Background: To explain the concept of the Castrop lens power calculation formula and show the application and results from a large dataset compared to classical formulae. Methods: The Castrop vergence formula is based on a pseudophakic model eye with 4 refractive surfaces. This was compared against the SRKT, Hoffer-Q, Holladay1, simplified Haigis with 1 optimized constant and Haigis formula with 3 optimized constants. A large dataset of preoperative biometric values, lens power data and postoperative refraction data was split into training and test sets. The training data were used for formula constant optimization, and the test data for cross-validation. Constant optimization was performed for all formulae using nonlinear optimization, minimising root mean squared prediction error. Results: The constants for all formulae were derived with the Levenberg-Marquardt algorithm. Applying these constants to the test data, the Castrop formula showed a slightly better performance compared to the classical formulae in terms of prediction error and absolute prediction error. Using the Castrop formula, the standard deviation of the prediction error was lowest at 0.45 dpt, and 95% of all eyes in the test data were within the limit of 0.9 dpt of prediction error. Conclusion: The calculation concept of the Castrop formula and one potential option for optimization of the 3 Castrop formula constants (C, H, and R) are presented. In a large dataset of 1452 data points the performance of the Castrop formula was slightly superior to the respective results of the classical formulae such as SRKT, Hoffer-Q, Holladay1 or Haigis.

Background: Surgically induced astigmatism (SIA) has been widely discussed in the literature as the change in corneal astigmatism resulting from corneal incision. The purpose of this study was to investigate the change in corneal refractive power preoperative to postoperative using a vector analysis of keratometry, total keratometry, and corneal back surface data from a modern optical biometer Methods: The analysis was based on a dataset of 122 eyes of 122 patients with preoperative and 1 month postoperative measurements performed with the IOLMaster 700 biometer from 1 clinical centre and a standardised surgical technique involving a corneal 2.5 mm 45°-incision made from the superior direction. Keratometry, total keratometry and corneal back surface data were processed in 3 vector components (spherical equivalent power SEQ and astigmatism considered in 0°/90° (C_0°) and in 45°/135° (C_45°) meridian), and the changes in corneal power vectors were analysed, comparing preoperative to postoperative values. Results: The mean corneal power of total keratometry reduced slightly after cataract surgery (-0.05 dpt), resulting mostly from a decrease in back surface power (-0.04 dpt). The astigmatism vector component C_0° of total keratometry reduced by -0.28 dpt, mostly due to a decrease at the corneal front surface (-0.26 dpt). With the corneal incision at 12 o’clock position this flattening in the 90° meridian refers to a SIA of around ¼ dpt. The change in C_0° and the C_45° astigmatic vector components for both keratometry and total keratometry show a large variation ranging between 0.24 and 0.33 dpt (standard deviations), indicating a poor predictability of the change in astigmatism due to cataract surgery. Conclusion: Cataract surgery locally flattens the cornea in the incision meridian. This flattening shows a large individual variation and therefore a poor predictability. Our study indicates that SIA in modern cataract surgery with standardised corneal incision is in a range of 1/4 dpt.

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.

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 S_w.Methods: This Monte–Carlo simulation study used a large dataset containing 16 667 preoperative IOLMaster 700 biometric measurements. Based on literature S_w 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.

Background: The corneal back surface is known to add some against the rule astigmatism, with implications in cataract surgery with toric lens implantation. This study aimed to set up and validate a deep learning algorithm to predict corneal back surface power from the corneal front surface power and biometric measures. Methods: This study was based on a large dataset of IOLMaster 700 measurements from two clinical centres. N = 19,553 measurements of 19,553 eyes with valid corneal front (CFSPM) and back surface power (CBSPM) data and other biometric measures. After a vector decomposition of CFSPM and CBSPM into equivalent power and projections of astigmatism to the 0°/90° and 45°/135° axes, a multi‐output feedforward neural network was derived to predict vector components of CBSPM from CFSPM and other measurements. The predictions were compared with a multivariate linear regression model based on CFSPM components only. Results: After pre‐conditioning, a network with two hidden layers each having 12 neurons was derived. The dataset was split into training (70%), validation (15%) and test (15%) subsets. The prediction error (predicted corneal back surface power CBSPP – CBSPM) of the network after training and crossvalidation showed no systematic offset, narrower distributions for CBSPP – CBSPM and no trend error of CBSPP – CBSPM vs. CBSPM for any of the vector components. The multivariate linear model also showed no systematic offset, but broader distributions of the prediction error components and a systematic trend of all vector components vs. CFSPM components. Conclusion: The neural network approach based on CFSPM vector components and other biometric measures outperforms the multivariate linear model in predicting corneal back surface power vector components. Modern biometers can supply all parameters required for this algorithm, enabling reliable predictions for corneal back surface data where direct corneal back surface data are unavailable.

Purpose: Validating a new nomogram for low to moderate astigmatism (0.75 D to 2.5 D) correction with epithelium‐ and Bowman‐penetrating femtosecond laser‐assisted arcuate incisions.Methodology: Prospective, interventional case series at the Augen‐ und Laserklinik, Castrop‐Rauxel, Germany. Cataract patients with low to moderate corneal astigmatism were treated with femtosecond laser‐assisted arcuate incisions. Patients with previous refractive corneal treatment were excluded. Outcome assessment was based on manifest refraction, astigmatic vector analysis and visual acuity.Results: The study analysed 43 eyes of 33 patients after three months and 35 eyes of 27 patients after 12 months. After 12 months, 100% of all eyes treated had ≤1.0 D and 97% ≤0.5 D of subjective residual astigmatism. Mean residual astigmatism was 0.27 D. 90% of all eyes were within one line of difference between UDVA and CDVA. SEQ Mean Absolute Error was 0.26 D and SEQ. Mean error was −0.08 ± 0.32 D. CI was 0.98 ± 0.2 D, and Index of Success, 0.20 ± 0.18 D.Conclusion: The Castrop nomogram showed results that are comparable to or better than results presented in the literature for existing nomograms. Our results for astigmatic reduction are comparable to published results for TIOL implantation. It seems to be a predictable and safe measure to reduce manifest astigmatism.