Coordinating bracket torque and incisor inclination : Part 1: The development of widely applicable equations

The objective of this investigation was to develop equations to describe the geometric relations among "targeted incisor inclinations" (tU1, tL1) accommodating different cephalometric norms (Ricketts, Bergen, etc.) with the "expected inclinations" (eU1, eL1), as they can be generated by bracket torque values according to Andrews, Roth, Ricketts, and MBT.

In its key parts, the present study is a theoretical work in which structural relationships are described using standard mathematical and geometric methodologies.

The "targeted norm-inclinations" (tU1, tL1) were calculated relative to a single reference plane (BOP according to Downs), thus, allowing for a direct comparison of different cephalometric values. Referring to the "expected inclinations" (eU1, eL1), it was found that in addition to bracket torque (BT) morphological and structural parameters also have to be taken into account. These are the "torque coordination angle" (TCA) representing the variation in dental morphology and, the correction angles between BOP and the upper (uOP) (α₁) or the lower (lOP) occlusal plane (β₁). Moreover, the angles α₂ between an upper (uBPP) and β₂ between a lower bracket positioning plane (lBPP) and the occlusal planes (uOP, lOP) have to be considered. As a consequence, suitable equations were developed (eU1_(BOP) = 90° - BT_(U1) - TCA_(U1) + α₁ - α₂, and eL1_(BOP) = 90° - BT_(L1) - TCA_(L1) + β₁ - β₂), allowing the calculation of expected torque-dependent inclinations (eU1, eL1) and representing the prerequisite for a comparison with the cephalometric targeted values (tU1, tL1).

By developing suitable equations, it became possible to name and quantify those parameters that are responsible for incisor inclinations and enable a comparison with targeted cephalometric values.