A New Method to Minimize the Standard Deviation and Root Mean Square of the Prediction Error of Single-Optimized IOL Power Formulas

Abstract

Purpose: The purpose of this study was to develop a simplified method to approximate constants minimizing the standard deviation (SD) and the root mean square (RMS) of the prediction error in single-optimized intraocular lens (IOL) power calculation formulas.

Methods: The study introduces analytical formulas to determine the optimal constant value for minimizing SD and RMS in single-optimized IOL power calculation formulas. These formulas were tested against various datasets containing biometric measurements from cataractous populations and included 10,330 eyes and 4 different IOL models. The study evaluated the effectiveness of the proposed method by comparing the outcomes with those obtained using traditional reference methods.

Results: In optimizing IOL constants, minor differences between reference and estimated A-constants were found, with the maximum deviation at −0.086 (SD, SRK/T, and Vivinex) and −0.003 (RMS, PEARL DGS, and Vivinex). The largest discrepancy for third-generation formulas was −0.027 mm (SD, Haigis, and Vivinex) and 0.002 mm (RMS, Hoffer Q, and PCB00/SN60WF). Maximum RMS differences were −0.021 and +0.021, both involving Hoffer Q. Post-minimization, the largest mean prediction error was 0.726 diopters (D; SD) and 0.043 D (RMS), with the highest SD and RMS after adjustments at 0.529 D and 0.875 D, respectively, indicating effective minimization strategies.

Conclusions: The study simplifies the process of minimizing SD and RMS in single-optimized IOL power predictions, offering a valuable tool for clinicians. However, it also underscores the complexity of achieving balanced optimization and suggests the need for further research in this area.

Translational Relevance: The study presents a novel, clinically practical approach for optimizing IOL power calculations.

Introduction

In cataract surgery, the determination of intraocular lens (IOL) power is primarily based on theoretical-optical equations used worldwide.^1–3 These equations require pre-surgical biometric measurements of the patient's eye. Additionally, they incorporate specific constants, designed to adapt the standard equation to the unique properties of the selected IOL.^4–6 These properties include the design of the optics and haptics, as well as the material composition. The interaction between these constants and variables, such as the effective lens position (ELP) or the IOL power, varies depending on the specific philosophy underlying each IOL power calculation formula.^7,8 Fundamentally, once a substantial amount of relevant clinical data has been gathered with a specific lens type, it is possible to conduct a post hoc constant optimization. The outcome of this optimization procedure can then be utilized to enhance the accuracy of future cataract surgeries through forward prediction. Theoretically, one can optimize for the mean, the mean absolute, the median, or the root mean square (RMS) of the prediction error in terms of the deviation of the achieved refraction after cataract surgery from the formula predicted refraction.^9–11 

Recently, we showed a concept to estimate the overall shift in ELP of a study population necessary to zero the mean of refractive prediction error (mean PE).¹² In this approach, the calculation necessitates solely the utilization of the mean keratometric power and the mean power of the intraocular lens post-implantation to ascertain the variation in the ELP. To calibrate the formula constant to achieve zero mean PE, it is imperative to examine the transformation process from an alteration in ELP to a corresponding change in the formula constant. This transformation can be deduced directly from the formula's definition, provided that the conceptual framework underpinning the IOL power computation is fully elucidated. In this previously published work, the specific goal was restricted to nullifying the mean prediction error (MPE). With this process called “zeroization,” the mechanisms for adjustment are used to minimize the average prediction error to nearly zero. Conversely, “precision” denotes the reliability of these estimations. It is measured by the standard deviation (SD) of the prediction error, with a lower SD signifying a more accurate calculation approach. This implies that the majority of prediction errors are tightly grouped near the average error. 

Building upon the framework of our preliminary research, which focused on reducing the mean PE, the current study aims to broaden the scope of the model. Our primary goal is to discover the optimal constant that can reduce both the SD and the RMS of the prediction error linked to the IOL power formula. Furthermore, this research will explore the inter-relationships between optimizing these metrics. In addition to these objectives, we will also conduct an empirical evaluation of the theoretical estimations of lens constant adjustments using observational data. This aspect will enable us to validate the practical applicability of our theoretical findings within real-world clinical settings. 

Materials and Methods

Before advancing to the derivation of formulas aimed at optimizing constants to minimize SD and RMS, it is essential to define the underpinning metrics of accuracy and precision. 

Standard Metrics for Accuracy and Precision

In the context of evaluating an IOL power calculation formula for a specific eye, the prediction error E_i represents the discrepancy between the achieved spherical equivalent (SE_a) and the predicted spherical equivalent (SE_p) calculated using the actual IOL power implanted:  

\begin{eqnarray} {E} = S{E_a} - S{E_p}\quad \end{eqnarray}

(1)

 

The mean prediction error of a formula on a set of N eyes, denoted \({\rm{\bar E}}\) is equal to:  

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

(2)

 

The SD of the error (σ) is given by the square root of the variance of the prediction error calculated as follows:  

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

(3)

 

The RMS of the error is equal to:  

\begin{eqnarray} RM{S^2} = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {\left( {{E_i}} \right)^2}\quad \end{eqnarray}

(4)

 

When the sum of all prediction errors for N eyes is non-zero (\(\mathop \sum \nolimits_{i = 1}^N {E_i} \ne 0),\) the lens constant can be adjusted to achieve the zeroization of the mean PE. This adjustment involves determining an increment (ΔELP) to be added to each predicted ELP, resulting in a refractive changes (ΔR_i) that satisfies:  

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

(5)

 

The ability to correlate positional adjustments of the lens with subsequent refractive changes is paramount. This relationship underpins the optimization process, where adjustments in the ELP are strategically implemented to achieve precise refractive outcomes. Assuming a certain relationship between certain parameters and the refractive change (ΔR) incurred by an incremental change in ELP (ΔELP), one can edit the following equation:  

\begin{eqnarray} \Delta {R_i} \approx {F_i} \times \;\Delta ELP\quad \end{eqnarray}

(6)

Where ΔELP is expressed in mm and F_i is the discrete value of function F. Crucially, F, representing a function that could embody any form, must be accurately identified or determined to facilitate this correlation between the ΔELP and the resultant refractive variation (ΔR) in a given eye. 

To elucidate this relationship, we utilized a paraxial thick pseudophakic eye model, which allowed for an in-depth investigation into how axial displacement of an IOL affects refraction, contingent upon the biometric characteristics of the eye in question. This model provides a comprehensive framework for analyzing the impact of IOL axial movements, offering insights into the dynamic interplay between ELP adjustments and their refractive consequences. In a previous work,¹² the F function was estimated through differential calculation methods, using a thick lens pseudophakic eye model:  

\begin{eqnarray} {F_i} = 0.0006\left( {{P_i}^2 + 2{K_i}{P_i}} \right)\quad \end{eqnarray}

(7)

where K_i and P_i, respectively, denote the corneal power and the implant power of the eye under consideration. 

This formula allows for the precise calculation of refractive changes resulting from incremental positional adjustments of the IOL within an individual eye. Once F_i is determined, it enables us to compute the impact of these positional changes on refraction using Equation 6. Given the analytical expression of F_i, where the IOL power P_i is raised to the square, it becomes evident that P_i plays a significant role in dictating the magnitude of refractive change resulting from a specific positional adjustment of the IOL. 

Formulas for Metric Minimization

Minimizing the Mean Prediction Error

Previous computation (12) led to the following approximation:  

\begin{eqnarray} \Delta EL{P_0} \approx - \frac{{\bar E}}{{\bar F\;}}\quad \end{eqnarray}

(8)

where ΔELP₀ is the incremental change in ELP which minimizes the mean PE, and \(\bar F = \frac{1}{N}\mathop \sum \nolimits_{i = 1}^N 0.0006( {{P_i}^2 + 2{K_i}{P_i}} )\) is the arithmetic mean of F over the N eyes of the dataset. 

Equation 8 is derived by aggregating the individual effects delineated in Equation 6 across the entire dataset, ensuring that the sum of individual variations (ΔR_i) achieves a value that effectively neutralizes the initial mean PE. Equation 8 emerges from integrating the individual effects described by Equation 6 across all eyes in the dataset so that the total of individual variations (ΔR_i) induced equals a value that neutralizes the initial mean PE. This equation signifies the adjustment to the lens position (ΔELP₀) that, on average, would neutralize the mean prediction error (\(\bar E\)), effectively optimizing the IOL power formula for the collective dataset. The derivation of Equation 8 is thus a direct consequence of understanding and applying the individualized effects of ΔELP on ΔR through F, culminating in a formula that optimizes the outcome for a broader population.¹² Equation 8 can be re-arranged to provide an estimate of the mean PE induced by an adjustment of the constant intended to nullify the mean PE or minimize the SD or RMS (ΔELP_SD/RMS).  

\begin{eqnarray} \vphantom{\sum} \bar E \approx - {\rm{\;}}\bar F{\rm{\;\Delta }}EL{P_{0/SD/RMS}}\quad \end{eqnarray}

(9)

 

The value of \(\bar F = \overline {0.0006( {{P_i}^2 + 2{K_i}{P_i}} )} \) can be determined for any dataset based on the IOL and corneal powers of the eyes within that dataset. 

In Equation 8, the increment sign is opposite to that of the average prediction error. In the postoperative context of minimizing the mean of the PE of a given formula based on the post hoc analysis of a set of documented eyes, the opposite of the computed increment must be added to the predicted ELP, which would have to be reduced for a negative mean PE and increased for a positive mean PE. 

Minimizing the SD of the PE

Leveraging the relationships established in Equations 3 and 6, we can deduce that for any specified value of ΔELP, the standard deviation (σ) of the prediction error can be calculated as follows:  

\begin{eqnarray} {\sigma ^2} = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {\left( {\left( {{E_i} + \Delta ELP{\rm{\;}}{F_i}} \right) - \left( {\bar E + \Delta ELP{\rm{\;}}\bar F} \right)} \right)^2}\quad \end{eqnarray}

(10)

 

We opt to utilize the expression for variance rather than the SD itself to minimize the SD, as the square of the SD can be derived more straightforwardly (derivative of a square is 2 times the original expression times the derivative of the original expression). To determine the value of ΔELP_SD which minimizes the value of σ², we will first differentiate its expression with respect to ΔELP, and then set the derivative equal to zero. This leads to the equation:  

\begin{eqnarray} &&\frac{2}{N}\mathop \sum \limits_{i = 1}^N \left( \left( {{F_i} - {\rm{\;}}\bar F} \right)\right.\nonumber\\ &&\,\,\, \left.\left( {{E_i} + \Delta EL{P_{SD}}\;{F_i} - \bar E - \Delta EL{P_{SD}}{\rm{\;}}\bar F} \right) \right) = 0\quad \end{eqnarray}

(11)

 

Equation 11 can be solved for ΔELP_SD.  

\begin{eqnarray} \Delta EL{P_{SD}} = - \frac{{\mathop \sum \nolimits_{i = 1}^N \left( {{F_i} - {\rm{\;}}\bar F} \right)\left( {{E_i} - \bar E} \right)}}{{\mathop \sum \nolimits_{i = 1}^N {{\left( {{F_i} - {\rm{\;}}\bar F} \right)}^2}}}\quad \end{eqnarray}

(12)

 

The obtained formula makes it possible to estimate the theoretical value of the ELP increment to be added to the postoperative theoretical position of the IOL, to obtain a minimum SD of the PE on the set of eyes considered, when using the starting constant in each formula. In the postoperative context of minimizing the SD or RMS of the PE of a given formula on a set of documented eyes, the opposite of the computed increment must be added to the predicted ELP. 

Minimizing the RMS of the PE

For any given value of ΔELP, the square of the RMS of the PE is expressed as:  

\begin{eqnarray} RM{S^2} = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {\left( {{E_i} + \Delta ELP{\rm{\;}}{F_i}} \right)^2}\quad \end{eqnarray}

(13)

 

This formulation of the RMS squared allows for a more straightforward differentiation with respect to ΔELP. 

Differentiating this expression with respect to ΔELP, and then setting the derivative equal to zero enables us to determine ΔELP_RMS, the value of ΔELP which minimizes the RMS:  

\begin{eqnarray} \frac{2}{N}\mathop \sum \limits_{i = 1}^N {F_i}\left( {{E_i} + \Delta EL{P_{RMS}}\;{F_i}} \right) = 0\quad \end{eqnarray}

(14)

 

\begin{eqnarray} \Delta EL{P_{RMS}} = - \frac{{\mathop \sum \nolimits_{i = 1}^N ({E_{i\;}}{F_i})}}{{\mathop \sum \nolimits_{i = 1}^N {F_i}^2}}\quad \end{eqnarray}

(15)

 

The obtained formula makes it possible to estimate the theoretical value of the ELP increment to be added to the postoperative theoretical position of the IOL, to obtain a minimum RMS of the prediction error on the set of eyes considered when using the starting constant in a specified formula. 

In the postoperative context of minimizing the SD or RMS of the PE of a given formula on a set of documented eyes, the opposite of the computed increment must be added to the predicted ELP. 

Testing the Theoretical Estimation of the Lens Constant Change to Optimize an IOL Power Formula on Observational Data

We analyzed five datasets containing measurements from eyes from a cataractous population. In each dataset, the same IOL model was inserted among the following: FineVision Micro F (BVI Medical, USA), Acrysof SN60WF (Alcon, USA), Tecnis PCB00 Monofocal (Johnson & Johnson Vision, USA), and Vivinex XC1/XY1 (Hoya, Japan). The study adhered to the tenets of the Declaration of Helsinki and was approved by the Rothschild Foundation Hospital institutional review board (IRB 00012801) under the number CE_20211123_6_GDE. The datasets were anonymized and contained pre-operative biometric data derived with the Lenstar 900 (Haag-Streit AG, Koeniz, Switzerland; EyeSuite software i8.2.2.0) for the Finevision dataset, and the IOL-Master 700 (Carl-Zeiss-Meditec, Jena, Germany; version 7.5.3.0084) for the 3 others, including axial length (AL), anterior chamber depth (ACD; from the front corneal apex to the front apex of the crystalline lens), lens thickness (LT), corneal thickness (CCT), corneal diameter (CD), and the front corneal surface flat (R₁) and steep (R₂) radii. The corneal power in the flat and the steep corneal meridian (K₁ = (n_K-1)/R₁ and K₂ = (n_K-1)/R₂) was calculated from the corneal front surface radii R₁ and R₂, with a keratometer index n_K as indicated in the formula definition. Postoperative refraction was measured at 6 meters 4 to 6 weeks after cataract surgery by an experienced optometrist and recorded in the dataset, which only included data with a postoperative Snellen decimal visual acuity of at least 0.8 (20/25) Snellen lines. 

For all open-source, third-generation formulas, except SRK/T (single-optimized Haigis, Hoffer Q, Holladay 1), the lens constant corresponds to an increment in lens position, added directly to the eye-specific predicted lens position value.^13–17 Hence, the minus ΔELP value calculated using our method can be directly added to the original lens constant (a0, pACD, SF for single optimized Haigis, Hoffer Q, Holladay 1, respectively). However, in the SRK/T formula, the A constant is transformed into an increment in lens position value (ACDconst) using the following equation:  

\begin{eqnarray} {base\, ACDconst} = {base\, A\, constant\, x\, }0.62467 - 68.747\nonumber\\ \end{eqnarray}

(16)

 

ACDconst is then added to the eye-specific predicted lens position value (H), and an offset equal to −3.336 is applied to obtain the final lens position value (ACDest)^18,19: 

Hence, when using our method, the original SRK/T A constant must first be converted to its ACDconst equivalent using Equation 16. Subsequently, minus ΔELP can be added to the latter, followed by converting this adjusted value back to an updated A constant. This can be achieved by solving Equation 16 for it:  

\begin{eqnarray} && {Updated\, A\, constant}\nonumber\\ && \,\,\,\, = \frac{{\left( {base\;ACDconst\; + \;minus\;\Delta ELP} \right) + 68.747\;}}{{0.62467}}\quad \end{eqnarray}

(17)

 

This methodology can be applied to any formula using a transformation of the SRK/T A constant, in order to determine its lens position increment. As an example, in the PEARL-DGS formula,²⁰ the change in theoretical internal lens position (TILP) related to the baseline TILP prediction is correlated to the A-constant via the following formula:  

\begin{eqnarray} \Delta TILP &=& \left( {A\;constant\;x\;0.00061311} \right)\nonumber\\ &&\,\,\,\, - \;0.072931129\quad \end{eqnarray}

(18)

 

For every dataset, the IOL constants respectively optimized for minimizing the mean PE, SD, and RMS were determined using the reference (iterative) method for the following formulas: SRK/T, Hoffer Q, Holladay 1, Haigis (single-optimized, with preset values for a1/a2 = 0.4/0.1), and PEARL-DGS. Starting from the value of the optimized constant (OC) zeroing the mean PE, the ELP increment leading to mean errors comprised between −0.5 diopters (D) and +0.5 D were calculated by 0.01 D steps. 

All calculations and analysis were performed using Python version 3.11.15. 

We first compared the linearly predicted values of the mean PE (\(\bar E\)) against the change in ELP (ΔELP₀) derived from Equation 9, against the values of mean PE obtained through the iterative process. This process allowed for a systematic evaluation of how closely the computed values from this linear model corresponded to the actual data point thereby facilitating a robust assessment of the model's predictive accuracy and efficacy in representing the observed data distribution. 

Then, for each ME step, the values of the SD and RMS of the PE were calculated. This allowed us to obtain the respective reference values of the constant that minimizes the SD and the RMS. We could obtain an estimation of these values by computing ΔELP_SD and ΔELP_RMS using Equations 12 and 15 and adding it to the constant zeroing out the mean PE. We then computed the difference between the reference and estimated constant values, and the differences between their corresponding SD and RMS values (i.e. the resulting values that would have been obtained by using the evaluated optimization method instead of the reference optimization method) were then determined. For each of the values of constants determined by the iterative method, we also indicated the corresponding values of ME, SD, RMS, whenever indicated. 

Results

The descriptive data on biometric data before cataract surgery for the 10,330 included eyes are summarized in Table 1. 

Table 1.

 

Descriptive Statistics of the Total Eye Dataset With Mean, SD, Minimum, Median, and Maximum

 

Table 2 presents data for each model of IOL, detailing the reference (determined with the iterative process) and estimated constant values (Equations 8, 12, and 15) along with the associated mean PE, SD, and RMS values for each of the five formulas tested. 

Table 2.

 

Comparison of IOL Models Using Five Distinct Formulas

 

Regarding the minimization of the SD, the maximum difference between the reference and estimated A-constants values was −0.086 (SRK/T and Vivinex). For the third-generation formulas where the lens constant corresponds to an increment in lens position, the largest difference in the reference versus estimated constant was −0.027 mm (Haigis and Vivinex). The largest difference between the corresponding reference and estimated SD values in the whole dataset was negligible (<0.0005). 

Regarding the minimization of the RMS, the maximum difference between the reference and estimated A-constants values was −0.003 (PEARL, DGS, and Vivinex). For the third-generation formulas where the lens constant corresponds to an increment in lens position, the largest difference in the reference versus estimated constant was 0.002 mm (Hoffer Q and PCB00, and Hoffer Q and SN60WF). The largest difference between the corresponding reference and estimated RMS values in the whole dataset were −0.021 (Hoffer Q and FineVision) and +0.021 (Hoffer Q and Vivinex). 

The largest mean PE after minimization of the SD was 0.726 D (Hoffer Q and PCB00). The largest mean PE after minimization of the RMS was 0.043 D (Hoffer Q and PCB00). 

The largest SD after zeroization was 0.529 D (SRK/T and PCB00). The largest SD after minimization of the RMS was 0.529 D (SRK/T and PCB00). 

The largest RMS after zeroization was 0.529 D (SRK/T and PCB00). The largest RMS after minimization of the SD was 0.875 D (Hoffer Q and PCB00). 

The Figure displays graphically the numerical results obtained for the SN60WF IOL and the five tested formulas. It underscores the reference and estimated constant values that are optimal for minimizing the SD and RMS. This material is designed to enhance the visualization of the relationships between the values of the constants and the corresponding values of the mean PE, SD, and RMS for each of the tested formula’s constant values. By graphically presenting the variations in these metrics across a range of constants for the SN60WF IOL, the figure provides a depiction of how different constant values influence the accuracy and precision of the considered IOL power formula, offering a visual guide to understanding the impact of constant adjustments on formula performance. 

Figure.

 

Visualization of metric variations across formula constants for SN60WF IOL. This figure graphically depicts the variation in mean, standard deviation (SD), and root mean square (RMS) of the prediction error across the range of constants for each tested formula, specifically focusing on the SN60WF intraocular lens (IOL). It also highlights the reference and estimated constant values that minimize the SD and RMS.

Analyzing the distribution of the mean PE values against the variation of the optimized constant, it has been observed that a linear relation disclosed in Equation 9 passing through the origin effectively explains this distribution. The slopes were −1.376 (SN60WF), −1.464 (FineVision), −1.376 (PCB), and −1.380 (Vivinex). Remarkably, for each comparison conducted, the Pearson correlation coefficient exceeded 0.99. This high coefficient value suggests an extremely strong linear relationship between the variables under consideration, underpinning the validity of the linear model used. 

Discussion

In our previous work,¹² we established a formula that facilitates the straightforward computation of the optimal lens constant value, thereby correcting systematic biases in formula calculations. This formula linearly approximates the variation in mean PE as a function of incremental changes in ELP. The Figure illustrates this linear relationship, as depicted by the blue lines. The negative slope of these lines indicates that an increase in ELP corresponds to an increase in the power of all IOLs. This increase in IOL power consequently leads to a shift in postoperative refractions toward the myopic (negative) spectrum. 

In a dataset characterized by an average keratometry of approximately 43 D and a mean IOL power around 22 D, the estimated mean value of \(\bar F\) is close to 1.4 D². This value represents an approximation of the absolute magnitude of the slope (approximately 1.4) observed in the blue lines in the Figure for the Hoffer Q, Haigis, and Holladay IOL power calculation formulas, so that an increment of the positional constant +/−0.1 changes the value of the mean PE by a magnitude of +/−0.14 D. For the formulas that incorporate the A constant, a modification of ±0.1 in this constant is projected to result in a corresponding shift in the mean refractive PE of ±0.224 D, calculated as ±0.1 × 1.4/0.62467. 

Enhancing an IOL power calculation formula may involve targeting metrics beyond the mean bias, such as reducing the SD or the RMS of the PE.^9–11 Our work presents a novel analytical approach designed to determine the optimal constant value for minimizing the SD and RMS of the PE in single optimized formulas. This method offers clinical practicality due to its ease of implementation. These estimations are deemed to possess adequate precision for clinical use, indicating their potential applicability in real-world ophthalmological settings dealing with single-optimization strategies. 

Our approach is constrained to utilizing a single constant. Langenbucher's application of nonlinear iterative algorithms has demonstrated effectiveness in minimizing not only the SD and RMS error but also the mean and median of the absolute error, including the 90th percentile of PEs, by using one or more optimization constants.^9–11,21,22 The simplicity and direct applicability of our proposed analytical method are primarily well-suited for reducing SD and RMS. However, it is evident that devising a simple analytical solution for minimizing the mean or median of the 90th percentile of prediction error is challenging, given the intricate nature of these statistical parameters. This observation highlights the necessity for employing more sophisticated computational strategies to address the full spectrum of optimization challenges IOL power formula calculations. This study is subject to some other limitations. Primarily, the scope of our analysis was confined to demonstrating the applicability of this straightforward concept in predicting the formula constants to five fully disclosed formulas. A notable constraint arises from the proprietary nature of the internal architecture of most contemporary formulas. This lack of disclosure precludes the possibility of discerning the precise relationship between a shift in the ELP and a corresponding adjustment in the formula constant. 

Despite these limitations, a significant observation of considerable relevance to the optimization constants applied in the realm of IOL power calculations has been made. Specifically, it has been noted that the optimization constant leading to the minimization of the RMS error is generally close to the constant that minimizes the mean PE. In the realms of engineering and mathematics, optimizing for the RMS of the PE is widely recognized for its robustness, particularly in its resistance to outliers. This approach is exemplified by its adoption as the standard for formula constant optimization on the IOLCon web platform (https://IOLCon.org). In stark contrast, the constant responsible for minimizing the SD exhibits a notably greater separation, often resulting in a non-zero mean PE. This outcome can be explored by delving into the mathematical relationship, where the square of the RMS is equivalent to the sum of the square of the SD (representing the variance of the PE) and the square of the mean bias (equivalent to the square of the MPE):  

\begin{eqnarray*} RM{S^2} = {\left( {SD} \right)^2} + {\left( {MPE} \right)^2} \end{eqnarray*}

 

Empirical analysis reveals that the SD displays minimal variation in response to changes in the constant, as evidenced by the almost flat curve delineating the relationship between the constant value and the SD. Notably, the minimum SD is achieved at a different constant value than the one producing a null mean PE or minimizing the RMS of the PE. The constant which minimizes the RMS lies intermediate but much closer to the one that nullifies the mean PE. Given that further reduction in SD is marginally impactful, attempting to minimize the SD beyond this equilibrium point leads to an increase in the mean PE. This increment in mean PE outweighs the minor gains achieved in SD reduction, consequently resulting in an elevation of the RMS. This observation highlights the intricate interplay between these statistical measures in the optimization of IOL power formulas and underscores the necessity of balancing the minimization of different error components to achieve optimal predictive accuracy. 

Langenbucher et al.²² identified the shifts in mean PEs when optimizing for the SD in IOL power formulas. This was highlighted in their study in which they demonstrated that optimizing for SD in IOL power calculations could result in a systematic offset in mean and median PEs.²³ In a recent study, we investigated how precision in IOL power calculation formulas is affected by zeroing the mean error through adjustments in the ELP value via the impact on the SD of their PE.²⁴ We also assessed the influence of these variations based on the source of the PE. The methods involved maintaining all variables constant while varying specific parameters individually, such as corneal curvature radius and axial length. Our results indicated that zeroing the mean error to correct inaccuracies in corneal power estimation leads to a significant and exponential increase in the SD, negatively impacting the formula's precision. However, when zeroing is used to address errors from AL measurements or predicted implant position, the effect on precision is minimal or potentially beneficial. This highlights the varied impact of zeroing based on the error source, emphasizing the importance of considering the source of error when analyzing changes in the SD of the PE after zeroing. In this study,¹⁹ we discovered that the evolution of the SD after zeroing depends on the sign and the magnitude of the covariance between the PE, referred to as E_i, and (P_i² + 2K_iP_i). Therefore, it is not surprising that Equation 12 has the expression for this covariance in the numerator. By definition, the constant minimizing SD in our model is reached for a value such that ΔELP_SD = 0, which is equivalent to zeroing the covariance between E_i, and (P_i² + 2K_iP_i). When this covariance is null, the PE of the formula can be related to factors extrinsic to the prediction of the effective position of the IOL. Indeed, if this were not the case, the induced error would necessarily be correlated with (P_i² + 2K_iP_i). 

Optimizing for one parameter (like minimizing SD) may inadvertently worsen other aspects of PE (like mean PE), depending on the primary sources of error in the IOL power formulas. In the present study, the relative robustness of SD to the variations of the optimization constant in all formulas might indicate that the sources of prediction error in IOL power formulas could be inherently linked to factors like corneal power or AL estimation rather than other variables that have a more significant impact on SD. This highlights the complexity of optimizing IOL power formulas and the need for a nuanced approach that considers the varied impact of different sources of error. 

In conclusion, our study contributes practical analytical formulas to approximate the constants that minimize SD and RMS, providing clinicians with accessible tools for optimizing IOL power calculations. Our study demonstrated that using a straightforward predictive model, based on the optimal formula constant needed to attain zero mean PE and including factors like mean keratometric power, and mean refractive power of the IOL, enables precise determination of the optimal formula constant for achieving the lowest SD and RMS error. This methodology is particularly pertinent in clinical environments where advanced formula constant optimization strategies are not accessible. The simplicity of the model, allowing for its implementation in commonly used spreadsheet software, offers a practical approach to enhancing postoperative refractive outcomes in patients who undergo cataract surgery. We observed that the constant minimizing the RMS usually mirrors within clinical precision, the one minimizing the mean PE. On the other hand, the constant minimizing the SD results in a notable, non-zero mean PE. This highlights the intricate interplay between different statistical measures and suggests that further study may be necessary to explore the complexity and critical nature of balanced optimization in IOL power formulas. We hope that this investigation may help to further refine the precision of IOL power calculations, thereby enhancing the accuracy of refractive outcomes in cataract surgery. 

Acknowledgments

Disclosure: D. Gatinel, None; G. Debellemanière, None; A. Saad, None; R. Rampat, None; A. Wallerstein, None; M. Gauvin, None; J. Malet, None