Introduction

Today, cataract surgery is not just a simple procedure to remove the opacity of the crystalline lens but has become an operation in which the final refractive result is significant. It has been considered a refractive surgery to offer patients independence from their glasses.

It implies the need for increasingly efficient and accurate calculations. To this end, calculation formulas have been evolving and adding more and more variables to their structures. Initially, first-generation formulas, such as Fyodorov, Colenbrander, Hoffer, or R. Binkhorst1, considered that the effective lens position (ELP) was fixed and independent of the anatomical characteristics of the eye and only varied according to the characteristics of the crystalline lens, which led to a lack of predictability of the position and refractive results. The second-generation formulas, such as SRK or SRKII2, introduced by the Hoffer ACD/AL adjustment followed by R. Binkhorst, substantially improve ELP prediction. Third-generation formulas such as Hoffer Q3, Holladay 14, and SRK/T5 added keratometry to the AL to adjust ELP. Finally, the fourth generation formulas started to add multiple variables in their formulas: Haigis6 incorporated the anterior chamber depth measured from epithelium to lens (ACD) preoperatively, Holladay II7, and Barrett Universal II8 added lens thickness (LT) and horizontal corneal diameter (CD) and Kane introduced corneal center thickness (CCT)9. Patient gender was first introduced by Hoffer & Savini with the Hoffer H-5 formula; Olsen10, in addition to introducing essential variables such as AL and K, using a C constant calculated with the preoperative ACD in the new generation formulas. Panacea11 included the corneal sphericity value and the ratio between corneal anterior and posterior curvature. Most of these variables are obtained from the anterior segment of the eye. However, we have yet to find publications that evaluate the influence of corneal variables as corneal aberrations and their influence on calculating the IOL power since optical biometry has been used. Therefore, this study aims to determine the importance of different biometric, topographic, and retinal variables for calculating IOL power and evaluates the statistical importance of new variables.

Methods

Study design and patient selection

A retrospective cross-sectional observational study used a database of 422 patients who underwent cataract surgery at the Oftalvist Center in Almeria between January 2021 and December 2022. The sample size was calculated using the Granmo calculator, which accepted an alpha risk of 0.05 and a beta risk of 0.2 in a bilateral contrast.

Patient information

The study complied with the declaration of Helsinki and obtained permission from the ethics committee of the Department of Nursing, Physiotherapy, and Medicine with code EFM 179/2022. Furthermore, all patients signed informed consent for the scientific use of their anonymized clinical data, which could be revoked.

Inclusion Criteria.

The inclusion criteria were:

  • Patients without previous refractive surgical procedures such as laser-assisted in situ keratomileusis (LASIK), photorefractive keratotomy (PRK), or incisional surgery.

  • Previous corneal diseases, such as keratoconus or corneal scarring, with a history of previous intraocular surgery.

  • Patients without macular or retinal abnormalities.

  • Uncomplicated surgery.

Only one eye was selected from each patient.

Variables collected

The qualitative variables were gender and age. Preoperative and postoperative corrected visual acuity (BCVA) ( decimal scale) was obtained with the Topcon IS-600 projector.

Postoperative residual refraction and preoperative and postoperative intraocular pressure (IOP) were measured with the Topcon TRK-2P auto-refractometer. Central corneal pachymetry, anterior-to-posterior corneal ratio, anterior chamber depth (ACD), white-to-white (WTW) equivalent to corneal diameter (CD), spherical aberration, astigmatic aberration, and coma aberration were obtained with Pentacam.

For the Haigis, Hoffer Q, Holladay 1, and SRK/T formulas, axial length, preoperative anterior keratometry (K1 and K2), implanted IOL power, and refractive residuals of each formula were obtained with IOL Master 500. Macular thickness was evaluated with Cirrus HD-OCT 500.

The KANE formula, refractive residual, and emmetropic lens power were obtained using the online calculator.

The datasets generated and analysed during the current study are available in : Castro de Luna, Gracia; Sanchez Liñan, Noelia (2024), “ OCULAR BIOMETRY DATA”, Mendeley Data, V1, doi: https://doi.org/10.17632/4twnn4546j.1.

Statistical analysis

Patient demographic and biometric data were described as frequencies (percentages) and means (SD). The normality of the data was assessed using the Kolmogorov-Smirnov test. Parametric tests, such as Pearson correlations, and non-parametric tests, such as Spearman correlations, were applied, as appropriate, to compare demographic and biometric data. A significance level of α = 0.05 was considered. The importance of each variable was analyzed using the random forest technique, which established a ranking of importance. It is essential to validate the model to ensure that the significance of variables is meaningful and robust across different datasets performing cross-validation.

Training the Random Forest model:

The dataset was splitted into training and test sets using the above cross-validation and fitted the Random Forest model to the training data. The evaluation metric was Mean Squared Error (MSE). Once the model was trained, its performance was evaluated using the MSE on the test set.

The MSE was calculated as the mean squares of the differences between the model’s predictions and the actual values of the target variable ( IOL power) in the test set.

The lower the MSE value, the better the model’s performance, as it indicated that the model predictions fit the test data well. The MSE provided a measure of the quality of the model predictions regarding the quadratic difference between the predictions and the actual values.

A low MSE indicated that the model can make accurate predictions and is generalizing well to unseen data. On the other hand, a high MSE suggested that the model may be under-fitting or over-fitting the data, meaning that it needs to be adjusted or that the set of features used may not be sufficient to capture the relationship between the features and the target variable. Statistical analysis was performed with SPSS software for Windows (V.27.0, SPSS) and R software version 3.5.1.

Results

A total of 422 patients were evaluated. The age mean was 69.82 ± 9.08. 58% were female. The database was divided into a database training (80%) and a database test (20%).to validate the model.

Table 1 evaluated the descriptive characteristics of the biometric variables. The axial length mean was 23.49 mm.

Table 1 Descriptive preoperative parameters.
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Figure 1 shows the % lnc MSE. The MSE has been transformed using the natural logarithm, and its interpretation is inverse to those mentioned above. So, the higher the %lnc MSE, the more adjusted the parameter is to the target variable, in this case, the IOL power.

Fig. 1
figure 1

Random Forests. Importance of different variables.

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Fig. 2
figure 2

Variable importance based on MIC.

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The importance of variables in Random Forests based on the MIC (Mutual Information Criterion) is an alternative measure used to assess the contribution of each variable in the model’s prediction (Fig. 2). Unlike information gain, which is based on reducing entropy at the tree nodes, MIC measures the amount of information shared between a variable and the target variable.

The MIC is a measure of the dependence between two variables. In the context of Random Forest, the MIC is calculated between each predictor variable and the target variable. The higher the MIC, the more information the variable shares with the target variable, in this case, IOL power, and therefore, the more important it is for the model. As with information gain (Fig. 3), variables are ordered according to their importance based on the MIC. Variables with a higher MIC are considered more important and are placed at the top of the list, while variables with a lower MIC are placed at the bottom.

The MIC-based significance of variables in Random Forest provides an alternative measure to assess the relevance of variables in model prediction. This measure can be helpful in conjunction with other variable importance assessment techniques to understand better how features contribute to model performance.

Fig. 3
figure 3

Variable Importance based on information gain.

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Information gain measures how much information a variable provides to predict the target variable. In the context of Random Forest, it calculates how much the entropy (or purity) of the tree nodes decreases when splitting them according to a specific variable. The greater the information gain, the more influential the variable is for the model (Fig. 3).

Variable importance based on MIC or information gain is useful in feature selection: As with information gain, the importance of variables based on the MIC can be used to select the most relevant features for the model. Eliminating features with a low MIC can help simplify the model and improve its generalisability.

Finally, we calculated the correlation between the variables of importance in the random forest and the implanted lens power (Table 2). Spearman’s non-parametric test was applied. A positive correlation was found with the following variables.: Al, K1 pre (high correlation with K2 preop), ACD, WTW, lens constant, and astigmatism aberration. The variables coma aberration (p. value = 0,12) and macular thickness (p. value = 0,10) were almost significant. If we increased the sample, their significance would increase.

Table 2 Correlations between random forest variables and IOL power.
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Discussion

Incorporating variables into the intraocular lens calculation formulas has improved the calculation of the IOL to be implanted for years. The introduction of AL and keratometry was fundamental. Variables have been added to improve the efficiency of the formulas they generate.

Improving the accuracy of IOL calculations is becoming increasingly important since the trifocal and extended depth of focus (EDOF) type lenses are implanted. These lenses require more excellent refractive outcome precision to maximize visual efficiency and avoid being penalized by unexpected refractive outcomes.

In the study by Ribeiro et al.12, they evaluated the changes in each optical parameter on the eye’s refractive state. They found that a 0.25 mm change in ACD measurement corresponds to an error of 0.10 D in an eye with an AL of 30.0 mm and 0.50 D in an eye with an AL of 20.0 mm. Following these results, studies were carried out assessing various ranges of ACD values to evaluate which formula was more effective13,14.

However, formulas such as Hoffer Q3 or SRK/T5, still very frequently used today, did not include ACD in their variables.

Thickening of the lens occurs throughout life, predominantly towards the anterior chamber15. In an older population, often in the cataract population, the lens is thickened, and therefore, the depth of the ACD is shallowest. Ribeiro et al.12 found that for every 1% increase in lens thickness, the refractive error changes by 0.097; a 0.104 mm change in LT means a 0.25 D change in refractive error. Studies have already been published relating the different ocular parameters, including LT, and the prediction error of the different formulas16,17.

However, no studies support incorporating new variables such as corneal aberrations and retina measurements. In our study, AL, ACD, preoperative K1 and K2, lens constant, and CD(WTW) are “important” variables, and they are correlated to the IOL power, but in addition, some new variables, as astigmatism aberration could be added to the new formulas. Other corneal aberrations, such as anterior coma aberration, and retina variables, such as preoperative central macular thickness, were slightly statistically significant variables. It is necessary to increase the sample size to determine whether they become statistically significant variables. It is the main limitation of our study. The introduction of corneal aberrations in biometric formulations could mean the customization of lenses and the improvement of the patient’s optical aberrations in the intraocular lens implanted in cataract surgery. Artificial intelligence introduces new formulas based on classification models that introduce unexpected variables by applying new techniques, not only from machine learning but also from neural networks, to determine if the importance of new variables not taken into account can improve the new generation of biometric formulas.

Conclusions

In cataract surgery, the implanted IOL power is mainly correlated with axial length, anterior chamber depth, corneal diameter, lens constant, and preoperative keratometry. New variables such as astigmatism and anterior coma aberration and retina variables such as preoperative central macular thickness could be included in the new generation of biometric formulas based on artificial intelligence techniques.