Translation model for CW chord to angle Alpha derived from a Monte-Carlo simulation based on raytracing.

BACKGROUND: The Chang-Waring chord is provided by many ophthalmic instruments, but proper interpretation of this chord for use in centring refractive procedures at the cornea is not fully understood. The purpose of this study is to develop a strategy for translating the Chang-Waring chord (position of pupil centre relative to the Purkinje reflex PI) into angle Alpha using raytracing techniques. METHODS: The retrospective analysis was based on a large dataset of 8959 measurements of 8959 eyes from 1 clinical centre, using the Casia2 anterior segment tomographer. An optical model based on: corneal front and back surface radius Ra and Rp, asphericities Qa and Qp, corneal thickness CCT, anterior chamber depth ACD, and pupil centre position (X-Y position: PupX and PupY), was defined for each measurement. Using raytracing rays with an incident angle IX and IY the CW chord (CWX and CWY) was calculated. Using these data, a multivariable linear model was built up in terms of a Monte-Carlo simulation for a simple translation of incident ray angle to CW chord. RESULTS: Raytracing allows for calculation of the CW chord CWX/CWY from biometric measures and the incident ray angle IX/IY. In our dataset mean values of CWX = 0.32±0.30 mm and CWY = -0.10±0.26 mm were derived for a mean incident ray angle (angle Alpha) of IX = -5.02±1.77° and IY = 0.01±1.47°. The raytracing results could be modelled with a linear multivariable model, and the effect sizes for the prediction model for CWX are identified as Ra, Qa, Rp, CCT, ACD, PupX, PupY, IX, and for CWY they are Ra, Rp, PupY, and IY. CONCLUSION: Today the CW chord can be directly measured with any biometer, topographer or tomographer. If biometric measures of Ra, Qa, Rp, CCT, ACD, PupX, PupY are available in addition to the CW chord components CWX and CWY, a prediction of angle Alpha is possible using a simple matrix operation.

Surgically Induced Astigmatism after Cataract Surgery - A Vector Analysis.

PURPORSE: 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 one month postoperative measurements performed with the IOLMaster 700 biometer from 1 clinical center and a standardized 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 three vector components (spherical equivalent power SEQ and astigmatism considered in 0°/90° (C0°) and in 45°/135° (C45°) meridian), and the changes in corneal power vectors were analyzed, 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 C0° 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 » dpt. The change in C0° and the C45° 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 standardized corneal incision is in a range of 1/4 dpt.

Prediction of total corneal power from measured anterior corneal power on the IOLMaster 700 using a feedforward shallow neural network.

BACKGROUND: The corneal back surface is known to add some astigmatism against-the-rule, which has to be considered in cataract surgery with toric lens implantation. The purpose of this study was to set up a deep learning algorithm which predicts the total corneal power from keratometry and biometric measures. METHODS: Based on a large data set of measurements with the IOLMaster 700 from two clinical centres, data from N = 21 108 eyes were included, each record containing valid data for keratometry K, total keratometry TK, axial length AL, central corneal thickness CCT, anterior chamber depth ACD, lens thickness LT and horizontal corneal diameter W2W from an individual eye. After a vector decomposition of K and TK into equivalent power (.EQ) and projections of astigmatism to the 0°/90° (.AST0° ) and 45°/135° (.AST45° ) axis, a multi-output feedforward shallow neural network was derived to predict TK from K, AL, CCT, ACD, LT, W2W and patient age. RESULTS: After some trial and error, the neural network having a Levenberg-Marquardt training function and three hidden layers (10/8/5 neurons) performed best and showed a fast convergence. The data set was split into training data (70%), validation data (15%) and test data (15%). The prediction error (predicted corneal power CPpred minus TK) of the network trained with the training and cross-validated with test data showed systematically narrower distributions for CPEQ-TKEQ, CPAST0° -TKAST0° and CPAST45° -TKAST45° compared with KEQ-TKEQ, KAST0° -TKAST0° and KAST45° -TKAST45° . There was no systematic offset in the components between CPpred and TK. CONCLUSION: Unlike any fixed correction term, which can compensate only for a static intercept of the astigmatic components TKEQ, TKAST0° and TKAST45° compared with KEQ, KAST0° and KAST45° , our trained neural network was able to reduce the variance in the prediction error significantly. This neural network could be used to account for the corneal back surface astigmatism for biometers where the corneal back surface measurement or total keratometry is not available.

Prediction of CW chord as a measure for the eye's orientation axis after cataract surgery from preoperative IOLMaster 700 measurement data.

BACKGROUND: The angles alpha and kappa are widely discussed for centring refractive procedures, but they cannot be determined with ophthalmic instruments. The purpose of this study is to investigate the Chang-Waring chord (position of the Purkinje reflex PI relative to the corneal centre) derived from an optical biometer before and after cataract surgery and to study the changes resulting from cataract surgery. METHODS: The analysis was based on a large dataset of 1587 complete sets of preoperative and postoperative IOMaster 700 biometry measurements from two clinical centres, each containing: valid data for pupil and corneal centre position, the position of the Purkinje reflex PI originated from a coaxial fixation target, keratometry (K), axial length (AL), anterior chamber depth (ACD), lens thickness (LT), central corneal thickness CCT, and horizontal corneal diameter W2W. The Chang-Waring chord CW was derived from pupil centre and Purkinje reflex PI analysed preoperatively and postoperatively, and a multilinear regression model together with a feedforward neural network algorithm was set up to predict postoperative CW chord from preoperative CW chord, K and biometric distances of the eye. RESULTS: The Y component of CW chord shows a slight shift in the inferior direction in both left and right eyes, before and after cataract surgery. The X component shows some shift in the temporal direction, which is more pronounced preoperatively and slightly reduced postoperatively but with a larger variation. The change in CW chord from preoperative to postoperative shows a slight shift in the superior and nasal directions. Our algorithms for prediction of postoperative CW chord using preoperative CW chord, keratometry and biometry as input data performed with a multilinear regression and a feedforward neural network approach were able to reduce the variance, but could not properly predict the postoperative CW chord X and Y components. CONCLUSION: The CW chord as the position of the Purkinje reflex PI with respect to the pupil centre can be directly measured with any biometer, topographer or tomographer with a coaxial fixation light. The mean Y component does not differ between right and left eyes or preoperatively and postoperatively, but the mean temporal shift of the X component preoperatively is slightly reduced postoperatively, but with a larger scatter of the values.

Prediction of corneal back surface power - Deep learning algorithm versus multivariate regression.

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.

The Castrop formula for calculation of toric intraocular lenses.

PURPOSE: To explain the concept behind the Castrop toric lens (tIOL) power calculation formula and demonstrate its application in clinical examples. METHODS: The Castrop vergence formula is based on a pseudophakic model eye with four refractive surfaces and three formula constants. All four surfaces (spectacle correction, corneal front and back surface, and toric lens implant) are expressed as spherocylindrical vergences. With tomographic data for the corneal front and back surface, these data are considered to define the thick lens model for the cornea exactly. With front surface data only, the back surface is defined from the front surface and a fixed ratio of radii and corneal thickness as preset. Spectacle correction can be predicted with an inverse calculation. RESULTS: Three clinical examples are presented to show the applicability of this calculation concept. In the 1st example, we derived the tIOL power for a spherocylindrical target refraction and corneal tomography data of corneal front and back surface. In the 2nd example, we calculated the tIOL power with keratometric data from corneal front surface measurements, and considered a surgically induced astigmatism and a correction for the corneal back surface astigmatism. In the 3rd example, we predicted the spherocylindrical power of spectacle refraction after implantation of any toric lens with an inverse calculation. CONCLUSIONS: The Castrop formula for toric lenses is a generalization of the Castrop formula based on spherocylindrical vergences. The application in clinical studies is needed to prove the potential of this new concept.

Considerations on the Castrop formula for calculation of intraocular lens power.

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

Back-calculation of keratometer index based on OCT data and raytracing - a Monte Carlo simulation.

PURPOSE: This study aims to develop a raytracing-based strategy for calculating corneal power from anterior segment optical coherence tomography data and extracting the individual keratometer index, which converts the corneal front surface radius to corneal power. METHODS: A large OCT dataset (10,218 eyes of 8,430 patients) from the Casia 2 (Tomey, Japan) was post-processed in MATLAB (MathWorks, USA). Radius of curvature, asphericity of the corneal front and back surface, central corneal thickness and pupil size (aperture) were used to trace a bundle of rays through the cornea and derive the best focus plane. Corneal power was calculated with respect to the corneal front vertex plane, and the keratometer index was back-calculated using corneal power and front surface radius. Keratometer index was analysed in a multivariate linear model. RESULTS: The averaged resulting keratometer index was 1.3317 ± 0.0017 with a median of 1.3317 and range from 1.3233 to 1.3390. In a univariate model, only the front surface asphericity affected the keratometer index. The multivariate model for modelling the keratometer index using all 6 input parameters performed very well (RMS error: 5.54e-4, R2 : 0.90, significance vs. constant model: <0.0001). CONCLUSIONS: In the classical calculation, the keratometer index used for converting corneal radius to dioptric power uses several model assumptions. As these assumptions are not generally satisfied, corneal power cannot be calculated from corneal front surface radius alone. Considering all 6 input variables, the linear prediction model performs well and can be used if all input parameters are measured with a tomographer.

Prediction model for best focus, power, and spherical aberration of the cornea: Raytracing on a large dataset of OCT data.

PURPOSE: To analyse corneal power based on a large optical coherence tomography dataset using raytracing, and to evaluate corneal power with respect to the corneal front apex plane for different definitions of best focus. METHODS: A large OCT dataset (10,218 eyes of 8,430 patients) from the Casia 2 (Tomey, Japan) was post-processed in MATLAB (MathWorks, USA). Using radius of curvature, corneal front and back surface asphericity, central corneal thickness, and pupil size (aperture) a bundle of rays was traced through the cornea. Various best focus definitions were tested: a) minimum wavefront error, b) root mean squared ray scatter, c) mean absolute ray scatter, and d) total spot diameter. All 4 target optimisation criteria were tested with each best focus plane. With the best-fit keratometer index the difference of corneal power and keratometric power was evaluated using a multivariate linear model. RESULTS: The mean corneal powers for a/b/c/d were 43.02±1.61/42.92±1.58/42.91±1.58/42.94±1.59 dpt respectively. The root mean squared deviations of corneal power from keratometric power (nK = 1.3317/1.3309/1.3308/1.3311 for a/b/c/d) were 0.308/0.185/0.171/0.209 dpt. With the multivariate linear model the respective RMS error was reduced to 0.110/0.052/0.043/0.065 dpt (R² = 0.872/0.921/0.935/0.904). CONCLUSIONS: Raytracing improves on linear Gaussian optics by considering the asphericity of both refracting surfaces and using Snell's law of refraction in preference to paraxial simplifications. However, there is no unique definition of best focus, and therefore the calculated corneal power varies depending on the definition of best focus. The multivariate linear model enabled more precise estimation of corneal power compared to the simple keratometer equation.

[Back-calculation of the keratometer index-Which value would have been correct in cataract surgery?] / Rückrechnung des Keratometerindex ­ Welcher Wert wäre bei der Kataraktchirurgie richtig gewesen?

BACKGROUND AND PURPOSE: In the clinical routine the conversion of corneal radii into corneal refractive power using a keratometer index is rarely discussed. The purpose of this study was to back-calculate the keratometer index in pseudophakic eyes based on the refractive power of the lens, biometric measurements and refraction, and to compare it to clinically established values. PATIENTS AND METHODS: In this retrospective case series 99 eyes of 99 patients without pathological alterations, previous diseases, comorbidities or history of ocular surgery apart from the uneventful cataract surgery were enrolled. In all eyes a CT Asphina 409M(P) (Carl-Zeiss Meditec, Berlin, Germany) had been implanted by two surgeons (EF and PE). For calculation we used shape and power data of the intraocular lens and data from optical biometry (axial length, pseudophakic anterior chamber depth, lens thickness, corneal radius; IOLMaster 700, Carl-Zeiss Meditec, Jena, Germany). The refraction was derived manually with a trial frame (measurement distance 5 m) and autorefractometry (iProfiler, Carl-Zeiss, Jena, Germany). For this three model eyes were used: a thin lens with the nominal refractive power positioned in the equatorial plane (model A) or in the secondary principal plane of the thick lens (model B) as well as a model considering the intraocular lens as a thick lens located at its measured position (model C). RESULTS: Back-calculation of the keratometer index using vergence formulas resulted in a keratometer index based on subjective refraction measurements considering lane distance correction of 1.3307 ± 0.0026/1.3312 ± 0.0026/1.332 ± 0.0027 for model A/model B/model C, respectively. Based on objective refraction measurements (autorefraction calibrated to infinity object distances) resulted in a keratometer index of 1.3301 ± 0.0021/1.3306 ± 0.0021/1.3315 ± 0.0021, for model A/model B/model C, respectively. The keratometer index did not show any trend in linear regression for axial length or corneal radius for any of the three models or for any refraction method. CONCLUSION: The keratometer index derived from back-calculation matched with the Zeiss index (1.332) but was much lower compared to other established indexes, e.g. the Javal index (1.3375). The missing trend for axial length or corneal radius implies that simple vergence formulas for intraocular lens refractive power calculation without correction terms or fudge factors perform best with a keratometer index slightly below 1.332, if the biometrically measured position of the intraocular lens is used as the effective lens position.