Onset and Risk Period of Functional Sound Tooth Loss: A Hyperbolic Tangent Model

Existing percentile analyses describe age-related tooth loss trajectories but do not quantify when accelerated loss begins or how long the high-risk period lasts. This study aimed to apply a hyperbolic tangent model to functional sound tooth loss data to derive clinically interpretable transition parameters.

A hyperbolic tangent regression was fitted to normalised functional sound tooth loss data from 27,215 adults across 5 cycles of the Korean National Health and Nutrition Examination Survey (2010-2015), covering the 10th, 25th, 50th, and 75th percentile groups. Goodness of fit was compared with linear, polynomial, and logistic alternatives using the coefficient of determination (R²) and the Akaike information criterion. Ninety-five percent confidence intervals for all model parameters were derived from the Hessian of the least squares objective function.

The hyperbolic tangent model achieved the highest goodness of fit across all percentile groups (R² = 0.963-0.996), outperforming all 3 alternatives. The peak age b (age of maximum tooth loss velocity, 50% cumulative loss) was 57.3, 64.9, 74.9, and 85.9 years, and the spreading coefficient c (width of the high-risk period) was 9.8, 12.8, 17.2, and 17.6 years for the 10th through 75th percentile groups, respectively. The onset age (b - c) was 47.4, 52.2, 57.6, and 68.3 years, with narrow 95% confidence intervals across all groups.

The hyperbolic tangent model provides a parsimonious 2-parameter characterisation of the functional tooth loss trajectory. The onset age and the high-risk duration (risky term, 2c) support percentile-specific risk stratification for preventive dental care and population-level oral health planning. Findings should be interpreted as population-level estimates from a cross-sectional design.