Toby Driscoll is a Unidel Chaired Professor in the Department of Mathematical Sciences at the University of Delaware. His research interests are in scientific computation, mathematical software, and applications of mathematics in the life sciences.
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PhD in Applied Mathematics, 1996
MS in Applied Mathematics, 1993
BS in Mathematics, 1991
Pennsylvania State University
BS in Physics, 1991
Pennsylvania State University
The contribution of different physical effects to tear breakup (TBU) in subjects with no self-reported history of dry eye are quantified. An automated system using a convolutional neural network is deployed on fluorescence (FL) imaging videos to identify multiple likely TBU instances in each trial. Once identified, extracted FL intensity data was fit by mathematical models that included tangential flow along the eye, evaporation, osmosis and FL intensity of emission from the tear film. The mathematical models consisted of systems of ordinary differential equations for the aqueous layer thickness, osmolarity, and the FL concentration. Optimizing the fit of the models to the FL intensity data determined the mechanism(s) driving each instance of TBU and produced an estimate of the osmolarity within TBU. Fits were produced for 467 instances of potential TBU from 15 non-DED subjects. The results showed a distribution of causes of TBU in these healthy subjects, as reflected by estimated flow and evaporation rates, which appear to agree well with previously published data. Final osmolarity depended strongly on the TBU mechanism, generally increasing with evaporation rate but complicated by the dependence on flow. The results suggest that it might be possible to classify individual subjects and provide a baseline for comparison and potential classification of dry eye disease subjects.
Motivated by problems arising in tear film dynamics, we present a model for the extensional flow of thin sheets of nematic liquid crystal. The rod-like molecules of these substances impart an elastic contribution to its response. We rescale a weakly elastic model due to Cummings et al. [European Journal of Applied Mathematics 25 (2014): 397-423] to describe a case of moderate elasticity. The resulting system of two nonlinear partial differential equations for sheet thickness and axial velocity is nonlinear and fourth order in space, but still represents a significant reduction of the full system. We analyze solutions arising from several different boundary conditions, motivated by the underlying application, with particular focus on dynamics and underlying mechanisms under stretching. We solve the system numerically, via collocation with either finite difference or Chebyshev spectral discretization in space, together with implicit time stepping. At early times, depending on the initial film shape, pressure either aids or opposes extensional flow, which changes the shape of the sheet and may result in the loss of a minimum or maximum at the moving end. We contrast this finding with the cases of weak elasticity and Newtonian flow, where the sheet retains all extrema from the initial condition throughout time.
The human tear film is rapidly established after each blink, and is essential for clear vision and eye health. This paper reviews mathematical models and theories for the human tear film on the ocular surface, with an emphasis on localized flows where the tear film may fail. The models attempt to identify the important physical processes, and their parameters, governing the tear film in health and disease.
Purpose: Fluorescence imaging is a valuable tool for studying tear film dynamics and corneal staining. Automating the quantification of fluorescence images is a challenging necessary step for making connections to mathematical models. A significant part of the challenge is identifying the region of interest, specifically the cornea, for collected data with widely varying characteristics. Methods: The gradient of pixel intensity at the cornea– sclera limbus is used as the objective of standard optimization to find a circle that best represents the cornea. Results of the optimization in one image are used as initial conditions in the next image of a sequence. Additional initial conditions are chosen heuristically. The algorithm is coded in open-source software. Results: The algorithm was first applied to 514 videos of 26 normal subjects, for a total of over 87,000 images. Only in 12 of the videos does the standard deviation in the detected corneal radius exceed 1% of the image height, and only 3 exceeded 2%. The algorithm was applied to a sample of images from a second study with 142 dry-eye subjects. Significant staining was present in a substantial number of these images. Visual inspection and statistical analysis show good results for both normal and dry-eye images. Conclusion: The new algorithm is highly effective over a wide range of tear film and corneal staining images collected at different times and locations.
Etiologies of tear breakup include evaporation-driven, divergent flow-driven, and a combination of these two. A mathematical model incorporating evaporation and lipid-driven tangential flow is fit to fluorescence imaging data. The lipid-driven motion is hypothesized to be caused by localized excess lipid, or “globs.” Tear breakup quantities such as evaporation rates and tangential flow rates cannot currently be directly measured during breakup. We determine such variables by fitting mathematical models for tear breakup and the computed fluorescent intensity to experimental intensity data gathered in vivo. Parameter estimation is conducted via least squares minimization of the difference between experimental data and computed answers using either the trust-region-reflective or Levenberg–Marquardt algorithm. Best-fit determination of tear breakup parameters supports the notion that evaporation and divergent tangential flow can cooperate to drive breakup. The resulting tear breakup is typically faster than purely evaporative cases. Many instances of tear breakup may have similar causes, which suggests that interpretation of experimental results may benefit from considering multiple mechanisms.
Many parameters affect tear film thickness and fluorescent intensity distributions over time; exact values or ranges for some are not well known. We conduct parameter estimation by fitting to fluorescent intensity data recorded from normal subjects’ tear films. The fitting is done with thin film fluid dynamics models that are nonlinear partial differential equation models for the thickness, osmolarity and fluorescein concentration of the tear film for circular (spot) or linear (streak) tear film breakup. The corresponding fluorescent intensity is computed from the tear film thickness and fluorescein concentration. The least squares error between computed and experimental fluorescent intensity determines the parameters. The results vary across subjects and trials. The optimal values for variables that cannot be measured in vivo within tear film breakup often fall within accepted experimental ranges for related tear film dynamics; however, some instances suggest that a wider range of parameter values may be acceptable.
The additive Schwarz method is usually presented as a preconditioner for a PDE linearization based on overlapping subsets of nodes from a global discretization. It has previously been shown how to apply Schwarz preconditioning to a nonlinear problem. By first replacing the original global PDE with the Schwarz overlapping problem, the global discretization becomes a simple union of subdomain discretizations, and unknowns do not need to be shared. In this way, restrictive-type updates can be avoided, and subdomains need to communicate only via interface interpolations. The resulting preconditioner can be applied linearly or nonlinearly. In the latter case, nonlinear subdomain problems are solved independently in parallel, and the frequency and amount of interprocess communication can be greatly reduced compared to global preconditioning of the sequence of linearized problems.
Experiments and theoretical models suggest that the performance of intermediate band solar cells based on quantum dots (QDs) will be enhanced by the formation of delocalized intermediate bands. However, reasonable device performance has only been achieved when the QD separation is large and energy states are localized to individual QDs. In this paper we analyze the formation of delocalized bands in a realistic QD material that has inhomogeneously distributed energy levels. We calculate the QD uniformity or barrier thickness necessary to create delocalized states in realistic materials and propose a design to create delocalized states while including strain balancing layers.