a significant role in defining its unique mechanical properties. Accordingly we analyzed the nanostructure and chemical composition of a group of teeth, and correlated it with the crack resistance of the same teeth. Here we show how the dimensions of apatite nanocrystals in enamel can affect its resistance to crack propagation. We conclude that the aspect ratio of apatite nanocrystals in enamel determines its resistance to crack propagation. According to this finding, we proposed a new model based on the Hall–Petch theory that accurately predicts crack propagation in enamel. Our new biomechanical model of enamel is the first model that can successfully explain the observed variations in the behavior of crack propagation of tooth enamel among different humans. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Enamel Microhardness Crack propagation Apatite Crystal size

1. Introduction Tooth enamel is the most highly mineralized and hardest tissue in mammals (Chai et al., 2009; Imbeni et al., 2005; Xu et al., 1998). Enamel inorganic consists of crystal key-hole shape structures composed of prisms (
6–8 mm) that are made of carbonate apatite (CAP) nanocrystals (Cui and Ge, 2007). Enamel organic is highly birefringent and is mainly composed of proteins and minor amounts of proteoglycans and lipoids that fill the spaces between the crystals (Cerny et al., 1996). Enamel suffers from continuous mechanical stress that causes the formation of cracks favoring bacteria growth, caries and tooth fracture (Huang et al., 2010; Imbeni et al., 2005). Enamel has certain damage tolerance to sustain cracks (Chai et al., 2009) that could be an important factor in the adaptation of tooth to diet in evolution (Ang et al., 2010; Barani et al., 2011; Janis and Fortelius, 1988). It is also of interest for biomaterials research in the development of biomimetic materials (Ang et al., 2010). It has been suggested that geometrical and microstructural characteristics of enamel have an

n Correspondence to: McGill University, Faculty of Dentistry, Strathcona Anatomy & Dent, 3640 University Street, Montreal, QC, Canada H3A 0C7. Tel.: þ 1 514 398 7203x09654; fax: þ 1 514 398 8900. E-mail address: [email protected] (F. Tamimi). 1 Both authors contributed equally to this work.

effect on its damage tolerance (Ang et al., 2010; Bajaj and Arola, 2009; Chai et al., 2009; Imbeni et al., 2005; Lawn and Lee, 2009; O’Brien et al., 2013; Xu et al., 1998). The fact that tooth enamel is tougher than geologic-hydroxyapatite seems to indicate that the specific characteristics of enamel such as the organic content, enamel prisms and crystals might have an effect on defining its mechanical properties (An et al., 2012; He and Swain, 2007; White et al., 2001; Yahyazadehfar et al., 2013). Removal of enamel organic content reduces its fracture resistance (Baldassarri et al., 2008; Zheng et al., 2013). Prisms degree of decussation and its crystal orientation affect tooth enamel mechanical properties (An et al., 2012; Bajaj and Arola, 2009; Chai et al., 2009; Xu et al., 1998; Yahyazadehfar et al., 2013). However, the specific contribution of enamel crystal size on crack propagation remains largely unknown. Crack propagation in polycrystalline materials is known to be regulated by their crystallographic dimensions. Increasing the crystal size in polycrystalline materials results in lower resistance to crack propagation (Mercer and Soboyejo, 1996; Wang and Shaw, 2009; Yusheng et al., 2004; Zhou et al., 2011), because cracks can propagate more easily around bigger crystals than around smaller ones (Yusheng et al., 2004). We hypothesize that crack propagation in enamel might be influenced by its crystallographic dimensions. This study was designed to analyze the associations of enamel crystallographic

http://dx.doi.org/10.1016/j.jbiomech.2014.04.005 0021-9290/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Ghadimi, E., et al., Regulated fracture in tooth enamel: A nanotechnological strategy from nature. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.04.005i

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nanostructure and chemical content with crack propagation in human teeth.

Crystallinity index of tooth enamel samples was analyzed using a Raman spectrometer coupled with a 785 nm diode laser (Senterra, Bruker, Karlsruhe, Germany) and an Olympus BX51 microscope (Olympus, Melville, NY). Seven different spots were analyzed half-way between DEJ and outer enamel surface. Crystallinity index was quantified relying on the bandwidth at FWHM of the phosphate peak (ν1PO4) at
960 cm  1 (Eimar et al., 2012b).

2. Materials and methods 2.1. Study sample After obtaining ethical approval from McGill University Health Center ethicalcommittee, 36 extracted human upper-anterior teeth were collected from McGill Undergraduate-Dental Clinic. Teeth were cleaned and stored as previously described (Eimar et al., 2012a, 2011; Ghadimi et al., 2013).

2.2. Hardness and enamel crack propagation A sagittal section was obtained from each tooth and fixed in clear resin (DP-Ortho-F, DenPlus, Montreal, QC). In our study and similar to previous studies (Baldassarri et al., 2008; Hassan et al., 1981; Hayashi-Sakai et al., 2012; Park et al., 2008a; Zheng et al., 2013), we evaluated cracks propagation in enamel using Vickers indenter (Clark-CM100AT, HT-CM-95605, Shawnee Mission, KS) since it provides direct measurements of cracks without causing a catastrophic failing of tooth enamel (Fett and Munz, 2006). Indentations were applied on tooth enamel incisal third because enamel is thicker there than areas cervical two thirds and its prisms follow roughly a straight path from the DEJ towards the tooth surface (Roberson et al., 2002). All pyramidshape indentation were conducted in regions that are not close to DEJ or tooth surfaces since previous studies have shown that these regions have toughening effects that resist cracks propagation (Imbeni et al., 2005; Jia and Xuan, 2012). The diagonals (d1 and d2) of each indentation were parallel and perpendicular to DEJ, respectively. For hardness, 7 indentations of 100 g with 10 s loading time and a minimum distance of 50 μm between the successive indentations were conducted. This low level of load was applied to enhance the efficiency in measuring enamel hardness by preventing microcrack formation (Quinn and Quinn, 1997). For crack measurement, another high load (500 g) 7 indentations with 20 s loading time and a minimum distance of 100 μm between successive indentations were applied. Each indentation created measurable cracks that propagated from its corners. Then, samples were sputter-coated with gold and images of the cracks were captured at 500  magnification using a low vacuum VP-SEM (Hitachi S-3000N VP, Japan) to minimize the influence of dehydration on crack growth (Kruzic and Ritchie, 2008). We also followed a previous protocol for sample dehydration conducted on bone in which it was demonstrated that the followed protocol did not influence the crack behavior (Burr and Hooser, 1995; Mullins et al., 2008). Crack length was measured using the ImageJ software (US National Institutes of Health, Bethesda, MD). The average crack length for each indentation was calculated by summing up the length of cracks and dividing by the number of cracks (Ang et al., 2011; Chicot et al., 2009). Two kinds of cracks were observed around each indentation, cracks emanating from the indenter-pyramid-diagonal (d1) that was parallel to DEJ and cracks originated at the indenter-pyramid-diagonal (d2) that was perpendicular to DEJ. The average of the cracks lengths extended from d1 and d2 were calculated separately.

2.3. XRD

kλ β cos θ

2.5. FTIR Enamel chemical composition was investigated by a Perkin-Elmer FTIR Spotlight-400 (Waltham, MA) equipped with an ATR imaging accessory. The machine was adjusted at 6.25  6.25 mm pixel size, and 64 spectra were accumulated within a surface of 50  50 mm entirely inside the enamel region. FTIR studies were carried out in the range 750–1800 cm  1 with a spectral resolution of 2 cm  1. Collected spectra were normalized according to the absorbance of ν3PO4 at 1013 cm  1. Organic content of enamel was estimated from the Amide I-to-ν3PO4 ratio (Bartlett et al., 2004). Carbonate content was estimated from the following ratios of ν2CO3 type A (
878 cm  1) and B (
872 cm  1) to the ν3PO4 and ν1PO4 (
960 cm  1) absorption bands (Eimar et al., 2012a). 2.6. Statistical analysis The data obtained was used to analyze the correlation of crack propagation within the tooth enamel with the characteristics of enamel apatite crystals and enamel protein content. The correlation coefficient “R”, the regression coefficient “B” and the significance of the correlation “P” were calculated for each correlation analysis. The statistical significance was set at Po 0.05.

3. Results and discussion 3.1. Enamel crystallographic dimension, obtained by XRD, and protein content, obtained by FTIR In this study, XRD data indicated that enamel apatite crystal size ranged between 9.4 and 22.2 nm along the a-axis and between 17.0 and 28.1 nm along the c-axis (Supplementary information, Fig. S2a and b). The crystal size dimensions reported in our study are similar to those measured by dark field electron microscopy (Grove et al., 1972). Similar to previous studies, it was found that the length of crystal lattice parameters along the a-axis and c-axis were
9.43 Å and
6.86 Å, respectively (Eimar et al., 2012a; Glas and Omnell, 1960; Legeros et al., 1983) (Supplementary information, Fig. S2c and d). The protein content of the tooth enamel was estimated by FTIR spectroscopy. The relative protein content was highly varied among the examined teeth and followed a distribution that was not normal (Supplementary information, Fig. S3). 3.2. Cracks propagation

X-ray diffraction (XRD) (D8-Discover/GADDS, Bruker, Karlsruhe, Germany) was used to determine the crystallographic dimension of enamel apatite as previously described (Eimar et al., 2012a). Average crystal dimensions along c-axis and a-axis for each enamel sample were calculated using Scherrer's-formula on the (0 0 2) and (3 1 0) Bragg peaks. D¼

2.4. Raman spectroscopy

ð1Þ

where D is the average diameter, k is the shape factor, λ is the X-ray wavelength β is the line broadening at half the maximum intensity (FWHM) and θ is the Bragg angle. Enamel crystal cell lattice parameters, a-axis and c-axis, were calculated from the XRD (002) and (300) Bragg peaks, relying on the following equation (Hong et al., 2006): ! !
 2 2 2 1 4 h þ hk þ k l ¼ þ 2 ð2Þ  2 2 3 a c d where d is the spacing between adjacent planes in the crystal, hkl are the miller indices that are the reciprocal intercepts of the plane on the unit cell axes, a is the a-axis and c is the c-axis.

The average crack length in each enamel indentation varied between 10.9 and 66.93 mm (mean ¼31.08 7 15.69 mm) (Supplementary information, Fig. S1). Average length of cracks parallel to d1 varied between 0 and 57.88 mm (mean¼ 19.95 715.17 mm) and the average length of the cracks parallel to d2 varied between 21.83 and 92.14 mm (mean¼ 42.147 19.18 mm). Differences in crack propagation along the directions of indenter pyramid-two-diagonals can be attributed to the anisotropic structure of the enamel detailed below. Enamel prisms are arranged in rows originating from DEJ to tooth surface (Daculsi et al., 1984), and are bound together across weak interfaces consisting of organic material (e.g., protein-rich rod sheaths) (Bajaj and Arola, 2009) as shown in Fig. 1c. During enamel fracture, cracks propagate along these weak interfaces following a straight path along the direction parallel to the prisms (indicated by red line in Fig. 1d), and a zigzag/bifurcated path perpendicular to the prisms (indicated by the blue line in Fig. 1d) (Bajaj and Arola, 2009;

Please cite this article as: Ghadimi, E., et al., Regulated fracture in tooth enamel: A nanotechnological strategy from nature. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.04.005i

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us to confirm the correlations obtained by XRD using different analytical techniques. It was found that there was a strong negative correlation between crystallinity index and crystal dimension along the c-axis (P-value ¼0.007, R ¼  0.47, Fig. 3) validating our XRD analysis. Crystallinity index had a strong and positive correlation with tooth enamel hardness (P-value ¼ 0.0004, R¼0.57, Fig. 4a) confirming our XRD results. In our study, we investigated the correlation between FWHM and crack length in tooth enamel samples. Enamel crystallinity index is inversely correlated with average crack length, average length of cracks perpendicular to DEJ and average length of cracks parallel to DEJ, respectively (P-value ¼0.014, R¼0.41; P-value ¼0.06, R¼0.32; P-value ¼0.007, R ¼0.44, respectively) (Fig. 4b–d).

3.5. Crystallographic dimensions and toughness For an indentation fracture with crack propagation along the direction η, the corresponding fracture toughness K η and crack length cη is related as follows (Bajaj and Arola, 2009; Lawn, 1993):
1=2 Kη 1 P pffiffiffi ¼ ; H cη 3=2 χ0 E

Fig. 1. (a) Diagram depicting an indentation on the tooth enamel. (b) The SEM image of Vicker's microindentation on enamel, showing cracks emanating from the indenter-pyramid-diagonal (d1) that was parallel to DEJ and cracks originated at the indenter-pyramid-diagonal (d2) that was perpendicular to DEJ. (c) 3-Dimensional diagram of tooth prisms showing the crack propagations parallel to the DEJ and perpendicular to the DEJ. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Bechtle et al., 2010). As a result, tooth enamel is 1.4 times tougher along the direction perpendicular to the prisms than along the direction parallel to the prisms (White et al., 2001). Consequently, upon indentation of tooth enamel, we could confirm that cracks along the direction d1, which is parallel to DEJ and roughly perpendicular to the prisms were shorter than cracks along the direction d2 which is perpendicular to DEJ and roughly parallel to the prisms (Hassan et al., 1981; He and Swain, 2007; Xu et al., 1998; Yahyazadehfar et al., 2013). 3.3. Crystallographic dimensions and cracks propagation Simple linear regression has revealed a significant correlation between crystal dimension along the c-axis (XRD) and hardness in tooth enamel (P-value ¼0.033, R¼  0.71) (Fig. 2a). The correlations between crystal dimension along the c-axis and the following, the average crack length, average length of cracks perpendicular to DEJ and average length of cracks parallel to DEJ, were positive (P-value ¼ 0.04, R ¼0.69; P-value ¼0.01, R¼0.47; P-value ¼0.013, R ¼0.45, respectively) (Fig. 2b–d): 3.4. Crystallinity and cracks propagation To confirm the correlation between crystal dimension (measured by XRD) and mechanical properties of enamel, we measured the crystallinity index of the enamel samples by Raman spectroscopy (Supplementary information, Fig. S2e) and correlated it with the crystal dimension along the c-axis assessed with XRD as well as hardness and crack length of the enamel samples. This allowed

η ¼ ? or J ;

ð3Þ

where P is the force on indenter, E and H are the elastic modulus and hardness of the enamel, and χ0 is a dimensionless constant that depends on the nature of the deformation. The direction η is perpendicular to DEJ ðη ¼ ? Þ or parallel to DEJ (η ¼||). Considering that the elastic modulus of enamel varies through the thickness, E represents the effective elastic modulus averaged through the indentation thickness (Park et al., 2008b). In the present study, the parameter E is not directly measured, and for simplicity we assume it to be constant. pffiffiffi From Eq. (3) K η (normalized by χ 0 E) is directly measurable from experiments. Meanwhile, K η is related to the energy pfracture ffiffiffiffiffiffi of the protein matrix G0 through E  1=2 K η ¼ αη βη G0 with αη and βη being toughening prefactors (Lawn, 1993; Ravi-Chandar, 2004) due to crack deflection and branching, respectively. From this we can note that the fracture toughness is orientation dependent, and for the two directions (i.e., η ¼ ? or ||) considered the difference resides only in αηβη. For the straight crack propagation parallel to DEJ, α┴ ¼ β┴ ¼ 1. For the zigzag/bifurcated propagation perpendicular to DEJ, α|| is a function of the crack deflection angle θ (Lawn, 1993) as a J ¼ ½ cos 2 ðθ=2Þ  1 where θ can be approximately related to the aspect ratio of the apatite nanocrystal through θ ¼ arctan ðdc =da Þ with dc and da being the crystal dimensions along the c-axis and a-axis respectively. A schematic is provided in the Supplementary materials to illustrate how the crack deflection angle is approximated based on the apatite geometry. The α|| is calculated to be between 1.17 and 1.47 for the tooth samples examined. On the other hand, β|| usually ranges from 1 to 3 (Ravi-Chandar, 2004). Therefore the fracture toughness would be 1.17 to 4.41 times higher along η ¼|| than η ¼ ?. In Fig. 5a, we plot K J vs. K ? , from which we can see that the scatter data fall beautifully in the region bounded by K J ¼ 1:17K ? and K J ¼ 4:41K ? . This is exactly what we expect from the above analysis, confirming that crack propagation is highly anisotropic in enamel. The anisotropy can be nicely captured by the toughening arising from crack deflection and branching. In the text that follows, for simplicity we focus on the crack propagation perpendicular to DEJ sinceα ? β ? ¼ 1 is precisely known. pffiffiffiffiffiffi Using the relation E  1=2 K ? ¼ Go we can also rewrite Eq. (3) as

χ 0  2 G0 ¼

P2 ; Hc ? 3

ð4Þ

Please cite this article as: Ghadimi, E., et al., Regulated fracture in tooth enamel: A nanotechnological strategy from nature. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.04.005i

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Fig. 2. Plots of stratified data of crystallinity index vs. (a) enamel microhardness, (b) average crack length of tooth enamel, (c) average length of cracks perpendicular to DEJ, (d) average length of cracks parallel to DEJ.

3.6. Protein content and cracks propagation

Fig. 3. (a) Plot of crystal size along c-axis vs crystallinity index. (b) Plot of crystal size along c-axis vs protein content.

to eliminate the dependence on E (which is not measured in the experiment) and show the crack propagation in terms of the material properties H and G0. In a previous study, we had shown that Hall–Petch equation explained the association between enamel hardness and its crystallographic dimensions (Eimar et al., 2012a), which were similar to those observed in previous studies of polycrystalline materials (Hall, 1951; Tipper et al., 1953). In this study, we find the hardness data also follows a Hall–Petch type formula and can be fit to the crystal dimension, dc (in the unit of nm) along the c-axis (see Supplementary information, Fig. S6). H ¼ 0:84 þ 20:91  dc

 1=2

ðGPaÞ

ð5Þ

Simple linear regression was used to find the correlation between protein content and crystal size along the c-axis in tooth enamel. It was found that the correlation between them was significant (P-value o0.05, R¼  0.41) (Fig. 2b, Supplementary information Table S1). The correlations between protein content and the following, hardness, average length of cracks perpendicular to DEJ and average length of cracks parallel to DEJ, were not significant (Supplementary information, Fig. 5 and Table S2). Our findings indicated that even though the protein content, a known major contributor for enamel toughness (Baldassarri et al., 2008), was slightly associated with enamel crystal size, however, it seems that the variation in enamel protein content among the studied teeth was too little to reveal an association between the protein content and enamel toughness (Supplementary information Table S2). As seen in Eq. (4), the fracture energy of the protein matrix G0 (normalized by a general, non-material-dependent constant χ20) can be obtained from the experimental data, allowing us to quantitatively examine the role of protein along with its interplay with other parameters. In Fig. 5b G0 and the protein content measured from FTIR spectrum (i.e., amide I/ν3PO4, Fig. 3S) are  1=2 investigated as functions of dc . We can note that the fracture energy G0 increases as the crystal dimension dc decreases and can be empirically fitted as G0 ¼  98:9 þ553:8  dc

 1=2

ðχ 0 2  104 N=mÞ

ð6Þ

This trend in fracture energy is likely due to the presence of more organic content in enamel with crystals of smaller

Please cite this article as: Ghadimi, E., et al., Regulated fracture in tooth enamel: A nanotechnological strategy from nature. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.04.005i

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Fig. 4. Plots of crystal dimension along c-axis vs. (a) enamel microhardness, (b) average crack length of tooth enamel, (c) average length of cracks perpendicular to DEJ, (d) average length of cracks parallel to DEJ.

Fig. 5. (a) Plot of K ? vs. K||, for all the tooth samples examined. The dashed and dash-dotted lines correspond to the expected curves if the toughness perpendicular to enamel prisms (i.e., K ? ) is 1.17 and 4.41 times the toughness parallel to the enamel prisms (i.e., K||); (b) the stratified data of measured fracture energy of the protein matrix G0 normalized by a constant χ20 (solid square symbols) and the organic content (open square symbols in the inserted figure) vs. the inverse square root of the crystal dimension along c-axis dc. The red line is the curve obtain from empirically linear fitting of the G0 w.r.t. dc 1/2, while the dashed line in the inserted figure is drawn to guide the eye; (c) the stratified data of the observed (solid square symbols), and predicted values from Eq. (6) of the crack length vs. dc 1/2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 6. XRD analysis of 2 representative enamel samples depicts the variation in the crystal size: large (a) and small (b) crystal size. The red frames represent (002) Bragg peak used to calculate the crystal size along their c-axis. The red circle and the red two-sided arrow represent the top of (0 0 2) Bragg peak and the full width of half height (FWHH) that was used to determine enamel crystal size, respectively. Raman analysis of 2 representative enamel samples depicts the variation in the crystallinity index: large (c) and small (d) crystal size. (e) and (f) are 2-dimensional schematics of the crack propagation path illustrating the role of enamel crystal size on crack propagation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

dimension along the c-axis (Simmer and Hu, 2001) (see the insetin Fig. 5b). One particular interesting fact we can note from the above analysis is that both the hardness and the fracture energy are highly dependent on the underlying nanostructure, in particular the crystal dimension along the c-axis (i.e., Eqs. (5) and (6)). Consequently the cracking in enamel presumably would also depend on the crystal dimension. In our experiments we found that the crystal dimensions along the c-axis were significantly correlated to average crack length (see Supplementary information, Table S2 and Fig. S4). For instance, enamel samples composed of large CAP crystals demonstrated lower resistance to crack propagation compared to those composed of smaller CAP crystals

(Fig. 6). Below we show this trend can also be captured using a simple analytical formula. Eq. (4) shows that the crack length is explicitly related to the hardness H and fracture energy G0. Combing their dependences on the crystal dimension along the c-axis into Eq. (4), we arrive at 1

c ? ¼ ðχ 0  2 P  2 G0 HÞ  1=3  ½ð463:2dc þ3:3Þ  10  3   1=3 ðmmÞ

 1=2

 101:3dc

ð7Þ

Eq. (7) provides a quantitative prediction of the crack propagation in enamel directly from its nanostructure. The predicted crack length c┴ is plotted together with the stratified experimental data in Fig. 5c, showing excellent agreement.

Please cite this article as: Ghadimi, E., et al., Regulated fracture in tooth enamel: A nanotechnological strategy from nature. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.04.005i

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All models in the literature described enamel toughness based on enamel prisms orientation, crystal orientation or protein content (Bechtle et al., 2010; Hassan et al., 1981; Jia and Xuan, 2012). There are also models that describe the elastic behaviors of the tooth enamel during nanoindentation (Hainsworth et al., 1993; Jia et al., 2013; Li and Chou, 1997; Saha and Nix, 2002). The model formulated in our study is not only able to predict the behavior of crack propagation within enamel based on its crystallographic dimensions, but it is the first epidemiological study that can successfully explain the observed variations in the behavior of crack propagation of tooth enamel among different humans. In addition, our results could explain why the inner layer of the enamel that composed of small crystals is tougher than the outer enamel that is composed of relatively larger crystals (Bajaj and Arola, 2009; Imbeni et al., 2005; Jiang et al., 2005; LeGeros et al., 1996). Also, our new model provides explanation for previously unexplained observations such as why the tooth enamel is 3 times tougher than the geologic apatite (White et al., 2001). In fact, tooth enamel is highly anisotropic structure that consists of crystals that are smaller than geological apatite crystals (Boskey, 2007; White et al., 2001). Also, our study provides explanation why cracks and fractures in enamel are usually observed in older patient (de Noronha et al., 2012; Hartmann and Muller, 2004; Lawn and Lee, 2009; Zheng et al., 2013). Aging is associated with a decrease in protein and an increase in mineral contents of tooth enamel (Zheng et al., 2013), major contributors for toughness degradation with aging. Enamel crystals in permanent teeth grow larger with age (Legeros et al., 1983). Therefore, the age-regulated increase in the mean size of enamel crystals might be one of the reasons behind the reduced tooth damage tolerance in elder individuals. However, future studies are required to confirm this hypothesis. One limitation of this study is that the exact orientation of enamel prisms and crystals was not precisely measured in correspondence to cracks propagation. In fact, enamel prisms do not form straight-line from the pulp chamber to the tooth surface (An et al., 2012; Park et al., 2008b; Xu et al., 1998). However, our indentation measurements were conducted at the incisal third where the enamel rods follow roughly a straight path from the DEJ towards the tooth surface (Roberson et al., 2002). Moreover, even-though our study did not assess the precise orientation of enamel prisms, a strong association between cracks propagation and the size of enamel crystals existed. In our study, indentations were applied in the middle area of enamel layer, in which enamel prisms exhibit decussation that retards crack growth (Bajaj and Arola, 2009; Chai et al., 2009; Yahyazadehfar et al., 2013). Indeed, this fact would strengthen our results since we were able to observe the influence of enamel crystal size in the presence of a major regulator (which is the decussation of enamel prisms) of cracks propagation in enamel. However, future studies are required to determine the association of crack growth, size of enamel crystal and orientation of enamel prisms. In our study, cracks propagation within the tooth enamel was created with Vickers indenter. The use of this technique to precisely quantify the materials hardness and toughness has been questioned before due to the large testing error (25%) (Jia and Xuan, 2012; Kruzic et al., 2009; Kruzic and Ritchie, 2008; Munz, 2007). Even with this large testing error, we were able to detect the association between crystal dimensions and enamel hardness and crack propagation, which indicates that these associations were very strong. Moreover, in our study we minimized the testing error by calculating the average 7 indentations to assess the enamel hardness and cracks propagations for each tooth sample. The testing error using Vickers indenter in our study ranged between 2 and 11%.

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4. Conclusion Cracks propagation in tooth enamel is associated with the variability in its apatite crystal dimensions along the c-axis. This association between the crack length and the enamel crystal dimensions along the c-axis followed the Hall–Petch model.

Conflict of interest statement We declare that we have no conflict of interest.

Acknowledgements The authors would like to acknowledge the “Fondation de l’Ordre des dentistes du Québec” (FODQ), “Le Réseau de recherche en santé buccodentaire et osseuse” (RSBO), the Faculty of Dentistry of McGill University and Natural Sciences and Engineering Research Council of Canada (NSERC-Discovery; F.T.) for their financial support. We also thank Prof. E. Lopez Cabarcos for his valuable advices and constructive criticism.

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Please cite this article as: Ghadimi, E., et al., Regulated fracture in tooth enamel: A nanotechnological strategy from nature. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.04.005i