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Residual shrinkage stress distributions in molars after composite restoration
Dental Materials (2004) 20, 554–564
http://www.intl.elsevierhealth.com/journals/dema
Residual shrinkage stress distributions in molars after composite restoration Antheunis Versluisa,*, Daranee Tantbirojnb, Maria R. Pintadoa, Ralph DeLonga, William H. Douglasa a
Department of Oral Science, Minnesota Dental Research Center for Biomaterials and Biomechanics, University of Minnesota, 16-212 Moos Tower, 515 Delaware Street SE, Minneapolis, MN 55455, USA b Department of Operative Dentistry, Chulalongkorn University, Bangkok, Thailand Received 3 December 2002; received in revised form 16 May 2003; accepted 29 May 2003
KEYWORDS Restorative composite; Polymerization shrinkage; Strain gauge; Microhardness; Post-gel shrinkage; Composite restoration; Shrinkage stress; Residual stress; Deformation
Summary Objective. Experimental measurements on various restoration configurations have shown that restored teeth deform under the influence of polymerization shrinkage, but actual residual stresses could not be determined. The purpose of this study was to calculate and validate shrinkage stresses associated with the reported tooth deformations. Methods. Three different restoration configurations were applied in a finite element model of a molar. The composite properties were based on experimentally determined composite behavior during polymerization. The occlusal deformation pattern and the residual stress states of the tooth, restoration, and tooth-restoration interface were calculated using a polymerization model based on the post-gel shrinkage concept. Reported strain gauge measurements and occlusal deformation patterns were used for validation. Results. The shrinkage stresses depended on the configuration and size of the restorations. The tooth’s resistance against polymerization shrinkage diminished with loss of dental hard tissue. Larger restorations resulted in lower stress levels in the restoration and tooth-restoration interface, but increased stresses in the tooth. The maximum stress values found for different configurations were not decisively different. Significance. The validated model indicated that shrinkage stress cannot be based on composite properties or restoration configuration alone, but has to be approached as a distributed pattern that depends on the location and on the properties of tooth and restoration, geometry, constraints, and restoration procedures. Tooth deformation was indicative of stresses in the tooth rather than in the restoration or across the tooth-restoration interface. Q 2003 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
Introduction *Corresponding author. Tel.: þ1-612-625-0950; fax: þ 1-612626-1484. E-mail address: [email protected]
It is widely accepted that polymerization shrinkage of current restorative composites cause residual stresses in restored teeth. These stresses are called
0109-5641/$ - see front matter Q 2003 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.dental.2003.05.007
Residual shrinkage stress distributions in molars after composite restoration
residual because after curing a restored tooth is left under stress even when there is no functional loading. The presence of residual stresses results in a changed behavior of the restored tooth, which may become evident in its clinical performance. Clinical symptoms associated with residual shrinkage stresses are inadequate adaptation, microcrack propagation, marginal loss, post-operative sensitivity, microleakage, and secondary caries.1 – 5 Stress is not a material property, but a local physical state that is derived from the combination of material properties, geometry, boundary conditions, and history. Since residual shrinkage stress is not a straightforward property that can be measured directly, its quantification in restorations remains a source of controversy. Various methods have been used to estimate residual shrinkage stresses, ranging from extrapolated shrinkage or load measurements in vitro to stress analyses in tooth shaped anatomies using photoelastic or finite element methods.6,7 Calculation of shrinkage stresses in a tooth-restoration complex is not trivial. One challenge is the description of the sequence of changes that take place in composite during polymerization. Another challenge is that stress depends on the geometry and mechanical properties of surrounding tissues. Many intricacies of the biomechanical manifestations of polymerization processes are still not well understood. As a result the development of a residual shrinkage stress model (i.e. the expression of our understanding of the event) is an interactive
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process with experimental observations. Deformation patterns of occlusal surfaces for restored molars due to shrinkage stresses have been determined and observed in a series of experiments.8 While deformation could be measured experimentally, the stress distribution must still be calculated using transient composite properties determined from another series of experiments.9 The purpose of this study was to develop and validate a polymerization model for the calculation of residual shrinkage stress distributions associated with the reported deformations, using the determined transient properties distributions. The validated model can consequently be used to investigate the shrinkage stresses in restored teeth, and study factors that affect them.
Material and methods Deformations of occlusal surfaces have been reported for a successive range of cavity preparations in extracted human molars:8 Class I, small Class II OM, large Class II OM, and Class II MOD. The deformation due to polymerization shrinkage was quantified by comparing the digitized occlusal surfaces before and after restoration (Fig. 1). To calculate corresponding residual stresses in the tooth, a finite element simulation was carried out for a similar combination of geometry, boundary conditions, material properties, and restoration procedures (history).
Figure 1 Cuspal deformation patterns (mm) for a molar determined in in vitro experiments for four consecutive restorations (Class I, small Class II OM, large Class II OM, and Class II MOD) using Cumulus software.8
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Figure 2 Finite element model of molar, indicating four simulated restorations (Class I, small Class II OM, large Class II OM, and Class II MOD), increments, and the direction of light curing (black arrows).
Four consecutive restorations were applied in a finite element model of a molar: Class I, small Class II OM, large Class II OM, and Class II MOD (Fig. 2). Additionally, a large Class I and small Class II MOD were created. The external molar shape was digitized using a contact stylus profilometer,10 while the internal enamel – dentin and pulp surfaces were estimated based on general anatomy. The molar root was fixed, simulating the embedded specimens of the corresponding experimental design.8 Orthotropic elastic properties were modeled for the enamel. The elastic moduli were 84 GPa (principal direction, perpendicular to the pulp surface) and 42 GPa, with a Poisson’s ratio of 0.30. The dentin elastic modulus was 18 GPa, with a Poisson’s ratio of 0.23. The composite properties used were based on experimental data9 obtained for Z100 Incisal shade (Lot 6BN, 3M ESPE, St Paul, MN, USA), the same as used for the companion study.8 Z100 is a commercially available universal restorative composite, containing bisGMA and TEGDMA resins with zirconia/silica fillers (66% inorganic filler loading by volume, particle size range of 3.5 – 0.01 mm). The relatively translucent composite was chosen to obtain good light penetration in the large restorations. The maximum elastic modulus and Poisson’s ratio for a fully cured composite were 20 GPa and 0.24, respectively. In this study a linear-elastic approach was validated for the simulation of residual shrinkage
stress development. A linear elastic approach allows a significant reduction of the stress calculation time, while the required properties that have to be determined are reduced to the elastic modulus and post-gel shrinkage. Both properties depend on the curing light intensity, exposure time, and storage time. In order to apply the properties in the finite element analysis, they have to be expressed in mathematical forms. The local light intensity I in the restoration was approximated using the light intensity of the curing light source, the effect on the light intensity due to the distance between the restoration and light guide, and the exponential relationship for light attenuation in the composite9 I ¼ I0 exp½2ðw=10Þ3 exp½21:19d 0:72
ð1Þ
where I0 is the light intensity of the curing light source (mW/cm2); w is the distance from lightguide to the restoration surface (mm); d is the depth in the composite (mm). In each composite element integration point (where stresses are determined by the finite element software) local properties were calculated as a function of local light intensity and time (light exposure and dark cure time). The reported correlation between Knoop microhardness and elastic modulus11 can be employed to calculate the local elastic modulus of the composite as a function of light energy and time. Local light energy density Y (mJ/cm2) is
Residual shrinkage stress distributions in molars after composite restoration defined as12,13 Y ¼ Itc
ð2Þ
where tc is the curing light exposure time (s). The empirical relationship found for the microhardness KMH is9 KMH¼ð66211exp½2t=3000Þð12exp½2ðY=2500Þ1:1 Þ ð3Þ where t is the total time since the start of curing (s). The first term describes how the microhardness values increase during the storage time t after light cure. The second term describes the exponential relationship between microhardness and light energy. The local elastic modulus E can be obtained by E ¼Emax KMH=KMHmax
ð4Þ
where Emax is the maximum attainable elastic modulus value (Emax ¼20 GPa), and KMHmax is the maximum microhardness value used to normalize KMH in Eq. (3) ðKMHmax ¼66Þ: Post-gel shrinkage values were obtained by the strain gauge method.9 Unlike microhardness, postgel shrinkage cannot be fully described by the applied light energy because it depends on the light intensity at which the polymerization reaction was initiated. The maximum value for the post-gel shrinkage amax can be expressed by the empirical relationship ðR . 0:999Þ
amax ¼ 5000ð1 2 exp½2ðI0 =500Þ0:22 Þ
ð5Þ
where I0 is the intensity of the light source with which the polymerization was initially initiated (mW/cm2). The polymerization process can be divided into two periods: light exposure and dark cure (after the light cure). Post-gel shrinkage during ðalight Þ and after ðadark Þ light exposure can be expressed by ðR . 0:97Þ
alight ¼ amax ð1 2 exp½2Y00:7 =ð450 þ 0:35I0 ÞÞ
ð6Þ
adark ¼ amax ð8 þ 0:06I0 Þexp½2ðY0 =1:161Þ0:186 ð1 2 exp½2{Y0 =ð858 þ 4:1I0 Þ}1:5 Þ
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(Eqs. (6) and (7)), using the intensity output of the curing light source rather than local intensity levels. A bulk property means that the acquired strain gauge data represents the response of the whole sample. Since post-gel shrinkage depends on the light intensity and light intensity is location dependent, the post-gel shrinkage also depends on the location. Therefore, the bulk value for post-gel shrinkage cannot be used in the finite element analysis. Considering that the strain gauge records the local deformation at the gauge surface, the post-gel shrinkage results (Eqs. (6) and (7)) can be easily adjusted by correction of the terms that contain the light intensity I0 ¼ FI Y0 ¼ FItc ¼ FY
ð9Þ ð10Þ
where F is the correction factor for the actual light intensity at the gauge surface. Using the light attenuation Eq. (1) the correction factor F is 6.1 (w ¼ 3 mm, d ¼ 1:75 mm). To validate the derived elastic modulus and post-gel shrinkage formulations the strain gauge design was also simulated in a finite element analysis. The strain gauge configuration was modeled by a 1.75 mm high composite sample placed on an 80 mm thick strain gauge (elastic modulus 6 GPa and Poisson’s ratio 0.3, manufacturer’s data) (Fig. 3). Light curing from above (light guide positioned 3 mm above composite sample) was simulated for the four curing light intensities used in the experimental setup9 (I0 ¼ 100; 300, 500, and 700 mW/cm2). The plane strain in a 1 mm2 area at the center of the strain gauge was recorded for validation with the experimental shrinkage strain results. In the restoration models, light cure was applied occlusally to the composite restorations (188 off for the proximal boxes) for 60 s per increment at 650 mW/cm2, mimicking the experimental study.8 The diameter of the simulated light guide was 12 mm. The Class II OM cavities were filled in two increments, the Class II MOD in three (Fig. 2).
ð7Þ
where I0 is the light intensity of the light source at which the composite polymerization is initiated (mW/cm2), and Y0 is the applied light energy ðI0 tc Þ: Note the difference between applied and local light intensity or energy. The applied values are the output of the curing light source. The post-gel shrinkage value a determined by the strain gauge becomes:
a ¼ alight þ adark
ð8Þ
Unlike the microhardness data, which were determined as local properties (Eq. (3)), the post-gel shrinkage results were determined as bulk properties
Figure 3 Finite element model of the strain gauge experiment,9 showing the element distribution for the quarter mesh.
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Additionally, a large Class I and a small Class II MOD were modeled. The time between increments was 60 s. It is important that the successive filling increments are not part of the analysis before they are placed. Eight-node isoparametric arbitrary hexahedral reduced integration elements were used. A custom program calculated and applied the material property relationships for enamel and composite, performed the restorative actions of filling and light cure, defined and submitted the restoration models to a finite element solver for calculation of stresses (MSC.Marc, MSC Software Co., Los Angeles, CA, USA), processed the results and created a convenient output format. The analyses were performed on a Compaq AlphaServer ES40 (Compaq, Houston, TX, USA). Modified von Mises equivalent stresses were used to express the stress conditions. The advantage of using equivalent von Mises stress is that a multidimensional stress distribution (e.g. six Cartesian stress components or three principal stresses) is expressed into one value, which eases the interpretation of the overall stress condition. The used equivalent stress is based on the well-known von Mises formulation, modified to take into account the difference between compressive and tensile strength for enamel, dentin, and the composite.14 Since cuspal deflection is often reported for in vitro studies, the inter-cuspal distance change were determined for the mesial and distal cusp pairs. The calculated occlusal deformations were compared with the experimentally determined occlusal deformation8 (Fig. 1), using a locally developed new
generation software package (Cumulus, copyright Regents of the University of Minnesota). Surface changes in Cumulus are determined perpendicular to the baseline surface taken before restoration. In order to import the finite element geometry in Cumulus, the before and after surfaces were isolated, refined and converted to object file format (Alias Wavefront Object Files). From the dense surface point-clouds, Cumulus created rendered surfaces. Differences between the before and after occlusal enamel surfaces were determined and visualized using a linear color scale. The quantitative color patterns were used to compare the experimental and finite element results. Cusp displacements (bucco-lingual direction) were determined to be compared with values reported in the literature.
Results To validate the polymerization modeling procedures, deformation was calculated in two finite element simulations (strain gauge simulation and occlusal deformation of a tooth crown), using the equations for light intensity, elastic modulus, and post-gel shrinkage derived from experimental data. The response of the strain gauge is not solely caused by post-gel shrinkage, but also by development of elastic modulus and distribution of properties due to the light attenuation in the composite. Results (microstrain) of the experimental data9 were compared with the results obtained by the finite element simulation (Table 1). The table
Table 1 Comparison of shrinkage strains measured by the strain gauge experiments9 (mean and standard deviation; n ¼ 5) and calculated by the numerical polymerization simulation of the strain gauge setup. Light intensity, I0 (mW/cm2)
100 100 100 100 300 300 300 300 500 500 500 500 700 700 700 700
Exposure time, tc (s)
10 20 40 60 10 20 40 60 10 20 40 60 10 20 40 60
Light energy, Y0 (mJ/cm2)
1000 2000 4000 6000 3000 6000 12,000 18,000 5000 10,000 20,000 30,000 7000 14,000 28,000 42,000
Shrinkage strain (mstrain) Experimental
Finite element analysis
451(89) 1227(31) 1703(16) 1844(40) 1825(60) 2280(65) 2560(75) 2641(49) 2224(95) 2664(69) 2849(68) 2907(73) 2602(80) 2889(86) 3060(44) 3136(81)
865 1315 1693 1926 1962 2283 2606 2777 2433 2665 2941 3077 2684 2868 3122 3241
The results were determined 2 min after polymerization for different combinations of light intensities ðI0 Þ and light exposure times ðtc Þ:
Residual shrinkage stress distributions in molars after composite restoration
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Figure 4 Cuspal deformation patterns (mm) for the simulated molar for four restorations (Class I, small Class II OM, large Class II OM, and Class II MOD) using the numerical polymerization model and Cumulus software.
shows a good agreement between the experimental and finite element data. Applying the material relationships to a tooth model, the occlusal deformations were calculated and visualized by color coding (Fig. 4) for comparison with the experimental results (Fig. 1) using the Cumulus software. Positive values are defined as higher surfaces, and negative values indicate lower surfaces compared to the baseline. The deformation pattern and values were similar in magnitude and distribution to the results found by the experimental method. The finite element analysis calculated occlusal deformation lower than 10 mm for the smallest restoration (Class I), and increased deformation with increasing restoration size, up to 20 mm for the Class II MOD. The calculated inter-cuspal distance changes (mean value of mesial and distal cusp pairs) were 5.3, 27.1 (10.1 distal), 46.8 (16.0 distal), and 45.5 mm for the Class I, Class II OM (small), Class II OM (large), and Class II MOD restorations, respectively. Note that in the OM-configurations the values for the distal cusps are given separately because the inter-cuspal distance changes were much smaller between the distal cusps compared with the mesial cusps. The cuspal deflections for the large Class I and small Class II MOD were 10.8 and 25.5 mm, respectively. Residual stress distributions (modified von Mises equivalent stresses) at the tooth surfaces were visualized for four configurations using a linear
color scale to indicate their intensity (Fig. 5). The restorations were rendered invisible to view the tooth-restoration interface. Note that the shown interfacial stresses are extrapolated across the tooth-restoration interface. The distributions show areas of high stress concentrations (yellow) and low stress areas (blue). The color patterns illustrate that stress values were local conditions. In general, the Class I restoration showed a larger area of high stresses at the tooth-restoration interface than the Class IIs. However, the Class II MOD showed higher stresses in the enamel surfaces than the other restorations. The residual stresses inside the tooth, composite, and along the tooth-composite interface were also plotted in distribution graphs that visualize all stresses (Fig. 6). These graphs plot the total range of internal stresses (modified von Mises equivalent stresses), ranked in ascending order, for all restoration variations. Note that the graphs do not contain information about local stress patterns. The ratio between bonded and free surfaces are also indicated in the figure. The graphs show that in general the stress level in the tooth (enamel and dentin) is the highest for the Class II MODs, followed by the Class II OMs, while the lowest overall stress level in the tooth was found for the Class I restoration (Fig. 6A). The order is reversed if general stress levels in the restoration or along the tooth-restoration interface are considered (Fig. 6B and C).
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Figure 5 Residual shrinkage stress distributions (modified von Mises equivalent stresses) for the four restorations (Class I, small Class II OM, large Class II OM, and Class II MOD) calculated by the numerical polymerization model. To show the tooth-restoration interface, the restoration was rendered invisible.
Discussion The large number of publications about polymerization shrinkage of restorative composites indicates that it is still considered a serious clinical concern. Shrinkage stress has been associated with clinical
symptoms such as microfracture, microleakage and post-operative sensitivity. None of these symptoms, however, are direct measures of shrinkage stress. The alleged presence of shrinkage stresses is thus only known through indirect manifestations. One of them, tooth deformation, is often not
Figure 6 Stress distribution plots representing all internal stresses (modified von Mises equivalent stresses) ranked in ascending order in: (A) the tooth, (B) the restoration, and (C) the tooth-restoration interface. The stresses are displayed for two restoration sizes (small and large) of three configurations (Class I, Class II OM, and Class II MOD). The ratios between bonded and free restoration surfaces are listed.
Residual shrinkage stress distributions in molars after composite restoration
specified among the clinical concerns, possibly because it is too small to be perceived under normal clinical conditions. It is well accepted, however, that teeth deform under loads.15 Among all clinical symptoms, deformation offers the closest relationship with shrinkage stresses. Furthermore, deformation can be quantified, unlike many of the other symptoms that remain substantially subjective. Deformation in teeth is often quantified by measuring the flexure of cusps.16 – 20 Cusp flexure indicates the deformation at particular locations on the tooth crown. It has been shown, however, that cusp flexure may not be a reliable indication of actual shrinkage stress levels.21 As part of this study the complete pattern of occlusal tooth deformations were measured and visualized (Fig. 1)8 The deformation patterns registered how a total crown deforms for different restoration configurations. However, the expectation fueled by dental literature that Class I configurations have higher stresses than Class IIs seems to be reversed if residual stresses are correlated directly with the tooth deformation. Obviously, there is more to the assessment of residual shrinkage stresses in restored teeth than simply measuring deformation on a restored tooth (let alone extrapolation of stress conditions from simplified laboratory specimens). Stress is a physical condition which depends on the combination of all material properties involved, geometry, and boundary conditions. The interactions between them obey universal mechanical laws. Unfortunately, shrinkage stress considerations have thus become much more complicated because they must involve comprehensive computations. Residual stresses can be studied using finite element analysis, which is a widely used contemporary computational technique.22 Finite element analysis provides the framework to combine material properties, with geometry and boundary conditions according to universal mechanical laws. Using computational techniques is not merely an option, but has become inevitable to express and test our current insight of dental materials behavior. This contemporary modeling approach also has consequences for the design of experimental studies. Supporting experiments can be divided into two categories (as demonstrated by the design of this study): experimental determination of basic properties9 and experimental simulation for model validation.8 Basic property experiments are used to obtain material behavior relationships required in a computational model, such as elastic modulus and post-gel shrinkage in the current study. They often have to be determined as theoretical or empirical functions of their controlling factors,
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such as time. Simulation experiments do not determine single properties but rather a bulk response that incorporates the interaction between various basic properties, such as the deformation of a restored tooth which involves the interaction between dental tissues and restorative properties, anatomy, fixation, and restorative sequence. One of the complexities for understanding (and thus modeling) polymerization shrinkage is knowing what exactly happens to the mechanical properties during polymerization. The realization that not all shrinkage contributes to shrinkage stresses is fundamental. The concept of pre- and post-gel shrinkage or time-dependent viscosity was noted by Bowen in one of the earliest publications about silica-reinforced polymer properties.23 Although this concept has thus been known in restorative dentistry since the first introduction of composites, its consequences are only recently gaining more recognition. This may be explained by the increasing acceptance of polymerization stress modeling and because of observations that polymerization kinetics affect post-gel shrinkage and bulk stresses. The post-gel concept is based on a division of the total shrinkage into a component that does not cause stress development due to relief by viscous flow (pre-gel), and a component that causes stress development due to elastic storage of contraction deformation (post-gel). From a physical point of view, polymerization is a process of simultaneous resin densification and composite modulus change. The time-dependent relationship between contraction on the one hand and the viscous (loss) and elastic (storage) modulus on the other hand determines the potential contribution of the composite properties for the generation of residual shrinkage stresses. For the tooth model used in this study, the calculation using a time-dependent viscoelastic model proved prohibitively time-consuming for current hardware specifications because of the large number of time-steps required to manage the distributed time-dependent nonlinear property changes. Using the post-gel concept, however, allowed for shrinkage stresses and tooth deformations to be approached in a linear elastic analysis. A linear elastic analysis requires considerably less time-steps, and thus a significantly reduced solver time. The validity of the linear elastic approach was supported by the good agreement between numerical and experimental results. The shrinkage stress model was experimentally validated by the results of the strain gauge and the restored molar. At first sight, the strain gauge experiments9 may not seem an independent validation, since they also provided the data on which the used post-gel shrinkage relationship was based.
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However, the shrinkage model also contained light attenuation and elastic modulus equations. These were derived from different experiments, and were combined with the post-gel shrinkage in the numerical polymerization model. The very close results of the measured and calculated strains not only validated the numerical polymerization model, they also indicated that the experimental strain gauge data actually represented local strain values at the gauge surface (Table 1). This is an important observation because it supports the notion that, when corrected for the local light intensity at the gauge surface, the strain gauge measures post-gel shrinkage as a local rather than a bulk value as was initially presumed. The calculated tooth deformations (Fig. 4) also compared well in patterns and magnitudes with the experimentally determined deformations (Fig. 1). Deformation can be defined in different ways. In order to ensure identical definition and measurement for the numerical and experimental approaches, the deformation for both was quantified using the same software (Cumulus). Additionally, the calculated cuspal flexure data was in the same range as has been reported in the literature (16 – 45 mm).16 – 20 An important conclusion that can be drawn from the positive validation of the numerical polymerization model is that post-gel shrinkage must be used to calculate residual shrinkage stresses (and deformations) rather than total shrinkage. Using total shrinkage values, typically about five to ten times higher than postgel shrinkage, would have grossly overestimated the stresses (and deformations).21 Consequently, shrinkage stress cannot be approximated from shrinkage that is determined by displaced volume or weight methods. Since a correlation between post-gel and total shrinkage is improbable (post-gel shrinkage depends mainly on reaction kinetics), total shrinkage values are unsuitable for qualitative shrinkage stress estimations (e.g. in shrinkage properties rankings). Note that it is not our intention to nullify the value of total shrinkage measurements. For example, total shrinkage plays a decisive role in the direction of shrinkage;24 however, total shrinkage does not reflect shrinkage stress expectations. Strictly speaking, post-gel shrinkage cannot be extrapolated to predict shrinkage stresses either. As discussed before, stress is not a material property. The stresses that were calculated by the shrinkage model were the result of the combination of all properties (tooth and restoration), the whole anatomy (tooth and restoration), all boundary conditions (root and interfaces), and the complete restoration history (each incremental filling and
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polymerization action). Considering that each of these conditions affected the final stress and deformation pattern, it is unrealistic to expect an exact fit between the finite element model and the experiments. For example, although in both cases a mandibular molar was used, their shapes were not exactly identical. Note that this is not a finite element versus ‘reality’ issue, because the five molars investigated in the experiments were not exactly identical either. Other factors that could have caused variations in the results were the enamel and dentin distribution, the properties for the dental tissues, the shape of the cavities, the applied light intensity, the exact distance between light guide and restoration during curing, etc. Taking into consideration the inherent variation in experiments, validation is successfully achieved when a general similarity in deformation patterns and deformation values is demonstrated. The goal of polymerization shrinkage research is to understand and predict the development of residual shrinkage stresses. The calculated deformation was caused by residual stresses predicted by the numerical polymerization model. Having validated the deformation, the calculated stress distributions associated with them can be considered validated too. It was found that the stress levels in the tooth increased with increasing restoration size, while stresses in the restoration and along the tooth-restoration interface decrease (Figs. 5 and 6). This result can be explained by the increasing loss of dental hard tissues for increasing restoration sizes. Removal of dental hard tissue decreases the stiffness of the tooth. Residual shrinkage stresses are caused by the resistance of the tooth structure to shrinkage deformation of the bonded restoration. Therefore, a decreased stiffness of the supporting tooth structure decreases residual stresses in the restoration and toothrestoration interface. However, it also results in more tooth deformation and consequently higher stresses in the tooth. Deformation measurements of the tooth crown are therefore more indicative of residual stresses in the tooth structure than of shrinkage stresses in the composite restoration or across their interfaces. Note that the stresses at the enamel surfaces and the stresses across the tooth-restoration interface should not be compared with each other in absolute terms, since they involve different materials. Furthermore, the interfacial stress values were extrapolated across the tooth-restoration interface. Since the bonding layer has not been modeled separately, the interfacial stresses should only be viewed as a qualitative indication of an interfacial stress distribution that a bonding layer would encounter.
Residual shrinkage stress distributions in molars after composite restoration
Areas of high stresses are at a higher risk for failure. Although the overall stress level along the tooth-restoration interface was generally higher for the Class I configuration, the maximum stresses of the other configurations approached or met the maximum stress values of the Class I (Fig. 6). When composites were introduced, Bowen25 predicted that restoration type will affect the resulting stress levels. He noted that cavities with a large free surface in proportion to bonded cavity walls would develop stresses of lower magnitudes. The current study confirms Bowen’s early observation as a general rule-of-thumb for the stress state in the composite and the tooth-restoration interface. However, extrapolation of the ratio between bonded and free surfaces as a prediction for shrinkage stresses, as has been proposed by others, cannot hold, even for stress levels in the restoration and across the interface. Fig. 6 shows that the ratio between bonded and free surface areas does not consistently predict the stress levels. For example (Fig. 6C), the small versus large Class I restorations with almost identical ratios show large differences in the interfacial stress levels. Moreover, the bondedfree surface ratio of the large Class I is much higher than the ratio for the Class II OM, but their interfacial stress levels are similar. A last example is the large Class II OM, which although it has a higher bondedfree surface ratio than the small Class II MOD, seems to have higher overall interfacial stress levels. The validated residual shrinkage stress distributions that were determined in this study confirm that stress is a local property, which within one restored tooth may have high values in one location and low values in another. Shrinkage stresses cannot be captured by single values, such as cusp flexure or restoration configuration, but must be studied as a location dependent distribution that takes the whole restored tooth into consideration. In the introduction, various clinical symptoms were listed that have been associated with shrinkage stresses. Those symptoms also involve all aspects of the restored tooth, i.e. the restoration (marginal loss), the interface (adaptation, microleakage, secondary caries), and the tooth structure (propagation of enamel microcracks, post-operative sensitivity). Any discussion of residual shrinkage stresses should, therefore, involve the whole tooth, not just the composite restoration or composite properties.
Conclusions This study used deformation patterns—measured experimentally—to validate a biomechanical
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shrinkage model that was based on experimental polymerization data. Although more has to be done to unravel remaining less-understood polymerization sequences, the current study confirmed the validity of a linear elastic approach based on the post-gel shrinkage concept for the calculation of residual stresses in a restored tooth. It was shown that tooth deformation indicated the stress levels in the tooth, rather than in the restoration. Shrinkage stress could not be expressed into a single average value based on composite properties or restoration configuration alone, but had to be approached as a distribution that depended on the location and properties of the tooth and restoration, tooth and restoration geometry, tooth and restoration fixture, and the consecutive restorative procedures.
Acknowledgements Based in part on abstract No. 498, presented at the 80th IADR meeting in San Diego, March 6 –9, 2002. The authors thank Iryna B. Olson for her assistance with the Cumulus analysis. This study was supported by the Minnesota Dental Research Center for Biomaterials and Biomechanics, a Faculty Development Grant from Chulalongkorn University, and NIH/NIDR Grant R01-DE12225.
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restorative composite. Dent Mater 2003;update with PII: S0109-5641(03)00159-3. DeLong R, Pintado MR, Douglas WH. Measurement of change in surface contour by computer graphics. Dent Mater 1985; 1:27—30. Marshall DB, Noma T, Evans AG. A simple method for determining elastic-modulus-to-hardness ratios using Knoop indentation measurements. J Am Ceram Soc 1982;65: C175—6. Nomoto R, Uchida K, Hirasawa T. Effect of light intensity on polymerization of light-cured composite resins. Dent Mater J 1994;13:198—205. Suh BI. Controlling and understanding the polymerization shrinkage-induced stresses in light-cured composites. Compendium 1999;20(Suppl 25):S34—S41. Versluis A, Tantbirojn D, Douglas WH. Why do shear bond tests pull out dentin? J Dent Res 1997;76:1298—307. Morin D, DeLong R, Douglas WH. Cusp reinforcement by the acid-etch technique. J Dent Res 1984;63:1075—8. Pearson GJ, Hegarty SM. Cusp movement of molar teeth with composite filling materials in conventional and modified MOD cavities. Br Dent J 1989;166:162—5. Suliman AA, Boyer DB, Lakes RS. Cusp movement in premolars resulting from composite polymerization shrinkage. Dent Mater 1993;9:6—10.
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18. Suliman AA, Boyer DB, Lakes RS. Interferometric measurements of cusp deformation of teeth restored with composite. J Dent Res 1993;72:1532—6. 19. Meredith N, Setchell DJ. In vitro measurement of cuspal strain and displacement in composite restored teeth. J Dent 1997;25:331—7. 20. Alomari QD, Reinhardt JW, Boyer DB. Effect of liners on cusp deflection and gap formation in composite restorations. Oper Dent 2001;26:406—11. 21. Versluis A, Douglas WH, Cross M, Sakaguchi RL. Does an incremental filling technique reduce polymerization shrinkage stresses? J Dent Res 1996;75:871—8. 22. Korioth TWP, Versluis A. Modeling the mechanical behavior of the jaws and their related structures using finite element (FE) analysis. Crit Rev Oral Biol Med 1997; 8:90—104. 23. Bowen RL. Properties of a silica-reinforced polymer for dental restorations. J Am Dent Assoc 1963;66:57—64. 24. Versluis A, Tantbirojn D, Douglas WH. Do dental composites always shrink toward the light? J Dent Res 1998;77: 1435—45. 25. Bowen RL. Adhesive bonding of various materials to hard tooth tissues. VI. Forces developing in direct-filling materials during hardening. J Am Dent Assoc 1967;74:439—45.
http://www.intl.elsevierhealth.com/journals/dema
Residual shrinkage stress distributions in molars after composite restoration Antheunis Versluisa,*, Daranee Tantbirojnb, Maria R. Pintadoa, Ralph DeLonga, William H. Douglasa a
Department of Oral Science, Minnesota Dental Research Center for Biomaterials and Biomechanics, University of Minnesota, 16-212 Moos Tower, 515 Delaware Street SE, Minneapolis, MN 55455, USA b Department of Operative Dentistry, Chulalongkorn University, Bangkok, Thailand Received 3 December 2002; received in revised form 16 May 2003; accepted 29 May 2003
KEYWORDS Restorative composite; Polymerization shrinkage; Strain gauge; Microhardness; Post-gel shrinkage; Composite restoration; Shrinkage stress; Residual stress; Deformation
Summary Objective. Experimental measurements on various restoration configurations have shown that restored teeth deform under the influence of polymerization shrinkage, but actual residual stresses could not be determined. The purpose of this study was to calculate and validate shrinkage stresses associated with the reported tooth deformations. Methods. Three different restoration configurations were applied in a finite element model of a molar. The composite properties were based on experimentally determined composite behavior during polymerization. The occlusal deformation pattern and the residual stress states of the tooth, restoration, and tooth-restoration interface were calculated using a polymerization model based on the post-gel shrinkage concept. Reported strain gauge measurements and occlusal deformation patterns were used for validation. Results. The shrinkage stresses depended on the configuration and size of the restorations. The tooth’s resistance against polymerization shrinkage diminished with loss of dental hard tissue. Larger restorations resulted in lower stress levels in the restoration and tooth-restoration interface, but increased stresses in the tooth. The maximum stress values found for different configurations were not decisively different. Significance. The validated model indicated that shrinkage stress cannot be based on composite properties or restoration configuration alone, but has to be approached as a distributed pattern that depends on the location and on the properties of tooth and restoration, geometry, constraints, and restoration procedures. Tooth deformation was indicative of stresses in the tooth rather than in the restoration or across the tooth-restoration interface. Q 2003 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
Introduction *Corresponding author. Tel.: þ1-612-625-0950; fax: þ 1-612626-1484. E-mail address: [email protected]
It is widely accepted that polymerization shrinkage of current restorative composites cause residual stresses in restored teeth. These stresses are called
0109-5641/$ - see front matter Q 2003 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.dental.2003.05.007
Residual shrinkage stress distributions in molars after composite restoration
residual because after curing a restored tooth is left under stress even when there is no functional loading. The presence of residual stresses results in a changed behavior of the restored tooth, which may become evident in its clinical performance. Clinical symptoms associated with residual shrinkage stresses are inadequate adaptation, microcrack propagation, marginal loss, post-operative sensitivity, microleakage, and secondary caries.1 – 5 Stress is not a material property, but a local physical state that is derived from the combination of material properties, geometry, boundary conditions, and history. Since residual shrinkage stress is not a straightforward property that can be measured directly, its quantification in restorations remains a source of controversy. Various methods have been used to estimate residual shrinkage stresses, ranging from extrapolated shrinkage or load measurements in vitro to stress analyses in tooth shaped anatomies using photoelastic or finite element methods.6,7 Calculation of shrinkage stresses in a tooth-restoration complex is not trivial. One challenge is the description of the sequence of changes that take place in composite during polymerization. Another challenge is that stress depends on the geometry and mechanical properties of surrounding tissues. Many intricacies of the biomechanical manifestations of polymerization processes are still not well understood. As a result the development of a residual shrinkage stress model (i.e. the expression of our understanding of the event) is an interactive
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process with experimental observations. Deformation patterns of occlusal surfaces for restored molars due to shrinkage stresses have been determined and observed in a series of experiments.8 While deformation could be measured experimentally, the stress distribution must still be calculated using transient composite properties determined from another series of experiments.9 The purpose of this study was to develop and validate a polymerization model for the calculation of residual shrinkage stress distributions associated with the reported deformations, using the determined transient properties distributions. The validated model can consequently be used to investigate the shrinkage stresses in restored teeth, and study factors that affect them.
Material and methods Deformations of occlusal surfaces have been reported for a successive range of cavity preparations in extracted human molars:8 Class I, small Class II OM, large Class II OM, and Class II MOD. The deformation due to polymerization shrinkage was quantified by comparing the digitized occlusal surfaces before and after restoration (Fig. 1). To calculate corresponding residual stresses in the tooth, a finite element simulation was carried out for a similar combination of geometry, boundary conditions, material properties, and restoration procedures (history).
Figure 1 Cuspal deformation patterns (mm) for a molar determined in in vitro experiments for four consecutive restorations (Class I, small Class II OM, large Class II OM, and Class II MOD) using Cumulus software.8
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Figure 2 Finite element model of molar, indicating four simulated restorations (Class I, small Class II OM, large Class II OM, and Class II MOD), increments, and the direction of light curing (black arrows).
Four consecutive restorations were applied in a finite element model of a molar: Class I, small Class II OM, large Class II OM, and Class II MOD (Fig. 2). Additionally, a large Class I and small Class II MOD were created. The external molar shape was digitized using a contact stylus profilometer,10 while the internal enamel – dentin and pulp surfaces were estimated based on general anatomy. The molar root was fixed, simulating the embedded specimens of the corresponding experimental design.8 Orthotropic elastic properties were modeled for the enamel. The elastic moduli were 84 GPa (principal direction, perpendicular to the pulp surface) and 42 GPa, with a Poisson’s ratio of 0.30. The dentin elastic modulus was 18 GPa, with a Poisson’s ratio of 0.23. The composite properties used were based on experimental data9 obtained for Z100 Incisal shade (Lot 6BN, 3M ESPE, St Paul, MN, USA), the same as used for the companion study.8 Z100 is a commercially available universal restorative composite, containing bisGMA and TEGDMA resins with zirconia/silica fillers (66% inorganic filler loading by volume, particle size range of 3.5 – 0.01 mm). The relatively translucent composite was chosen to obtain good light penetration in the large restorations. The maximum elastic modulus and Poisson’s ratio for a fully cured composite were 20 GPa and 0.24, respectively. In this study a linear-elastic approach was validated for the simulation of residual shrinkage
stress development. A linear elastic approach allows a significant reduction of the stress calculation time, while the required properties that have to be determined are reduced to the elastic modulus and post-gel shrinkage. Both properties depend on the curing light intensity, exposure time, and storage time. In order to apply the properties in the finite element analysis, they have to be expressed in mathematical forms. The local light intensity I in the restoration was approximated using the light intensity of the curing light source, the effect on the light intensity due to the distance between the restoration and light guide, and the exponential relationship for light attenuation in the composite9 I ¼ I0 exp½2ðw=10Þ3 exp½21:19d 0:72
ð1Þ
where I0 is the light intensity of the curing light source (mW/cm2); w is the distance from lightguide to the restoration surface (mm); d is the depth in the composite (mm). In each composite element integration point (where stresses are determined by the finite element software) local properties were calculated as a function of local light intensity and time (light exposure and dark cure time). The reported correlation between Knoop microhardness and elastic modulus11 can be employed to calculate the local elastic modulus of the composite as a function of light energy and time. Local light energy density Y (mJ/cm2) is
Residual shrinkage stress distributions in molars after composite restoration defined as12,13 Y ¼ Itc
ð2Þ
where tc is the curing light exposure time (s). The empirical relationship found for the microhardness KMH is9 KMH¼ð66211exp½2t=3000Þð12exp½2ðY=2500Þ1:1 Þ ð3Þ where t is the total time since the start of curing (s). The first term describes how the microhardness values increase during the storage time t after light cure. The second term describes the exponential relationship between microhardness and light energy. The local elastic modulus E can be obtained by E ¼Emax KMH=KMHmax
ð4Þ
where Emax is the maximum attainable elastic modulus value (Emax ¼20 GPa), and KMHmax is the maximum microhardness value used to normalize KMH in Eq. (3) ðKMHmax ¼66Þ: Post-gel shrinkage values were obtained by the strain gauge method.9 Unlike microhardness, postgel shrinkage cannot be fully described by the applied light energy because it depends on the light intensity at which the polymerization reaction was initiated. The maximum value for the post-gel shrinkage amax can be expressed by the empirical relationship ðR . 0:999Þ
amax ¼ 5000ð1 2 exp½2ðI0 =500Þ0:22 Þ
ð5Þ
where I0 is the intensity of the light source with which the polymerization was initially initiated (mW/cm2). The polymerization process can be divided into two periods: light exposure and dark cure (after the light cure). Post-gel shrinkage during ðalight Þ and after ðadark Þ light exposure can be expressed by ðR . 0:97Þ
alight ¼ amax ð1 2 exp½2Y00:7 =ð450 þ 0:35I0 ÞÞ
ð6Þ
adark ¼ amax ð8 þ 0:06I0 Þexp½2ðY0 =1:161Þ0:186 ð1 2 exp½2{Y0 =ð858 þ 4:1I0 Þ}1:5 Þ
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(Eqs. (6) and (7)), using the intensity output of the curing light source rather than local intensity levels. A bulk property means that the acquired strain gauge data represents the response of the whole sample. Since post-gel shrinkage depends on the light intensity and light intensity is location dependent, the post-gel shrinkage also depends on the location. Therefore, the bulk value for post-gel shrinkage cannot be used in the finite element analysis. Considering that the strain gauge records the local deformation at the gauge surface, the post-gel shrinkage results (Eqs. (6) and (7)) can be easily adjusted by correction of the terms that contain the light intensity I0 ¼ FI Y0 ¼ FItc ¼ FY
ð9Þ ð10Þ
where F is the correction factor for the actual light intensity at the gauge surface. Using the light attenuation Eq. (1) the correction factor F is 6.1 (w ¼ 3 mm, d ¼ 1:75 mm). To validate the derived elastic modulus and post-gel shrinkage formulations the strain gauge design was also simulated in a finite element analysis. The strain gauge configuration was modeled by a 1.75 mm high composite sample placed on an 80 mm thick strain gauge (elastic modulus 6 GPa and Poisson’s ratio 0.3, manufacturer’s data) (Fig. 3). Light curing from above (light guide positioned 3 mm above composite sample) was simulated for the four curing light intensities used in the experimental setup9 (I0 ¼ 100; 300, 500, and 700 mW/cm2). The plane strain in a 1 mm2 area at the center of the strain gauge was recorded for validation with the experimental shrinkage strain results. In the restoration models, light cure was applied occlusally to the composite restorations (188 off for the proximal boxes) for 60 s per increment at 650 mW/cm2, mimicking the experimental study.8 The diameter of the simulated light guide was 12 mm. The Class II OM cavities were filled in two increments, the Class II MOD in three (Fig. 2).
ð7Þ
where I0 is the light intensity of the light source at which the composite polymerization is initiated (mW/cm2), and Y0 is the applied light energy ðI0 tc Þ: Note the difference between applied and local light intensity or energy. The applied values are the output of the curing light source. The post-gel shrinkage value a determined by the strain gauge becomes:
a ¼ alight þ adark
ð8Þ
Unlike the microhardness data, which were determined as local properties (Eq. (3)), the post-gel shrinkage results were determined as bulk properties
Figure 3 Finite element model of the strain gauge experiment,9 showing the element distribution for the quarter mesh.
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Additionally, a large Class I and a small Class II MOD were modeled. The time between increments was 60 s. It is important that the successive filling increments are not part of the analysis before they are placed. Eight-node isoparametric arbitrary hexahedral reduced integration elements were used. A custom program calculated and applied the material property relationships for enamel and composite, performed the restorative actions of filling and light cure, defined and submitted the restoration models to a finite element solver for calculation of stresses (MSC.Marc, MSC Software Co., Los Angeles, CA, USA), processed the results and created a convenient output format. The analyses were performed on a Compaq AlphaServer ES40 (Compaq, Houston, TX, USA). Modified von Mises equivalent stresses were used to express the stress conditions. The advantage of using equivalent von Mises stress is that a multidimensional stress distribution (e.g. six Cartesian stress components or three principal stresses) is expressed into one value, which eases the interpretation of the overall stress condition. The used equivalent stress is based on the well-known von Mises formulation, modified to take into account the difference between compressive and tensile strength for enamel, dentin, and the composite.14 Since cuspal deflection is often reported for in vitro studies, the inter-cuspal distance change were determined for the mesial and distal cusp pairs. The calculated occlusal deformations were compared with the experimentally determined occlusal deformation8 (Fig. 1), using a locally developed new
generation software package (Cumulus, copyright Regents of the University of Minnesota). Surface changes in Cumulus are determined perpendicular to the baseline surface taken before restoration. In order to import the finite element geometry in Cumulus, the before and after surfaces were isolated, refined and converted to object file format (Alias Wavefront Object Files). From the dense surface point-clouds, Cumulus created rendered surfaces. Differences between the before and after occlusal enamel surfaces were determined and visualized using a linear color scale. The quantitative color patterns were used to compare the experimental and finite element results. Cusp displacements (bucco-lingual direction) were determined to be compared with values reported in the literature.
Results To validate the polymerization modeling procedures, deformation was calculated in two finite element simulations (strain gauge simulation and occlusal deformation of a tooth crown), using the equations for light intensity, elastic modulus, and post-gel shrinkage derived from experimental data. The response of the strain gauge is not solely caused by post-gel shrinkage, but also by development of elastic modulus and distribution of properties due to the light attenuation in the composite. Results (microstrain) of the experimental data9 were compared with the results obtained by the finite element simulation (Table 1). The table
Table 1 Comparison of shrinkage strains measured by the strain gauge experiments9 (mean and standard deviation; n ¼ 5) and calculated by the numerical polymerization simulation of the strain gauge setup. Light intensity, I0 (mW/cm2)
100 100 100 100 300 300 300 300 500 500 500 500 700 700 700 700
Exposure time, tc (s)
10 20 40 60 10 20 40 60 10 20 40 60 10 20 40 60
Light energy, Y0 (mJ/cm2)
1000 2000 4000 6000 3000 6000 12,000 18,000 5000 10,000 20,000 30,000 7000 14,000 28,000 42,000
Shrinkage strain (mstrain) Experimental
Finite element analysis
451(89) 1227(31) 1703(16) 1844(40) 1825(60) 2280(65) 2560(75) 2641(49) 2224(95) 2664(69) 2849(68) 2907(73) 2602(80) 2889(86) 3060(44) 3136(81)
865 1315 1693 1926 1962 2283 2606 2777 2433 2665 2941 3077 2684 2868 3122 3241
The results were determined 2 min after polymerization for different combinations of light intensities ðI0 Þ and light exposure times ðtc Þ:
Residual shrinkage stress distributions in molars after composite restoration
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Figure 4 Cuspal deformation patterns (mm) for the simulated molar for four restorations (Class I, small Class II OM, large Class II OM, and Class II MOD) using the numerical polymerization model and Cumulus software.
shows a good agreement between the experimental and finite element data. Applying the material relationships to a tooth model, the occlusal deformations were calculated and visualized by color coding (Fig. 4) for comparison with the experimental results (Fig. 1) using the Cumulus software. Positive values are defined as higher surfaces, and negative values indicate lower surfaces compared to the baseline. The deformation pattern and values were similar in magnitude and distribution to the results found by the experimental method. The finite element analysis calculated occlusal deformation lower than 10 mm for the smallest restoration (Class I), and increased deformation with increasing restoration size, up to 20 mm for the Class II MOD. The calculated inter-cuspal distance changes (mean value of mesial and distal cusp pairs) were 5.3, 27.1 (10.1 distal), 46.8 (16.0 distal), and 45.5 mm for the Class I, Class II OM (small), Class II OM (large), and Class II MOD restorations, respectively. Note that in the OM-configurations the values for the distal cusps are given separately because the inter-cuspal distance changes were much smaller between the distal cusps compared with the mesial cusps. The cuspal deflections for the large Class I and small Class II MOD were 10.8 and 25.5 mm, respectively. Residual stress distributions (modified von Mises equivalent stresses) at the tooth surfaces were visualized for four configurations using a linear
color scale to indicate their intensity (Fig. 5). The restorations were rendered invisible to view the tooth-restoration interface. Note that the shown interfacial stresses are extrapolated across the tooth-restoration interface. The distributions show areas of high stress concentrations (yellow) and low stress areas (blue). The color patterns illustrate that stress values were local conditions. In general, the Class I restoration showed a larger area of high stresses at the tooth-restoration interface than the Class IIs. However, the Class II MOD showed higher stresses in the enamel surfaces than the other restorations. The residual stresses inside the tooth, composite, and along the tooth-composite interface were also plotted in distribution graphs that visualize all stresses (Fig. 6). These graphs plot the total range of internal stresses (modified von Mises equivalent stresses), ranked in ascending order, for all restoration variations. Note that the graphs do not contain information about local stress patterns. The ratio between bonded and free surfaces are also indicated in the figure. The graphs show that in general the stress level in the tooth (enamel and dentin) is the highest for the Class II MODs, followed by the Class II OMs, while the lowest overall stress level in the tooth was found for the Class I restoration (Fig. 6A). The order is reversed if general stress levels in the restoration or along the tooth-restoration interface are considered (Fig. 6B and C).
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Figure 5 Residual shrinkage stress distributions (modified von Mises equivalent stresses) for the four restorations (Class I, small Class II OM, large Class II OM, and Class II MOD) calculated by the numerical polymerization model. To show the tooth-restoration interface, the restoration was rendered invisible.
Discussion The large number of publications about polymerization shrinkage of restorative composites indicates that it is still considered a serious clinical concern. Shrinkage stress has been associated with clinical
symptoms such as microfracture, microleakage and post-operative sensitivity. None of these symptoms, however, are direct measures of shrinkage stress. The alleged presence of shrinkage stresses is thus only known through indirect manifestations. One of them, tooth deformation, is often not
Figure 6 Stress distribution plots representing all internal stresses (modified von Mises equivalent stresses) ranked in ascending order in: (A) the tooth, (B) the restoration, and (C) the tooth-restoration interface. The stresses are displayed for two restoration sizes (small and large) of three configurations (Class I, Class II OM, and Class II MOD). The ratios between bonded and free restoration surfaces are listed.
Residual shrinkage stress distributions in molars after composite restoration
specified among the clinical concerns, possibly because it is too small to be perceived under normal clinical conditions. It is well accepted, however, that teeth deform under loads.15 Among all clinical symptoms, deformation offers the closest relationship with shrinkage stresses. Furthermore, deformation can be quantified, unlike many of the other symptoms that remain substantially subjective. Deformation in teeth is often quantified by measuring the flexure of cusps.16 – 20 Cusp flexure indicates the deformation at particular locations on the tooth crown. It has been shown, however, that cusp flexure may not be a reliable indication of actual shrinkage stress levels.21 As part of this study the complete pattern of occlusal tooth deformations were measured and visualized (Fig. 1)8 The deformation patterns registered how a total crown deforms for different restoration configurations. However, the expectation fueled by dental literature that Class I configurations have higher stresses than Class IIs seems to be reversed if residual stresses are correlated directly with the tooth deformation. Obviously, there is more to the assessment of residual shrinkage stresses in restored teeth than simply measuring deformation on a restored tooth (let alone extrapolation of stress conditions from simplified laboratory specimens). Stress is a physical condition which depends on the combination of all material properties involved, geometry, and boundary conditions. The interactions between them obey universal mechanical laws. Unfortunately, shrinkage stress considerations have thus become much more complicated because they must involve comprehensive computations. Residual stresses can be studied using finite element analysis, which is a widely used contemporary computational technique.22 Finite element analysis provides the framework to combine material properties, with geometry and boundary conditions according to universal mechanical laws. Using computational techniques is not merely an option, but has become inevitable to express and test our current insight of dental materials behavior. This contemporary modeling approach also has consequences for the design of experimental studies. Supporting experiments can be divided into two categories (as demonstrated by the design of this study): experimental determination of basic properties9 and experimental simulation for model validation.8 Basic property experiments are used to obtain material behavior relationships required in a computational model, such as elastic modulus and post-gel shrinkage in the current study. They often have to be determined as theoretical or empirical functions of their controlling factors,
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such as time. Simulation experiments do not determine single properties but rather a bulk response that incorporates the interaction between various basic properties, such as the deformation of a restored tooth which involves the interaction between dental tissues and restorative properties, anatomy, fixation, and restorative sequence. One of the complexities for understanding (and thus modeling) polymerization shrinkage is knowing what exactly happens to the mechanical properties during polymerization. The realization that not all shrinkage contributes to shrinkage stresses is fundamental. The concept of pre- and post-gel shrinkage or time-dependent viscosity was noted by Bowen in one of the earliest publications about silica-reinforced polymer properties.23 Although this concept has thus been known in restorative dentistry since the first introduction of composites, its consequences are only recently gaining more recognition. This may be explained by the increasing acceptance of polymerization stress modeling and because of observations that polymerization kinetics affect post-gel shrinkage and bulk stresses. The post-gel concept is based on a division of the total shrinkage into a component that does not cause stress development due to relief by viscous flow (pre-gel), and a component that causes stress development due to elastic storage of contraction deformation (post-gel). From a physical point of view, polymerization is a process of simultaneous resin densification and composite modulus change. The time-dependent relationship between contraction on the one hand and the viscous (loss) and elastic (storage) modulus on the other hand determines the potential contribution of the composite properties for the generation of residual shrinkage stresses. For the tooth model used in this study, the calculation using a time-dependent viscoelastic model proved prohibitively time-consuming for current hardware specifications because of the large number of time-steps required to manage the distributed time-dependent nonlinear property changes. Using the post-gel concept, however, allowed for shrinkage stresses and tooth deformations to be approached in a linear elastic analysis. A linear elastic analysis requires considerably less time-steps, and thus a significantly reduced solver time. The validity of the linear elastic approach was supported by the good agreement between numerical and experimental results. The shrinkage stress model was experimentally validated by the results of the strain gauge and the restored molar. At first sight, the strain gauge experiments9 may not seem an independent validation, since they also provided the data on which the used post-gel shrinkage relationship was based.
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However, the shrinkage model also contained light attenuation and elastic modulus equations. These were derived from different experiments, and were combined with the post-gel shrinkage in the numerical polymerization model. The very close results of the measured and calculated strains not only validated the numerical polymerization model, they also indicated that the experimental strain gauge data actually represented local strain values at the gauge surface (Table 1). This is an important observation because it supports the notion that, when corrected for the local light intensity at the gauge surface, the strain gauge measures post-gel shrinkage as a local rather than a bulk value as was initially presumed. The calculated tooth deformations (Fig. 4) also compared well in patterns and magnitudes with the experimentally determined deformations (Fig. 1). Deformation can be defined in different ways. In order to ensure identical definition and measurement for the numerical and experimental approaches, the deformation for both was quantified using the same software (Cumulus). Additionally, the calculated cuspal flexure data was in the same range as has been reported in the literature (16 – 45 mm).16 – 20 An important conclusion that can be drawn from the positive validation of the numerical polymerization model is that post-gel shrinkage must be used to calculate residual shrinkage stresses (and deformations) rather than total shrinkage. Using total shrinkage values, typically about five to ten times higher than postgel shrinkage, would have grossly overestimated the stresses (and deformations).21 Consequently, shrinkage stress cannot be approximated from shrinkage that is determined by displaced volume or weight methods. Since a correlation between post-gel and total shrinkage is improbable (post-gel shrinkage depends mainly on reaction kinetics), total shrinkage values are unsuitable for qualitative shrinkage stress estimations (e.g. in shrinkage properties rankings). Note that it is not our intention to nullify the value of total shrinkage measurements. For example, total shrinkage plays a decisive role in the direction of shrinkage;24 however, total shrinkage does not reflect shrinkage stress expectations. Strictly speaking, post-gel shrinkage cannot be extrapolated to predict shrinkage stresses either. As discussed before, stress is not a material property. The stresses that were calculated by the shrinkage model were the result of the combination of all properties (tooth and restoration), the whole anatomy (tooth and restoration), all boundary conditions (root and interfaces), and the complete restoration history (each incremental filling and
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polymerization action). Considering that each of these conditions affected the final stress and deformation pattern, it is unrealistic to expect an exact fit between the finite element model and the experiments. For example, although in both cases a mandibular molar was used, their shapes were not exactly identical. Note that this is not a finite element versus ‘reality’ issue, because the five molars investigated in the experiments were not exactly identical either. Other factors that could have caused variations in the results were the enamel and dentin distribution, the properties for the dental tissues, the shape of the cavities, the applied light intensity, the exact distance between light guide and restoration during curing, etc. Taking into consideration the inherent variation in experiments, validation is successfully achieved when a general similarity in deformation patterns and deformation values is demonstrated. The goal of polymerization shrinkage research is to understand and predict the development of residual shrinkage stresses. The calculated deformation was caused by residual stresses predicted by the numerical polymerization model. Having validated the deformation, the calculated stress distributions associated with them can be considered validated too. It was found that the stress levels in the tooth increased with increasing restoration size, while stresses in the restoration and along the tooth-restoration interface decrease (Figs. 5 and 6). This result can be explained by the increasing loss of dental hard tissues for increasing restoration sizes. Removal of dental hard tissue decreases the stiffness of the tooth. Residual shrinkage stresses are caused by the resistance of the tooth structure to shrinkage deformation of the bonded restoration. Therefore, a decreased stiffness of the supporting tooth structure decreases residual stresses in the restoration and toothrestoration interface. However, it also results in more tooth deformation and consequently higher stresses in the tooth. Deformation measurements of the tooth crown are therefore more indicative of residual stresses in the tooth structure than of shrinkage stresses in the composite restoration or across their interfaces. Note that the stresses at the enamel surfaces and the stresses across the tooth-restoration interface should not be compared with each other in absolute terms, since they involve different materials. Furthermore, the interfacial stress values were extrapolated across the tooth-restoration interface. Since the bonding layer has not been modeled separately, the interfacial stresses should only be viewed as a qualitative indication of an interfacial stress distribution that a bonding layer would encounter.
Residual shrinkage stress distributions in molars after composite restoration
Areas of high stresses are at a higher risk for failure. Although the overall stress level along the tooth-restoration interface was generally higher for the Class I configuration, the maximum stresses of the other configurations approached or met the maximum stress values of the Class I (Fig. 6). When composites were introduced, Bowen25 predicted that restoration type will affect the resulting stress levels. He noted that cavities with a large free surface in proportion to bonded cavity walls would develop stresses of lower magnitudes. The current study confirms Bowen’s early observation as a general rule-of-thumb for the stress state in the composite and the tooth-restoration interface. However, extrapolation of the ratio between bonded and free surfaces as a prediction for shrinkage stresses, as has been proposed by others, cannot hold, even for stress levels in the restoration and across the interface. Fig. 6 shows that the ratio between bonded and free surface areas does not consistently predict the stress levels. For example (Fig. 6C), the small versus large Class I restorations with almost identical ratios show large differences in the interfacial stress levels. Moreover, the bondedfree surface ratio of the large Class I is much higher than the ratio for the Class II OM, but their interfacial stress levels are similar. A last example is the large Class II OM, which although it has a higher bondedfree surface ratio than the small Class II MOD, seems to have higher overall interfacial stress levels. The validated residual shrinkage stress distributions that were determined in this study confirm that stress is a local property, which within one restored tooth may have high values in one location and low values in another. Shrinkage stresses cannot be captured by single values, such as cusp flexure or restoration configuration, but must be studied as a location dependent distribution that takes the whole restored tooth into consideration. In the introduction, various clinical symptoms were listed that have been associated with shrinkage stresses. Those symptoms also involve all aspects of the restored tooth, i.e. the restoration (marginal loss), the interface (adaptation, microleakage, secondary caries), and the tooth structure (propagation of enamel microcracks, post-operative sensitivity). Any discussion of residual shrinkage stresses should, therefore, involve the whole tooth, not just the composite restoration or composite properties.
Conclusions This study used deformation patterns—measured experimentally—to validate a biomechanical
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shrinkage model that was based on experimental polymerization data. Although more has to be done to unravel remaining less-understood polymerization sequences, the current study confirmed the validity of a linear elastic approach based on the post-gel shrinkage concept for the calculation of residual stresses in a restored tooth. It was shown that tooth deformation indicated the stress levels in the tooth, rather than in the restoration. Shrinkage stress could not be expressed into a single average value based on composite properties or restoration configuration alone, but had to be approached as a distribution that depended on the location and properties of the tooth and restoration, tooth and restoration geometry, tooth and restoration fixture, and the consecutive restorative procedures.
Acknowledgements Based in part on abstract No. 498, presented at the 80th IADR meeting in San Diego, March 6 –9, 2002. The authors thank Iryna B. Olson for her assistance with the Cumulus analysis. This study was supported by the Minnesota Dental Research Center for Biomaterials and Biomechanics, a Faculty Development Grant from Chulalongkorn University, and NIH/NIDR Grant R01-DE12225.
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