Identification of Apical and Cervical Curvature Radius of Human Molars

determine the frequency of apical and cervical curvatures in human molars using the radius method and cone-beam computed tomography (CBCT) images. Four hundred images of mandibular and maxillary first and second molars were selected from a database of CBCT exams. The radius of curvature of curved root canals was measured using a circumcenter based on three mathematical points. Radii were classified according to the following scores: 0-straight line; 1-large radius (r>8 mm, mild curvature); 2-intermediate radius (r>4 and r<8 mm, moderate curvature); and 3-small radius (r≤4 mm, severe curvature). The frequency of curved root canals was analyzed according to root canal, root thirds, and coronal and sagittal planes, and assessed using the chi-square test (significance at α=0.05). Of the 1,200 evaluated root canals, 92.75% presented curved root canals in the apical third and 73.25% in the cervical third on coronal plane images; sagittal plane analysis yielded 89.75% of curved canals in the apical third and 77% in the cervical third. Root canals with a large radius were significantly more frequent when compared with the other categories, regardless of root third or plane. Most root canals of maxillary and mandibular first and second molars showed some degree of curvature in the apical and cervical thirds, regardless of the analyzed plane (coronal or sagittal).


Introduction
Effective root canal preparation (RCP) is a determinant factor for the success of all subsequent procedures, e.g., root canal debridement, placement of intracanal medication and canal geometry optimization for filling (1).Peters (1) summarizes the main challenges of RCP in three aspects: 1) identification, access and enlargement of the main canals without procedural errors; 2) establishment and maintenance of adequate working lengths throughout the shaping procedure; and 3) selection of preparation sizes and overall geometries that allow adequate disinfection and subsequent filling.In this sense, knowledge of root canal morphology is the first requirement for a successful root canal treatment (2,3).Several anatomical characteristics, e.g., number of canals, cross-sections, root canal curvatures, shape and position of apical foramina, ramifications and development disorders should all be carefully analyzed by the clinician to avoid unexpected events during RCP (2)(3)(4)(5)(6)(7).
Up to a few years ago, the great challenge was to prepare a curved root canal with no ledge formation, foramen transportation, loss of working length or deviation from the original canal path, in addition to preventing accidents such as root perforation or instrument fracture.All these aspects motivated the performance of several studies of canal curvatures (8)(9)(10)(11)(12)(13)(14)(15)(16).However, in the last few years, endodontics has moved towards new directions, with a focus on new technologies in RCP, in particular nickel-titanium instruments (1).Today, undergraduate students in clinical training use nickel-titanium rotary instruments as routine and perceive better results in root canal treatment (17).
Periapical radiographs are routinely used in root canal treatment.The anatomical complexities of human molars, their nuances and peculiarities, highlight the care required during RCP of these teeth, especially in view of the frequent presence of curved canals.For many years, researchers have studied and proposed methods to determine the curvature of root canals, using either the angle or the radius of curvature method in periapical radiographs (4,(8)(9)(10)(11)(12)(13)(14)(15)(16).However, important limitations of this diagnostic imaging method have been described, especially because it generates a two-dimensional image of a three-dimensional structure (3,7,(16)(17)(18).Conebeam computed tomography (CBCT), in turn, has been increasingly used to identify anatomical and pathological alterations and has opened new perspectives when compared to conventional imaging exams (18).
The objective of the present study was to determine the frequency of apical and cervical curvatures in human molars using the radius method and CBCT images.

Image Selection
This study was developed using images from the database of private radiology clinics (CROIF, Cuiabá, MT, Brazil; and CIRO, Goiânia, GO, Brazil).Patients were referred to the clinics for different diagnostic reasons.A consecutive sample of 400 maxillary and mandibular first and second molars was selected among CBCT images from 328 patients (112 males and 216 females; mean age: 39.9 years).
Inclusion criteria for image selection were: highresolution CBCT images of maxillary and mandibular first and second molars longer than 20 mm, with no previous endodontic treatment, no post and core placement, no calcified root canals or internal/external root resorption, and presence of a fully formed apex, with no history of orthodontic treatment, developmental disorders, or pathological processes.
A total of 1,200 root canals were evaluated.In mandibular molars, mesiobuccal (MB), mesiolingual (ML), and distal (D) root canals were considered, while in the maxillary molars, MB, distobuccal (DB), and palatal (P) root canals were analyzed.In D root canals showing bifurcation into two canals, only the most curved root canal was selected for analysis.C-shaped root canals were excluded.The study design was approved by UFG Ethics Committee (Process #7968214.8.0000.5083).

Determination of Radicular Curvature Radius
The frequency of curved root canals was determined on CBCT images using a method previously described (16).Briefly, the curvature radius was determined in two regions of the root canal, namely the apical third (starting from the apical foramen) and the cervical third (starting from the canal entrance), and in two planes (coronal and sagittal).
First, the curvature radius of the apical region was calculated, followed by the curvature radius of the cervical region.This method uses two semi-straight lines (line segments) superimposed to the root canal, where the primary line corresponds to the longer continuity of the apical region, and the secondary line to the middle and cervical thirds.Regardless of the length of the secondary line, only the 6 mm closest to the primary line are taken into consideration and the midpoint of each semi-straight line is determined.From this point, two lines perpendicular to the semi-straight lines are drawn until they meet at a central point, which is named the circumcenter (19).The distance between the circumcenter and the center of each semi-straight line constitutes the radius of the circumference and defines the magnitude of the curve.The semi-straight lines may be shorter or greater, depending on the curvature of each root canal.
The curvature length evaluated was 6 mm, but in some cases it was necessary to draw an imaginary semi-straight line to establish the circumcenter, particularly when the root canal showed more than one curve or lower arches than established (Fig. 1).
All CBCT exams were analyzed by two examiners (one endodontist and one radiologist, both with over 10 years of experience), which were previously calibrated using 10% of the sample.When disagreements were observed, consensus was reached by discussing results with a third observer.

Statistical Analysis
Data were analyzed using the Statistical Package for the Social Sciences (IBM Co., New York, NY, USA), including frequency distribution and cross-tabulation.Curvature scores were compared using the chi-square test.Significance was set at 5%.

Results
Among the 1,200 assessed root canals, 92.75% presented curved root canals in the apical third, and 73.25% in the cervical third as assessed using coronal plane images.Sagittal plane analysis yielded 89.75 and 77% of curved root canals in the apical and cervical thirds, respectively.Root canals with a large radius (mild curvature) were significantly more frequent when compared with the other categories (p<0.05),regardless of root third or analyzed image plane.
Curvature values in the apical and cervical thirds of Curvature radius determination using CBCT MB and ML root canals of mandibular first and second molars did not show significant differences (p>0.05) when analyzed in either the coronal or the sagittal plane.
Conversely, the root canals of maxillary first molars showed differences (p<0.05) in the cervical third in both the assessed planes (Tables 1 and 2).Figures 1 and  2 illustrate the method of determining curvature radii used in this study.

Discussion
The great majority of maxillary and mandibular molars showed root canals with some degree of curvature in both the apical and cervical thirds (Tables 1 and 2), a finding that is compatible with the literature (8,13).High resolution of images, possibility to obtain images in different planes and segments, and dynamic navigation are some of the technological advancements brought forward by CBCT.
The frequency, degree of curvature, and configuration of MB and ML root canals of mandibular molars determined by radiographs in clinical and proximal directions showed curvature in both views.Secondary curvature, in a direction opposite to that of the principle curve, was seen more frequently in proximal view images.Proximal radiographs revealed greater mean curvatures than clinical images 38% of the time (8).Schäfer et al. (13) assessed canal curvatures in 700 extracted human permanent teeth by measuring the angle and radius of curvature and the length of the curved part of the canal, also in clinical and proximal views.All radiographs were analyzed using a computerized digital image processing system.The results showed that among the 1,163 root canals examined, 84% were curved.
The mesial root of mandibular molars has been well studied (2,6,7,8,13).In the study sample, in coronal plane the apical thirds of MB and ML root canals of first and second mandibular molars were classified as showing moderate curvature (intermediate radius: r>4 and r<8 mm) in 21/22% and 24/19% of the specimens, respectively.In conventional imaging exams such as periapical radiography the ML root canal is usually poorly visualized, particularly in the sagittal plane.
One aspect worth mentioning is the higher frequency of cervical curvature found on sagittal images in the mesial root canals of mandibular molars and in MB and DB canals of maxillary molars (Tables 1 and 2).The negative influence of curved root canals has been a long-time concern in endodontics (1,2,(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16).One way of dealing with this finding is to stabilize the apical curvature by cervical enlargement, which may avoid common drawbacks such as loss of working length, ledge formation, root perforation, and instrument fracture.In this sense, substantial advancements have been achieved in the quality of RCP after the advent of the new generation of nickel-titanium instruments with recognized flexibility (20)(21)(22)(23).In addition, an innovative technique has been increasingly used to study root canal anatomy (24,25) changes in canal geometry after RCP (21,22), namely, microcomputed tomography.
The comparison of different methods used to determine root curvatures was beyond the scope of this study.Rather, the aim was to use a previously described radius determination method (16,23) and already adopted in clinical practice to analyze the frequency of curved root canals in different root thirds and planes using CBCT.

Teeth Scores
Coronal Plane Sagittal Plane      According to those authors, informing only the angle of curvature of root canals yields ambiguous results, as the curvature angle is independent of the curvature radius (canals with the same angle of curvature may have different radii).Therefore, in order to improve comparison across studies, curvatures should be described using both the angle of curvature, according to Schneider's method, and the radius and length of the curve.We agree with the conclusions of Schäfer et al. (13).
The focus of our study was to determine the frequency of curved root canals based on curvature radius in CBCT images.The method employed to determine curvature radius allows the analysis of a given segment of the root canal, and not necessarily its entire length.This is interesting because one root canal may present more than one curve, preventing correct radius calculation for the entire root length.In the present method, semi-straight lines can be shorter or longer, depending on the size of the curve analyzed in each root canal.
In sum, the curvature radius method used in the present study was easy, reproducible and can improve the planning of RCP in curved canals.The majority of root canals of maxillary and mandibular first and second molars showed some degree of curvature in the apical and cervical thirds, regardless of the analyzed plane (coronal or sagittal).

Figure 1 .
Figure 1.The root curvature radius based on 3 mathematical points can be determined in both apical and coronal directions.Curvature radius considering the two 6-mm semistraight lines are classified as (16): small radius (r≤4 mm) -severe curvature (A); intermediary radius (r>4 and r≤8 mm) -moderate curvature (B); and large radius (r>8 mm) -mild curvature (C).

Figure 2 (
Figure 2 (A-I) Illustrations of curvature radii method showing cervical and apical curvature in sagittal and coronal planes in mandibular and maxillary molars.
canal morphology was suggested(4), with the following categories: I (straight), J (apical curve), C (completely curved), or S (multicurved).Nagy et al.(10) described root canal curvatures mathematically and suggested a standard model with the help of differentiated geometrical pattern analysis and computer graphics.Those authors analyzed 433 root canals by approximating measured points of the same radiographs using fourth degree polynomial functions describing the imaginary axis of canals.Schäfer et al. (13) examined 1,163 root canals and found a secondary curvature (S-shaped) in 17.5% (n=204; 12.3% of maxillary teeth and 23.3% of mandibular teeth).

Table 2 .
Frequency (%) of root canal curvature (scores 0-3) in cervical (C) and apical (A) thirds of first and second maxillary molars, in coronal and sagittal planes