Simple Models of Tear Break Up (TBU) and Fluorescence

Abstract

Purpose : Simplified mathematical models for TBU dynamics and it fluorescent (FL) imaging are developed. The models capture some essential dynamics of some common types of TBU. Etiologies of TBU include evaporation driven (Type I; Figure 1a) and divergent flow driven (Type II; Fig 1b); a mix of these two appears to occur in vivo (Type III). We propose models for all three types to aid understanding of TBU visualization.

Methods : Ordinary differential equation (ODE) models are derived for the tear film thickness h(t), osmolarity c(t) and FL concentration f(t) where t is time. In each case, the ODEs are solved for h(t) using a custom Matlab code; solute concentrations are computed from separate equations for solute mass conservation. FL intensity (I) is computed as described by Nichols et al (IOVS 2012, 53:5426).

Results : For Type I, there is evaporation and osmosis but no divergent flow (Braun et al, IOVS 2014, 55:1133). Osmolarity is elevated during thinning. For imaging, this theory works best concentrations near of above the critical FL concentration where self-quenching occurs (f₀=0.7 or 2; Fig. 2). For Type II, there is divergent flow but no evaporation or osmosis. Thinning follows h(t)=e^-at, while c and f remain at their initial values (vertical dashed lines, Fig. 2). For Type III, all three of divergent flow, evaporation and osmosis are active. A single ODE for h(t) is solved, and solute conservation requires f=f₀e^-at/h. The resulting intensity I falls between Types I and II (Fig. 2). For f₀ near or above the critical concentration, intensity drops rapidly like Type I. Unlike Type II, both and change for small (dilute, f₀=0.1). The FL intensity from the math models is consistent with experimental results when time is not too large.

Conclusions : Simple models of TBU can capture dynamics of evaporation, osmosis, divergent flow, and imaging. Previous results for quenching are recovered, and new results for flow are confirmed by more complex models (Zhong et al, Bull. Math. Biol. 2018, to appear).

This abstract was presented at the 2019 ARVO Annual Meeting, held in Vancouver, Canada, April 28 - May 2, 2019.

 

Sketches of Type I (a) and II (b) TBU.

 

Sample results for I as a function of f (normalized with f_cr for Type I (solid curves), Type II (dashed), and Type III (dash-dot) TBU. Constant thickness curves are in green (5, 3.5 and 2 μm, from top).