Evaporation-driven tear film thinning and breakup in two space dimensions

TL;DR

The paper Develops a two-dimensional thin-film model of tear film thinning and breakup driven by spatially varying evaporation $J(x,y)$, capturing spot, streak, and intermediate patterns in a localized region of the cornea. It solves the non-dimensional PDEs for thickness $h$, pressure $p$, osmolarity $c$, and fluorescein concentration $f$ using a Fourier spectral collocation method and advances the solution with a DAE solver, while applying proper orthogonal decomposition (POD) to project onto low-dimensional bases and accelerate computation. The results show that TBU dynamics cannot be represented as a simple sum of 1D solutions; the shape of the evaporation distribution continuously interpolates between circular spots and elongated streaks, with diffusion, evaporation, and osmosis balancing to determine thinning, osmolarity, and fluorescence patterns. POD-based acceleration yields fourfold or greater speedups with tolerable error, enabling efficient exploration of multi-spot interactions and paving the way for inverse problem applications and parameter estimation of unobservable quantities such as local osmolarity in vivo.

Abstract

Evaporation profiles have a strong effect on tear film thinning and breakup (TBU), a key factor in dry eye disease (DED). In experiments, TBU is typically seen to occur in patterns that locally can be circular (spot), linear (streak), or intermediate . We investigate a two-dimensional (2D) model of localized TBU using a Fourier spectral collocation method to observe how the evaporation distribution affects the resulting dynamics of tear film thickness and osmolarity, among other variables. We find that the dynamics are not simply an addition of individual 1D solutions of independent TBU events, and we show how the TBU quantities of interest vary continuously from spots to streaks for the shape of the evaporation distribution. We also find a significant speedup by using a proper orthogonal decomposition to reduce the dimension of the numerical system. The speedup will be especially useful for future applications of the model to inverse problems, allowing the clinical observation at scale of quantities that are thought to be important to DED but not directly measurable in vivo within TBU locales.

Figures (19)

• Figure 1: At top left, bottom left and bottom right are fluorescence images from three different video capture of STARE trials awisi-gyauChangesCornealDetection2019a. The top right shows a time series of the evolution of the dark spot highlighted in the top left image. The highlights in the bottom left and bottom right images show respectively an indeterminate shape and a formation of connected spots.(Image courtesy of D.Awisi-Gyau.)
• Figure 2: Top left: Computation time in hours with respect to the Fourier grid resolution $N$. The other three figures show the relative error at TBUT in the dependent variables $h,$$p,$ and $c$.
• Figure 3: Nondimensional solution for a single circular evaporation spot, shown as functions of distance $r$ from the center of the spot and time $t$. The peak evaporation rate is $a_1=1$ over a baseline rate of $v_b=0.1$, and the spot width is $x_w=y_w=0.5$.
• Figure 4: FL intensity and osmolarity at different time levels for three different one-spot solutions. The TBUTs for cases (a), (b), and (c) in \ref{['eq:threespots']} are $1.1$, $1.7$, and $2.2$, respectively.
• Figure 5: Left: Central values of tear film thickness, osmolarity and FL intensity for three different spots \ref{['eq:threespots']}. Right: Dominant terms in the osmolarity equation \ref{['eq:termtypes']}.