INTRODUCTION

In recent years, the introduction of new intraocular lenses (IOLs), in particular toric, multifocal, aspheric, phakic and accommodative, accompanied by an increasing refractive requirement, led necessarily to the evolution of IOL power calculation formulas in order to achieve a greater effectiveness^1-3. Within the various third-generation formulas that combine regressive methods with theoretical models of the eye, SRK-T is considered the standard formula and, while providing excellent refractive results in patients whose eyes have normal anterior segment anatomy and axial lengths within normal values, it is not so predictable in eyes with unusual anatomy⁴. Although no formula is suitable for all eyes, the Barrett Universal II formula has been shown to be more accurate in calculating IOL⁵, not only for different types of lenses but also for eyes with any axial length dimension⁶. In addition to the formulas, new calculation methodologies have also been developed, such as the PhacoOptics (IOL Innovations Aps, Aarhus, Denmark), which uses ray tracing⁷, and Radial Basis Functions (RBF), which uses artificial intelligence for pattern recognition⁸. There are also online calculators available, such as Panacea, that combine several of these methodologies.

The integration of these new formulas and methodologies, many of them with free access, in daily clinical practice requires their previous validation in different populations.

3^rd GENERATION FORMULAS AND THE “SUPER FORMULA”

The 3^rd generation formulas, with SRK-T being one of the most used formulas, use only the axial length and the keratometry values ​​to estimate the effective position of the IOL. It is well known that certain 3^rd generation formulas have shown greater accuracy under certain conditions related to the variables used, such as axial length and corneal power. But since there is no single formula that works well in every eye, Ladas created the so-called "Super Formula" that incorporates the ideal segments of the SRK-T, Hoffer-Q, Holladay I (with Koch’s adjustment), and Haigis formulas, and developed a new method for representing formulas in 3 dimensions⁹. The "Super Formula" is freely accessible and can be used at http://www.iolcalc.com (Figure 1).

RAY TRACING

Ray tracing, based on Snell's Law, is a method for calculating rays that cross an optical system. This methodology has already been used in pseudophakic eye models to calculate IOL power¹⁰. Unlike conventional formulas, ray tracing does not contain approximations since it uses a measured, not deduced geometry, and does not include keratometric indices but only refractive indexes of the cornea, which avoids overestimation of corneal power when using habitual keratometric index of 1.3375, as well as the variability caused by the use of different keratometric indices used in the various measuring instruments.

One of the lightning-based platforms is PhacoOptics, by Thomas Olsen – which is available through purchase of a license. It requires six IOL constants and therefore is not optimized for all lenses on the market.

ARTIFICIAL INTELLIGENCE AND THE RADIAL BASE FUNCTION (RBF)

Radial Base Function (RBF) is a new methodology, recently developed by a team of surgeons and engineers coordinated by Warren Hill. RBF is a mathematical function used in artificial neural networks. It is used, for example, in facial recognition software, ECG interpretation and forecasting systems for financial markets. It uses artificial intelligence in recognition of patterns and performs so much better as more data is introduced. With this method the eyes are considered as a standard regardless of their axial length. It also uses a validation boundary model, indicating to the user when it is working in a defined area of ​​precision. Conversely, if the calculator does not know the type of measurements that are introduced, it transmits an out-of-bounds message. This methodology was optimized for the Lenstar LS900 (Haag-Streit) biometer and the SN60WF IOL (Alcon). It is freely accessible at http://www.rbfcalculator.com. Very recently, the RBF database has been expanded, including more extreme cases of high myopias and axial hyperopia.

PANACEA

Panacea is a calculator that combines theoretical bases, including statistical analysis for each variable, with the use of artificial intelligence. It was developed by David Flickier and includes the axial length, keratometry, anterior chamber depth and lens thickness as variables to estimate the effective position of IOL. It can also take into account topometric data, considering the relation between the posterior and anterior radii and corneal asphericity.

This calculator is available as a free application for iPad and Mac (Figure 2).

COMPARISON OF METHODOLOGIES

In the Implant-Refractive Department of the Hospital da Luz, Lisbon, the authors performed a retrospective study with the objective of comparing the subjective refractive error determined by four methods of calculating intraocular lenses (IOLs) – SRK-T and Barrett Universal II formulas, and PhacoOptics and RBF methodologies – with the residual subjective spherical equivalent, in a Portuguese population sample of 188 pseudophakic eyes with three types of monofocal lenses¹¹: Acrysof SA60AT (Alcon Labs, Fort Worth, USA), Acrysof SN60WF (Alcon Labs, Fort Worth, USA), or Tecnis ZCB00 (Abbot Medical Optics, Santa Ana, USA). Subsequently we also included in the analysis the results of calculations with the Super Formula and with the Panacea calculator.

RESULTS

Demographic and biometric data, such as age, sex, laterality, axial length, anterior chamber depth and mean keratometry, are presented in Table 1. Figure 3 shows the absolute median error between the predicted refractive error and the subjective spherical equivalent for each methodology. The RBF had a lower mean absolute error (0.26 D), followed by Barrett (0.27 D), SRK-T (0.29 D), PhacoOptics (0.29 D), and Panacea (0.29 D), all with very similar dispersions (Panacea with the lowest dispersion). SRK-T presented the highest dispersion of values, with a maximum error of 1.66 D. The Super Formula was the one with the highest absolute median error (0.31 D), with a statistically significant difference in relation to the RBF.

Results according to axial length

An analysis of the mean errors according to the axial length was made. For the axial length were considered small eyes (axial length ≤22 mm), medium (22-24.5 mm), and medium long/long (≥24.5 mm). Figure 4 shows the absolute median error according to the axial length. The RBF was the methodology that presented a lower absolute mean error for small (0.29 D) ​​and medium long/long (0.20 D) eyes, with Barrett's formula presenting a lower absolute mean error (0.27) for eyes with average biometric parameters in the population (axial lengths between 22 and 24.5 mm). These differences were not, however, statistically significant. The absolute mean error was higher in all methods for eyes with axial length ≤22.0mm.

For eyes with an axial length equal to or greater than 24.5 mm the differences between Super Formula (0.34 D) and RBF (0.20 D) and between Super Formula (0.34 D) and PhacoOptics (0.23 D) were statistically significant.

PERCENTAGE OF ABSOLUTE ERROR CONCERNING MAGNITUDE OF REFRATIVE ERROR

Figure 5 shows the percentage of cases for each methodology in which the absolute error was less than 0.25 D, less than 0.50 D, less than 1.00 D and greater than 1.00 D. In 97.9% of cases PhacoOptics had an error less than 1.00 D, followed by Barrett and Panacea with 97.3%, RBF with 96.3% and finally with SRK-T with 93.6%. For these differences the methodological comparisons were not statistically significant.

WHICH TO CHOOSE?

The six methodologies evaluated in the study had a good overall performance and revealed low mean absolute errors ranging from 0.26 D to 0.31 D.

The RBF, in the analysis of the absolute median error, was the one that most approached zero, with a statistically significant difference from Super Formula. It was also the methodology that allowed achieving a higher percentage of cases with absolute errors less than 0.50 D and less than 0.25 D.

The main limitation of the RBF is that it does not allow the calculation when the values ​​entered are outside the data base used. In fact, in the Monte Carlo analysis, due to infrequent combinations of biometric parameters, RBF was unable to calculate IOL potency in 36.7% of the cases.

PhacoOptics was the methodology that resulted in a greater percentage of cases with absolute error lower than 1.00 D. Together with Panacea, these were the methodologies that obtained the lowest maximum error and the lowest dispersion of results. In addition, the patient's topometric data were not considered in the Panacea. Probably the inclusion of this data can further improve the performance of this calculator.

In the study, the Barrett formula was the one that showed the lowest error variability for eyes with average biometric parameters in the population (axial lengths between 22 and 24.5 mm), which is in agreement with the good results reported for this formula¹². The Barrett Universal II formula is based on a theoretical model in which the anterior chamber depth (ACD) is related to axial length and keratometry⁶. It only requires a single constant and can be accessed online for free, which makes it a good option for IOL calculation under these conditions.

In small eyes (with an axial length of less than 22 mm) all methodologies had worse results, as previously reported^13,14, with an absolute mean error higher than that observed for mean axial lengths (between 22 and 24 mm) and medium long/long (≥24.5 mm).

In the larger eyes the RBF showed the lowest absolute median error, followed by PhacoOptics. However, it is important to note that the sample of smaller and larger eyes was reduced, making it difficult to draw robust conclusions about the different methodologies in these extreme cases. And extreme eyes are expected to perform better with the new beta version of RBF.

CONCORDANCE BETWEEN FORMULAS AND METHODOLOGIES

To analyze the impact in clinical practice of the differences between the various methodologies in the choice of intraocular lens power, the maximum difference between the potencies determined by the four methodologies was calculated for the totality of the sample to obtain more results near emmetropia.

Figure 6 shows the percentage of cases according to the magnitude of the maximum difference between calculated IOL and emmetropia. It was found that the differences between the IOL estimates for emmetropia are small, with 69.7% of the cases with differences equal to or less than 0.5 D.

In cases of average biometric parameters there is less variability between methodologies and discrepancies occur in extreme eyes with infrequent combinations of parameters.

COMBINATIONS OF INFREQUENT BIOMETRIC PARAMETERS

Effectively, the linearity and the regressive factors assumed by the SRK-T are a limitation to the calculation in combinations of more infrequent biometric parameters, where theoretically RBF will have an advantage. However, this methodology still has limitations in these situations because the database that it uses is already limited. The new beta version may come to optimize performance in these cases.

In the Authors' analysis, a Monte Carlo pseudo-population was also used with infrequent combinations of biometric parameters and the IOL power determination was compared. The Monte Carlo analysis uses the nominal value of each biometric parameter and the standard errors associated with the measurement technique. 500 Monte Carlo cycles were performed so that, in each cycle, all parameters varied randomly and simultaneously, using a Gaussian distribution. This analysis showed that, due to infrequent combinations, the discrepancy between the choice of lenses tends to be higher, with more than 50% of cases with more than 1.00 D of difference, and in 36.7% of cases the RBF methodology was unable to calculate IOL potency (Figure 7). In 39.4% of cases, SRK-T and RBF had exactly the same IOL calculation values.