**Purpose**:
To develop a machine learning regression model of subjective refractive prescription from minimum ocular biometry and corneal topography features.

**Methods**:
Anterior corneal surface parameters (Zernike coefficients and keratometry), axial length, anterior chamber depth, and age were posed as features to predict subjective refractions. Measurements from 355 eyes were split into training (75%) and test (25%) sets. Different machine learning regression algorithms were trained by 10-fold cross-validation, optimized, and tested. A neighborhood component analysis provided features’ normalized weights in predictions.

**Results**:
Gaussian process regression algorithms provided the best models with mean absolute errors of around 1.00 diopters (D) in the spherical component and 0.15 D in the astigmatic components.

**Conclusions**:
The normalized weights showed that subjective refraction can be predicted by only keratometry, age, and axial length. Increasing the topographic description detail of the anterior corneal surface implied by a high-order Zernike decomposition versus adjustment to a spherocylindrical surface is not reflected as improved subjective refraction prediction, which is poor, mainly in the spherical component. However, the highest achievable accuracy differs by only 0.75 D from that of other works with a more exhaustive eye refractive elements description. Although the chosen parameters may have not been the most efficient, applying machine learning and big data to predict subjective refraction can be risky and impractical when evaluating a particular subject at statistical extremes.

**Translational Relevance**:
This work evaluates subjective refraction prediction by machine learning from the anterior corneal surface and ocular biometry. It shows the minimum biometric information required and the highest achievable accuracy.

**RESUMEN**:

**Objetivo**:
El desarrollo de un modelo de regresión de aprendizaje automático prescripción refractiva subjetiva a partir de las características mínimas de la biometría ocular y la superficie corneal.

**Métodos**:
Los parámetros de la superficie corneal anterior (coeficientes de Zernike y queratometría), además de longitudes axiales y de cámara anterior, edades y las refracciones subjetivas no ciclopléjicas de 355 ojos se dividieron en un conjunto de entrenamiento (75%) y otro de test (25%) y se entrenaron diferentes algoritmos de regresión de aprendizaje automático mediante validación cruzada 10 veces, se optimizaron y se probaron sobre el conjunto test.

**Resultados**:
Los algoritmos de regresión del proceso gaussiano proporcionaron los mejores modelos con un error absoluto medio fue de alrededor de 1.00 D en el componente esférico y de 0.25 D en los componentes astigmáticos.

**Conclusiones**:
Los pesos normalizados mostraron que la refracción subjetiva puede predecirse utilizando únicamente la queratometría, la edad y la longitud axial como características. El aumento del detalle de la descripción topográfica de la superficie corneal anterior que supone una descomposición de Zernike de alto orden frente al ajuste a una superficie esferocilíndrica realizado por queratometría no se refleja en una mejora de la predicción de la refracción subjetiva, que es pobre, en cualquier caso, principalmente en el componente esférico. Sin embargo, la máxima precisión alcanzada difiere en sólo 0,75 D de la de otros trabajos con una descripción más exhaustiva de los elementos refractivos del ojo. De todos modos, el aprendizaje automático y los datos masivos aplicados a la predicción de la refracción subjetiva pueden ser arriesgados y poco prácticos cuando se evalúa a un sujeto concreto en los extremos estadísticos, aunque los parámetros elegidos puedan no haber sido los más ineficaces.

**Relevancia Traslativa**:
El trabajo evalúa la predicción de la refracción subjetiva mediante aprendizaje automático a partir de la superficie corneal anterior y la biometría ocular, mostrando la mínima información biométrica requerida y la máxima precisión alcanzable.

^{1}

^{,}

^{2}The two fundamental elements are the cornea and crystalline lens. Corneal topography can be assessed by different techniques like interferometry, ultrasonography, profile photography, holography, Placido disk principle, and Scheimpflug photography. Corneal surfaces are usually described with the keratometry

^{3}and/or Zernike coefficients,

^{4}

*c*, which result from fitting a series of Zernike polynomials to heights maps. The Zernike decomposition of a surface

_{j}*W*can be expressed as polar coordinates

^{5}:

*n*is the radial order,

*m*is the azimuthal frequency,

*j*is the single index for the Zernike polynomial,

*p*is the number of terms in the expansion,

*c*are the Zernike coefficients associated with their Zernike polynomial, δ

_{j}_{m0}is the Kronecker delta function, and ⌈ · ⌉ denotes the ceiling (round-up) operator. Regarding the crystalline lens, on the one hand, phakometry,

^{6}Scheimpflug imaging,

^{7}

^{–}

^{10}magnetic resonance imaging,

^{11}and optical coherence tomography

^{12}

^{,}

^{13}have been used to assess the in vivo optical properties of lens shape and lens thickness. On the other hand, in vitro techniques

^{14}

^{–}

^{18}have also been followed to evaluate lens shape and power.

^{19}

^{,}

^{20}photorefraction images,

^{21}retinal fundus images,

^{22}other ophthalmologic devices,

^{23}and intraocular lenses characteristics.

^{24}The aim of this work is to develop a machine learning regression model that predicts patients’ subjective refractive prescription from the anterior corneal surface and ocular biometry. The first step consists of choosing physiologic descriptors as the model's features. It is a key point because using too many features can degrade prediction performance, even if all features are relevant.

^{25}The strong correlation between the spherical power components and astigmatic components

^{26}of the anterior and posterior corneal surfaces, as well as the fact that the latter contributes only about one-eighth of the eye's refractive power,

^{27}leads to the hypothesis that the anterior surface suffices to assess the whole corneal refractive effect on an eye model. Apart from anterior corneal topography, other physiologic parameters that are expected to be involved in patients’ subjective refraction were added to the model: axial length (AL) and then constructing a similar approach to Emsley schematic eye; the patient's age, on which refractive indices and crystalline lens morphology depend; and, finally, anterior chamber depth (ACD), which can be related to lens location. Compared to previous works,

^{20}this proposal offers the advantages of using simpler measurement devices, requiring fewer descriptive data and, therefore, faster performance. However, disregarding the crystalline lens effect will probably provide poor results. In fact, the distribution of some aberrations between the cornea and lens appears to be auto-compensated.

^{28}Next, the selection and tuning of machine learning models must be performed with training and test population subsets. The Results section shows an analysis of the main indispensable characteristics and the evaluation of the final predictive accuracy of the selected models.

*S*,

*C*, α) was transformed through equation (5) to power vector notation

^{29}(M, J

_{0}, J

_{45}), which consists of components that are independent of one another. Figure 1 shows the histograms of the age and power vector components of the population sample herein used.

*C*,α

_{k}_{k}) were selected from the measurements taken by the Sirius Topographer as corneal physiologic descriptors. As the device does not provide defocus coefficient

*c*

_{4}, we used the keratometry equivalent sphere (

*M*) instead, which was computed from the keratometry for the same pupil diameter as

_{k}*n*is the keratometric index and

_{k}*R*and

_{f}*R*are, respectively, the flattest and the steepest anterior corneal curvature radius. The conventional keratometric index (1.3375) was used in this work. However, its value is not relevant for machine learning algorithms because it is constant. Keratometry data (

_{s}*C*,α

_{k}_{k}) were also transformed into standard power vector notation

^{29}according to equation (5).

*c*

_{0}) and defocus (

*c*

_{4}), that is, 34 coefficients from the anterior corneal surface decomposition that are above the maximum limit of 23 features. Bearing in mind that a patient’s age,

*M*, ACD, and AL are all features that characterize the eye apart from cornea, a thus-detailed anterior corneal surface description can be foregone to meet that limit. Therefore, those coefficients, and excluding tilts (

_{k}*c*

_{1}and

*c*

_{2}) for their little relevance as they can be naturally compensated by eye movements, were selected in ascending order without exceeding that limit. The fifth order includes 17 Zernike coefficients that, together with age,

*M*, ACD, and AL, make up a set of 21 selected features as a first approach, to characterize each eye and to train the models. As part of the preprocessing data step, a filter-type feature selection algorithm

_{k}^{30}that used a diagonal adaptation of the neighborhood component analysis

^{31}was applied to determine the features’ normalized weights (NWs). This reports about the importance of each feature in the regression models of the power vector components and allows different feature selections to train and tune the new models to be run. An extra parameter with a random value (

*rv*) was added to check the significance of the 21 selected features in the model. The NW of the extra random parameter is expected to be zero. If not, any feature with an NW that equals or goes below that can be discarded because its significance would be equal or worse than that of a random variable.

^{32}of MATLAB (version R2021b; MathWorks, Inc., Natick, MA, USA), which provides linear regression models,

^{33}regression trees,

^{34}support vector machines (SVMs),

^{35}Gaussian process regression (GPR) models,

^{36}ensembles of trees,

^{37}

^{–}

^{39}and neural networks.

^{40}Algorithms were trained to predict each refraction vector component separately because of their mathematical independence. The models that obtained the best root mean squared errors (RMSEs) were later hyperparameter-tuned by Bayesian optimization. Finally, the best models were tested with the test subset data. Otherwise, a model might only perform well with the training data but may fail to predict anything useful in yet unseen data.

^{32}with the 266 eyes from the training subset characterized by the 21 features and by following the above-described procedure. Table 2 shows the RMSE and the coefficient of determination (

*R*

^{2}) values obtained for each power vector component with the trained models, which gave better results.

*R*

^{2}compares the trained model to the model with a constant response, and it equalled the training response mean. If the model is worse than this constant model, then

*R*

^{2}is negative and the model is discarded.

*R*

^{2}obtained from the test subset for each power vector through the optimized GPR model trained for 21 features.

*c*

_{6}(vertical trefoil),

*c*

_{7}(vertical coma),

*c*

_{12}(primary spherical),

*c*

_{13}(vertical secondary astigmatism), and

*c*

_{19}(oblique secondary trefoil). Hence, the M component features were consecutively in NW importance terms AL, the keratometry equivalent sphere, and age. Both J

_{0}and J

_{45}were respectively modeled using the primary astigmatism Zernike coefficient (

*c*

_{3}and

*c*

_{5}). The independency of the three components was evidenced because they showed different NWs of features. Hence, a different subset of features could be selected for each one to perform the regression models. However, common selections of features have been sought for three components for simplicity's sake and to obtain only a set of features for each threshold. Table 4 shows the results, for the test subset, of the trained and optimized GPR models by considering features according to these two different thresholds for the three power vector components.

*R*

^{2}of component J

_{45}improved. The results of the optimized GPR model with all features and of that with those of NW >20% (five features) were similar. Consequently, a regression model with only these five parameters can achieve maximum accuracy. Disregarding high-order aberrations and ACD did not significantly make the results worse.

*R*

^{2}from the test subset for the models trained with keratometry, age, and AL. One again, the results were not different from those obtained using the models with all the features.

_{0}and J

_{45}was good (around 0.25 diopters [D] of RMSE), but the coefficient of determination for J

_{45}was poor. For spherical component M, although

*R*

^{2}was good, the error (MAE around 1.00 D) seemed to be too high to employ it as a technique to predict subjective refraction. Notwithstanding, the achieved accuracies were not much worse than those reported in previous works,

^{20}although the presented proposal used simpler measurement devices (a keratometer would work) and required fewer descriptive data.

_{0}and J

_{45}. This could indicate the age dependence of astigmatism.

^{42}

^{,}

^{43}The ACD effect, which was hypothesized to be related to the lens location in the eye, and the higher-order Zernike coefficients effect were assessed from the results obtained with the parameters of NW >0 in the M component model. The result obtained considering these effects was not significantly better (only about 0.1 D of MAE) than those obtained in the model with the features of NW >20%, which excluded these parameters in relation to the previous one

_{0}component prediction but with an NW below 20% and, hence, its poor significance.

_{45}and J

_{0}showed other features apart from c

_{3}and c

_{5}(oblique and vertical astigmatism Zernike coefficients, respectively), albeit with lower NWs. The models that optimized GPR and only employed the astigmatic components were also trained and tested. The results in Table 5 are practically no different from those obtained using all the parameters with NWs other than zero, which implies that anterior corneal surface topography determines astigmatism. These results, together with the fact that the errors of models for astigmatic components were lower than the error of the spherical ones, confirmed the strong correlation between the astigmatic components

^{26}of the anterior and the posterior corneal surfaces.

^{20}who contemplated whole-eye aberrations. Therefore, the contribution of the posterior corneal surface and the crystalline lens to the eye's astigmatism falls within those errors. For the spherical component, the difference with that work

^{20}was below 0.75 D. Therefore, by assuming the strong correlation between the spherical power

^{26}of corneal surfaces, disregarding the lens would be partially, but not completely, compensated by considering age. The difference in patients’ demographics between both works lay in them having a refractive spherical equivalent between −6.75 D and 6.13 D, which, in this work, is between −11 D and 6.25 D. This could also be a cause of the worse results herein obtained for the widest range of ametropies.

^{44}by a well-trained eye care professional, and its subjective nature implies deviations in the spherical component around ±0.25 D

^{45}

^{,}

^{46}and in the astigmatic component above 0.75 D

^{47}in both intra- and interoptometrist variability.

^{20}offers the advantages of using simpler measurement devices, requiring fewer descriptive data, and, therefore, having faster performance. It fits in with the unsupervised methods’ trend of minimizing misunderstandings between the clinician and the patient and the measurement variability and time. However, we must consider that the proposed approaches statistically predict subjective refraction within a tolerance range for a population, and, individually, the errors that they may make can be intolerable. So although the choice of input parameters in this work might not be the most predictive efficient, a general model should clearly show its extreme errors and not merely its average performance. Otherwise, establishing a standard predictive method based on machine learning algorithms and big data can be risky and impractical when evaluating a particular patient.

**J. Espinosa**, None;

**J. Pérez**, None;

**A. Villanueva**, None

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