Considerations on the Castrop formula for calculation of intraocular lens power

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 Comprehensive Evaluation of 16 Old and New Intraocular Lens Power Calculation Formulas in Pediatric Eyes

Purpose

To compare the accuracy of 16 intraocular lens (IOL) power calculation formulas in pediatric cataract eyes.

Patients and Methods

The data records of pediatric patients who had been implanted with three IOL models (SA60AT, MA60AC, and enVista-MX60) between 2012 and 2018 were analyzed. The accuracy of 16 IOL power calculation methods was evaluated: Barrett Universal II (BUII), Castrop, EVO 2.0, Haigis, Hill-RBF 3.0, Hoffer Q, Hoffer QST, Holladay 1, Kane, LSF AI, Naeser 2, Pearl-DGS, SRK/T, T2, VRF, and VRF-G. The non-optimized (ULIB/IOLcon) and optimized constants were used for IOL power calculation. The mean prediction error (PE), Performance Index (FPI), and all descriptive statistics were calculated.

Results

Ninety-seven eyes of 97 pediatric patients aged 13.2 (IQR 11.2-17.1) were included. No statistically significant difference (HS-test) was observed (p > 0.818) except for the Hoffer Q, and Naeser 2 (P = 0.014). With optimized lens constants, the best FPI indices were obtained by Hoffer Q (0.256) and VRF-G (0.251) formulas, followed by Hill-RBF 3.0 and BUII, with an index of 0.248. The highest FPI indices with non-optimized constants showed SRK/T and T2 formulas (0.246 and 0.245, respectively), followed by VRF-G and Holladay 1, with an index of 0.244. The best median absolute error values (MedAE) were achieved by Hoffer Q (0.50 D), VRF-G (0.53 D), and Hill-RBF 3.0 (0.54 D), all P ≥ 0.074.

Conclusion

Our results place the Hoffer Q, VRF-G, Hill-RBF 3.0, and BUII formulas as more accurate predictors of postoperative refraction in pediatric cataract surgery.

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.

Prediction of spectacle refraction uncertainties with discrete IOL power steps and manufacturing tolerances according to ISO using a Monte Carlo model

PURPOSE

The purpose of this study was to develop a concept for predicting the effects of both discrete intraocular lens (IOL) power steps (PS) and power labelling tolerances (LT) on the uncertainty of the refractive outcome (REFU).

DESIGN

Retrospective non-randomised cross-sectional Monte Carlo simulation study.

METHODS

We evaluated a dataset containing 16 669 IOLMaster 700 preoperative biometric measurements. The PS and the delivery range of two modern IOLs (Bausch and Lomb enVista and Alcon SA60AT) were considered for this Monte Carlo simulation. The uncertainties from PS or LT were assumed to be normally distributed according to ±½ the IOL PS or the ISO 11979 LT. REFU was recorded and analysed for all simulations.

RESULTS

With both lenses the REFU from discrete PS ranged from 0.11 to 0.12 dpt. Due to the larger PS for low/high power lenses with the enVista/SA60AT, REFU is more dominant in initially myopic/hyperopic eyes. REFU from LT ranged from 0.18 to 0.19 dpt for both lenses. Since LT increases stepwise with IOL power, REFU is more prevalent in initially hyperopic eyes requiring high IOL power values, and for lenses with a wide delivery range towards higher powers.

CONCLUSIONS

Since surgeons and patients are typically aware of the effect of discrete PS on REFU, these might be tolerated in cataract surgery. However, REFU resulting from LT is inevitable while the true measured IOL power is not reported on the package, leading to background noise in postoperative achieved refraction.

Impact of uncertainties in biometric parameters on intraocular lens power formula predicted refraction using a Monte-Carlo simulation

PURPOSE

The purpose of this study was to investigate the uncertainty in the formula predicted refractive outcome REFU after cataract surgery resulting from measurement uncertainties in modern optical biometers using literature data for within-subject standard deviation Sw.

METHODS

This Monte-Carlo simulation study used a large dataset containing 16 667 preoperative IOLMaster 700 biometric measurements. Based on literature Sw values, REFU was derived for both the Haigis and Castrop formulae using error propagation strategies. Using the Hoya Vivinex lens (IOL) as an example, REFU was calculated both with (WLT) and without (WoLT) consideration of IOL power labelling tolerances.

RESULTS

WoLT the median REFU was 0.10/0.12 dpt for the Haigis/Castrop formula, and WLT it was 0.13/0.15 dpt. WoLT REFU increased systematically for short eyes (or high power IOLs), and WLT this effect was even more pronounced because of increased labelling tolerances. WoLT the uncertainty in the measurement of the corneal front surface radius showed the largest contribution to REFU, especially in long eyes (and low power IOLs). WLT the IOL power uncertainty dominated in short eyes (or high power IOLs) and the uncertainty of the corneal front surface in long eyes (or low power IOLs).

CONCLUSIONS

Compared with published data on the formula prediction error of refractive outcome after cataract surgery, the uncertainty of biometric measures seems to contribute with ⅓ to ½ to the entire standard deviation. REFU systematically increases with IOL power and decreases with axial length.

Comparison of Pre- and Post-DMEK Keratometry and Total Keratometry Values for IOL Power Calculations in Eyes Undergoing Triple DMEK

PURPOSE

To evaluate prediction accuracy of pre- and post-DMEK keratometry (K) and total keratometry (TK) values for IOL power calculations in Fuchs endothelial corneal dystrophy (FECD) eyes undergoing DMEK with cataract surgery (triple DMEK).

METHODS

Retrospective cross-sectional multicenter study of 55 FECD eyes (44 patients) that underwent triple DMEK between 2019 and 2022 between two centers in USA and Europe. Swept-source optical coherence tomography biometry (IOLMaster 700) was used for pre- and post-DMEK measurements. K and TK values were used for power calculations with ten formulae (Barrett Universal II (BUII), Castrop, Cooke K6, EVO 2.0, Haigis, Hoffer Q, Hoffer QST, Holladay I, Kane, and SRK/T). Mean error, mean absolute error (MAE), standard deviation, and percentage of eyes within ±0.50/±1.00 diopters (D) were calculated. Studied formulae were additionally adjusted using a method published previously (IOLup1D Method), which increases the IOL power by 1D. While both eyes from the same patient were considered for descriptive statistics, we restricted to one eye per individual (44 eyes for statistical comparisons.

RESULTS

MAEs for all formulae were lower for post-DMEK K and TK than pre-DMEK K and TK by an average of 0.24 and 0.47 D, respectively. The lowest MAE was 0.49 D for Kane using post-DMEK TK, and the highest MAE was 1.05 D for BUII using pre-DMEK TK. Most IOLup1D formulae had lower MAEs than pre-DMEK K and TK formulae.

CONCLUSIONS

The IOLup1D Method should be used instead of pre-DMEK K and TK values for triple DMEK in FECD eyes. Using post-DMEK TK values for cataract surgery after DMEK provides better refractive accuracy than any of the three studied methods used for triple DMEK procedures.

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.

The effect of corneal power on the accuracy of 14 IOL power formulas

BACKGROUND

This study evaluates the impact of corneal power on the accuracy of 14 newer intraocular lens (IOL) calculation formulas in cataract surgery. The aim is to assess how these formulas perform across different corneal curvature ranges, thereby guiding more precise IOL selection.

METHODS

In this retrospective case series, 336 eyes from 336 patients who underwent cataract surgery were studied. The cohort was divided into three groups according to preoperative corneal power. Key metrics analyzed included mean prediction error (PE), standard deviation of PE (SD), mean absolute prediction error (MAE), median absolute error (MedAE), and the percentage of eyes with PE within ± 0.25 D, 0.50 D, ± 0.75 D, ± 1.00 D and ± 2.00 D.

RESULTS

In the flat K group (Km < 43 D), VRF-G, Emmetropia Verifying Optical Version 2.0 (EVO2.0), Kane, and Hoffer QST demonstrated lower SDs (± 0.373D, ± 0.379D, ± 0.380D, ± 0.418D, respectively) compared to the VRF formula (all P < 0.05). EVO2.0 and K6 showed significantly different SDs compared to Barrett Universal II (BUII) (all P < 0.02). In the medium K group (43 D ≤ Km < 46 D), VRF-G, BUII, Karmona, K6, EVO2.0, Kane, and Pearl-DGS recorded lower MAEs (0.307D to 0.320D) than Olsen (OLCR) and Castrop (all P < 0.03), with RBF3.0 having the second lowest MAE (0.309D), significantly lower than VRF and Olsen (OLCR) (all P < 0.05). In the steep K group (Km ≥ 46D), RBF3.0, K6, and Kane achieved significantly lower MAEs (0.279D, 0.290D, 0.291D, respectively) than Castrop (all P < 0.001).

CONCLUSIONS

The study highlights the varying accuracy of newer IOL formulas based on corneal power. VRF-G, EVO2.0, Kane, K6, and Hoffer QST are highly accurate for flat corneas, while VRF-G, RBF3.0, BUII, Karmona, K6, EVO2.0, Kane, and Pearl-DGS are recommended for medium K corneas. In steep corneas, RBF3.0, K6, and Kane show superior performance.

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.

Comparison of the formula accuracy for calculating multifocal intraocular lens power: a single center retrospective study in Korean patients

This study evaluated the accuracy of newer formulas (Barrett Universal II, EVO 2.0, Kane, Hoffer QST, and PEARL-DGS) and the Haigis formula in Korean patients with the Alcon TFNT multifocal intraocular lens. In total, 3100 randomly selected eyes of 3100 patients were retrospectively reviewed. After constant optimization, the standard deviation (SD) of the prediction error was assessed for the entire group, and the root mean square error was compared for short and long axial length (AL) subgroup analysis. The Cooke-modified AL (CMAL) was experimentally applied to the Haigis formula. All the newer formulas performed well, but they did not significantly outperform the Haigis formula. In addition, all the newer formulas exhibited significant myopic outcomes (- 0.23 to - 0.29 diopters) in long eyes. Application of the CMAL to the Haigis formula with single constant optimization produced similar behavior and higher correlation with the newer formulas. The CMAL-applied triple-optimized Haigis formula yielded a substantially smaller SD, even superior to the Barrett and Hoffer QST formulas. The AL modification algorithms such as the CMAL used in newer formulas to cope with optical biometry's overestimation of the AL in long eyes seemed to overcompensate, particularly in the long eyes of the East Asian population.

Monte-Carlo simulation of a thick lens IOL power calculation

BACKGROUND

The purpose of this Monte-Carlo study is to investigate the effect of using a thick lens model instead of a thin lens model for the intraocular lens (IOL) on the resulting refraction at the spectacle plane and on the ocular magnification based on a large clinical data set.

METHODS

A pseudophakic model eye with a thin spectacle correction, a thick cornea (curvatures for both surfaces and central thickness) and a thick IOL (equivalent power PL derived from a thin lens IOL, Coddington factor CL (uniformly distributed from -1.0 to 1.0), either preset central thickness LT = 0.9 mm (A) or optic edge thickness ET = 0.2 mm, (B)) was set up. Calculations were performed on a clinical data set containing 21 108 biometric measurements of a cataractous population based on linear Gaussian optics to derive spectacle refraction and ocular magnification using the thin and thick lens IOL models.

RESULTS

A prediction model (restricted to linear terms without interactions) was derived based on the relevant parameters identified with a stepwise linear regression approach to provide a simple method for estimating the change in spectacle refraction and ocular magnification where a thick lens IOL is used instead of a thin lens IOL. The change in spectacle refraction using a thick lens IOL with (A) or (B) instead of a thin lens IOL with identical power was within limits of around ±1.5 dpt when the thick lens IOL was placed with its haptic plane at the plane of the thin lens IOL. In contrast, the change in ocular magnification from considering the IOL as a thick lens instead of a thin lens was small and not clinically significant.

CONCLUSION

This Monte-Carlo simulation shows the impact of using a thick lens model IOL with preset LT or ET on the resulting spherical equivalent refraction and ocular magnification. If IOL manufacturers would provide all relevant data on IOL design data and refractive index for all power steps, this would make it possible to perform direct calculations of refraction and ocular magnification.

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.

Best fit formula approach in delayed sequential bilateral cataract surgery

BACKGROUND

We evaluated whether the best-fit intraocular lens (IOL) power formula for the first operated eye (BF1) was also the most accurate formula for the second eye.

METHODS

This was a retrospective study of 152 patients who underwent uncomplicated delayed bilateral cataract surgery with a minimum delay of 3 weeks using only one 1-piece IOL (HOYA, Vivinex) at the Medical University of Vienna, Austria. Seven different formulae (Barrett Universal II, Castrop, Haigis, Hoffer Q, Holladay 1, Kane, and SRK/T) were investigated to test the formula selection approach with regard to the calculated mean and median absolute prediction errors (MAE/MedAE).

RESULTS

The mean intraindividual difference in axial length was 0.2 mm (±0.3 mm). BF1 coincided with the best-fit formula for the second eye (BF2) in 56% of patients (p < 0.05). Using BF1 for the second eye led to a lower MedAE (0.22 dioptre, D) than using a formula at random (0.33 D) and was less accurate than using the best-fit formula for each eye separately (0.1 D). The MedAEs of all formulae were generally low, ranging from 0.28 to 0.35 D.

CONCLUSION

Using BF1 for the second eye led to a lower MedAE than the random selection of a formula. Therefore, BF1 can be used for the second eye if the surgeon is unsure of the choice of formula.

Influence and predictive value of optional parameters in new-generation intraocular lens formulas

PURPOSE

To evaluate the accuracy of various variations of new-generation multivariate intraocular lens (IOL) power calculation using the Barrett Universal II, Castrop, Emmetropia Verifying Optical 2.0, Hill-Radial Basis Function 3.0, Kane, and PEARL-DGS formulas with and without optional biometric parameters.

SETTING

Tertiary care academic medical center.

DESIGN

Retrospective case series. Single-center study.

METHODS

Inclusion of patients after uneventful cataract surgery implanting AU00T0 IOLs. Data from one eye per patient were randomly included. Eyes with a corrected distance visual acuity worse than 0.1 logMAR were excluded. IOLCON-optimized constants were used for all formulas other than the Castrop formula. The outcome measures were prediction error (PE) and absolute prediction error (absPE) for the 6 study formulas.

RESULTS

251 eyes from 251 patients were assessed. Excluding lens thickness led to statistically significant differences in absPE in several formulas. Leaving out horizontal corneal diameter did not impact absPE in several formulas. Differences in PE offset were observed between the various formula variations.

CONCLUSIONS

When using multivariate formulas with an A-constant, including certain optional parameters is vital for optimal refractive results. Formula variations excluding certain biometric parameters need specifically optimized constants and do not perform similarly when using the constant of the respective formula using all parameters.

Theoretical Impact of Intraocular Lens Design Variations on the Accuracy of IOL Power Calculations

To ascertain the theoretical impact of optical design variations of the intraocular lens (IOL) on the accuracy of IOL power formulas based on a single lens constant using a thick lens eye model. This impact was also simulated before and after optimization. We modeled 70 thick-lens pseudophakic eyes implanted with IOLs of symmetrical optical design and power comprised between 0.50 D and 35 D in 0.5-step increments. Modifications of the shape factor resulting in variations in the anterior and posterior radii of an IOL were made, keeping the central thickness and paraxial powers static. Geometry data from three IOL models were also used. Corresponding postoperative spherical equivalent (SE) were computed for different IOL powers and assimilated to a prediction error of the formula due to the sole change in optical design alone. Formula accuracy was studied before and after zeroization on a uniform and non-uniform realistic IOL power distribution. The impact of the incremental change in optic design variability depended on the IOL power. Design modifications theoretically induce an increase in the standard deviation (SD), Mean Absolute Error (MAE), and Root Mean Square (RMS) of the error. The values of these parameters reduce dramatically after zeroization. While the variations in optical design can affect refractive outcomes, especially in short eyes, the zeroization of the mean error theoretically reduces the impact of the IOL's design and power on the accuracy of IOL power calculation.

Bootstrap Outlier Identification in Clinical Datasets for Lens Power Formula Constant Optimization

PURPOSE

Bootstrapping is a modern technique widely used in statistics to evaluate the performance of model parameters. The purpose of this study was to develop a strategy to identify and eliminate outliers in a dataset used for optimizing formula constants for lens power calculation.

METHODS

In a dataset with N = 888 clinical cases treated with a monofocal aspherical intraocular lens (XC1/XY1, Hoya) constants for the SRKT, Haigis and Castrop formula were optimized and the prediction error PE calculated. The PE was bootstrapped NB = 1000 times, and the mean and trimmed mean of the bootstrapped PE were derived to generate the Bootlier plot showing the probability density function of the mean minus trimmed mean. With outliers this Bootlier plot shows some multimodality, and a Bootlier Index was extracted as a measure for multimodality. Outliers were removed from the tails of the PE distribution in a stepwise fashion until the Bootlier Index fell below a threshold of 0.001.

RESULTS

With the entire dataset the mean/SD/median/mean absolute/root mean squared PE using the optimized formula constants were -0.0045/0.44415/0.0134/0.3406/0.4412 dpt with SRKT, 0.0065/0.3711/-0.0056/0.2830/0.3710 dpt with Haigis, and 0.0034/0.3452/0.0023/0.2683/0.3451 dpt with the Castrop formula. After identifying and removing outliers the respective metrics for the PE were -0.0036/0.4028/0.0134/0.3205/0.4026 dpt for the SRKT (13 cases removed), 0.0050/0.3375/-0.0056/0.2656/0.3373 dpt with Haigis (11 cases removed), and 0.0035/0.3168/0.0023/0.2531/0.3166 dpt with Castrop (11 cases removed). The multimodality in the Bootlier plots was reduced from 0/0.1567/0.0587/0.0258/0.0007/0 with SRKT, 0/0.0981/0.0261/0.0202/0.0003/0 with Haigis, and 0.0006/0.0006/0.0161/0.0191/0.0005/0 with Castrop for the entire dataset to values below 1e-3 for trimming both tails of the PE distribution by ⅛, ¼, ½, 1, 2.5, and 5% respectively.

CONCLUSION

We were able to prove that bootstrapping with outlier identification based on Bootlier plots and the Bootlier Index is a powerful tool to clean a dataset of outliers for formula constant optimization.

Formula constant optimisation techniques including variation of keratometer or corneal refractive index and consideration for classical and modern IOL formulae

BACKGROUND

To investigate whether variation of the keratometer/corneal refractive index nK/nC improves the performance (prediction error PE) of classical and a modern intraocular lens (IOL) power calculation formula and further, to establish whether any trend error of PE for corneal radius R could be eliminated using formula constant and nK/nC optimisation.

METHODS

Based on 2 large datasets (1: N = 888 Hoya Vivinex aberration-correcting and 2: N = 822 Alcon SA60AT spherical lens) a classical formula constant optimisation has been performed for the Hoffer Q, Holladay 1, Haigis and Castrop formulae, to minimise the root mean squared (rms) PE (situation A). In two further optimisations, the formula constants and the formula specific nK/nC value were optimised to minimise the rms PE (situation B) or rms PE and trend error of PE for R (situation C). Nonlinear iterative optimisation strategy was applied according to Levenberg-Marquardt.

RESULTS

Optimising for rms PE and trend error (C) mainly improved the performance of the Holladay 1. The Haigis formula also showed a slight improvement compared to (A). The Hoffer Q formula shows no relevant trend error of PE for R. In contrast, the Holladay shows a positive and the Haigis (and the Castrop a slight) negative trend error of PE for R. The trend error could be fully eliminated by optimising formula constants and nK/nC in (B), but this was at the cost of overall performance in the case of the Holladay 1 formula.

CONCLUSION

Classical IOL calculation concepts should be critically examined for potential improvement of formula performance by variation of the empirical nK/nC value defined in the formula. With additional degrees of freedom additional optimisation terms such as trend errors might be considered in new intelligent optimisation strategies.

Preconditioning of clinical data for intraocular lens formula constant optimisation using Random Forest Quantile Regression Trees

PURPOSE

To implement a fully data driven strategy for identifying outliers in clinical datasets used for formula constant optimisation, in order to achieve proper formula predicted refraction after cataract surgery, and to assess the capabilities of this outlier detection method.

METHODS

2 clinical datasets (DS1/DS2: N = 888/403) of eyes treated with a monofocal aspherical intraocular lens (Hoya XY1/Johnson&Johnson Vision Z9003) containing preoperative biometric data, power of the lens implant and postoperative spherical equivalent (SEQ) were transferred to us for formula constant optimisation. Original datasets were used to generate baseline formula constants. A random forest quantile regression algorithm was set up using bootstrap resampling with replacement. Quantile regression trees were grown and the 25% and 75% quantile, and the interquartile range were extracted from SEQ and formula predicted refraction REF for the SRKT, Haigis and Castrop formulae. Fences were defined from the quantiles and data points outside the fences were marked and removed as outliers before recalculating the formula constants.

RESULTS

NB = 1000 bootstrap samples were derived from both datasets, and random forest quantile regression trees were grown to model SEQ versus REF and to estimate the median and 25% and 75% quantiles. The fence boundaries were defined as being from 25% quantile - 1.5·IQR to 75% quantile + 1.5·IQR, with data points outside the fence being marked as outliers. In total, for DS1 and DS2, 25/27/32 and 4/5/4 data points were identified as outliers for the SRKT/Haigis/Castrop formulae respectively. The respective root mean squared formula prediction errors for the three formulae were slightly reduced from: 0.4370 dpt;0.4449 dpt/0.3625 dpt;0.4056 dpt/and 0.3376 dpt;0.3532 dpt to: 0.4271 dpt;0.4348 dpt/0.3528 dpt;0.3952 dpt/0.3277 dpt;0.3432 dpt for DS1;DS2.

CONCLUSION

We were able to prove that with random forest quantile regression trees a fully data driven outlier identification strategy acting in the response space is achievable. In a real life scenario this strategy has to be complemented by an outlier identification method acting in the parameter space for a proper qualification of datasets prior to formula constant optimisation.

Considerations of a thick lens formula for intraocular lens power calculation

BACKGROUND

In recent years, some lens manufacturers have committed to providing lens shape data for some of their lens models. The purpose of this study is to present a strategy for prediction of intraocular lens power and residual refraction based on a pseudophakic model eye containing 5 refractive surfaces and to show its applicability using worked examples.

METHODS

A pseudophakic model eye with a thin spectacle correction, a thick cornea (radius of curvatures for both surfaces and central thickness) and a thick IOL (either radius of curvatures RLa and RLp for front and back surface or equivalent power PL and Coddington factor CL; and either central thickness LT or edge thickness and optic diameter) was set up. Calculations were performed based on linear Gaussian optics (vergence formulae). Formulae were provided to derive the lens power/shape and the residual equivalent spectacle refraction SEQ. From the lens shape the location of the haptic plane HP, the image sided principal plane of the lens HL, and the ocular magnification OM were extracted.

RESULTS

The calculation of a thick intraocular lens and the prediction of residual refraction is presented with reference to 3 working examples: A) lens varied in PL and shifted with its haptic plane keeping the CL constant, B) lens varied in CL and shifted with its haptic plane keeping PL constant, and C) CL and PL of the lens varied keeping its haptic plane position in the eye constant. For each combination of parameters (PL, CL, or haptic plane shift) the parameters influencing SEQ, OM and HL-HP were analysed.

CONCLUSION

Some modern optical biometers currently on the market provide the radii of curvature of both corneal surface and all relevant distances in the eye. With additional data on the lens shape, it would be possible to improve lens power calculations by switching from thin to thick lens models for the cornea and for the lens. This would overcome one of the major drawbacks of current lens power calculation methods.

Comparison of 2 modern swept-source optical biometers—IOLMaster 700 and Anterion

Purpose

To compare biometric measures from 2 modern swept-source OCT biometers (IOLMaster700 (Z, Carl-Zeiss-Meditec) and Anterion (H, Heidelberg Engineering)) and evaluate the effect of measurement differences on the resulting lens power (IOLP).

Methods

Biometric measurements were made on a large study population with both instruments. We compared axial length (AL), central corneal thickness (CCT), anterior chamber depth (ACD), lens thickness (LT) and corneal front and back surface curvature measurements. Corneal curvature was converted to power vectors and total power derived using the Gullstrand formula. A paraxial lens power calculation formula and a prediction for the IOL axial position according to the Castrop formula were used to estimate differences in IOLP targeting for emmetropia.

Results

There were no systematic differences between measurements of AL (− 0.0146 ± 0.0286 mm) and LT (0.0383 ± 0.0595 mm), whereas CCT yielded lower (7.8 ± 6.6 µm) and ACD higher (0.1200 ± 0.0531 mm) values with H. With H, CCT was lower for thicker corneas. The mean corneal front surface radius did not differ (− 0.4 ± 41.6 µm), but the corneal back surface yielded a steeper radius (− 397.0 ± 74.6 µm) with H, giving lower mean total power (− 0.3469 ± 0.2689 dpt). The astigmatic vector components in 0°/90° and 45°/135° were the same between both instruments for the front/back surface or total power.

Conclusion

The biometric measures used in standard formulae (AL, corneal front surface curvature/power) are consistent between instruments. However, modern formulae involving ACD, CCT or corneal back surface curvature may yield differences in IOLP, and therefore, formula constant optimisation customised to the biometer type is required.

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.

Biometry and Intraocular Lens Power Calculation in Eyes with Prior Laser Vision Correction (LVC) - A Review

BACKGROUND

An intraocular lens (IOL) calculation in eyes that have undergone laser vision correction (LVC) poses a significant clinical issue in regards to both patient expectation and accuracy. This review aims to describe the pitfalls of IOL power calculation after LVC and give an overview of the current methods of IOL power calculation after LVC.

REVIEW

Problems after LVC derive from the measurement of anterior corneal radii, central corneal thickness, asphericity, and the predicted effective lens position. A central issue is that most conventional 3rd generation formulas estimate lens position amongst other parameters on keratometry, which is altered in post-LVC eyes.

CONCLUSION

An IOL power calculation results in eyes with prior LVC that are notably impaired in eyes without prior surgery. Effective corneal power including anterior corneal curvature, posterior corneal curvature, CCT (central corneal thickness), and asphericity is essential. Total keratometry in combination with the Barrett True-K, EVO (emmetropia verifiying optical formula), or Haigis formula is relatively uncomplicated and seems to provide good results, as does the Barrett True-K formula with anterior K values. The ASCRS ( American Society of Cataract and Refractive Surgery) calculator combines results of various formulae and averages results, which allows a direct comparison between the different methods. Tomography-based raytracing and the Kane and the Castrop formulae need to be evaluated by future studies.

Update on Biometry and Lens Calculation - A Review of the Basic Principles and New Developments

These days, accurate calculation of artificial lenses is an important aspect of patient management. In addition to the classic theoretical optical formulae there are a number of new approaches, most of which are available as online calculators. This review aims to explain the background of artificial lens calculation and provide an update on study results based on the latest calculation approaches. Today, optical biometry provides the computational basis for theoretical optical formulae, ray tracing, and also empirical approaches using artificial intelligence. Manufacturer information on IOL design and IOL power recorded as part of quality control could improve calculations, especially for higher IOL powers. With modern measurement data, there is further potential for improvement in the determination of the axial length to the retinal pigment epithelium and by adopting a sum-of-segment approach. With the available data, the cornea can be assumed to be a thick lens. The Kane formula, the EVO 2.0 formula, the Castrop formula, the PEARL-DGS, formula and the OKULIX calculation software provide consistently good results for artificial lens calculations. Excellent refractive results can be achieved using these tools, with approximately 80% having an absolute prediction error within 0.50 dpt, at least in highly selected study populations. The Barrett Universal II formula also produces excellent results in the normal and long axial length range. For eyes with short axial lengths, the use of Barrett Universal II should be reconsidered; in this case, one of the methods mentioned above is preferable. Second Eye Refinement can also be considered in this patient population, in conjunction with established classic third generation formulae.

Meridional ocular magnification after cataract surgery with toric and non-toric intraocular lenses

Background

Overall ocular magnification (OOM) and meridional ocular magnification (MOM) with consequent image distortions have been widely ignored in modern cataract surgery. The purpose of this study was to investigate OOM and MOM in a general situation with an astigmatic refracting surface.

Methods

From a large dataset containing biometric measurements (IOLMaster 700) of both eyes of 9734 patients prior to cataract surgery, the equivalent (P_IOLeq) and cylindric power (P_IOLcyl) were derived for the HofferQ, Haigis, and Castrop formulae for emmetropia. Based on the pseudophakic eye model, OOM and MOM were extracted using 4 × 4 matrix algebra for the corrected eye (with P_IOLeq/P_IOLcyl (scenario 1) or with P_IOLeq and spectacle correction of the residual refractive cylinder (scenario 2) or with P_IOLeq remaining the residual uncorrected refractive cylinder (blurry image) (scenario 3)). In each case, the relative image distortion of MOM/OOM was calculated in %.

Results

On average, P_IOLeq/P_IOLcyl was 20.73 ± 4.50 dpt/1.39 ± 1.09 dpt for HofferQ, 20.75 ± 4.23 dpt/1.29 ± 1.01 dpt for Haigis, and 20.63 ± 4.31 dpt/1.26 ± 0.98 dpt for Castrop formulae. Cylindric refraction for scenario 2 was 0.91 ± 0.70 dpt, 0.89 ± 0.69 dpt, and 0.89 ± 0.69 dpt, respectively. OOM/MOM (× 1000) was 16.56 ± 1.20/0.08 ± 0.07, 16.56 ± 1.20/0.18 ± 0.14, and 16.56 ± 1.20/0.08 ± 0.07 mm/mrad with HofferQ; 16.64 ± 1.16/0.07 ± 0.06, 16.64 ± 1.16/0.18 ± 0.14, and 16.64 ± 1.16/0.07 ± 0.06 mm/mrad with Haigis; and 16.72 ± 1.18/0.07 ± 0.05, 16.72 ± 1.18/0.18 ± 0.14, and 16.72 ± 1.18/0.07 ± 0.05 mm/mrad with Castrop formulae. Mean/95% quantile relative image distortion was 0.49/1.23%, 0.41/1.05%, and 0.40/0.98% for scenarios 1 and 3 and 1.09/2.71%, 1.07/2.66%, and 1.06/2.64% for scenario 2 with HofferQ, Haigis, and Castrop formulae.

Conclusion

Matrix representation of the pseudophakic eye allows for a simple and straightforward prediction of OOM and MOM of the pseudophakic eye after cataract surgery. OOM and MOM could be used for estimating monocular image distortions, or differences in overall or meridional magnifications between eyes.

Similarity of eyes in a cataractous population—How reliable is the biometry of the fellow eye for lens power calculation?

Background

In some situations it is necessary to use biometry from the fellow eye for lens power calculation prior to cataract surgery. The purpose of this study was to analyse the lateral differences in biometric measurements and their impact on the lens power calculation.

Methods

The analysis was based on a large dataset of 19,472 measurements of 9736 patients prior to cataract surgery with complete biometric data of both left and right eyes extracted from the IOLMaster 700. After randomly indexing the left or right eye as primary (P) and secondary (S), the differences between S and P eye were recorded and analysed (Keratometry (RSEQ), total keratometry (TRSEQ) and back surface power (BRSEQ)), axial length AL, corneal thickness CCT, anterior chamber depth ACD, lens thickness LT). Lens power was calculated with the Castrop formula for all P and S eyes, and the refraction was predicted using both the P and S eye biometry for the lens power calculation.

Results

Lateral differences (S-P, 90% confidence interval) ranged between -0.64 to 0.63 dpt / -0.67 to 0.66 dpt / -0.12 to 0.12 dpt for RSEQ / TRSEQ / BRSEQ. The respective difference in AL / CCT / ACD / LT ranged between -0.46 to 0.43 mm / -0.01 to 0.01 mm / -0.20 to 0.20 mm / -0.13 to 0.14 mm. The resulting difference in lens power and predicted refraction ranged between -2.02 to 2.00 dpt and -1.36 to 1.30 dpt where the biometry of the S eye is used instead of the P eye. The AL and RSEQ were identified as the most critical parameters where the biometry of the fellow eye is used.

Conclusion

Despite a strong similarity of both eyes, intraocular lens power calculation with fellow eye biometry could yield different results for the lens power and finally for the predicted refraction. In 10% of cases, the lens power derived from the S eye deviates by 2 dpt or more, resulting in a refraction deviation of 1.36 dpt or more.

Prediction of ocular magnification and aniseikonia after cataract surgery

BACKGROUND

Ocular magnification and aniseikonia after cataract surgery has been widely ignored in modern cataract surgery. The purpose of this study was to analyse ocular magnification and inter-individual differences in a normal cataract population with a focus on monovision.

METHODS

From a large dataset containing biometric measurements (IOLMaster 700) of both eyes of 9734 patients prior to cataract surgery, eyes were indexed randomly as primary (P) and secondary (S). Intraocular lens power (IOLP) was derived for the HofferQ, Haigis and Castrop formulae for emmetropia for P and emmetropia or myopia (-0.5 to -2 dpt) for S to simulate monovision. Based on the pseudophakic eye model in addition to these formulae, ocular magnification was extracted using matrix algebra (refraction and translation matrices and a system matrix describing the optical property of the entire spectacle corrected or uncorrected eye).

RESULTS

With emmetropia for P and S the IOLP differences (S-P) showed a standard deviation of 0.162/0.156/0.157 dpt and ocular magnification differences yielded a standard deviation of 0.0414/0.0405/0.0408 mm/mrad for the HofferQ/Haigis/Castrop setting. Simulating monovision, the myopic eye (S) showed a systematically smaller mean absolute spectacle corrected ocular magnification than the emmetropic eye (-0.0351/-0.0340/-0.0336, respectively, relative magnification around 2%). If myopia in the S eye remains uncorrected, the reduction of ocular magnification is much smaller (around 0.2-0.3%).

CONCLUSION

Vergence formulae for IOLP calculation sometimes implicitly define a pseudophakic eye model which can be directly used to predict ocular magnification after cataract surgery. Despite a strong similarity of both eyes, ocular magnification does not fully match between eyes and the prediction of ocular magnification and aniseikonia might be relevant to avoid eikonic problems in the pseudophakic eye.

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.

Prediction of the axial lens position after cataract surgery using deep learning algorithms and multilinear regression

BACKGROUND

The prediction of anatomical axial intraocular lens position (ALP) is one of the major challenges in cataract surgery. The purpose of this study was to develop and test prediction algorithms for ALP based on deep learning strategies.

METHODS

We evaluated a large data set of 1345 biometric measurements from the IOLMaster 700 before and after cataract surgery. The target parameter was the intraocular lens (IOL) equator plane at half the distance between anterior and posterior apex. The relevant input parameters from preoperative biometry were extracted using a principal component analysis. A selection of neural network algorithms was tested using a 5-fold cross-validation procedure to avoid overfitting. The results were then compared with a traditional multilinear regression in terms of root mean squared prediction error (RMSE).

RESULTS

Corneal radius of curvature, axial length, anterior chamber depth, corneal thickness, lens thickness and patient age were identified as effective predictive parameters, whereas pupil size, horizontal corneal diameter and Chang-Waring chord did not enhance the model. From the tested algorithms, the Gaussian prediction regression and the Support Vector Machine algorithms performed best (RMSE = 0.2805 and 0.2731 mm), outperforming the multilinear prediction model (0.3379 mm). The mean absolute prediction error yielded 0.1998, 0.1948 and 0.2415 mm for the respective models.

CONCLUSION

Modern prediction techniques may have the potential to outperform traditional multilinear regression techniques as they can deal easily with nonlinearities between input and output parameters. However, in all cases a cross-validation is mandatory to avoid overfitting and misinterpretation of the results.

Pre-clinical estimation of the intraocular lens A-constant, and its relationship to power, shape factor, and asphericity

Calculating the intraocular lens power for a particular patient requires an empirical "lens constant" to estimate the final axial location after surgery. This is normally calculated from clinical results for each new lens style, but it can also be estimated without clinical data by comparing a new style to an existing style. The lenses are axially positioned in a model eye at comparable locations, and image distances are used to estimate the change in lens constant. The A-constant used by the SRK/T calculation method is evaluated here, but this can be easily converted for other calculations using an average eye. Raytrace calculations demonstrate the method, and also illustrate the effects that refractive index, shape factor, and asphericity have on the refractive error. Actual lens measurements at 35°C in saline are preferable if details of the reference lens are uncertain.