This study followed the tenets of the Declaration of Helsinki and the study protocol was approved by the Institutional Review Board and Ethics Committee of the Fundación Jiménez Díaz.

Patients with causes of endothelial dysfunction other than FECD (such as herpetic endothelitis or iridocorneal endothelial syndrome) or with history of corneal injuries, ocular infection, ocular inflammation or refractive surgery were excluded. Cases with preoperative ultrasound biometry or with low visual acuity that made difficult performing postoperative subjective refraction were also excluded.

Some cases in the sample of this study contain incomplete or missing data. To clarify and prevent errors in the analysis, different study groups were created for hypothesis testing. For instance, when comparing errors in the biometric formulas, patients with manually entered preoperative keratometry data (manual group) were separated from those with keratometric biometric inputs. Similarly, when analyzing corneal changes, only the cases with available anterior corneal surface information (pre- and postoperative Simk) were used for SimK change calculation. On the other hand, only those patients with available pre- and postoperative information of both corneal surfaces (pre- and postoperative SimK and corneal ratio) were included for TK change analysis. Finally, to develop the fitting model based on corneal change, hereafter referred to as the linear corneal change model (LCCM), only the cases with pre- and postoperative measurements with information on both corneal surfaces and labelled as “Successful” by the biometer were used. In addition, for the comparison between the optimization method and the LCCM, only data from those patients with Bi-Flex (Medicontur) IOL and available preoperative corneal ratio were used.

The collected data from patient's medical records included the following: demographic information (age and sex), best corrected visual acuity (BCVA) in the latest clinical review prior to triple-DMEK, surgery date, specific hydrophobic monofocal IOL model implanted and clinical data obtained at 3 months or later after the surgery (including date, autorefractor measurements, subjective refraction and BCVA).

The remaining variables used in this study were obtained from the IOL Master 700 swept-source optical biometer (Carl Zeiss Meditec AG, Jena, Germany). Pre- and postoperative measurements were collected, for a minimum of 3 months after the surgery in the latter case. These measurements included AL, anterior chamber depth (ACD) and the principal meridians of both the anterior and posterior corneal surfaces (flat K and steep K). For convenience and clarity, the mean radius of the anterior and posterior corneal surfaces were converted from millimetres to power expressed in diopters (D). The front corneal power determined through simulated keratometry (SimK), was derived from the front radius and the standard keratometric refractive index of 1.3375. The power of the whole cornea, encompassing both the anterior and posterior surfaces, was assessed through TK measurements. To describe those variables, the median and interquartile range was used due to skew distribution.

The corneal ratio was calculated as the ratio between the front and back corneal radius in both the pre- and postoperative scenarios. The change in this variable was determined by subtracting the preoperative ratio from the postoperative ratio.

Triple-DMEKs were performed by 3 experienced corneal surgeons (NAA, BGS and IJA) following the same standardized surgical technique that we have previously specified in a previously published article.11 One or the other IOL was chosen according to the refractive target (between −0.50D and −1.00D) and the surgeon's preference, as several models of hydrophobic IOLs were available.

The variables used for IOL calculations included the AL, the ACD and the mean SimK, obtained from the IOLMaster 700 (1.90.33.04 version). Classical formulas algorithms (SRK/T, Hoffer Q, Holladay I and Haigis) were programmed using routine algorithms in Python language and cross-checked with those given by IOL Master 700.12–15 New-generation formulas results were computed using on-line free calculators at date April 2023: Barrett Universal II formula V1.05 (https://calc.apacrs.org/barrett_universal2105/), Emmetropia Verifying Optical (EVO) IOL calculator V2.0 (https://www.evoiolcalculator.com/calculator.aspx), Kane formula V1.42 (https://www.iolformula.com/) and PEARL-DGS formula for regular eyes (https://iolsolver.com/regular). The initial constants for each formula were obtained from the IOLCon optimized constants database website (https://iolcon.org/). For the new formulas the optimization procedure was computed using A constant. The selection of these specific formulas and sources was based on their widespread acceptance and validation in the ophthalmology community, as well as their availability for free and easy access, ensuring comprehensive and up-to-date results for IOL calculations.

The prediction error of each formula was analyzed by subtracting the predicted refraction (biometric target) from the residual subjective refraction, measured 3 months or more after Triple-DMEK, using SE notation. In addition, we also evaluated the mean error (ME) and mean absolute error (MAE), as suggested by best practice guidelines.16 The ME provides insight into the average difference between the predicted refraction by each formula and the postoperative residual subjective refraction, considering both positive and negative deviations. On the other hand, the MAE calculates the average absolute difference between the predicted and observed refraction, disregarding the direction of the deviation. Both measures are widely used in the literature to describe the formulas performance and reliability in IOL calculations. MAE values are expressed as mean (95% confidence interval).

Another commonly used method for evaluating the error, that we have also carried out, is to assess the number or percentage of eyes with a prediction error within various error intervals in D and visually represent the results using histograms. This approach provides a clear depiction of the distribution and frequency of errors across the sample, allowing for a comprehensive understanding of the accuracy of the formulas.

The constant optimization was assessed adding the ME to the initial constant (obtained from IOLCon). Then, this new prospective constant was used to obtain a new ME. This iterative procedure was repeated until the ME reached a threshold of <0.01D At this point, we determined that the constant had been optimized. The optimized constants were only assessed for the Bi-Flex HB-877FAB (Medicontur Medical Engineering Ltd., Inc., Zsambek, Hungary) and AcrySof-IQ SN60WF (Alcon Laboratories, Inc., Fort Worth, TX, USA) IOL models using routine algorithms.17

Firstly, we conducted a univariate analysis using the Wilcoxon signed-ranked test to investigate the changes in SimK and TK separately. A p-value < 0.05 was considered statistically significant.

Astigmatism was analysed using the power vectors method (J0 and J45 vectors). J0 represents the horizontal cylindrical component of astigmatism and, if it is positive, it means that the astigmatism is with-the-rule (WTR), while if it is negative is against-the-rule (ATR). J45 represents the oblique cylindrical component of astigmatism, a positive value indicates oblique astigmatism with its principal axis at 45°, while a negative value indicates principal axis at 135°. For the analysis of the results, we used statistical description of those mentioned vectors on the pre- and postoperative stage. Also, the astigmatic change was computed as the subtraction (postoperative - preoperative) and it was visually represented on the double angle vector diagrams (DAVD) using the positive cylinder notation with the 95% confidence ellipse of the data.18,19 The same analysis were performed for a filtered database which included only those measurements labelled as “Successful” by the IOLMaster 700 device.

The modelling process was obtained utilizing the filtered database that exclusively included "Successful" biometric measurements of eyes with pre- and postoperative data of both corneal surfaces, ensuring the validity of the data. To analyse the correlation between variables that describe the corneal change (TK and SimK) and the preoperative variables (preoperative corneal ratio (CRpreop)), a least square linear model was tested. The robustness of the model was evaluated using the coefficient of determination (R2), with a result > 0.4 considered a moderate correlation. By means of this, we aimed to gain insights into the underlying factors affecting corneal changes. The formulas to obtain the corneal change prediction are depicted on Fig. 1. The difference in correlation between the middle graph for TK (R² = 0.39) and the right graph for SimK (R² = 0.28) suggests that the posterior corneal surface plays a significant role in corneal change. For this reason, the formula chosen to estimate corneal change was: estimated TK change = +12.88 × CRpreop - 14.74. A negative value will indicate that the cornea loses power after the surgery, and this value is the one that will indicate the target we should aim for with the selected biometric formula. To see an example that helps to understand its application, consult the Supplementary material.

Fig. 1.

Linear least squares of the preoperative variables that showed strongest correlation with the corneal change.aa

Total Keratometry (TK), Simulated Keratometry (SimK).

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To compute the change in corneal power the pre- and postoperative corneal ratio are needed (real TK change = CRpostop - CRpreop). The corneal ratio is obtained by dividing the front corneal radius by the back corneal radius, and it provides information about the relative curvatures of the anterior and posterior corneal surfaces. In the literature, the normal value for the corneal ratio has been commonly defined around 1.2.6,20 Changes in the corneal ratio can indicate alterations in one or both corneal curvatures, thus considering only the SimK to determine the corneal refractive power might be an important source of error in patients with an altered corneal ratio.

To assess the LCCM error for each formula, the corneal change prediction (estimated TK change) is needed. As mentioned above, this value is obtained by a linear regression model using the CRpreop and the formula that is located in the upper left corner of the middle graphs in Fig. 1. Finally, this change is subtracted from the formula expected refraction with the initial constant (not optimized), and the result of this difference is subtracted from the real postoperative subjective refraction in SE notation. Thus, the error for the IOL calculation considering the corneal change can be assessed for the different formulas, as presented in the example showed in Supplementary material.

To evaluate the accuracy of the LCCM, which considers the corneal change in IOL calculation with the IOL constants from IOLCon, a comparative analysis was performed in eyes implanted with the Bi-Flex IOLs that exhibited corneal ratio data available. This involved comparing the LCCM results with those obtained by constant optimization for each formula. Descriptive statistical analysis was applied to evaluate absolute errors and error intervals.

Demographical and clinical data from the sample, including data availability, implanted hydrophobic IOL model and preoperative ocular biometry measurements are depicted in Table 1.

When comparing the group of patients in which biometry inputs were available (measured group, N = 88) with the group of patients in which keratometric values from the autorefractometer were manually entered into the biometer due to measurement failures (manual group, N = 9), no notable differences were observed between them in the preoperative values. Also, the residual refraction in SE notation for both groups was similar, −0.05 ± 0.84 D for those eyes measured and −0.35 ± 1.81 D for the manual group. Selecting the Bi-Flex and SN60WF IOL models, the mean target refraction across all the formulas was −0.61 ± 0.3 D for the measured group (N = 63) and −0.95 ± 0.6 D for the manual group (N = 9). A notable MAE difference was found between the measured group and the manual group considering the average of the values obtained with the initial constants for the different formulas, 0.75 D (−0.62, 2.12) versus 1.35 D (−1.04, 3.75) respectively. No statistical tests were performed for these comparisons between both groups due to the difference in sample sizes. The value improves even further when analysing only the eyes in the measured group labeled as successful by the biometer: 0.69 D (−0.07, 1.45).

For the analysis of the first corneal surface, we obtained the pre- and postoperative SimK from 84 eyes, as 4 eyes of the measured group (N = 88) had no postoperative information. In this group, the SimK change was −0.21 ± 0.68 D Similarly, considering only those eyes where the quality of the keratometry measurement was “Successful” (N = 50), the change was −0,21 ± 0.51 D In both cases the SimK change was statistically significant (p < 0.001). On the other hand, the manual group (N = 9) showed a SimK change of −0.3 ± 0.82 D, not reaching statistical significance (p = 0.31). Astigmatic power vectors J0 and J45, changed from 0.17 ± 0.68 D and 0.02 ± 0.45 D to −0.15 ± 0.65 D and −0.11 ± 0.48 D, respectively. More information is shown in Fig. 2a and b.

Fig. 2.

Double angle plotted diagrams showing front corneal astigmatism (2.a, N = 84) and total corneal astigmatism (2.c, N = 61) induced by Triple-DMEK considering all eyes with available postsurgical data and front corneal astigmatism (2.b, N = 50) and total corneal astigmatism (2.d, N = 38) considering only "Successful" measurements determined by the IOLMaster 700 biometer.bb

Simulated Keratometry (SimK), Total Keratometry (TK), Against-The-Rule astigmatism (ATR), With-The-Rule astigmatism (WTR), Oblique astigmatism (OBL), Diopters (D).

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The pre- and postoperative data for the whole cornea were available for 61 eyes. In this group, the TK change was −0.62 ± 1.09 D For the subgroup with optimal keratometry readings (N = 38), this change was −0.63 ± 0.78 D In both cases the change was also statistically significant (p < 0.001). Consistent with the astigmatism in SimK, the preoperative TK power vectors J0 and J45 were 0.18 ± 0.58 D and −0.07 ± 0.53 D, while postoperatively changed to −0.23 ± 0.67 D and −0.12 ± 0.49 D More information is available in Fig. 2c and d.

• In both SimK and TK DAVDs (Fig. 2), a trend toward against the rule (ATR) astigmatism induction was observed. Moreover, a reduction of around 0.5 D in the magnitude of both vertical and horizontal axis of the confidence ellipse when considering only the successfully measured eyes, suggests a more accurate value of the surgical induced astigmatism (SIA). To evaluate the changes in the SimK and TK cylinders we conducted a univariate and multivariate analysis. In the univariate analysis, when comparing the SimK cylinder change, the preoperative and postoperative J0 vector showed statistically significant differences using the paired-sample Student's t-test (p < 0.001, T = 5.39), while the oblique component J45 did not (p = 0.97, T=−0.04). Similarly, the TK cylinder J0 component changed significantly (p < 0.001, T = 6.54), whereas J45 did not (p = 0.92, T = 0.09). The multivariate analysis (preoperative J0 and J45 versus their postoperative counterparts) confirmed the previous findings. The Hotelling's T-squared test showed statistically significant differences for both SimK (p < 0.001, T²=40.27) and TK (p < 0.001, T²=44.41). These results suggest that Triple-DMEK could significantly induce ATR astigmatism.

To assess a regression model that could predict the corneal power change, only those eyes with optimal quality measurements from both corneal surfaces were selected (N = 38). The preoperative feature with the highest determination coefficient was the corneal ratio. There was an inverse weak correlation between the corneal ratio and the SimK change (R2 = 0.28). This correlation increases when considering the TK change (R2 = 0.39). Also, the change in corneal ratio and preoperative corneal ratio showed a moderate correlation (R2 = 0.42). The equations are depicted in Fig. 1.

First, we analysed the results obtained with the initial (non-optimized) constants and the selection of the Bi-Flex (N = 45) or SN60WF (N = 18) IOL power for a target of −0.61 ± 0.30 D The MAE was similar across all formulas (0.75 D (−0.62, 2.12)). Before constant optimization, the formula with the lowest MAE was EVO (0.69 D (−0.69, 2.06)) while Haigis and Holladay I obtained the highest MAE values (0.80 D (−0.62, 2.22) and 0.80 D (−0.55, 2.16), respectively) (p < 0.001). For a thorough analysis segregated by IOL model, please consult the table 2. The manual group showed higher values of MAE than the measured group, being the SRK/T formula the one that exhibited the lowest value (1.23 D (−1.13, 3.60) and the Haigis the one that displayed the highest value (1.50 D (−0.78, 3.77)). In contrast, the eyes in the measured group labeled as successful by the biometer showed lower MAE values, with the Kane formula yielding the best value (0.60 D (−0.13, 1.34)) and the Holladay I formula displaying the highest value (0.81 D (0.08, 1.54)).

Subsequently, slight differences were observed when the constant optimization was performed for the same eyes as before. The MAE was reduced (0.65 D (−0.59, 1.88)), but the differences remained consistent across the different formulas. After constant optimization, the lowest MAE was obtained by the SRK/T (0.60 D (−0.53, 1.74)) and the highest MAE by the Hoffer Q and Haigis (0.70 D (−0.62, 2.03) and 0.69 (−0.58, 1.96), respectively) (p = 0.005). For a detailed breakdown based on the IOL model, please refer to table 3. If we analyze only the eyes in the measured group labeled as successful by the biometer, the formula with the best MAE value was SRK/T (0.32 D (−0.33, 0.98)), while the worst value was yielded by the Hoffer Q (0.42 D (−0.42, 1.26)).

The MAE differences considering all formulae with initial constants versus optimised constants were significant (p < 0.001).

Finally, regarding the LCCM applied to IOL calculation, only Bi-Flex IOLs with preoperative CR data available were considered (N = 39). No statistical differences were found between the optimization procedure and the performance of the LCCM analysed by formula (p = 0.38). Holladay I LCCM outperformed the other formulas in terms of MAE with 0.61 D (−0.36, 1.57), while Hoffer Q LCCM showed the highest MAE at 0.68 D (−0.34, 1.69) (p = 0.53). The mean of the maximum absolute error (MaxAE) assessed with all formulas and the optimization method was 2.08 D (1.77, 2.39), while for the LCCM method was 1.87 D (1.62, 2.12). More information is available in table 4. Prediction error distribution analysed by formula and method is shown in Fig. 3. The Kane formula adjusted with LCCM showed the best results for hyperopic surprises, with only 2.33% of eyes within the 1 to 1.5 D range.

Fig. 3.

Error intervals for the different intraocular lens calculation formulas after constant optimization and considering corneal change approaches in the subgroup of eyes with Bi-Flex intraocular lenses with successful biometric measurements of both corneal surfaces.cc

Prediction error (PE).

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