Impact of Single Constant Optimization on the Precision of IOL Power Calculation

Abstract

Purpose: The primary objective of this research is to examine how precision in intraocular lens calculation formulas can be impacted by zeroing the mean error through adjustments in the effective lens position value. Additionally, the study aims to evaluate how this modification influences outcomes differently based on the source of the prediction error.

Methods: In order to analyze the impact of individual variables on the standard deviation, the study maintained all variables constant except for one at a time. Subsequently, variations were introduced to specific parameters, such as corneal curvature radius, keratometric refractive index, axial length, and predicted implant position.

Results: According to our findings, when zeroing the mean error is applied to correct for inaccuracies in corneal power estimation, it results in a significant and exponential rise in standard deviation, thus adversely affecting the formula's precision. However, when zeroing is employed to compensate for prediction errors stemming from axial length measurements or predicted implant position, the effect on precision is minimal or potentially beneficial.

Conclusions: The study highlights the potential risks associated with the indiscriminate but necessary zeroing of prediction errors in implant power calculation formulas. The impact on formula precision greatly depends on the source of the error, underscoring the importance of error source when analyzing variations in the standard deviation of the prediction error after zeroing.

Translational Relevance: Our study contributes to the ongoing effort to enhance the accuracy and reliability of these formulas, thereby improving the surgical outcomes for cataract patients.

Introduction

One of the key determinants of postoperative satisfaction after cataract surgery is the accurate calculation of intraocular lens (IOL) power to induce the desired postoperative refraction. To achieve optimal results, these calculations must be both accurate and precise. A more accurate calculation method will have a mean prediction error, defined as the difference between achieved and predicted refraction, close to zero. This concept is the driving factor behind the process known as “zeroization,” in which adjustment mechanisms are used to bring the mean prediction error as close to zero as possible. “Precision,” on the other hand, refers to the consistency of these calculations. Precision is quantified as the standard deviation (SD) of the prediction error. A smaller SD indicates a more precise calculation method, suggesting that most prediction errors are clustered closely around the mean error. 

Table.

 

Demographic and Biometric Parameters of Group 1 (Eyes With a PE > 0.05 D, n = 4042) and Group 2 (Eyes With a PE < 0.05 D, n = 541)

 

These metrics are commonly used in clinical studies and research to evaluate the performance of different IOL power formulas and assess their accuracy and precision in predicting the appropriate lens power for cataract surgery.^1,2 

The initial step of such analysis involves setting the arithmetic mean of the prediction error (PE) to zero for each formula. This is achieved by adjusting the lens constant. The sources of error in implant power calculation are multiple and may increase in some contexts. For an accurate calculation of corneal power, it is necessary to know both the anterior and posterior corneal radii. Traditional keratometry and topography, however, estimate the corneal refractive power from a single measurement of the anterior radius, using a keratometric index value assuming a fixed ratio between anterior and posterior corneal radii. The calculation of corneal power, or keratometry (K) values, has been determined for over a hundred years using only measurements from the anterior corneal surface. There have been assumptions made regarding the posterior corneal curvature in these calculations. The anterior radius of curvature, quantified in millimeters, is transformed into the entire cornea's dioptric power using an assumed refractive index. This is commonly known as the keratometric refractive index (n_k), typically set at 1.3375, also referred to as the Javal Index. This method of computing corneal power has found utility in various applications, including IOL power estimations. The continued usage of the Javal Index is primarily due to its compatibility with established IOL constants in most calculation formulas. However, the Javal Index may overestimate the corneal power, resulting in inaccuracies in the calculated IOL power.^3–8 Some IOL power calculation formulas use a specific value for the refractive index (Haigis: 1.3315, Holladay: 4/3). Similarly, systematic errors can arise when a new or different device or technique is used for ocular biometry measures, such as estimating the axial length using the group refractive index or sum of segments to optimize axial length (AL).⁹ Any of these variations lead to potential errors contributing to the sign and magnitude of the average prediction error by IOL power calculation formulas, whose constants may need to be adjusted to restore their accuracy. 

Holladay et al.¹⁰ provided convincing arguments that PE's SD of the prediction error is the single most accurate measure of outcomes. This metric provides an accurate assessment and forecasts other measures like the percentage of cases within a given range (e.g., ±0.50), the mean absolute deviation, and the median. The computation of the SD is performed after constant optimization to cancel any systematic bias.¹¹ The SD of the error is based on the distance of each data point from the mean, not the actual value of the mean. Hence, the impact of the zeroization of the mean prediction error on the SD of the error is not predictable without additional information because the SD depends on the individual values of the error, not the mean of the error. 

In this study, we aimed to assess the impact of zeroization on the SD value. To the best of our knowledge, no previous studies have delved into the effects of reducing prediction errors, which arise from systematic inaccuracies in various biometric parameters. In a recent publication, we proposed a method allowing easy computation of the optimum lens constant value, correcting the systematic formula calculation bias.¹² From this work, we derived analytical methodology from identifying relationships and predicting the impact of systematic errors on keratometry and AL measurement, the keratometric index estimation, and the effective implant position prediction. We focused on how these factors influenced the change in SD following the zeroization of the predicted error. 

Predicting the postoperative spherical equivalent relies on certain inputs, including preoperative ocular biometry data and the power of the implanted lens. In this study, we are conducting a series of theoretical simulations designed to assess the predictive accuracy of an IOL power calculating formula (Haigis single-optimized), having demonstrated a near-zero prediction error (<0.05 diopters [D]) in a preselected group of eyes. We aim to rerun the predictions by intentionally modifying one of the input variables by a consistent positive or negative increment. This simulation is intended to mimic a scenario where the data acquisition method for this variable is altered, such as introducing a new measurement instrument that might yield different values from the same eyes. For instance, we might adjust the anterior corneal curvature radius value, uniformly adding or subtracting a particular amount from all radii. 

In doing so, the formula will predict a new postoperative refractive value, which will differ from the original prediction. This difference represents the theoretical prediction error induced by the inaccuracies in the altered biometric variable. Subsequently, we explore the impact of nullifying the prediction error on the SD of this error by adjusting the value of the optimization constant designed to offset the mean prediction error. 

Material and Methods

Metrics of IOL Power Formula Accuracy and Precision

For a given eye, the prediction error E_i of an implant power calculation formula is equal to the difference between the achieved and predicted spherical equivalent:  

\begin{eqnarray}{E_i} = S{E_a} - S{E_p}\end{eqnarray}

(1)

 

The mean prediction error of a formula on a set of N eyes is equal to  

\begin{eqnarray}\bar{E} = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {E_i}\end{eqnarray}

(2)

 

The SD of the error (σ) is the square root of the error variance. It is given by  

\begin{eqnarray}\sigma = \sqrt {\frac{1}{N}\mathop \sum \limits_{i = 1}^N {{\left( {{E_i} - \bar{E}} \right)}^2}} \end{eqnarray}

(3)

 

Zeroization of the Mean Prediction Error

When (\(\mathop \sum \nolimits_{i = 1}^N {E_i} \ne 0\)), the lens constant can be adjusted to achieve the zeroization of the mean PE. This method leads to determining the value of an increment, added to each IOL's predicted effective lens position (ELP) (ΔELP) so that the refractive change produced (ΔR_i) is such that  

\begin{eqnarray}\mathop \sum \limits_{i = 1}^N \left( {{E_i} + \Delta {R_i}} \right) = 0\end{eqnarray}

(4)

 

Consequences of Zeroization on the SD of the PE

Using previous work, the refractive change (ΔR_i) incurred by an incremental change in ELP (ΔELP) can be estimated using a differential calculation and a thick lens pseudophakic eye model:  

\begin{eqnarray}\Delta {R_i} \approx - 0.0006\ \left( {{P_i}^2 + 2{K_i}{P_i}} \right)\ \times \ \Delta ELP\end{eqnarray}

(5)

where P_i is the IOL power, and K_i is the corneal power of a given eye of the data set. 

The relation between the mean PE and the increment (ΔELP₀) necessary to cancel the systematic bias \(\bar{E}\) is given by  

\begin{eqnarray}\Delta EL{P_0} = \frac{{\bar{E}}}{{0.0006\ \overline {\left( {{P^2} + 2KP} \right){\rm{\ }}} }}\end{eqnarray}

(6)

where \(\overline {( {{P^2} + 2KP} )} \) is the arithmetic mean of (P_i² + 2K_iP_i) over the N eyes of the data set. 

Since,  

\begin{eqnarray}\bar{E} \approx 0.0006\ \overline {\left( {{P^2} + 2KP} \right){\rm{\ }}} \times \Delta EL{P_0}\end{eqnarray}

(7)

 

the SD σ of the PE is given by  

\begin{eqnarray}{\sigma ^2} &=& \frac{1}{N}\mathop \sum \limits_{i = 1}^N \Bigg( {E_i} - 0.0006\left( {{P_i}^2 + 2{K_i}{P_i}} \right) \nonumber \\ && \times \,\frac{{\bar{E}}}{{\overline {0.0006\left( {P + 2KP} \right)} \ }}\Bigg)^2\end{eqnarray}

(8)

 

This simplifies to  

\begin{eqnarray}{\sigma ^2} = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {\left( {{E_i} - \bar{E}\frac{{\left( {{P_i}^2 + 2{K_i}{P_i}} \right)}}{{\overline {{P^2} + 2KP} \ }}} \right)^2}\ \end{eqnarray}

(9)

which can be rewritten as  

\begin{eqnarray}{\sigma ^2} = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {\left( {\left( {{E_i} - \bar{E}} \right) + \bar{E}\ \left( {1 - \ \frac{{\left( {{P_i}^2 + 2{K_i}{P_i}} \right)}}{{\overline {{P^2} + 2KP} \ }}} \right)} \right)^2}\ \nonumber \\ \end{eqnarray}

(10)

 

After expanding, this expression is equivalent to σ² = (A + B + C), where  

\begin{eqnarray}A = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {\left( {{E_i} - \bar{E}} \right)^2}\end{eqnarray}

(11)

is the variance of the prediction error before optimization, 

\begin{eqnarray} B = \frac{1}{N}{\left( {\frac{{\bar{E}}}{{\overline {{P^2} \,{+}\, 2KP} }}} \right)^2}\! \mathop \sum \limits_{i = 1}^N {\left( {\left( {{P_i}^2 \,{+}\, 2{K_i}{P_i}} \right) {-} \overline {{P^2} \,{+}\, 2KP\ } } \right)^2} \nonumber \\ \end{eqnarray}

(12)

is proportional to \(\overline{E } \) and the variance of (P_i² + 2K_iP_i), and  

\begin{eqnarray} && C = - \frac{2}{N}\left( {\frac{{\bar{E}}}{{\overline {{P^2} + 2KP} }}} \right) \nonumber \\ \mathop \sum \limits_{i = 1}^N {E_i}\left( {( {{P_i}^2 + 2{K_i}{P_i}} ) - \overline {{P^2} + 2KP\ } } \right)\end{eqnarray}

(13)

is proportional to the covariance between E_i and (P_i² + 2K_iP_i). 

Impact on the SD of Zeroization of the Mean Prediction Error

General Setting

We analyzed a data set containing measurements from the eyes of a cataractous population. The study adhered to the tenets of the Declaration of Helsinki and was approved by the Ethics Review Board of the Canadian Ophthalmic Research Center. The data were anonymized and contained preoperative biometric data derived with the Lenstar 900 equipped with the EyeSuite software i8.2.2.0 (Haag Streit, Koeniz, Switzerland), including AL, anterior chamber depth (from front corneal apex to the front apex of the crystalline lens), lens thickness, corneal thickness, corneal diameter, and the front corneal surface flat (R1) and steep (R2) radii. The same IOL was inserted in all eyes (Finevision Micro F; BVI Medical, Waltham, MA, USA). Postoperative refraction was measured 4 to 6 weeks after cataract surgery by an experienced optometrist and recorded in the data set, which only included data with a postoperative Snellen decimal visual acuity of at least 0.8 (20/25) Snellen lines. 

Selection of Eyes with Negligible Prediction Error

We selected eyes from the data set for which the prediction error made by single optimized Haigis formula¹³ was less than 0.05 D. This ensures that the mean prediction error and its SD are negligible at baseline. This configuration is assumed to correspond to the situation obtained thanks to a “perfect” theoretical calculation formula (negligible prediction error) postoperatively. In such a situation, a systematic change in any of the numerical values of the variables or indices used by the formula would alter the value of the predicted refraction of each eye. This variation in predicted refraction can be likened to a prediction error caused by an inaccurate estimation or measurement of one of these variables and enables computation of its specific impact on the formula's accuracy and precision before and after zeroization. 

Theoretical Influence of the Variation of Biometric Measurements and Estimators on the Mean Prediction Error

Simulations were carried out by altering the value of four variables used in predicting the postoperative spherical equivalent by various step increments. Two of these variables, the cornea's anterior radius of curvature (ARC) and the AL, are computed preoperatively by the Lenstar 900, using various methods and assumptions. The simulations emulate a theoretical scenario in which a flawlessly accurate formula is applied to the same set of eyes now measured with a new biometric device. This new instrument introduces systematic measurement deviations in variables such as keratometry or AL compared to the original device for which the formula had been optimized. Thus, our study investigates the theoretical impact of such deviations on the formula's predictive accuracy, assuming a shift from the baseline measurement conditions. Increasing the R value by a positive increment is analogous to a situation where a new biometer would overestimate the anterior corneal radius. The formula's underestimation of corneal power leads to a positive prediction error, resulting in a more positive or less negative refraction being predicted. Decreasing the AL by a constant increment (underestimation of the AL) can also cause the formula to generate a positive prediction error. Note that in the Haigis formula, the value of the AL affects the predicted value of the ELP. 

In the present study, two estimated variables—namely, the keratometric index (n_k) and the effective lens position (ELP)—are examined. Systematic variations in the ELP are performed to investigate the influence of lens design changes, such as a constant shift in haptic position relative to the IOL's optic. Additionally, by introducing incremental adjustments to the initial keratometric index (n_k = 1.3315) utilized in the Haigis formula, prediction errors are generated for each alteration. This approach allows us to evaluate the theoretical impact of systematic shifts on the accuracy of the implant power prediction formula and gain valuable insights into the precision of the formula concerning changes in this particular variable, which directly affects corneal power estimation. 

Estimation of the Variation of the Haigis Formula's Mean Error and SD Before and After Zeroization

Equation (8) was used to compute the offset to add to the a0 constant of the Haigis formula to zero out the mean prediction error. Using Equations (1), (2), and (3), we can estimate the theoretical impact of the departure of these values on the prediction errors, their mean and SD before and after zeroization. Using Equations (11), (12), and (13) enables us to isolate the contribution of the variance of (P_i² + 2K_iP_i) and its covariance with (E_i) to the postoptimization total variance of the PE. 

Results

General Settings

A total of 4583 pseudophakic eyes, all implanted with the Finevision IOL, were split in two groups according to the value of the prediction error: group 1 comprised the eyes with an absolute value of the PE > 0.05 D (n = 4042), and group 2 comprised the eyes with an absolute value of the PE < 0.05 D (n = 541). The Mann–Whitney U test was used to compare the distributions of the main variables of interest. A P value of less than 0.05 indicated a statistically significant difference between the groups (see Table). 

Theoretical Influence of the Variation of Biometric Measurements and Estimators on the Mean and SD of Prediction Error Before and After Zeroization

The values of the mean prediction errors before and after zeroization and the offset and constant a0 values are presented in the appendix for each of the performed simulations (Supplementary Table S1). 

To explore the influence of the studied variables on the SD, we analyzed the variations in the total variance, which was decomposed into three terms (A + B + C), both before and after zeroization. The results are presented in the form of graphs. 

Systematic ARC Increment

The anterior corneal radius of curvature was adjusted in increments ranging between −0.05 mm and +0.05 mm, with changes made in steps of ±0.01 mm. 

The mean prediction error displayed a linear increase corresponding to the increment value added to the variables of interest variation, where negative increments led to more negative values, and positive increments led to more positive values. However, in contrast, the SD exhibited an exponential increase with these increments (Fig. 1A). 

Figure 1.

 

Impact of anterior corneal radius misestimation. (A) Mean PE and SD values before optimization. (B) The anticipated SD versus the actual measurement, and the total variance in contrast to the broken-down variance of the average PE following optimization (A + B + C). A.O., after optimization; B.O., before optimization.

Neutralizing the average PE leads to an exponential expansion of the SD, which is directly proportional to the absolute value of the increment. This growth is driven by the value of B, while the term C remains negative but is almost zero within the tested range (see Fig. 1B). 

Systematic Keratometric Index Value Increment

The value of n_k was adjusted in increments that ranged from −0.005 to +0.005, with each step involving a change of ±0.001. 

The average prediction error displayed a linear increase corresponding to the increment added to the variation in the refractive index value, where positive increments led to more positive values, and negative increments led to more negative values. The increase in the SD was exponential with the absolute value of the increments (Fig. 2). The zeroization of the mean PE causes an increase in the SD that grows exponentially with the value of the increment. This increase follows the value of B, with the term C being negative but very close to zero on the interval tested (Fig. 2B). 

Figure 2.

 

Impact of keratometric index misestimation. (A) Mean PE and SD values before optimization. (B) The anticipated SD versus the actual measurement, and the total variance in contrast to the broken-down variance of the average PE following optimization (A + B + C).

Systematic ELP Increment

The value of the ELP was adjusted in increments ranging from −0.5 mm to +0.5 mm, with each step involving a change of 0.1 mm. The mean PE increased linearly with the ELP variation (toward more negative versus positive values with negative versus positive increments). The SD values increased exponentially with the absolute value of the increments (Fig. 3A). 

Figure 3.

 

Impact of ELP misestimation. (A) Mean PE and SD values before optimization. (B) The anticipated SD versus the actual measurement and the total variance in contrast to the broken-down variance of the average PE following optimization (A + B + C).

Neutralizing the average PE leads to a decrease in the SD, which reverts back to its original value (prior to the increment). This reduction stems from the negative value of the C term, which in absolute terms grows more rapidly than the B term (Fig. 3B). 

Systematic AL Increment

The value of the AL was altered between −0.05 mm and +0.05 mm by 0.01-mm increments. 

The mean PE increased linearly with the value of the increment added to the refractive index value variation, where positive increments led to more positive values, and negative increments led to more negative values. In contrast, the increase in the SD was exponential with the absolute value of the increment (Fig. 4A). 

Figure 4.

 

Impact of AL misestimation. (A) Mean PE and SD values before optimization. (B) The anticipated SD versus the actual measurement and the total variance in contrast to the broken-down variance of the average PE following optimization (A + B + C).

Neutralizing the average PE does not result in a substantial shift in the SD. The persistence of the SD can be attributed to the contrasting values of terms B and C (see Fig. 4B). 

Discussion

This research study is primarily concerned with the evaluation and comparison of the precision of IOL power calculation formulas. In the framework of this investigation, we simulate an ideal scenario where a formula provides a perfect result (i.e., a postoperative refractive error less than 0.05 D). We then deliberately modify certain parameters like the corneal curvature radius, the keratometric refractive index, the AL, and the predicted value of the implant position to analyze the generated prediction error before and after zeroing. This is distinctly different from the real-world application of these formulas where the primary goal is to predict the power of an implant that should be used to achieve a targeted postoperative refraction. This study does not directly aim to improve the prediction of the implant power in real-life surgical scenarios but rather seeks to provide a more objective way of comparing the potential performance of different power calculation formulas. 

We evaluated how isolated changes in certain variables implicated in biometric computation influence the SD of the mean prediction error, both before and after neutralization. We derived formulas that anticipate the trajectory of the SD after zeroization. We found noticeable discrepancies depending on which variable was contributing to the prediction error. Variables involved in determining corneal power (ARC, n_k) cause a minor increase in the SD prior to neutralization, yet this grows significantly after zeroization. Conversely, positional variables (ELP, AL) show a reversed trend. 

The implementation of a systematic error on ELP leads to a substantial surge in the SD, which is neutralized subsequently. This outcome aligns with expectations, as the zeroization process effectively compensates for the systematic error through a counterbalancing offset. In a previous work, we investigated the theoretical impact of variations in the Coddington factor, which describes the bending of the lens, on the accuracy and precision of the IOL power calculation.¹⁴ These design variations cause a displacement of the principal planes of the implant and are therefore analogous to an ELP prediction error. We found that zeroization was an important step to reduce the impact of optical design changes, as it induced a marked reduction in the value of the SD. Our results show that the beneficial effect of zeroization cannot, however, be generalized to all sources of prediction errors. Introducing a systematic error of the AL produces an increase in the SD, which is minimally influenced by the zeroization of the mean prediction error. 

Equations (12) and (13) tell us about the parameters that influence the evolution of SD after zeroization. The value of the term B is always positive. It increases with the square of the mean prediction error and the variance of (P_i² + 2K_iP_i). The term C appears to calculate a form of covariance between (E_i) and (P_i² + 2K_iP_i). Covariance measures how two variables move together—if they tend to increase and decrease together, the covariance is positive, and if one tends to increase while the other decreases, the covariance is negative. So if mean PE is positive, and P and K are always positive, the covariance could be positive if higher values of (E_i) often correspond to higher values of (P_i² + 2K_iP_i) and negative if lower values of (E_i) often correspond to lower values of (P_i² + 2K_iP_i). 

The sign of C can theoretically be positive or negative, depending on whether the covariance between (E_i) and (P_i² + 2K_iP_i). has the same sign as the mean error (C < 0) or not (C > 0). In our simulations, the sign of C was negative for the variables tested. This is explained by the impact of the systematic change in the variables on the sign of the mean prediction error and that of the covariance between (E_i) and (P_i² + 2K_iP_i). For the variables studied, the greater the covariance between (E_i) and (P_i² + 2K_iP_i), the more negative the value of C, which reduces the impact of the increase in variance related to B. 

In our study, we discovered that the evolution of the SD after zeroing depends on the sign and the magnitude of the covariance between the prediction error, referred to as E_i, and (P_i² + 2K_iP_i), where P represents the implant power and K the corneal power. The SD of (P_i² + 2K_iP_i) remains invariant for a given data set. Despite this, it is challenging to link the covariance to the correlation between (E_i) and (P_i² + 2K_iP_i) as the variation of the SD of (E_i) cannot be easily anticipated. While we can generally anticipate some correlation between covariance and the correlation coefficient, it is essential to understand that this is not a direct relationship. Covariance measures the degree to which two variables move together, but it does not account for the individual variances of the variables as the correlation coefficient does. Thus, although both statistical measures describe the relationship between two variables, they are not directly interchangeable, and any presumptions between them should be made with a comprehensive understanding of their unique characteristics and limitations. 

Figure 5 shows the intensity relationship between the (E_i) and (P_i² + 2K_iP_i) for the eyes of our data set as a function of the variable causing the error. The sign of the covariance and the intensity of the correlation between these variables illustrate the observed trends. 

Figure 5.

 

(A) Overestimating the ELP results in a more significant negative prediction error, especially as the lens power (P) is larger (equation = FM). The mean error and the covariance between (E_i) and (P_i² + 2K_iP_i) share the same negative sign, leading to a negative covariance (C < 0). (B) In our simulations, when the ARC is overestimated, the value of P² + 2KP diminishes due to the inverse relationship between R and the corneal power K. Consequently, Haigis's formula predicts a more hyperopic refraction, resulting in a negative prediction error. Both the prediction error E and (P² + 2KP) have the same negative sign, leading to a negative covariance (C < 0). (C) When the refractive index of the cornea (n_k) is overestimated, the value of P² + 2KP rises due to the corresponding increase in K. As a result, the formula predicts a more myopic refraction. The mean error and the covariance between (E_i) and (P_i² + 2K_iP_i) bear the same positive sign, implying that C is negative (C < 0). (D) Overestimating the AL generates a prediction error that becomes more positive as P² + 2KP increases. The mean error and covariance share the same positive sign, resulting in a negative covariance (C < 0).

In cases where a constant prediction deviation of ELP escalates the total prediction error and its SD, it is unsurprising that our model predicts a return to the initial values upon zeroing. The compensation for a persistent AL error through an ELP offset does not substantially impact the SD. On the other hand, the zeroing of an average error resulting from a corneal power measurement mistake triggers a significant and exponential rise in the SD. It is important to note that adjusting for an optical power error through positional modifications can impair the formula's accuracy. This highlights the need for precision in determining corneal power, as the consequential error, if nullified, can considerably amplify the SD. Unlike a systematic measurement error of AL, the accuracy of the measurement of ARC is fundamental because it cannot be neutralized by the adjustment without affecting the precision of the formula. The importance of an accurate estimation of corneal power is particularly critical for eyes that have undergone corneal refractive surgery (CRS). Our results suggest that zeroing a systematic error related to the measurement of corneal power can significantly impact the precision of the formula. This impact is even more pronounced as the average prediction error is significant (Equation (12)). A comprehensive systematic review conducted by Wang and Koch¹⁵ in 2021 revealed that frequently used IOL formulas in contemporary clinical practice were unable to consistently generate precise postoperative refractive outcomes in eyes with a history of CRS. With the majority of formulas accessible on the American Society of Cataract and Refractive Surgery (ASCRS) IOL calculator, less than 70% of the eyes achieved refractive prediction errors within ±0.5 D. Recently, advanced technologies and methods such as total keratometry, ray tracing, intraoperative aberrometry, and machine learning–based algorithms have shown substantial potential in enhancing IOL power computations in eyes previously subjected to corneal refractive surgery. This demonstrated improvement is particularly evident when compared to traditional vergence-based formulas like SRK/T, Haigis, Holladay, Hoffer Q, Potvin–Hill, and Barrett.^16–19 These conventional methods, which estimate corneal power based on the anterior corneal radius and keratometric index, appear less effective in comparison. In our study, we strategically chose to keep all variables constant, with the exception of the one. This strategy was specifically devised to facilitate the examination of its impact on the standard error, considering both variance and bias components. This approach is especially beneficial when dealing with intricate models such as pseudophakic eyes with thick lenses, where numerous variables interact. By simplifying the model this way, we aimed to isolate the influence of some variables of interest. Understanding the effects of individual variables can yield valuable insights before introducing further complexity. The approach outlined in our research provides a solid foundation for future studies. A subsequent study could harness the power of computational tools like Monte Carlo simulations. This technique, known for its robustness in addressing complex stochastic systems, would be especially valuable in exploring the intertwined effects of multiple variables. It would enable us to examine the interactive behavior of the variables in a probabilistic framework, providing a more nuanced understanding of their impact on the outcome. A recent study used a Monte Carlo simulation approach to analyze a large data set of preoperative IOLMaster 700 (Jena, Germany) biometric measurements, consisting of 16,667 observations, to assess the uncertainty in the predicted refractive outcome (REFU) following cataract surgery.²⁰ This uncertainty was evaluated in light of measurement uncertainties in modern optical biometers. The study employed the Haigis and Castrop formulas, using error propagation strategies to derive REFU. Results showed that without considering lens power labeling tolerances, the median REFU was 0.10 dpt for the Haigis formula and 0.12 dpt for the Castrop formula. The REFU systematically increased for short eyes or high-power IOLs. The largest contributor to REFU, particularly in long eyes and low-power IOLs, was the uncertainty in measuring the corneal front surface radius when not considering lens power labeling tolerances. The study concluded that the uncertainty of biometric measures contributes to about one-third to one-half of the SD compared with the published data on the formula prediction error of refractive outcome after cataract surgery. 

The increase in the SD of the prediction error forecasted in our simulations, both before and after zeroization, is of an order of magnitude lower than what is typically observed in a clinical setting when evaluating the performance of implant power calculation formulas. This suggests that the elevation in SD likely stems from the cumulative effect of various potential sources of prediction error. Each of these factors, alone or in combination, can influence the outcome, underscoring the complexity of accurate prediction in the clinical context. Therefore, a comprehensive understanding of these variables and their interactions is essential in enhancing the precision of implant power prediction formulas. 

While it is acknowledged that in real-world clinical scenarios, the effects of individual variables may be blurred due to the influence of other variables affecting refractive outcomes, our theoretical model aims to specifically isolate those aspects related to optical design under paraxial conditions and assess the impact of optimization. We can infer that the weaker the covariance between the prediction error (E_i) and (P_i² + 2K_iP_i), the more zeroization is likely to degrade the formula's accuracy through an increase in SD. Still, Equations (12) and (13) can be used in a real clinical situation to estimate the impact of zeroization on the formula's precision. Calculating the covariance between the prediction error and the IOL power as well as comparison of the relative values of B and C could provide interesting insight into the origin of the variables causing the prediction error. In essence, the prediction of a marked increase of the SD after zeroization suggests that the source of the prediction errors could be mainly caused by inaccurate estimation of the corneal power or any other variable causing prediction errors that are not correlated with (P_i² + 2K_iP_i). Several studies suggest that the value of the keratometric index (n_k) used by the formula for computing the IOL power is overestimated, leading to an overestimation of corneal power.^3–8 Compensation for all or part of this error can be accomplished by adjusting constants, but our results suggest that this comes at the cost of degradation in the precision of the formula. This trend is likely to increase for eyes undergoing corneal refractive surgery, for which the correct estimation of corneal power raises a challenge. 

The findings of our study hold significant relevance in the context of comparing IOL implant calculation formulas. Specifically, the zeroing technique, which involves optimizing a single constant linked to the predicted implant position, has the potential to influence the precision of a particular formula. This is particularly true if the optimization addresses adjustments required for a new measuring device, which estimates corneal power differently from the instrument on which the formula was initially optimized. Consequently, studies focusing on specific subgroups like long or short eyes should be cautiously interpreted. The average error and variance of the term (P_i² + 2K_iP_i) could exhibit substantial fluctuations within a specific subgroup, and this variation could potentially distort the findings and interpretations. Therefore, a thorough comprehension of these intricacies is essential for accurately evaluating the performance and practicality of various implant power calculation formulas. 

In conclusion, our findings underscore a key insight: the zeroing or neutralization of the mean error in IOL implant calculation formulas can potentially compromise their precision. This effect is especially pronounced when the genesis of prediction errors is tied to an imprecise estimation of corneal power. The imprecision in corneal power estimation can result in substantial prediction errors, and when these errors are zeroed, the precision of implant calculation formulas may significantly decrease. Conversely, compensating for prediction errors arising from measurements related to AL or predicted implant position seems to have little impact or even a positive effect on the precision of the formulas. This suggests that the source of the error plays a critical role in how its adjustment would affect the formula's precision, especially for the formulas that would mainly compensate for a systematic keratometry error through an adjustment of their zeroization constant. 

These observations hold important implications for the design and optimization of implant power calculation formulas. They underline the necessity to distinguish between different sources of error and their respective impacts on formula precision, thus providing valuable guidance for future improvements in this field. Ultimately, our study contributes to the ongoing effort to enhance the accuracy and reliability of these formulas, thereby improving the surgical outcomes for cataract patients. 

Acknowledgments

Disclosure: D. Gatinel, BVI Medical (C); G. Debellemanière, None; A. Saad, None; A. Wallerstein, None; M. Gauvin, None; R. Rampat, None; J. Malet, None