The Expression and Comparison of Healthy and Ptotic Upper Eyelid Contours Using a Polynomial Mathematical Function

Abstract

Purpose: To derive a mathematical expression for the healthy upper eyelid (UE) contour and to use this expression to differentiate the normal UE curve from its abnormal configuration in the setting of blepharoptosis.

Methods: The study was designed as a cross-sectional study. Fifty healthy subjects (26M/24F) and 50 patients with blepharoptosis (28M/22F) with a margin–reflex distance (MRD₁) of ≤2.5 mm were recruited. A polynomial interpolation was used to approximate UE curve. The polynomial coefficients were calculated from digital eyelid images of all participants using a set of operator defined points along the UE curve. Coefficients up to the fourth-order polynomial, iris area covered by the UE, iris area covered by the lower eyelid and total iris area covered by both the upper and the lower eyelids were defined using the polynomial function and used in statistical comparisons. The t-test, Mann–Whitney U test and the Spearman’s correlation test were used for statistical comparisons.

Results: The mathematical expression derived from the data of 50 healthy subjects aged 24.1 ± 2.6 years was defined as y = 22.0915 + (−1.3213)x + 0.0318x²+ (−0.0005x)³. The fifth and the consecutive coefficients were <0.00001 in all cases and were not included in the polynomial function. None of the first fourth-order coefficients of the equation were found to be significantly different in male versus female subjects. In normal subjects, the percentage of the iris area covered by upper and lower lids was 6.46 ± 5.17% and 0.66% ± 1.62%, respectively. All coefficients and mean iris area covered by the UE were significantly different between healthy and ptotic eyelids.

Conclusions: The healthy and abnormal eyelid contour can be defined and differentiated using a polynomial mathematical function.

• Blepharoptosis
• eyelid contour
• polynomial function

Notes

*The study was conducted at the Department of Ophthalmology and Department of Mathematics at Hacettepe University, Ankara, Turkey