Ocular Biometric Components in Hyperopic Children and a Machine Learning-Based Model to Predict Axial Length

Abstract

Purpose: The purpose of this study was to investigate the development of optical biometric components in children with hyperopia, and apply a machine-learning model to predict axial length.

Methods: Children with hyperopia (+1 diopters [D] to +10 D) in 3 age groups: 3 to 5 years (n = 74), 6 to 8 years (n = 102), and 9 to 11 years (n = 36) were included. Axial length, anterior chamber depth, lens thickness, central corneal thickness, and corneal power were measured; all participants had cycloplegic refraction within 6 months. Spherical equivalent (SEQ) was calculated. A mixed-effects model was used to compare sex and age groups and adjust for interocular correlation. A classification and regression tree (CART) analysis was used to predict axial length and compared with the linear regression.

Results: Mean SEQ for all 3 age groups were similar but the 9 to 11 year old group had 0.49 D less hyperopia than the 3 to 5 year old group (P < 0.001). With the exception of corneal thickness, all other ocular components had a significant sex difference (P < 0.05). The 3 to 5 year group had significantly shorter axial length and anterior chamber depth and higher corneal power than older groups (P < 0.001). Using SEQ, age, and sex, axial length can be predicted with a CART model, resulting in lower mean absolute error of 0.60 than the linear regression model (0.76).

Conclusions: Despite similar values of refractive errors, ocular biometric parameters changed with age in hyperopic children, whereby axial length growth is offset by reductions in corneal power.

Translational Relevance: We provide references for optical components in children with hyperopia, and a machine-learning model for convenient axial length estimation based on SEQ, age, and sex.

Introduction

Hyperopia is a common pediatric eye condition in Western countries,^1–3 with refractive errors greater than +3.25 diopters (D) impacting up to 11.9% of non-Hispanic White children in the United States aged 3 to 5 years.⁴ Among hyperopic children, accommodative esotropia and amblyopia are often reported, particularly among children with hyperopia exceeding +4.00 D. Over 60% of children with hyperopia exceeding +4.00 D have accommodative esotropia.^5–7 Among preschoolers who failed automated vision screening due to hyperopia ≥+3.75 D, 47% had esotropia at their initial ophthalmologic examination or developed esotropia during 3.5 years follow-up and 74% had or developed amblyopia.⁷ 

Unlike those with mild refractive error or emmetropia,^8–10 children with moderate or high hyperopia show minimal changes in their refractive error as they age.^11–15 This suggests that ocular growth in hyperopic children may significantly differ from that in emmetropic or myopic children. It is worth noting that hyperopic adults tend to have shorter axial lengths (ALs),¹⁶ which implies that hyperopic eyes may stop growing at a younger age. Alternatively, ocular growth may continue, but changes in corneal and lens powers may compensate for it, resulting in little change in refractive error with age. 

Although ocular biometric components have been widely studied in emmetropic and myopic children,^17,18 data on ocular biometric components for hyperopic children are sparse, particularly from children with moderate to high hyperopia. Some studies have focused on emmetropization during infancy and preschool years, and included primarily low hyperopia (+1.00 to +3.00 D) in their cohort.^8–10 Additionally, accommodative esotropia and anisometropia, which are prevalent among children with moderate to high hyperopia, were either not included or were explicitly excluded from most studies.^8–10,19 

Of the few studies that have investigated ocular biometric components in children with moderate to high hyperopia, most focused solely on AL. Uretmen et al. investigated children with accommodative esotropia and observed that AL and age were correlated but only weakly.²⁰ Debert et al.²¹ focused on children with esotropic amblyopia, and reported that the amblyopic eyes had higher power and shorter AL than the fellow eyes. Among hyperopic children aged 4 to 12 years with hyperopia of +2.25 D to +8.25 D, Khan²² proposed prediction of AL from spherical equivalent (SEQ) and corneal radius (CR) with a linear regression model: ([−0.04 *SEQ] +2.98) *CR. However, the Khan study included only Saudi children with accommodative esotropia and did not include sex or age as a variable. A recent retrospective longitudinal study involving Chinese hyperopic children (≥+5.00 D) observed an increase in AL alongside a decrease in SEQ²³; however, it remains uncertain whether these findings can be generalized to the US population. In summary, ocular biometric components data for American children with moderate to high hyperopia are sparse and their relationship with SEQ refractive error is still unclear. 

Of note, AL is an important parameter of interest because it is significantly related to the magnification of retinal images. When analyzing an optical coherence tomography (OCT) or OCT angiography image, magnification needs to be corrected for AL.^24,25 In school-age children, there is a linear relationship between AL and SEQ; each diopter difference in SEQ correlates with 0.28 mm of AL increase for 6 to 7year olds and 0.32 mm for 12 to 13 year olds.²⁶ As a result, a linear relationship of SEQ and AL derived from cohorts that included primarily emmetropic and myopic children can be used as a proxy to derive an estimate of AL when the AL is not available for the interpretation of OCT images. Unfortunately, we do not have sufficient data to support extending this proxy approach to predicting AL from SEQ for children with moderate to high hyperopia. Moreover, AL in children is related to age and sex,^11,27–29 and the growth of AL is likely to be nonlinear. Thus, predicting AL from SEQ for hyperopic children is complex and a linear regression may be insufficient to predict AL accurately. 

We chose to evaluate a recursive partitioning machine-learning model, specifically a classification and regression tree (CART) model, to understand the relationship between AL and SEQ in hyperopic children. 

The CART model is a commonly used machine-learning analysis. Algorithms follow a simple general rule, including partitioning the observations by univariate splits in a recursive way and fitting a constant model in each cell of the resulting partition until a stopping criterion is fulfilled. The advantage of CART analysis is that it is applicable to both parametric and non-parametric data and to nonlinear relationships; there is no need to make assumptions of normality or linearity. CART analysis can be implemented in two ways: (1) regression trees, which use the average value of examples at leaf nodes to make numeric predictions, and (2) model trees, which build a regression model at each leaf node in a hybrid approach. Both implementations yield a simple straightforward tree structure that facilitates an easy prediction even for non-experts. Hence, we applied CART analysis to predict AL with SEQ. 

This prospective cross-sectional study aimed (1) to investigate ocular biometric components in children with a wide range of hyperopic errors as a function of age and sex; (2) to investigate associations between hyperopia and ocular biometric components; and (3) to develop a machine-learning – CART model to predict AL from hyperopic SEQ for use when AL is not available. 

Methods

Participants

The research protocol observed the tenets of the Declaration of Helsinki and was approved by the Institutional Review Board of the University of Texas Southwestern Medical Center. Policies and procedures conformed to the requirements of the United States Health Insurance Portability and Privacy Act (HIPPA). Participants were referred to the Retina Foundation of the Southwest in Dallas, Texas, by Dallas-area pediatric ophthalmologists. Informed consent was obtained from each participant's parent or guardian after explanation of the nature and possible consequences of the study. 

Eligibility criteria included: (1) 3 to <12 years old; (2) a cycloplegic refraction (1% cyclopentolate) performed by the referring ophthalmologists within 6 months of the optical component measurements; and (3) SEQ ≥ +1.00 D and < +10.00 D. 

Exclusion criteria were gestational age at birth of ≤32 weeks, developmental delay, and co-existing ocular or systemic conditions. 

Procedures

Medical records were obtained from the referring pediatric ophthalmologists to extract cycloplegic retinoscopy refraction, which was recorded in conventional form as a sphere, plus the cylinder, and axis. To analyze refractive error, we converted the cylinder and axis into power vector format: the sphere and cylinder were converted into SEQ value: SEQ = sphere + 0.5*cylinder, J₀ (positive J₀ indicates with-the-rule [WTR] astigmatism, negative J₀ indicates against-the-rule [ATR] astigmatism), and J₄₅ (oblique astigmatism; positive J₄₅ indicates 135 degrees astigmatism, whereas negative J₄₅ indicates 45 degrees astigmatism).³⁰ 

At the study visit, ocular biometric components were measured by examiners with LenStar LS 900 (Haag Streit Diagnostics, Bern, Switzerland); all examiners were trained and certified on the measurement procedures. The participants were measured under their natural pupil without dilation and the following 5 parameters were recorded: (1) AL (mm): the distance between the front surface of the cornea, including the tear film, and the inner limiting membrane. (2) Anterior chamber depth (ACD, mm): the distance between the posterior corneal (endothelial) surface to the anterior lens surface. (3) Lens thickness (LT, mm): the thickness of the crystalline lens. (4) Central cornea thickness (CCT, microns). (5) Anterior radius of the corneal power (K1 and K2, D): for the flattest and steepest meridian, respectively. 

Data Analysis and Statistics

The following parameters were calculated: 

• • Mean corneal power (K, D): K = (K1 + K2)/2.
• • The mean anterior corneal curvature (CR, mm): (CR = 1000*(1.3375-1)/K.
• • The axial length-corneal radius ratio (AL/CR): AL/CR (reported to have a stronger correlation with SEQ than AL).³¹
• • Corneal toricity (K2-K1, D): corneal power in the vertical minus the horizontal meridian (positive values are WTR).³²

Data analysis was performed and plotted using R 4.2 Statistics (R Core Team, Vienna, Austria). Descriptive statistics are presented as mean ± standard deviation (SD). Comparisons of prevalence were conducted by the Chi-squared test. 

To compare age groups and sex, we conducted analysis on both refractive errors and biometric components using a mixed-effect model to adjust for inter-eye correlation.³³ Age and sex were treated as fixed effects; individual and eye as random effects. To predict AL from SEQ, we used CART analysis from the R 4.2 rpart (recursive partitioning) package. (1) Data sets were randomized to avoid any impact from testing order. (2) Data were partitioned; 160 (75%) were used as training data and the remaining 52 (25%) were used to test for the accuracy of the prediction CART analysis. (3) The model was trained with rpart using a 10-fold cross-validation analysis to assess the generalizability of the model. (4) The tree was pruned based on cross-validated prediction error to choose different numbers of splits of the tree. The “10-fold cross-validation” is a common choice that strikes a balance between robust evaluation and computational efficiency so that a model can perform well even on new unseen data. It works by dividing the data into 10 equal parts, or “folds,” then goes through 10 rounds of testing. In each round, nine folds are used for training the model and one fold is used for testing the model's performance. After 10 rounds, average results of the model performance provided an evaluation of the model. (5) The CART model was evaluated by comparing predicted and measured refractive error by calculating the mean absolute error (MAE) and root mean square error (RMSE). (6) The CART model was compared with the common linear regression model. We used the same data set to train a linear regression model and compared the prediction of CART model with the linear regression model by Bland-Altman analysis. 

Results

Demographics and Refractive Error

A total of 212 children were enrolled. Table 1 summarizes the demographic and refractive error data for the overall cohort and for each of the three age groups separately. Female children comprised 53% of the cohort; the percentage of female children in the three age groups were similar (χ² = 0.5,  P = 0.80). Most participants (73%) were Non-Hispanic White, with no significant difference in the percentages among the three ages (χ² = 1,  P = 0.50). Overall, 59% had moderate or high hyperopia (≥+4.00 D), 54% had anisometropia, and 59% had accommodative esotropia. 

Table 1.

 

Demographic Characteristics of the Hyperopic Participants

 

Table 2A provides cycloplegic refraction data among the three age groups. SEQ ranged from +1.00 to +9.38 D; there was no significant difference in SEQ between sexes (P = 0.43), but the 9 to 11 year group had lower SEQ than the 3 to 5 year group by about 0.49 D (P < 0.001). Overall, 48% of the cohort had significant astigmatism (≥1 D), primarily WTR astigmatism (J₀ > 0). There was no significant difference in cylinder magnitude, J₀, or J₄₅ among the three age groups, or between sexes (all P > 0.05). 

Table 2A.

 

Comparison of Cycloplegic Refractive Errors Among Three Age Groups (OD, the right eye; OS, the left eye)

 

Ocular Biometric Components by Sex and Age Group

Ocular biometric components are summarized by age group and sex in Figure 1, with statistical details provided in Table 2B. With the exception of CCT, there was a significant sex difference in all other measured components (AL, ACD, LT, K1, and K2, P < 0.01). Compared with girls, boys in general had longer AL, longer ACD, thicker LT, and flatter K1 and K2. As for the calculated parameters, K or CR was significantly different between boys and girls (P < 0.001); boys had flatter K and lower CR than girls. Note, there was no sex difference in the AL/CR ratio (P = 0.38), or corneal toricity (P = 0.99). The older age groups had significantly longer AL, longer ACD, and flatter K than the 3 to 5 year group (P < 0.001). The older age groups had higher AL/CR ratios (all P < 0.01). 

Figure 1.

 

Boxplots of SEQ (A) and ocular biometric components AL (B), ACD (C), LT (D), CCT (E), K1 (F), K2 (G), K (H), AL/CR ratio (I) in three age groups and sex. Note: K and AL/CR ratio were calculated.

Table 2B.

 

Comparison of Ocular Biometric Components (Measured and Calculated) Among Three Age Groups

 

Adjusting for sex, AL, ACD, K1, K2, and AL/CR ratio had significant differences among the age groups; whereas CCT and LT only showed a difference in one age group. The 3 to 5 year group had significantly shorter AL, shorter ACD, and steeper K1 and K2 (P < 0.01), consistent with significant growth of the eyes (AL and ACD) along with a flattening of K2 in the older age groups. The calculated parameters, AL/CR ratio, increased with age (P < 0.01). In addition, the correlation of AL/CR ratio with SEQ was R = −0.86 (P < 0.001). 

Classification and Regression Tree Model to Predict AL With SEQ

Figure 2 shows the CART model. Accuracy of the CART model predictions, (MAE = 0.60 and RMSE = 0.75) was better than those of the linear regression model (MAE = 0.76 and RMSE = 1.01). Correlation with test data was R = 0.73 for the CART model and only R = 0.47 for the linear regression model. 

Figure 2.

 

Classification and regression tree (CART) model to predict axial length (AL) with spherical equivalent (SEQ), age and sex. CART analysis identified SEQ = 3.94 D, SEQ = 7.06 D, Age = 6.83 years, and Age = 6 years as threshold values for axial length classification.

In a CART model, each node in the tree represents a decision based on a feature (or predictor variable) and a threshold value, reflecting how the data should be partitioned based on the selected features. Here, the selected features are SEQ, sex, and age. All three input variables of SEQ, age, and sex were critical forks in the model. The first split was at SEQ = +3.94 D. For children with SEQ ≥ +3.94 D, the next split was SEQ = +7.06 D and, if SEQ was ≥ +7.06 D, the model predicted AL = 20.3 mm. If SEQ < +7.06 D the model classifies on the basis of age, with AL = 21.5 mm for children ≥6.83 years, and if the age is <6.83 years old, the model incorporates sex as the final spilt, predicting AL = 20.7 for female children and AL = 21.2 for male children. On the other limb of the classification tree, for children with SEQ < +3.94 D, the next split was sex. Girls <6 years of age are predicted to have AL = 21.5 mm and, if ≥6 years, AL = 22.1 mm. For boys with SEQ < +3.31 D, the prediction is AL = 21.8 mm and for boys with SEQ ≥3.31 and AL = 22.9 mm. 

To compare with the CART model, we also fitted a linear regression model. Figures 3A and 3B show Bland-Altman plots of prediction versus data from the CART and the linear regression model. Table 3 summarizes statistics derived from Bland-Altman analysis. Compared with the CART model, the linear regression model had less bias but a larger limit of agreement (LOA); whereas the mean bias was low, the linear regression model overestimated when AL was shorter and underestimated when AL was longer. 

Figure 3.

 

Bland-Altman plot prediction vs testing data of two models. (A) CART model; (B) linear regression model. The mean difference between the model prediction and the test data of each individual is plotted against the mean of the model predicted data and the test data. The dashed line in the middle shows the bias, and the dashed lines on the top and bottom show the upper and lower 95% of limit of agreement (LOA); whereas ±1.96 SD were plotted in the fine dashed lines with the shading area corresponding to upper 95% LOA, bias, and lower 95% LOA, respectively.

Astigmatism and Corneal Power

Cylinder magnitude was negatively correlated with K1 (R = −0.25, P < 0.001) and positively correlated with K2 (R = 0.30, P < 0.001). We also calculated corneal toricity (K2 and K1), which is highly correlated to cylinder astigmatism (R = 0.845, t = 20, P < 0.001). This indicates that the corneal toricity significantly contributed to cylinder magnitude. 

Discussion

To our knowledge, this is the first paper to study ocular biometric components of a large cohort of children with a broad range of hyperopic refractive errors; 59% of participating children had hyperopia exceeding +4.00 D. Whereas CCT did not show significant sex differences, other ocular biometric components such as AL, ACD, K2, and LT did exhibit significant differences between boys and girls. Although SEQ was similar among the three age groups, children with hyperopia aged 9 to 11 years had longer AL and longer ACD and flatter K than the 3 to 5 year group. We developed a CART model to determine AL from SEQ and provide a valuable reference for clinical practice. Moreover, instead of arbitrarily defining the severity of hyperopia, it allows us to classify hyperopia severity based on AL. 

Ocular Biometric Components and Sex

Several studies have reported that ocular components differ between sexes and change with age. Ocular components are significantly different for boys and girls during infancy¹¹ and for school-aged children.²⁷ Rauscher et al. found clear sex differences and age differences in 4 to 17 year old children.²⁸ Consistent with the existing literature,²⁹ our cohort of children with hyperopia also exhibited significantly longer AL and longer ACD in boys than girls. In addition, CCT and K2 were significantly different between boys and girls. Boys had greater central cornea thickness and flatter K2. As a result, sex differences are important to consider when predicting ocular biometric parameters and analyzing related data. 

Ocular Biometric Components and Age

Notably, AL and ACD on children with hyperopia increased with age. According to results from school-age children, the most rapid changes in ocular biometric components took place between 6 and 9 years.²⁷ In our cohort, AL and ACD were significantly lower in the 3 to 5 year old age group than in the older age groups. Despite being derived from cross-sectional data, these findings suggest that AL and ACD increase with age. 

Comparison of the AL data from 3 to 5 year old children in our study (21.10 ± 0.97 mm) with the Mutti et al.¹¹ results from emmetropic children at 4.5 years (21.96 ± 0.70 mm) highlights that the hyperopic cohort had significantly shorter AL almost by 0.86 mm. The AL data from our 9 to 11 year old children with hyperopia (21.85 mm) are similar to those previously reported by Little et al. for 9 to 10 year old children with hyperopia > +3.00 D (N = 11, 21.64 mm).³⁴ 

The LT data from 3 to 5 year old children in our study (3.73 ± 0.38 mm) are similar to the Mutti et al.¹¹ results from emmetropic children at 4.5 years (3.70 ± 0.17 mm), indicating that the different SEQ in the 2 cohorts had little relation with LT. 

AL/CR Ratio

Our AL/CR ratio showed a high correlation with SEQ (R = −0.86). The AL/CR ratio in our study was similar between sexes, but significantly lower in the 3 to 5 year old group than in the other age groups. This may indicate that AL and cornea curvature compensate well for each other in boys and girls. Our findings align with the suggestion that the AL/CR ratio serves as the key determinant of the refractive error state in hyperopic children’s eyes.³¹ Moreover, our findings indicate that AL/CR ratio may serve as a superior predictor compared to AL alone, given its independence from sex. 

Compared with the literature on myopic children or a normal population (Table 4), the AL/CR for children with hyperopia in this study was significantly lower. AL/CR ratio in the myopic studies was 3.0 to 3.1.^17,18 Even compared with population-based studies that included myopia, emmetropia, and hyperopia, our AL/CR ratio for 6 to 8 year old children with hyperopia is lower than their reports of 2.8 to 2.9.^26,29 Our AL/CR is similar to Khan's AL/CR report (on average 2.8), which included a relatively broad range of hyperopia (2.25 D to 8.25 D) in 4 to 12 year old children.²² We observed that AL increases with age and hypothesized that other ocular components, especially CR, compensate for increased AL in children with hyperopia to maintain a relatively constant SEQ. Our findings align with those of Sun et al., who, based on retrospective longitudinal data from Chinese children, also reported an increase in AL/CR with age.²³ 

Table 3.

 

Bland-Altman Comparison of Model Prediction and Testing Data Set

 

Table 4.

 

Comparison of AL/CR With Those From Previous Studies During Childhood

 

Moderate-High Hyperopia and the CART Model

When compared with the CART model, the linear regression model shows a wider LOA range of approximately 0.9 mm. Despite slightly more bias, the CART model outperforms the linear regression model with fewer prediction errors (MAE or RMSE) and a higher correlation with the testing data. 

Previously, defining cut-points for moderate and high hyperopia were based on either arbitrary criteria or clinical experience.^35–37 An alternative approach is to use the CART model. Based on the +3.94 D and +7.06 D spilt points of our CART model, with reference to the AL growth pattern, we suggest classifying moderate hyperopia as +4.00 D to +7.00 D and high hyperopia as > +7.00 D. Others have suggested +4.00 D as the cut-point for moderate-high hyperopia previously.^13,14,38 Our CART model provides support for this cut-point. Note, this is a simplified CART model based on only SEQ, age, and sex. As is common with any modeling approach, it is important to recognize that there may be various alternative models available. As we utilize this CART model, it is crucial to remain mindful of its limitations. 

Limitation of this study include that the LenStar biometer was not able to measure lens thickness in a few children due to clear lenses in the youngest age group. Although there is no significant difference in AL, ACD, and K between LenStar and IOLMaster,³⁹ lens thickness measurements with LenStar can differ from ocular ultrasound,⁴⁰ so results obtained with different instruments should be interpreted with caution. In addition, a previous study reported that ACD was 0.05 to 0.06 mm shorter before cycloplegia.⁴¹ Our study measured ocular biometric parameters without cycloplegia. Finally, our study presents cross-sectional data and does not delve into longitudinal growth curves for ocular components. We are currently collecting longitudinal data on the majority of these children and intend to publish it in the near future. 

Clinical Significance

Results of this paper define the relationship between AL and SEQ for children with a broad range of hyperopia, especially those with moderate and high hyperopia. In cases where an estimate of AL is needed, such as performing an OCT measurement when an ocular biometer is not available, our CART model can provide a good estimate of AL using SEQ, age, and sex, which are all easily obtained in medical records. Thus, this model could be easily and conveniently applied in clinical practice to estimate AL for hyperopic eyes. Furthermore, our CART analysis also provides insight that, if we consider the eye growth pattern, SEQ at +4.00 D and +7.00 D are threshold values for partition, and the age of 7 years old is a threshold value. Thus, we suggest to classify moderate hyperopia as +4.00 D to +7.00 D and high hyperopia as > +7.00 D. 

Conclusions

In children with hyperopia, AL increased with age and differs between sexes. The major contribution components to hyperopia include AL, ACD, and K. The AL/CR ratio is highly correlated with SEQ. We developed a CART model to predict AL from SEQ for children with hyperopia. 

Acknowledgments

Supported by a grant from the National Eye Institute EY022313 (Birch), Hoffman Foundation (Wang), and American Academy of Optometry Foundation Pilot Grant Program (Wang). 

Disclosure: J. Wang, None; R.M. Jost, None; E.E. Birch, None