Reduced order modelling of air puff test for corneal material characterisation

Muting Hao; Osama M. Maklad

REVIEW 3 major objections 5 minor 25 references

Reduced order modelling of air puff test for corneal material characterisation

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A machine-learning surrogate that predicts patient-specific air puff pressure loads can replace the fluid solver in a coupled FSI simulation, cutting runtime from roughly 28 hours to 12 minutes while matching clinical corneal deformation.

desk verdict Useful GBR surrogate for air puff pressure, but the off-axis generalization claim needs a proper validation before the speedup claim is fully credible. read the letter →

arxiv 2505.22495 v1 pith:WIED7EUI submitted 2025-05-28 physics.flu-dyn physics.bio-phphysics.comp-ph

Osama M. Maklad , Muting Hao This is my paper

classification physics.flu-dynphysics.bio-phphysics.comp-ph

keywords airpufftestcornealbiomechanicsgradientboostingregressionfluid-structureinteractionreducedordermodellingGaussian-modulatedwaveformintraocularpressuresurrogatemodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that the expensive fluid solver in a coupled air-puff and cornea simulation can be replaced by a machine-learned surrogate. A Gradient Boosting Regressor, trained on 17 full fluid-structure interaction simulations, predicts patient-specific air puff pressure loads on the cornea in seconds. Applying those loads to the finite element eye model cut the total simulation time from about 101,000 seconds (28 hours) to 720 seconds (12 minutes), a roughly 99.2% reduction, while keeping the accuracy of the full FSI model when compared with clinical deformation data. The paper also proposes a neural network that maps IOP, corneal central thickness, and material stiffness to five coefficients of a Gaussian-modulated waveform, reconstructing both pressure and deformation distributions over the corneal surface and time. If the claims hold, patient-specific inverse estimation of IOP and corneal material properties becomes practical enough for large clinical datasets.

What carries the argument

The carrying object is the Gaussian-modulated waveform function $$p_{\text{pred}}=A\, (1-\$\alpha$\,\$\beta$\,$x^{2}$)\,\exp(-\$\beta$\,$x^{2}$)\,\exp\left(-\tfrac{(t-\mu)^2}{2\$sigma^{2}$}\right),$$ with $A$ amplitude, $\beta$ spatial scaling, $\alpha$ spatial attenuation, $\mu$ temporal center, and $\sigma$ temporal spread. The neural network maps the biomechanical inputs (CCT, IOP, $\mu_0$) to the five coefficients, while the Gradient Boosting Regressor, an additive model $\hat{y}=\sum_m \gamma_m f_m(x)$ built by sequential gradient descent in function space, maps inputs directly to the full pressure field. The GBR is what removes the CFD solver from the loop; the waveform function is what gives the NN its interpretable, compact representation of pressure and deformation dynamics.

What would settle it

Run a full FSI air puff simulation at an off-grid combination such as IOP=17.5 mmHg, CCT=560 µm, mu=0.061, which no training run used, and compare the GBR-predicted pressure-time field against the CFD field; the off-axis accuracy claim fails if the error at that interior point substantially exceeds the RMSE range of 0.01 to 0.09 reported for the axis-aligned cases.

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Extended reading notes

Core claim

The central claim is that the air puff pressure distribution on the cornea, normally computed by the CFD component of an FSI model, is a smooth function of three biomechanical parameters namely intraocular pressure, central corneal thickness, and a material stiffness coefficient, together with time, and that a Gradient Boosting Regressor trained on axis-aligned simulations can reproduce that pressure field. When the predicted pressure loads were applied to the finite element eye model, the apical deformation matched four clinical cases with RMSE values between 0.0533 and 0.1305, while the same FE model driven by a generic pressure profile gave RMSE values between 0.3655 and 0.6393, so patient-specific pressure loading carries most of the deformation signal. The proposed neural network with an embedded Gaussian-modulated waveform reconstructs pressure and deformation across the corneal surface and time, with deformation amplitude approximately negative one percent of the pressure amplitude and shared temporal coefficients but distinct spatial attenuation and scaling.

Load-bearing premise

The pressure field learned from 17 simulations, in which only one of IOP, CCT, or stiffness changes at a time, is assumed to vary smoothly enough between those sampled lines that the surrogate predicts patient-specific combinations lying off the sampled grid with the reported accuracy.

Editorial extensions

If this is right

Patient-specific air puff pressure loads, rather than a generic pressure profile, explain most of the variation in corneal deformation; the reported apical-deformation RMSE improves by factors of roughly 5 to 12 across the four clinical cases.
With the CFD solver removed, an individualized air puff simulation can run in about 12 minutes instead of 28 hours on the same hardware, making population-scale parametric studies feasible.
The coefficient-based neural network reveals that pressure and deformation share temporal behavior through similar $\mu$ and $\sigma$ but differ in spatial character, with deformation amplitude opposite in sign and about one percent of the pressure amplitude.
A 99.2% runtime reduction enables building much larger training sets for inverse finite element analysis, which the paper identifies as the route to improved IOP and stress-strain index equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the reported 99.2% speedup applies to the described simulation workflow on one machine, and the axis-aligned training design may under-sample parameter interactions, so off-grid clinical combinations should be tested before the surrogate is used to derive tonometry equations.
Beyond the paper: the Gaussian-modulated form suggests the corneal surface response behaves like a damped, spatially scaled Gaussian in time; if confirmed on a broader parameter set, it could support closed-form corrections to clinical air puff readings.
Beyond the paper: the learned ratio between deformation and pressure amplitudes offers a cheap surrogate for the inverse problem, potentially estimating IOP and stiffness from measured deformation alone.
Beyond the paper: extending the approach to keratoconus or post-refractive-surgery corneas would require constraining the spatial attenuation and scaling coefficients on non-healthy geometries, which the current axis-aligned healthy dataset does not provide.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the GBR surrogate for air puff pressure is a practical contribution, and the four-case clinical validation is a real point in its favor. But the accuracy claim relies on extrapolation from axis-aligned training data, and the internal error metrics don't test what you want them to test. The NN section is currently a curve-fitting exercise.

What's new: applying gradient boosting to predict the spatial/temporal pressure distribution on the cornea from IOP, CCT, and stiffness is new, and the runtime reduction from ~28 hours to ~12 minutes is substantial. The authors also propose a parametric Gaussian-modulated waveform model for pressure/deformation; that's an interesting idea, though it's borrowed from signal processing and lacks a physical derivation.

What's done well: the clinical validation using four patient cases is the most convincing part. Feeding GBR-predicted loads into the FE model and comparing apical deformation to clinical measurements gives an independent check, and the improvement over the FE-only baseline is large. The paper is clearly structured and the GBR methodology is standard and correctly described.

Where it's soft: - The 17 training simulations all vary one parameter at a time, so the training points lie on the axes of the (IOP, CCT, mu) box. The four clinical cases are all off-axis. The authors never show that GBR can generalize off-axis; the internal MAE/RMSE in Section 3.2 are computed from space-time splits within the same simulations, so they measure interpolation in x-t, not generalization to new parameter combinations. This is the load-bearing issue. - The NN section (2.4, 3.4) has no independent validation. The coefficients are fit by minimizing MSE against the same pressure/deformation data shown in the figures, so the good agreement is expected. No error bars, no held-out cases. - Minor: abstract says 48 hours, text says 28; Eq. (1) writes sigma instead of sigma^2 in the Gaussian exponent.

Bottom line: the GBR part is a genuine engineering contribution, but the generalization claim needs a proper hold-out evaluation (e.g., leave-one-out across the 17 simulations, or a few off-axis training runs). I'd send it to peer review with a request for major revision; a good referee will push on exactly this. The paper is probably of interest to the ocular biomechanics and surrogate-modeling communities.

Referee Report

3 major / 5 minor

Summary. The paper proposes a reduced-order modelling pipeline for the Corvis ST air puff test. A Gradient Boosting Regressor (GBR) is trained on 17 fully coupled FSI simulations in which IOP, CCT, and the stiffness parameter μ are varied one at a time; the GBR predicts the time- and space-dependent air puff pressure load on the cornea. This predicted pressure is then applied to a standalone finite element model of the eye and the resulting apical deformation is compared with four clinical cases. The authors report that this GBR+FE workflow reduces computation from about 101,000 seconds (28 hours) to 720 seconds (12 minutes), a 99.2% reduction, while improving the RMSE against clinical deformation compared with an FE-only baseline (Table 6). The paper also proposes a neural network with an embedded Gaussian-modulated waveform that maps (IOP, CCT, μ0) to five coefficients (A, α, β, μ, σ) to reconstruct pressure and deformation fields, supported by qualitative 3D surface plots and correlation analyses.

Significance. If the GBR surrogate truly preserved the accuracy of the FSI model for patient-specific parameters, the 99.2% runtime reduction would be practically valuable for inverse FEA and for building larger datasets for IOP and material-property estimation. The authors deserve credit for including clinical validation: Table 6 shows a consistent RMSE improvement of the GBR+FE model over the FE-only baseline across all four clinical cases, which is the strongest independent evidence in the paper. The NN parametric-embedding idea is also interpretable, linking biomechanical inputs to explicit spatial and temporal coefficients. However, the central claim is currently supported mainly by in-sample fits and by clinical deformation comparisons that are not fully controlled; the NN part is presented without quantitative error metrics. The significance is real but conditional on additional validation of off-axis generalization.

major comments (3)

[§2.1, §3.2] The GBR training design does not support the claimed generalization to patient-specific off-axis parameter combinations. Section 2.1 states that the 17 FSI simulations vary IOP, CCT, or μ one parameter at a time, so every training point lies on an axis of the three-dimensional parameter box. The four clinical validation cases in Table 3 are all off-axis (e.g., Case 1 has IOP=17.5 mmHg, CCT=560 μm, μ=0.061). The internal MAE/RMSE reported in §3.2 (0.0258/0.0673) is computed after splitting spatiotemporal samples, apparently within the same simulations, so it measures space-time interpolation rather than generalization to new corneal parameter sets. The claim that the surrogate 'maintains the accuracy offered by the FSI model' therefore rests on an unverified extrapolation from axis data to the interior of the parameter box. I ask the authors to add a whole-simulation holdout evaluation, or ideally a small set of off-axis FSI simulations, and to report errors for those held-out parameter combinations separately.
[§3.3, Table 6] The clinical validation as reported is not fully controlled. The comparison in Table 6 is between 'GBR+FE model' and 'FE model only', but the text does not specify what pressure profile the FE-only baseline uses. If the FE-only model uses a generic or population-averaged pressure load while the GBR+FE model uses the patient's IOP, CCT, and μ, then the RMSE improvement could reflect the effect of patient-specific parameters in the FE model rather than the accuracy of the GBR pressure prediction. Furthermore, the μ values in Table 3 are assigned from the age-based relationship of reference [19], and the same μ is used both to generate the GBR pressure input and to set the FE material stiffness. The authors should clarify exactly which inputs differ between the two columns of Table 6 and should state whether μ was calibrated against the clinical deformation or fixed a priori.
[§2.4, §3.4] The neural network section is presented without quantitative validation. Section 3.4 reports no train/test split, no error metrics, and no held-out cases; Figure 8 provides qualitative overlays only, and Figure 9 shows fitted correlations. The abstract's claim that the network 'accurately reconstructs' the high-fidelity CFD data is therefore not supported by the reported evidence. In addition, Eq. (1) contains an error: the temporal Gaussian exponent is written as -(t-μ)^2/(2σ), but the gradient equations (6)-(7) and standard Gaussian form require -(t-μ)^2/(2σ^2). The authors should fix Eq. (1), add quantitative reconstruction errors (e.g., MAE/RMSE on held-out cases), and state how the network training data were split.

minor comments (5)

[Abstract vs. §3.3, §4] The abstract states that the FSI model takes '48 hours', while §3.3 and §4 consistently state 28 hours (101,000 seconds); this discrepancy should be resolved.
[§2.1] The stated dataset of 17 simulations appears inconsistent with the listed parameter sets: 8 IOP values plus 5 CCT values plus 5 μ values, with the overlap of (IOP=15, CCT=445, μ=0.0541) and (IOP=15, CCT=545, μ=0.0541), give 16 unique combinations, not 17. Please clarify the counting.
[§3.4] The text in §3.4 says 'Figure 1 demonstrates that the NN ...', but the relevant figure appears to be Figure 8; the figure cross-reference should be corrected.
[Nomenclature and text] The material parameter is denoted inconsistently as μ, μ0, and 'Mu' in different places; please use one symbol throughout.
[§2.1] The phrase 'six-nodded 3D fluid continuum elements' should read 'six-noded', and the element type FC3D6 is otherwise not explained in a way that a reader can verify the mesh quality.

Circularity Check

1 steps flagged · score 4.0 of 10

Central GBR claim has independent clinical grounding, but the NN reconstruction accuracy is an in-sample fit presented as agreement with CFD data.

fitted input called prediction [Section 2.4 (Eqs. 1-2) and Section 3.4 (Figure 8)]
"The loss function is designed to minimize the Mean Squared Error (MSE) between the predicted and actual pressure or deformation values. Taking pressure for example: Loss = 1/N sum (p_pred,i - p_true,i)^2. ... The agreement between the NN-predicted and CFD original surfaces demonstrates the accuracy of the proposed neural network in reconstructing both pressure and deformation distributions."

Equation (1) defines p_pred entirely through the five parametric coefficients (A, alpha, beta, mu, sigma). Equation (2) optimizes those coefficients by minimizing the MSE against p_true from the CFD data. Figure 8 then plots p_predict against p_original for the same CFD cases used to fit the coefficients, so the reported 'agreement' is the in-sample training loss, not an out-of-sample prediction. The statement that the NN 'accurately reconstructs' the CFD result is therefore a restatement of the objective function rather than an independent result.

full rationale

The paper's main claimed contribution—the GBR surrogate replacing the CFD component of the coupled FSI model—is not circular: it is trained on 17 FSI simulations and then validated on four independent clinical cases, where GBR-predicted pressure loads are applied to an FE model and the resulting corneal deformations are compared with measured clinical deformation. That external comparison gives the central claim independent support. The internal GBR MAE/RMSE values may be weak evidence of off-axis generalization because the training parameter combinations are axis-aligned and the split metrics may share simulations, but that is a correctness and data-coverage concern, not a circular reduction. The one clear circular element is the neural-network section: its coefficients are fitted by minimizing MSE against p_true, and the same p_true is used in the figures as 'CFD original' to demonstrate accuracy, making that demonstration equivalent to reporting the fit error. Since the NN framework is secondary to the GBR-based reduced-order model, the overall circularity score is moderate rather than high.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

The central GBR claim rests on the fidelity of the FSI training data from prior work, the adequacy of the Gaussian-modulated parametric form (introduced ad hoc for the NN), and the sufficiency of the sparse axis-aligned dataset to interpolate to clinical parameter combinations. No new physical entities are introduced.

free parameters (6)

A (amplitude coefficient) = varies per case, fitted by NN
Governing coefficient in Eq. (1) for pressure and deformation; optimized to minimize MSE against CFD data.
alpha (spatial attenuation coefficient) = varies per case, fitted by NN
Governing coefficient in Eq. (1) for pressure and deformation; optimized to minimize MSE against CFD data.
beta (spatial scaling coefficient) = varies per case, fitted by NN
Governing coefficient in Eq. (1) for pressure and deformation; optimized to minimize MSE against CFD data.
mu (temporal mean) = varies per case, fitted by NN
Temporal center of the Gaussian-modulated waveform in Eq. (1); fitted by the NN.
sigma (temporal standard deviation) = varies per case, fitted by NN
Temporal spread of the Gaussian-modulated waveform in Eq. (1); fitted by the NN.
GBR hyperparameters (learning rate, n_estimators, max_depth, min_samples_split) = 0.3, 2700, 6, 5
Chosen by hand in Section 2.2; they affect the GBR model performance but are not fitted to the data in a statistical sense.

assumptions (3)

domain assumption The FSI model of [9] accurately reproduces the physical air puff test, providing valid training data.
Section 2.1 relies on the previously published FSI model as ground truth for pressure and deformation.
ad hoc to paper The Gaussian-modulated waveform in Eq. (1) is an adequate functional representation of the true pressure and deformation distributions.
Section 2.4 introduces this parametric form citing a wireless signal processing reference [18]; no physical derivation is given.
domain assumption The 17 simulations sample the parameter space densely enough to interpolate to clinical combinations.
The training set varies only one parameter at a time (Section 2.1), yet the GBR is applied to off-axis clinical cases.

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Cite this review

Pith. "Pith review of Reduced order modelling of air puff test for corneal material characterisation." pith.science (2026). https://pith.science/paper/WIED7EUI

@misc{pith2026250522495,
  author       = {Pith},
  title        = {Pith review of: Reduced order modelling of air puff test for corneal material characterisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WIED7EUI}},
  note         = {Machine review of arXiv:2505.22495}
}

read the original abstract

Models of the fluid-structure interaction (FSI) model for the air puff test were analysed. Using Abaqus, the air puff test is applied to eyes with varying biomechanical parameters, such as material properties, corneal thickness, and radius. A reduced order model of the air puff (a turbulent impinging jet) has been acquired to decrease simulation time from 48 hours for the FSI model to approximately 12 minutes for the finite element analysis (FEA) model alone. To further accelerate simulations and improve model accuracy, Physics-Informed Neural Networks (PINNs) will be integrated with the reduced-order model. This hybrid approach will help expand the model to a larger dataset, enhancing intraocular pressure (IOP) estimation accuracy and the corneal material properties algorithm through inverse FEA. Additionally, a neural network (NN) framework with embedded Gaussian-modulated waveforms is proposed to model the pressure and deformation distributions on the corneal surface as functions of spatial and temporal parameters. By learning the relationship between corneal biomechanical inputs such as Corneal Central Thickness (CCT), Intraocular Pressure (IOP), and baseline properties (Mu), and the governing coefficients of pressure and deformation, the network accurately reconstructs the result that matches well with the high-fidelity CFD data. This approach can quickly capture the distribution of pressure and deformation. It can also provide insights into the distinct spatial and temporal dynamics of pressure and deformation, giving a more comprehensive understanding of fluid-structure interaction phenomena in the air puff test.

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Works this paper leans on

25 extracted references · 23 canonical work pages

[19]

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Elsheikh,A.,Geraghty,B.,Rama,P.,Campanelli,M.,andMeek,K.M.,“Characterizationofage-related variation in corneal biomechanical properties,”Journal of the Royal Society Interface, Vol. 7, No. 51, 2010, pp. 1475–1485. https://doi.org/10.1098/rsif.2010.0108

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [19]

Characterizationofage-related variation in corneal biomechanical properties,

Elsheikh,A.,Geraghty,B.,Rama,P.,Campanelli,M.,andMeek,K.M.,“Characterizationofage-related variation in corneal biomechanical properties,”Journal of the Royal Society Interface, Vol. 7, No. 51, 2010, pp. 1475–1485. https://doi.org/10.1098/rsif.2010.0108

work page arXiv 2010

[2] [1]

Glaucoma,

Weinreb, R. N., Leung, C. K., Crowston, J. G., et al., “Glaucoma,”The Lancet, Vol. 388, No. 10043, 2016, pp. 2183–2193

work page 2016

[3] [2]

Determining in vivo biomechanical properties of the cornea with an ocular response analyzer,

Luce, D. A., “Determining in vivo biomechanical properties of the cornea with an ocular response analyzer,”Journal of Cataract & Refractive Surgery, Vol. 31, No. 1, 2005, pp. 156–162

work page 2005

[4] [3]

Determination of corneal biomechanical behavior in vivo for different ocular conditions using CorVis ST,

Eliasy,A.,Chen,K.-J.,Vinciguerra,R.,Lopes,B.,Abass,A.,Vinciguerra,P.,Roberts,C.,andElsheikh, A., “Determination of corneal biomechanical behavior in vivo for different ocular conditions using CorVis ST,”Journal of Cataract & Refractive Surgery, Vol. 45, No. 6, 2019, pp. 1040–1050

work page 2019

[5] [4]

Keratoconus,

Rabinowitz, Y. S., “Keratoconus,”Survey of Ophthalmology, Vol. 42, No. 4, 1998, pp. 297–319

work page 1998

[6] [5]

Corneal biomechanical properties in different ectatic diseases,

Ambrosio,R.,Alonso,R.,Luz,A.,Correia,F.F.,Lopes,B.,Faria-Correia,F.,Salomao,M.Q.,andLopes, A. S., “Corneal biomechanical properties in different ectatic diseases,”Investigative Ophthalmology & Visual Science, Vol. 58, No. 8, 2017, pp. 3083–3091

work page 2017

[7] [6]

The influence of corneal thickness on intraocular pressure measurement accuracy,

McMonnies, C. W., “The influence of corneal thickness on intraocular pressure measurement accuracy,” Ophthalmic & Physiological Optics, Vol. 36, No. 1, 2016, pp. 47–59

work page 2016

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Topographic and biomechanical indices for the detection of keratoconus,

Saad, A., and Gatinel, D., “Topographic and biomechanical indices for the detection of keratoconus,” Journal of Cataract & Refractive Surgery, Vol. 38, No. 9, 2012, pp. 1395–1403

work page 2012