Abstract: Abstract: Objective To establish a mathematical model to describe the skeletal class Ⅲ malocclusion of patient dental and basal bone arch form, for providing a data reference and basis for further study. Methods Thirty-five patients with skeletal class Ⅲ malocclusion were selected in this study for computed tomography CBCT. The data of 3-D image were analyzed, and dental arch marker (Fa) and base bone arch marker (Ba) were determined. The reference plane was determined by least square method. Software Matlab 7.0 was used to calculate two- dimensional coordinate system. Based on this, a mathematical model was established to describe the dental and basal bone arch form and then to validate the mathematical model. Results (1) The mathematical model can be used to describe the dental arch form of skeletal class Ⅲ malocclusion, maxillary: Y=46.12 [1- (2X/70.99)2]1.052; mandibular: Y=39.16 [1- (2X/64.51)2]1.038. (2) The mathematical model can be used to describe the basal bone arch form of skeletal class Ⅲ malocclusion, maxillary: Y=43.14 [1- (2X/75.09)2]1.061; mandibular: Y=39.03 [1- (2X/60.63)2]1.021. (3) Fa was located at Ba labial side in the maxilla, the distance was positive. Fa was located at Ba lingual side in the mandibular, and the distance was negative. (4) The fitting correlation coefficient of beta-function curve and each tooth on the dental and basal bone arch of skeletal class Ⅲ malocclusion were greater than 0.7 (P＜0.05). Conclusion In this study, the mathematical model can be used to describe the dental and basal bone arch form of the skeletal class Ⅲ malocclusion, which can guide further research.

Key words: skeletal class Ⅲ malocclusion, dental arch, basal bone arch, mathematical model, cone-beam computed tomography, Beta-function