Abstract
Purpose:
In cataract surgery, accurate intraocular lens (IOL) power calculations are crucial for optimal postoperative refractive outcomes. This study explores the impact of prioritizing the reduction of the standard deviation (SD) of prediction errors before mean prediction error (PE) adjustment on IOL calculation formula precision and accuracy.
Methods:
We conducted a retrospective analysis of 4885 eyes from 2611 patients, all implanted with the same IOL model, comparing four traditional IOL power calculation formulas: SRK/T, Holladay 1, Haigis, and Hoffer Q. We introduced new constants aiming to minimize the SD of PE (new_const) against traditionally optimized constants (classic_const), using a heteroscedastic statistical method for comparison. Validation of precision improvements used a secondary dataset of 262 eyes from 132 patients.
Results:
We observed significant reductions in mean absolute error (MAE) across training and test sets for Hoffer Q, Holladay, and Haigis formulas, indicating accuracy enhancements. Optimized constants significantly reduced SDs for Haigis from 0.3255 to 0.3153 and for Hoffer Q from 0.3521 to 0.3387. These optimizations also increased the proportion of eyes achieving PE within ±0.25 D. SRK/T showed improved SD from 0.3596 to 0.3585. However, Holladay 1 showed minimal change with no significant improvement. In the test dataset, significant reductions in SD were observed for Haigis and Hoffer Q.
Conclusions:
Prioritizing SD minimization before adjusting mean PE significantly improves the precision of selected IOL power formulas, enhancing postoperative refractive outcomes. The effectiveness varies among formulas, underscoring the need for formula-specific adjustments.
Translational Relevance:
The study presents a novel two-step approach for optimizing IOL power calculations.
Accurate intraocular lens (IOL) power calculation is crucial for achieving satisfactory postoperative outcomes in cataract surgery, particularly in attaining the intended postoperative refraction. Precision and accuracy in these calculations are essential.
Accuracy is reflected in the mean prediction error (PE), the discrepancy between the achieved, and anticipated refraction, which ideally should approximate zero. This principle underpins the strategy of “zeroization,” where adjustments are made to minimize the mean PE. Precision, on the other hand, is gauged by the standard deviation (SD) of the PE, with a lower SD signifying greater precision, indicating a tight clustering of PE around the mean.
These parameters are frequently used in clinical research to evaluate the efficacy of various IOL calculation power formulas, specifically in terms of their accuracy and precision in determining the correct lens power for cataract procedures.
1–6 The preliminary stage in such analyses typically involves normalizing the arithmetic mean of the PE to zero for each formula by adjusting the lens constant.
7
Holladay et al.
8 have suggested that the SD of the PE is essential for determining the accuracy and reliability of IOL power calculation formulas. We recently showed that the neutralization of the mean PE in IOL power calculation formulas can inadvertently impair their precision.
9 This phenomenon is particularly evident when PEs arise from inaccurate corneal power estimations, leading to significant PE and potentially diminishing the precision of IOL calculation formulas. Conversely, rectifying PEs from axial length measurements or predicted implant position estimations seems to have minimal or beneficial effects on precision. This underscores the importance of identifying error sources when evaluating their impact on formula accuracy, especially in formulas that correct systematic errors from keratometry inaccuracies by adjusting their zeroization constant.
On the other hand, a uniform alteration of keratometric power by a constant value across all eyes would shift the predictive refraction for a given IOL power formula, maintaining the consistency of the error distribution and leaving the SD unchanged. Therefore a mean shift in refractive PE, not affecting the SD, can be achieved through uniform keratometric power modification. This suggests that optimizing an IOL power calculation formula involves a two-step process: first, adjusting constants to minimize the SD of predictive errors; second, neutralizing the residual average PE in the corneal plane by adjusting keratometric power with an increment opposite to the mean refractive bias at the corneal plane (
Fig. 1). This approach aims to enhance both precision (minimized SD) and accuracy (neutralized mean predictive error) of IOL power calculations in clinical applications.
Our study investigates this hypothesis and evaluates the potential benefits in terms of precision enhancement using a well-documented dataset. We applied standard metrics to assess various IOL power calculation formulas, conducting a rigorous comparison of their accuracy and precision in predicting postoperative refraction. The proposed two-step optimization process was tested on both training and test sets to determine its robustness and generalizability in modern cataract surgery.
Prediction error was defined by the subtraction of the predicted spherical equivalent (SE) from the real postoperative SE. A training set of 4885 eyes of 2611 patients implanted with the Finevision IOL (PhysIOL, Liège, Belgium) was used to determine the usual optimized constants (classic_const) (i.e., the constants leading to a mean PE equal to zero) for four classical formulas (SRK/T, Holladay 1, Haigis single-optimized, Hoffer Q). Constants leading to the smallest SD of PE (new_const) were then iteratively determined, and the resulting mean PE e (new_ME) and the difference between the two constants (delta_const = classic_const − new_const) were recorded for each formula. Predicted SE were then calculated using the usual classic_const values, and a second calculation was performed by using new_const values and then adding the new_ME value of the formula to each resulting predicted SE to obtain an ME equal to zero for the whole set for each formula.
A second set of 262 eyes of 132 patients from another center implanted with the same IOL model was used to simulate real-life use of the methodology. In this setting, delta_const and new_ME values determined using the first set were used. Optimized constants for this test set (classic_const_2) were first determined for each formula using the standard methodology, and related formula predictions were calculated. New constants for the second set (new_const_2) were then determined by subtracting delta_const from classic_const_2; predicted SE was then calculated for each formula and subsequently corrected by adding new_ME to each prediction.
The SD of the PE, mean absolute PE (MAE), median absolute PE (MedAE), minimum PE (MinE), maximum PE (MaxE), as well as the percentage of eyes with PEs of ±0.25, ±0.50, ±1.00, and ± 1.50 D of each method were calculated for each formula, for both sets. MEs were equal to zero, by definition, for both predictions of the first set and the classical prediction of the second set. ME was different from zero in the second set when using the evaluated optimization method because delta_const and new_ME values from the first set were deliberately used.
Each couple of formulas (optimized using the standard method versus optimized using the evaluated method) were compared to absolute PE using the Wilcoxon signed-rank test. McNemar test was used to compare the percentage of eyes with a PE within 0.25 D. A
P value ≤ 0.05 was considered significant. Statistical comparison of SDs between pairs of formulas was conducted according to the heteroscedastic method described by Holladay et al.
8 using the
R programming language version 4.3.3.
10 All pairwise comparisons of the IOL power calculation formulas were conducted using a slight extension of the HC4 version of the Morgan-Pitman test (file “Rallfun-v43.txt”).
Retrospective data query was obtained from the Ethics Review Board of the Canadian Ophthalmic Research Center and by the Regional Committee for Medical and Health Research Ethics, Norway. The study was conducted according to the tenets of the Declaration of Helsinki.
Demographics of both sets are presented in
Table 1.
Table 2 lists standard optimized constants (classic_const) and SD-optimized constants (new_const), the difference between the two constants observed on the first set (delta_const), and the mean PE obtained on the first set when using the SD-optimized constants (new_ME).
Table 3 presents prediction outcomes obtained on the first set with both optimization methods and the
P value of the Wilcoxon test for the mean absolute error.
Table 4 presents prediction outcomes obtained on the second set with both optimization methods and the
P value of the Wilcoxon test for the mean absolute error. Both sets observed significant reductions in the mean absolute error (MAE) for the Hoffer Q, Holladay, and Haigis formulas.
Table 1. Demographics of the First and Second Sets
Table 1. Demographics of the First and Second Sets
Table 2. Standard and SD-Optimized Constants for the Tested Formulas
Table 2. Standard and SD-Optimized Constants for the Tested Formulas
Table 3. Metrics of Precision and Accuracy of the Tested Formula for the Standard and Evaluated Optimization (First Set)
Table 3. Metrics of Precision and Accuracy of the Tested Formula for the Standard and Evaluated Optimization (First Set)
Table 4. Metrics of Precision and Accuracy of the Tested Formula for the Standard and Evaluated Optimization (Second Set)
Table 4. Metrics of Precision and Accuracy of the Tested Formula for the Standard and Evaluated Optimization (Second Set)
The percentage of eyes with a PE within 0.25 D was significantly different for the Haigis and the Hoffer Q formula on the first set (
P < 0.005, McNemar test) (
Fig. 2). It was not significantly different otherwise (
Fig. 3).
In the first (training) dataset, the standard optimization produced SD values of 0.3914 for Haigis, 0.4006 for Holladay 1, 0.4107 for Hoffer Q, and 0.4228 for SRK/T. With the evaluated optimization, significant reductions in SD were observed for Haigis to 0.3846 and Hoffer Q to 0.4033, accompanied by statistically significant improvements with p-values of 0.009 and 0.001, respectively. Holladay 1 and SRK/T, with SD values at 0.4006 and 0.4174, respectively, did not show significant improvements in precision, as indicated by P values of 0.604 and 0.785.
For the second (test) set, the standard optimization strategy yielded SD values for Haigis at 0.3255, Holladay 1 at 0.3363, Hoffer Q at 0.3521, and SRK/T at 0.3596. Under the evaluated optimization, SD values improved for Haigis to 0.3153, Hoffer Q to 0.3387, and SRK/T to 0.3585, with P values showing significant enhancements for Haigis (2.981e-10), Hoffer Q (2.824e-10), and SRK/T (1.552e-06), indicating a meaningful reduction in the SD of prediction errors. However, Holladay 1 showed minimal change in SD from 0.3363 to 0.3362 without a statistically significant improvement (P = 0.747).
Although attaining a mean PE of zero enhances the formula's accuracy, this alone does not guarantee its precision, particularly in situations where there is a significant dispersion of errors. When comparing IOL power formulas, the SD computation usually follows constant optimization to eliminate systematic bias, focusing on the deviation of each data point from the zeroed mean. The effect of zeroing the mean PE on the error's SD is not straightforwardly predictable, because the SD hinges on individual error values rather than the error's mean.
Machine learning–based IOL power calculation formulas show great promise in enhancing postoperative refraction outcomes and can be integrated into existing formulas to improve their accuracy.
11,12 However, despite these advancements, they do not fully resolve the issue of optimization.
Although the statistical significance of the observed changes is evident, it is important to consider their clinical and refractive impact. The proposed optimization methodology, which prioritizes minimizing the standard deviation (SD) before adjusting the mean prediction error (PE), has the potential to significantly enhance the precision of intraocular lens (IOL) power calculations. This improvement in precision leads to more consistent and predictable postoperative refractive outcomes, thereby reducing the incidence of extreme refractive errors and enhancing patient satisfaction. The reduction in SD indicates that a higher percentage of patients are likely to achieve their target refractions within clinically acceptable ranges (e.g., ±0.25 D, ±0.50 D).
Figure 1 illustrates the principles of this method, demonstrating how optimizing SD first, followed by adjusting the mean PE, can lead to both high precision and accuracy in IOL power calculations. By systematically addressing these two aspects, our approach not only improves the precision (as shown by reduced variability) but also ensures the accuracy of the refractive outcomes.
The findings from this study provide a promising insight into the optimization of IOL power calculation formulas through the prioritization of minimizing the SD of prediction errors before adjusting the mean PE (zeroization). Notably, the evaluated optimization strategy resulted in significant improvements in the precision of Haigis, Hoffer Q, and SRK/T formulas in the training set, as evidenced by reduced SD values and statistically significant P values. These enhancements suggest that adjusting the constants to reduce variability in prediction errors can lead to more accurate and reliable IOL power calculations.
The improvements observed in the Haigis and Hoffer Q formulas were consistent across both training and test datasets, demonstrating the reproducibility and potential clinical relevance of the new optimization approach. These formulas showed marked precision improvements, indicating that the methodology is not only statistically significant but also practically meaningful in clinical settings. However, the results for the Holladay 1 and SRK/T formulas were mixed, with significant improvements seen in the SRK/T formula in the training set but not in the test set and minimal changes observed in the Holladay 1 formula. This variability underscores the need for formula-specific optimizations and suggests that different IOL formulas may respond differently to the same optimization strategies.
In the proposed optimization method for IOL power calculation formulas, a distinctive approach is used that inherently minimizes the SD of prediction errors, thereby enhancing the precision of refractive outcomes. This method diverges from traditional optimization techniques, which primarily aim to nullify the mean PE without directly addressing the variability of outcomes. By focusing on minimizing the SD, our approach specifically targets the reduction of outcome variability, which is crucial for improving the consistency of postoperative refractive results. The importance of the SD in comparing IOL calculation formulas lies in its ability to measure the dispersion or variability of prediction errors around the mean. Utilizing the SD as a metric for comparison not only highlights the precision of different formulas but also emphasizes the consistency of the outcomes they predict. This measure effectively evaluates and anticipates other metrics, such as the percentage of cases within specific ranges (e.g., ±0.50), mean absolute deviation, and median.
8 Hence, the consideration of SD in the comparison of IOL calculation formulas is vital for evaluating their accuracy and reliability. However, to achieve a fair and insightful comparison, statistical analyses must be carefully designed to account for the potential impact of heteroscedasticity on the results.
Several factors can contribute to errors in IOL power calculations. In comparative analysis involving normal subjects, it has been demonstrated that the measurements obtained through conventional keratometry are significantly greater than those derived from total keratometry, which uses Gaussian optics or ray-tracing methodologies to account for the anterior and posterior surfaces of the cornea.
13–15 This methodology, despite its potential overestimation of corneal power and resultant inaccuracies in IOL power calculations, continues to be widely used because of its compatibility with existing IOL constants in most formulas. Variations in the refractive index values used in different IOL power calculation formulas (e.g., Haigis: 1.3315, Holladay: 4/3) can introduce systematic errors, as can discrepancies in ocular biometry measurements, such as different techniques for estimating axial length. These variations can significantly influence the sign and magnitude of the average PE by IOL power calculation formulas, necessitating adjustments to their constants to maintain accuracy.
In a recent publication, we introduced a method for easy computation of the optimal lens constant value to correct systematic biases in formula calculations.
16 This approach helped us develop an analytical methodology to identify and predict the effects of systematic errors in keratometry, axial length measurement, keratometric index estimation, and effective lens position prediction. We investigated the impact of zeroization on the SD value and explored the influence of minimizing prediction errors originating from systematic inaccuracies in various biometric parameters.
9 We found that a systematic change in the estimation of corneal power would increase the SD after the zeroization of the predicted error. As a result, our method makes the calculation formulas more robust to variability in the estimation of corneal power related to instrumental variations.
A critical distinction in our proposed method lies in its handling of positional errors versus keratometric power estimation errors. Systematic positional errors related to the effective lens position (ELP) are addressed by adjusting a positional constant to minimize the SD of prediction errors. To nullify the non-zero mean PE after minimizing the SD, we subtracted its value from the predicted refraction values obtained through the IOL calculation formulas. In the context of formula optimization, the value of this refractive offset can be converted to the corneal plane and serve as an adjustment constant for the estimation of corneal power. This adjustment is particularly effective because it directly correlates with the primary source of systematic errors, allowing for a more precise correction that is tailored to the underlying issue. Given that the mean PE can be controlled without altering the SD, prioritizing the minimization of the SD becomes a strategic approach in formula optimization. This focus on reducing the SD ensures that the errors are not only centered around zero but are also closely clustered, indicating a high level of consistency and reliability in the predictions made by the formula.
There is no standard consensus on how to optimize target parameters or choose the appropriate norm for optimization. Different optimization criteria and target parameters can yield varied results, and there is no agreed-upon method for determining the best approach.
17 One method involves back-calculating an individual formula constant for each data point in a dataset, using the mean or median as an optimized constant. However, this approach does not ensure that the mean or median of refraction errors will be zeroed. Alternatively, our method, which avoids the use of complex algorithms, focuses on minimizing the SD and zeroing out the mean PE, which in turn results in a minimized root mean square error (RMSE) since RMSE is defined by the equation RMSE
2 = mean PE
2 + SD
2. This approach can be simpler yet effective in achieving lower RMSE values compared to using nonlinear optimization algorithms to minimize metrics like the MAE or RMSE. A notable limitation of our advanced IOL calculation formula optimization method is the need for a straightforward mechanism for identifying the constant that minimizes SD, necessitating reliance on complex algorithms. Algorithmic processes, although effective, demand considerable computational effort and a deep understanding of optimization techniques, which may not be universally available within the ophthalmic community. To address this complexity, we introduced in a recently accepted publication an innovative analytical method designed to compute constants that minimize the SD and RMS of the PE in single-optimized IOL power calculation formulas.
18 Validated against a robust dataset comprising 10,330 eyes and various IOL models, this method has proven both accurate and practical. By enabling the prioritization of SD and RMS minimization in the formula optimization process, this new approach significantly enhances the precision of IOL power calculations. The introduction of this method marks a pivotal advancement in refractive surgery, offering clinicians a tool for achieving more personalized and precise refractive outcomes. This recent contribution to the field underscores the ongoing evolution and refinement of techniques aimed at optimizing IOL calculations to meet specific clinical requirements.
Another limitation of the proposed optimization approach is the necessity of incorporating an additional constant to adjust the estimated corneal power value. This step introduces an extra layer of complexity in the optimization process, potentially complicating the application of the method across diverse clinical settings and IOL models. However, this approach also presents a unique advantage: it facilitates the comparison of the corneal power adjustment constant across different IOL models. Should these constants converge towards a common mean value, this could indicate the presence of a systematic bias in the estimation of corneal power by the utilized formula and/or biometer. Identifying such a bias is crucial, as it would offer insights into inherent inaccuracies in current measurement or calculation practices, prompting further refinement of IOL calculation formulas and potentially leading to more accurate and consistent refractive outcomes post-cataract surgery. The Castrop formula includes a constant,
R, which is specifically aimed at addressing the effects of lane distance in refraction measurement or systematic errors, impacting the calculation of the predicted spherical equivalent.
19 This adjustment could be used to zero out the mean PE after the SD is minimized using our method.
This aspect of our method not only addresses a specific optimization challenge but also contributes to the broader understanding of factors influencing IOL power calculations. This nuanced approach acknowledges the different impacts of positional and keratometric errors on prediction accuracy. By attempting to address these error sources separately with targeted adjustments, the proposed method offers a more refined optimization strategy that not only aims for zero mean error but also prioritizes reducing outcome variability, which is paramount for achieving consistent and reliable postoperative refractive results.
Our study demonstrates the potential benefits of optimizing IOL power calculation formulas to enhance precision and accuracy in postoperative refractive outcomes. However, the substantial discrepancy between the training set (4885 eyes) and the testing set (262 eyes) is an important consideration. Although the smaller testing set provides valuable insights into the real-life applicability of our methodology, a larger testing sample would enhance the reliability and generalizability of our findings. Future studies should aim to increase the size of the testing set to validate our assumptions more robustly, ensuring that the optimized constants are clinically reliable across diverse patient populations. This approach will strengthen the statistical power of our comparisons and provide more comprehensive evidence for the efficacy of the proposed optimization method.
In conclusion, this study's results are preliminary yet encouraging, pointing to the potential for extending this optimization approach to other types of IOLs and different clinical scenarios. The consistent improvements in some formulas across independent datasets affirm the robustness of the evaluated optimization method and highlight its potential for broader applicative value in ophthalmic practice. Further research is warranted to explore the application of this optimization strategy to other IOL types. Such investigations could provide deeper insights into the generalizability of the optimization techniques and their effectiveness across a wider range of IOL power calculation formulas. Additionally, studies involving larger and more diverse patient populations could help to validate these findings and refine the optimization process to enhance postoperative refractive outcomes universally.
Disclosure: D. Gatinel, None; G. Debellemanière, None; A. Saad, None; L.F. Brenner, None; M. Gauvin, None; A. Wallerstein, None; J. Malet, None