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arxiv: 1906.09149 · v2 · pith:Z6MOW4H4new · submitted 2019-06-20 · ⚛️ physics.med-ph

Mean shape of the human limbus

Alejandra Consejo , Clara Llorens-Quintana , Hema Radhakrishnan , D. Robert Iskander This is my paper

Pith reviewed 2026-05-25 18:46 UTC · model grok-4.3

classification ⚛️ physics.med-ph
keywords limbus shapeFourier seriescorneo-scleral topographywhite-to-white diametereye asymmetryrefractive erroranterior eye height
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The pith

The human limbus is asymmetrically shaped and best modeled by a second-order Fourier series, independent of age and refractive error.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This study measures the limbus of 74 eyes aged 20 to 84 using a corneo-scleral topographer. It fits the topographical boundary to a circle, an ellipse, and Fourier series, determining that the second-order Fourier series gives the lowest root mean square error. No significant differences appear in the horizontal or vertical meridians when comparing young versus older subjects or myopes versus emmetropes. The topographical limbus diameter exceeds the white-to-white diameter by 0.33 mm on average, and quadrant differences are statistically significant. These results indicate that while limbus shape varies between individuals, it does not depend on age or refractive power.

Core claim

Second order Fourier series resulted in the most accurate model to describe the shape of the human limbus. No statistically significant difference in horizontal and vertical meridian between age groups or between myopes and emmetropes. Human limbus is not symmetrical and its shape is subject dependent but not related to age and eye refractive power.

What carries the argument

Second-order Fourier series fitting to the demarcated topographical limbus from anterior eye height data, compared against circle and ellipse models via root mean square error.

Load-bearing premise

The demarcation of the topographical limbus from the raw anterior eye height data accurately represents the true anatomical limbus without bias from the measurement device or fitting choices.

What would settle it

Direct comparison of the fitted limbus boundary against histological sections from donor eyes would show if the topographical demarcation matches the anatomical boundary.

Figures

Figures reproduced from arXiv: 1906.09149 by Alejandra Consejo, Clara Llorens-Quintana, D. Robert Iskander, Hema Radhakrishnan.

Figure 1
Figure 1. Figure 1: shows an example of the limbal demarcation for a randomly chosen subject. Given the demarcated limbal points (indicated by small yellow overlapping circles), calculated using an appropriate algorithm,23 for each acquired measurement, the three types of fit (circular, elliptical and Fourier series) were performed over each of the four measurements acquired per subject. The resulting fitting parameters were … view at source ↗
Figure 2
Figure 2. Figure 2: Mean radial distance in each sector of the eye, for 74 subjects. Error bars indicate +/- one standard deviation. Limbus is not symmetrical. As [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Model of the mean shape of the human topographical limbus (green line) and its range of variability calculated as one standard deviation from the mean (yellow lines), overlapping an image in which the eye is open sufficiently wide for the eyelids not to interfere with the estimated values. Cyan lines represent the average eyelids position in natural gaze position.29 Note that the superior part of the corne… view at source ↗
Figure 4
Figure 4. Figure 4: Population distribution of horizontal limbal diameter (left) and vertical limbal diameter (right) for 74 eyes. Further, we investigated whether age-related differences in limbal shape exist, using Fourier series fitting, since it is the fitting with smallest error from the proposed ones. We analyzed the horizontal and vertical meridian of each subject and related it with their age. No correlation was found… view at source ↗
read the original abstract

Purpose: To characterize the mean topographical shape of the human limbus of a normal eye and ascertain whether it depends on age and refractive power. Setting: Academic institution. Design: Prospective case series. Methods: 74 subjects aged from 20 to 84 years and with no previous ocular surgeries were included in this study. The left eye was measured four times with a corneo scleral topographer (Eye Surface Profiler). From the raw anterior eye height data of each measurement, topographical limbus was demarcated and fitted in three dimensions to a circle, an ellipse and a Fourier series. Root mean square error (RMSE) was calculated to evaluate the goodness of fit. In addition, white to white (WTW) corneal diameter was taken from the readings of the measuring device and compared with the topographical limbus. For statistical analysis, subjects were grouped as young and older, and also according their equivalent sphere correction. Results: From the considered models, second order Fourier series resulted in the most accurate model to describe the shape of the human limbus. The difference between the topographical limbus diameter and the WTW corneal diameter amounted on average and standard deviation to 0.33 mm and 0.24 mm. Statistically significant differences among eye quadrants were found. No statistically significant difference in horizontal and vertical meridian between age groups (P=0.71 and P=0.082, respectively) or between myopes and emmetropes (P=0.78 and P=0.68, respectively) were found. Conclusions: Human limbus is not symmetrical and its shape is subject dependent but not related to age and eye refractive power.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript reports a prospective case series in which the left eyes of 74 subjects (ages 20-84) were measured four times each with the Eye Surface Profiler. From the raw anterior-eye height data the topographical limbus was demarcated and fitted in 3-D to a circle, an ellipse and a second-order Fourier series; RMSE was used to rank the models. The study also compares the fitted limbus diameter with the device's WTW reading, tests for quadrant differences, and reports no statistically significant effects of age or refractive error on the horizontal and vertical meridians.

Significance. If the model-ranking result survives proper complexity penalization, the work supplies a quantitative description of limbus shape variability that could inform contact-lens design and anterior-segment surgery. The repeated-measures design and direct statistical comparisons are methodological strengths.

major comments (1)
  1. [Results] Results section (model comparison paragraph): the claim that the second-order Fourier series is the most accurate rests on direct RMSE comparison among models with unequal numbers of free parameters (circle: 3, ellipse: 5, Fourier series: at least 5-7). Because RMSE decreases monotonically with added degrees of freedom, the ranking does not establish that the Fourier model captures the underlying shape better than the simpler models; AIC, BIC or cross-validation should be applied.
minor comments (2)
  1. The exact number of Fourier coefficients retained and the precise demarcation protocol (automated vs manual, exclusion criteria) are not stated, limiting reproducibility.
  2. Table or figure reporting the actual RMSE values for each model and each subject group would allow readers to judge the magnitude of improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. Below we respond to the major comment.

read point-by-point responses

  1. Referee: [Results] Results section (model comparison paragraph): the claim that the second-order Fourier series is the most accurate rests on direct RMSE comparison among models with unequal numbers of free parameters (circle: 3, ellipse: 5, Fourier series: at least 5-7). Because RMSE decreases monotonically with added degrees of freedom, the ranking does not establish that the Fourier model captures the underlying shape better than the simpler models; AIC, BIC or cross-validation should be applied.

    Authors: We agree that direct RMSE comparison without penalizing for the differing number of free parameters is not the most rigorous method for model selection. In the revised manuscript we will add AIC (and optionally BIC) values for the three models to allow a complexity-adjusted comparison. This will test whether the second-order Fourier series remains preferred after penalization. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical measurement and model fitting

full rationale

The paper reports direct topographic measurements of 74 eyes, manual/automated limbus demarcation from raw height data, and RMSE comparison among circle, ellipse, and second-order Fourier fits. No derivation chain, first-principles prediction, or self-referential step is present; the central claim that the Fourier model yields lowest RMSE is a direct numerical comparison of fitted residuals, not a quantity forced by construction from its own inputs. No self-citations appear in the load-bearing sections, and the study is self-contained against external benchmarks of surface fitting accuracy.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the domain assumption that the Eye Surface Profiler height data and the subsequent demarcation step faithfully represent the anatomical limbus. Standard least-squares fitting and basic statistical tests are used without additional free parameters beyond model order selection. No new physical entities are postulated.

axioms (1)
  • domain assumption The raw anterior eye height data from the Eye Surface Profiler can be used to demarcate the topographical limbus in a manner that corresponds to the anatomical structure.
    Invoked in the Methods description of how the limbus was extracted from each measurement.

pith-pipeline@v0.9.0 · 5846 in / 1465 out tokens · 30286 ms · 2026-05-25T18:46:29.824033+00:00 · methodology

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