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Advances in Materials Science and Engineering

Volume 2015, Article ID 189703, 6 pages

http://dx.doi.org/10.1155/2015/189703

## Survival Analysis of Factors Influencing Cyclic Fatigue of Nickel-Titanium Endodontic Instruments

^{1}Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University in Olomouc, 17 Listopadu 12, 771 46 Olomouc, Czech Republic^{2}Department of Mathematics, Faculty of Applied Informatics, Tomas Bata University in Zlín, Nám. T.G. Masaryka 5555, 760 05 Zlín, Czech Republic^{3}Institute of Clinical and Experimental Dental Medicine, First Faculty of Medicine, Charles University General Teaching Hospital, Prague, Czech Republic^{4}Onedent, Příkop 8, 602 00 Brno, Czech Republic^{5}Communication and Information Systems Agency, Czech Army, Tychonova 1, 160 01 Prague, Czech Republic

Received 27 May 2015; Accepted 9 September 2015

Academic Editor: Luigi Nicolais

Copyright © 2015 Eva Fišerová et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

*Objective*. The aim of this study was to validate a survival analysis assessing the effect of type of rotary system, canal curvature, and instrument size on cyclic resistance. *Materials and Methods*. Cyclic fatigue testing was carried out in stainless steel artificial canals with radii of curvature of 3 or 5 mm and the angle of curvature of 60 degrees. All the instruments were new and 25 mm in working length, and ISO colour coding indicated the instrument size (yellow for size 20; red for size 25). *Wizard Navigator* instruments, *Mtwo* instruments, *ProTaper* instruments, and *Revo-S* instruments were passively rotated at 250 rotations per minute, and the time fracture was being recorded. Subsequently, fractographic analysis of broken tips was performed by scanning electron microscope. The data were then analysed by the Kaplan-Meier estimator of the survival function, the Cox proportional hazards model, the Wald test for regression covariates, and the Wald test for significance of regression model. *Conclusion*. The lifespan registered for the tested instruments was *Mtwo* > *Wizard Navigator* > *Revo-S* > *ProTaper*; 5 mm radius > 3 mm radius; and yellow > red in ISO colour coding system.

#### 1. Introduction

Nickel-titanium (Ni-Ti) instruments are increasingly used in the present endodontics due to the outstanding combination of mechanical properties superior to those of stainless steel. Yet, despite increased flexibility, an unexpected failure during root canal treatment is still a matter of great concern, as these instruments can undergo fracture within their elastic limit without any visible sign of previous permanent deformation [1–3]. The fracture of rotary Ni-Ti instruments can occur as a consequence of torsional overload or as a result of flexural fatigue [3, 4]. Clinically, Ni-Ti rotary instruments are subjected to both torsional load and cyclic fatigue [5]. Torsional fracture occurs when the tip of the instrument is stuck in a canal while the rest of the tool is being rotated. When the elastic limit of the metal is exceeded by the torque exerted by the handpiece, a fracture of the tip becomes inevitable [6]. The cyclic fatigue fracture is caused by alloy fatigue. The instrument freely rotates in a curved canal, generating tensions/compressions cycles at the point of maximum flexure until the fracture occurs [2, 5, 7]. Cyclically fatigued instruments show no macroscopic evidence of plastic deformation, but instruments that fracture as a result of torsional overload demonstrate variable deformation, such as unwinding, straightening, reverse winding, and twisting [3, 8]. The incorrect clinical use [9] and several other factors contribute to instrument fracture [10].

There are currently many rotary endodontic systems of various designs and dimensions, which are used for cleaning and shaping of root canals. The development of engine-driven rotary endodontic instruments, along with the absence of adequate testing instrument standards, necessitates further analysis in all areas. The resistance of rotary instruments to cyclic fatigue, for instance, is affected by the angle and radius of canal curvature. Instrument lifespan also decreases with the increased severity in the angle and the radius of the curves around which the instrument rotates [1, 2, 7]. Testing instruments in canals with radii of curvature of 2, 5, and 10 mm showed that the smaller the radius, the shorter the life of the instrument when rotating. The resistance of rotary instruments to cyclic fatigue is also affected by the diameter of the instrument. Several studies have shown that an increased diameter, which is determined by the tip size and taper, reduces the time to fracture at the point of maximum curvature of the instrument [1, 2, 7]. The fatigue resistance caused by rotational speed in metal-simulated canals was studied for instruments of sizes 30 and 40, and it was concluded that the speed is not a significant factor [2]. Although the morphology of a Ni-Ti rotary file on its performance has been investigated [11–14], the influence of file design on the cyclic fatigue stress remains unclear, so the effect of a Ni-Ti rotary system during root canal treatment still necessitates further study.

The cyclic fatigue resistance is traditionally analysed by one-way ANOVA. The time to failure is multiplied by rotation per minute (rpm) in order to obtain the number of cycles to failure. In this paper, we analyse fatigue resistance from a different point of view. Survival analysis is a collection of statistical procedures for analysing the duration until the occurrence of an event of interest, the event in this study being the cyclic fatigue fracture of a rotary file. In the survival approach, Ni-Ti instruments are compared by analysing their times to failure. The survival analysis approach allows us to estimate the probability that failure of rotary system does not occur beyond a specific time and to assess the hazard function, which gives the instantaneous potential per unit time for the failure to occur, provided that the instrument has survived up to that time.

The data-set comes from the Department of Mechanical Engineering at the University of Defence in Brno, where the cyclic fatigue resistance of four Ni-Ti rotary systems was tested. This study assesses the times to failure in relation to the type of Ni-Ti rotary system (*ProTaper*,* Revo-S, Wizard Navigator*, and* Mtwo*), radius of curvature to which the instruments were subjected during preparation (3 and 5 mm), and instrument size identified by ISO standards (yellow for size 20 and red for size 25).

#### 2. Survival Analysis of Cyclic Fatigue Lifetime of the Rotary System/Statistical Background

The primary interest is usually given to the survival function, also known as reliability function, which is the probability that the time of the cyclic fatigue fracture is longer than a specified time . In other words, the survival function indicates the probability that the rotary system resists beyond a specified time. The Kaplan-Meier estimator [15], also known as the product-limit, is a well-known estimator for the survival function. The Kaplan-Meier estimator of the survival function iswhere denote the ordered distinct failure times of the fracture due to cyclic fatigue in a group of instruments used when a fixed category was set (e.g., all instruments rotated in stainless-steel blocks with artificial canals with a mm radius of curvature), is the number of the cyclic fatigue fractures at a time , and is the number of the cyclic fatigue fractures before the time . The Kaplan-Meier estimate is a step function with the points of discontinuity (jumps) at the observed times of the cyclic fatigue fractures.

The most widely used method to investigate several variables at a time is the Cox proportional hazards model [16]. The hazard function at a time , also known as the failure function, is the probability that, during a very short time interval, an event will occur, conditional on not having the event up to a time . The hazard function is nonnegative; it assesses the instantaneous risk of the cyclic fatigue fracture for a rotary system which has resisted a time . The Cox proportional hazards model specifies the hazard function to covariates for individual instruments. In our study, the hazard function is related to the radius of curvature, type of Ni-Ti rotary system, and instrument size, which are categorical variables. We distinguish two categories for the radius (3 and 5 mm), four categories for the type (*Mtwo*,* Wizard Navigator*,* Revo-S*, and* ProTaper*), and two categories for the instrument size (ISO colour coded yellow and red). For each categorical variable one category is excluded and is considered as the reference category. Our selection includes a 3 mm radius,* Wizard Navigator*, and red ISO colour. When modelling with the categorical variables, the so-called dummy variables (indicators of categories) are usually used; for example, denotes a dummy variable coded 1 for all* ProTaper* instruments and 0 for the other systems. The model can thus be formally expressed aswhere are unknown regression coefficients, is a vector containing the values of dummy variables ordered according to the model, and is the hazard function for the reference set. That means that for all dummy variables in the model are zeros, that is, . When times in the continuous time model are grouped, ties in failure times can be observed. If the number of observations of ties is tolerable with respect to computing time, the unknown regression coefficients can be estimated, for example, by the Breslow result [17, 18] or the Efron result [19]. Generally, since effects of covariates could vary by categories, the model can be considered with interactions, which means that covariates can interact in affecting the hazard function. It should be noted that the quantitative variables may also be included in the Cox proportional hazards model as the explanatory variables. That means that when different rotational speeds are used in the study, rpm can be included into the model as the quantitative variable. Finally, the ratio of hazard functions, known as the hazard ratio, is a useful tool for comparing the risk of fracture between different categories of one categorical variable. The hazard ratio can be directly obtained from the Cox proportional hazards model. Due to the proportionality, the hazard ratio is the same at any time . Note that the proportions of the hazard functions do not depend on the choice of the reference set in the model without interactions.

#### 3. Materials and Methods

Four Ni-Ti endodontic instrument systems were tested:* Wizard Navigator* (Medin, Nové Město na Moravě, Czech Republic);* Mtwo* (Sweden and Martina, Padova, Italy);* Revo-S* (Micro-Mega, Besancon, France); and* ProTaper* (Dentsply-Maillefer, Ballaigues, Switzerland). The* ProTaper* group consisted of 17 F1 files of size 20, 0.07 taper (yellow ISO colour) and 18 F2 files of size 25, 0.08 taper (red ISO colour);* Mtwo* of 17 files of size 20, 0.06 taper (yellow ISO colour) and 17 files of size 25, 0.06 taper (red ISO colour);* Wizard Navigator of *20 W-3 files of size 20, 0.06 taper (yellow ISO colour) and 20 W-4 files of size 25, 0.06 taper (red ISO colour); and Revo-S of 15 files of size 20, 0.04 taper (yellow ISO colour) and 18 files of size 25, 0.06 taper (red ISO colour). All instruments were new and 25 mm in working length. Seventy-three of Ni-Ti rotary systems were tested in a steel block with a simulated canal with a 3 mm radius of curvature, and sixty-nine of Ni-Ti rotary systems were tested in a steel block with a 5 mm radius of curvature and a 60-degree angle of curvature. Numbers of instruments tested in simulated canals of a specified radius of curvature according to the instruments’ size are listed in Table 1. All 142 instruments had been firstly checked by a microscope for visible defects and deformations.