A naturalistic study of brushing patterns using powered toothbrushes.

Dental caries and periodontal disease are very common chronic diseases closely linked to inadequate removal of dental plaque. Powered toothbrushes are viewed as more effective at removing plaque; however, the conflicting evidence and considerable unexplained heterogeneity in their clinical outcomes does not corroborate the relative merits of powered tooth brushing. To explain the heterogeneity of brushing patterns with powered toothbrushes, we conducted a observational study of tooth brushing practices of 12 participants in their naturalistic setting. Integrated brush sensors and a digital data collection platform allowed unobtrusive and accurate capture of habitual brushing patterns. Annotated brushing data from 10 sessions per participant was chosen for scrutiny of brushing patterns. Analysis of brushing patterns from the total 120 sessions revealed substantial between- and within-participant variability in brushing patterns and efficiency. Most participants (91.67%) brushed for less than the generally prescribed two minutes; individual participants were also inconsistent in brushing duration across sessions. The time devoted to brushing different dental regions was also quite unequal. Participants generally brushed their buccal tooth surfaces more than twice as long as the occlusal (2.18 times longer (95% CI 1.42, 3.35; p < 0.001)) and lingual surfaces (2.22 times longer (95% CI 1.62, 3.10; p < 0.001); the lingual surfaces of the maxillary molars were often neglected (p < 0.001). Participants also varied in the epochs of excessive brushing pressure and the regions to which they were applied. In general, the occlusal surfaces were more likely to be brushed with excessive pressure (95% CI 0.10, 0.98; p = 0.015). Our study reveals that users of powered toothbrushes vary substantially in their use of the toothbrushes and diverge from recommended brushing practices. The inconsistent brushing patterns, between and within individuals, can affect effective plaque removal. Our findings underscore the limited uptake of generic oral self-care recommendations and emphasize the need for personalized brushing recommendations that derive from the objective sensor data provided by powered toothbrushes.

Extensions to the Proximal Distance Method of Constrained Optimization.

The current paper studies the problem of minimizing a loss f(x) subject to constraints of the form Dx ∈ S, where S is a closed set, convex or not, and D is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method of optimization with the proximal distance principle. The latter is driven by minimization of penalized objectives f(x)+ρ2dist(Dx,S)2 involving large tuning constants ρ and the squared Euclidean distance of Dx from S. The next iterate xn+1 of the corresponding proximal distance algorithm is constructed from the current iterate xn by minimizing the majorizing surrogate function f(x)+ρ2‖Dx-𝒫S(Dxn)‖2. For fixed ρ and a subanalytic loss f(x) and a subanalytic constraint set S, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare their results to those delivered by the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems. Julia code to replicate all of our experiments can be found at https://github.com/alanderos91/ProximalDistanceAlgorithms.jl.