A new mathematical process for the calculation of average forms of teeth

Qualitative visual inspections and linear metric measurements have been predominant methods for describing the morphology of teeth. No quantitative formulation exists for the description of dental features.

The aim of this study was to determine and validate a mathematical process for calculation of the average form of first maxillary molars, including the general occlusal features.

Stone replicas of 174 caries-free first maxillary molar crowns from young patients ranging from 6 to 9 years of age were measured 3-dimensionally with a laser scanning system at a resolution of approximately 100,000 points. Then, the average tooth was computed, which captured the common features of the molar's surface quantitatively. This new method adapts algorithms both from computer science and neuroscience to detect and associate the same features and same surface points (correspondences) between 1 reference tooth and all other teeth. In this study, the method was tested for 7 different reference teeth. The algorithm does not involve any prior knowledge about teeth and their features.

Irrespective of the reference tooth used, the procedure yielded average teeth that showed nearly no differences (less than +/-30 microm).

This approach provides a valid quantitative process for calculating 3-dimensional (3D) averages of occlusal surfaces of teeth even in the event of a high number of digitized surface points. Additionally, because this process detects and assigns point-wise feature correspondences between all library teeth, it may also serve as a basis for a more substantiated principal component analysis evaluating the main natural shape deviations from the 3D average.