Biomechanical Analysis of Restored Teeth with Cast Intra-Radicular Retainer with and Without Ferrule

The intra-radicular post is used to provide retention to a core, but would also have the function of distributing the functional load in a larger area of the remaining coronary structure and dental root. However, many authors (Assif & Gofil, 1994; Martinez-Insua et al., 1999; Rundquist & Versluis, 2006; Whitworth et al., 2002) have been demonstrating that this reconstruction method doesn't restore the original strength of a vital tooth, what is attributed to the wedge effect that this restoration type causes and to the stiffness difference between the post and the tooth.

The 3D model of the maxillary second premolar was built based on data presented by Shillinburg & Grace (1973), regarding the measures of the dentin thickness in the mesiodistal and vestibular-lingual axis in four horizontal slices (with 3.5 mm interval) accomplished along the main axis of the dental root starting from the cementoenamel junction. The dentin's thickness in the remnant of the root was built by the interpolation with a spline curve starting from the existent data, following the root anatomy. The pulp dimensions were built based on the data presented by Green & Brooklyn (1960), that it supplies the average diameter of the apical foramen and Shillinburg & Grace (1973), that it supplies the average dimensions in the cervical area.
For the making of the crown geometry the enamel thickness and coronary dentin were based on the data presented by Shillinburg & Grace (1973), through slices of interval of 1 mm, by Cantisano at al. (1987) and Ueti at al. (1997). The height of the vestibular cusp was considered larger than the lingual cusp in 0.9 mm (Shillinburg et al., 1972).
The lamina dura was represented with mechanical properties equal of the cortical bone and as a uniform layer of 0.25 mm thickness as well as the periodontal ligament (Lee et al., 2000).
The models that represented intra-radicular retainer were built from the natural healthy tooth model already. Specify modifications were accomplished in the area corresponding to the pulp and to the dentin, in the places corresponding to the core and porcelain-faced crown.
The dimension of the pulp was enlarged to reproduce the endodontic access stage (Cohen & Hargreaves 2005), the biomechanical prepare and the filling of the root canal.
The porcelain-faced crown was represented with the same outline of the enamel, being average thickness of 1.5 mm in the vestibular face, 1.2 mm in the lingual face, 1.0 mm in the proximal face and 2.0 mm in the occlusal face, with an end in chamfer around the core. The minimum thickness of the metal (NiCr) in the restoration porcelain-faced was 0.3 mm (Yamamoto, 1985).
The zinc phosphate cement was select because is usually used for cementation of the crown and of intra-radicular retainer with thickness from 50 to 100 m (Anusavice, 2003) and the gutta-percha height was of 5 mm (Morgano, 1996). In four models with intra-radicular retainer was varied the ending line of the core preparation (with ferrule geometry given by the core) and another model was simulated with simple core (without ferrule), staying the same coronary restoration.
The width and the height of the ferrule were determined as a proportion of the thickness of radicular dentine found in the cementoenamel junction region on the lingual side (1.85 mm), separated for three thirds. For other words, the ferrule with x height varying of 0.62 x 0.62 (T1H1), 0.62 x 1.34 (T1H2), 1.34 x 0.62 (T2H1) and 1.34 x 1.34 mm (T2H2), (Fig. 1).
All of the models with intra-radicular retainer presented the same characteristics in relation to the supporting structures, porcelain-faced crown, cement thickness and apical seal (Fig. 2). With the measures of the thickness in each one of the structures (pulp, dentin, enamel) in each face (mesial, distal, lingual, vestibular) a coordinates system was generated with the origin in the pulpal apex and all the values of thickness of the structures were transformed in coordinates. As many coordinates existed to generate the points in the Finite Element (FE) program, a routine in PCL (Patran Command Language) was created to read the spreadsheet and to generate the points. After the points generation (Fig. 3a), the curves were generated and, soon afterwards, the surfaces. Starting from the dental structures surfaces built, the superficial meshes were generated with triangular linear elements (Tri3). After www.intechopen.com Finite Element Analysis -From Biomedical Applications to Industrial Developments 136 that, volumetric meshes ( Fig. 3d) with tetrahedral linear elements (Tet4) were generated, following the procedures in Poiate et al. (2008Poiate et al. ( , 2009aPoiate et al. ( , 2009bPoiate et al. ( , 2011.  The degree of discretization of the FE models (Table 1)  Finally, boundary conditions or the model constraint and the loads are also applied. The constraint conditions applied were: in the maxillary sinus, translation in x, y and z directions and rotations in x, y and z axis, fully anchor; in the mesial and distal extremities of the cortical and spongy bone, translation in x direction (perpendicular to this faces) and rotations in y and z axis were anchor. Four load cases were built varying the site, inclination and the application area of a 291.36 N total static load, maximum masticatory force (Ferrario et al., 2004).
Initially, all of the models received the resultant load intensity of 291.36 N parallel to the tooth's long axis, with the aim to evaluate the wedge effect (Fig. 4). The load was applied on the lingual and vestibular cusp above the occlusal surface, at 45 o in relation to the tooth's long axis (Holmes et al. 1996), distributed in 19 nodal points of 0.85 mm 2 area in vestibular cusp and in 19 nodal points of 0.75 mm 2 area of in lingual cusp (Kumugai et al., 1999).

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After that, the model with ferrule that best minimized the wedge effect, as well as the models that represent natural healthy tooth and restored with simple core (T0H0, without ferrule, Fig. 1) were submitted to the following loads (Fig. 4): a. Oblique Load with 45 o in relation to the tooth's long axis distributed in 19 nodal points of 0.85 mm 2 area in the vestibular cusp, with the aim to evaluate the vestibular lever effect; b. Load parallel to the tooth's long axis distributed in 19 nodal points of 0.80 mm 2 area in the mesial marginal ridge, with the aim to evaluate the proximal lever effect; c. Oblique Load with 45 o in relation to the tooth's long axis pointed to vestibular cusp, distributed in 19 nodal points of 0.80 mm 2 area in the mesial marginal ridge, with the aim to evaluate the torsion effect. In this study, the maximum principal stress was used as the stress criterion to present the stress patterns distribution in the analyzed models. The cement and dentin's tensile strength of 8.3 MPa (Powers at al., 1976) and 103 MPa (Tanaka et al., 2003), respectively, as well as dentin´s compressive strength of 282 MPa (Tanaka et al., 2003), respectively, will serve as reference for results comparison to evaluate if the loads would be potentially harmful to the studied structures.

Wedge effect
Under load with parallel resultant to the long axis, all of the models answered equally. The differences were just in the dentin area in contact with the ferrule ( obtained with models T1H1 and T2H1 presented similar pattern of stress distribution in Maximum Principal Stress (MPS), but different from models T1H2 and T2H2 (that present low tensile stress under the ferrule). In the model H, the dentin area between the bone and the enamel presents some tensile stress, but the Fig. 6 show that the stress vector direction is not totally tangential to the transversal section and it wouldn't be deleterious, nor for the intensity nor for the direction. Similar stress appears in all the models with retainer immediately for apical of the ferrule and cannot are responsible for eventual root fracture.
The Fig. 7 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin. The results obtained with models T2H1, T0H0 and T1H1 presented a similar pattern of stress distribution in cement below the ferrula, but different from the model T1H2, which shows higher tensile stresses (up to 4 MPa), which by direction ( Fig. 6) suggests greater tendency to loosen or break the cement layer in the region, because the stresses are significantly higher, but the failure of the cement seems unlikely when comparing the absolute values of stress with the stress required to the cement cohesive failure (8.3 MPa).

Vestibular lever effect
Under load with inclination of 45 o in the vestibular cusp the T2H2, T0H0 and H models answered equally (Fig. 8), except in the radicular dentin that is being doubled and it is observed compressive stress in the vestibular, concentrated in the contact area with the cortical bone that acts as fulcrum (Fig. 9). The tensile stress concentrates and reaches the maximum on the opposite side, lingual, but, unexpectedly, it presents a larger extension in the model with ferrule, with maximum of 104, 110 and 109 MPa for H, T0H0 and T2H2 models, respectively.
www.intechopen.com  The Fig. 11 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig. 10 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the intersections with in M-D and V-L planes.

Proximal lever effect
The Fig. 12 shows the results under load parallel to the tooth's long axis in the mesial marginal ridge. The maximum tensile stress in dentin (Fig. 13) at the distal edge are also much seemed (68, 68 and 67 MPa), smaller than in the vestibular lever effect.  The MPS in dentin are present in Fig. 17. The Fig. 18 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig.19 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the intersections with in M-D and V-L planes.

Stress plot in wedge, vestibular lever, proximal lever and torsion effect
The Figs 20 to 28 shows the MPS in a ridge A to B, indicated in the drawing corresponding to each graph, in the surface of root dentin (interface with periodontal ligament), depending on the relative position of the node under every load, due to the abcissa axis is adimensional (Distance/Total length),.
The Fig. 20 shows the outside edge of the lingual surface. It is observed that the models that generate the wedge or proximal lever present in the lingual edge, very low stresses, all of the same order of magnitude. Since the models that generate vestibular lever and torsion have high tensile stresses in this edge, but below the dentin´s strenght limit. The Fig. 21 shows the outside edge of the vestibular surface. It is observed that the models that generate proximal wedge and proximal lever have the same order of magnitude of stresses acting on the vestibular edge with inflection behavior (tensile to compressive) from the apex to the cervical direction. In models that generate proximal lever, the tensile stresses near the apex are close to the wedge effect, however, near the neck, the stress tensile increase up to 5 MPa.

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Finite Element Analysis -From Biomedical Applications to Industrial Developments 152 Fig. 21. MPS at the edge of the root dentin on the vestibular surface, from the apex to the cervical direction.
The Fig. 22 shows the outside edge of the distal surface. We observed lowest stresses in wedge models, followed by vestibular lever models, torsion models and reaching the maximum stresses in proximal lever models. In models of proximal lever and torsion effect are generated stress peaks on the same nodes of the same models; the lower stress in torsion models could be due to the lower longitudinal component of the load in this case. The Fig. 23 shows the outside edge of the mesial surface. We observed lowest stresses in wedge models. The inversion of tensile to compression stress occurs so evident in the proximal lever models and so milder in torsion models. The Fig. 24 shows the inner edge of the lingual surface. We observed lowest stresses in the models of wedge and tensile stresses up to 45 MPa in the models of vestibular lever, followed by models of torsion effect, but in both types of load, in models of healthy tooth the stresses are smaller. It seems important to note that the order of values is much lower than those in Fig. 20, corresponding to the outer surface, which leads to think that the fracture is expected to start on the outside, apparently motivated by folding of the tooth as a whole. Discard the possibility of fracture driven by stress concentration on the inside face, which could be motivated by the wedging effect promoted by the retainer.
The Fig. 20 shows that the highest stress on the outside edge was achieved by the healthy tooth model. However, it is unlikely that the healthy tooth had a greater tendency to fracture: this is not what is observed in practice.
On the other hand Fig. 24 shows the stress almost doubles in inner edge, for the cases of vestibular lever with retainer, compared to the healthy tooth. This suggests that the failure criterion should not be simply the value of the maximum principal stress, but must be influenced by the stress gradient. The Fig. 25 shows the inner edge of the vestibular surface. Tensile stresses are observed up to 27 MPa in the region near the retainer apex in the models that generate vestibular lever, followed by models that generate torsion, but in both types of loading, the stresses are smaller in the healthy tooth models. Figure 11 shows that the direction of the peak tensile is radial and therefore not likely to cause longitudinal fracture but delamination in the dentin and detachment of the retainer apex. In fact, presents a direction parallel to the tensile stress on the edge of the external vestibular root, which appears to be motivated by the apical ligament region, to oppose the rotation of the tooth that is supported by the vestibular cortical bone. The Fig. 26 shows the inner edge of the distal surface. We observed lowest stresses in wedge models. Except the loading wedge models, all have tensile stresses, which shows that practically the entire thickness of the wall works in traction. Moreover, none of the models exceeded the stress developed by the model of healthy tooth with proximal lever load, which leads to the hypothesis that in no case has reached the stress level compatible with the fracture. The Fig. 27 shows the inner edge of the mesial surface. We observed lowest stresses in wedge models. The highest tensile stress occur to the torsion effect, regardless of the presence or not of the ferrula, which also does not help to reduce the stress on the lever vestibular.
www.intechopen.com In Figure 28 shows the circular edge of the inner dentin around the apex of the retainer. The abscissa axis is adimensional (Distance/Total length), where the 0 and 1 represents the vestibular, 0.25 the mesial, 0.5 the lingual and 0.75 the distal aspect. We observed lowest stresses in wedge models, followed by models of proximal lever, torsion and vestibular lever models. The region that remains the most tensioned is the vestibular.

Discussion
Several studies (Pierrisnard et al., 2002;Zhi-Yue & Yu-Xing, 2003) show that the ferrule creates a positive effect in the reduction of the stress concentration in the dentin-core junction and helps to maintain the integrity of the cement seal in the crown (Libman & Nicholls 1995).
However, in spite of some authors recommend a coronary minimum height of ferrule (Tan et al., 2005;Pereira et al., 2006;Morgano, 1996;Morgano & Bracket, 1999;Aykent, 2006) to increase the fracture strength values, in this study the ferrule didn't influence on the stress distribution. An important observation should be made: for the perfect adhesion among the structures, it was not possible to notice a wedge effect appreciable.
Under load with parallel resultant to the long axis, all of the models present compression stress in the root apex, but with very low intensity when compared with the compression strength of dentin. The light stress concentration found around of the post apex cannot are responsible by longitudinal fracture, because the orientation and the intensity would not justify the fracture.
The results of models T1H1 and T2H1 show that the increase in the width of ferrule, keeping the height, it was important to the root protection with the ferrule effect, because the compressive stress increased four times. The model T2H2 presented compressive stress in the radicular dentin on the ferrule (0.5 to 2.0 MPa) and in the radicular dentin above the periodontal ligament (0.5 to 20 MPa), mesial and distal face (Fig. 4).
On the other hand, the results of models T1H2 and T2H2 show that tensile stresses disappears in the dentine in contact with the ferrule, what can be attributed to the increase in the ferrule width, keeping the height. This characteristic confers to the ferrule design some superiority, because it can be inferred that it will be more difficult than the cement is unstuck in that area, since the tensile stress isn't submitted. The model T2H2 seems also to minimize the tensile stress in the cervical dentine (compressive stress from 0.5 to 2.0 MPa) and in the radicular dentine near to the periodontal ligament, following by the model T2H1, T1H1 and T1H2.
Posteriors teeth can be subject also to vestibular lever effects whenever requested eccentric efforts in lateral excursion. The contact oclusal on the work side in lateral excursion can reach the vestibular cusp in posteriors teeth, generating lever force on the involved roots that serve as guide for those movements.
Under load with inclination of 45 o in the vestibular cusp, the maximum stress value is compatible with the fracture occurrence. If the fracture begins at that place can follow a perpendicular plan to the tensile vector and to spread tending the uprighting that suggests a vertical radicular fracture would pass exactly in the bone crest limit.
Fractures vertical appear as a result of stresses generated inside the root canal (Lertchirakarn et al., 2003), but the models showed tensile stress concentrated in the post apex, although of smaller magnitude that in the height of the bone crest.
In all of the models, the root canal stress in the mesial-distal slice presented tensile stress (except for the area near to the post apex) and near to the bone crest presents change in the vector orientation due to the rotation of the same ones, that it seems related with the presence of cortical bone. Therefore, the fracture area not just depends on the post and ferrule, but also of location of the tooth in the alveolus.
An interesting finding was the relatively small difference among the three models that could be attributed to the fact of the interfaces between the structures of the models was simulated with perfect adhesion. The perfect adhesion can losing in the course of time, what would explain that the failures didn't use to happen in retainer recently cemented. On the other hand, the models showed high tensile stresses concentration in more than half of the cement layer, which could be responsible for cohesive fracture.
Under the load area the tensile stresses on the ferrule generates 40 MPa, exceeding the cement's tensile strength of 8.3 MPa and associated with the stress orientation (Fig. 11) suggests the tendency to the cement layer failure in this area.
Under load parallel to the tooth's long axis in the mesial marginal ridge, the tooth restored with post would present a tendency to the rupture similar to the one of the natural, what is not supported by the clinical observations. The explanation for this discrepancy could be again in the fact that the cement layer presented stresses much larger than the necessary ones for cohesive fracture. After the cement failure the stress distribution change and could propitiate the root fracture.
The intra-radicular retainers can be subject also to rotation force or torsion, whenever requested clinically through contacts functional cusp in the ridge, in other words, tooth relationship to two teeth (Hemmings et al., 1991).
The occlusal contacts in the marginal ridge of the premolar could produce a torsion loads that tends to rotate the root along the tooth's axis. For this reason is extremely important have in mind the use of retainers that offer larger safety to the radicular remnant in any situation of mechanical effort what the root is submitted.
Under oblique load with 45 o in relation to the tooth's long axis pointed to vestibular cusp in the mesial marginal ridge, the maximum stress and orientation evidences again that cement layer is prone to failure and that the stress concentration is more serious in the ferrule case.
When the cement around of the most coronary portion deteriorates, the fulcrum migrates to apical, what increases the lever arm progressively (Cohen et al., 1993).
The modeling in this study considered all the components without setting out eventual contact problems, in other words, without cement failure, what couldn't happen in the clinic reality.
Regarding the clinical significance of the results, it seems that the best way to protect the radicular remnant when restored with post would be to guarantee the occlusal adjustment, that don't happen loads different from the longitudinal. The most vulnerable part of the whole restorative system is the cement layer, that doesn't resist to the tensile stresses result from the loads application different from the longitudinal load. The post perfectly adhered doesn't seem lead to the occurrence of longitudinal fracture (below the bone crest), but probably to fracture that would begin in the vestibular in the height of the bone crest.

Conclusions
Within the limitations of this study, the following conclusions were drawn: 1. In spite of none of the simulated cases to evidence the wedge effect, the longitudinal load produced stresses that don't justify the rupture nor of the dentine nor of the cement layer and the stresses are very inferior to developed by all the other load types; 2. The ferrule isn't necessary to improve the stresses distribution, except for longitudinal load, in that presents discreet beneficial effect; 3. In the vestibular load were found, in the lingual face, tensile stresses guided parallel to the longitudinal axis and magnitude enough to induce vertical fractures above the bone crest. These stresses are associated to the tooth's bend, leaning in the cortical bone that it acts as a fulcrum; 4. In the cement layer were found stresses that are prone to fracture in all of the loads, except for the case of longitudinal load; 5. A concerted effort to develop other FE models to improve the results may provide more reliable data about the non-linearly of the structures and the cement failures in the clinic reality; 6. The analyses help the field better understand the biomechanical analysis of restored teeth with cast intra-radicular retainer with and without ferrule.

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Acknowledgements
This study was partially based on a thesis submitted to the University of São Paulo, in fulfillment of the requirements for PhD degree. The authors are grateful to the CAPES (Coordenação de Aperfeicoamento de Pessoal de Nível Superior, Coordination for the Improvement of Higher Education Personnel) that supported this research project. Finite Element Analysis represents a numerical technique for finding approximate solutions to partial differential equations as well as integral equations, permitting the numerical analysis of complex structures based on their material properties. This book presents 20 different chapters in the application of Finite Elements, ranging from Biomedical Engineering to Manufacturing Industry and Industrial Developments. It has been written at a level suitable for use in a graduate course on applications of finite element modelling and analysis (mechanical, civil and biomedical engineering studies, for instance), without excluding its use by researchers or professional engineers interested in the field, seeking to gain a deeper understanding concerning Finite Element Analysis.

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