Determination of Personalized IOL-Constants for the Haigis Formula under Consideration of Measurement Precision

A Novel Method to Optimize Personal IOL Constants

OBJECTIVE

To describe a novel method called "three variable optimization" that entails a process of doing just one calculation to zero out the mean prediction error of an entire dataset (regardless of size), using only 3 variables: (1) the constant used, (2) the average intraocular lens (IOL) power, and (3) the average prediction error (PE as actual refraction - predicted refraction).

DESIGN

Development, evaluation, and testing of a method to optimize personal IOL constants.

METHODS

A dataset of 876 eyes was used as a training set, and another dataset of 1,079 eyes was used to test the method. The Barrett Universal II, Cooke K6, Haigis, RBF 3.0, Hoffer Q, Holladay 1, Holladay 2, SRK/T, and T2 were analyzed. The same dataset was also divided into 3 subgroups (short, medium, and long eyes). The three variable optimization process was applied to each dataset and subset, and the obtained optimized constants were then used to obtain the mean PE of each dataset. We then compared those results with those obtained by zeroing out the mean PE in the classical method.

RESULTS

The three variable optimization showed similar results to classical optimization with less data needed to optimize and no clinically significant difference. Dividing the dataset into subsets of short, medium and long eyes, also shows that the method is useful even in those situations. Finally, the method was tested in multiple formulas and it was able to reduce the PE with no clinically significant difference from classical optimization.

CONCLUSION

This method could then be applied by surgeons to optimize their constants by reducing the mean prediction error to zero without prior technical knowledge and it is available online for free at http://wwww.ioloptimization.com.

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.

A New Method to Minimize the Standard Deviation and Root Mean Square of the Prediction Error of Single-Optimized IOL Power Formulas

Purpose

The purpose of this study was to develop a simplified method to approximate constants minimizing the standard deviation (SD) and the root mean square (RMS) of the prediction error in single-optimized intraocular lens (IOL) power calculation formulas.

Methods

The study introduces analytical formulas to determine the optimal constant value for minimizing SD and RMS in single-optimized IOL power calculation formulas. These formulas were tested against various datasets containing biometric measurements from cataractous populations and included 10,330 eyes and 4 different IOL models. The study evaluated the effectiveness of the proposed method by comparing the outcomes with those obtained using traditional reference methods.

Results

In optimizing IOL constants, minor differences between reference and estimated A-constants were found, with the maximum deviation at -0.086 (SD, SRK/T, and Vivinex) and -0.003 (RMS, PEARL DGS, and Vivinex). The largest discrepancy for third-generation formulas was -0.027 mm (SD, Haigis, and Vivinex) and 0.002 mm (RMS, Hoffer Q, and PCB00/SN60WF). Maximum RMS differences were -0.021 and +0.021, both involving Hoffer Q. Post-minimization, the largest mean prediction error was 0.726 diopters (D; SD) and 0.043 D (RMS), with the highest SD and RMS after adjustments at 0.529 D and 0.875 D, respectively, indicating effective minimization strategies.

Conclusions

The study simplifies the process of minimizing SD and RMS in single-optimized IOL power predictions, offering a valuable tool for clinicians. However, it also underscores the complexity of achieving balanced optimization and suggests the need for further research in this area.

Translational Relevance

The study presents a novel, clinically practical approach for optimizing IOL power calculations.

Limitations of constant optimization with disclosed intraocular lens power 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.

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.

Ptosis effects on intraocular lens power calculation

PURPOSE

To evaluate quantitatively ocular biometric parameters and intraocular lens (IOL) power measurements after ptosis surgery.

SETTING

Adiyaman University Hospital, Adiyaman, Turkey.

DESIGN

Comparative prospective clinical study.

METHODS

This study comprised involutional ptosis patients divided into droopy eyelid severity groups: Group 1: >4 mm, Group 2: 3 to 4 mm, and Group 3: 1 to 2 mm. The patients underwent anterior levator resection, and preoperative and postoperative biometry measurements at 3 months postoperatively were obtained.

RESULTS

The Group 1 sample size was 19, Group 2 was 22, and Group 3 was 16. The mean flattest keratometry (K 1 ), steepest keratometry (K 2 ), and mean keratometry (K m ) values significantly decreased at 3 months postoperatively in Group 1 ( P < .001 for all). The mean K 1 , K 2 , and K m values nonsignificantly decreased at 3 months postoperatively in Groups 2 and 3 ( P > .05 for all). The mean corneal astigmatism magnitude decreased at 3 months postoperatively in Group 1 ( P < .01), Group 2 ( P = .186), and Group 3 ( P = .952). The mean recommended IOL powers targeting emmetropia increased postoperatively in Group 1 and were similar preoperatively and postoperatively in Groups 2 and 3. In Group 1, the mean changes after ptosis surgery by the formula were 0.47 diopters (D) for SRK/T, 0.52 D for Hoffer Q, 0.55 D for Haigis, 0.50 D for Barrett Universal II, and 0.55 D for Holladay 2.

CONCLUSIONS

Ptosis >4 mm significantly affects corneal curvature values and IOL power calculations when cataract surgery is planned. Surgeons might consider altering their lens power choice accordingly if cataract surgery is to be sequentially followed by ptosis repair.

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.

Theoretical Relationship Among Effective Lens Position, Predicted Refraction, and Corneal and Intraocular Lens Power in a Pseudophakic Eye Model

Purpose

To ascertain the theoretical impact of anatomical variations in the effective lens position (ELP) of the intraocular lens (IOL) in a thick lens eye model. The impact of optimization of IOL power formulas based on a single lens constant was also simulated.

Methods

A schematic eye model was designed and manipulated to reflect changes in the ELP while keeping the optical design of the IOL unchanged. Corresponding relationships among variations in ELP, postoperative spherical equivalent refraction, and required IOL power adjustment to attain target refractions were computed for differing corneal powers (38 diopters [D], 43 D, and 48 D) with IOL power ranging from 1 to 35 D.

Results

The change in ELP required to compensate for various systematic biases increased dramatically with low-power IOLs (less than 10 D) and was proportional to the magnitude of the change in refraction. The theoretical impact of the variation in ELP on postoperative refraction was nonlinear and highly dependent on the optical power of the IOL. The concomitant variations in IOL power and refraction at the spectacle plane, induced by varying the ELP, were linearly related. The influence of the corneal power was minimal.

Conclusions

The consequences of variations in the lens constant mainly concern eyes receiving high-power IOLs. The compensation of a systematic bias by a constant increment of the ELP may induce a nonsystematic modification of the predicted IOL power, according to the biometric characteristics of the eyes studied.

Translational Relevance

Optimizing IOL power formulas by altering the ELP may induce nonsystematic modification of the predicted IOL power.

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.

Influence of the invariant refraction assumption in studies of formulas for monofocal and multifocal intraocular lens power calculation

PURPOSE

To assess the influence in paired design studies of formulae comparison for intraocular lens (IOL) power calculation of using a single formula for deciding the implanted power with monofocal (mIOL) and multifocal (MIOL) lenses.

DESIGN

Retrospective observational.

METHODS

Ninety-six right eyes were retrospectively analyzed. Eyes were assigned in two independent groups, SG and HG, depending on the formula used for deciding the implanted power, SRK-T (n = 54) and Haigis (n = 42), respectively. Median absolute prediction error (MedAE) was evaluated between independent samples (SRK-T in SG vs Haigis in HG) and between paired samples (SRK-T vs Haigis in both SG and HG). Percentages of eyes within a specific range of prediction error (PE) were also calculated for both, the standard steps and the clinically relevant steps.

RESULTS

MedAE difference was lower than 0.09 D between both formulas for the comparison of independent samples in the mIOL (p = 0.62) and MIOL (p = 0.83) groups. However, paired samples resulted in better MedAE for SRK-T in the SG (0.14 D lower, p = 0.003) and for Haigis in the HG (0.07 D lower, p = 0.015), but only in the mIOL group. These small differences were also manifested, but not reaching statistical significance (p > 0.05), in the percentage of eyes achieving a specific range of PE, especially in the mIOL group.

CONCLUSIONS

A small superiority for the formula used for selecting the final implanted IOL power can appear in studies following current standards. These studies should clearly specify which formula was used for selecting the implanted power.

Refractive Outcomes after Cataract Surgery

A post-operative manifest refractive error as close as possible to target is key when performing cataract surgery with intraocular lens (IOL) implantation, given that residual astigmatism and refractive errors negatively impact patients’ vision and satisfaction. This review explores refractive outcomes prior to modern biometry; advances in biometry and its impact on patients’ vision and refractive outcomes after cataract surgery; key factors that affect prediction accuracy; and residual refractive errors and the impact on visual outcomes. There are numerous pre-, intra-, and post-operative factors that can influence refractive outcomes after cataract surgery, leaving surgeons with a small “error budget” (i.e., the source and sum of all influencing factors). To mitigate these factors, precise measurement and correct application of ocular biometric data are required. With advances in optical biometry, prediction of patient post-operative refractory status has become more accurate, leading to an increased proportion of patients achieving their target refraction. Alongside improvements in biometry, advancements in microsurgical techniques, new IOL technologies, and enhancements to IOL power calculations have also positively impacted patients’ refractory status after cataract surgery.

Optimal Dataset Sizes for Constant Optimization in Published Theoretical Optical Formulae

Purpose: To determine the optimal number of data points required for optimization of formulae for classical lens power calculation.Methods: A large dataset of preoperative biometric values was used to assess the convergence of formula constants in a number of established intraocular lens power calculation formulae.Results: In formulae with a single constant, 80-100 clinical data points are sufficient to obtain convergence. The Haigis formula (three constants) requires 200-300 data points although refractive error converges more rapidly.Conclusions: In all formulae, 80-100 clinical data points are sufficient to achieve a stable mean refractive error.

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.

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.

Relationship between effective lens position and axial position of a thick intraocular lens

PURPOSE

To discuss the impact of intraocular lens-(IOL)-power, IOL-thickness, IOL-shape, corneal power and effective lens position (ELP) on the distance between the anterior IOL vertex (ALP) of a thick IOL and the ELP of its thin lens equivalent.

METHODS

We calculated the ALP of a thick IOL in a model eye, which results in the same focal plane as a thin IOL placed at the ELP using paraxial approximation. The model eye included IOL-power (P), ELP, IOL-thickness (Th), IOL-shape-factor (X), and corneal power (DC). The initial values were P = 10 D (diopter: 1 D = 1 m-1), 20 D, 30 D, Th = 0.9 mm, ELP = 5 mm, X = 0, DC = 43 D. The difference between ALP and the ELP was illustrated as a function of each of the model parameters.

RESULTS

The ALP of a thick lens has to be placed in front of the ELP for P>0 IOLs to achieve the same optical effect as the thin lens equivalent. The difference ALP-ELP for the initial values is -0.57 mm. Minus power IOLs (ALP-ELP = -0.07 mm, for IOL-power = -5 D) and convex-concave IOLs (ALP-ELP = -0.16 mm, for X = 1) have to be placed further posterior. The corneal power and ELP have less influence, but corneal power cannot be neglected.

CONCLUSION

The distance between ELP and ALP primarily depends on IOL-power, IOL-thickness, and shape-factor.