Purpose: The aim of this study is to present strategies for optimization of lens power (IOLP) formula constants and to show options how to present the results adequately. Methods: A dataset of N = 1,601 preoperative biometric values, IOLP data and postoperative refraction data was split into a training set and a test set using a random sequence. Based on the training set, we calculated the formula constants for established lens calculation formulae with different methods. Based on the test set, we derived the formula prediction error (PE) as difference of the achieved refraction from the formula predicted refraction. Results: For formulae with 1 constant, it is possible to back-calculate the individual constant for each case using formula inversion. However, this is not possible for formulae with >1 constant. In these cases, more advanced concepts such as non-linear optimization strategies are necessary to derive the formula constants. During cross-validation, measures such as the mean absolute or the root mean squared PE or the ratio of cases within mean absolute PE (MAE) limits could be used as quality measures. Conclusions: Different constant optimization concepts yield different results. To test the performance of optimized formula constants, a cross-validation strategy is mandatory. We recommend performance curves, where the ratio of cases within absolute PE limits is plotted against the MAE.

Background: To investigate modern nonlinear iterative strategies for formula constant optimisation and show the application and results from a large dataset using a set of disclosed theoretical-optical lens power calculation concepts. Methods: Nonlinear iterative optimisation algorithms were implemented for optimising the root mean squared (SoSPE), the mean absolute (SoAPE), the mean (MPE), the standard deviation (SDPE), the median (MEDPE), as well as the 90% confidence interval (CLPE) of the prediction error (PE), defined as the difference between postoperative achieved and formula predicted spherical equivalent power of refraction. Optimisation was performed using the Levenberg-Marquardt algorithm (SoSPE and SoAPE) or the interior point method (MPE, SDPE, MEDPE, CLPE) for the SRKT, Hoffer Q, Holladay 1, Haigis, and Castrop formulae. The results were based on a dataset of measurements made on 888 eyes after implantation of an aspherical hydrophobic monofocal intraocular lens (Vivinex, Hoya). Results: For all formulae and all optimisation metrics, the iterative algorithms showed a fast and stable convergence after a couple of iterations. The results prove that with optimisation for SoSPE, SoAPE, MPE, SDPE, MEDPE, and CLPE the root mean squared PE, mean absolute PE, mean PE, standard deviation of PE, median PE, and confidence interval of PE could be minimised in all situations. The results in terms of cumulative distribution function are quite coherent with optimisation for SoSPE, SoAPE, MPE and MEDPE, whereas with optimisation for SDPE and CLPE the standard deviation and confidence interval of the PE distribution could only be minimised at the cost of a systematic offset in mean and median PE. Conclusion: Nonlinear iterative techniques are capable of minimising any statistical metrics (e.g. root mean squared or mean absolute error) of any target parameter (e.g. PE). These optimisation strategies are an important step towards optimising for the target parameters which are used for evaluating the performance of lens power calculation formulae.

Purpose: To investigate the effect of formula constants on predicted refraction and limitations of constant optimization for classical and modern intraocular lens (IOL) power calculation formulae. Setting: Tertiary care center. Design: Retrospective single-center consecutive case series. Methods: This analysis is based on a dataset of 888 eyes before and after cataract surgery with IOL implantation (Hoya Vivinex). Spherical equivalent refraction predSEQ was predicted using IOLMaster 700 data, IOL power, and formula constants from IOLCon (https://iolcon.org). The formula prediction error (PE) was derived as predSEQ minus achieved spherical equivalent refraction for the SRKT, Hoffer Q, Holladay, Haigis, and Castrop formulae. The gradient of predSEQ (gradSEQ) as a measure for the effect of the constants on refraction was calculated and used for constant optimization. Results: Using initial formula constants, the mean PE was −0.1782 ± 0.4450, −0.1814 ± 0.4159, −0.1702 ± 0.4207, −0.1211 ± 0.3740, and −0.1912 ± 0.3449 diopters (D) for the SRKT, Hoffer Q, Holladay, Haigis, and Castrop formulas, respectively. gradSEQ for all formula constants (except gradSEQ for the Castrop R) decay with axial length because of interaction with the effective lens position (ELP). Constant optimization for a zero mean PE (SD: 0.4410, 0.4307, 0.4272, 0.3742, 0.3436 D) results in a change in the PE trend over axial length in all formulae where the constant acts directly on the ELP. Conclusions: With IOL power calculation formulae where the constant(s) act directly on the ELP, a change in constant(s) always changes the trend of the PE according to gradSEQ. Formulae where at least 1 constant does not act on the ELP have more flexibility to zero the mean or median PE without coupling with a PE trend error over axial length.

Purpose: The purpose of this study was to investigate the repeatability of biometric measures and also to assess the interactions between the uncertainties in these measures for use in an error propagation model, using data from a large patient cohort. Methods: In this cross-sectional non-randomised study we evaluated a dataset containing 3379 IOLMaster 700 biometric measurements taken prior to cataract surgery. Only complete scans with at least 3 successful measurements for each eye performed on the same day were considered. The mean (Mean) and standard deviations (SD) for each sequence of measurements were derived and analysed. Correlations between the uncertainties were assessed using Spearman rank correlations. Results: In the dataset with 677 eyes matching the inclusion criteria, the within subject standard deviation and repeatability for all parameters match previously published data. The SD of the axial length (AL) increased with the Mean AL, but there was no noticeable dependency of the SD of any of the other parameters on their corresponding Mean value. The SDs of the parameters are not independent of one another, and in particular we observe correlations between those for AL, anterior chamber depth, aqueous depth, lens thickness and corneal thickness. Conclusions: The SD change over Mean for AL measurement and the correlations between the uncertainties of several biometric parameters mean that a simple Gaussian error propagation model cannot be used to derive the effect of biometric uncertainties on the predicted intraocular lens power and refraction after cataract surgery.

To investigate the performance of a simple prediction scheme for the formula constants optimized for a mean (MPE), standard deviation (SDPE) or root-mean-squared refractive prediction error (RMSPE). Retrospective cross-sectional study. Using IOLMaster 700 biometric data from 888 eyes treated with the Hoya Vivinex lens and 821 eyes treated with the Alcon SA60AT lens, plus the power of the implanted lens and postoperative spherical equivalent refraction, optimized constants for SRKT, Hoffer Q, Holladay 1, Haigis, and K6 formulae were calculated using an iterative nonlinear optimization for zero MPE and minimal SDPE and RMSPE. Start values were detuned by ±1.5 from the MPE optimized constants and formula constants generated using the simple prediction scheme were compared to the corresponding directly optimized constants. For all 5 formulae under test and with both datasets, constants optimized using the simple scheme showed excellent agreement with those from the iterative method with either MPE or RMSPE used as the optimization metric and good agreement with SDPE as the metric. Constants optimized for zero MPE or minimal RMSPE agreed within 0.05, whereas constants for minimal SDPE could be systematically off by up to 0.6 from the MPE values, making SDPE unsuitable as an optimization metric. This simple formula constant optimization scheme performs excellently for 4 disclosed formulae and one nondisclosed formula in our 2 monocentric datasets with zero MPE or minimal RMSPE as metrics. Multicentric studies with other study populations and biometers are required to further investigate the clinical applicability.