Maxillary and Mandibular Arch Perimeter Prediction Using Ramanujan's Equation for the Ellipse-In vitro Study

Introduction: Arch expansion in the molar region or incisor proclination is an effective method to resolve arch length discrepancy. Predicting the change in arch perimeter by arch expansion in the molar region or incisor proclination is important in deciding during treatment planning if the removal of teeth will be required.

Purpose: The primary objective of this study was to determine if there is a mathematical correlation between Ramanujan's equation for the perimeter of an ellipse and the maxillary and mandibular arch perimeter. The secondary goal was to predict the gain in arch perimeter by arch expansion in molar region and incisor proclination using same equation for non-extraction cases.

Materials and Methods: 55 maxillary and 55 mandibular well-aligned diagnostic casts of untreated patients with class I molar relationships were used. Arch perimeter was measured using light wire and was compared with the calculated arch perimeter obtained by using Ramanujan’s equation.

Results: A strong correlation was found between measured and calculated arch perimeters with 1.5% error in maxillary arch and 1.7% error in mandibular arch. The average gain in arch perimeter by 1 mm molar expansion was 0.73 mm in maxilla and 0.74 mm in the mandibular arch. Average gain in arch perimeter by 1mm incisor proclination was 1.67 mm in maxilla and 1.65 mm in the mandible.

Conclusion: Ramanujan’s equation for an ellipse can be used to calculate the arch perimeter in the maxillary arch with a 1.5% chance of error and a 1.7% chance of error in mandibular arch. The equation can also be used to predict change in arch perimeter by molar expansion and incisor proclination.