The Relation between Orthodontic Force and Subsequent Tooth Movement, a Mathematical Model Based on Beagle Dog Data

Objective: To describe the relation between the magnitude of an orthodontic force and the rate of subsequent tooth movement in a mathematical model. Methods: Data from different experimental studies in beagle dogs were used. In all cases mandibular second premolars were moved distally by well-controlled standardised forces of 10, 25, 50, 100, 200, 300, 600, or 1200 cN . In total 120 individual time-displacement curves were constructed. From these curves, the mean rate of experimental tooth movement was determined after hyalinisation had ceased. The mathematical modelling was based on the following assumptions: (a) no tooth movement without force, (b) a positive correlation between force and velocity at low forces, (c) a maximum or plateau at higher forces, (d) the velocity may remain stable or decrease with further increase of force, and (e) the velocity will never become negative. The following equation was used: V(F) = (Vmax)*(F/Fmax)^(Fmax/Fs)*e^((Fmax-F)/Fs) Iterative non-linear regression techniques were used to achieve a best fit. Results: The best fit of this equation to the data resulted in an explained variance (R2) of 43%. The mean Vmax = 0.27 mm/wk (95% CI = 0.23 – 0.30 mm/wk), Fmax = 248 cN (95% CI = 104 – 454 cN), and Fb = 2550 cN (95% CI = 40 – 9120). The model fits well to the data, a decrease of the rate at high force levels is present, but its slope is rather uncertain. Conclusion: This mathematical model shows that in dogs the maximum rate of experimental tooth movement is between 0.23 and 0.30 mm/wk, and that it is achieved using forces between 104 and 454 cN. The magnitude of the applied force, however, is not the most important factor. Individual differences in morphology and physiology of bone and periodontal ligament may be more important for the rate of tooth movement.