# Mathematical modeling of the human dental arch and its usefulness in longitudinal analysis of treatment effects

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In the practice of orthodontics, the shape of the dental arches is important in the planning and implementation of treatment. Many mathematical functions have been proposed for the characterization of arch form including catenary, p~lynomials, beta, conic sections, and cubic splines. The purpose of this study was to use linear and nonlinear least squares estimation to fit polynomial, catenary-like and beta-like curves to a longitudinal dataset and evaluate both the curve fits and the longitudinal information obtained. A longitudinal dataset was obtained from a private orthodontist. Dental casts of the upper and lower arches were made at three time points for each of 20 subjects: before treatment, immediately following treatment, and following a post-retention follow-up period of at least two years. Each cast and a calibration strip was scanned into a separate image computer file. Image analysis software was used to mark the (x,y) coordinates of buccal landmarks on each tooth from first molar to first molar. The (x, y) coordinates from each cast were collected into a central database for analysis. It was desired to use least squares for curve fitting due to its wide availability and well known properties. In order to use least squares, the casts were required to have consistent x-axis and y-axis orientation. This was done by orienting the x-axis parallel to the line connecting the centroids of the posterior teeth on the right and left sides of each cast. Eight functions were used in the curves fitting. The linear least squares method was used to fit polynomials of 2nd, 3rd, 4th, and 5th order. The nonlinear least squares method was used to fit a generalized 5-parameter beta function and generalized inverse catenary functions with 3, 4, and 5 parameters. Each of the eight functions was fit to each of the 120 dental casts in the study. Curve fits were examined for each function and each subject, arch, and time point. The 4th and 5th order polynomials, the generalized beta, and the 4-parameter and 5-parameter generalized inverse catenary functions fit well. For the 4th and 5th order polynomials, the R2 values ranged from xxx to xxx with acceptable visual fits. For the nonlinear models, the model sum of squares approximated the total sum of squares and the curves yielded good visual fits to the data points. Longitudinal analysis was done using Euclidean distance as the metric in the parameter space of each model. In order to assess the parameter metric in terms of physical measurements, the Euclidean distances in the parameter spaces were correlated with intercanine width, intermolar width, and molar-incisor distance. Consistent correlations were not identified though the curve fits were excellent. A comparison of arch form change between upper and lower arches was also done. Since the upper arches changed more, checking the ability of the parameter metrics of the various models to detect the change was of interest. and 3rd order polynomials, All of the models except 2nd and molar-incisor distance measures were capable of detecting the difference in change between the upper and lower arches (ANOVA p-values ~ 0.05). In summary, this study shows a successful method of orienting the casts for curve fitting by least squares. The models with at least 4 parameters generally fit well across the range of dental casts studied with the 5- parameter models slightly superior. The longitudinal analysis indicates that traditional linear measurements such as intercanine width may not adequately measure the multidimensional aspects of arch form change. The parameter space metrics were able to discriminate between upper and lower arch form changes.