#### Abstract

To investigate the accuracy of intraocular lens (IOL) power calculation formulas using swept-source optical coherence tomography (SS-OCT).

Eyes with biometry measurement by IOLMaster 700 (Carl Zeiss Meditec AG), uncomplicated phacoemulsification, and IOL implantation were enrolled in this retrospective study. Newly released artificial intelligence–based formulas including Hill-Radial Basis Function (RBF) 2.0, Kane, and PEARL-DGS were compared with Gaussian optics-based standard formulas. The refraction predicted by each formula was compared with the actual refractive outcome in spherical equivalent.

A total of 410 eyes of 410 patients were included in this study. Using optimized constants for SS-OCT biometry led to a significant decrease in median absolute error (MedAE) for Barrett, Haigis, and Hoffer Q formulas compared with using User Group for Laser Interference Biometry constants (*P* < .05). Overall, Olsen (0.283 diopters [D]) and Kane (0.286 D) formulas had significantly lower MedAEs than RBF 2.0 (0.314 D), Haigis (0.322 D), SRK/T (0.371 D), Holladay 1 (0.376 D), and Hoffer Q (0.379 D) formulas under constant optimization (*P* < .05). The first four formulas with the lowest standard deviations of prediction error were Kane (0.451 D), Olsen (0.456 D), EVO 2.0 (0.460 D), and Barrett (0.470 D). Olsen (47.1%), Barrett (45.9%), Kane (45.4%), and EVO 2.0 (45.1%) formulas had greater proportions of eyes within ±0.25 D of the predicted refraction than Hoffer Q (35.9%), SRK/T (35.9%), and Holladay 1 (33.4%) formulas (*P* < .05).

Constant optimization for SS-OCT biometry further improves the performance of formulas. The most accurate prediction of postoperative refraction can be achieved with Barrett, EVO 2.0, Kane, and Olsen formulas.

**[ J Refract Surg. 2020;36(7):466–472.]**

Modern optical biometric measurements, updated intraocular lens (IOL) calculation methods, and refined surgical techniques have contributed to the steadily improving accuracy in predicting postoperative refraction for cataract surgery.^{1} On the other hand, patients now have higher expectations for realizing the target refraction than ever before, which requires more accurate measurement of ocular parameters.^{2} The IOLMaster 700 (Carl Zeiss Meditec AG), a new optical biometer that integrates swept-source optical coherence tomography (SS-OCT) to provide enhanced biometric measurement, has been developed.^{3} The SS-OCT technology scans ocular structures with a rapid cycle, tunable wavelength laser source, giving lower signal-to-noise ratio and better image quality.^{4} It yields a significantly higher success rate for axial length measurement compared with partial coherence interferometry.^{5} Additionally, new formulas have been developed in an effort to increase the accuracy and applicability of IOL power selection to a wide range of eye measurements.

Artificial intelligence is the fourth industrial revolution and has been extended to the field of IOL power calculation.^{6} The Kane formula (available at www.iol-formula.com), which incorporates regression and artificial intelligence to refine its accuracy, has been shown to be more accurate than all other existing formulas using optical low-coherence reflectometry (OLCR) biometry.^{7} Another artificial intelligence formula, Hill-Radial Basis Function (RBF), was derived from OLCR measurements using pattern recognition and data interpolation (available at http://rbfcalculator.com). Although the RBF 2.0 formula has been shown to be inferior to other modern formulas using OLCR,^{7} its accuracy using SS-OCT biometry has not been well validated. The PEARL-DGS formula with precision enhancement using artificial intelligence and output linearization was optimized for values obtained with the IOLMaster 700 and was first introduced in 2019 (available at https://www.pearldgscalculator.com/). Before the introduction of artificial intelligence, vergence formulas based on Gaussian optics were the standard choice for IOL power calculation. The newer vergence formula, Barrett Universal II (Barrett), outperformed its predecessors when using partial coherence interferometry^{8} and OLCR.^{9} However, we are unaware of comparative studies including Barrett, Kane, PEARL-DGS, and RBF 2.0 formulas, using SS-OCT biometry. With the increasing popularity of the IOLMaster 700, it is imperative to evaluate existing formulas with measurements taken by this new biometer.

The objective of the current study was to compare new artificial intelligence–based formulas (Kane, PEARL-DGS, and RBF 2.0) with other existing formulas, including Barrett,^{10} Emmetropia Verifying Optical (EVO) 2.0 (available at www.evoiolcalculator.com), Haigis,^{11} Hoffer Q,^{12} Holladay 1,^{13} Holladay 2, Olsen,^{14} STK/T,^{15} and T2,^{16} using SS-OCT biometry.

### Patients and Methods

The current study was conducted with the approval of the Institutional Review Board of Zhongshan Ophthalmic Center (IRB approval number 2019KYPJ124) and adhered to the tenets of the Declaration of Helsinki. Patients undergoing uncomplicated phacoemulsification and intracapsular insertion of the enVista MX60 IOL (Bausch & Lomb, Inc) between August 2018 and April 2019 were eligible. All patients were successfully measured with the IOLMaster 700 (software version 1.80). Subjective refraction was performed at least 1 month postoperatively and only patients with corrected distance visual acuity of 20/40 or better were enrolled. Six-meter refractive lanes were used to measure postoperative distance visual acuity. Patients with previous ocular surgeries, corneal pathology, and retinal diseases that impaired visual acuity, astigmatism greater than 4.00 D, and biometry data outside the boundary of the RBF 2.0 formula were excluded. When both eyes of a patient were eligible, only one eye was chosen at random.

Lens constant optimization was implemented to produce a mean prediction error equal to zero according to a previous protocol.^{17} For proprietary IOL calculation formulas, the optimized lens constants were achieved with repeated trial and error. To evaluate the benefits of constant optimization for SS-OCT biometry on the accuracy of formulas, Haigis, Hoffer Q, Holladay 1, and SRK/T formulas were calculated with both optimized constants and the User Group for Laser Interference Biometry (ULIB) constants (available at http://ocusoft.de/ulib). Calculation with the Holladay 2 formula was performed using both the optical biometer with default ACD constant and the IOL Consultant Software with optimized constant. Both the ULIB SRK/T A constant and the optimized lens factor for the Barrett formula were used with its online calculator.

The prediction error was calculated as the refractive outcome in spherical equivalent (SE) minus the refraction predicted by each formula for the IOL power actually implanted. The mean prediction error (ME), median absolute error (MedAE), mean absolute error (MAE), and the proportions of eyes within different prediction error ranges (±0.25, ±0.50, and ±1.00 D) were also calculated. Eyes were separated into four subgroups according to axial lengths: short (≤ 22 mm), medium (> 22 to < 24.5 mm), medium-long (≤ 24.5 to < 26 mm), and long (≥ 26 mm).

Statistical analysis was performed using SPSS software (version 22.0; IBM Corporation). Data distribution was assessed with the Kolmogorov-Smirnov test. The non-parametric Friedman and Wilcoxon tests (with Bonferroni correction for multiple comparisons) were used to compare the differences in MedAE between formulas. The Cochran Q test was used to compare the proportions of eyes within different prediction error groups between formulas. The Levene's test was used to determine the equality in the variances of absolute error. A *P* value of less than .05 was considered statistically significant.

### Results

The current study comprised 410 eyes of 410 patients in total. A total of 262 patients (63.9%) were female and 241 eyes (58.8%) were right eyes. The mean age was 66 ± 12 years (range: 25 to 90 years). The pre-operative optical properties of all included eyes are shown in **Table 1**. The mean power of the implanted IOLs was 18.96 ± 5.19 D (range: 0.00 to 27.00 D). **Table A** (available in the online version of this article) presents the optimized lens constants based on SS-OCT biometry. The manufacturer's constant and the constants for standard formulas converted from this constant are also presented for reference. The outcomes for intraocular lens–based refractive surgeries of all included patients are displayed in **Figure 1**.

Table 1: Characteristics of Eyes (N = 410) |

Table A: Optimized Constants for Each Formula (N = 410) |

The MedAE of the Haigis formula calculated with the three optimized constants for SS-OCT biometry was significantly lower than that calculated with the single ULIB a0 constant (*P* < .001). Similar findings were made for the Barrett (*P* < .001) and Hoffer Q (*P* = .020) formulas, but no significant difference was found for the SRK/T (*P* = .050) and Holladay 1 (*P* = .146) formulas. Additionally, despite the trivial difference in the constants for the Holladay 2 formula, the MedAE calculated with the standalone software was significantly lower than that calculated with the biometer (*P* = .014) (**Figure 2**).

As shown in **Table 2**, the Olsen and Barrett formulas had the lowest MedAE with the Kane formula the third lowest. The Kane and Olsen formulas had significantly lower MedAEs than the RBF 2.0 and earlier formulas (*P* < .05). The Barrett formula also produced a significantly lower MedAE in comparison to the SRK/T, Holladay 1, and Hoffer Q formulas (*P* < .05). The lowest MAE was obtained with the Kane formula, followed by the Olsen and EVO 2.0 formulas. A significant difference in the variance of absolute error was also detected (*P* < .001), with the variance of the Kane formula being the lowest.

Table 2: Prediction Errors for Each Formula Over the Entire Axial Length Range (N = 410) |

The proportions of eyes with a prediction error within the specified ranges for various formulas are shown in **Figure A** (available in the online version of this article). Olsen had the highest proportion of eyes within ±0.25 D of the predicted SE (47.1%), which was significantly higher compared with the Holladay 2 (38.5%), Hoffer Q (35.9%), SRK/T (35.9%), and Holladay 1 (33.4%) formulas (*P* < .05). The Barrett (45.9%), EVO 2.0 (45.1%), and Kane (45.4%) formulas had significantly higher proportions of eyes within ±0.25 D than the Hoffer Q, SRK/T, and Holladay 1 formulas (*P* < .05). The PEARL-DGS (42.7%) and RBF 2.0 (41.2%) formulas also had higher values for the ±0.25 D endpoint compared with the Holladay 1 formula (*P* < .05). The Kane formula had the highest value for the ±0.50 D endpoint (77.1%) and the lowest risk of refractive surprise (> 1.00 D from the intended refraction, 3.7%).

Figure A. Stacked histogram for the percentages of eyes within different prediction error groups over the entire axial length range. |

The prediction errors of formulas in different axial length subgroups are presented in **Table 3**. In all axial length subgroups, statistically significant differences in MedAE were found between formulas (*P* < .01). In short eyes, the Hoffer Q formula produced the lowest MedAE, whereas the PEARL-DGS formula had the lowest MAE. Interestingly, contrary to expectations, the RBF 2.0 formula had the highest MedAE and MAE. The RBF 2.0 formula yielded a significantly higher MedAE than did the Hoffer Q, PEARL-DGS and Holladay 1 formulas (*P* < .05). Only the PEARL-DGS formula met the benchmark criteria of reaching at least 71% and 93% of eyes within ±0.50 and ±1.00 D of the predicted SE.

Table 3: Prediction Errors of Different Formulas in Axial Length Subgroups |

In medium eyes, the Olsen formula produced the lowest MedAE, followed by the Kane and EVO 2.0 formulas. Olsen produced a significantly lower MedAE compared with the Holladay 1 (*P* = .004), Haigis (*P* = .004), and Hoffer Q formulas (*P* < .001). The Hoffer Q formula also had a significantly higher MedAE than the EVO 2.0 (*P* = .003), Kane (*P* = .006), PEARL-DGS (*P* = .016), and Barrett (*P* = .049) formulas. All formulas met the criteria that at least 93% of eyes should have a prediction error of ±1.00 D or less and all formulas except the Haigis, Hoffer Q, Holladay 1, and SRK/T formulas had at least 71% of eyes within ±0.50 D.

In medium-long eyes, the Barrett formula had the lowest MedAE and the Kane formula had the lowest MAE. The Holladay 1 formula had a significantly higher MedAE than the Barrett (*P* = .001), Kane (*P* < .001), Olsen (*P* < .001), and EVO 2.0 (*P* < .001) formulas. All formulas met the criteria of having at least 93% of eyes with a prediction error within ±1.00 D and all formulas except the Haigis, Hoffer Q, Holladay 1, SRK/T, and T2 formulas had at least 71% of eyes within ±0.50 D.

In long eyes, the Barrett formula yielded the lowest MedAE, whereas the Kane formula produced the lowest MAE. The Kane, EVO 2.0, Olsen, Barrett, RBF 2.0, Holladay 2, T2, and Haigis formulas had significantly lower MedAEs than the Hoffer Q and Holladay 1 formulas (*P* < .01). In addition, the PEARL-DGS and SRK/T formulas had significantly lower MedAEs than the Holladay 1 formula (*P* = .001 and .003, respectively). All formulas except Hoffer Q, Holladay 1, PEARL-DGS, and SRK/T met the criteria of having at least 71% and 93% of eyes within ±0.50 and ±1.00 D of the predicted SE.

### Discussion

Lens constant optimization is essential to clinicians and mandatory for clinical studies, whether it is performed individually or derived externally from a large dataset.^{18} The current study optimized lens constants for the MX60 IOL using a larger number of cases with IOLMaster 700 measurements, which led to better performance for the Haigis, Barrett, and Hoffer Q formulas compared with using ULIB constants.

The current study mainly focused on the accuracy comparison of artificial intelligence–based formulas with well validated standard formulas. To our knowledge, this is the first study evaluating new IOL calculation formulas using SS-OCT biometry and one single IOL model. Overall, the Kane formula had the lowest standard deviation of prediction error and MAE, and the highest proportions of eyes within ±0.50 and ±1.00 D of the intended SE. Further analysis revealed that the Kane formula yielded the lowest proportion of refractive surprise in eyes longer than 22 mm. A previous study showed that the Kane formula had the lowest MedAE and MAE, and the highest proportions of eyes within ±0.25 D, ±0.50 D and ±1.00 D of the predicted SE using the IOLMaster, predominantly model 3 in comparison to 8 other formulas.^{19} Our study had similar findings except that using IOLMaster 700 biometry, the Olsen formula had the highest value for the ±0.25 D endpoint and the same lowest MedAE with the Barrett formula.

Refractive outcomes remain variable due to differences in preoperative measurement, population cohort, surgical technique, and experience.^{20} Since the introduction of partial coherence interferometry technology, the benchmark criteria that 71% and 93% of eyes should be within ±0.50 and ±1.00 D of prediction error has been proposed.^{21} Overall, the Barrett, EVO 2.0, Holladay 2, Kane, Olsen, PEARL-DGS, RBF 2.0, and T2 formulas met the benchmark criteria, suggesting that good refractive predictability could be achieved using newer formulas and the T2 formula in this study group. As a modification to the algorithm of SRK/T, the T2 formula was developed using a regression formula for corneal height.^{16} In the current study, the T2 formula had a lower MedAE than the SRK/T formula in all axial length subgroups, although it was not statistically significant. Despite its potential superiority over the SRK/T formulas, the T2 formula did not outperform other modern formulas in any axial length subgroup, which impedes its wide adoption.

Before the introduction of the Kane formula, the Barrett formula had been shown to be the best-performing formula.^{9} In the current study, the Barrett formula also had a significantly smaller MedAE compared to the Hoffer Q, Holladay 1, and SRK/T formulas, which was consistent with previous studies.^{8} Across all axial length subgroups, no significant difference was detected in MedAE between the Barrett, EVO 2.0, Kane, and Olsen formulas. The EVO is a new thick lens formula developed with the theory of emmetropization. The original version of the EVO formula was found to have compromised accuracy in short and long eyes,^{7} whereas in this study the EVO 2.0 formula had the second smallest MAE and the third smallest MedAE in long eyes, suggesting improved accuracy with the EVO 2.0 formula. In short eyes, being the only formula meeting the benchmark criteria, the PEARL-DGS formula might be more accurate than other new formulas, although the difference was only significant when comparing the PEARL-DGS to the RBF 2.0 formula. No other published study has evaluated the accuracy of the PEARL-DGS and EVO 2.0 formulas; thus, we were unable to compare our results with others.

Kane et al^{22} showed that the RBF 1.0 formula was inferior to the Barrett and Holladay 1 formulas in the whole axial length range. We found that overall the RBF 2.0 formula was as accurate as the Barrett, EVO 2.0, and Holladay 2 formulas, more accurate than the Holladay 1 formula, but less accurate than the Olsen and Kane formulas. A recent study reported that the Barrett and RBF 2.0 formulas had equivalent refractive predictability in long eyes.^{23} In our long eye subgroup, the RBF 2.0 formula had an equally low MedAE compated with the Barrett, EVO 2.0, and Olsen formulas, indicating that the accuracy of the RBF 2.0 formula was improved compared with its previous version, especially in long eyes. The RBF 1.0 formula was shown to provide the lowest MAE compared with the Barrett formula and readily available third generation formulas in short eyes with implantation of the SN60WF IOL.^{24} However, in the current study, the RBF 2.0 formula failed to produce similar outcomes for short eyes with the MX60 IOL. SN60WF is the default IOL for the development of the RBF formulas. Thus, the accuracy of the RBF 2.0 formula in short eyes remains to be validated using different IOL models.

The Holladay 2 formula in the standalone software was updated with axial length adjustment, which was more accurate than its counterpart preinstalled in the IOLMaster 700 in the current study, indicating improved accuracy of the Holladay 2 formula after axial length adjustment. In a previous study, the updated Holladay 2 formula was less accurate than the Olsen formula and failed to show superiority over the Haigis, Holladay 1, or SRK/T formulas.^{19} Our study had similar findings except that the Holladay 2 formula yielded a significantly lower MedAE compared with the Holladay 1 formula.

The main limitation of this study was the relatively limited number of eyes with short axial lengths. Although the Hoffer Q formula had the lowest MedAE, no significant difference was found between the Hoffer Q and other formulas except the RBF 2.0. Among the three artificial intelligence–based formulas, the PEARL-DGS formula yielded the highest percentages of eyes for all investigated prediction error ranges, although not statistically significant. Based on sample size calculation with the PASS software (version 15.0.5; NCSS, LLC), a minimum of 81 short eyes might be required to detect a difference of 0.12 D in MedAE between the Hoffer Q and Barrett formulas (standard deviation of 0.43 D) found in this study for a test power of 80% and a significance level of 5%. The shortest eye was 21.28 mm in this study, which compromised the analysis of eyes shorter than 22 mm. Whether the same findings can be made in shorter eyes requires further investigation. The inaccuracy of formulas in short eyes has been well documented in the literature.^{25} A previous study found that even the most accurate formula still failed to yield 75% of eyes within ±0.50 D of the predicted SE, and newer generation formulas like the Barrett were not always better than old two-variable vergence formulas,^{26} which was consistent with our findings.

We found that lens constant optimization further improved the accuracy of IOL calculation formulas using SS-OCT biometry. The highest prediction accuracy might be achieved with the Barrett, EVO 2.0, Kane, and Olsen formulas. The Kane formula might provide the lowest risk of refractive surprise in eyes longer than 22 mm.

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Characteristics of Eyes (N = 410)

Axial length (mm) | 24.62 ± 2.42 | 21.28 to 34.80 |

Anterior chamber depth (mm)^{a} | 3.15 ± 0.47 | 1.71 to 4.70 |

Keratometry (D) | 44.16 ± 1.58 | 40.16 to 48.31 |

Lens thickness (mm) | 4.47 ± 0.49 | 2.75 to 6.05 |

Horizontal corneal diameter (mm) | 11.83 ± 0.47 | 10.60 to 13.60 |

Central cornea thickness (µm) | 541.87 ± 32.70 | 456 to 650 |

Prediction Errors for Each Formula Over the Entire Axial Length Range (N = 410)

Kane | 0.000 | 0.286 | 0.348 | 0.451 | −1.453 to 1.244 | 0.081 |

Olsen | 0.000 | 0.283 | 0.349 | 0.456 | −1.503 to 1.507 | 0.085 |

EVO 2.0 | 0.000 | 0.293 | 0.354 | 0.460 | −1.565 to 1.595 | 0.086 |

Barrett | 0.000 | 0.283 | 0.362 | 0.470 | −1.523 to 1.257 | 0.089 |

Holladay 2 | 0.000 | 0.325 | 0.378 | 0.482 | −1.490 to 1.480 | 0.089 |

RBF 2.0 | 0.000 | 0.314 | 0.385 | 0.492 | −1.746 to 1.544 | 0.093 |

T2 | 0.000 | 0.317 | 0.391 | 0.500 | −1.697 to 1.693 | 0.097 |

PEARL-DGS | 0.000 | 0.305 | 0.388 | 0.515 | −1.520 to 1.973 | 0.114 |

Haigis | 0.000 | 0.322 | 0.404 | 0.521 | −1.553 to 1.504 | 0.108 |

SRK/T | 0.000 | 0.371 | 0.426 | 0.548 | −2.076 to 1.706 | 0.120 |

Holladay 1 | 0.000 | 0.376 | 0.478 | 0.611 | −1.845 to 2.010 | 0.144 |

Hoffer Q | 0.000 | 0.379 | 0.465 | 0.612 | −1.852 to 2.338 | 0.158 |

Prediction Errors of Different Formulas in Axial Length Subgroups

Short (n = 24) | ||||||

Hoffer Q | 0.273 | 0.409 | −1.852 to 0.919 | 45.8 | 70.8 | 91.7 |

PEARL-DGS | 0.278 | 0.378 | −1.520 to 0.580 | 45.8 | 70.8 | 95.8 |

Holladay 1 | 0.352 | 0.420 | −1.732 to 1.040 | 45.8 | 70.8 | 91.7 |

Kane | 0.417 | 0.472 | −1.420 to 1.244 | 29.2 | 62.5 | 87.5 |

RBF 2.0 | 0.579 | 0.608 | −1.346 to 1.544 | 12.5 | 41.7 | 83.3 |

Medium (n = 234) | ||||||

Olsen | 0.255 | 0.347 | −1.473 to 1.112 | 48.7 | 76.5 | 95.7 |

Kane | 0.271 | 0.351 | −1.453 to 1.128 | 47.4 | 76.9 | 95.7 |

EVO 2.0 | 0.280 | 0.353 | −1.565 to 1.230 | 47.4 | 75.2 | 95.3 |

Barrett | 0.281 | 0.361 | −1.523 to 1.252 | 46.6 | 75.6 | 96.2 |

PEARL-DGS | 0.292 | 0.356 | −1.470 to 1.080 | 45.3 | 74.4 | 95.7 |

Medium-long (n = 65) | ||||||

Barrett | 0.308 | 0.357 | −1.088 to 0.777 | 40.0 | 72.3 | 96.9 |

RBF 2.0 | 0.334 | 0.368 | −1.141 to 0.794 | 35.4 | 76.9 | 96.9 |

Olsen | 0.337 | 0.353 | −1.093 to 0.677 | 43.1 | 75.4 | 96.9 |

Kane | 0.338 | 0.350 | −1.039 to 0.727 | 35.4 | 78.5 | 98.5 |

EVO 2.0 | 0.355 | 0.358 | −1.090 to 0.655 | 36.9 | 72.3 | 96.9 |

Long (n = 87) | ||||||

Barrett | 0.247 | 0.341 | −1.223 to 1.112 | 51.7 | 77.0 | 96.6 |

Kane | 0.248 | 0.306 | −1.249 to 0.968 | 51.7 | 80.5 | 98.9 |

EVO 2.0 | 0.250 | 0.315 | −1.125 to 1.095 | 49.4 | 78.2 | 97.7 |

RBF 2.0 | 0.251 | 0.345 | −1.396 to 0.894 | 49.4 | 74.7 | 97.7 |

PEARL-DGS | 0.325 | 0.475 | −0.960 to 1.973 | 42.5 | 60.9 | 86.2 |

Optimized Constants for Each Formula (N = 410)

SRK/T | A constant | 119.32 | 119.2 | 118.2 |

Haigis | a0 (a1, a2) | −0.12 (0.25, 0.20) | 1.46 (0.40, 0.10) | 1.40 (0.40, 0.10) |

Hoffer Q | pACD | 5.95 | 5.68 | 5.08 |

Holladay 1 | Surgeon factor | 2.13 | 1.91 | 1.34 |

Holladay 2 | ACD | 5.68 | 5.67^{a} | – |

Barrett | Lens factor | 2.15 | – | – |

EVO 2.0 | ACD | 119.38 | – | – |

Kane | A constant | 119.41 | – | – |

Olsen | ACD | 4.94 | – | – |

PEARL-DGS | Surgeon factor | 2.25 | – | – |

RBF 2.0 | A constant | 119.32 | – | – |

T2 | A constant | 119.32 | – | – |