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 $9.6^{\circ}$ and rotated counterclockwise by $5.4^{\circ}$. These angles were almost similar to those predicted by the beam theory. When the amount of tip-back and toe-in angles were $11.5^{\circ}$ and $9.9^{\circ}$, respectively, bodily movement of the molar was achieved. When the bend angles were increased to $14.2^{\circ}$ and $18.7^{\circ}$, the molar tipped distally by $4.9^{\circ}$ and rotated clockwise by $1.5^{\circ}$. 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.