## Abstract

## Objective

To evaluate the effects of clinically relevant variables on the maximum principal stress (MPS) in the veneer layer of an anatomically correct veneer–core–cement–tooth model.

## Methods

The average dimensions of a mandibular first molar crown were imported into CAD software; a tooth preparation was modeled by reducing the proximal walls by 1.5 mm and the occlusal surface by 2.0 mm. ‘Crown systems’ were composed by varying characteristics of a cement layer, structural core, and veneer solid, all designed to fit the tooth preparation. The main and interacting effects of proximal wall height reduction, core material, core thickness, cement modulus, cement thickness, and load position on the maximum stress distribution were derived from a series of finite element models and analyzed in a factorial analysis of variance.

## Results

The average MPS in the veneer layer over the 64 models was 488 MPa (range = 248–840 MPa). MPS increased significantly with the addition of horizontal load components and with increasing cement thickness. In addition, MPS levels varied as a function of interactions between: proximal wall height reduction and load position; load position and cement thickness; core thickness and cement thickness; cement thickness and proximal wall height reduction; and core thickness, cement thickness and proximal wall height reduction.

## Conclusion

Rational design of veneered structural ceramics must consider the complex geometry of the crown–tooth system and integrate the influence of both the main effects and interactions among design parameters.

## 1

## Introduction

Over the past 2 decades, improvements in materials and manufacturing processes have resulted in structural ceramics with increased mechanical properties supporting veneering porcelains, thus giving way for fabrication of all ceramic crowns. All-ceramic crowns have become increasingly popular because they esthetically mimic the original color and translucency of the tooth and show higher biocompatibility compared to metallic structures .

Even as recent technological improvements in ceramic manufacturing has improved on both strength and esthetic aspects of all-ceramic crown systems, their extension into posterior usage has not positively affected performance and failure rates reported for posterior all ceramic crowns are in the range of 1–4% per year . However, instead of bulk fractures observed in monolithic crown systems, clinical evaluation of zirconia core layered structures have shown cohesive fracture modes in the veneer layer .

Recently, in vitro laboratory testing has been able to reproduce the clinically observed cohesive failures of all-ceramic crown veneer layers when supported by zirconia cores . However, in vitro laboratory testing of anatomically correct all-ceramic crowns considering a large number of clinically relevant variables is costly and time consuming . Thus, the utilization of the finite element analysis (FEA) method may allow insight as to how variations in materials properties and geometric configurations affect stress levels providing an informed landscape regarding which variables should be considered for laboratory validation .

Simulations have been previously used in crown–tooth models for a variety of purposes . However, most models were simplified and do not accurately portray the interplay of system components and geometry as well as the effects of loading components . With any finite element model, the more complex and accurate the geometry is to its true anatomical state, the more likely the research will uncover unexpected results. In general, as the geometry becomes more complex, the stress fields also increase in complexity .

While substantial work has been done concerning the mechanical behavior of tooth-crown systems, these studies primarily included variations in materials and component physical dimensions. To date, the only comprehensive study has included a large number of variable and their interplay . Rekow et al. found that crown thickness and material are of the utmost importance regarding the magnitude of stress throughout the system. Notwithstanding, other variables such as cement thickness and modulus, load position, and supporting tooth core have roles in the magnitude of the overall stress and cannot be ignored. Even though the relative importance of different components and loading scenario was evaluated by Rekow et al. , the physical dimensions and anatomy utilized was simplified in an axis symmetric model. Thus, multivariate analysis incorporating an anatomically correct tooth-crown system configuration is lacking in the literature.

The objective of this investigation was to use a set of clinical variables to evaluate their relative contribution to maximum principal stress (MPS) in an anatomically correct veneer–core–cement–tooth model.

## 2

## Methods

An anatomically based mandibular crown was investigated utilizing three dimensional finite element analysis (FEA). For modeling, the dimensions of a typical mandibular first molar crown were imported into CAD software (ProEngineer Wildfire, Needham, MA, USA). A tooth preparation was simulated by reducing the occlusal surface by 2.0 mm following the anatomy of the occlusal table cusps and ridges and presented uniform axial wall reduction in all flanges or with interproximal axial walls 1.5 mm shorter than the buccal and lingual axial walls. Convergence of 12 degrees was created between the buccal and lingual walls as well as between the mesial and distal walls. A 0.5 mm deep reduction chamfer margin was designed ( Fig. 1 a ).

A crown was designed to occupy the space between the original tooth form (plus a cement layer space) and the prepared tooth design. The crown was subdivided to create a veneer and core layer ( Fig. 1 b and c) and to fit the underlying cement layer ( Fig. 1 d). The veneer layer solid internal aspects were changed in order to accommodate variations in other components’ geometry (allowing the same final crown size irrespective of geometric variation ( Fig. 1 e).

The core was designed to be either 0.5 mm or 1 mm constant thickness (variable between zirconia and alumina cores). The 5-cusp porcelain veneer layer ( *E *= 70 GPa, *ν *= 0.19) was 1 mm thick on the axial walls and 1.5 mm thick on the occlusal surface, and its thickness was reduced according to core thickness variation of either 0.5 mm or 1 mm. A resin cement layer either 50 or 200 μm thickness separated the crown from the tooth preparation. Dentin properties were assigned for the preparation ( *E *= 16 GPa, *ν *= 0.33) .

Overall, the following geometric and materials variables were included in the analysis: (1) interproximal wall reduction was modeled as either no interproximal wall height reduction or as a 1.5 mm reduced interproximal wall height; (2) core thickness was modeled as either 0.5 mm or 1 mm; (3) core material was modeled as either Y-TZP ( *E *= 205 GPa, *ν *= 0.19) or as alumina ( *E *= 300 GPa, *ν *= 0.21) ; (4) cement layer thickness was modeled as either 50 or 200 μm; and (5) cement modulus was modeled as either 5 or 40 GPa . The final variable was (6) load direction, which was modeled either as a vertical load of 1200 N or as a vertical load of 1200 N plus a horizontal load of 300 N in the central occlusal direction (in this case a vertical and a horizontal loads were placed in the loading region). The load magnitude was based upon in vitro single load to fracture results obtained for an all ceramic system . Both loading conditions were applied over a 1 mm diameter circle area near the tip of the buccal cusp ridge, simulating typical occlusal contact area loading ( Fig. 1 e).

The following assumptions were included in the finite element model: (1) all solids are homogeneous, isotropic, and linear elastic; (2) no slip was permitted between components (perfect bonding), (3) the cement layer is of uniform thickness; (4) there are no flaws in any component, (5) the tooth root is fully constrained. The boundary conditions constrained all 6 degrees of freedom in the tooth preparation root surface located at 1.5 mm below the most cervical cement-tooth preparation boundary. The number of tetrahedral elements for the models ranged from ∼23,000 to 25,000 (changes in the number of elements were due to changes different model components other than the veneer layer, which presented the same mesh throughout the study). The veneer layer MPS regions and values for each of the 64 model combinations were determined through 3D graphs and software output values. The highest tensile stress was utilized for statistical analysis.

Data were submitted for analysis using the general linear model (GLM) procedure (SPSS v.14, SPSS Inc., Chicago, IL) to compute a reduced model that included main effects, as well as 2- and 3-way interactions. All interactions accounted for 98.2% of variance in the crown stress values. Higher order interactions and residual variance were summed to form a pooled error term. In this way, contrasts between the two levels within each of the six main effects were based on an average of 32 observations, 2-way interactions on an average of 16 observations within each of four cells, and 3-way interactions on the basis of 8 observations within each of the eight cells. Significance was indicated by statistical tests with type 1 error rates less than 5%.

## 2

## Methods

An anatomically based mandibular crown was investigated utilizing three dimensional finite element analysis (FEA). For modeling, the dimensions of a typical mandibular first molar crown were imported into CAD software (ProEngineer Wildfire, Needham, MA, USA). A tooth preparation was simulated by reducing the occlusal surface by 2.0 mm following the anatomy of the occlusal table cusps and ridges and presented uniform axial wall reduction in all flanges or with interproximal axial walls 1.5 mm shorter than the buccal and lingual axial walls. Convergence of 12 degrees was created between the buccal and lingual walls as well as between the mesial and distal walls. A 0.5 mm deep reduction chamfer margin was designed ( Fig. 1 a ).

A crown was designed to occupy the space between the original tooth form (plus a cement layer space) and the prepared tooth design. The crown was subdivided to create a veneer and core layer ( Fig. 1 b and c) and to fit the underlying cement layer ( Fig. 1 d). The veneer layer solid internal aspects were changed in order to accommodate variations in other components’ geometry (allowing the same final crown size irrespective of geometric variation ( Fig. 1 e).

The core was designed to be either 0.5 mm or 1 mm constant thickness (variable between zirconia and alumina cores). The 5-cusp porcelain veneer layer ( *E *= 70 GPa, *ν *= 0.19) was 1 mm thick on the axial walls and 1.5 mm thick on the occlusal surface, and its thickness was reduced according to core thickness variation of either 0.5 mm or 1 mm. A resin cement layer either 50 or 200 μm thickness separated the crown from the tooth preparation. Dentin properties were assigned for the preparation ( *E *= 16 GPa, *ν *= 0.33) .

Overall, the following geometric and materials variables were included in the analysis: (1) interproximal wall reduction was modeled as either no interproximal wall height reduction or as a 1.5 mm reduced interproximal wall height; (2) core thickness was modeled as either 0.5 mm or 1 mm; (3) core material was modeled as either Y-TZP ( *E *= 205 GPa, *ν *= 0.19) or as alumina ( *E *= 300 GPa, *ν *= 0.21) ; (4) cement layer thickness was modeled as either 50 or 200 μm; and (5) cement modulus was modeled as either 5 or 40 GPa . The final variable was (6) load direction, which was modeled either as a vertical load of 1200 N or as a vertical load of 1200 N plus a horizontal load of 300 N in the central occlusal direction (in this case a vertical and a horizontal loads were placed in the loading region). The load magnitude was based upon in vitro single load to fracture results obtained for an all ceramic system . Both loading conditions were applied over a 1 mm diameter circle area near the tip of the buccal cusp ridge, simulating typical occlusal contact area loading ( Fig. 1 e).

The following assumptions were included in the finite element model: (1) all solids are homogeneous, isotropic, and linear elastic; (2) no slip was permitted between components (perfect bonding), (3) the cement layer is of uniform thickness; (4) there are no flaws in any component, (5) the tooth root is fully constrained. The boundary conditions constrained all 6 degrees of freedom in the tooth preparation root surface located at 1.5 mm below the most cervical cement-tooth preparation boundary. The number of tetrahedral elements for the models ranged from ∼23,000 to 25,000 (changes in the number of elements were due to changes different model components other than the veneer layer, which presented the same mesh throughout the study). The veneer layer MPS regions and values for each of the 64 model combinations were determined through 3D graphs and software output values. The highest tensile stress was utilized for statistical analysis.

Data were submitted for analysis using the general linear model (GLM) procedure (SPSS v.14, SPSS Inc., Chicago, IL) to compute a reduced model that included main effects, as well as 2- and 3-way interactions. All interactions accounted for 98.2% of variance in the crown stress values. Higher order interactions and residual variance were summed to form a pooled error term. In this way, contrasts between the two levels within each of the six main effects were based on an average of 32 observations, 2-way interactions on an average of 16 observations within each of four cells, and 3-way interactions on the basis of 8 observations within each of the eight cells. Significance was indicated by statistical tests with type 1 error rates less than 5%.

## 3

## Results

In all of the 64 models run, the mean (SD) maximum principal stress (MPS) in tension in the porcelain crown was 488 (152) MPa. The range of values was from 248 to 834 MPa. The minimal veneer layer MPS in tension was observed in the configuration which had no interproximal axial wall height reduction, no horizontal loading components, a core thickness of 0.5 mm, a cement thickness of 0.05 mm, cement elastic modulus of 5 GPa and a zirconia core ( Fig. 2 ). The maximal veneer layer MPS occurred in the configuration that had no interproximal axial wall height reduction, a horizontal loading component, a core thickness of 0.5 mm, a cement thickness of 0.2 mm, a cement elastic modulus of 40 GPa, and a zirconia core ( Fig. 3 ). Thus, the most extreme values appear attributable to horizontal loading, cement modulus and cement thickness. Irrespective of model configuration, tensile fields developed at the buccal cusp bulk region extending throughout the occlusal surface of the veneer layer, at the veneer–core interface region below the indenter, and at the preparation margins. Compressive fields were observed primarily below the loading area ( Figs. 2 and 3 ).

Descriptive data for the veneer layer MPS as a function of each variable is presented in Table 1 . As shown in Table 2 , MPS was significantly greater with than without horizontal loads, and in thicker than thinner cement layers. However, these findings were not consistent across levels of other variables, as described below. Table 2 also shows several significant 2-way interactions: between interproximal wall reduction and load position; between load position and cement thickness; between core thickness and cement thickness; between cement thickness and interproximal wall height reduction; and a 3-way interaction among core thickness, cement thickness and interproximal wall height reduction.

Parameter studied | Variables | Average MPS (MPa) | Standard deviation (MPa) | Range | |
---|---|---|---|---|---|

Minimum | Maximum | ||||

Proximal axial wall height | 1.5 mm axial wall height reduction | 493.09 | 103.08 | 321.80 | 619.80 |

No interproximal axial wall height reduction | 482.73 | 190.12 | 243.00 | 839.50 | |

Load position | Without horizontal components | 366.69 | 82.05 | 243.00 | 473.00 |

With horizontal components | 609.13 | 98.79 | 450.40 | 839.50 | |

Core thickness | 0.5 mm | 484.13 | 164.10 | 243.00 | 839.50 |

1 mm | 491.69 | 140.96 | 246.60 | 684.10 | |

Cement thickness | .200 mm | 504.95 | 173.24 | 246.60 | 839.50 |

.050 mm | 470.87 | 127.35 | 243.00 | 684.10 | |

Cement modulus | 40 GPa | 487.92 | 155.55 | 243.10 | 839.50 |

5 GPa | 487.90 | 150.44 | 243.00 | 798.80 | |

Core material | Alumina | 487.76 | 151.73 | 243.40 | 795.10 |

Zirconia | 488.06 | 154.29 | 243.00 | 839.50 |