Journal of Indian Society of Periodontology
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Year : 2013  |  Volume : 17  |  Issue : 1  |  Page : 68-71  

Mathematical analysis of furcation angle in extracted mandibular molars

Department of Periodontology, Ragas Dental College and Hospitals, Uthandi, Chennai, India

Date of Submission30-Jan-2011
Date of Acceptance17-Aug-2012
Date of Web Publication21-Feb-2013

Correspondence Address:
Avaneendra Talwar
Department of Periodontics, Ragas Dental College and Hospital, 2/102, East Coast Road, Uthandi, Chennai
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Source of Support: None, Conflict of Interest: None

DOI: 10.4103/0972-124X.107477

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Background: Multi-rooted teeth with furcation involvement exhibit a poorer prognosis when compared to single rooted teeth. The furcation angle (formed by the divergent roots and the roof) may exert a considerable influence on the accessibility for both home care maintenance and instrumentation during periodontal therapy. As there are few anatomy based reports, the furcation angle has not yet been delineated. Materials and Methods: Furcation angle (FA) was mathematically evaluated in extracted mandibular first and second molar teeth, using the Computer-aided design - computer-aided manufacturing technology. Results: The furcations were divided into three groups (Group I: <30°, Group II: 30°-60°, Group III: >60°) based on the furcation angle and their prevalence. The first molar showed greater prevalence of group II FA, while second molar showed a greater prevalence of group III FA. Conclusion: Linear, two dimensional measurements may not accurately reflect the complexities of the furcation area which exhibits considerable intermolar and intramolar (buccal and lingual furcations of second molar) variation.

Keywords: Computer-aided design - computer-aided manufacturing, furcation angle, prevalence

How to cite this article:
James JR, Arun K V, Talwar A, Kumar T. Mathematical analysis of furcation angle in extracted mandibular molars. J Indian Soc Periodontol 2013;17:68-71

How to cite this URL:
James JR, Arun K V, Talwar A, Kumar T. Mathematical analysis of furcation angle in extracted mandibular molars. J Indian Soc Periodontol [serial online] 2013 [cited 2021 Jun 17];17:68-71. Available from:

   Introduction Top

Periodontal disease is a chronic inflammatory condition which is multifactorial in etiology. Although plaque is recognized to be the primary etiological factor, several environmental, genetic and developmental factors have been identified to modify disease progression. [1],[2],[3],[4] Anatomical areas such as the furcations favour plaque retention due to limited access for routine oral hygiene procedures. [5] Consequently, it has been reported that furcation involvement is a reliable prognostic indictor of periodontal disease. [6],[7]

The difficulties in early detection, management and maintenance of furcation areas have been extensively documented. [5],[8],[9] These difficulties arise in part due to the complex topography of the furcation area. [4] The tortuous course of the roots, often associated with concavities, complicates instrumentation in these areas for both diagnostic and therapeutic purposes. Mandibular furcations are however, considered good candidates for regenerative therapy. [9],[10] Several authors have reported that site related anatomical factors may significantly impact the success of these regenerative procedures. [4],[11],[12],[13]

In spite of the prominent role played by topographical characteristics of the furcation in both diagnosis and therapeutic outcomes, most of the widely used classification systems are based on assessment of hard and soft tissue. [14] A few authors have attempted to classify furcations based on its anatomy. [4],[15],[14],[15],[16],[17] Hou and Tsai [4] proposed a classification based on the differences in root trunk length and suggested that molars with short root trunks have a greater predisposition to early furcation involvement. More recently, they further classified furcations based on the furcation entrance dimension (FED) and correlated it to disease progression. [16]

The limitation with using the FED as the sole parameter is that it does not take into account the area subjacent to the entrance of the furcation. Root divergence and the degree of separation are important variables that have been reported to influence the outcome of both regenerative and resective therapy. [18],[19] Regardless of the form of therapy, it has been documented that roots with minimal root separation are difficult to instrument and maintain. [4],[20],[21],[22],[23] Previous studies have reported that resective surgeries are difficult to perform in fused or convergent roots, [24] while regeneration is limited in widely divergent roots. [18] However, root divergence has been described in a somewhat arbitrary manner in previous literature, with very little mathematical delineation. [16],[25]

It has been recently reported that 3 dimensional analysis of furcation may be required for accurate diagnosis and therapeutic assessment. [26] In the present study, the nonlinear angle formed at the roof of the furcation by the diverging rootsdesignated the furcation angle (FA), was evaluated to obtain an accurate three dimensional topography of mandibular first and second molars.

The aim of the present study was to estimate the FA mathematically in mandibular first and second molars.

   Materials and Methods Top

The study sample consisted of a random collection of 53 Mandibular I st molars and 55 Mandibular II nd molars. Permanent I st and II nd molars extracted due to caries, periodontitis or periapical pathology which were morphologically intact in area 2 mm coronal to CEJ and 6 mm apical to the fornix of furcation were included in the study. Teeth with fused roots, evidence of extraction damage near the furcation or caries, restoration or resorptions extending apical to the CEJ were excluded. Soft and hard tissues adhering to the teeth were carefully removed with curettes and ultrasonic scalers, without damaging root surface.

Computer-aided design (CAD)-computer-aided manufacturing (CAM) alignment

Considering the furcal area and roots are non-linear and allowing for concavities, the furcation angles were measured using mathematic formulas that are employed for studying angle of a curve.

A 3D Digitizer and Coordinate machine (The Faro-Arm, FARO Technologies Inc., Florida) was used to obtain the coordinates on the root surface. A 3D Caliper (Ver 2.11) software was used to convert these coordinates into digital images. All measurements were made by a single investigator.

When the instrument tip was applied on the tooth surface, x, y, z coordinates are derived. The z coordinate was kept a constant so as to derive the other variables along the same plane as the fornix (x 2 , y 2 ).

Three points (approx. 3 mm apart) forming arbitrary curves along the possible line angle followed by the curette were picked up for the mesial and distal roots [Figure 1].
Figure 1: Schematic representation of points 3 mm apart (along x, y coordinates with z coordinate kept constant) on the mesial and distal root surface of the extracted mandibular molar. The points on the root form a curve. The angle of the curve is mathematically derived

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Curves I and II were approximated using respective quadratic polynomials;

(xi, yi), were i = 0, 1, 2, 3, 4

(x 2 , y 2 ) is the common point representing the fornix.

Then, each of these polynomials was differentiated and the values of first derivatives were evaluated at the common point.

The slopes m 1 and m 2 of the tangents to curve I and curve II respectively at common point (x 2 , y 2 ) were obtained by the mathematical equation;

Slope m 1 :{(x 2 − x 1 )/(x o − x 1 )(x o − x 2 )*y 2}+{(x 2 − x o )/(x 1 − x o )(x 1 − x 2 )*y 1}+{(x 2 − x o − x 1)/ (x 2 − x o )(x 2 − x 1 )*y 2

Slope m 2 :{(x 2 − x 3 − x 4 /(x 2 − x 3 )(x 2 − x 4 )* y 2}+{(x 2 − x 4 )/(x 3 − x 2 )(x 3 − x 4 )*y 3}+{(x 2 − x 3 )/(x 4 − x 2 )(x 4 − x 3 )*y 4.

Angle between two curves (furcation angle) was computed as:

Ø = Tan-1 (m 1 - m 2 /1 + m 1 m 2 )

Standardization and statistical analysis

Reproducibility of measurements was tested by repeated measurement of 40 samples (10 from each group). Inter-measurement differences were analyzed using χ2 homogeneity test. Repeated procedures of taking coordinates and measurement of angles did not show a statistically significant difference (P>0.05). To check the validity of the methodology, a similar process was used to measure known angles from 10°-80°. A correlation co-efficient of 0.9992, i.e., an accuracy of 99.92% was achieved.

The prevalence of furcation angle in mandibular 1 st and 2 nd molars were analyzed using the Chi-square test.

   Results Top

Furcation angles (FA) obtained from this study using CAD-CAM analysis were plotted on a scatter diagram [Figure 2] and based on the strength of prevalence; the FAs were grouped into three categories:
Figure 2: Scatter diagram representing the distribution of furcation angles of extracted mandibular molars

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  1. Group I: <30°
  2. Group II: 30°-60°
  3. Group III: >60°.
Distribution of FA in extracted mandibular molar tooth

[Table 1] represents the prevalence of FA at furcation sites on mandibular molars. For the first molar, the buccal furcation sites showed that the group II FA was more prevalent (45.5 %) when compared to group III FA (29.5%) and group I FA (25%). Group II FA occurred more frequently in the first molar (45.5%) than in the second molar (15.9%). The difference in FA between buccal and lingual surfaces of the first molar was not statistically significant (P>0.05).
Table 1: Prevalence and distribution of furcation angle on buccal and lingual surfaces of extracted mandibular molars

Click here to view

For the second molar, the buccal furcation sites showed that the group III FA (84.1 %) was more prevalent than group II FA (15.2%) while there was no furcation in the group I FA.

Group III FA occurred more frequently in the second molar (84.1 %) than in the first molar (29.5%). On the lingual furcation sites, group III FA (51.1%) was more prevalent than group II FA (42.9%) and group I FA (6.1%). The difference in FA between buccal and lingual surfaces in the second molar was statistically significant (P<0.05).

   Discussion Top

Anatomic factors such as furcation areas that favor plaque accumulation are known to contribute to progression of periodontal disease. Regeneration/healing of the involved furcation is complicated by the presence of developmental anamolies in the furcation areas such as cervical enamel projection, bifurcational ridge, root fusion, enamel pearls and root concavities. [27] However, there is still inadequate literature documenting the normal anatomy of the furcation. Existing anatomic classifications are based on two dimensional measurements that may not truly reflect the complexity of the furcation area. [16],[25]

Our results showed that the furcation angles (FA) in the first molars tended to be either group II (45.5%) or III (29.5%) on both buccal and lingual aspects. This uniform distribution may clinically translate itself to greater ease of instrumentation as there would not be any non-negotiable convergence or divergence. The importance of and the difficulty associated with thorough debridement have both been equally well documented, [4],[20] and the presence of anatomically intractable areas would render the process virtually impossible. [22] In this context, it is not surprising that first molars have shown good predictability to both resective and regenerative therapy. [9],[28]

On the other hand, the second molars showed an unequal distribution of FAs on the lingual and buccal aspects. There was a significantly greater distribution of group III (84.1%) on the buccal aspect when compared to the lingual aspect. As a result, several furcations in the second molar would exhibit a group III FA in the buccal side and a group II FA on the lingual surface. Accessibility of instrumentation from buccal to lingual aspect may thus be compromised due to this anatomical constriction. Long term evidence suggests that the 1 st molar may respond more favorably when compared to the 2 nd molar, [18] and this anatomical discrepancy could be one contributory factor.

In all, there was significantly greater distribution of Group II and Group III FA in both the molars when compared to group I FA. Previous reports have shown that the access for instrumentation, e.g., with curettes, may be limited when furcation entrance dimension (FED) is minimal. [11],[22] As the FED is influenced by the furcation angle, those results may be applicable to the present study. [16] Group I FA would therefore, result in a furcal dimension that would be too small to permit adequate instrumentation, thereby, affecting both resective and regenerative procedures. Excessive divergence of roots as observed in group III FA would ease instrumentation but limit the extent of regeneration achieved after new attachment procedures. [29] Group II FA, would thus, be expected to respond most favorably to periodontal therapy. As group I FA showed the least prevalence, our study indicates that mandibular molars may, in terms of anatomy, not be unfavorable to treatment.

The results of the present study differed from to those reported by Hou, et al., [16] who did not report any significant differences in FED grade between buccal and lingual surfaces in both mandibular molars. This disparity may have occurred because they used linear measurements to determine the FED. These results demonstrate the inadequacy of using FED as the only variable to characterize nonlinear curved roots. The present study also suggests that previous assessment of regenerated tissue in furcation defects may be volumetrically inaccurate as a result of two dimensional measurements.

These results indicate that the existing furcations classifications based on loss of hard and soft tissue in the furcations areas may be inadequate to deal with the complexities of the furcation challenge in a comprehensive manner. Three dimensional analysis of the furcations area using CT based assessment may provide clinically relevant predictive information regarding the onset and progression of furcation involvement in periodontal disease.

A future classification that incorporates the furcation's anatomy, along with existing classification systems may provide useful clinical information regarding prognosis and treatment outcomes in multi-rooted teeth.

   Acknowledgments Top

The authors thank Mr. Illangovan M.E. (Department of CAD/CAM, Central Institute of Plastic Engineering and Tools (CIPET), Chennai for his valuable guidance in the CAD/CAM procedures. We thank Dr. R. Ravanan Ph.D. Professor, Department of Statistics, Presidency College, Chennai for his competent help in statistical evaluation.

   References Top

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  [Figure 1], [Figure 2]

  [Table 1]


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