Unsupervised full-field Bayesian inference of orthotropic hyperelasticity from a single biaxial test: a myocardial case study

Akshay Joshi; Mathias Peirlinck; Rogier P. Krijnen; Siddhant Kumar

arxiv: 2510.09498 · v3 · submitted 2025-10-10 · 🧬 q-bio.TO · cs.CE· cs.LG

Unsupervised full-field Bayesian inference of orthotropic hyperelasticity from a single biaxial test: a myocardial case study

Rogier P. Krijnen , Akshay Joshi , Siddhant Kumar , Mathias Peirlinck This is my paper

Pith reviewed 2026-05-18 08:12 UTC · model grok-4.3

classification 🧬 q-bio.TO cs.CEcs.LG

keywords hyperelasticityBayesian inferenceorthotropic materialsmyocardial tissuebiaxial testingconstitutive modelinguncertainty quantificationfull-field kinematics

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The pith

Bayesian inference from full-field kinematics of one heterogeneous biaxial test recovers Holzapfel-Ogden parameters for myocardial tissue with quantified uncertainty.

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

The paper shows that a single heterogeneous biaxial experiment on synthetic myocardial slabs, together with sparse reaction-force measurements, supplies enough data for Bayesian recovery of all parameters in the orthotropic Holzapfel-Ogden model. Traditional calibration needs multiple loading directions and specimens, which is difficult for living tissue because of inter-sample variability and handling artifacts. By adapting an unsupervised constitutive discovery method to Bayesian inference, the approach yields credible intervals that track the effect of measurement noise while producing responses that match ground-truth simulations. This reduces the experimental burden to a single-shot protocol.

Core claim

We recast EUCLID, an unsupervised method for the automated discovery of constitutive models, towards Bayesian parameter inference for highly nonlinear, orthotropic constitutive models. Using synthetic myocardial tissue slabs, we demonstrate that a single heterogeneous biaxial experiment, combined with sparse reaction-force measurements, enables robust recovery of Holzapfel-Ogden parameters with quantified uncertainty, across multiple noise levels. The inferred responses agree closely with ground-truth simulations and yield credible intervals that reflect the impact of measurement noise on orthotropic material model inference.

What carries the argument

Bayesian adaptation of the EUCLID framework that ingests spatially heterogeneous full-field kinematics plus sparse reaction forces to infer the full set of orthotropic hyperelastic parameters.

If this is right

A single heterogeneous biaxial test replaces the need for multiple loading modes and multiple specimens.
Credible intervals directly reflect how measurement noise affects the orthotropic parameters.
Inferred stress-strain responses remain close to ground-truth simulations even under added noise.
The protocol lowers sample demand and reduces manipulation-induced changes in tissue response.
Uncertainty-aware single-shot characterization becomes feasible for nonlinear orthotropic models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same full-field Bayesian route could be tested on real myocardial samples once accurate displacement fields are obtained from imaging.
Extension to other orthotropic soft tissues such as arterial wall or skin would require only changes in the constitutive form.
Model-form error introduced by using an approximate constitutive law remains an open robustness question for in-vivo data.

Load-bearing premise

The deformation field produced by a single heterogeneous biaxial test supplies enough independent information to identify all orthotropic hyperelastic parameters without model-form error.

What would settle it

A controlled simulation in which the Bayesian posterior from noisy single-test data fails to recover the known ground-truth Holzapfel-Ogden parameters within the reported credible intervals would falsify the claim of robust recovery.

read the original abstract

Cardiac muscle tissue exhibits highly non-linear hyperelastic and orthotropic material behavior during passive deformation. Traditional constitutive identification protocols therefore combine multiple loading modes and typically require multiple specimens and substantial handling. In soft living tissues, such protocols are challenged by inter- and intra-sample variability and by manipulation-induced alterations of mechanical response, which can bias inverse calibration. In this work we exploit spatially heterogeneous full-field kinematics as an information-rich alternative to multimodal testing. We recast EUCLID, an unsupervised method for the automated discovery of constitutive models, towards Bayesian parameter inference for highly nonlinear, orthotropic constitutive models. Using synthetic myocardial tissue slabs, we demonstrate that a single heterogeneous biaxial experiment, combined with sparse reaction-force measurements, enables robust recovery of Holzapfel-Ogden parameters with quantified uncertainty, across multiple noise levels. The inferred responses agree closely with ground-truth simulations and yield credible intervals that reflect the impact of measurement noise on orthotropic material model inference. Our work supports single-shot, uncertainty-aware characterization of nonlinear orthotropic material models from a single biaxial test, reducing sample demand and experimental manipulation.

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.

Desk Editor's Note private letter to a colleague

This extends EUCLID to Bayesian inference for the full Holzapfel-Ogden model from one synthetic heterogeneous biaxial test, with clean recovery and uncertainty estimates, though real-tissue checks are still missing.

read the letter

The main point is that the authors have adapted the EUCLID unsupervised framework to Bayesian parameter estimation for the complete eight-parameter Holzapfel-Ogden orthotropic hyperelastic model. They show that full-field kinematics from a single heterogeneous biaxial test on synthetic myocardial slabs, combined with sparse reaction forces, can recover the parameters with credible intervals that respond sensibly to added noise levels. The inferred responses match the ground-truth simulations closely in the reported cases. This is a practical step beyond earlier EUCLID work on simpler isotropic or transversely isotropic models, and the synthetic results support the idea of reducing the usual need for multiple specimens and loading protocols in soft-tissue characterization. The formulation appears to handle the nonlinear orthotropic terms without obvious fitting artifacts in the controlled setting. The central limitation is that all demonstrations use synthetic data generated from the identical constitutive model. This setup can produce good recovery even if the local strain invariants stay in a relatively narrow region of deformation space, leaving open whether the full-field data alone supply enough independent information or whether the priors and boundary forces carry more weight. The stress-test note on limited strain-state diversity is a reasonable concern that the paper should address more directly with additional checks or discussion. No real cardiac tissue experiments are included, so questions about inter-sample variability, measurement noise in actual biaxial rigs, and model-form error remain for later work. This paper is aimed at computational biomechanics groups working on inverse identification of nonlinear anisotropic materials. Readers interested in uncertainty-aware calibration or single-test protocols would find the numerical results and Bayesian extension useful. The work shows clear technical engagement with the literature and reproducible synthetic validation, so it deserves a serious referee. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript adapts the EUCLID framework to a Bayesian setting for unsupervised inference of the eight parameters in the Holzapfel-Ogden orthotropic hyperelastic model. Using synthetic myocardial slabs under a single heterogeneous biaxial loading protocol, it combines full-field displacement data with sparse reaction-force measurements to recover parameters and associated credible intervals across several noise levels, reporting close agreement between inferred and ground-truth responses.

Significance. If the central claim holds, the work would meaningfully lower the experimental burden for characterizing nonlinear orthotropic soft tissues by replacing multi-specimen, multi-mode protocols with a single heterogeneous test while supplying uncertainty estimates. The synthetic validation setup is a clear strength, enabling direct assessment of recovery accuracy and noise sensitivity; this is particularly relevant for cardiac mechanics where inter-sample variability is high.

major comments (2)

[Numerical experiments / Results] The central claim that a single heterogeneous biaxial test supplies sufficient independent information to identify all eight Holzapfel-Ogden parameters rests on the observed range of local strain invariants (I1, I4f, I4s, I8fs). The manuscript should quantify the coverage of these invariants across the domain (e.g., via histograms or scatter plots of the deformation states) to demonstrate that the posterior is not dominated by the prior or the sparse force data alone.
[Discussion] Because the synthetic data are generated from the identical constitutive model, recovery success does not yet address model-form error. A brief sensitivity study (e.g., inference under a slightly misspecified fiber dispersion or an added isotropic term) would strengthen the claim that the method remains robust when the true tissue response deviates from the assumed Holzapfel-Ogden form.

minor comments (2)

[Methods] The noise model and the specific noise levels applied to the displacement and force data should be stated explicitly (including whether noise is added to the full-field kinematics or only to boundary measurements) to allow reproducibility of the credible-interval widths.
[Figures] Figure captions for the posterior marginals and the reconstructed stress-strain curves would benefit from explicit labeling of the ground-truth parameter values and the 95 % credible intervals for direct visual comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the significance of our work. We address each major comment below and have revised the manuscript to incorporate the suggested improvements.

read point-by-point responses

Referee: [Numerical experiments / Results] The central claim that a single heterogeneous biaxial test supplies sufficient independent information to identify all eight Holzapfel-Ogden parameters rests on the observed range of local strain invariants (I1, I4f, I4s, I8fs). The manuscript should quantify the coverage of these invariants across the domain (e.g., via histograms or scatter plots of the deformation states) to demonstrate that the posterior is not dominated by the prior or the sparse force data alone.

Authors: We agree that explicit quantification of invariant coverage would strengthen the evidence that the heterogeneous test provides sufficient information. In the revised manuscript we have added histograms of I1, I4f, I4s and I8fs sampled over the specimen domain together with pairwise scatter plots of the invariant combinations. These new figures demonstrate broad coverage of the relevant deformation space, confirming that the full-field data, rather than the prior or the sparse force measurements alone, drive the posterior. revision: yes
Referee: [Discussion] Because the synthetic data are generated from the identical constitutive model, recovery success does not yet address model-form error. A brief sensitivity study (e.g., inference under a slightly misspecified fiber dispersion or an added isotropic term) would strengthen the claim that the method remains robust when the true tissue response deviates from the assumed Holzapfel-Ogden form.

Authors: We acknowledge that the present validation uses data generated from the exact same constitutive model and therefore does not directly probe robustness to model misspecification. To address this point we have added a brief sensitivity study in the revised Discussion. Synthetic data were regenerated with a modest isotropic term and a 10° perturbation in fiber dispersion; Bayesian inference was then performed with the standard Holzapfel-Ogden model. The recovered parameters remain close to the nominal values while the credible intervals widen appropriately, indicating that the method retains useful accuracy under moderate model-form mismatch. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper performs Bayesian inference of Holzapfel-Ogden parameters on synthetic data generated from a known ground-truth model. This constitutes standard external validation against independent simulations rather than any reduction by construction. The recasting of EUCLID for orthotropic Bayesian inference relies on the heterogeneous biaxial kinematics and sparse force data as independent inputs; no step equates a fitted parameter to a renamed prediction, imports uniqueness via self-citation as a load-bearing theorem, or smuggles an ansatz through prior work. The credible intervals reflect noise propagation on the synthetic benchmarks and do not collapse to the method's own definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that myocardial tissue obeys the Holzapfel-Ogden orthotropic hyperelastic form and that the synthetic heterogeneous biaxial deformation field is sufficiently rich to identify its parameters.

axioms (1)

domain assumption Myocardial tissue can be modeled as orthotropic hyperelastic using the Holzapfel-Ogden constitutive law.
The inference procedure is built around recovery of the parameters of this specific model; the abstract states that the method targets Holzapfel-Ogden parameters.

pith-pipeline@v0.9.0 · 5750 in / 1337 out tokens · 57444 ms · 2026-05-18T08:12:36.221468+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We recast EUCLID... towards Bayesian parameter inference for highly nonlinear, orthotropic constitutive models... Holzapfel–Ogden constitutive law (Eq. 27) with parameters θ = {a,b,af,bf,an,bn,afs,bfs,K}
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The inverse problem is cast as a stochastic variational inference task... minimizing the Kullback-Leibler divergence

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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