Validation of a mathematical-geometrical model to calculate the length of an individual anterior arch.

Purpose: For resolving anterior dental crowding or spacing, it is of key interest in personalised orthodontic diagnostics and treatment planning to predict the extent of space gained or lost in the anterior dental arch by changing incisor inclination or position. To facilitate the determination of anterior arch length (AL) and to predict its alterations following tooth movements, a mathematical-geometrical model, based on a third-degree parabola, was established. The aim of this study was to validate this model and assess its diagnostic precision.

Methods: This retrospective diagnostic study evaluated 50 randomly chosen dental casts taken before (T0) and after (T1) orthodontic treatment with fixed appliances. Plaster models were digitally photographed, allowing two-dimensional digital measurements of arch width, depth and length. A computer programme based on the mathematical-geometrical model to be validated was created to calculate AL for any given arch width and depth. Mean differences and correlation coefficients as well as Bland-Altman plots were used to compare the measured and the calculated (predicted) AL, evaluating the precision of the model.

Results: Inter- and intrarater reliability tests showed reliable measurements of arch width, depth and length. Measured and calculated (predicted) AL revealed high concordance according to concordance correlation coefficient (CCC), intraclass correlation coefficient (ICC), and Bland-Altman analyses and negligible differences between the mean values.

Conclusions: The mathematical-geometrical model calculated anterior AL without significant difference to the measured AL, indicating its validity. The model can thus be used clinically for predicting alterations of AL following therapeutic changes of incisor inclination/position.