A mathematical model for predicting toothbrush stiffness

Both cleaning efficiency and gingival damage depend on the access of toothbrush bristles to sheltered areas and their ability to deliver sufficient force to remove plaque as they travel over tooth surfaces. A mathematical expression was therefore developed which relates bristle properties and features of brush construction to overall brush stiffness, in order to provide a framework for the prediction and systematic investigation of brush performance:. Brush stiffness = 0.125E(D_BD_T)²N_TP_f/L³. This shows that brush stiffness is predominantly affected by bristle modulus (E), bristle and tuft diameter (D_B and D_T), the number of tufts (N_T), the number of bristles per unit area packed into a tuft hole (P_f = packing factor), and the trim length of bristles (L). Bristle composition and shape had no measurable effect. The 0.125 factor is empirically derived and probably depends on visco-elastic, frictional, and other dynamic effects which were not examined. Thus, this is a first-order approximation, and further work must be done to account for bristle interactions and brushing rate, and to correlate stiffness with a measure of cleaning efficiency.