Refractive changes after descemet stripping endothelial keratoplasty: a simplified mathematical model.

PURPOSE: To develop a mathematical model that can predict refractive changes after Descemet stripping endothelial keratoplasty (DSEK). METHODS: A mathematical formula based on the Gullstrand eye model was generated to estimate the change in refractive power of the eye after DSEK. This model was applied to four DSEK cases retrospectively, to compare measured and predicted refractive changes after DSEK. RESULTS: The refractive change after DSEK is determined by calculating the difference in the power of the eye before and after DSEK surgery. The power of the eye post-DSEK surgery can be calculated with modified Gullstrand eye model equations that incorporate the change in the posterior radius of curvature and change in the distance between the principal planes of the cornea and lens after DSEK. Analysis of this model suggests that the ratio of central to peripheral graft thickness (CP ratio) and central thickness can have significant effect on refractive change where smaller CP ratios and larger graft thicknesses result in larger hyperopic shifts. This model was applied to four patients, and the average predicted hyperopic shift in the overall power of the eye was calculated to be 0.83 D. This change reflected in a mean of 93% (range, 75%-110%) of patients' measured refractive shifts. CONCLUSIONS: This simplified DSEK mathematical model can be used as a first step for estimating the hyperopic shift after DSEK. Further studies are necessary to refine the validity of this model.

Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine.We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky). © Institute of Mathematical Statistics, 2013.

Locomotor head movements and semicircular canal morphology in primates.

Animal locomotion causes head rotations, which are detected by the semicircular canals of the inner ear. Morphologic features of the canals influence rotational sensitivity, and so it is hypothesized that locomotion and canal morphology are functionally related. Most prior research has compared subjective assessments of animal "agility" with a single determinant of rotational sensitivity: the mean canal radius of curvature (R). In fact, the paired variables of R and body mass are correlated with agility and have been used to infer locomotion in extinct species. To refine models of canal functional morphology and to improve locomotor inferences for extinct species, we compare 3D vector measurements of head rotation during locomotion with 3D vector measures of canal sensitivity. Contrary to the predictions of conventional models that are based upon R, we find that axes of rapid head rotation are not aligned with axes of either high or low sensitivity. Instead, animals with fast head rotations have similar sensitivities in all directions, which they achieve by orienting the three canals of each ear orthogonally (i.e., along planes at 90° angles to one another). The extent to which the canal configuration approaches orthogonality is correlated with rotational head speed independent of body mass and phylogeny, whereas R is not.