Best fit formula approach in delayed sequential bilateral cataract surgery

We evaluated whether the best‐fit intraocular lens (IOL) power formula for the first operated eye (BF1) was also the most accurate formula for the second eye.


| INTRODUCTION
Intraocular lens (IOL) power calculations continue to evolve, and retaining an overview of the increasing number of formulae is difficult. Most new-generation formulae have not been fully disclosed; therefore, we have limited insight into the calculation concepts. 1 The thirdgeneration formulae Hoffer Q, 2-4 Holladay, 5 SRK/T, 6,7 and Haigis 8 are incorporated into all biometers and are therefore widely used. Although they provide good predictions, the newer formulae present more precise predictions. [9][10][11][12][13] These newer formulae, for example, the Barrett Universal II formula 14 (hereafter referred to as Barrett formula) and Kane formula, 15 focus on a different approach than the regression formulae (thick lens theory and artificial intelligence, respectively). In formulae comparison studies, newer-generation formulae outperform the most commonly used thirdgeneration formulae. 11,12,[16][17][18] The Castrop formula is a recently introduced paraxial vergence formula with a thick lens model for the cornea, incorporating three formula constants. 19 Generally, refractive cataract surgeons who maintain an overview of the published formulae and use them appropriately achieve a prediction error within 0.5 dioptre (D) in 80-87%. 12,20,21 Nevertheless, most cataract surgeons have experienced predictions of different formulae spreading by more than 1 D when looking at an IOL calculation sheet (such as the 4-in-1 print). If the eye shows certain characteristics, such as a short axial length (or even more complex with flat corneas), the surgeon will rely on the Hoffer Q formula rather than the SRK/T formula. However, in most cases, no conspicuous attributes guide the formula selection. Several studies have tested other approaches for IOL power selection by considering the refractive error of the first operated eye. Therefore, the first operated eye was a test run to achieve an emmetropic outcome in the second eye. These studies suggest adjusting the target refraction. According to these studies, an adjustment of 30%-80% 22 of the prediction error, depending on both the formula used and the corneal condition, is recommended. Most studies advise a second eye adjustment rate of 50%. 23,24 The success of this approach depends on the anatomical symmetry and comparability of the eyes, 22,[24][25][26][27] as well as the formula used. 25 Generally, the eyes of individuals are similar in terms of biometry. 27 Considering this factor, a new approach was developed based on the similarity between the eyes of the patient. The concept of formula selection was demonstrated in this retrospective study. This study aimed to assess whether the best-fit formula for the first eye showed the lowest prediction error for the second eye in delayed sequential bilateral cataract surgery.

| METHODS
This retrospective data analysis included 304 eyes from 152 patients. Each patient underwent uncomplicated delayed bilateral cataract surgery with implantation of one 1-piece acrylic IOL (Vivinex XC1/XY1; Hoya Surgical Optics, Tokyo, Japan) in both eyes. Patients with a bestcorrected visual acuity of less than 20/40, preceding ocular surgery, ocular pathology influencing the accuracy of biometry, or intraoperative or postoperative complications were excluded. Institutional ethics approval (Approval No. 2144/2021) was obtained before initiating the study. The study protocol was designed in accordance with the tenets of the Declaration of Helsinki.
An experienced specialist performed biometry of all patients before surgery using the IOL Master 700 (Carl Zeiss Meditec AG, Jena, Germany, software version 1.5 or higher). IOL power calculations were performed using the Barrett Universal II, Castrop, Haigis, Hoffer Q, Holladay 1, Kane, and SRK/T formulae. All surgeries were performed by two experienced surgeons (R.M. and C.L.). Manifest subjective refraction and visual acuity were measured at least 3 weeks after surgery using Snellen acuity charts at a refraction lane distance of 6 m.
For this retrospective study, the following approach was evaluated: 1. Cataract surgery of the first cataract eye, 2. Postoperative examination including subjective refraction and visual acuity, 3. Calculating a ranking according to the absolute value of the prediction error to define the "best-fit" formula, 4. Use of this "best-fit" formula to determine the IOL power for the fellow cataract eye, 5. Cataract surgery and postoperative examinations included subjective refraction and visual acuity of the second eye.
Biometric data (e.g., axial length, corneal curvature) was recorded as mean ± standard deviation, median and 90% confidence interval (5% and 95% quantiles). Categorical variables (e.g., laterality of the eye or sex) were documented based on frequency. The left or right eye of each individual was randomly selected as the first or second operated eye to avoid selection bias. The optimised constants published in IOLCON.org were used for the SRK/T (A-constant: + 119.22), Hoffer Q (pACD: + 5.71), Holladay 1 (SF: +1.94), and Haigis (a0: À0.278, a1: +0.215, a2: +0.201) formulae. Predetermined constants were used for all other formulae (Barrett: lens factor: +1.99, Castrop: C: +0.34, H: + 0.0124, R: +0.0119). Constant optimisation is only possible for the disclosed formulae. Therefore, to avoid unfair conditions for comparison and simulate a "real-world scenario", constant optimisation was not performed specifically for this dataset. This would only be possible for the formulae Haigis, Hoffer Q, Holladay 1, and SRK/T, but not for the formulae Barrett, Castrop, and Kane.
The ranking of formula performance for each eye was derived. Mean absolute error (MAE) and median absolute error (MedAE) were calculated for each formula. The best-fit formula was defined as that with the lowest MedAE scores. After determining the best-fit formula for the first eye, the best-fit formula for the second eye was calculated. The percentage of cases that resulted in the same best-fit formula for both the first and second eyes was computed. A tolerance of |0.1 D| difference between the formula predictions was used to define the best-fit formula for the second eye. Additionally, subgroup analysis was performed for all eyes with a formula prediction disparity of >0.5 D for one eye. Data processing and statistical analyses was performed using MATLAB (MathWorks, Natick, USA, Version 2019b).

| RESULTS
Three-hundred-four eyes from 152 patients were included in this retrospective study. The demographic characteristics of the study cohort are summarised in Table 1. Of the 152 patients, 57.9% were female and 42.1% male.
The MedAE was generally low, ranging from 0.28 to 0.35 D (Figure 1), the MAE ranged from 0.34 to 0.44 D.    p < 0.05). Without tolerance limits, the agreement of the best-fit formula for both eyes was 33.5% (51 patients, χ 2 = 71.37, p = 0.07). The best-fit formula for the first eye showed a MedAE of 0.11 D (0-1.03 D), the best-fit formula of the second eye showed a MedAE of 0.09 D (0-1.50 D). Figure 1 shows the absolute prediction error of each formula for all the eyes. The best-fit formula of the first eye showed a MedAE of 0.22 D (0-1.50 D) in the second eye. In contrast, if a random formula was selected for the second eye, excluding the best-fit formula for the first eye (excluding BF1), the MedAE for the second eye was 0.33 D (0.04-1.60 D) (Figure 2). There was no statistically significant difference between the MedAE of the best-fit formula of the first eye (BF1) and that of the second eye (BF2) (p = 0.9; boxes 1 and 2 in Figure 2). The difference in the MedAEs of BF2 and of BF1 used for the second eye was statistically significant (p < 0.01; boxes 2 and 3 in Figure 2). The difference in the MedAE of BF2 and of using any formula except BF1 for the second eye was statistically significant ( p < 0.01, boxes 2 and 4 in Figure 2). The difference in the MedAE of BF1 used for the second eye and of any formula except BF1 was also statistically significant (p < 0.01; boxes 3 and 4 in Figure 2).
The lowest MedAE was observed for the Castrop formula (0.28 D), closely followed by the Barrett Universal II and Kane formulas (0.29 D). The Haigis formula (0.31 D) outperformed the third-generation formulae (0.34 and 0.35 D) ( Table 2). Table 3 summarises the cross-tabulations of BF1 and BF2. For example, the SRK/T formula showed the lowest prediction error in the first and second eyes of three patients. The Kane formula showed the lowest prediction error in the first and second eyes of the 24 patients, as shown on the diagonal of the cross-tabulation. Consequently, in 51 patients, the best-fit formulas in the first and second eyes matched.
The results show that the best-fit formula approach is formula-dependent. If the SRK/T formula was determined as the best-fit formula for the first eye, it was also identified as the best-fit formula for the second eye in 18% of cases. Similarly, the Hoffer Q formula was used in 24%, Holladay 1 formula in 0%, Haigis formula in 15%, Castrop formula in 36%, Barrett formula in 33%, and Kane formula in 49% of cases. Overall, 95 patients showed a formula prediction disparity of >0.5 D within one eye. Subgroup analysis demonstrated that the agreement of the best-fit formula in both eyes was 57%.

| DISCUSSION
In terms of refractive outcomes after cataract surgery, cataract surgeons try to achieve minimal refractive error. To achieve this goal, numerous studies have compared different formulae and tested different formula-choice strategies. 13 The formula selection for each eye is quite complex, mostly because of the high number of published formulae and divergent study results. For example, only vague recommendations are available in the NICE guidelines. 28 Formula selection for intraocular lens power calculation based solely on axial length is no longer sufficient. Current knowledge suggests that proper formula selection requires the consideration of additional factors such as corneal curvature and anterior segment dimensions. Neglecting these factors can result in inaccurate calculations and suboptimal outcomes. [29][30][31] Furthermore, different strategies, integrating the prediction error of the first eye for choosing the needed IOL power in the second eye have been proposed, such as the application of a polynomial regression formula. 32 In comparison, the advantage of our strategy is its simple implementation in the daily clinical routine without requiring additional formulae by simply determining the formula with the lowest prediction error of the first operated eye.
This study showed that the best-fit formulae for the first and second eyes matched in 56% of all patients, which was statistically significant. Using BF1 for the second eye resulted in a lower MedAE than using a random formula; however, using only the best-fit formula for each individual eye led to an even lower MedAE. Furthermore, the applicability of the formula selection approach was formula-dependent. The MedAEs of all formulae were generally low. The average patient presented with two similar eyes with similar biometry. Consequently, the IOL power calculation formula should be applied equally for both eyes. Our study demonstrates that using the best-fit formula leads to better outcomes than choosing a formula randomly; however, it should be noted that this approach cannot be "blindly trusted". Nevertheless, detecting the best-fit formula for each eye led to a lower MedAE than using the best-fit formula for the first eye ( Figure 2).
The Castrop formula had the lowest MedAE value (0.28 D). Because this formula has only been reported recently, there are only a few published studies reporting its use. 19 Wendelstein et al. reported a significantly lower MAE of the Castrop and Kane formulae compared to thirdgeneration formulae in a population with short eyes and lens powers of 28 D or more. 33 Similar to our results, the Kane formula performed better than most other formulae reported in the literature. While only a few independent studies have evaluated the use of Kane's formula, consistent with our results, these independent studies have reported excellent results. 9,20,34 The Barrett Universal II formula was the third formula used in this study, showing an MAE lower than 0.3 D. The use of this formula is already widespread and has been incorporated into several biometers; however, the formula has not yet been disclosed. 9,[35][36][37] In accordance with our results, Olsen et al. reported that the success of power adjustment based on the refractive outcome of the first operated eye was formula-dependent. 25 The Kane formula is more reliable for formula selection of the second eye, whereas the success of the SRK/T formula in the first eye does not necessarily translate to its efficacy in the second eye.
A potential limitation of the study is that patients were not excluded because of potential disparities in the eyes of one patient. In this study, only three patients had axial length differences >1 mm. Because the difference between the performances of the formulae was small, they might not be clinically relevant. Nevertheless, subgroup analysis demonstrated comparable results when investigating all eyes with clinically significant differences in formula predictions of more than >0.5 D. A subsequent study, investigating the applicability of the "best-fit approach" with greater differences in formula prediction would be interesting. Furthermore, the outliers were not "trimmed" (meaning excluding eyes that fall outside two standard deviations from the mean in any relevant observation) to report the results unbiased.
In conclusion, our findings suggest that to achieve better refractive results in second eye surgery, the best-fit formula of the first eye can be used for the second eye especially if the surgeon is unsure about the choice of formula.