0 Title: Influence of cycloplegia on the axial length prediction models in a peadiatric cohort. OriginalPaper 1 1 2018 44402 Ivo Soares +351 918111609 isoares@ubi.pt António Baptista Oscar Torrado Pedro Serra Department of Physics University of Beira Interior Covilhã Portugal Health Sciences Research Center (CICS-UBI) University of Beira Interior Covilhã Portugal Clinical and Experimental Center in Vision Sciences (CCECV) University of Beira Interior Covilhã Portugal Centre of Physics University of Minho Braga Portugal Ophthalmology Clinic Vista Sánchez Trancón Badajoz Spain Department of Physics University of Beira Interior, Rua Marquês D’Ávila e Bolama Covilhã, Covilhã Portugal, Portugal Authors: Ivo Soares1,2,3*, António Baptista4, Oscar Torrado5, Pedro Serra5 1. Department of Physics, University of Beira Interior, Covilhã, Portugal 2. Health Sciences Research Center (CICS-UBI), University of Beira Interior, Covilhã, Portugal 3. Clinical and Experimental Center in Vision Sciences (CCECV), University of Beira Interior, Covilhã, Portugal 4. Centre of Physics, University of Minho, Braga, Portugal 5. Ophthalmology Clinic Vista Sánchez Trancón, Badajoz, Spain Corresponding Author: Ivo Soares Department of Physics, University of Beira Interior, Covilhã, Portugal Address: Rua Marquês D'Ávila e Bolama, Covilhã, Portugal Email: isoares@ubi.pt Phone number: +351 918111609 Word count: 3572 words Abstract Clinical Relevance Accurate axial length (AL) estimation is vital for monitoring myopia progression in children, especially in primary care where optical biometers are often unavailable. Prediction models adjusted for cycloplegic effects may offer a reliable alternative. Purpose To assess the effect of cycloplegia on the accuracy and repeatability of several AL prediction models in a paediatric cohort, and to identify which models maintain minimal bias under both cycloplegic and non-cycloplegic conditions. Methods Ninety-six children (mean age 12.5 ± 2.4 years-old) underwent repeated measurements pre- and post-cycloplegia of spherical equivalent (SE), anterior corneal curvature (Kmean), and AL using the Myopia Master. Seven published prediction models incorporating SE, Kmean, age, and sex were evaluated. Agreement, bias, limits of agreement (LoA), coefficient of repeatability (CR), intraclass correlation coefficient (ICC), and regression analyses were used to assess performance and repeatability. Results Cycloplegia induced a hyperopic shift (mean + 0.79 D), most pronounced in emmetropic and hyperopic eyes. Measured AL and all models showed improved repeatability post-cycloplegia (measured AL CR decreased from ~ 0.14 mm to ~ 0.09 mm; ICC > 0.99). Pre-cycloplegia, models overestimated AL (mean differences MD from − 0.87 to − 0.24 mm); these biases reduced post-cycloplegia (MD from − 0.56 to + 0.10 mm). Models by Morgan, Queirós, and Lingham had the smallest bias (< 0.10 mm) and narrowest LoA (< 0.84 mm). Variation in SE accounted for ~ 97–99% of change in predicted AL; Kmean contributed ≤ 1.2%. Conclusion Cycloplegic refraction significantly enhances both accuracy and repeatability of AL prediction models in children. Models by Morgan et al., Queirós et al., and Lingham et al. performed best. Predictive models may be a valuable substitute in settings without access to optical biometers, provided cycloplegic measurements are used when possible. Word count: 278 Introduction Myopia is a rapidly increasing global ocular condition, largely attributed to prolonged engagement in near-work activities,1 reduced outdoor exposure,2 and genetic predisposition.3 Its escalating prevalence represents a major public health concern, not only because more individuals experience visual impairment requiring frequent ophthalmic care, but also due to of the lifelong risk of sight-threatening pathologies associated with myopia.4 These complications can ultimately lead to severe visual loss and reduced quality of life.5,6 Myopia progression is intrinsically linked to axial elongation of the eye,7 which strongly correlates with visual impairment.4 Consequently, axial length (AL) measurement is a key biometric parameter for monitoring refractive development.8 The ratio of AL to anterior corneal radius also correlates strongly with spherical equivalent (SE) refractive error9 and may serve as an indicator for detecting myopia onset.10 Axial length can be measured using ultrasound or optical coherence biometers.11 While ultrasound requires corneal contact and local anaesthesia, optical biometers—based on interferometric or coherence-based technologies—offer superior precision and accuracy, achieving approximately ± 0.1 mm (≈ 0.25 D) even in paediatric populations.12 Despite their utility, particularly in cataract surgery planning, optical biometers are mainly available in tertiary centres. In contrast, myopia detection and monitoring typically occur in primary care settings where such devices are less accessible.13 To address the lack of direct AL measurement in primary care, several studies have developed theoretical14 and statistical13,15–21 models to predict AL using routinely obtained parameters. Common predictors include SE and corneal curvature (keratometry), with some models incorporating demographic or biometric variables such as age, sex,16,17,20 weight19 and anterior chamber depth.21 However, differences in population characteristics, measurement devices, and testing conditions (e.g., cycloplegic vs. non-cycloplegic refraction) can affect model accuracy and agreement with measured AL. Among these predictors, SE is particularly sensitive to accommodation. When measured without cycloplegia, accommodative effort induces a transient myopic shift that may lead to overestimated AL predictions.22,23 Morgan et al. reported only a marginal difference in the limits of agreement when comparing predictions based on cycloplegic (± 0.73 mm) and non-cycloplegic (± 0.75 mm) data,13 while Tang et al. reported a decrease in the difference between predicted and measured AL as well as an improvement in the range of agreement.16 To the best of our knowledge, no previous study has evaluated these predictive models within the same population. The present study assesses the accuracy of multiple AL prediction models under cycloplegic and non-cycloplegic conditions in a shared paediatric cohort. Additionally, it examines the intrasession repeatability of these models when refraction and keratometry are measured sequentially. Methods Literature Search A comprehensive literature search was conducted in PubMed, Web of Science, Scopus, and Cochrane databases for studies published between 2000 and 2025 on AL prediction formulas. Models were selected if they used input variables routinely measured in clinical practice — SE, keratometry, age, and sex. The search strategy combined terms related to AL, prediction or modelling, and clinical parameters. Only models based on optical coherence interferometry were included. The selected prediction models are summarised in Table 1. Author (year) Model He**15 (2015) Kim*14 (2019) Tang**16 (2020) Morgan**13 (2020) Queirós*17 (2022) Dutt**18 (2022) Lingham**20 (2024) SE – spherical equivalent; Kmean – Mean curvature; *- Non-cycloplegic SE; **- Cycloplegic SE

Table 1 Formulas for predicting axial length using the input variables SE, Kmean, Sex or Age.

Participants This study included the same cohort of ninety-six paediatric participants previously described in detail in a previous study.24 That study evaluated the repeatability and agreement of Myopia Master measurements under cycloplegic and non-cycloplegic conditions. The present work addresses a distinct question — the evaluation of AL prediction models using this dataset. The original measurement procedures, including the cycloplegia protocol, measurement sequence, fixation target, and quality control criteria, followed the validated methodology reported by Peñaranda et al. and are summarised below. The ninety-six participants (48 females; mean age 12.5 ± 2.4 years-old, range 7–16 years-old) underwent a comprehensive ophthalmologic examination at Clínica Oftalmológica Vista Sánchez-Trancón (Spain). Inclusion criteria were refractive astigmatism < 2.50 D under cycloplegia, distance-corrected visual acuity of 6/6 or better, and absence of ocular pathology or strabismus. The examination protocol included assessments of distance visual acuity (with and without correction), autorefraction, anterior corneal curvature measurements, AL measurement, objective refraction (with and without cycloplegia), subjective refraction, cover test, slit-lamp biomicroscopy, and fundus examination. Autorefraction, keratometry, and AL were measured using the Myopia Master (version 7.2 R3; Oculus, Germany), in "Myopia Mode". The device incorporates a fixation target simulating optical infinity with a fogging system to control accommodation. Measurements were performed automatically and sequentially, following manufacturer's protocol. Only measurements meeting predefined quality thresholds were included (quality index ≥ 7 for autorefraction and keratometry; signal-to-noise ratio ≥ 6.0 for AL). Before cycloplegia, autorefraction, anterior corneal curvature, and AL were each measured twice within a five-minute interval to assess intrasession repeatability. Cycloplegia was induced with one drop of cyclopentolate (1% cyclopentolate Colircusí, Alcon), followed by a second drop after 10 minutes. Post-cycloplegia measurements were performed 30 minutes after the initial instillation and were repeated twice within five minutes. All measurements were performed by the same experienced optometrist to ensure procedural consistency. The SE was calculated as sphere + (–cylinder)/2 and mean anterior corneal curvature (Kmean) as the average of the flat and steep meridians. Only right-eye data were analysed. The study adhered to the Declaration of Helsinki, was approved by the local ethics committee (Comité Ético para Investigación Clínica de Badajoz), and written informed consent was obtained from parents or legal guardians. Data and Statistical Analysis All statistical analyses were performed using JupyterLab (v4.0.11; https://jupyter.org). Descriptive statistics are reported as mean ± standard deviation (M ± SD), together with range and 95% confidence interval (95% CI), same for mean difference (MD). Data normality was assessed using the Kolmogorov–Smirnov test. Participants were classified as myopic (SE ≤ − 0.50 D), emmetropic (–0.50 D < SE < + 1.00 D), or hyperopic (SE ≥ + 1.00 D) based on cycloplegic SE. Repeatability Analysis Intrasession repeatability of AL measurements—both directly measured and model-predicted using models — was evaluated using three complementary metrics. First, the within-subject standard deviation (Sw) quantified the variability between repeated measurements for each participant. Second, the coefficient of repeatability (CR) was calculated as 2.77 × Sw, with its 95% CI estimated from the chi-squared distribution. This coefficient represents the smallest detectable change that can be interpreted as exceeding expected measurement noise under identical conditions. Third, the intraclass correlation coefficient (ICC) was computed using a two-way random-effects model for absolute agreement [ICC(2,1)], accounting for variability attributable to both participants and repeated measures. ICC were interpreted as follows: <0.50, poor; 0.50–0.75, moderate; 0.75–0.90, good; and > 0.90, excellent).25 This approach is particularly suitable for assessing the reliability of repeated measurements obtained with the same device under consistent testing conditions. Differences between repeated measurements were analysed using two-tailed paired t-tests. To the inflation of type I error arising from multiple comparisons across the seven predictive models and the measured AL, the significance threshold was adjusted using Bonferroni correction (α = 0.05/8 = 0.006). Agreement Analysis Agreement between measured and model-predicted AL was evaluated under both cycloplegic and non-cycloplegic conditions. For each participant, the reference AL was defined as the mean of two repeated measurements. Predicted AL values were derived using each model’s formula, with SE and Kmean from the same measurement condition. Agreement was quantified by the mean difference between measured and predicted AL values, representing systematic bias. The 95% limits of agreement (LoA) were defined as the mean difference ± 1.96 times the SD of the paired differences, indicating the interval within which 95% of the differences are expected to lie. Confidence intervals for the LoA were also computed to reflect estimate precision. The intraclass correlation coefficient [ICC(3,1)] was used to assess relative reliability—that is, the consistency with which measured and predicted AL values rank across participants, irrespective of systematic bias. The coefficient of variation within subjects (CVWS) expressed as a percentage, was calculated as the within-subject SD divided by the mean AL. Together, these metrics provided a scale-independent assessment of prediction reliability. Paired t-tests were used to detect systematic bias between predicted and measured AL. Significance was again adjusted using the Bonferroni correction (α = 0.05/7 = 0.007). Influence of Input Variation on Model-Predicted AL The effect of variability in pre-cycloplegic inputs on predicted AL was examined using multiple linear regression, with post-cycloplegic measurements serving as the reference standard. The dependent variable was the difference between pre- and post-cycloplegic predicted AL (ΔAL). Independent variables included changes in SE (ΔSE) and mean corneal curvature (ΔKmean), while static factors (e.g., age, gender) were excluded. These ΔAL were then regressed against the corresponding differences in SE (ΔSE) and Kmean (ΔKmean), allowing to determine the extent to which variability in each parameter contributed to deviations in model-predicted AL relative to its cycloplegic baseline. Statistical Power Calculation The study had 88% power to detect a mean difference smaller than 0.2 mm between measured and predicted AL, assuming a standard deviation of 0.5 mm and a Bonferroni-adjusted significance level of 0.007 (0.05/7). A difference of 0.2 mm corresponds to approximately 0.50D, which is considered clinically relevant in paediatric myopia management. Results The group comprised 35 myopes, SE = − 2.48 ± 1.24 D (range: −5.64 to − 0.79 D), 30 emmetropes, SE = + 0.29 ± 0.48 D (range: −0.66 to + 0.95 D), and 31 hyperopes, SE = + 2.48 ± 1.56 D (range: +1.00 to + 7.25 D), Table 2. Comparison of pre- and post-cycloplegic data showed statistically significant SE shifts toward more positive refractions in emmetropes, hyperopes, and the combined group. Mean differences were − 0.59 ± 0.12 D (95% CI: −0.83 to − 0.34, p < 0.001) for emmetropes, − 1.47 ± 0.39 D (95% CI: −2.26 to − 0.68, p < 0.001) for hyperopes, and + 0.79 ± 0.32 D (95% CI: 0.16 to 1.42, p = 0.014) for the overall sample. Myopes showed no significant change (MD = 0.38 ± 0.30 D; 95% CI: −0.98 to 0.22; p = 0.204). Anterior corneal curvature and axial length differences were not statistically significant (p > 0.203). Pre-cycloplegia* M ± SD (range) Post-cycloplegia* M ± SD (range) Patients n SE(D) Kmean(mm) AL (mm) SE(D) Kmean(mm) AL(mm) Myopes 35 -2.71 ± 1.30 (-6.27 to -0.72) 7.77 ± 0.25 (7.20 to 8.35) 24.40 ± 0.97 (22.05 to 26.68) -2.33 ± 1.29 (-5.64 to -0.54) 7.78 ± 0.25 (7.19 to 8.37) 24.40 ± 0.96 (22.05 to 26.70) Emmetropes 30 -0.20 ± 0.47‡ (-1.36 to 0.59) 7.77 ± 0.26 (7.31 to 8.36) 23.26 ± 0.69 (22.26 to 24.40) 0.39 ± 0.41‡ (-0.42 to 0.95) 7.77 ± 0.25 (7.31 to 8.39) 23.28 ± 0.69 (22.27 to 24.46) Hyperopes 31 1.01 ± 1.48‡ (-1.54 to 5.84) 7.82 ± 0.32 (7.11 to 8.46) 22.34 ± 0.84 (20.43 to 23.94) 2.48 ± 1.56‡ (1.00 to 7.25) 7.82 ± 0.33 (7.11 to 8.50) 22.35 ± 0.87 (20.41 to 24.07) All 96 -0.80 ± 2.00† (-6.27 to 5.84) 7.79 ± 0.27 (7.11 to 8.46) 23.41 ± 1.21 (20.43 to 26.68) -0.01 ± 2.37† (-5.64 to 7.25) 7.79 ± 0.28 (7.11 to 8.50) 23.42 ± 1.22 (20.41 to 26.70) * - Average of two measurements; M – Mean; SD – Standard deviation; SE – Spherical equivalent; Kmean – Mean anterior corneal curvature; AL – Axial length. † - Paired t-test; p = 0.014; ‡ - Paired t-test; p < 0.001.

Table 2 Demographic, refractive and biometric data pre- and post-cycloplegia, represented by mean ± standard deviation and range.

Intrasession repeatability of measured and predicted AL Table 3 shows that the MD between consecutive AL measurements and predicted AL derived from consecutive SE and Kmean was close to zero, pre- and post-cycloplegia. Measured AL demonstrated the lowest CR pre-cycloplegia (0.14 mm, 95% CI: 0.13 to 0.17mm) and post-cycloplegia (0.09 mm, 95% CI: 0.08 to 0.11mm), outperforming all AL predictive models. Among the models, Tang et al. achieved the best repeatability pre-cycloplegia (0.24 mm, 95% CI: 0.21 to 0.28mm) and post-cycloplegia (0.16 mm, 95% CI: 0.14 to 0.18), whereas He et al. showed the poorest pre-cycloplegia (0.48 mm, 95% CI: 0.42 to 0.56 mm) and post-cycloplegia (0.27 mm, 95% CI: 0.24 to 0.32mm). Overall, both measured and predicted AL improved in CR repeatability after cycloplegia. Intraclass correlation for measured and predicted AL were excellent under both conditions (ICC > 0.99, with a range of the 95%LoA of 0.98 to 1.00). Pre-Cycloplegia Method Measure 1 M ± SD (mm) Measure 2 M ± SD (mm) MD ± SD (mm) (95% CI) CR (mm) (95% CI) Measured AL 23.41 ± 1.22 23.41 ± 1.21 -0.00 ± 0.07 * (-0.02 to 0.01) 0.14 (0.13 to 0.17) He et al. 23.88 ± 1.67 23.89 ± 1.65 -0.01 ± 0.24 * (-0.06 to 0.04) 0.48 (0.42 to 0.56) Kim et al. 24.28 ± 1.11 24.29 ± 1.11 -0.01 ± 0.14 * (-0.04 to 0.02) 0.28 (0.25 to 0.33) Tang et al. 24.11 ± 0.91 24.11 ± 0.91 -0.01 ± 0.12 * (-0.03 to 0.02) 0.24 (0.21 to 0.28) Morgan et al. 23.65 ± 0.95 23.66 ± 0.95 -0.01 ± 0.13 * (-0.03 to 0.02) 0.25 (0.22 to 0.29) Queirós et al. 23.69 ± 1.05 23.70 ± 1.04 -0.01 ± 0.15 * (-0.04 to 0.03) 0.30 (0.26 to 0.35) Dutt et al. 23.97 ± 0.97 23.97 ± 0.96 -0.01 ± 0.14 * (-0.03 to 0.02) 0.27 (0.24 to 0.32) Lingham et al. 23.87 ± 1.06 23.88 ± 1.05 -0.01 ± 0.14 * (-0.04 to 0.02) 0.28 (0.25 to 0.33) Post-Cycloplegia Method Measure 1 M ± SD (mm) Measure 2 M ± SD (mm) MD ± SD (mm) (95% CI) CR (mm) (95% CI) Measured AL 23.42 ± 1.22 23.42 ± 1.22 0.00 ± 0.05 * (-0.01 to 0.01) 0.09 (0.08 to 0.11) He et al. 23.32 ± 1.86 23.32 ± 1.87 -0.01 ± 0.14 * (-0.04 to 0.02) 0.27 (0.24 to 0.32) Kim et al. 23.97 ± 1.20 23.98 ± 1.21 -0.01 ± 0.10 * (-0.03 to 0.01) 0.20 (0.17 to 0.23) Tang et al. 23.83 ± 1.00 23.84 ± 1.01 -0.01 ± 0.08 * (-0.02 to 0.01) 0.16 (0.14 to 0.18) Morgan et al. 23.36 ± 1.02 23.37 ± 1.03 -0.01 ± 0.08 * (-0.02 to 0.01) 0.16 (0.14 to 0.18) Queirós et al. 23.35 ± 1.17 23.36 ± 1.18 -0.01 ± 0.09 * (-0.02 to 0.01) 0.18 (0.160 to 0.21) Dutt et al. 23.65 ± 1.08 23.65 ± 1.09 -0.01 ± 0.09 * (-0.02 to 0.01) 0.17 (0.15 to 0.20) Lingham et al. 23.54 ± 1.16 23.54 ± 1.17 -0.01 ± 0.09 * (-0.02 to 0.01) 0.17 (0.15 to 0.20) *p > 0.492 for all; M –mean; SD – standard deviation; MD – mean difference; CR – coefficient of repeatability; CI –confidence intervals;

Table 3 Intrasession repeatability of measured and predicted AL.

Agreement between measured and predicted AL Pre-cycloplegia, all predictive models significantly overpredicted measured AL (p < 0.001 for all; Table 4 and Fig. 1). The largest bias was observed with the Kim et al. model (MD=-0.87 ± 0.60 mm, 95% CI: -0.99 to − 0.75 mm), and the smallest with Morgan et al. predictive model (MD=-0.24 ± 0.55 mm, 95% CI: -0.35 to − 0.13 mm). Kim et al. also showed the widest LoA (-2.05 to + 0.30 mm) and the highest CVws (3.13%), whereas Queirós et al. exhibited the narrowest LoA (-1.31 to -0.74 mm) and the lowest CVws (1.78%). Intraclass correlations indicated good-to-excellent reliability (all ICC ≧ 0.81). Pre-Cycloplegia Prediction Model AL Predicted AL Measured MD ± SD (mm) (95% CI) Lower LoA (mm) (95% CI) Upper LoA (mm) (95% CI) CVWS‡ (%) ICC§ (95%LoA) M ± SD (mm) He et al. 23.88 ± 1.65 23.41 ± 1.21 -0.47 ± 0.78 * (-0.63 to − 0.31) -1.99 (-2.26 to -1.81) 1.05 (0.87 to 1.32) 2.70 0.90 (0.76 to 0.95) Kim et al. 24.28 ± 1.11 -0.87 ± 0.60 * (-0.99 to − 0.75) -2.05 (-2.25 to − 1.90) 0.30 (0.16 to 0.51) 3.13 0.81 (-0.12 to 0.94) Tang et al. 24.11 ± 0.90 -0.70 ± 0.55 * (-0.81 to − 0.59) -1.78 (-1.96 to-1.65) 0.39 (0.25 to 0.56) 2.64 0.84 (0.02 to 0.95) Morgan et al. 23.65 ± 0.95 -0.24 ± 0.55 * (-0.35 to − 0.13) -1.33 (-1.50 to − 1.19) 0.84 (0.71 to 1.02) 1.81 0.92 (0.85 to 0.95) Queirós et al. 23.69 ± 1.04 -0.28 ± 0.52 * (-0.39 to-0.18) -1.31 (-1.47 to − 1.18) 0.74 (0.62 to 0.91) 1.78 0.93 (0.85 to 0.96) Dutt et al. 23.96 ± 0.96 -0.56 ± 0.55 * (-0.67 to − 0.45) -1.63 (-1.82 to − 1.59) 0.51 (0.39 to 0.70) 2.32 0.88 (0.34 to 0.95) Lingham et al. 23.87 ± 1.05 -0.46 ± 0.52 * (-0.57 to − 0.35) -1.49 (-1.65 to − 1.36) 0.57 (0.44 to 0.73) 2.08 0.91 (0.59 to 0.96) Post-Cycloplegia Prediction Model AL Predicted AL Measured MD ± SD (mm) (95% CI) Lower LoA (mm) (95% CI) Upper LoA (mm) (95% CI) CVWS‡ (%) ICC§ (95%LoA) M ± SD (mm) He et al. 23.32 ± 1.86 23.42 ± 1.22 0.10 ± 0.81** (-0.06 to + 0.26) -1.49 (-1.76 to -1.30) 1.69 (1.50 to 1.96) 2.47 0.93(0.89 to 0.95) Kim et al. 23.98 ± 1.21 -0.56 ± 0.43 * (-0.65 to -0.47) -1.40 (-1.55 to -1.30) 0.28 (0.18 to 0.43) 2.09 0.92 (0.22 to 0.98) Tang et al. 23.42 ± 1.22 -0.41 ± 0.38* (-0.49 to − 0.33) -1.15 (-1.28 to − 1.07) 0.33 (0.25 to 0.46) 1.68 0.94 (0.53 to 0.98) Morgan et al. 23.36 ± 1.03 0.06 ± 0.40 ** (-0.02 to 0.14) -0.72 (-0.86 to − 0.63) 0.84 (0.75 to 0.98) 1.21 0.97 (0.95 to 0.98) Queirós et al. 23.35 ± 1.17 0.07 ± 0.36 ** (0.0 to 0.14) -0.64 (-0.76 to − 0.55) 0.78 (0.69 to 0.90) 1.11 0.98 (0.96 to 0.98) Dutt et al. 23.65 ± 1.08 -0.23 ± 0.38 * (-0.31 to -0.15) -0.89 (-1.10 to − 0.97) 0.51 (0.43 to 0.64) 1.33 0.96 (0.91 to 0.98) Lingham et al. 23.54 ± 1.17 -0.12 ± 0.35* (-0.19 to -0.05) -0.81 (-0.92 to − 0.72) 0.57 (0.48 to 0.68) 1.12 0.98 (0.96 to 0,98) *Statistically significant at p < 0.007 (Bonferroni-adjusted significance level); ** p > 0.05; §Two-way random-effects model. M ± SD - Mean ± standard deviation; MD ± SD - mean difference ± SD of the differences; LoA – Limits of Agreement; CVWS - coefficient of variation; ICC- Intraclass correlation coefficients; CI – Confidence Intervals

Table 4 Agreement between measured and estimated AL.

Post-cycloplegia, mean differences between predicted and measured AL decreased across all models. Kim et al retained the largest bias (MD = − 0.56 ± 0.43 mm, 95% CI: − 0.65 to − 0.47mm), while Morgan et al. showed the smallest (MD = 0.06 ± 0.40 mm, 95% CI: − 0.02 to 0.14 mm). He et al. had the widest LoA (–1.49 to 1.69 mm) and highest CVws (2.47%), while Lingham et al. demonstrated the narrowest LoA (–0.81 to 0.57 mm) and one of the lowest CVws (1.12%). Reliability improved further (ICC ≥ 0.92; 95% LoA: − 0.12 to 0.94 mm), post-cycloplegia. A consistent proportional bias was found, shorter eyes (hyperopic and emmetropic) were overpredicted, while myopic eyes showed near-accurate estimates, except in the He et al. model, which displayed the opposite pattern.

Fig. 1 Analysis of AL prediction against measured AL (n = 96). Red, green and blue dots indicate myopic, emmetropic and hyperopic participants, respectively. The first and second column shows the Bland-Altman plots for pre- and post-cycloplegia, respectively. The black solid line represents the mean difference between measured and predicted values (measured – predicted). The black dashed lines are the 95% limits of agreement. The third and fourth column shows the Pearson correlation analysis of predicted and measured AL values for pre- and post-cycloplegia, respectively. The solid black lines represent the regression model and the dashed line the identity line (y = x).

Grouped AL analysis confirmed that cycloplegia reduced overestimation in hyperopic and emmetropic eyes, while myopes tended to maintain the underestimation, Fig. 2. The proportion of eyes with prediction errors < 0.5 mm, increased from 23% to 43% for Kim et al. and from 66% to 83% for Queirós et al.; post-cycloplegia, the Tang, Morgan, Queirós, and Lingham models predicted AL within ± 1.00 mm in > 97% of eyes, Fig. 3.

Fig. 2 Estimated AL error for each predictive model in pre- (top row) and post-cycloplegia (bottom row) for several AL intervals (negative values indicate model overestimation). Red, green and blue dots indicate myopic, emmetropic and hyperopic participants, respectively.

Fig. 3 – Percentage of eyes with axial length prediction below thresholds, pre-cycloplegia (top row) and post-cycloplegia (bottom row).

Influence of SE and Kmean Cycloplegia Variation on Model-Predicted AL Figure 4 illustrates the relationship between cycloplegia-induced changes in spherical equivalent (ΔSE) and mean corneal curvature (ΔKmean) with variation in predicted axial length (ΔAL). Across all models, ΔSE— defined as post- minus pre-cycloplegic SE —was the main driver of ΔAL variation, whereas ΔKmean had minimal impact. Multivariate linear regression (Table 5) confirmed this trend, with all models statistically significant, explaining 98.8–99.7% of the variance in ΔAL. The contribution of ΔSE accounted for 97.3–99.1% of the explained variance, while ΔKmean contributed only 0.5–1.2%. These findings demonstrate that cycloplegia-induced changes in SE are the principal determinant of variation in predicted AL, whereas alterations in corneal curvature play a negligible role.

Fig. 4 Relationship between SE measurement error (ΔSE = pre - post) and AL prediction error (ΔAL = pre - post) (top row) and between Kmean measurement error (ΔKmean = pre- post) and ΔAL (bottom row).

Model β 1 β 2 Error Adjusted R² p-value He et al. -0.997 -0.073 0.036 0.996 < 0.001 Kim et al. -0.989 -0.139 0.030 0.992 < 0.001 Tang et al. -0.994 -0.111 0.020 0.995 < 0.001 Morgan et al. -0.990 -0.102 0.033 0.987 < 0.001 Queirós et al. -0.996 -0.093 0.022 0.996 < 0.001 Dutt et al. -0.996 -0.093 0.020 0.996 < 0.001 Lingham et al. -0.995 -0.084 0.027 0.994 < 0.001 β1 and β2 are the regression coefficients of ΔSE (pre-cycloplegia – post-cycloplegia); ΔKmean (pre-cycloplegia - post-cycloplegia)

Table 5 Multivariate linear regression parameters for AL prediction error (ΔAL = AL pre-cycloplegia – post-cycloplegia)., respectively.

Discussion This study compared the performance of several AL prediction models in a paediatric population, before and after cycloplegia. The models were based on linear regression incorporating ocular (SE and Kmean) and demographic parameters (age and sex), with SE showing a strong association with AL.9 It was hypothesized that variability in SE measurements could affect model performance. The findings indicate that, across all models, AL prediction accuracy and repeatability improved following cycloplegia, with five of the seven models showing LoA for AL prediction within 1.0 mm of the measured values after cycloplegia. Cycloplegia shifted the refractive error towards more positive values (+ 0.79 D), with the effect being more pronounced in emmetropes (+ 0.59 D) and hyperopes (+ 1.47 D).24 This shift is consistent with values reported by Hu et al. (+ 0.78 ± 0.79 D) in children aged 4–18 years26 and large-scale studies in 12-year-olds (+ 0.84 D).22 This behaviour confirms the greater accommodative effort of emmetropic and hyperopic eyes during autorefraction.23 Differences in Kmean and AL remained within instrument repeatability limits, indicating that these anatomical parameters are unaffected by cycloplegia.24,27 Before cycloplegia, all models overpredicted AL, with mean differences ranging from − 0.87 mm (Kim et al.)14 to − 0.24 mm (Morgan et al.)13, and LoA extending up to 1.18 mm from the mean. In the He et al.15 model, LoA were even broader, spanning 1.53 mm. After cycloplegia, prediction errors decreased: the Kim et al.14 model overpredicted by − 0.56 mm, whereas the He et al.15 model slightly underpredicted (+ 0.10 mm). Models by Morgan et al.13 and Queirós et al.17 showed mean differences below 0.10 mm, corresponding to refractive errors under 0.25 D according to the Gullstrand model eye. LoA also narrowed (< 0.84 mm) for all models except He et al.,15 whose precision remained similar to pre-cycloplegia conditions. Using the relationship of 0.4 mm axial elongation per 1.0 D,28 the LoA still represent potential differences of about 2.0 D —highlighting that AL estimations from predictive models should be interpreted cautiously. Bland-Altman and regression analyses (Fig. 1) show that most models (except He et al.)15 tended to overpredict AL in emmetropic and hyperopic eyes and underpredict in myopic eyes. Grouping eyes by AL range (Fig. 2) revealed that prediction errors were greatest in shorter eyes (hyperopic and emmetropic) and reduced post-cycloplegia. Myopic eyes showed smaller, more stable errors. Post-cycloplegia, especially for the Queirós et al. and Lingham et al. models, 50% of eyes with AL between 21–26 mm clustered around the zero-error line, with 95% of cases within 1.0 mm. The reduction in proportional bias post-cycloplegia observed in the Bland–Altman analysis appears particularly relevant in eyes with more extreme axial lengths. The performance of a prediction model depends on its developmental methodology, the population used, and the independent variables selected. These factors influence the model’s generalizability. All reviewed models share a strong dependence of AL on SE and anterior corneal curvature, grounded in the principles of ocular refraction. He, Tang, Queirós and Lingham et al.15–17,20 further incorporated age and sex, which improved model fit and are known to influence AL. He et al. demonstrated that ~ 83% of SE variance is explained by the AL/ Kmean ratio.15 They developed an SE prediction model based on AL, Kmean, and sex, later reformulated here to predict AL. Their sample comprised 3922 Asian children aged 6–12 years-old with cycloplegic refraction. The model showed underprediction in hyperopes and overprediction in myopes—opposite to other models—likely because their SE model estimated more positive SE in hyperopes and more negative SE in myopes. Sex-related differences in AL (males: +0.44 mm) and Kmean (females: +0.13 mm) were consistent with our data (AL: +0.70 mm; Kmean: +0.19 mm, data not shown), suggesting that including sex can enhance performance. Kim et al. derived a model from the simplified Gullstrand eye, using Kmean and SE with a correction constant for SE variation.14 Their 382-participant Asian cohort (ages 7–77 years) was measured without cycloplegia. They reported an AL overprediction of 0.18 ± 0.47 mm (95% LoA: − 0.75 to + 1.10 mm), with myopes showing the largest errors. In our cohort, pre-cycloplegia overprediction was higher (0.87 ± 0.60 mm; 95% LoA: − 2.05 to + 0.30 mm), especially in hyperopic and emmetropic eyes. Kim et al. reported 75.5% and 95.5% of predictions within 0.5 mm and 1.0 mm, respectively—substantially higher than our findings (23% ≤ 0.5 mm; 61% ≤ 1.0 mm). These differences likely reflect cohort characteristics, since age and high ametropia reduce model precision.20 Tang et al. developed an AL prediction model using linear regression and machine learning on 1011 myopic Asian children aged 6–18 years-old, with cycloplegic refraction. Their predictors (Kmean, SE, sex, and age) explained 81% of AL variance—comparable to 82% (pre-) and 92% (post-cycloplegia) in this study. The inclusion of age accounted for AL elongation during childhood and adolescence and the concurrent reduction in lens power.29–31 Morgan et al. proposed a linear regression model based on cycloplegic SE and Kmean in 144 Caucasian participants aged 8–12 years-old, later validated in 1046 individuals aged 6–22 years.13 They reported an AL underestimation of 0.13 mm (95% LoA: − 0.73 to + 0.99 mm), consistent with our post-cycloplegia results (0.06 mm; 95% LoA: − 0.72 to + 0.84 mm). Queirós et al. applied this model to 1783 participants aged 6–25 years-old (SE measured with an open-field autorefractor) and found an AL overestimation of 0.25 ± 0.48 mm (95% LoA: +0.70 to + 1.20 mm), aligning with our pre-cycloplegia data.17 Queirós et al proposed a new model, adding age as a predictor, which explained ~ 80% of AL variability. In the preset study, this model showed an 0.28 mm overprediction (one of the lowest) pre-cycloplegia and an 0.08 mm underprediction after cycloplegia—supporting the role of accommodation control in improving AL prediction. Dutt et al. developed a regression model using cycloplegic SE and Kmean from 1301 Caucasian adults aged 18–22 years-old.18 Under non-cycloplegic and cycloplegic conditions, their model overestimated AL by 0.10 ± 0.52 mm (95% LoA: − 0.92 to + 0.11 mm) and 0.01 ± 0.49 mm (95% LoA: − 0.94 to + 0.97 mm), respectively. In our cohort, overestimations were greater (pre-cycloplegia: − 0.56 ± 0.55 mm; post-cycloplegia: − 0.23 ± 0.38 mm), reaffirming that cycloplegia enhances precision and accuracy. Lingham et al. proposed a regression model based on cycloplegic SE, Kmean, sex, and age, trained on 1068 Caucasians (6–20 years-old), 3429 Asians (5–18 years-old), and 240 Caucasian myopes (6–19 years-old). Their model underpredicted AL by 0.08 ± 0.40 mm (95% LoA: − 0.71 to + 0.87 mm) in a myopic test set. In our cohort after cycloplegia, it slightly overpredicted AL (–0.12 ± 0.35 mm; 95% LoA: − 0.81 to + 0.57 mm), likely due to inclusion of hyperopic and emmetropic eyes. Regression analysis of AL prediction errors against cycloplegic variations in Kmean and SE showed that changes in spherical equivalent (ΔSE) explained 97.3%–99.1% of the variance in predicted AL across all models, whereas mean corneal curvature (ΔKmean) contributed minimally (0.5%–1.2%). Considering the reported repeatability of approximately 0.65 D for non-cycloplegic objective refraction, 0.32 D after cycloplegia,24 and 0.35 D for subjective refraction,32 the resulting prediction error may reach ~ 0.15 mm, limiting the detection of subtle AL changes. These findings confirm that precise SE measurement is the primary determinant of axial length estimation accuracy and repeatability. A major strength of this study lies in the direct comparison of multiple AL prediction models on the same paediatric cohort, both before and after cycloplegia. Since these models rely on routinely acquired clinical parameters, the findings offer practical insights into their clinical applicability. The inclusion of a balanced distribution of refractive error types allows for a broader and more representative comparison than those of Tang et al. and Lingham et al., which were limited to myopic eyes. In the context of myopia progression, monitoring AL in emmetropic and low-hyperopic eyes is clinically relevant, as a reduction in hyperopia may indicate axial elongation.30 However, some limitations should be acknowledged. The sample size was smaller than those in the original model development studies. Nonetheless, this study aimed to compare rather than validate or train new models, for which the sample size was adequate. Additionally, only paediatric Caucasian participants were included, whereas most existing models were derived from Asian or mixed cohorts. Although ethnic differences in AL have been,33 they appear to stem mainly from ocular biometric relationships rather than ethnicity itself,20 supporting the validity of our comparisons while underscoring the need for broader cross-population assessments. Conclusions This study highlights the importance of cycloplegic refraction for improving the accuracy and repeatability of AL predictive models. The models of Morgan et al.,13 Queirós et al.,17 and Lingham et al.20 demonstrated minimal bias and superior repeatability under cycloplegic conditions. SE was the primary factor influencing prediction errors, while corneal curvature had negligible impact. Overall, AL predictions were more accurate in myopic eyes than in hyperopic or emmetropic eyes, where mild overprediction persisted. These models provide a useful alternative in primary care settings without optical biometers, particularly for paediatric myopia management, though they should not replace direct AL measurements when available. AcknowledgementThe authors want to thank Agustin Peñaranda for his role in the clinical data collection. Declarations Ethics approval This study was carried out in accordance with the principles of the Declaration of Helsinki, and was approved by the local ethics committee (Comité Ético para Investigación Clínica de Badajoz). Consent to participate A written informed consent for each participant was obtained from parents or legal guardians. Funding Open access funding provided by FCT|FCCN (b-on). The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. Disclosure statement No potential conflict of interest was reported by the author(s) ORCID Ivo Soares http://orcid.org/0000-0001-6712-7514 António Baptista https://orcid.org/0000-0002-7304-8756 Oscar Torrado https://orcid.org/0000-0001-5808-7602 Pedro Serra https://orcid.org/0000-0003-0471-0213 Author ContributionI. S. and P.S. performed the conceptualization, formal analysis, methodology, software, supervision, prepared figures and wrote the main manuscript text. A.B. performed formal analysis, supervision, methodology and wrote the main manuscript text. O.T. performed the conceptualization, data curation and validation. All authors reviewed the manuscript References 1. Gajjar S, Ostrin LA. A systematic review of near work and myopia: measurement, relationships, mechanisms and clinical corollaries. Acta Ophthalmol. 2022;100(4):376–387. doi:10.1111/aos.15043 2. Xiong S, Sankaridurg P, Naduvilath T, et al. Time spent in outdoor activities in relation to myopia prevention and control: a meta-analysis and systematic review. Acta Ophthalmol. 2017;95(6):551–566. doi:10.1111/aos.13403 3. Jiang X, Tarczy-Hornoch K, Cotter SA, et al. Association of Parental Myopia With Higher Risk of Myopia Among Multiethnic Children Before School Age. JAMA Ophthalmol. 2020;138(5):501. doi:10.1001/jamaophthalmol.2020.0412 4. Tideman JWL, Snabel MCC, Tedja MS, et al. Association of Axial Length With Risk of Uncorrectable Visual Impairment for Europeans With Myopia. JAMA Ophthalmol. 2016;134(12):1355. doi:10.1001/jamaophthalmol.2016.4009 5. Han X, Ruan X, Zhang Y, et al. Effect of Myopia Severity on the Health-Related Quality of Life in Children and Their Parents. Curr Eye Res. 2023;48(12):1189–1194. doi:10.1080/02713683.2023.2250581 6. Łazarczyk JB, Urban B, Konarzewska B, et al. The differences in level of trait anxiety among girls and boys aged 13–17 years with myopia and emmetropia. BMC Ophthalmol. 2016;16(1):201. doi:10.1186/s12886-016-0382-2 7. Meng W, Butterworth J, Malecaze F, Calvas P. Axial Length of Myopia: A Review of Current Research. Ophthalmologica. 2011;225(3):127–134. doi:10.1159/000317072 8. Németh J, Tapasztó B, Aclimandos WA, et al. Update and guidance on management of myopia. European Society of Ophthalmology in cooperation with International Myopia Institute. Eur J Ophthalmol. 2021;31(3):853–883. doi:10.1177/1120672121998960 9. Grosvenor T, Scott R. Role of the Axial Length/Corneal Radius Ratio in Determining the Refractive State of the Eye. Optometry and Vision Science. 1994;71(9):573–579. doi:10.1097/00006324-199409000-00005 10. Fan Y, Huang Y, Huang X. Association between Axial Length to Corneal Curvature Radius Ratio and Myopia in Adult Patients. J Ophthalmol. 2024;2024:1–7. doi:10.1155/2024/4981095 11. Buckhurst PJ, Wolffsohn JS, Shah S, Naroo SA, Davies LN, Berrow EJ. A new optical low coherence reflectometry device for ocular biometry in cataract patients. British Journal of Ophthalmology. 2009;93(7):949–953. doi:10.1136/bjo.2008.156554 12. Garcia Ardoy E, Mateos N, Roda L, Torrado Sierra O, Baptista AM, Serra PM. Repeatability and agreement of swept-source optical coherence tomography and partial coherence interferometry biometers in myopes. Clin Exp Optom. Published online 2022. doi:10.1080/08164622.2022.2152312 13. Morgan PB, McCullough SJ, Saunders KJ. Estimation of ocular axial length from conventional optometric measures. Contact Lens and Anterior Eye. 2020;43(1):18–20. doi:10.1016/j.clae.2019.11.005 14. Kim HS, Yu DS, Cho HG, Moon BY, Kim SY. Comparison of predicted and measured axial length for ophthalmic lens design. PLoS One. 2019;14(1):e0210387. doi:10.1371/journal.pone.0210387 15. He X, Zou H, Lu L, et al. Axial Length/Corneal Radius Ratio: Association with Refractive State and Role on Myopia Detection Combined with Visual Acuity in Chinese Schoolchildren. PLoS One. 2015;10(2):e0111766. doi:10.1371/journal.pone.0111766 16. Tang T, Yu Z, Xu Q, et al. A machine learning-based algorithm used to estimate the physiological elongation of ocular axial length in myopic children. Eye and Vision. 2020;7(1):50. doi:10.1186/s40662-020-00214-2 17. Queirós A, Amorim-de-Sousa A, Fernandes P, Ribeiro-Queirós MS, Villa-Collar C, González-Méijome JM. Mathematical Estimation of Axial Length Increment in the Control of Myopia Progression. J Clin Med. 2022;11(20):6200. doi:10.3390/jcm11206200 18. Dutt DDCS, Yazar S, Charng J, Mackey DA, Chen FK, Sampson DM. Correcting magnification error in foveal avascular zone area measurements of optical coherence tomography angiography images with estimated axial length. Eye and Vision. 2022;9(1):29. doi:10.1186/s40662-022-00299-x 19. Noya-Padin V, Nores-Palmas N, Castro-Giraldez A, Giraldez MJ, Pena-Verdeal H, Yebra-Pimentel E. A Mathematical Model for Axial Length Estimation in a Myopic Pediatric Population Based on Easily Obtainable Variables. Photonics. 2024;11(7):664. doi:10.3390/photonics11070664 20. Lingham G, Loughman J, Panah DS, et al. The long and short of it: a comprehensive assessment of axial length estimation in myopic eyes from ocular and demographic variables. Eye. 2024;38(7):1333–1341. doi:10.1038/s41433-023-02899-w 21. Martínez-Plaza E, López‐de la Rosa A, Molina‐Martín A, Bataille L, Piñero DP. Axial length association with corneoscleral sagittal height and scleral asymmetry. Ophthalmic and Physiological Optics. 2025;45(1):152–159. doi:10.1111/opo.13402 22. Fotedar R, Rochtchina E, Morgan I, Wang JJ, Mitchell P, Rose KA. Necessity of Cycloplegia for Assessing Refractive Error in 12-Year-Old Children: A Population-Based Study. Am J Ophthalmol. 2007;144(2). doi:10.1016/j.ajo.2007.03.041 23. Choong YF, Chen AH, Goh PP. A Comparison of Autorefraction and Subjective Refraction With and Without Cycloplegia in Primary School Children. Am J Ophthalmol. 2006;142(1). doi:10.1016/j.ajo.2006.01.084 24. Peñaranda A, Torrado O, Márquez A, Baptista AM, Serra PM. The effect of cycloplegia in the accuracy of autorefraction, keratometry and axial length using the Myopia Master. BMC Ophthalmol. 2024;24(1):321. doi:10.1186/s12886-024-03529-z 25. Koo TK, Li MY. A Guideline of Selecting and Reporting Intraclass Correlation Coefficients for Reliability Research. J Chiropr Med. 2016;15(2):155–163. doi:10.1016/j.jcm.2016.02.012 26. Hu YY, Wu JF, Lu TL, et al. Effect of Cycloplegia on the Refractive Status of Children: The Shandong Children Eye Study. PLoS One. 2015;10(2):e0117482. doi:10.1371/journal.pone.0117482 27. Momeni-Moghaddam H, Maddah N, Wolffsohn JS, et al. The Effect of Cycloplegia on the Ocular Biometric and Anterior Segment Parameters: A Cross-Sectional Study. Ophthalmol Ther. 2019;8(3). doi:10.1007/s40123-019-0187-5 28. Rubin ML. The induction of refractive errors by retinal detachment surgery. Trans Am Ophthalmol Soc. 1975;73:452–490. 29. Atchison DA, Markwell EL, Kasthurirangan S, Pope JM, Smith G, Swann PG. Age-related changes in optical and biometric characteristics of emmetropic eyes. J Vis. 2008;8(4):29. doi:10.1167/8.4.29 30. Zadnik K, Mutti D, Mitchell G, Jones L, Burr D, Moeschberger M. Normal Eye Growth in Emmetropic Schoolchildren. Optometry and Vision Science. 2004;81(11):819–828. doi:10.1097/01.OPX.0000145028.53923.67 31. Tideman JWL, Polling JR, Vingerling JR, et al. Axial length growth and the risk of developing myopia in European children. Acta Ophthalmol. 2018;96(3):301–309. doi:10.1111/aos.13603 32. Rosenfield M, Chiu NN. Repeatability of subjective and objective refraction. Optom Vis Sci. 1995;72(8):577–579. 33. Naduvilath T, He X, Saunders K, et al. Regional/ethnic differences in ocular axial elongation and refractive error progression in myopic and non-myopic children. Ophthalmic and Physiological Optics. 2025;45(1):135–151. doi:10.1111/opo.13401 Yes