The limitations of centile curves for evaluating myopic eye growth

Abstract

SIGNIFICANCE:

Pediatric growth charts are widely used to track height and weight. Recently, axial length growth charts have been developed. Unfortunately, they underestimate the rate of normal myopic eye growth, making it challenging to evaluate the benefits of myopia control interventions, due to the conflation of myopes and nonmyopes.

PURPOSE:

The aim is to assess the value of axial length centile curves in the management of childhood myopia.

METHODS:

Papers reporting centile curves were identified by searching PubMed. For comparison, axial length values for a representative selection of baseline values (21 to 24 mm at 6 years) were calculated as a function of age and ethnicity using published meta-analyses of myopic and emmetropic eye growth data.

RESULTS:

Six published centile curves, largely based on cross-sectional data, were identified: three from European populations, two from China, and one from India. The trajectory of the emmetropic eye growth model generally tracks the European and Indian centile curves at lower centiles. This is not the case for the Chinese centile curves, likely due to the significant numbers of myopic children even at lower centiles. In contrast, the trajectory of the myopic eye growth model is steeper than that of the centile curves, even at higher centiles. This suggests that the higher centiles contain substantial numbers of nonmyopic children. Only in the centile curves for Chinese children, who have a higher prevalence of myopia, do they approach myopic eye growth, and then only for older children and at higher centiles.

CONCLUSIONS:

Centile curves do not accurately represent myopic eye growth, are not the best tool to monitor myopia progression and treatment, do not accurately represent growth in incident myopes, and are not the best way to predict myopia onset. Separate centile curves for myopic eyes do not alleviate the problem because of incident myopia. Annualized growth models may provide a better approach to assessing axial elongation relative to population norms.

The well-known concept of growth curves to plot changes in body measurements of children dates back to Montbeillard in the 18th century.¹ It was in 1875 that Galton indirectly introduced the concept of centiles to describe these anthropometric characteristics, recognizing that the data often departed from standard statistical operations of the time and observing that they were “much simpler in conception, more convenient in certain cases, and of incomparably wider applicability.”² Pediatric growth charts have been used by physicians, nurses, and parents to track the growth of infants, children, and adolescents in the United States since 1977.³

The idea of percentile curves in the field of optometry is not new. In 1952, Hirsch, noting that refractive error is not normally distributed, published a table of refractive error percentiles by sex and age for a group of children aged 5 to 14 years.⁴ More recently, interest in the rising prevalence of myopia^5,6 has underlined the urgent need for improved myopia management tools. This has prompted researchers to revisit centile refractive error curves in the interest of monitoring progression of myopia.^7–9 While these representations are informative, refractive error is less valuable for monitoring progression than axial length when measured using an optical biometer. Furthermore, axial length measurements do not require the use of cycloplegia, facilitating more frequent measurements.¹⁰ To this end, axial length growth curves have also been developed.^11–16 These have been incorporated into clinical instruments and acclaimed by some as a useful method to predict the onset and progression of myopia¹⁴ and to evaluate the efficacy of treatment.¹⁷

One recognized challenge of centile curves is their dependence on the incidence and prevalence of myopia in the population used to generate them. For example, Truckenbrod et al. overlaid their curves derived from German children on previously published curves for Chinese children.^12,14 The 95th percentile for older German children corresponded to the 50th percentile for Chinese due to the higher prevalence of myopia in the latter. This is even evident within a given ethnicity with curves based on two large samples of Chinese children showing meaningful differences at higher percentiles.^12,15

The true utility of axial length centile curves has not been subjected to detailed scrutiny. Here, we consider the value of these curves with emphasis on the extent to which they:

• accurately represent emmetropic and myopic eye growth,

• are useful for monitoring myopia progression,

• have value in assessing the efficacy of interventions to slow myopic progression, and

• can be used to predict myopia onset.

METHODS

Papers reporting centile curves were identified by searching PubMed using the following terms: (myopi* OR refract*) AND (percentile OR centile OR growth) AND axial AND (curve OR chart).

For comparison, axial length values for a representative selection of baseline values (21 to 24 mm at 6 years) were calculated using published meta-analyses of myopic¹⁸ and emmetropic¹⁹ eye growth models. These analyses were chosen as they both include age and ethnicity in their models. The myopic growth data have been shown to be consistent with other reported values.¹⁸ The emmetropic growth data arise from cross-sectional data and are consistent with a previous report that did not provide separate models for different ethnicities.²⁰ The use of cross-sectional data also avoids the overestimation of emmetropic eye growth due to myopic drift in longitudinal studies of so-called stable emmetropes.²¹

The relevant values from these eye growth models were superimposed on each set of centile axial length curves to qualitatively compare the trajectory of axial elongation across the full range of centiles. For clarity, data drawn from the meta-analyses of axial growth are referred to as growth models, and centile curves for axial length simply as centile curves. Although the latter are often referred to as growth curves,^11–14,16 they do not plot growth per se but rather axial length. Plots for the growth models are assumed to be independent of all covariates except age and ethnicity. While other covariates, such as sex or baseline refractive error, may have a statistically significant impact on axial elongation, these effects have been shown to be small.²¹ In this regard, Yii’s¹⁹ emmetropic growth model describes a mean axial length as a function of age, giving values of 22.74 and 22.56 mm at 6 years of age in East Asians and non-East Asians, respectively, but the rate of growth at a given age is considered invariant with baseline axial length.

RESULTS

On October 24, 2024, the search yielded 56 results. Five papers with relevant data were identified. The most common reasons for exclusion were 18 reports of ocular growth models that did not include centiles.^19,22,23 Other common reasons included evaluations of myopia control interventions (7), intraocular lenses and pseudophakia (6), and animal models (6). An additional paper that did not appear in the results of the search but was previously known to the authors was also included.¹³ An additional citation search using the six publications identified no further papers.

Table 1 lists these six papers with relevant details. Three of the six papers were from European populations, two were from China, and the most recent was from India. The centile curves were based largely on cross-sectional data although some included limited longitudinal data.^11–14

TABLE 1.

Summary of published axial length centile curves

Population	Ages (y)	n	Longitudinal data?	Cycloplegia used?
Netherlands11	6, 9, and 15	6084 at 6 y 5295 at 9 y 2495 at 15 y	4787 at both 6 and 9 y	Only at 9 y
Northern Ireland13	6–22	390 at 6–7 y 657 at 12–13 y	Followed for up to 9 y (125 of 390 and 113 of 657)	Yes
Germany14	3–18	1965	4511 examinations	No
China12	5–16	12,554	226 with longitudinal data (2.6 y)	Yes
China15	4–18	14,127	No	Yes
India16	6–12	4514	No, but small validation set to test predictions	No

Fig. 1 shows the six published centile curves. Four are reproductions of the original published figures^11,13,15 and two are replotted from the published data.^14,16 Five show separate data for girls and boys, while the sixth combined data for the sexes, but shows two separate age cohorts.¹³

FIGURE 1.

Six published axial length centile curves. (A), (B), (D), and (E) are original published figures.^11–13,15 (A), (B), and (D) are reproduced under the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/) and (E) under the Creative Commons Attribution NonCommercial license (https://creativecommons.org/licenses/by-nc/4.0/). (C) and (F) are replotted from the published data.^14,16 Five studies present separate centile curves for girls (left) and boys (right), while (B) combined data for the sexes, but shows two separate age cohorts.¹³ The text annotations within (A) and (B), for example, myopia 71%, myopia 53%, etc., are the percentage of myopes at a particular age. Axial length growth from published myopic¹⁸ and emmetropic¹⁹ models are superimposed on each panel. Three parallel red plots anchored at 22, 23, and 24 mm at 6 years of age represent myopic growth and three blue curves anchored at 21, 22, and 23 mm at 6 years of age represent emmetropic growth.

Axial length growth from published myopic¹⁸ and emmetropic¹⁹ models are superimposed on each set of centile curves for a representative selection of baseline values to compare the trajectory of axial elongation across the full range of centiles. Three parallel plots represent myopic growth¹⁸ anchored at 22, 23, and 24 mm at 6 years of age and three represent emmetropic growth anchored at 21, 22, and 23 mm at 6 years of age.

The trajectory of the emmetropic eye growth model generally tracks the European and Indian centile curves at lower percentiles (Fig. 1A–C, F). This is to be expected as these represent predominantly nonmyopic eyes. The parallel trajectory of the lower percentiles also supports our assumption regarding the consistency of axial elongation across a range of baseline axial lengths. This is not the case for the two sets of Chinese centile curves (Fig. 1D, E).^12,15 The most plausible explanation is the significant numbers of myopic children even at lower percentiles given, for example, that over 85% of children 11 years and older are myopic in one of these samples.¹²

The trajectory of the myopic eye growth model is steeper than that of the centile curves, even at higher percentiles. This is particularly evident for the curves from the European^11,13,14 and Indian studies (Fig. 1A–C, F).¹⁶ This suggests that the higher percentiles contain substantial numbers of nonmyopic children. Only in the centile curves for Chinese children,^12,15 who have a higher prevalence of myopia, do the curves reflect myopic eye growth, and then only for older children and at higher percentiles (Fig. 1D, E).

DISCUSSION

Centile curves may not accurately represent myopic eye growth

Given that juvenile-onset myopia is associated with increased axial length, it is expected that higher percentiles on growth curves are associated with an increased likelihood of myopia. Likewise, it is intuitive that the curves for the higher percentiles should be steeper and more closely represent myopic growth. Conversely, the curves for the lower percentiles should be flatter and more closely represent emmetropic growth. To test this hypothesis, recently published growth models were leveraged. For myopic eye growth, a recent systematic review and meta-regression were used based on an appropriate exponential slowing of annual elongation,^24,25 which includes ethnicity as a variable.¹⁸ Other models show similar results.¹⁸ For emmetropic eye growth, the recent model of Yii¹⁹ was used because he also produced separate curves for East Asians and non-East Asians.

It is clear from Fig. 1 that, in most instances, the trajectory of the lower percentiles accurately reflects emmetropic eye growth. It is also evident that for non-East Asian children, the trajectory of the higher percentiles is flatter than expected for myopic eye growth, and thus centile curves underestimate average axial elongation in myopic children. For example, the myopic growth model of Brennan et al. predicts an average cumulative axial elongation of 1.29 mm between 9 and 15 years in a European myopic child. In contrast, the data shown in Fig. 1B show that Northern Irish children at the 90th percentile will show 0.69 mm of cumulative axial elongation. The other centile curves predict even slower cumulative axial elongation: 0.45 mm for boys and 0.48 mm for girls in the Netherlands (Fig. 1A) and 0.36 mm for boys and 0.46 mm for girls in Germany (Fig. 1C). In summary, myopic axial elongation is underestimated by a factor or two or more, even at higher centiles. The fact that these centiles contain data from both myopes and emmetropes is clearly responsible. Indeed, inspection of Fig. 1A, B demonstrates that fewer than 30% of 10-year-old children at the 90th percentile are myopic. This range of values also serves to emphasize the variation with gender and country of origin. Although axial length is clearly higher in boys than girls, it is unclear whether rates are different between the two sexes,^21,26 although the trajectory of the upper centile would be influenced by both myopia incidence²⁷ and axial elongation rates in existing myopes.

An obvious potential solution might be to construct centile curves specifically for myopes. This is also problematic. With increasing age, new incident myopes are added to the charts, and they will most likely enter with shorter axial lengths than their early-onset peers, that is, at the lower end of the percentile rankings. This will elevate the percentile ranking of existing myopes without them necessarily departing from a normal myopic growth pattern.

A further limitation of centile curves is a general lack of sensitivity to changes in growth patterns. The charts track axial length. A more sensitive method is to use annualized axial elongation, which is the change, or first derivative, of axial length,¹⁸ although in an individual child, the year-to-year variation in axial elongation may be considerable.^28,29

Centile curves are not the best tool to monitor myopia control treatment

The fact that centile curves can underestimate average eye growth in myopic children by a large degree severely limits their value in evaluating myopia control interventions in an individual child. Consider the data from the pivotal clinical trial of the Food and Drug Administration–approved dual-focus soft contact lens.³⁰ The mean 3-year axial elongation (mean age at baseline = 10.1 years) was 0.30 and 0.62 mm in the treated and control groups, respectively. Using the growth curves in Fig. 1A, the predicted 3-year axial elongation for the 90th percentile, around which we would expect many myopes to fall, is 0.27 and 0.29 mm for boys and girls, respectively. In other words, the typical child wearing this myopia control lens for 3 years would ostensibly show little treatment benefit if success were based on the centile curves, in spite of significant slowing of axial elongation. Fig. 2 superimposes the 3-year change in axial length in children wearing either dual-focus soft contact lenses or single-vision soft contact lenses on centile curves. The first uses the curves from Fig. 1A and anchors the aforementioned 3-year clinical trial data at the 75th, 90th, and 98th percentiles—as myopes are most likely to reside at these levels.¹¹ At all three percentiles, the treated children follow the trajectory of the centile curve, which is the expectation for untreated myopes. Paradoxically, untreated children in the trial move to higher centiles. The second uses the curves from Fig. 1C and anchors the 3-year data at the 75th and 98th percentiles.¹⁴ At the 75th percentile, the treated children follow the trajectory of the centile, but at the 98th percentile, the untreated children follow the trajectory of the centile.

FIGURE 2.

Three-year change in axial length in children wearing either dual-focus soft contact lenses³⁰ (solid green lines) or single-vision soft contact lenses (dashed red lines) superimposed on centile curves. The left panel uses the curves from Fig. 1A and anchors the aforementioned 3-year clinical trial data at the 75th, 90th, and 98th percentiles.¹¹ The right panel uses the curves from Fig. 1C and anchors the 3-year data at the 75th and 98th percentiles.¹⁴

In the same way that centile curves fail to reflect untreated myopic eye growth, they also do not appropriately map the efficacy of established myopia control treatments. This again is due to the conflation of emmetropic and myopic eye growth. Even at the upper percentiles, the curves frequently underestimate the rate of axial elongation in European myopic children by a factor of two or more. An experienced practitioner may be able to explain to a parent that their treated child moving to a higher centile still represents a positive outcome, but the expectation propagated by examples in the literature is that successful treatment will transition a child to lower centiles.¹⁷ Again, the use of annualized axial elongation provides a clearer basis for interpretation.¹⁸

Centile curves do not accurately represent growth in incident myopes

Most young children are nonmyopic. Their axial length is increasing but in most cases, their rate of axial elongation is slowing steadily.¹⁹ Those who are myopic show greater rates of axial elongation that slow over time. Those destined to become myopic show acceleration of axial elongation some 2 to 4 years before myopia onset.^31,32 Their axial elongation is most rapid at or just before onset,^31,32 and is followed by an average slowing of 15% per year once myopia has set in.¹⁸ Thus, as well as the conflation of myopes and emmetropes in centile curves, there is a clear trichotomy among persistent emmetropes, persistent myopes, and incident myopes that these curves do not capture. Fig. 3 shows these three distinct axial length growth patterns. Fig. 3A illustrates theoretical annual axial elongation starting at 6 years of age for three children: a myope, an emmetrope, and a third who develops myopia at 12 years of age. The plots for the myope and emmetrope are mean values for East Asian children from the previously described myopic¹⁸ and emmetropic¹⁹ models. The plot for the incident myope initially reflects emmetropic growth, followed by the acceleration described by Mutti et al.³¹ and Rozema et al.,³² whereafter there is growth equivalent to the pre-existing myope. Fig. 3B replots these data as axial length, assuming all three children have an axial length of 22 mm at 6 years of age and would be at the same percentile on a centile curve (although, admittedly, the myope would likely already have a longer eye). A notable difference between the two representations in Fig. 3 is the greater sensitivity of annual axial elongation to the change in growth pattern at ages 9 and 10 for the incident myope. Nonetheless, the acceleration in axial elongation preceding the onset of myopia should be easily visualized as an increase in centile.

FIGURE 3.

(A) Annual axial elongation for three representative children: a myope, an emmetrope, and a third who develops myopia at 12 years of age. (B) Change in axial length assuming all three children have an axial length of 22 mm at 6 years of age.

There is an opportunity for centile curves to have some value in predicting onset if a nonmyopic child is monitored over a period of time and shows escalation from low centile numbers to higher ones. This approach and its limitations are further described in the following section.

Centile curves are not the best way to predict myopia onset

The authors of some centile curves have proposed their use for predicting myopia onset in young children. For example, Truckenbrod et al.¹⁴ state that they “can be used as a predictive measure for the likelihood of developing as well as the progression of myopia,” while Sanz Diez et al.¹² say that “children with longer eyes are more likely to develop myopia.” Certainly, emmetropes with longer axial length have a higher likelihood of becoming myopic but the predictive value is not well established. While some authors have evaluated the ability of their centile curves to predict myopia, few did the same for other potential predictors, notably refractive error.

Earlier work had demonstrated that baseline axial length was associated with subsequent myopia onset, but that cycloplegic refractive error was the best predictor. Zadnik et al.³³ reported data from the Collaborative Longitudinal Evaluation of Ethnicity and Refractive Error (CLEERE) Study. Data were analyzed from 4512 ethnically diverse, nonmyopic US children aged 6 to 11 years at baseline of whom 414 became myopic. Axial length was one of eight factors associated with the risk of myopia onset in multivariate models. Nonetheless, refractive error was the single best predictive factor and performed as well as all eight factors combined with an area under the receiver operating characteristic (ROC) curve between 0.87 and 0.93. Axial length was measured using ultrasound, known to be less repeatable than optical biometry, but their findings have been consistently replicated by subsequent studies in other populations using the more modern technology. For example, Ma et al.³⁴ evaluated the ability of various baseline parameters to predict 2-year myopia onset in 874 nonmyopic children in Shanghai. The area under the ROC curve was 0.63, 0.76, and 0.86 for axial length, axial length:corneal radius ratio, and cycloplegic refractive error, respectively.

Few of the developers of centile curves have subjected their data to the same rigorous analysis, due largely to the absence of robust longitudinal data. The exception is McCullough et al.¹³ who compared the predictive value of axial length and refractive error at 6 to 7 years of age in predicting myopia over the subsequent 9 years. While an axial length of 23.07 mm gave a sensitivity of 49% and a specificity of 80% (area under the ROC curve = 0.69), a refractive error of +0.63 D gave a sensitivity of 76% and a specificity of 83% (area = 0.87).

Finally, Gopalakrishnan et al.¹⁶ evaluated the ability of their centile curves to predict a 3-year incidence of myopia in 377 nonmyopic children. Baseline axial length was a relatively poor predictor of myopia onset (area under the ROC curve not given), but axial length:corneal radius ratio fared much better with the area under the ROC curve of 0.79 and 0.72 for prediction of myopia onset at 11 and 13 years, respectively. This ratio, of course, represents the two greatest determinants of refractive error and may thus be considered a surrogate measure.

In the absence of the cycloplegic refractive error, biometric data may be an alternative, but axial length alone is a poorer predictor of myopia onset than axial length:corneal radius ratio.^16,34 Longitudinal axial length measurements may prove to be a useful predictor. As shown in Fig. 3, axial elongation accelerates in the years before myopia onset,^31,32 and annual measurements may allow detection of this trend. For example, Tideman et al.¹¹ found that an increase in percentile between 6 and 9 years was associated with myopia. Of the 354 children who had 10-point or greater increase in percentile, 162 (46%) were myopic at 9 years of age. In contrast, only 85 of the 1781 children (5%) who had an increase of less than 10 points were myopic at 9 years of age. Of course, those who had at least a 10-point increase in percentile but were not yet myopic may have been accelerating toward myopia. Nonetheless, the parameters under which onset can be anticipated and differentiated from random variations in persistent emmetropes need further definition. Clinically, it would require multiple annual measurements of axial length in nonmyopic children, which is not common practice. Of course, seasonal variation in eye growth would likely confound the interpretation of more frequent measurements.^35–37 Incorporating the measurement of axial length into an annual school screening might enable this predictor to be used should it be shown to have predictive capacity when cycloplegic refraction is not possible.

Limitations

The nature of the material discussed herein and the unavailability of the raw data preclude rigorous statistical analysis of the differences in slopes between centile curves and models of eye growth. Furthermore, our critique of the published centile curves relies on the validity of the myopic and emmetropic eye growth models. There is good supporting evidence that these are representative for age and ethnicity^20–23 but we note that other covariates, such as sex and baseline refractive error/axial length, may influence their application. For example, most studies reviewed in our recent paper suggest faster axial elongation in girls than boys.¹⁸ Gender differences in the centile curves in Fig. 1 will be further impacted by a higher incidence of myopia in one gender.²⁷ The consistent divergence of the curves across age observable in Fig. 1 is most likely entirely attributable to increasing proportions of myopes at higher centile values and higher ages but it is not possible to rule out that higher baseline axial length is accompanied by slightly more rapid eye growth.

CONCLUSIONS

To the best of our knowledge, this is the first critical review of the value of centile curves of axial length. Centile charts have some value, if only in that the concept may be familiar to many parents. The introduction of such charts for axial length, based on a range of populations, has been met with considerable enthusiasm and incorporation into a number of commercially available biometers (e.g., https://www.oculus.de/us/products/myopia-master/). Unfortunately, this needs to be tempered by the issues identified in this paper, and axial eye growth based on centile curves should be interpreted more qualitatively than quantitatively. They provide a useful snapshot of where a child lies at a moment in time. The challenges stem primarily from their use in interpreting longitudinal data. In particular, centile curves substantially underestimate the rate of normal myopic growth in European children. In doing so they significantly undermine the ability to evaluate the effect of a myopia control intervention. The principal source of error is the conflation of myopes and nonmyopes. Annualized growth charts represent a more readily interpretable and valid method for monitoring eye growth in myopic children and evaluating treatment,¹⁸ although they may benefit from refinement to include baseline refractive error, gender, family history, and other covariates.