A Mathematical Model for Axial Length Estimation in a Myopic Pediatric Population Based on Easily Obtainable Variables

Abstract

Determining the axial length (AL) of the eye is of significant interest in the management of myopia. However, the devices that allow this value to be obtained are either expensive, for example, optical biometers, or inconvenient for use in pediatric population, such is the case with ultrasound biometers. Therefore, this study aimed to develop a mathematical model for estimating the AL value based on easily obtainable variables, with the novel addition of body height to the analysis. A total of 170 eyes of 85 myopic volunteers (mean age of 10.8 ± 1.45 years, ranging from 7 to 14 years) were included in the analysis. Participants underwent anamnesis, keratometry by NVISION-K 5001, subjective refraction by an optometrist, AL measurement by the Topcon MYAH biometer, and body height measurement. Spearman’s correlation test was employed to analyze the relationships between AL and keratometry, spherical equivalent, body height (Sperman’s correlation, all r ≥ 0.267, all p < 0.001), and age (Spearman’s correlation, p = 0.081). Subsequently, multiple regression analysis was conducted on the variables that demonstrated a previous correlation. The mathematical model obtained permits the estimation of AL based on average keratometry, spherical equivalent, and body height. This model is significant (p < 0.001) and explains 82.4% of AL variability.

1. Introduction

The study of myopia has become increasingly important in recent years due to its rising prevalence and earlier age of onset [1]. While some studies have already indicated that by 2050, approximately half of the world’s population may be affected by myopia [1,2], other regional studies offer more pessimistic estimates. For instance, the prevalence of myopia among the United States’ population is projected to reach 72% by that date [3], and it is projected to exceed 80% in Chinese children and young people [4]. Regarding the age of onset, myopia typically manifests when the human eye is still growing, between the ages of 8 and 13 years, although this may vary between populations [5]. This early onset allows for a prolonged period of progression, which may even extend beyond the age of 20 years [5,6], with the most significant peak increment occurring between the ages of 7 and 12 years [7]. The clinical implications of high myopia are substantial and can even result in blindness [8]. Consequently, given the trends mentioned above, pathological myopia is projected to become one of the leading causes of irreversible visual impairment and blindness worldwide [8]. Nevertheless, it must be emphasized that there is no safe level of myopia. Even low levels of this refractive error are associated with an increased risk of ocular complications, such as cataracts, glaucoma, and other conditions affecting the posterior eye segment [9,10]. In addition to the individual impact, myopia has a high socio-economic cost. In 2015, the global potential productivity loss due to myopia-related visual impairment was estimated to be USD 250 billion [11]. Annual spending on correction alone is estimated to range from USD 3.9 to USD 7.2 billion in the United States [12]. Meanwhile, in urban China, the combined annual costs of treating and preventing myopia, along with the associated productivity loss for individuals aged 5 to 50 years, amounted to USD 26.3 billion in 2016 [13]. The presented data highlight the considerable economic burden myopia imposes on both a global and national scale, which has already surpassed, in terms of economic cost, the burden of other diseases, such as Parkinson’s disease [10].

The growing concern at the clinical and economic level surrounding myopia has led to the search for techniques to slow its progression. These include pharmacological agents, such as atropine, and optical interventions, such as orthokeratology lenses and soft contact lenses or spectacle lenses based on the principle of defocusing the peripheral retina [14]. Additionally, numerous instruments have been developed to monitor the progression of myopia, with the ocular biometer emerging as one of the gold-standard devices in the management of this refractive error [15]. The significance of determining the length of the myopic eye lies in the fact that an excessive axial length (AL) is the principal cause of myopia and its associated risks. Greater AL of the myopic eye is associated with serious ocular complications due to the overstretching of the retina, choroid, and sclera [15]. Thus, the knowledge of the AL of the eye and the application (if necessary) of myopia control methods is of particular interest not only for the future individual well-being of each subject but also on a socio-economic level [9].

Currently, a wide variety of ocular biometers are available on the market, utilizing two primary methods for biometry measurement: ultrasound biometry or optical biometry [16,17]. Models using ultrasound have been the historical standard [18]. The AL measurement is achieved by using a high-frequency sound wave, which is captured after bouncing off the ocular structures [17]. However, these devices have a major disadvantage since they involve contact with the cornea [16], making their use in children difficult. In this context, the advances of optical devices have been of significant assistance, as they permit the acquisition of measurements with high sensitivity and resolution in a non-invasive manner [19]. For this purpose, partial coherence interferometry or optical low-coherence reflectometry are employed. Both techniques utilize a multimode laser diode to reflect light off the corneal surface and the retinal pigment epithelium, thereby determining the distance between the two points [17]. The primary disadvantage of the optical ocular biometers is their economic value. Given the high cost of these devices, many eye care professionals do not have access to them [19]. In order to address this problem, mathematical models have been developed to estimate AL based on easily obtainable biometric variables such as the refractive error, keratometry, or age [20,21,22]. Nevertheless, variables such as body height, which is easy to obtain and has been demonstrated to be related to AL [23,24,25], are not typically incorporated into these mathematical models.

Thus, the objective of the present study was to develop a mathematical model that utilizes easily obtainable variables for the estimation of AL. By assessing the incorporation of new variables, such as body height, the aim was to enhance the precision of estimations.

2. Materials and Methods

2.1. Sample

The a priori Sample Size Calculator for Multiple Regression of the Free Statistics Calculators software, version 4.0 [available from: https://www.danielsoper.com/statcalc/calculator.aspx?id=1 (accessed on 19 February 2024)], was used to calculate the sample size required for the present study [26]. This calculator allows one to obtain the minimum required sample size for conducting a multiple regression study. In order to perform this calculation, it is necessary to indicate the anticipated effect size, the desired statistical power level, the number of predictors, and the probability level. The values indicated for these variables were 0.15 (the medium effect size by convention, based on Cohen’s calculations [27]), 0.8, 3 (based on the results obtained in the analysis of correlations), and 0.05, respectively. The minimum sample size required for the analysis was ultimately determined to be 76.

The sample was recruited based on myopic pediatric patients in the optometry clinic of the center. The study included patients aged between 7 and 14 years and with a myopia of −0.25 D or greater who were not using or had not used any strategy for myopia control. Additionally, patients with a previous diagnosis of corneal ectasia, media opacity, or ocular infection or disease at the time of the measurements were excluded.

This research study adhered to the principles of the Declaration of Helsinki and was approved by the University Bioethics Committee (approval number: USC 04/2022). Prior to inclusion in the study, each participant’s legal guardian provided written informed consent.

2.2. Measurements

The tests were conducted in a single session, beginning with anamnesis. Subsequently, keratometry was conducted using the NVISION-K 5001 (Rexxam Co., Kagawa, Japan) [23,28]. Subjective refraction was then performed by an optometrist employing the fogging method based on the retinoscopy value [29]. Following this, the AL was measured with the MYAH biometer (Topcon, Tokyo, Japan) [23,30]. Finally, to measure body height, a tape measure with 1 cm increments was attached to the wall and used. To ensure accuracy, participants were required to remove their shoes and position their heels, back, shoulders, and head against the wall.

To facilitate an analysis of the data, the subjective refraction values were transformed to spherical equivalents using Equation (1), and for keratometry, the average value of curvature in mm of the two principal meridians was used.

Spherical Equivalent = Sphere power + (Cylinder power/2)

(1)

2.3. Statistical Analysis

The statistical package SigmaPlot for Windows Version 15.0 (Grafiti LLC, Palo Alto, CA, USA) was used to perform statistical analysis. The chosen level of statistical significance was p ≤ 0.05. Prior to the analysis, the normality of the sample was assessed with the Kolmogorov–Smirnov test [31]. Spearman’s correlation test was used to evaluate the correlations between the AL and the other variables: age, average keratometry (Km), spherical equivalent of the refractive error and body height. The correlations were classified as weak (0.20 to 0.40), moderate (0.41 to 0.60), good (0.61 to 0.80), or strong (0.81 to 1.00) [32]. Finally, a multiple linear regression analysis was conducted to predict the value of AL based on the variables that demonstrated a significant correlation in the previous analysis [31].

3. Results

A total of 170 eyes of 85 myopic volunteer participants (51 females and 34 males) were analyzed. The descriptive values of the sample were as follows: the mean age was 10.8 ± 1.45 years, ranging from 7 to 14 years; the Km was 7.7 ± 0.25 mm, ranging from 7.02 to 8.39 mm; the mean spherical equivalent value was −2.00 ± 1.16 D, ranging from −0.25 to −4.75 D, with anisometropia ranging from 0.00 to 2.00 D; and the mean body height was 145.6 ± 10.96 cm, ranging from 122.0 to 175.0 cm.

The analysis of the sample distribution revealed that the variables age, Km, and body height exhibited a normal distribution (Kolmogorov–Smirnov test, all p ≥ 0.103), whereas AL and spherical equivalent of refraction demonstrated a non-normal distribution (Kolmogorov–Smirnov test, both p < 0.001).

3.1. Correlations between AL and the Other Variables

The results indicated a positive correlation between AL and Km and body height, with the former relationship being classified as good (Spearman’s correlation, r = 0.673, p < 0.001) and the latter as weak (Spearman’s correlation, r = 0.267, p < 0.001). The AL was also found to correlate negatively and moderately with spherical equivalent (Spearman’s correlation, r = −0.569, p < 0.001). No significant correlations were found between AL and age (Spearman’s correlation, p = 0.081) (

Table 1. Correlation between AL and age, Km, spherical equivalent, or body height.

		Age (Months)	Km (mm)	Spherical Equivalent (D)	Body Height (cm)
AL (mm)	rs	0.134	0.673 *	−0.569 *	0.267 *
	p	0.081	<0.001	<0.001	<0.001

AL = axial length; Km = average keratometry; r_s = Spearman correlation. * statistically significant.

).

3.2. Multiple Linear Regression

A multiple linear regression analysis was employed to predict the dependent variable, AL, from the independent variables: Km, spherical equivalent of refraction, and body height. All variables introduced contribute significantly to the model (all p ≤ 0.004), from which Equation (2) is derived:

AL = 4.203 + (2.330 × Km) − (0.420 × Spherical Equivalent) + (0.008 × Body Height)

(2)

For this model, the coefficient of determination (R² = 0.824) shows that 82.4% of the variability of the AL variable can be explained by the variables used. The remaining 17.6% of variability can be attributed to unknown variables. Furthermore, the results of the normality test (Shapiro–Wilk test, p = 0.647) and the constant variance test (Spearman rank correlation, p = 0.216) indicate compliance with the assumptions of the linear regression model. Finally, the power of the model for an alpha value of 0.05 is 1.000, suggesting that the probability of type II error is low.

The equation obtained by this regression analysis is available in an interactive spreadsheet, which is provided as a Supplementary Materials (Suplementary Material S1).

4. Discussion

The integration of mathematical models for estimating the AL of the myopic eye into clinical practice was initiated with the introduction of calculators designed to predict future levels of myopia. These calculators utilize current biometric measurements and specific risk factors for myopia progression from the individual subjects to generate predictions [33,34]. The use of these predictive models is beneficial in assisting parents to understand their children’s situation [33], allowing them to make informed decisions about the child’s eye care. For eye care practitioners, while not essential, as periodic assessment of refractive status can alert one to increments in myopia, these tools provide Supplementary Information that can assist decisions regarding the implementation of myopia control strategies [34]. Given the interest generated by these predictive calculators, it is reasonable to suggest that clinicians could also benefit from mathematical models with other uses in myopia management, such as estimating current eye AL when direct measurement devices are not available. Therefore, the aim of the present study was to develop a model for predicting AL among myopic children using easily obtainable biometric variables such as ocular refractive error, average keratometry, and body height. The results indicate that these variables are indeed useful for estimating AL and support the incorporation of body height into the mathematical model for this purpose.

To the authors’ knowledge, Morgan et al.’s [21] work was the first publication to provide a mathematical estimate of AL based on easily obtainable variables. Their calculations were derived from a population of myopic children aged 8 to 12 years, as described in a study by Chamberlain et al. [35]. The authors estimated AL based on Km and the spherical equivalent of the refractive error obtained by cycloplegic refraction. AL and topography were determined with IOL Master 500 (Carl Zeiss, Oberkochen, Germany), while refraction was performed with a WR-5100 K or WAM-5500 autorefractometer (Grand Seiko Co., Hiroshima, Japan). It was found that a reasonable estimate of AL could be obtained from a mathematical model [21]. The authors emphasized the utility of their equation (Equation (3)) in categorizing the risk of visual impairment, in accordance with the assumptions of Tideman et al. [36], rather than in obtaining exact values of AL or in monitoring its progression in myopic children.

AL = 1/[(0.22273/Km) + (0.00070 × Spherical Equivalent) + 0.01368]

(3)

The next study to develop a mathematical model for AL estimation was conducted by Queirós et al. [20]. The sample included children and young people between the ages of 6 and 25 years, with a refractive spheric error between −8.25 D and +8.25 D and astigmatism up to −4.00 D. In their regression model (4), the authors included average keratometry, spherical equivalent refractive error measured with an open-field autorefractometer without cycloplegia, and age. The combination of these three variables explained 79.8% of the variability in AL [20]. As in Morgan’s study, AL was measured with the IOL Master 500 biometer, and refraction was performed with the WAM-5500 autorefractometer.

AL = (0.019 × Age) + (2.271 × Km) − (0.444 × Spherical Equivalent) + 5.414

(4)

Lingham et al.’s study [22] is the most recent to develop a mathematical model for estimating AL. The data used to develop the model were obtained from several studies carried out in Ireland [37,38,39] and China [40]. The devices used to measure the variables varied between studies. Regarding refractive error, in all studies, it was performed by autorefraction under cycloplegic conditions. Combined, these studies provided a sample of children aged between 4 and 22 years with a spherical equivalent of refractive error between −14.00 D and +12.00 D. The novelty of this study was the inclusion of the variable of sex in the mathematical model, encoded as 0 for males and 1 for females. In addition to this, Equation (5) incorporates the spherical equivalent of the refractive error, Km, and age, with the latter included both in absolute value and its logarithm in base 10 [22]. However, this increased complexity may negatively impact its practical application in daily clinical practice.

AL = 5.472 − (0.069 × Age) + (3.06 × log₁₀(Age)) − (0.265 × Sex) − (0.135 × Spherical Equivalent) + (2.019 × Km) − (0.072 × Spherical Equivalent × Km)

(5)

The above equations were developed in populations with different characteristics. Morgan’s model [21], like that of the present study, is based on a myopic population, whereas the other two studies cover a wider range of refractive errors [20,22]. Given the accelerated growth rate of the myopic eye [41], it may be advisable to develop separate models for myopic individuals. When myopic, emmetropic, and hyperopic eyes are combined in a single regression model, the distinct growth patterns of myopic eyes may be masked, potentially reducing the accuracy of the model for all refractive groups. Consequently, the creation of distinct models for myopes, emmetropes, or hyperopes could improve the accuracy of these equations. It should also be noted that these studies generated a model based on AL values measured with a specific ocular biometer. This may mean that each equation is contingent on the device used and that there may be differences when comparing results obtained with other ocular biometers. Equally, the equation may be conditioned by the refractive technique used. In the present study, subjective refraction was performed using the fogging technique, which controls accommodation to achieve maximum plus to maximum visual acuity. The use of autorefractors may yield different values, especially when using closed-field autorefractors and without cycloplegic instillation, as accommodation can introduce measurement errors.

A significant limitation of the current study is that the sample size is small, and thus, the findings should be considered preliminary and indicative of future trends. Further research is required to validate the model in a larger population. Additionally, the use of both eyes of each participant can be considered a limitation, as it applies the same body height measurement to two eyes with differing parameters. However, this approach is justified on the basis that the model is intended for use in real clinical settings, where inter-eye variability is a common occurrence. Similarly, the exclusion criteria were intentionally kept broad to enhance the model’s generalizability; for example, excluding certain levels of anisometropia may further induce bias in the applicability or the translational utility of the results. Furthermore, the variable age, which has been incorporated in previous models, was not a significant variable in the present analysis. This may be due to the sample size or the narrow age range selected, which limits age-related variability. While the Queirós and Lingham studies were conducted in a population with a wide age range of almost 20 years, the present study and Morgan’s study included children in a more limited age range, less than 10 years in both cases. It can be inferred that the narrow age range may account for the relevance of body height in the model, while age was not significant, despite the correlation between these parameters [23]. Future research in this field should include specific analyses of children with high astigmatism to determine the interchangeability of predictive models when the refractive errors are predominantly spherical and when there is a highly cylindrical component. Additionally, it may be pertinent to analyze other variables that could enhance the formula, such as the length of the anterior chamber, provided that such additions do not compromise its practical applicability in daily clinics. The applicability may be compromised if attempts to improve its accuracy result in excessive complexity or require the inclusion of variables that, despite having greater predictive power, are difficult to obtain due to economic costs or time constraints in clinical practice.

5. Conclusions

In conclusion, this study presents a novel approach to predicting AL by incorporating body height as a variable within a regression model. This model could be of significant utility in the daily clinical practice of health professionals in the absence of an ocular biometer. Nevertheless, it is important to note that the model should be regarded as an informative estimate rather than a criterion for treatment decisions, as outliers are present in daily clinical settings. For the purpose of precise monitoring, such as the control of AL elongation in the context of myopia, the utilization of ocular biometers is recommended.