Influence of dental restorations and mastication loadings on dentine fatigue behaviour: Image-based modelling approach.

JJOD-2428; No. of Pages 12 journal of dentistry xxx (2015) xxx–xxx

Available online at www.sciencedirect.com

ScienceDirect journal homepage: www.intl.elsevierhealth.com/journals/jden

Influence of dental restorations and mastication loadings on dentine fatigue behaviour: Image-based modelling approach Arso M. Vukicevic a,b, Ksenija Zelic c, Gordana Jovicic a,*, Marija Djuric c, Nenad Filipovic a,b a

Faculty of Engineering, University of Kragujevac, Sestre Janjic 6, 34000 Kragujevac, Serbia Bioengineering Research and Development Center Kragujevac, Prvoslava Stojanovica 6, 34000 Kragujevac, Serbia c University of Belgrade – School of Medicine, Institute of Anatomy, Laboratory for Anthropology, 4/2 Dr Subotica, 11000 Belgrade, Serbia b

article info

abstract

Article history:

Objectives: The aim of this study was to use Finite Element Analysis (FEA) to estimate the

Received 4 December 2014

influence of various mastication loads and different tooth treatments (composite restora-

Received in revised form

tion and endodontic treatment) on dentine fatigue. The analysis of fatigue behaviour of

9 February 2015

human dentine in intact and composite restored teeth with root-canal-treatment using FEA

Accepted 23 February 2015

and fatigue theory was performed.

Available online xxx

Methods: Dentine fatigue behaviour was analysed in three virtual models: intact, composite-

Keywords:

composite was modelled by thermal expansion in a heat transfer analysis. Low and high

restored and endodontically-treated tooth. Volumetric change during the polymerization of Human dentine

shrinkage stresses were obtained by varying the linear shrinkage of composite. Mastication

Image-based modelling

forces were applied occlusally with the load of 100, 150 and 200 N. Assuming one million

Finite element method

cycles, Fatigue Failure Index (FFI) was determined using Goodman’s criterion while residual

Fatigue

fatigue lifetime assessment was performed using Paris-power law.

Mastication

Results: The analysis of the Goodman diagram gave both maximal allowed crack size and maximal number of cycles for the given stress ratio. The size of cracks was measured on virtual models. For the given conditions, fatigue-failure is not likely to happen neither in the intact tooth nor in treated teeth with low shrinkage stress. In the cases of high shrinkage stress, crack length was much larger than the maximal allowed crack and failure occurred with 150 and 200 N loads. The maximal allowed crack size was slightly lower in the tooth with root canal treatment which induced somewhat higher FFI than in the case of tooth with only composite restoration. Conclusions: Main factors that lead to dentine fatigue are levels of occlusal load and polymerization stress. However, root canal treatment has small influence on dentine fatigue. Clinical significance: The methodology proposed in this study provides a new insight into the fatigue behaviour of teeth after dental treatments. Furthermore, it estimates maximal allowed crack size and maximal number of cycles for a specific case. # 2015 Elsevier Ltd. All rights reserved.

* Corresponding author at: Sestre Janjic 6, Kragujevac 34000, Serbia. Tel.: +381 34334379; fax: +381 34333192. E-mail address: [email protected] (G. Jovicic). http://dx.doi.org/10.1016/j.jdent.2015.02.011 0300-5712/# 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Vukicevic AM, et al. Influence of dental restorations and mastication loadings on dentine fatigue behaviour: Image-based modelling approach. Journal of Dentistry (2015), http://dx.doi.org/10.1016/j.jdent.2015.02.011

JJOD-2428; No. of Pages 12

2

journal of dentistry xxx (2015) xxx–xxx

1.

Introduction

As the third most common cause of tooth loss after dental caries and periodontal disease, tooth fracture has a profound influence on dental health care.1,2 According to the clinical reports, root canal treatment is specified as a major cause of tooth fracture.3,4 Furthermore, cavity preparation, followed by tooth restoration also influences the tooth strength. Since dental treatments lead to tissue loss, it is considered that changes in the geometry and replacement of tooth tissues with artificial materials are one of the major factors inducing increased risk of failure.4–6 From the biomechanical aspect, the failure may occur due to a single load that exceeds the strength of the tissue or due to the fatigue caused by a cyclic loading.7,8 The focus of this study was on the fatigue caused by habitual loading – mastication, where the intensity of a single load is much lower than the intensity of the critical breaking force.9 Cyclic mastication load causes cyclic stress changes which, over time, may cause degradation of mechanical properties, initiation and growing of micro-cracks and, consequently, tooth fracture – ‘‘fatigue failure’’. Tooth fracture is often related to dentine failure, since this tissue occupies the majority of the tooth.

Most of the studies focusing on dentine fatigue were based on in vitro experiments performed on standard test specimens cut from the bulk of dentine.10–14 These studies contributed to better understanding of dentine as a material, but they did not analyse dentine as a part of a complex tooth structure. Moreover, the influence of dentine structure deterioration and mechanical properties caused by fatigue still remains unclear. It is difficult to perform multiple physical tests on a single specimen. In addition, precise measuring of physical quantities (such as stress, strain, displacements, temperature, etc.) requires a very expensive equipment and usually may not provide results for a complex tooth structure. In such situations, Finite Element Analysis (FEA or FEM) is reported to be a very cost-effective tool.15,16 In literature, FEA has been widely used for: tooth stress analysis,17 implants design and optimization,18–20 modelling and optimization of restorations,21–24 evaluation of fatigue lifetimes before crack failure,25–27 to name just a few applications. In analysing the influence of composite restoration on tooth strength, it is important to take into consideration the occurrence of residual (shrinkage) stress.28 Shrinkage stress has been investigated both numerically and experimentally.29 Depending on the type of restoration and materials used for it,

Fig. 1 – Overall procedure. Please cite this article in press as: Vukicevic AM, et al. Influence of dental restorations and mastication loadings on dentine fatigue behaviour: Image-based modelling approach. Journal of Dentistry (2015), http://dx.doi.org/10.1016/j.jdent.2015.02.011


it has been found that values of shrinkage stress vary from low values to values which may influence the tooth strength.30,31 The aim of this study was to propose a procedure for numerical investigation of fatigue behaviour of human dentine in physiological conditions. Particularly, the goal was to use FEA to estimate the influence of various mastication loads and different tooth treatments (composite restoration and endodontic treatment) on dentine fatigue.

2.

Methods

2.1.

Obtaining teeth geometry

Finite element models of teeth were based on computerized tomography scans (Fig. 1a–d). With the institutional review board approval, two maxillary second premolars obtained from the same person with similar morphology (confirmed on dental retroalveolar radiograms) were included in this study. One was left intact while the other one underwent mesio-occlusal (MO) Class II preparation followed by an endodontic access opening and root canal enlargement. Both teeth were scanned using Computerized Tomography (Siemens Somatom Sensation 16, Munich, Germany). After the acquisition, Mimics 10.01 software

3

(Materialize, Leuven, Belgium) was used for image segmentation and generation of surface meshes for each of the tooth materials Fig. 2a–c. Additional mesh refinement and assembly of different parts of models were performed in Geomagic Studio 10 (Geomagic GmbH, Stuttgart, Germany). Model 1 was based on the CT scans of an intact tooth. It consisted of four parts: enamel, dentine, pulp tissue in the pulp chamber and periodontal ligament (Fig. 2a). Periodontal ligament (PDL) was modelled as a 250 mm thick shell which started from 2 mm apically to the cemento-enamel junction and surrounds the tooth root. Surrounding bone was not modelled since PDL’s external nodes were constrained. Models 2 and 3 represented two post-treatment cases. Model 2 included mesio-oclusal (MO) Class II cavity preparation restored with composite resin (Fig. 2b). Model 3, was based on CT scans of endodontically treated premolar and included MO Class II cavity preparation restored with composite resin, an access opening to the pulp chamber and widened root canals, filled with cold gutta percha (Fig. 2c).

2.2.

FEM procedures

For each part of the developed models, 3D volume discretization was carried out using the TetGen (Hang Si, WIAS, Berlin,

Fig. 2 – FEM procedures: (a) considered models, (b) boundary conditions and (c) sketch of stress changes caused by restoration and mastication. Please cite this article in press as: Vukicevic AM, et al. Influence of dental restorations and mastication loadings on dentine fatigue behaviour: Image-based modelling approach. Journal of Dentistry (2015), http://dx.doi.org/10.1016/j.jdent.2015.02.011


4


Table 1 – Material characteristics of the considered dental materials. Material

Pulp Dentine Enamel PDL Composite resin Gutta-percha

Young’s modulus [MPa]

Poison’s ratio

Thermal conductivity [J m1 s1 8C1]

Density [kg m3]

Specific heat [J kg1 8C1]

Thermal expansion [(m/m)1 8C1]

6.8 18.6e+3 84.10e+3 0.68 16.6e+3 70

0.45 0.31 0.3 0.45 0.24 0.40

0.67 0.59 0.93 – 1.087 332.4

1000 1960 2800 – 2100 2700

4200 1600 712 – 200 1042

1.01e5 1.01e5 1.15e5 – 2.70e5 16.20e5

Germany) meshing software. The obtained four-nodal tetrahedral elements were then split into the eight-node hexahedra (brick elements, Fig. 1e) and appropriate materials were assigned to each part according to Table 1.32–35 All the materials were assumed to be homogenous, isotropic and linear. The overall FEA analysis was performed using in-house PAK software (University of Kragujevac, Serbia), which has been used and validated on the various problems in biomechanics.16,36–39 The polymerization shrinkage was modelled as thermal expansion in a heat transfer analysis. Following the literature, Young’s modulus-curing depth relation was defined as E = EZ250 EZ250e0.722d, where E is the Young’s modulus after polymerization and d is the depth from the irradiated resin surface (Fig. 2d).21 The restoration with high shrinkage stress was obtained after reaching 0.5% linear shrinkage of the composite material.40 The restoration with low shrinkage stress was obtained after reaching 0.05% linear shrinkage. Mastication forces were assumed to be 100 N, 150 N and 200 N. The number of cycles was 1,000,000 cycles for each load intensity which represents an approximate number of chewing cycles during 4 years.41 The load was applied on the buccal and lingual cup surfaces simultaneously (the half of the load was applied on each site) to obtain the occlusal load parallel to the long axis of the tooth (Fig. 2e). Constraints were applied to the nodes on the outer surface of the PDL, restricting the displacements in all three directions (Fig. 2e).

smi, ði ¼ 1; 3Þ are principal nominal mean stresses smi = (smaxi + smini)/2. Regarding the ratio R = smin/smax, it is important to mention that realistic values were considered since smini corresponds to shrinkage stress (for intact tooth smini ffi 0) and smaxi corresponds to the different cases of mastication (Fig. 2f). Places with values of FFI less than 1 were assumed to be safe from the fatigue failure (allowable operating zone in Fig. 1i). Places with FFI values higher than 1 were assumed to be the places of potential crack initiation and growth (dangerous operating zone in Fig. 1i). Three distinctive phases of residual fatigue crack growth are shown in Fig. 3.45 The subject of this study was to estimate the period of dentine stable crack growth (region II) – particularly, the study aims to find the number of mastication cycles before dentine failure (which starts with an unstable crack growth – region III). From Fig. 3, stable crack growth may be defined as the period between crack initiation (region I) and unstable crack growth (region III). In this study, it was assumed that the period of stable crack growth should last at least one million mastication cycles (or approximately 4 years). For the numerical analysis of dentine stable fatigue crack growth, Paris law da/dN = C(DK)m was used.46 Parameters: da

2.3. Dentine fatigue assessment and residual lifetime prediction In this paper, dentine fatigue assessment was performed by using S–N approach with the assumption of no pre-existing flaws.42 The Goodman’s Fatigue Failure Index (FFI) was calculated as: sqa/se + sqm/su = FFI, where endurance strength se = 50 MPa and ultimate stress su = 160 MPa were experimentally determined material properties of dentine taken from the literature.11,43 Since the stresses caused by shrinkage and mastication are usually multiaxial with dominant tensile stresses in the dentine,34 the Equivalent Stress Theory (EST) was used as a reliable one for the multiaxial fatigue analysis of materials with ductile behaviour.44 According to the EST, equivalent nominal stress amplitude was computed as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s qa ¼ ½ðs a1 s a2 Þ2 þ ðs a2 s a3 Þ2 þ ðs a3 s a1 Þ2 =2, where sai, ði ¼ 1; 3Þ are principal alternating nominal stresses sai = (smaxi smini)/2. Equivalent nominal mean stress was calculated as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s qm ¼ ½ðs m1 s m2 Þ2 þ ðs m2 s m3 Þ2 þ ðs m3 s m1 Þ2 =2, where

Fig. 3 – Three characteristic phases of fatigue crack growth (region I – crack initiation, region II – stable crack growth, region III – unstable crack growth).



5


represents the incremental changes in crack length (Da); dN represents the number of cycles (DN); C and m are fatigue crack growth coefficient and exponent, respectively; DK is stress intensity range. The stress intensity range is defined as: DK = YDs(pa)1/2, where Y is correction factor; Ds is far-field stress range ahead the crack tip and a is crack length. Starting from the Paris power law, the number of remaining loading cycles N (in the phase of stable crack growth) was estimated as a function of the current crack length a as: 1m=2 a1m=2 Þ, where a0 repreN ¼ ð2=ðm 2ÞCYm ðDsÞm pm=2 Þða0 sents the initial crack size and Ds = (smax smin) is the stress range.47 The initial crack size was estimated as a0 = p1(Kth/ Yse)2 and it corresponds to the minimal value of crack size from which Paris power law may be applied. Parameter Kth is fatigue threshold; se is endurance limit. For dentine, the pffiffiffiffiffi fatigue threshold parameter has a value of Kth ¼ 1:06 MPa m, Y = 1.12 for a shallow flaw, fracture toughness Kc = 1.8 MPa m, while scaling constant parameters are C = 6.24 and m = 8.76.11 By using the previous equations and parameters, the value of initial crack size was: a0 = 107 mm, while the critical crack size was calculated from the fracture toughness as: aC = p1(KC/ Yse)2 = 312 mm. Therefore, unstable crack growth (which may cause fatigue fracture) will occur when the crack size reaches the critical value ac – and this moment will be reached after N cycles (which can be calculated from the equation given above).

3.

Results

Mesh-independency was reached at approximately 300,000 elements for all three models. The results obtained with the FEA are presented in terms of the Von Mises Stress (VMS), Fatigue Failure Index (FFI) and stress ratio (R). The cases considered to be representative for the results are given in Figs. 4–7 while overall results are given in Table 2. For the models given in Figs. 4–7 with FFI > 1, the diagrams for residual life-time prediction are shown in Fig. 8. Considering the Model 1 (intact tooth), the results showed that there was no risk of fatigue-failure (Fig. 4) even for the worst mastication scenario (loads of 200 N resulted with

FFI = 0.42 which is far away from the dangerous zone). Assuming that, it may be confirmed that dentine in healthy intact tooth is not susceptible to failure under normal oral cyclic loading conditions (100, 150 and 200 N). Regarding the two treated cases (Models 2 and 3), residual shrinkage stress (Fig. 5) was considered to be the minimum mastication stress, while the maximum stress corresponded to the different levels of mastication loadings (Figs. 6 and 7d–f, k). In this study, shrinkage stress was observed as a pre-stress caused by restoration shrinkage and it existed in models with composite restorations (Models 2 and 3) regardless of the occlusal loads. On this basis, the values of stress ratios were obtained (Figs. 6 and 7g–i, l) as quotient of the shrinkage stress caused by restorative treatments and mastication stress. On the other hand, minimum mastication stress for Model 1 was assumed to be close to zero smin ffi 0, so that obtained value of stress ratio was also R ffi 0. In this study we assumed that, before restorations, teeth had zero-stress from which fatigue was measured. From the results, it may be observed that in treated teeth (Models 2 and 3), failure occurred in the case of high shrinkage stress and mastication loads of 150 N (Figs. 6j–l and 7j–l). On the other hand, in the case of low shrinkage stress, both models were far away from the failure even for mastication loads of 200 N (Figs. 6c, f, i and 7c, f, i), similarly as for the intact tooth. By comparing FFI of the two postoperative models with low shrinkage stress and mastication loads of 150 N, it was found that Model 3 (Fig. 7b) showed increased risk of failure in comparison to the Model 2 (Fig. 6b). However, both models were far away from the unsafe zone and it may be concluded that, for low shrinkage stress, root canal treatment does not significantly contribute to fatigue failure. On the other side, if we considered the same mastication conditions (loads 150 N) the risk of fatigue failure increased to FFI = 1.08 for Model 2 (Fig. 6j) and FFI = 1.18 for Model 3 (Fig. 7j) with high shrinkage stress. Under these conditions, tooth with only two-surface restoration and no root canal treatment entered the dangerous zone although the FFI is just a bit above 1 (FFI = 1.08). Thus, only in situations when restorations resulted in a significant level of shrinkage stress, mastication loads and root canal treatment may significantly contribute to the fatigue failure of dentine.

Table 2 – Overall FEA results. Model 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3

Loads [N]

Shrinkage stress [MPa]

Mastication Stress [MPa]

R

FFI

ac (106 cycles) [mm]

100 150 200 100 150 200 100 150 200 100 150 200 100 150 200

0 0 0 6.5 6.5 6.5 105.3 105.3 105.3 10 10 10 117.2 117.2 117.2

19.4 28.9 36.2 26.6 35.3 44.6 130.4 138.1 149.5 37.5 52.7 66.9 139.3 159.9 165.8

0 0 0 0.29 0.26 0.17 0.78 0.73 0.64 0.31 0.27 0.19 0.87 0.83 0.78

0.23 0.31 0.45 0.31 0.40 0.52 0.97 1.08 1.23 0.35 0.57 0.64 1.03 1.18 1.30

148 118 97 144 115 94 125 110 79 131 108 86 123 96 62



6


Fig. 4 – FEA results for Model 1 (VMS – Von Mises Stress, FFI – Fatigue Failure Index).

For the Models 2 and 3 with FFI > 1, whose results are given in Figs. 6 and 7, Goodman diagrams were constructed and presented in Fig. 8a and b. As it was previously mentioned, the triangle on Goodman diagram represents the nodes safe from fatigue failure (FFI > 1). The points outside the triangle indicate places where fatigue failure may occur. It may be noted that slightly higher number of such points appear in Model 3 compared to Model 2, which is in accordance with the

obtained FFI values in Figs. 6 and 7. Moreover, for the models with FFI > 1 the residual life-time prediction was calculated following the equations in Section 2.3 considering that the initial flaw had the stable crack-growth (Fig. 8c). From the obtained results, in the region with FFI > 1 stress range for Model 2 (F = 150 N) was approximately 30.5 MPa and for Model 3 (F = 150 N) was 36.7 MPa. Assuming that cracks existed in the regions with FFI 1 (or that such regions were places of

Fig. 5 – Developed shrinkage stresses for the considered restoration cases (a – low shrinkage, Model 2, c – low shrinkage stress Model 3, b – high shrinkage stress, Model 2, d – high shrinkage stress Model 3. Please cite this article in press as: Vukicevic AM, et al. Influence of dental restorations and mastication loadings on dentine fatigue behaviour: Image-based modelling approach. Journal of Dentistry (2015), http://dx.doi.org/10.1016/j.jdent.2015.02.011


7

Fig. 6 – FEA results for Model 2; a–i low shrinkage stress cases, j–l high shrinkage stress cases (VMS – Von Mises Stress, FFI – Fatigue Failure Index, R – stress ratio) (For interpretation of the references to color in the text citation, the reader is referred to the web version of the article.).

potential crack initiation and growth), the remaining number of cycles until fatigue failure could be estimated from Fig. 8 from the length of these regions. According to the diagram, maximal allowed size of crack (which will not result in fatigue failure for one million cycles) is 110 mm for Model 2 and 96 mm for Model 3 (Table 2). For the crack lengths measured in vivo, in vitro or on virtual models, the number of remaining mastication cycles may be measured on the vertical axis if the

stress range is known. The stress range is given by the FE analysis. For the two cases which lifetime diagrams are shown in Fig. 8, damage lengths (regions with FFI > 1) were 693 mm for Model 2 and 753 mm for Model 3 (Figs. 6j and 7j). Since the measured damage lengths were bigger than the maximal allowed, it can be concluded that maximal allowed shrinkage stress which (under considered mastication conditions) will not cause dentine fatigue failure should be less than the high



8


Fig. 7 – FEA results for Model 3; a–i low shrinkage stress cases, j–l high shrinkage stress cases (VMS – Von Mises Stress, FFI – Fatigue Failure Index, R – stress ratio).

shrinkage stress adopted in this study (Fig. 5b and d). Regarding the cases of high shrinkage stress which are not presented in Figs. 6–8 (Models 2 and 3, loads F = 100 N and F = 200 N), for the developed shrinkage stress above 100 MPa variations in mastication loads (100–200 N) do not cause major changes in term of fatigue safety, since all cases are in the dangerous zone (Table 2). For that reason only cases of F = 150 N were shown as representative examples while the rest are given in Table 2.

4.

Discussion

As for the most biological materials, teeth (dentine) fatigue is manifested through degradation of its mechanical properties over a long time of exploitation. Starting from the characteristics of dentine adopted from the literature and assuming habitual mastication, this paper presents the procedures for numerical analysis of dentine fatigue in restored teeth. The procedure is carried out through the stress-based fatigue



9

Fig. 8 – Fatigue failure diagrams: (a, b) Goodman’s diagrams for Models 2 and 3 (F = 150 N and increased shrinkage stress); (c) residual fatigue lifetime vs. crack size for different levels of fatigue loading.

total-life assessment and residual lifetime prediction for fatigue crack growth. The results indicated that the key contribution in dentine fatigue was not attributed to mastication (amplitude) stress solely, but also to the shrinkage stress in the cases of composite restorations. High shrinkage stress was found to be very influential on dentine fatigue. It is also shown that there are only small differences in fatigue behaviour between tooth with only two-surface composite restoration and tooth model with root canal treatment and the same composite restoration. Previous studies focusing on tooth and dentine fatigue used different approaches-experimental fatigue analysis of whole teeth, experimental fatigue analysis of dentine and numerical fatigue analysis of whole teeth. Batalha-Silva et al. investigated fatigue resistance in teeth with large MOD composite restorations under loads of 200 N (5000 cycles) followed by 400, 600, 800, 1000, 1200 and 1400 N (30,000 cycles for each – 18,500 cycles altogether).48 They used artificial mouth to simulate the masticatory process and the cracks of enamel were detected by photographing the teeth with 1.5 magnification and transilluminator. They recorded enamel cracks after the restoration and for direct restoration they reported the occurrence of cracks larger than 3 mm in 40% while in teeth with inlay there were no cracks. After the fatigue testing, all teeth with MOD inlay restoration survived all 185,000 cycles but survival in teeth with direct composite restoration was only 13%. This study showed the importance of preexisting cracks which are the consequence of the shrinkage stress in

direct composite restoration. This is in agreement with the conclusions of our study indicating that shrinkage stress has an important role in the process of fatigue failure. However, according to the literature, the number of cycles in the mentioned study48 corresponds to less than 1 year of mastication31,49 which is not appropriate because restorations are considered to be successful if they last at least 4 years (one million cycles) or more.49 Thus, as in vitro fatigue testing is complicate and long lasting, the use of FEA analysis for this purpose is very convenient. Another study used the same study design with the aim to detect cracks in the restorations placed on the occlusal surface of highly eroded posterior teeth (so called ‘‘occlusal veneers).50 They recorded cracks larger than 2 mm. However, in this study, simulation of the periodontal ligament was omitted because elastomers and silicone which are usually used for this purpose show accelerated degradation. Without the PDL, results of stress distribution in alveolar bone and tooth could be very different comparing to the real values and this can have an impact on tooth fatigue behaviour. Again, FEA has an advantage because PDL could be modelled making the results more reliable. Moreover, in both studies the smallest cracks which could be detected using the proposed study design were 2 mm while in our study using FEA the size of cracks was less than 1 mm. In addition, the approach of incremental increasing of force during the fatigue test used in both studies48,49 may hide the answer to the question ‘‘Was the specimen fractured as a consequence of a high occlusal load (critical breaking force)



10


or it was a fatigue-fracture?’’ The present study is the continuation of previously reported researches where FEA was used to analyse the stress behaviour of the same teeth models under critical breaking force.34,35 The results showed that in Model 1 (intact tooth), under critical breaking force of 1025 N, developed stress was 187 MPa; For the Model 3 (restored tooth with root canal treatment) critical force was 710 N and developed stress was 179 MPa.34,35 Thus, the approach in our study, which used FEA, more clearly identified the cause of the tooth failure since it enables the testing of various causes on the same model. The question which is of particular interest to clinicians is ‘‘Assuming normal mastication, what size of the cracks may be tolerated without the risk of failure and for how long?’’ The procedure proposed in this study gives an answer to this question. The analysis of the residual lifetime diagram (Fig. 8) gives both maximal allowed crack size and maximal number of cycles for known stress range (which is given by the FE simulation). Moreover, the size of the crack is measured on virtual models (the red region Fig. 6j). This way, our approach provides the estimation of maximal allowed crack size for a given case, and measuring the crack produced during the fatigue simulation. Another group of studies solely focused on experimental fatigue analysis of dentine. Commonly, experiments were performed on specimens cut from the bulk of dentine and analysed under controlled conditions described with standardized procedures for fatigue testing.51 Nalla et al. reported ‘‘Metal-like’’ stress-life behaviour of dentine, concluding that fatigue lifetimes: (1) decrease with increasing stress amplitudes (at constant stress ratio); (2) decrease with increasing stress ratio (at a given stress-amplitude) and (3) decrease with increasing stress ratio (at constant stress-amplitude).11,43 In our study stress-amplitude was in range of 0–36 MPa while stress-ratio and mean-stress through dentine varied a lot. From Figs. 6 and 8, one may note that there is no correlation between high stress-ratio and FFI. For the cases with FFI > 1 (Fig. 8a and b) the correlation between high mean stress (dominated by shrinkage-stress) and FFI is obvious. Therefore, we report that in restored teeth the increased mean-stress caused by shrinkage process is a driving factor for fatiguefailure of dentine. Despite the fact that experimental studies are time consuming and require expensive equipment, they are valuable since they reveal the key mechanical characteristics crucial for applying FEA. Magne52 demonstrated how virtual prototyping could be used for analysis of endodontically treated molars. Reported stress levels of 24–46 MPa for loads of 200 N are in accordance with the results of our research.53–55 However, studies focusing on numerical fatigue analysis of whole teeth using FEA are rare. In the recent study focusing on numerical fatigue 3D-FE modelling of indirect compositerestored teeth, the authors reported the stress of 90100 MPa for very high occlusal loadings of 600 N.25 As in our study, the authors analysed indirect composite restoration, the influence of polymerization stress was not considered. However, in case of direct composite restoration, the simulation of shrinkage is very important. Some recent studies combined FEA with experimental measuring of shrinkage in order to analyse and validate such approach.59,60 Due to the

heterogeneity of dental materials, various levels of shrinkagestress could be developed and the reported values of the shrinkage stress are varying between the values of several MPa and several hundred MPa.24 Our study revealed that in composite restored teeth (assuming normal mastication loads of 100200 N) developed stresses are similar to the stresses for intact tooth only in the case where shrinkage-stress is low. In the cases of high polymerization stress, the fatigue failure of teeth with composite restoration and teeth with root canal treatment is likely to occur within 4 years of habitual mastication. The present study also analysed the differences between tooth with root canal treatment and tooth with only composite restoration. It has been previously stated that the root canal treatment is specified as a major cause of tooth fracture3,4 Furthermore, it is considered that changes in the geometry and replacement of tooth tissues with artificial materials are one of the major factors inducing increased risk of failure.4,5 However, maximal allowed crack size is slightly lower in tooth with root canal treatment which induces somewhat higher FFI than for tooth with only composite restoration. Thus, this result indicates that fatigue behaviour does not change significantly after root canal treatment. On the other hand, in case of high occlusal load these teeth did show significant differences.34 Thus, based on this paper and the previous research, for this specific case, it can be concluded that root canal treatment and widening of the root canals do not have influence on dentine fatigue but can reduce tooth fracture resistance on high occlusal load. Therefore, other changes in tooth and dentine can be considered as more important factors in increased tooth fragility after root canal treatment.6 In addition, we would like to report a few features which represent a challenge from the aspect of biomechanics and modelling tooth fatigue fracture. First of all, the results of the presented study referred to this specific scenario. Therefore, the results and conclusions can only suggest what can be expected in other cases of composite restoration with or without root canal treatment. Moreover, the values used for critical stress definitions (endurance limit and threshold stress intensity range) are varying in literature – we remind that one may find more liberal or restrictive values in literature which was not covered with this study. Furthermore, dentine is highly heterogeneous material, consisting of fibre-like tubules whose material properties and biomechanical behaviour are varying depending on its direction.61 Fatigue crack growth resistance of human dentine decreases with both age of the tissue and dehydration.12 In the study focusing on determination of ultimate tensile strength (UTS) of enamel, dentine and enamel-dentine junction, Giannini et al. reported that UTS varied according to its nature and location.62 The same authors reported that measured UTS varied with the frequency of loading. The stress-ratio contributes to fatigue crack growth in dentine increasing the C (fatigue crack growth coefficient) while decreasing m (fatigue crack growth exponent) and DK (stress intensity range).62 However, the authors considered only stress-ratio in range of 0.5 to 0.5, while our study showed that stress-ratio may be higher than 0.5 in the case of increased shrinkage stress. For that reason, in our study the constant values were adopted for C and m,9 and this



is the possible limitation of this study. It means that diagrams in Fig. 8c should be taken cautiously before performing additional experimental studies.

5.

Conclusion

Results of the study indicated that the main influence on dentine fatigue is attributed to the high shrinkage stress of the composite filling. The level of occlusal load in the simulation of one million cycles does not produce high fatigue of dentine when the residual stress is close to zero. Keeping this in mind, it is important to minimize the shrinkage stress of the composite materials when restoring vital or devitalized teeth.

Conflict of interest The authors declare no conflict of interest.

Acknowledgement This work is supported partly by the Ministry of Education and Science in Serbia with projects III41007, ON174028 and III45005.

references

1. Wanchek TN, Rephann TJ. Economics of dentistry. In: Cylier AJ, editor. Encyclopedia of health economics. Oxford: Elsevier; 2014. p. 175–82. 2. Baelum V. Dentistry and population approaches for preventing dental diseases. Journal of Dentistry 2011;39:S9–19. 3. Burke FJT. Tooth fracture in vivo and in vitro. Journal of Dentistry 1992;20:131–9. 4. Tang W, Wu Y, Smales RJ. Identifying and reducing risks for potential fractures in endodontically treated teeth. Journal of Endodontics 2010;36:609–17. 5. Sathorn C, Palamara J, Palamara D, Messer H. Effect of root canal size and external root surface morphology on fracture susceptibility and pattern: a finite element analysis. Journal of Endodontics 2005;31:288–92. 6. Zelic K, Milovanovic P, Rakocevic Z, Askrabic S, Potocnik J, Popovic M, et al. Nano-structural and compositional basis of devitalized tooth fragility. Dental Materials 2014;30:476–86. 7. Kishen A. Mechanisms and risk factors for fracture predilection in endodontically treated teeth. Endodontic Topics 2006;13:57–83. 8. Teoh SH. Fatigue of biomaterials: a review. International Journal of Fatigue 2000;22:825–37. 9. van der Bilt A, Engelen L, Pereira LJ, van der Glas HW, Abbink JH. Oral physiology and mastication. Physiology & Behavior 2006;89:22–7. 10. Kruzic JJ, Ritchie RO. Fatigue of mineralized tissues: cortical bone and dentin. Journal of the Mechanical Behavior of Biomedical Materials 2008;1:3–17. 11. Nalla RK, Imbeni V, Kinney JH, Staninec M, Marshall SJ, Ritchie RO. In vitro fatigue behavior of human dentin with implications for life prediction. Journal of Biomedical Materials Research Part A 2003;66:10–20.

11

12. Bajaj D, Sundaram N, Nazari A, Arola D. Age, dehydration and fatigue crack growth in dentin. Biomaterials 2006;27:2507–17. 13. Arola D, Bajaj D, Ivancik J, Majd H, Zhang D. Fatigue of biomaterials: hard tissues. International Journal of Fatigue 2010;32:1400–12. 14. Kruzic JJ, Nalla RK, Kinney JH, Ritchie RO. Mechanistic aspects of in vitro fatigue-crack growth in dentin. Biomaterials 2005;26:1195–204. 15. Bathe KJ. Finite element procedures. New Jersey: PrenticeHall; 1996. 16. Kojic M, Filipovic N, Stojanovic B, Kojic N. Computer modeling in bioengineering – theoretical background, examples and software. 1st ed. London: Wiley and Sons; 2008. 17. Murakami N, Wakabayashi N. Finite element contact analysis as a critical technique in dental biomechanics: a review. Journal of Prosthodontic Research 2014;58:92–101. 18. Changa CL, Chenc CS, Huangd CH, Hsua ML. Finite element analysis of the dental implant using a topology optimization method. Medical Engineering & Physics 2012;34:999–1008. 19. Garriga-Majo D, Paterson RJ, Curtis RV, Said R, Wood RD, Bonet J. Optimisation of the superplastic forming of a dental implant for bone augmentation using finite element simulations. Dental Materials 2004;20:409–18. 20. Lu S, Li T, Zhang Y, Lue C, Sun Y, Zhang J, et al. Biomechanical optimization of the diameter of distraction screw in distraction implant by three-dimensional finite element analysis. Computers in Biology and Medicine 2013;43:1949–54. 21. Chuanga SF, Changb CH, Chenc TYF. Contraction behaviors of dental composite restorations – finite element investigation with DIC validation. Journal of the Mechanical Behavior of Biomedical Materials 2011;4:2138–49. 22. Couegnata G, Fokb SL, Cooperb JE, Qualtroughc AJE. Structural optimization of dental restorations using the principle of adaptive growth. Dental Materials 2006;22:3–12. 23. Dejak B, Młotkowski A. 3D-Finite element analysis of molars restored with endocrowns and posts during masticatory simulation. Dental Materials 2013;29:e309–e317. 24. Rodrigues FP, Silikas N, Watts DC, Ballester RY. Finite element analysis of bonded model Class I ‘restorations’ after shrinkage. Dental Materials 2012;28:123–32. 25. Ausiello P, Franciosa P, Martorelli M, Watts DC. Numerical fatigue 3D-FE modeling of indirect composite-restored posterior teeth. Dental Materials 2011;27:423–30. 26. Mutluay M, Yahyazadehfar M, Ryou H, Majd H, Do D, Arola D. Fatigue of the resin–dentin interface: a new approach for evaluating the durability of dentin bonds. Dental Materials 2013;29:437–49. 27. Corazza PH, Feitosa SA, Borges ALS, Bona AD. Influence of convergence angle of tooth preparation on the fracture resistance of Y-TZP-based all-ceramic restorations. Dental Materials 2013;29:339–47. 28. Schneider LPJ, Cavalcante LM, Silikas N. Shrinkage stresses generated during resin-composite applications: a review. Journal of Dental Biomechanics 2010;2010:131630. 29. Giachetti L, Scaminaci Russo D, Bambi C, Grandini R. A review of polymerization shrinkage stress: current techniques for posterior direct resin restorations. Journal of Contemporary Dental Practice 2006;7:79–88. 30. Yamamoto T, Kubota Y, Momoi Y, Ferracane JL. Polymerization stresses in low-shrinkage dental resin composites measured by crack analysis. Dental Materials 2012;28:e143–9. 31. Borkowski K, Kotousov A, Kahler B. Effect of material properties of composite restoration on the strength of the restoration–dentine interface due to polymerization



12

32.

33.

34.

35.

36.

37. 38.

39.

40.

41.

42. 43.

44.


shrinkage, thermal and occlusal loading. Medical Engineering & Physics 2007;29:671–6. Arola D, Huang MP. The influence of simultaneous mechanical and thermal loads on the stress distribution in molars with amalgam restorations. Journal of Material Science: Materials in Medicine 2000;11:133–40. Arola D, Zheng W, Sundaram N, Rouland JA. Stress ratio contributes to fatigue crack growth in dentin. Journal of Biomedical Materials Research Part A 2005;73:201–12. Zelic K, Vukicevic A, Jovicic G, Aleksandrovic S, Filipovic N, Djuric M. Mechanical weakening of devitalized teeth: three dimensional Finite Element Analysis and prediction of tooth fracture. International Endodontic Journal 2014. http:// dx.doi.org/10.1111/iej.12381. Zelic K, Vukicevic A, Jovicic G, Aleksandrovic S, Filipovic N, Djuric M. Finite element analysis of devitalized teeth. Journal of the Serbian Society for Computational Mechanics 2013;7:38–44. Filipovic N. PAK software for static and dynamics finite element analysis. Kragujevac: University of Kragujevac; 2008. Kojic M, Bathe KJ. Inelastic analysis of solids and structures. Springer; 2005. Vukicevic AM, Stepanovic NM, Jovicic GR, Apostolovic SR, Filipovic ND. Computer methods for follow-up study of hemodynamic and disease progression in the stented coronary artery by fusing IVUS and X-ray angiography. Medical & Biological Engineering & Computing 2014; 52:539–56. Jovicic G, Vukicevic A, Filipovic N. Computational assessment of stent durability using fatigue to fracture approach. Journal of Medical Devices 2014. http://dx.doi.org/ 10.1115/1.4027687. Kwon YH, Jeon GH, Jang CM, Seol HJ, Kim HI. Evaluation of polymerization of light-curing hybrid composite resins. Journal of Biomedical Materials Research Part B: Applied Biomaterials 2006;76:106–13. DeLong R, Douglas WH. An artificial oral environment for testing dental materials. IEEE Transactions on Biomedical Engineering 1991;38:339–45. Goodman J. Mechanics applied to engineering. London: Longman Greens; 1899. Nalla RK, Kinney JH, Marshall SJ, Ritchie RO. On the in vitro fatigue behavior of human dentin: effect of mean stress. Journal of Dental Research 2004;83:211–5. Li B, de Freitas M. A procedure for fast evaluation of high-cycle fatigue under muliaxial random loading. Journal of Mechanical Design 2002;124:558–63.

45. Jovicic G. An extended finite element method for fracture mechanics and fatigue analysis. [PhD theses] Faculty of Mechanical Engineering, University of Kragujevac; 2005. 46. Paris PC, Erdogan F. A critical analysis of crack propagation laws. Journal of Fluids Engineering 1963;85:528–33. 47. Ritchie RO. Mechanisms of fatigue-crack propagation in ductile and brittle solids. International Journal of Fracture 1999;100:55–83. 48. Batalha-Silva S, de Andrada MA, Maia HP, Magne P. Fatigue resistance and crack propensity of large MOD composite resin restorations: direct versus CAD/CAM inlays. Dental Materials 2013;29:324–31. 49. Downer MC, Azli NA, Bedi R, Moles DR, Setchell DJ. How long do routine dental restorations last? A systematic review. British Dental Journal 1999;187:432–9. 50. Magne P, Schlichting LH, Maia HP, Baratieri LN. In vitro fatigue resistance of CAD/CAM composite resin and ceramic posterior occlusal veneers. Journal of Prosthetic Dentistry 2010;104:149–57. 51. ASTM fatigue standards and fracture standards, http:// www.astm.org/Standards/fatigue-and-fracture-standards. html [accessed 17.07.14]. 52. Magne P. Virtual prototyping of adhesively restored, endodontically treated molars. Journal of Prosthetic Dentistry 2010;103:343–51. 53. Anderson DJ. Measurement of stress in mastication I. Journal of Dental Research 1956;35:664–70. 54. Anderson DJ. Measurement of stress in mastication II. Journal of Dental Research 1956;35:671–3. 55. Gibbs C, Mahan P, Lundeen H, Brehnan K, Walsh E, Holbrook W. Occlusal forces during chewing and swallowing as measured by sound transmission. Journal of Prosthetic Dentistry 1981;46:443–9. 59. Boaro LC, Brandt WC, Meira JB, Rodrigues FP, Palin WM, Braga RR. Experimental and FE displacement and polymerization stress of bonded restorations as a function of the C-Factor, volume and substrate stiffness. Journal of Dentistry 2013;42:140–8. 60. Pabis LV, Xavier TA, Rosa EF, Rodrigues FP, Meira JB, Lima RG, et al. A method to investigate the shrinkage stress developed by resin-composites bonded to a single flat surface. Dental Materials 2012;28:e27–34. 61. Chen TYF, Huang PS, Chuang SF. Modeling dental composite shrinkage by digital image correlation and finite element methods. Optics and Lasers in Engineering 2014;61:23–30. 62. Giannini M, Soares CJ, de Carvalho RM. Ultimate tensile strength of tooth structures. Dental Materials 2004;20:322–9.


Measurements and modelling of the influence of dentine colour and enamel on tooth colour.

Modelling Influence and Opinion Evolution in Online Collective Behaviour.

Modelling the longevity of dental restorations by means of a CBR system.

Mechanical fatigue degradation of ceramics versus resin composites for dental restorations.

Influence of sleep disturbance, fatigue, vitality on oral health and academic performance in indian dental students.

Dentist-specific effects on the longevity of dental restorations.

Dental Implants Fatigue as a Possible Failure of Implantologic Treatment: The Importance of Randomness in Fatigue Behaviour.

Reasons for replacement of amalgam dental restorations.

Fatigue of dental ceramics.

Mechanical and chemical characterisation of demineralised human dentine after amalgam restorations.

Implant-abutment connections: influence of the design on the microgap and their fatigue and fracture behavior of dental implants.

Effects of low and high fluoride levels on rat dental caries and simultaneous dentine apposition.

Fracture load of ceramic restorations after fatigue loading.

Systematic approach to preparing ceramic-glass composites with high translucency for dental restorations.

In vitro marginal microleakage associated with five dentine bonding systems and associated composite restorations.

Modeling the biomechanics of swine mastication--an inverse dynamics approach.

Influence of menstrual cycle on volunteering behaviour.

Mastication and its influence on human salivary flow and alpha-amylase secretion.

Influence of electrotaxis on cell behaviour.

Mixed Class V restorations: the potential of a dentine bonding agent.

Fatigue behaviour of young ostrich pericardium.

Modelling the stochastic behaviour of primary nucleation.

Influence of maximum bite force on jaw movement during gummy jelly mastication.

Quantification of finger joint loadings using musculoskeletal modelling clarifies mechanical risk factors of hand osteoarthritis.