Evaluating intraocular lens power formula constant robustness using bootstrap algorithms

Abstract

Background: Bootstrapping is a modern technique mostly used in statistics to evaluate the robustness of model parameters. The purpose of this study was to develop a method for evaluation of formula constant uncertainties and the effect on the prediction error (PE) in intraocular lens power calculation with theoretical-optical formulae using bootstrap techniques.

Methods: In a dataset with N = 888 clinical cases treated with the monofocal aspherical intraocular lens (Vivinex, Hoya) constants for the Haigis, the Castrop and the SRKT formula were optimised for the sum of squared PE using nonlinear iterative optimisation (interior point method), and the formula predicted spherical equivalent refraction (predSEQ) and the PE were derived. The PE was bootstrapped NB = 1000 times and added to predSEQ, and formula constants were derived for each bootstrap. The robustness of the constants was calculated from the NB bootstrapped models, and the predSEQ was back-calculated from the NB formula constants.

Results: With bootstrapping, the 90% confidence intervals for the a0/a1/a2 constants of the Haigis formula were -0.8317 to -0.5301/0.3203 to 0.3617/0.1954 to 0.2100, for the C/H/R constants of the Castrop formula they were 0.3113 to 0.3272/0.1237 to 0.2149/0.0980 to 0.1621, and for the A constant of the SRKT formula they were 119.2320 to 119.3028. The back-calculated PE from the NB bootstrapped formula constants standard deviation for the mean/median/mean absolute/root mean squared PE were 5.677/5.735/0.401/0.318 e-3 dpt for the Haigis formula, 5.677/5.735/0.401/0.31829 e-3 dpt for the Castrop formula and 14.748/14.790/0.561/0.370 e-3 dpt for the SRKT formula.

Conclusion: We have been able to prove with bootstrapping that nonlinear iterative formula constant optimisation techniques for the Haigis, the Castrop and the SRKT formulae yield consistent results with low uncertainties of the formula constants and low variations in the back-calculated mean, median, mean absolute and root mean squared formula prediction error.

Keywords: bootstrap techniques; constant optimisation; formula prediction error; lens formula evaluation; lens power calculation; performance metrics.