**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.

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^{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.

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^{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}

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^{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}

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^{11}

^{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.

_{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:

_{i}) that satisfies:

*R*) incurred by an incremental change in ELP (Δ

*ELP*), one can edit the following equation:

_{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.

^{12}the F function was estimated through differential calculation methods, using a thick lens pseudophakic eye model:

_{i}and P

_{i}, respectively, denote the corneal power and the implant power of the eye under consideration.

_{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.

*ELP*

_{0}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.

_{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 (ΔELP

_{0}) 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}).

*ELP*which minimizes the value of σ

_{SD}^{2}, we will first differentiate its expression with respect to Δ

*ELP*, and then set the derivative equal to zero. This leads to the equation:

*ELP*.

_{SD}*ELP*.

*ELP*, and then setting the derivative equal to zero enables us to determine Δ

*ELP*

_{RMS }, the value of Δ

*ELP*which minimizes the RMS:

_{1}) and steep (R

_{2}) radii. The corneal power in the flat and the steep corneal meridian (K

_{1}= (n

_{K}-1)/R

_{1}and K

_{2}= (n

_{K}-1)/R

_{2}) was calculated from the corneal front surface radii R

_{1}and R

_{2}, 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.

^{13}

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^{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:

^{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:

_{0}) 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.

_{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.

^{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.

^{2}. 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.

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^{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.

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^{11}

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^{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.

^{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

_{i}

^{2}+ 2K

_{i}P

_{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

_{i}

^{2}+ 2K

_{i}P

_{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

_{i}

^{2}+ 2K

_{i}P

_{i}).

**D. Gatinel**, None;

**G. Debellemanière**, None;

**A. Saad**, None;

**R. Rampat**, None;

**A. Wallerstein**, None;

**M. Gauvin**, None;

**J. Malet**, None

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