Effect of lens constants optimization on the accuracy of intraocular lens power calculation formulas for highly myopic eyes

Performance of a simplified strategy for formula constant optimisation in intraocular lens power calculation

PURPOSE

To investigate the performance of a simple prediction scheme for the formula constants optimised for a mean refractive prediction error.

METHODS

Analysis based on a dataset of 888 eyes before and after cataract surgery with IOL implantation (Hoya Vivinex). IOLMaster 700 biometric data, power of the implanted lens and postoperative spherical equivalent refraction were used to calculate the optimised constants (.)opt for SRKT, HofferQ, Holladay and Haigis formula with an iterative nonlinear optimisation. For detuning start values by ±1.5 from (.)opt, the predicted formula constants (.)pred were calculated and compared with (.)opt. Formula performance metrics mean (MPE), median (MEDPE), mean absolute (MAPE), median absolute (MEDAPE), root mean squared (RMSPE) and standard deviation (SDPE) of the formula prediction error were analysed for (.)opt and (.)pred.

RESULTS

(.)pred - (.)opt showed a 2nd order parabolic behaviour with maximal deviations up to 0.09 at the tails of detuning and a minimal deviation up to -0.01 for all formulae. The performance curves of different metrics of PE as functions of detuning variations show that the formula constants for zeroing MPE and MEDPE yield almost identical formula constants, optimisation for MAPE, MEDAPE and RMSPE yielded formula constants very close to (.)opt, and optimisation for SDPE could result in formula constants up to 0.5 off (.)opt which is unacceptable for clinical use.

CONCLUSION

This simple prediction scheme for formula constant optimisation for zero mean refraction error performs excellently in our monocentric dataset, even for larger deviations of the start value from (.)opt. Further studies with multicentric data and larger sample sizes are required to investigate the performance in a clinical setting further.

Surrogate optimisation strategies for intraocular lens formula constant optimisation

PURPOSE

To investigate surrogate optimisation (SO) as a modern, purely data-driven, nonlinear adaptive iterative strategy for lens formula constant optimisation in intraocular lens power calculation.

METHODS

A SO algorithm was implemented for optimising the root mean squared formula prediction error (rmsPE, defined as predicted refraction minus achieved refraction) for the SRKT, Hoffer Q, Holladay, Haigis and Castrop formulae in a dataset of N = 888 cataractous eyes with implantation of the Hoya Vivinex hydrophobic acrylic aspheric lens. A Gaussian Process estimator was used as the model, and the SO was initialised with equidistant datapoints within box constraints, and the number of iterations restricted to either 200 (SRKT, Hoffer Q, Holladay) or 700 (Haigis, Castrop). The performance of the algorithm was compared to the classical gradient-based Levenberg-Marquardt algorithm.

RESULTS

The SO algorithm showed stable convergence after fewer than 50/150 iterations (SRKT, HofferQ, Holladay, Haigis, Castrop). The rmsPE was reduced systematically to 0.4407/0.4288/0.4265/0.3711/0.3449 dioptres. The final constants were A = 119.2709, pACD = 5.7359, SF = 1.9688, -a0 = 0.5914/a1 = 0.3570/a2 = 0.1970, C = 0.3171/H = 0.2053/R = 0.0947 for the SRKT, Hoffer Q, Holladay, Haigis and Castrop formula and matched the respective constants optimised in previous studies.

CONCLUSION

The SO proves to be a powerful adaptive nonlinear iteration algorithm for formula constant optimisation, even in formulae with one or more constants. It acts independently of a gradient and is in general able to search within a (box) constrained parameter space for the best solution, even where there are multiple local minima of the target function.

Insight into small eyes: a practical description from phenotypes presentations to the management

This narrative review aimed to have an algorithmic approach to microphthalmos by a systematic search. The definition can be related to a number of special phenotypes. In the more challenging cases of complex microphthalmos, relative anterior microphthalmos, and nanophthalmos, the surgeon can approach these cases more safely if they have a deep understanding of the anatomical variations and ideal formulae for intraocular lens computation and knows how to avoid intra- and post-operative complications. In this article, we review the criteria by which we recognize and describe pre-, intra-, and post-operative considerations, as well as discuss the ideal intraocular lenses for microphthalmos, given the intricate varieties of small eye phenotypes.

Intraocular Lens Power Calculation Formulas-A Systematic Review

PURPOSE

The proper choice of an intraocular lens (IOL) power calculation formula is an important aspect of phacoemulsification. In this study, the formulas most commonly used today are described and their accuracy is evaluated.

METHODS

This review includes papers evaluating the accuracy of IOL power calculation formulas published during the period from January 2015 to December 2022. The articles were identified by a literature search of medical and other databases (PubMed/MEDLINE, Crossref, Web of Science, SciELO, Google Scholar, and Cochrane Library) using the terms "IOL formulas," "Barrett Universal II," "Kane," "Hill-RBF," "Olsen," "PEARL-DGS," "EVO," "Haigis," "SRK/T," and "Hoffer Q." Twenty-nine of the most recent peer-reviewed papers in English with the largest samples and largest number of formulas compared were considered.

RESULTS

Outcomes of mean absolute error and percentage of predictions within ±0.5 D and ±1.0 D were used to evaluate the accuracy of the formulas. In most studies, Barrett achieved the smallest mean absolute error and PEARL-DGS the highest percentage of patients with ±0.5 D in short eyes, while Kane obtained the highest percentage of patients with ±0.5 D in long eyes.

CONCLUSIONS

The third- and fourth-generation formulas are gradually being replaced by more accurate ones. The Barrett Universal II among vergence formulas and Kane and PEARL-DGS among artificial intelligence-based formulas are currently most often reported as the most precise.

Particle swarm optimisation strategies for IOL formula constant optimisation

PURPOSE

To investigate particle swarm optimisation (PSO) as a modern purely data driven non-linear iterative strategy for lens formula constant optimisation in intraocular lens power calculation.

METHODS

A PSO algorithm was implemented for optimising the root mean squared formula prediction error (rmsPE, defined as achieved refraction minus predicted refraction) for the Castrop formula in a dataset of N = 888 cataractous eyes with implantation of the Hoya Vivinex hydrophobic acrylic aspheric lens. The hyperparameters were set to inertia: 0.8, accelerations c1 = c2 = 0.1. The algorithm was initialised with NP  = 100 particles having random positions and velocities within the box constraints of the constant triplet parameter space C = 0.25 to 0.45, H = -0.25 to 0.25 and R = -0.25 to 0.25. The performance of the algorithm was compared to classical gradient-based Trust-Region-Reflective and Interior-Point algorithms.

RESULTS

The PSO algorithm showed fast and stable convergence after 37 iterations. The rmsPE reduced systematically to 0.3440 diopters (D). With further iterations the scatter of the particle positions in the swarm decreased but without further reduction of rmsPE. The final constant triplet was C/H/R = 0.2982/0.2497/0.1435. The Trust-Region-Reflective/Interior-Point algorithms showed convergence after 27/17 iterations, respectively, resulting in formula constant triplets C/H/R = 0.2982/0.2496/0.1436 and 0.2982/0.2495/0.1436, both with the same rmsPE as the PSO algorithm (rmsPE = 0.3440 D).

CONCLUSION

The PSO appears to be a powerful adaptive nonlinear iteration algorithm for formula constant optimisation even in formulae with more than 1 constant. It acts independently of an analytical or numerical gradient and is in general able to search for the best solution even with multiple local minima of the target function.

Evaluating intraocular lens power formula constant robustness using bootstrap algorithms

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.

Accuracy of Six Intraocular Lens Power Calculations in Eyes with Axial Lengths Greater than 28.0 mm

The purpose of this study was to compare the accuracy of several intraocular (IOL) lens power calculation formulas in long eyes. This was a single-site retrospective consecutive case series that reviewed patients with axial lengths (AL) > 28.0 mm who underwent phacoemulsification. The Wang−Koch (WK) adjustment and Cooke-modified axial length (CMAL) adjustment were applied to Holladay 1 and SRK/T. The median absolute error (MedAE) and the percentage of eyes with prediction errors ±0.25 diopters (D), ±0.50 D, ±0.75 D, and ±1.00 D were used to analyze the formula’s accuracy. This study comprised a total of 35 eyes from 25 patients. The Kane formula had the lowest MedAE of all the formulas, but all were comparable except Holladay 1, which had a significantly lower prediction accuracy with either AL adjustment. The SRK/T formula with the CMAL adjustment had the highest accuracy in predicting the formula outcome within ±0.50 D. The newer formulas (BU-II, EVO, Hill-RBF version 3.0, and Kane) were all equally predictable in long eyes. The SRK/T formula with the CMAL adjustment was comparable to these newer formulas with better outcomes than the WK adjustment. The Holladay 1 with either AL adjustment had the lowest predictive accuracy.

Assessment of the influence of keratometry on intraocular lens calculation formulas in long axial length eyes

PURPOSE

Hyperopic surprises tend to occur in axial myopic eyes and other factors including corneal curvature have rarely been analyzed in cataract surgery, especially in eyes with long axial length (≥ 26.0 mm). Thus, the purpose of our study was to evaluate the influence of keratometry on four different formulas (SRK/T, Barrett Universal II, Haigis and Olsen) in intraocular lens (IOL) power calculation for long eyes.

METHODS

Retrospective case series. A total of 180 eyes with axial length (AL) ≥ 26.0 mm were divided into 3 keratometry (K) groups: K ≤ 42.0 D (Flat), K ≥ 46.0 D (Steep), 42.0 < K < 46.0 D (Average), and all the eyes were underwent phacoemulsification cataract surgery with Rayner (Hove, UK) 920H IOL implantation. Prediction errors (PE) were compared between different formulas to assess the accuracy of different formulas. Multiple regression analysis was performed to investigate factors associated with the PE.

RESULTS

The mean absolute error was higher for all evaluated formulas in Steep group (ranging from 0.66 D to 1.02 D) than the Flat (0.34 D to 0.67 D) and Average groups (0.40 D to 0.74D). The median absolute errors predicted by Olsen formula were significantly lower than that predicted by Haigis formula (0.42 D versus 0.85 D in Steep and 0.29 D versus 0.69 D in Average) in Steep and Average groups (P = 0.012, P < 0.001, respectively). And the Olsen formula demonstrated equal accuracy to the Barrett II formula in Flat and Average groups. The predictability of the SRK/T formula was affected by the AL and K, while the predictability of Olsen and Haigis formulas was affected by the AL only.

CONCLUSIONS

Steep cornea has more influence on the accuracy of IOL power calculation than the other corneal shape in long eyes. Overall, both the Olsen and Barrett Universal II formulas are recommended in long eyes with unusual keratometry.

Comparison of accuracy of intraocular lens power calculation for eyes with an axial length greater than 29.0 mm

PURPOSE

To evaluate and compare the accuracy of six different formulas (Emmetropia Verifying Optical version 2.0, Kane, SRK/T, Barrett Universal II, Haigis and Olsen) in intraocular lens (IOL) power calculation for extremely long eyes.

METHODS

Retrospective case-series. Seventy-three eyes with axial length (AL) ≥ 29.0 mm and underwent phacoemulsification cataract surgery with Rayner (Hove, UK) 920H IOL implantation from January 2018 to March 2020 were included. Prediction errors (PE) were calculated and compared between different formulas to evaluate the accuracy of formulas. Multiple regression analysis was performed to investigate factors associated with the PE.

RESULTS

The Kane formula had mean prediction error close to zero (- 0.01 ± 0.51 D, P = 0.841), whereas the EVO 2.0, SRK/T, Barrett Universal II, Haigis and Olsen formulas produced hyperopic outcomes (all P < 0.001). The median absolute error [inter-quartile range] produced by the EVO 2.0, Kane, Barrett Universal II and Olsen formulas showed no significant difference (0.33 D [0.48], 0.30 D [0.44], 0.34 D [0.39], 0.29 D [0.37], respectively, pairwise comparison P > 0.05), but was significantly lower than that of the SRK/T and Haigis formulas (0.85 D [0.66], 0.80 D [0.54], respectively, pairwise comparison P < 0.001). The AL and the PE produced by the SRK/T formula were significantly positively correlated in extremely myopic eyes (β = 0.248, P < 0.001), whereas the trend was not demonstrated in other formulas.

CONCLUSIONS

For cataract patients with axial length greater than 29.0 mm, the accuracy of the EVO 2.0, Kane, Barrett Universal II and Olsen formulas is comparable and significantly better than that of the SRK/T and Haigis formulas.

Strategies for formula constant optimisation for intraocular lens power calculation

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.

Optimizing lens constants specifically for short eyes: Is it essential?

Purpose:

Optimization of lens constants is a critically important step that improves refractive outcomes significantly. Whether lens constants optimized for the entire range of axial length would perform equally well in short eyes is still a matter of debate. The aim of this study was to analyze whether lens constants need to be optimized specifically for short eyes.

Methods:

This retrospective observational study was conducted at a tertiary care hospital in Central India. Eighty-six eyes of eighty-six patients were included. Optical biometry with IOLMaster 500 was done in all cases and lens constants were optimized using built-in software. Barrett Universal II, Haigis, Hill-RBF, Hoffer Q, Holladay 1, and SRK/T formulae were compared using optimized constants. Mean absolute error, median absolute error (MedAE), and percentage of eyes within ±0.25, ±0.50, ±1.00, and ±2.00 diopter of the predicted refraction, of each formula were analyzed using manufacturer’s, ULIB, and optimized lens constants. MedAE was compared across various constants used by Wilcoxon signed-rank test and among optimized constants by Friedman’s test. Cochran’s Q test compared the percentage of eyes within ± 0.25, ±0.50, ±1.00, and ± 2.00 diopter of the predicted refraction. A value of P < 0.05 was considered statistically significant.

Results:

Optimized constant of Haigis had significantly lower MedAE (P < 0.00001) as compared to manufacturers. However, there was no statistically significant difference between ULIB and optimized constants. Postoptimization, there was no statistically significant difference among all formulae.

Conclusion:

Optimizing lens constants specifically for short eyes gives no added advantage over those optimized for the entire range of axial length.

Considerations on the Castrop formula for calculation of intraocular lens power

Background

To explain the concept of the Castrop lens power calculation formula and show the application and results from a large dataset compared to classical formulae.

Methods

The Castrop vergence formula is based on a pseudophakic model eye with 4 refractive surfaces. This was compared against the SRKT, Hoffer-Q, Holladay1, simplified Haigis with 1 optimized constant and Haigis formula with 3 optimized constants. A large dataset of preoperative biometric values, lens power data and postoperative refraction data was split into training and test sets. The training data were used for formula constant optimization, and the test data for cross-validation. Constant optimization was performed for all formulae using nonlinear optimization, minimising root mean squared prediction error.

Results

The constants for all formulae were derived with the Levenberg-Marquardt algorithm. Applying these constants to the test data, the Castrop formula showed a slightly better performance compared to the classical formulae in terms of prediction error and absolute prediction error. Using the Castrop formula, the standard deviation of the prediction error was lowest at 0.45 dpt, and 95% of all eyes in the test data were within the limit of 0.9 dpt of prediction error.

Conclusion

The calculation concept of the Castrop formula and one potential option for optimization of the 3 Castrop formula constants (C, H, and R) are presented. In a large dataset of 1452 data points the performance of the Castrop formula was slightly superior to the respective results of the classical formulae such as SRKT, Hoffer-Q, Holladay1 or Haigis.

Optimizing the intraocular lens formula constant according to intraocular lens diameter

AIM

To determine whether the different diameters of a specific intraocular lens (IOL) have significantly different optimized SRK/T A constants and whether these new A constants can improve refractive outcomes.

METHODS

Data were collected prospectively from Jan. 2011 to Dec. 2012 on all patients undergoing routine cataract surgery at a district general hospital in the UK. Patients were divided into three groups according to the size of the Akreos AO MI60 IOL used. A constants for the SRK/T formula were optimized according to the size of the IOL. These optimized A constants were then used to select future refractive outcomes.

RESULTS

A total of 2398 cataract operations were performed during the study period of which 1131 met the inclusion criteria. The three optimized A constants for the different sized IOLs were 118.98, 119.13, 119.32. The difference between them was highly significant (P≤0.0001). Two optimized A constants for three sizes of IOL led to an improvement in refractive outcomes (from 93.4% to 94.6% of refractive outcomes within 1.00 D of predicted spherical equivalent). The optimized A constant for the largest IOL was based on a small number of cases and was not used.

CONCLUSION

Optimizing the A constant for the three distinct sizes of the Bausch & Lomb Akreos MI60 lens lead to three significantly different A constants. In our practice, using two different optimized A constants for three different sized IOLs give the least refractive error, however, using three optimized A constants may give better results with a larger dataset.

IOL formula constants - strategies for optimization and defining standards for presenting data

PURPOSE

To present strategies for optimization of lens power formula constants and to show options how to present the results adequately.

METHODS

A dataset of N=1601 preoperative biometric values, lens power 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 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 more than 1 constant. In these cases, more advanced concepts such as nonlinear optimization strategies are necessary to derive the formula constants. During cross-validation, measures such as the mean absolute or the root mean squared prediction error or the ratio of cases within mean absolute prediction error 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 prediction error limits is plotted against the mean absolute prediction error.

Correlation between changes in intraocular pressure and refractive error indices post Cataract Surgery

Objective:

To evaluate the correlation between refractive errors and change in intraocular pressure in patients undergoing cataract surgery.

Methods:

This interventional retrospective case study was carried out from September 2018 to April 2019 at Bodla Eye Care and Multan Medical and Dental College, Multan. A total of 127 eyes were recruited in the study among which six were excluded. Out of remaining 121, 53 eyes were emmetropes, 41 were mild myopes and 27 were high myopes. Single surgeon performed the procedure. Pre-operative investigations of IOP and refractive error were done by goldmann tonometry and auto refractometry. IOP was reviewed at day 1, 7, 14 and 28 post cataract surgeries.

Results:

Out of 121 eyes, 53 eyes were emmetropes, 41 were mild myopes and 27 were high myopes, who underwent phacoemulsification. There was an elevation of 2-3mm Hg at Day-1, in emmetropes and mild myopes, and further on, a constant drop was noticed on follow ups. In high myopes a significant fluctuation of IOP was noted in first fourteen days followed by an unremarkable gradual decline afterwards.

Conclusion:

Cataract surgery helps lowering the IOP in patients with refractive errors. Mild myopic and emmetropic patients showed a linear swift pattern while high myopes presented instable and gradual reduction in IOP. A total decrease of 1-2mm Hg was seen at the end of the study depicting that relation between IOP and cataract surgery is insignificant.