University of Texas - Department of Biostatistics, MD Anderson Cancer Center
Glaucoma is an ocular condition involving damage to the optic nerve that can lead to blindness if not treated. Its complete etiology is not known, but it is hypothesized that biomechanics in regions of the sclera most adjacent to the optic nerve head under interocular pressure (IOP) may play a role. It is proposed that as people age, their scleral surface may lose elasticity, causing less displacement in response to IOP and thus increased pressure affecting the optic nerve head (ONH). In a study involving 19 pairs of healthy donor eyes, researchers have developed and used a novel system for inducing IOP and precisely measuring induced displacement and maximum principal strain (MPS) on a fine grid on the outer scleral surface of the eye for a range of pressures. These yield multi-level functional data of MPS for a fine grid on a (partially) spherical manifold, with functional observations from each of 9 IOPs for each eye, two eyes per subject. The goal is to assess which regions of the sclera experience the highest MPS, and how MPS changes with age in different regions of the sclera.
Their typical analysis approach would be to bin the functional data into several fixed regions and then model using linear mixed models, which does not use all of the information in the data nor carefully account for the interocular correlations. Here, we introduce Bayesian functional data analysis techniques to model the entire functional data on the fine grid of the sphere as a response, from which regional and sectional summaries can be computed or results interpreted on the space of the entire sclera. We propose to use multiresolution basis functions on the sphere in our modeling to capture the interocular correlations in a nonstationary and flexible way. We capture the correlation between left and right eyes using random effect functions, and we account for the correlation of the MPS functions for different pressures on the same eye by extending growth curve ideas to the spherical functional data setting.
Most existing work on functional response regression and functional mixed models assumes linear relationships between the functional response and predictors. Here, we extend the nonparametric additive models idea to our setting to allow a nonparametric effect of age on MPS through penalized smoothing splines. This nonparametric effect is allowed to freely vary over the scleral surface, and in our approach borrows strength from nearby positions on the sphere in terms of estimation, variance, and degrees of freedom of the nonparametric fit. Further, our model also includes an IOP-age interaction that allows the nonparametric age effect to vary over IOP, and a growth curve component to capture the longitudinal correlations across different IOPs for a given eye. Our analysis provides substantially detailed inferential summaries on the scleral space, and demonstrates MPS is greatest near the optic nerve, where there is a clear decreasing trend in MPS with age that accelerates around age 65. This work greatly extends the functional mixed model framework to incorporate additive model and growth curve ideas for regression analyses involving responses that are functions or objects on some fixed domain.