2.1

Finite elements models (FEM) generation

The solid model generation first step consisted in the definition of the volumes associated to different material properties of a human central maxillary incisor (CMI), the periodontal ligament (PDL) and the corresponding alveolar bone process. The external shape of the maxillary central incisor was obtained by laser-based 3D digitizing (Cyberware, Inc., Monterey, California, USA) of a plaster cast (Thanaka Manufacturer, Tokyo, Japan) while dentin and enamel internal volumes were generated using horizontal and vertical profiles from Wheeler’s anthropometric studies . The scanned and literature profiles were assembled in a 3D wire frame structure using a 3D CAD (Autocad 12, Autodesk, Inc., San Rafael, California, USA) and exported into a 3D parametric solid modeler (Pro-Engineering 16.0 Parametric Technologies, Needham, Massachusetts, USA). Fitting the horizontal and vertical profiles then generated the tooth different model volumes. The average anatomical dimensions of the bone in adults were generated according to literature data . The maxillary bone was modeled as a spongy core surrounded by 2.0 mm thick cortical bone. The buccal–lingual and the mesial–distal dimensions of bone were 7.3 mm and 13 mm respectively at the root apex level. An average thickness of 0.7 mm and 0.2 mm were considered for the alveolar cortex and for the periodontal ligament (PDL) respectively .

As described in a previous work , the finite element (FE) base model was obtained by importing the solid models into ANSYS rel. 9.0 FEM software (Ansys, Inc., Canonsburg, Pennsylvania, USA) using IGES format. The volumes were then redefined in the new environment and meshed with 8 nodes brick with 3 degrees of freedom per node , finally resulting in a 3D FE model made up of 311,623 elements and 53,902 nodes ( Fig. 1 ). The model was constrained at the top surface of the maxillary bone. Accuracy of the model elements was checked by convergence tests .

3D models of incisor’s alveolar process and incisor. The models are presented in exploded view (A) and in the assembled configuration used for the finite element analysis (B). In red is shown the elements network created to simulate the periodontal ligament.

After the set-up of the base model of the sound incisor, a second modeling step was performed to obtain the volumes of a tooth with a fractured incisal edge restored by mean of resin composite. Such model consists of 4 main volumes made leaving the same overall volume of the sound model: the dentin and enamel volumes geometrically re-configured for the restoration placement, the adhesive layer with an average thickness of 50 μm, the incisal margin made-up of composite resin ( Fig. 2 A) . The fracture area of the tooth was a plane modeled perpendicularly to the long axis of the tooth in the mesial–distal direction for all the tested models. Three limiting different angulations of the fracture plane in buccal–palatal direction were then modeled: the fracture plane perpendicular to the long axis in the buccal–palatal direction (90°); the fracture plane inclined bucco-palatally in apical–coronal direction (−30°); the fracture plane inclined palatal-buccally in apical–coronal direction (+30°) ( Fig. 2 B). The composite resin restored incisor volumes were meshed with the same type of elements and element size of the sound incisor base model. The alveolar bone structure configured for the sound incisor model was also coupled with the composite resin restored incisor models.

3D models representing the three types of tested fractured incisors configurations. (A) Exploded view of the 0° fracture with its restoration (dark yellow) and the adhesive layer with thickness of 10 μm (in light blue). (B) Models simulating the −30°, 0° and 30° fracture plane orientation with their respective composite restorations. In dark yellow is presented the composite restoration, in light blue is presented the adhesive interposed along the fracture–restoration interface.

2.2

Tested models

All the tested models were CMI with PDL and surrounding alveolar process.

• Model 0 (control model): Sound tooth with no fracture planes

• Model 1: Fracture plane perpendicular to the long axis of the tooth Fig. 2 B, restored with Gradia Forte.

• Model 2: Fracture plane +30° to the long axis of the tooth Fig. 2 B, restored with Gradia Forte.

• Model 3: Fracture plane −30° to the long axis of the tooth Fig. 2 B, restored with Gradia Forte.

• Model 4: Fracture plane perpendicular to the long axis of the tooth Fig. 2 B, restored with Gradia Direct Anterior.

• Model 5: Fracture plane +30° to the long axis of the tooth Fig. 2 B, restored with Gradia Direct Anterior.

• Model 6: Fracture plane −30° to the long axis of the tooth Fig. 2 B, restored with Gradia Direct Anterior.

2.3

Tissues material properties

Enamel and spongy bone were considered as isotropic linear materials. The Young’s modulus and Poisson’s ratio of spongy bone and enamel are showed in Table 1 .

Cortical bone was assumed orthotropic and divided into 2 regions, the buccal side and palatal side, according to the mean statistical mechanical properties reported for a dentate mandible by Schwartz-Dabney and Dechow . These data have already been used by the Authors for the experimental validation of a full mandible FEA model . The actual model subdivision is reported in Fig. 3 B . Each area has its own orthotropic constants values and orientation of the maximum stiffness direction with respect to the global coordinate system. Orthotropic elastic constants and orientation for buccal and palatal side of cortical bone are reported in Table 2 . Local orientation of the maximum stiffness ( E ₃ ) and the other two orthogonal stiffness directions ( E ₁ , E ₂ ) have been reproduced dividing the shell of external elements (compact bone structure) in orientation sites according to the proposed experimental mechanical characterization. The orientation of the maximum stiffness has been reproduced for each site by defining a local coordinate system and by orienting the site’s elements coordinate systems accordingly. The view of the orientation sites is reported in Fig. 3 B, the maximum stiffness ( E ₃ ) direction, which is expressed in degree and referred to the occlusal plane on buccal and palatal side, are reported in Table 2 . In all cortical site E ₁ direction is normal to the cortical surface.

(A) Areas where strain-gauges were positioned on the alveolar process of the dry skull. (B) Exploded view of the overall finite elements model showing elements subdivision of simulated tissues. The axis indicates the directions of the elastic moduli of cortical bone considered as an orthotropic material.

Dentin, as well, was considered orthotropic linear material with the maximum stiffness direction perpendicular to the axis of the tubules (as explicatively shown in Fig. 4 A for a simplified FEA model) . E ₁₁ and E ₃₃ represent the elastic moduli along the x and z -directions while G ₁₂ and G ₂₃ and ν ₂₁ and ν ₃₁ are the shear moduli and the Poisson’s ratios in the relative orthogonal planes. Dentin orthotropic mechanical properties are reported in Table 3 . In order to simulate the tubules orientation along the incisor model each element was oriented with respect to the defined local coordinate systems ( Fig. 4 A).

(A) Managing of dentin orthotropicity in an incisors simplified FEM. (B) PDL stress–strain curves at different straining rates adopted for the PDL FEM.

2.4

Time dependency of PDL

Available clinical and experimental evidences indicate that the response of the PDL is both elastic and viscous . Time dependency characterizes the viscosity in contrast with elasticity in which all deformations are instantaneous. It is commonly accepted that the PDL elastic component is non-linear and, recently, some authors demonstrated that its viscous component is also governed by a non-linear behavior . The effect of the elastic component non-linearity on the stress–strain curve is a stiffening at higher deformation levels; conversely, the effect of non-linearity of the viscous component on PDL stress–strain curves is a stiffening at higher straining rates, that could not be described by a constant strain-rate hardening parameter. The method used to account for the strain-rate dependency of stress required the definition of four visco-elastic models based on four different stress–strain curves performed at different straining rates . Literature data about bovine PDL stress–strain curves performed at different straining rates were used to set up the visco-elastic models; these models consist of “elastic modulus vs time” curves. Once the strain rate of a given stress–strain curve is known, it is possible to evaluate for a discrete number of its points a series of elastic moduli corresponding to the different time step of the performed experiment according to the material’s constitutive equation. In the present study, a “elastic modulus vs time” curve was evaluated for each stress–strain curve. Then, data were uploaded in the FEA software. A macro was set up in order to enable a time dependent analysis that computed at each step time the strain rate of a given element. The strain value and the corresponding strain rate were interpolated between the “nearest” visco-elastic models table to evaluate the corresponding stress value according to the following logic flux:

• 1

  n ← n + 1

• 2

  k ←0

• 3

  Increment load and time until load history ends→ΔFn, Δtn

• 4

  k ← k + 1

• 5

  Solve→

• 6

  Loop on element “ e ”

• 7

  Retrive

• 8

  Modify element Young modulus: Ee ← E ( )

• 9

  Solve→

• 10

  Check equilibrium:

• 11

  If equilibrium go to 4 then go to 1

Literature stress–strain curves at different straining rates are shown in Fig. 4 B. Although, the bovine PDL is wider than the human PDL, their compositions seem to be comparable . In previous studies , the bovine model has been considered as a promising approach to define the main parameters responsible for the mechanical behavior of human PDL .

The above algorithm and mechanical properties accounted for the response of PDL fibers in the direction parallel to fibers main axis (normal to incisor root dentin). PDL stiffness in the direction normal to fibers main axis (parallel to incisor root dentin) depending by PDL shear modulus ( G ) was evaluated through the procedure reported by Maceri et al. .

2.5

Restorative materials properties

The restoration volumes in Models 1–3 and Models 4–6 were coupled with the mechanical properties reported in Table 1 for Gradia Forte and Gradia Direct Anterior isotropic elastic composite resin, respectively.

2.6

Interfaces and adhesive layers

The adhesive layers along the interface between the incisor fracture surface and the restoration were modeled with a layer of brick solid elements with an average thickness of 10 μm . Adhesive layer interface elements were coupled to the mechanical properties of an adhesive material .

2.7

Dynamic loading conditions in FEA

In order to reproduce a dynamic loading effect of the experimental tests, a transient analysis with time integration was used in the FEA simulation. Each load reached the maximum assigned value through defined load step at different times.

2.7.1

Short term intrusive load

An intrusive displacement was calculated referring to a chewing activity of 70 cycles per minute ( Fig. 5 ) . Parfitt’s experimental test was reproduced in the FE environment by applying an intrusive load of 3 N. The intrusive load was gradually increased up to 3 N (0.5 N for each load step of 0.14 s) within an overall time interval of 0.85 s.

Loading conditions applied to the models. Black arrow: load applied in the in vivo experiments and on the incisor FE-model. Red arrow: load applied on the incisor FE-model for PDL model validation, 15 N in a time range of 75.00 s. Orange arrow: load applied on the incisor FE-model for PDL model validation, 3 N in a time range of 0.85 s.

2.7.2

Horizontal load

The horizontal load was gradually increased up to 15 N (1 N for each load step of 5 s) within an overall interval of 75 s ( Fig. 5 ). The displacement in the direction of the load was measured and compared to experimental data and literature computed data .

2.7.3

Study load

In order to simulate the tearing function in the final analysis, load was applied at 60° angle with tooth longitudinal axis on the palatal surface of the crown 2 mm apically to the incisal margin and progressively increased up to 50 N in a time interval of 0.5 s ( Fig. 5 ).

2.8

Dynamic experimental strain gauges measurements on cortical bone of a dry human maxilla

The FEA simulated case was reproduced on a real human dry maxillary bone.

Thin, flexible four rosette gauges (CEA-06-032UW-120, Vishy Measurements Group, Inc, NC, USA) with a cast polyimide backing and encapsulation featuring copper-coated solder tabs, Constantan alloy in self-temperature-compensated form at 6 ppm/°F, with UW linear grid and tab geometry with an active gage length 32 mils and 120 ohms resistance were attached with an adhesive (Type 200 Vishy Measurement Group, Inc, NC, USA) to the alveolar cortical bone in the incisor are of the maxilla on the palatal and facial side as indicated in Fig. 3 A. Switch-selectable potentiometer and Wheatstone bridge output circuits have been used. Deformations over the time were recorded by an acquisition data software (System 4000 with 20 input channels by Vishay Measurements Group, Inc, NC, USA) at 2 points per second.

The human maxilla was positioned in a dynamometric test machine INSTRON 4204 (INSTRON Inc., UK) and constrained on the orbital surface of the corpus and on the palatal surface of the palatal process. The maxilla is oriented in order that the INSTRON knife is positioned at 60° angle with central incisor longitudinal axis on the palatal surface of the crown 2 mm apically to the incisal margin. The load was increased up to 50 N in order to simulate a tearing function. The temperature and humidity of the room in which the measurements were performed were maintained at fixed levels by 24-h central air-conditioning. The measurements were started after establishing the stability of the electric circuits with approximately a 1-h warm-up. The experimental data were elaborated by rosette analysis using Microsoft Excel 2000 (Microsoft, Seattle, Wash). Maximum principal strain, minimum principal strain, maximum principal strain angle and minimum principal strain angle were calculated.

2.9

In vivo strain measurements for FE-models validation

The study protocol was approved by the institutional review board (IRB) of the University of Siena with clinical trial number CT01532947. The right of the enrolled participants have been protected by the IRB of the University of Siena, Italy, and written informed consent was provided by the participants

Ten healthy volunteers were selected according to the following inclusion criteria:

• –

  Presence of the central maxillary incisors.

• –

  Vitality of the CMI.

• –

  Absence of dental crowding involving the CMI.

• –

  Absence of diastemata involving the CMI.

• –

  Absence of decay on the CMI.

• –

  Absence of conservative and/or prosthetic restorations on the CMI.

• –

  Absence of gingivitis nearby the CMI.

• –

  Absence of periodontal disease to the periodontal tissue of CMI.

• –

  Absence of periodontal pockets involving the root and the alveolus of CMI.

• –

  Absence of attachment loss/alveolar support loss.

• –

  A physiological distance between the cement-enamel-junction (CEJ) and the alveolar ridge of the CMI.

• –

  A standardized periapical radiograph (70 kV and 0.08 s) was taken by mean of the long cone paralleling technique and incisal silicon index (o key) for each CMI to examine root dimensions; only CMI with a minimum crown-to-root ratio of 2/3 were included in this in vivo validation.

• –

  Teeth with root anomalies or root resorption were excluded from the study.

• –

  Only teeth with an average length of 22 ± 1 mm, buccal–lingual coronal dimension of 7 ± 1 mm and mesial–distal coronal dimensions of 9 ± 1 mm were chosen, according to Wheeler’s anthropometric studies .

• –

  The patients age ranged from 18 to 30 years old; only caucasic individuals were included in this in vivo study.

• –

  Absence of dental wear, erosion and abfraction of enamel. The absence of abfractions was detected by the trans-illumination test.

The apparatus was designed to record the surface strain of CMI crown subjected to a constant load. In each patient the strain-gauge was bonded in the central area of the clinical crown of the CMI. CMI was isolated with a rubber dam and its buccal aspect etched with ortho-phosporic acid. A linear strain-gauge self-compensated in temperature (C-980204, Micro group, Inc., Raleigh, North Carolina, USA) was bonded with a surgical cyanoacrylate (Histoacryl, Braun, Italy) on the buccal aspect of the CMIs ( Fig. 8 ). Rubber dam was removed after the strain-gauge bonding procedure.

Strain-gauge with a recording grid of 4 mm by 4 mm was connected to a digital strain measuring hardware designed for biomedical applications (Omicron-T, Battipaglia, Italy) interfaced to a personal computer equipped with a software providing data visualization and storage. The strain state was recorded as a function of time.

During the tests, the CMIs were subjected to 10 loading cycles. In each loading cycle, a force was applied on the buccal surface of the CMI nearby the incisal margin by a linear dynamometer (MD1000, Pesola, Italy) up to 5 N and then kept constant for 10 s. Reproducibility of the positioning of dynamometric gauge was guaranty by the use of a customized silicon index placed onto the buccal surface of the CMI at level of the incisal margin. Such silicon index was provided with a slot to place the dynamometric gauge in a standard position and direct contact with the buccal enamel at 2 mm apical to the incisal margin. The load was applied 90° with respect to the CMI main axis, in the palatal to buccal versus. The strain-gauge was debonded from the incisor crown after each single force application, and then glued again for the next force application. Finally, 10 measuring strain tests were recorded for each Patient. The strain state was reported as the mean of the 10 loading cycles.

Data were statistically analyzed with the software SPSS 15.0 (SPSS Inc. for Windows, Chicago, IL, USA). Having checked both normality of data distribution (Kolmogorov–Smirnov test) and homogeneity of variances (Levene test) of the two groups, the Student’s t -test for independent samples was adopted to assess the statistical significance of the difference between the subject that partake the experiments. The level of significance was set at α = 0.05 ( Table 4 ).

2.1

Finite elements models (FEM) generation

The solid model generation first step consisted in the definition of the volumes associated to different material properties of a human central maxillary incisor (CMI), the periodontal ligament (PDL) and the corresponding alveolar bone process. The external shape of the maxillary central incisor was obtained by laser-based 3D digitizing (Cyberware, Inc., Monterey, California, USA) of a plaster cast (Thanaka Manufacturer, Tokyo, Japan) while dentin and enamel internal volumes were generated using horizontal and vertical profiles from Wheeler’s anthropometric studies . The scanned and literature profiles were assembled in a 3D wire frame structure using a 3D CAD (Autocad 12, Autodesk, Inc., San Rafael, California, USA) and exported into a 3D parametric solid modeler (Pro-Engineering 16.0 Parametric Technologies, Needham, Massachusetts, USA). Fitting the horizontal and vertical profiles then generated the tooth different model volumes. The average anatomical dimensions of the bone in adults were generated according to literature data . The maxillary bone was modeled as a spongy core surrounded by 2.0 mm thick cortical bone. The buccal–lingual and the mesial–distal dimensions of bone were 7.3 mm and 13 mm respectively at the root apex level. An average thickness of 0.7 mm and 0.2 mm were considered for the alveolar cortex and for the periodontal ligament (PDL) respectively .

As described in a previous work , the finite element (FE) base model was obtained by importing the solid models into ANSYS rel. 9.0 FEM software (Ansys, Inc., Canonsburg, Pennsylvania, USA) using IGES format. The volumes were then redefined in the new environment and meshed with 8 nodes brick with 3 degrees of freedom per node , finally resulting in a 3D FE model made up of 311,623 elements and 53,902 nodes ( Fig. 1 ). The model was constrained at the top surface of the maxillary bone. Accuracy of the model elements was checked by convergence tests .

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