Evaluating keratometry and corneal astigmatism data from biometers and anterior segment tomographers and mapping to reconstructed corneal astigmatism.

BACKGROUND: To compare results from different corneal astigmatism measurement instruments; to reconstruct corneal astigmatism from the postimplantation spectacle refraction and toric intraocular lens (IOL) power; and to derive models for mapping measured corneal astigmatism to reconstructed corneal astigmatism. METHODS: Retrospective single centre study involving 150 eyes treated with a toric IOL (Alcon SN6AT, DFT or TFNT). Measurements included IOLMaster 700 keratometry (IOLMK) and total keratometry (IOLMTK), Pentacam keratometry (PK) and total corneal refractive power in 3 and 4 mm zones (PTCRP3 and PTCRP4), and Aladdin keratometry (AK). Regression-based models mapping the measured C0 and C45 components (Alpin's method) to reconstructed corneal astigmatism were derived. RESULTS: Mean C0 components were 0.50/0.59/0.51 dioptres (D) for IOLMK/PK/AK; 0.2/0.26/0.31 D for IOLMTK/PTCRP3/PTCRP4; and 0.26 D for reconstructed corneal astigmatism. All corresponding C45 components ranged around 0. The prediction models had main diagonal elements lower than 1 with some crosstalk between C0 and C45 (nonzero off-diagonal elements). Root-mean-squared residuals were 0.44/0.45/0.48/0.51/0.50/0.47 D for IOLMK/IOLMTK/PK/PTCRP3/PTCRP4/AK. CONCLUSIONS: Results from the different modalities are not consistent. On average IOLMTK/PTCRP3/PTCRP4 match reconstructed corneal astigmatism, whereas IOLMK/PK/AK show systematic C0 offsets of around 0.25 D. IOLMTK/PTCRP3/PTCRP4. Prediction models can reduce but not fully eliminate residual astigmatism after toric IOL implantation.

Vector analysis of corneal astigmatism in cataractous eyes based on IOLMaster 700 biometry.

PURPOSE: The purpose of this study was to investigate the effect of the corneal back surface by comparing the keratometric astigmatism (K, derived from the corneal front surface) of a modern optical biometer against astigmatism of Total Keratometry (TK, derived from both corneal surfaces) in a large population with cataractous eyes. The results were then used to define linear prediction models to map K to TK. METHODS: From a large dataset containing bilateral biometric measurements (IOLMaster 700) in 9736 patients prior to cataract surgery, the total corneal astigmatism was decomposed into vectors for K, corneal back surface (BS), and TK. A multivariate prediction model (MV), simplified model with separation of vector components (SM) and a constant model (CM) were defined to map K to TK vector components. RESULTS: The K centroid (X/Y) showed some astigmatism with-the-rule (0.1981/-0.0211 dioptre (dpt)) whereas the TK centroid was located around zero (-0.0071/-0.0381 dpt against-the-rule) and the BS centroid showed systematic astigmatism against-the-rule (-0.2367/-0.0145 dpt). The respective TK-K centroid was located at -0.2052/-0.0302 dpt. The MV model showed the same performance (i.e. mean absolute residuum) as the SM did (0.1098 and 0.1099 dpt respectively) while the CM performed only slightly worse (0.1121 dpt mean absolute residuum). CONCLUSION: In cases where tomographic data are unavailable statistical models could be used to consider the overall contribution of the back surface to the total corneal astigmatism. Since the performance of the CM is sufficiently close to that of MV and SM we recommend using the CM which can be directly considered e.g. as surgically induced astigmatism.

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.

Repeatability of biometric measures from the IOLMaster 700 in a cataractous population.

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

Technical variability of cornea parameters derived from anterior segment OCT fitted with Fringe Zernike polynomials.

BACKGROUND: This study uses bootstrapping to evaluate the technical variability (in terms of model parameter variation) of Zernike corneal surface fit parameters based on Casia2 biometric data. METHODS: Using a dataset containing N = 6953 Casia2 biometric measurements from a cataractous population, a Fringe Zernike polynomial surface of radial degree 10 (36 components) was fitted to the height data. The fit error (height - reconstruction) was bootstrapped 100 times after normalisation. After reversal of normalisation, the bootstrapped fit errors were added to the reconstructed height, and characteristic surface parameters (flat/steep axis, radii, and asphericities in both axes) extracted. The median parameters refer to a robust surface representation for later estimates of elevation, whereas the SD of the 100 bootstraps refers to the variability of the surface fit. RESULTS: Bootstrapping gave median radius and asphericity values of 7.74/7.68 mm and -0.20/-0.24 for the corneal front surface in the flat/steep meridian and 6.52/6.37 mm and -0.22/-0.31 for the corneal back surface. The respective SD values for the 100 bootstraps were 0.0032/0.0028 mm and 0.0093/0.0082 for the front and 0.0126/0.0115 mm and 0.0366/0.0312 for the back surface. The uncertainties for the back surface are systematically larger as compared to the uncertainties of the front surface. CONCLUSION: As measured with the Casia2 tomographer, the fit parameters for the corneal back surface exhibit a larger degree of variability compared with those for the front surface. Further studies are needed to show whether these uncertainties are representative for the situation where actual repeat measurements are possible.

Prediction of IOL decentration, tilt and axial position using anterior segment OCT data.

BACKGROUND: Intraocular lenses (IOLs) require proper positioning in the eye to provide good imaging performance. This is especially important for premium IOLs. The purpose of this study was to develop prediction models for estimating IOL decentration, tilt and the axial IOL equator position (IOLEQ) based on preoperative biometric and tomographic measures. METHODS: Based on a dataset (N = 250) containing preoperative IOLMaster 700 and pre-/postoperative Casia2 measurements from a cataractous population, we implemented shallow feedforward neural networks and multilinear regression models to predict the IOL decentration, tilt and IOLEQ from the preoperative biometric and tomography measures. After identifying the relevant predictors using a stepwise linear regression approach and training of the models (150 training and 50 validation data points), the performance was evaluated using an N = 50 subset of test data. RESULTS: In general, all models performed well. Prediction of IOL decentration shows the lowest performance, whereas prediction of IOL tilt and especially IOLEQ showed superior performance. According to the 95% confidence intervals, decentration/tilt/IOLEQ could be predicted within 0.3 mm/1.5°/0.3 mm. The neural network performed slightly better compared to the regression, but without significance for decentration and tilt. CONCLUSION: Neural network or linear regression-based prediction models for IOL decentration, tilt and axial lens position could be used for modern IOL power calculation schemes dealing with 'real' IOL positions and for indications for premium lenses, for which misplacement is known to induce photic effects and image distortion.

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.

Evaluation of statistical correction strategies for corneal back surface astigmatism with toric lenses: a vector analysis.

PURPOSE: To compare actual and formula-predicted postoperative refractive astigmatism using measured posterior corneal power measurements and 4 different empiric posterior corneal astigmatism correction models. SETTING: Tertiary care center. DESIGN: Single-center retrospective consecutive case series. METHODS: Using a dataset of 211 eyes before and after tIOL implantation (Hoya Vivinex), IOLMaster 700 (IOLM) or Casia2 (CASIA) keratometric and front/back surface corneal power measurements were converted to power vector components C0 (0/90 degrees) and C45 (45/135 degrees). Differences between postoperative and Castrop formula predicted refraction at the corneal plane using the labeled parameters of the tIOL and the keratometric or front/back surface corneal powers were recorded as the effect of corneal back surface astigmatism (BSA). RESULTS: Generally, the centroid of the difference shifted toward negative C0 values indicating that BSA adds some against the rule corneal astigmatism (ATR). From IOLM/CASIA keratometry, the average difference in C0 was 0.39/0.32 diopter (D). After correction with the Abulafia-Koch, Goggin, La Hood, and Castrop nomograms, it was -0.18/-0.24 D, 0.27/0.18 D, 0.13/0.08 D, and 0.17/0.10 D. Using corneal front/back surface data from IOLM/CASIA, the difference was 0.18/0.12 D. CONCLUSIONS: The Abulafia-Koch method overcorrected the ATR, while the Goggin, La Hood, and Castrop models slightly undercorrected ATR, and using measurements from the CASIA tomographer seemed to produce slightly less prediction error than IOLM.

Evaluation of corneal power from an AS-OCT thick lens model and ray tracing: reliability of the keratometer index.

PURPOSE: To investigate and compare different strategies of corneal power calculations using keratometry, paraxial thick lens calculations and ray tracing. SETTING: Tertiary care center. DESIGN: Retrospective single-center consecutive case series. METHODS: Using a dataset with 9780 eyes of 9780 patients from a cataractous population the corneal front (Ra/Qa) and back (Rp/Qp) surface radius/asphericity, central corneal thickness (CCT), and entrance pupil size (PUP) were recorded using the Casia 2 tomographer. Beside keratometry with the Zeiss (PK Z ) and Javal (PK J ) keratometer index, a thick lens paraxial formula (PG) and ray tracing (PR) was implemented to extract corneal power for pupil sizes from 2 mm to 5 mm in steps of 1 mm and PUP. RESULTS: With PUP PK Z /PK J overestimates the paraxial corneal power PG in around 97%/99% of cases and PR in around 80% to 85%/99%. PR is around 1/6 or 5/6 diopters (D) lower compared with PK Z or PK J . For a 2 mm pupil PR is around 0.20/0.91 D lower compared with PK Z /PK J and for a 5 mm pupil PR is comparable with PK Z (around 0.03 D lower) but around 0.70 to 0.75 D lower than PK J . CONCLUSIONS: "True" values of corneal power are mostly required in lens power calculations before cataract surgery, and overestimation of corneal power could induce trend errors in refractive outcome with axial length and lens power if compensated with the effective lens position.

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.

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.

Monte-Carlo simulation for calculating phakic supplementary lenses based on a thick and thin lens model using anterior segment OCT data.

BACKGROUND: Phakic lenses (PIOLs, the most common and only disclosed type being the implantable collamer lens, ICL) are used in patients with large or excessive ametropia in cases where laser refractive surgery is contraindicated. The purpose of this study was to present a strategy based on anterior segment OCT data for calculating the refraction correction (REF) and the change in lateral magnification (ΔM) with ICL implantation. METHODS: Based on a dataset (N = 3659) containing Casia 2 measurements, we developed a vergence-based calculation scheme to derive the REF and gain or loss in ΔM on implantation of a PIOL having power PIOLP. The calculation concept is based on either a thick or thin lens model for the cornea and the PIOL. In a Monte-Carlo simulation considering, all PIOL steps listed in the US patent 5,913,898, nonlinear regression models for REF and ΔM were defined for each PIOL datapoint. RESULTS: The calculation shows that simplifying the PIOL to a thin lens could cause some inaccuracies in REF (up to ½ dpt) and ΔM for PIOLs with high positive power. The full range of listed ICL powers (- 17 to 17 dpt) could correct REF in a range from - 17 to 12 dpt with a change in ΔM from 17 to - 25%. The linear regression considering anterior segment biometric data and the PIOLP was not capable of properly characterizing REF and ΔM, whereas the nonlinear model with a quadratic term for the PIOLP showed a good performance for both REF and ΔM prediction. CONCLUSION: Where PIOL design data are available, the calculation concept should consider the PIOL as thick lens model. For daily use, a nonlinear regression model can properly predict REF and ΔM for the entire range of PIOL steps if a vergence calculation is unavailable.

EDOF intraocular lens design: shift in image plane vs object vergence.

BACKGROUND: To compare 2 different design scenarios of EDOF-IOLs inserted in the Liou-Brennan schematic model eye using raytracing simulation as a function of pupil size. METHODS: Two EDOF IOL designs were created and optimized for the Liou-Brennan schematic model eye using Zemax ray tracing software. Each lens was optimized to achieve a maximum Strehl ratio for intermediate and far vision. In the first scenario, the object was located at infinity (O1), and the image plane was positioned at far focus (I1) and intermediate focus (I2) to emulate far and intermediate distance vision, respectively. In the second scenario, the image plane was fixed at I1 according to the first scenario. The object plane was set to infinity (O1) for far-distance vision and then shifted closer to the eye (O2) to reproduce the corresponding intermediate vision. The performance of both IOLs was simulated for the following 3 test conditions as a function of pupil size: a) O1 to I1, b) O1 to I2, and c) O2 to I1. To evaluate the imaging performance, we used the Strehl ratio, the root-mean-square (rms) of the spot radius, and the spherical aberration of the wavefront for various pupil sizes. RESULTS: Evaluating the imaging performance of the IOLs shows that the imaging performance of the IOLs is essentially identical for object/image at O1/I1. Designed IOLs perform dissimilarly to each other in near-vision scenarios, and the simulations confirm that there is a slight difference in their optical performance. CONCLUSION: Our simulation study recommends considering the difference between object shift and image plane shift in design and test conditions to achieve more accurate pseudoaccommodation after cataract surgery.

Prediction of corneal power vectors after cataract surgery with toric lens implantation-A vector analysis.

BACKGROUND: Intraocular lenses are typically calculated based on a pseudophakic eye model, and for toric lenses (tIOL) a good estimate of corneal astigmatism after cataract surgery is required in addition to the equivalent corneal power. The purpose of this study was to investigate the differences between the preoperative IOLMaster (IOLM) and the preoperative and postoperative Casia2 (CASIA) tomographic measurements of corneal power in a cataractous population with tIOL implantation, and to predict total power (TP) from the IOLM and CASIA keratometric measurements. METHODS: The analysis was based on a dataset of 88 eyes of 88 patients from 1 clinical centre before and after tIOL implantation. All IOLM and CASIA keratometric and total corneal power measurements were converted to power vector components, and the differences between preoperative IOLM or CASIA and postoperative CASIA measurements were assessed. Feedforward neural network and multivariate linear regression prediction algorithms were implemented to predict the postoperative total corneal power (as a reference for tIOL calculation) from the preoperative IOLM and CASIA keratometric measurements. RESULTS: On average, the preoperative IOLM keratometric / total corneal power under- / overestimates the postoperative CASIA keratometric / real corneal power by 0.12 dpt / 0.21 dpt. The prediction of postoperative CASIA real power from preoperative IOLM or CASIA keratometry shows that postoperative total corneal power is systematically (0.18 dpt / 0.27 dpt) shifted towards astigmatism against the rule, which is not reflected by keratometry. The correlation of postoperative CASIA real power to the corresponding preoperative CASIA values is better than those as compared to the preoperative IOLM keratometry. However, there is a large variation from preoperative IOLM or CASIA keratometry to the postoperative CASIA real power of up to 1.1 dpt (95% confidence interval). CONCLUSION: One of the challenges of tIOL calculation is the prediction of postoperative total corneal power from preoperative keratometry. Keratometric power restricted to a front surface measurement does not fully reflect the situation of corneal back surface astigmatism, which typically adds some extra against the rule astigmatism.

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.

Bootstrapping of Corneal Optical Coherence Tomography Data to Investigate Conic Fit Robustness.

BACKGROUND: Fitting of parametric model surfaces to corneal tomographic measurement data is required in order to extract characteristic surface parameters. The purpose of this study was to develop a method for evaluating the uncertainties in characteristic surface parameters using bootstrap techniques. METHODS: We included 1684 measurements from a cataractous population performed with the tomographer Casia2. Both conoid and biconic surface models were fitted to the height data. The normalised fit error (height-reconstruction) was bootstrapped 100 times and added to the reconstructed height, extracting characteristic surface parameters (radii and asphericity for both cardinal meridians and axis of the flat meridian) for each bootstrap. The width of the 90% confidence interval of the 100 bootstraps was taken as uncertainty and quoted as a measure of the robustness of the surface fit. RESULTS: As derived from bootstrapping, the mean uncertainty for the radii of curvature was 3 µm/7 µm for the conoid and 2.5 µm/3 µm for the biconic model for the corneal front/back surface, respectively. The corresponding uncertainties for the asphericity were 0.008/0.014 for the conoid and 0.001/0.001 for the biconic. The respective mean root mean squared fit error was systematically lower for the corneal front surface as compared to the back surface (1.4 µm/2.4 µm for the conoid and 1.4 µm/2.6 µm for the biconic). CONCLUSION: Bootstrapping techniques can be applied to extract uncertainties of characteristic model parameters and yield an estimate for robustness as an alternative to evaluating repeat measurements. Further studies are required to investigate whether bootstrap uncertainties accurately reproduce those from repeat measurement analysis.

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.

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.

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.