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

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 (PIOLeq) and cylindric power (PIOLcyl) 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 PIOLeq/PIOLcyl (scenario 1) or with PIOLeq and spectacle correction of the residual refractive cylinder (scenario 2) or with PIOLeq 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, PIOLeq/PIOLcyl 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

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

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

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

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 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

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

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). 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. 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. 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

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