Estimation of three-dimensional long axes of the maxillary and mandibular first molars with regression analysis

The purpose of this study was to determine the long axes of molars with multiple roots through ordinary least squares regression (LSR) and to compare them with the axes defined by principal component analysis (PCA). Three-dimensional radiological images of 20 dry skulls were obtained by cone-beam computed tomography (CBCT). Data from maxillary and mandibular first molars were extracted from the CBCT DICOM data with a three-dimensional image visualization system. The obtained data were reconstructed, converted to STL files, and three-dimensional coordinate values were extracted. The long axes were estimated by an algorithm to synchronize the LSR line with the horizontal axis which was translated to the vertical axis. The axes of the molars defined by LSR were compared with the axes of the molars defined by PCA. The coordinate point number of each molar was 5400-5800. The algorithm for determining the tooth axes in this study consisted of four stages containing three steps each. The distance between the two axes calculated by the two methods (LSR and PCA) on the horizontal plane through the origin was less than 10^-12 mm and the deviations between them were less than 0.003°. The long axes of the molars estimated by LSR agree almost exactly with the axes estimated by PCA, and the accuracy is sufficient for clinical usage; however, the distance between them would shorten with a more severe convergence condition of the α value at each stage of this LSR system.