A mathematic-geometric model to calculate variation in mandibular arch form

Area of Science:

• Orthodontics
• Biomechanical Engineering
• Dental Morphology

Background:

• Understanding mandibular dental arch length variation is crucial for effective orthodontic treatment planning.
• Incisor inclination and tipping are key factors influencing arch dimensions.
• Previous models often simplified the complex geometric relationships within the dental arch.

Purpose of the Study:

• To mathematically model and quantify the changes in mandibular dental arch length based on incisor inclination.
• To compare the effects of controlled versus uncontrolled incisor tipping on arch length.
• To analyze how different dental arch forms (parabolic, elliptical, hyperbolic, circular, catenary) influence these changes.

Main Methods:

• Development of a mathematic-geometric model to represent the lower dental arch.
• Analytical study of curve equations describing the arch before and after simulated incisor inclination (1 mm and 1 degree).
• Calculation of arch length variations for different tipping control (controlled vs. uncontrolled) and arch forms.

Main Results:

• Mandibular dental arch length changes varied significantly across different arch forms for each millimeter of incisor inclination.
• Catenary arch forms showed the greatest increase in length (2.07 mm/mm inclination), while elliptical and circular forms showed the least (1.21 mm/mm inclination).
• Controlled tipping resulted in a longer inter-incisive arch length compared to uncontrolled tipping across all analyzed forms.

Conclusions:

• The specific form of the mandibular dental arch influences the amount of space gained or required during orthodontic tooth movement.
• Controlled incisor tipping is a more efficient method for increasing arch length than uncontrolled tipping.
• Orthodontic treatment planning should incorporate the analysis of dental arch type to optimize space management and treatment outcomes.