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Aim To analyse the prevalence of periapical lesions and their association with previous root canal treatment, root canal filling length and type of coronal restoration using in vivo cone‐beam computed tomographic (CBCT) assessment. Methodology A global sample of 20 836 teeth, with a combined total of 27 046 roots, from 1160 patients, was analysed via CBCT assessment in eight health centres. Each tooth was evaluated by one out of five examiners after having performed a defined calibration procedure on the basis of 319 teeth. Intra‐ and inter‐rater reliability tests were performed. Each tooth was classified according the tooth number, presence/absence of periapical lesions, presence/absence of previous root canal treatment, length of root canal filling (short, good or overfilling) and type of coronal restoration. The z‐test for proportions was used to analyse differences between tooth subgroups, and an odds ratio was determined in order to analyse the association between treatment status and periapical lesions. A P < 0.05 was considered significant. Results At a tooth level, the overall prevalence of periapical lesions in the sample was 10.4%. Maxillary teeth were associated with a significantly larger percentage of lesions (13.1%), whilst maxillary first molars had the greater proportion of lesions (21.2%). The prevalence of periapical lesions was significantly larger in root filled teeth (55.5%), short root canal fillings (72.7%) and in teeth restored with crowns (46.1%). At a root level, the mesiobuccal roots of both maxillary first molars had a tendency for a larger percentage of periapical lesions. Conclusion History of root canal treatment, root canal filling length and type of coronal restoration influenced the presence of periapical lesions. Molars were more commonly associated with periapical lesions on root filled teeth, particularly those with short root fillings and those with crowns.

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Summary In this article, we study the existence of solutions for the problem of interaction of linear water waves with an array of three-dimensional fixed structures in a density-stratified multi-layer fluid, where in each layer the density is assumed to be constant. Considering time-harmonic small-amplitude motion, we present recursive formulae for the coefficients of the eigenfunctions of the spectral problem associated with the water-wave problem in the absence of obstacles and for the corresponding dispersion relation. We derive a variational and operator formulation for the problem with obstacles and introduce a sufficient condition for the existence of propagating waves trapped in the vicinity of the array of obstacles. We present several (arrays of) structures supporting trapped waves and discuss the possibility of approximating the continuously stratified fluid by a multi-layer model.

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