Prediction of optimal bending angles of a running loop to achieve bodily protraction of a molar using the finite element method

Objective The purpose of this study was to predict the optimal bending angles of a running loop for bodily protraction of the mandibular first molars and to clarify the mechanics of molar tipping and rotation. Methods A three-dimensional finite element model was developed for predicting tooth movement, and a mechanical model based on the beam theory was constructed for clarifying force systems. Results When a running loop without bends was used, the molar tipped mesially by 9.6° and rotated counterclockwise by 5.4°. These angles were almost similar to those predicted by the beam theory. When the amount of tip-back and toe-in angles were 11.5° and 9.9°, respectively, bodily movement of the molar was achieved. When the bend angles were increased to 14.2° and 18.7°, the molar tipped distally by 4.9° and rotated clockwise by 1.5°. Conclusions Bodily movement of a mandibular first molar was achieved during protraction by controlling the tip-back and toe-in angles with the use of a running loop. The beam theory was effective for understanding the mechanics of molar tipping and rotation, as well as for predicting the optimal bending angles.