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REVIEW 4 major objections 6 minor 39 references

Physics-informed neural network estimation of active material properties in time-dependent cardiac biomechanical models

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A physics-informed neural network can reconstruct the spatially varying contractility of heart muscle—including the location and shape of fibrotic scars—from sparse, noisy displacement and strain data alone, with no stress measurements.

desk verdict Solid in-silico methods paper on PINN-based active contractility estimation; the central time-dependent heterogeneity claim is not yet tested because all time-dependent runs use a spatially uniform parameter and the assumed space-time factorization excludes local activation delays. read the letter →

arxiv 2505.03382 v1 pith:RBLSIIQR submitted 2025-05-06 cs.LG cs.NAmath.NA

Matthias Höfler , Francesco Regazzoni , Stefano Pagani , Elias Karabelas , Christoph Augustin , Gundolf Haase , Gernot Plank , Federica Caforio This is my paper · ORCID
classification cs.LGcs.NAmath.NA
keywords physics-informedneuralnetworkscardiacbiomechanicsactivestressestimationinverseproblemsscardetectionFourierfeaturesparametertime-dependentelasticity
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 establish that one can infer the local strength of cardiac muscle contraction from the kind of data medical imaging actually provides: displacement and strain fields sampled at limited points, possibly on parallel slices and corrupted by noise, without ever measuring stress. It does this by fitting two neural networks—one for the displacement field and one for the contractility field—to an energy-minimisation problem that balances data misfit against the residual of the cardiac mechanics equations. If the claim holds, a single training run can produce a spatially resolved map of contractility from cine- or tagged-MRI-like data, and that map can be thresholded into healthy tissue, scar core, and grey zone. The study is in-silico: all ground truths come from finite-element simulations on a cuboid tissue sample, so the claim is about feasibility of the inversion strategy rather than clinical validation.

What carries the argument

The central mechanism is the factorisation of active stress into a space-dependent amplitude and a known time profile, $S_a(t;\mathbf{x})=\sigma_0(\mathbf{x})\,S^1_a(t)$, where $S^1_a$ is the solution of the scalar active-stress rate equation with unit amplitude. Because the parameter enters linearly, the inverse problem reduces to recovering one scalar field instead of a full spatio-temporal activation history. That field is represented by a second neural network trained jointly with the displacement network; Fourier feature embeddings on the parameter network counteract spectral bias, a distance-function layer enforces the Dirichlet boundary exactly, and a gradient-based regularisation weighted away from the Dirichlet face restores identifiability of $\sigma_0$ near that boundary. Residual-based attention on the PDE loss stabilises the resulting multi-objective optimisation.

What would settle it

A decisive test: simulate the same contraction model with two regions that activate at different times, feed the network only displacement and strain data, and threshold the recovered $\sigma_0(\mathbf{x})$ field; if the late-activating region shows up as a spurious low-contractility scar, the single-time-profile assumption marks the boundary of the method’s validity.

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Extended reading notes

Core claim

The paper claims that a physics-informed neural network, given displacement measurements (and, for heterogeneous fields, additional strain measurements) sampled on a few parallel slices of cardiac tissue, can recover the spatially varying active-stress parameter $\sigma_0(\mathbf{x})$ that drives contraction, in both a quasi-static approximation and the full time-dependent momentum equation. The key reduction is that the active-stress rate equation has the rescaling property $S^{\sigma}_a(t)=\sigma\,S^1_a(t)$, so the temporal shape of contraction is computed once and the inverse problem becomes the estimation of the scalar field $\sigma_0(\mathbf{x})$. In the reported in-silico tests the field is recovered with relative errors of a few percent even at noise level $LD=0.05$, and thresholding the reconstructed field locates simulated fibrotic scars and their surrounding grey zones without stress data and without prior shape assumptions.

Load-bearing premise

The method assumes every point in the tissue follows the same known time course of contraction and differs only in one overall strength number, so the entire spatial pattern of contractility is captured by that single strength field; if contraction timing varies from region to region, the estimated field can be misleading.

Editorial extensions

If this is right

  • At the tested noise levels, up to an $LD=0.05$ limiting dispersion, the contractility parameter is recovered to within roughly 1–6% relative error in both quasi-static and time-dependent cases, with displacement fields recovered to 2–5% $L^2$ error.
  • Simulated scars, including a core surrounded by a grey zone, are located by thresholding the reconstructed contractility field at about 50 kPa, with only minor misclassification and a slight underestimation of the scarred area.
  • Stress data are never needed; the heterogeneous cases use displacement plus strain, and the paper notes that strain can be obtained from tagged-MRI or derived from displacement data in post-processing.
  • Because the temporal activation profile is precomputed once from a scalar ODE and the architectures are tailored to the problem, the method requires fewer training points and epochs than previous PINN-based inversions in elasticity.
  • Exactly enforcing the Dirichlet boundary and applying the weighted gradient regularisation mitigates a known failure of active-stress identifiability near the fixed boundary, and the results are robust to mis-specification of the boundary condition as a Robin condition with an uncertain spring stiffness.

Reading between the lines

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

  • A natural extension the paper does not run: allow the activation timing itself to vary spatially, and try to estimate $t_{\mathrm{sys}}(\mathbf{x})$ as an additional field; the same two-network construction could turn the single-time-profile limitation into a diagnostic for dyssynchrony.
  • Because the optimal classification threshold depends on the relative proportions of healthy and scarred tissue, clinical use would likely need an adaptive threshold chosen from the reconstructed field’s histogram rather than the fixed 50 kPa value used here.
  • The space–time factorisation of the parameter should transfer to any constitutive parameter that enters the stress linearly with a known temporal envelope, so the approach may generalise beyond cardiac active stress to other contractile materials.
  • The seed-to-seed spread of the reconstructed fields, which the paper reports as trajectory-spanned areas, could be converted into a pixel-wise confidence map for scar boundaries, giving clinicians an uncertainty estimate alongside the thresholded classification.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper develops a physics-informed neural network (PINN) methodology for estimating the active contractility of cardiac tissue from sparse, noisy displacement and strain measurements. The forward model is a three-dimensional, time-dependent hyperelastic biomechanical model with an active-stress term governed by a Bestel-Clément-Sorine ODE. The unknown maximum active stiffness σ0 is either a scalar trainable parameter or a neural-network field, and the displacement field is represented by a separate network. The reconstruction is formulated as an energy minimization with data, PDE-residual, and boundary losses, augmented with Fourier features, residual-based attention, exact Dirichlet boundary enforcement, and a complementary-weight regularization of the parameter network near the Dirichlet boundary. Experiments on an idealized 10 mm cube cover quasi-static and time-dependent homogeneous cases, and quasi-static heterogeneous cases with one or two spherical scars. The reported results show small L2 errors in displacement and parameter estimates at noise levels up to LD = 0.05, and qualitative scar reconstructions after thresholding.

Significance. If the results hold, the paper offers a useful step toward non-invasive scar identification from imaging-like data without stress measurements. The manuscript has concrete strengths: it systematically ablates weighting schemes (Section 2.4 and Appendix S-2), compares exact versus weak boundary enforcement (Appendix S-4), analyzes robustness to boundary-condition misspecification (Appendix S-5), and reports multi-seed runs with geometric-mean trajectories. The apparent-Pareto-front analysis for the scalar case is a practical contribution to PINN hyperparameter selection. However, the current evidence is a self-consistency check on an idealized cube geometry: the FEM ground truths are generated with the same constitutive and active-stress models used in the physics loss, and the central time-dependent spatial-parameter claim is not exercised by any experiment. The clinical relevance suggested by the Introduction and Conclusion therefore rests on additional, untested assumptions.

major comments (4)
  1. [§3.1.2, Tables 3–5] The parametrisation Sa(x,t)=σ0(x) S1_a(t), introduced in §2.1.1 and used throughout §3.1.2, is load-bearing for the time-dependent claim, but it is only exact if the activation time course a(t) is the same at every material point. Regional activation delays (e.g., around scars or in dyssynchrony) replace a(t) by a(t−τ(x)); because S1_a(t) varies strongly over the systolic window 160–350 ms, a delay τ(x) is confounded with an amplitude change, so a uniform σ0 could be reconstructed as a spurious low-contractility region. The time-dependent experiments in §3.1.2 treat homogeneous σ0 only, and the heterogeneous scar reconstructions in §3.2 are quasi-static, so no experiment exercises this confound. The Introduction's claim that the method 'enables the estimation of spatially-varying active contractility properties both in quasi-static and time-dependent scenarios' is therefore supported only under a physiologically non-trivial assumption that is not tested. I recommend adding a time-dependent heterogeneous test case with regional activation delay (or with different activation time courses per region) and, if the confound persists, tempering the claim accordingly.
  2. [§3.1.2, Tables 3–5] The robustness results are reported only for 'successful seeds' while the text states that each test case was run with ten seeds in total (e.g., Table 4: 7/10 noiseless and 6/10 noisy; Table 5: 8/10 and 9/10). Since the success criterion is not defined, the reader cannot tell whether the excluded seeds failed to converge, found a wrong local minimum, or were discarded for other reasons. Averaging over the survivors alone overstates the method's robustness and the reported uncertainty. Please define the success criterion a priori, report outcomes for all ten seeds (including failures), or provide a failure-mode analysis; this is needed to substantiate the paper's robustness claims.
  3. [Appendix S-6 and §3.2.1–3.2.2] The classification threshold T for scar detection is selected by minimising the total misclassification rate on the same test cases for which classification accuracy is reported (Figures S7–S8, with T≈50 kPa). This is an in-sample hyperparameter choice, so the reported false-positive/false-negative behaviour in Figures 7 and 9 is optimistically biased and does not estimate generalisation to new scar geometries or tissue proportions; Appendix S-6 itself notes that the optimal threshold depends on the relative proportions of healthy and scarred tissue. Please evaluate the threshold on a hold-out set (or via cross-validation) and report classification metrics as a function of T on unseen cases.
  4. [§3.1 and §3.2] Because the observation data are produced by FEM solutions of the same governing equations (Eq. (1)) and the same active-stress model (Eq. (2)) used in the PINN loss, the benchmarks are a self-consistency check rather than an external validation of the constitutive or active-stress modeling assumptions. This is an appropriate scope for a first methods paper, but the Introduction and Conclusion should state this limitation explicitly and avoid wording that suggests immediate applicability to clinical data.
minor comments (6)
  1. [Section 3.2 and Appendix S-6] The main text and Appendix S-6 state that the optimal classification threshold is approximately 50.00 kPa, but Figure S7's caption reports the optimal threshold as 34.3 kPa for the single-scar case. Please reconcile this inconsistency.
  2. [Tables 1, 3, 4, 5] The caption of Table 3 reports an 'L1 relative error on Sa,' while Tables 1, 2, 4, and 5 do not specify the norm used; please state consistently which norm is used for each reported parameter error.
  3. [Section 3.1.2] The handling of the initial condition for the time-dependent PDE is not fully specified: 100 data points at t = 160 ms are mentioned, but Eq. (1) requires both u(0) and v0; please state whether velocity data are used and how the implicit initial condition is enforced in the loss.
  4. [Section 2.3.2 and Appendix S-3] The main text says the modified boundary loss plus R2 regularization 'effectively reduces' the identifiability problem, while Appendix S-3 states that it mitigates the issue only 'to some extent' and that artifacts remain near the edges; please make these statements consistent.
  5. [Section 3.2] No quantitative classification metrics (e.g., Dice coefficient, FPR/FNR) are reported in the main text for the scar cases, so the central scar-detection claim rests on qualitative figures; please add numeric metrics.
  6. [General] The paper does not include a data/code availability statement. Since the study is fully in silico and uses the open-source carputils/carpentry framework, please state whether scripts and trained configurations will be released to support reproducibility.

Circularity Check

2 steps flagged · score 6.0 of 10

Scar-detection threshold and loss weights are selected on the very test cases whose accuracy is then reported; the underlying active-stress field inversion itself is not circular.

  1. fitted input called prediction [Appendix S-6, applied in Sections 3.2.1 and 3.2.2]
    "For the sake of completeness, Figures S7 and S8 plot FPR and FNR against different threshold values T for the single scar and two-scar test cases of Section 3.2.1 and Section 3.2.2, respectively. The optimal threshold for both test cases is approximately 50.00 kPa. For the sake of simplicity, we then consider this threshold value for both test cases as a satisfying educated guess of the optimal threshold."

    The scar-detection results in Section 3.2 (Figures 7 and 9) are obtained by thresholding the reconstructed active-stress field of the same two test cases whose ground-truth labels were used in Appendix S-6 to select T = 50 kPa by minimizing the total misclassification rate FPR+FNR. Therefore the reported false-positive/false-negative behavior and the claim of accurate scar-shape reconstruction are in-sample evaluations of a threshold fitted to the labels being classified, not independent predictions. The active-stress field estimation itself is not circular, but the classification step that supports the headline scar-detection claim reduces to a fit on the target data.

  2. fitted input called prediction [Section 3.1.1, Table 2 and Section 2.4.1]
    "The sweet spot according to the Pareto front is reached with the following relative weights (normalised by the sum) given in Table 2."

    Table 2 is titled 'Optimal weights obtained from the Pareto front analysis in Figure 4', and Figure 4 encodes the relative error epsilon_Sa;rel for each weight combination. The performance reported in Table 1 for the quasi-static homogeneous case is thus obtained with loss weights selected on the same test case by minimizing the very error metric being reported (e.g., 1.24e-2 noiseless, 4.93e-2 at LD=0.05). This is in-sample hyperparameter selection rather than a prediction under fixed a-priori weights. It does not make the PINN or the physics-loss derivation circular, but it inflates the reported accuracy by construction.

full rationale

The paper's central estimation chain is not circular: the PINN minimizes a physics loss based on the cardiac biomechanical model (Eq. 1-5) against sparse displacement/strain data, and the time-dependent factorization Sa(t;x) = sigma0(x) S1_a(t) is an exact consequence of the linearity of the Bestel-Clement-Sorine ODE (Eq. 2), not an ansatz that assumes the answer. Using FEM solutions of the same model as synthetic ground truth is a standard self-consistency benchmark for inverse problems and, by itself, does not make the derivation circular. However, two evaluation steps do reduce by construction. First, the scar-classification threshold is chosen in Appendix S-6 by minimizing misclassification on the same single-scar and two-scar test cases whose detection accuracy is then reported in Section 3.2, so the scar-detection claim is partly fitted to the labels it claims to predict. Second, the loss weights used for the headline homogeneous-case errors are selected via the apparent-Pareto-front analysis on that same test case, making the reported errors in-sample best-case numbers. Neither step invalidates the field-reconstruction methodology, but both mean that several reported accuracies are self-selected rather than out-of-sample predictions. Hence a partial-circularity score of 6 is appropriate.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

No new physical entities, forces, or conserved quantities are introduced. The neural networks and attention mechanisms are computational tools, not physical postulates. The free parameters are mostly hyperparameters tuned on the same in-silico test cases, which is the main circularity burden.

free parameters (6)
  • Loss weights (lambda_OBS, lambda_PDE, lambda_BC) = Table 2: lambda_OBS/sum = 0.9985, lambda_PDE/sum = 9.885e-4; Table 6: data 1e2, PDE 1e-1; Table 7: data 1e3, PDE 1e1
    Chosen via apparent Pareto front analysis on the same test case (Section 2.4.1) and reported per test case; reconstruction quality depends strongly on these values.
  • Classification threshold T for scar detection = 50.00 kPa (with Figure S7 caption reporting 34.3 kPa)
    Selected in Appendix S-6 by minimizing total misclassification on the same heterogeneous test cases; used for all scar detection claims.
  • Fourier feature frequency sigma_F = sigma_F = 3, feature dimension 24 or 18
    Chosen in Section 2.5.3 to avoid salt-and-pepper artifacts; the paper notes sensitivity to this parameter.
  • Sigmoid slope alpha_S in parameter network output = alpha_S = 8
    Set in Section 3.2 for interval constraints on the active stress field; results claimed robust for alpha in [1,10].
  • Regularization weight for NNSa = 1e-4 (single scar), 1e-3 (two scars)
    Hand-tuned in heterogeneous test cases (Tables 6 and 7) to stabilize boundary identifiability.
  • Moving average parameter alpha for adaptive weighting = alpha = 0.1
    Used in Appendix S-2 comparison; the final method uses RBA instead, but this tuning is reported as influencing the comparison.
assumptions (7)
  • domain assumption The Guccione transverse-isotropic passive constitutive law with the given parameter values (alpha_P = 0.8 kPa, kappa = 650 kPa, etc.) describes passive myocardial tissue.
    Invoked in Section 2.1.1 for all forward simulations and PINN physics losses. If the law is wrong, the estimated active parameters inherit the error.
  • domain assumption Active stress follows the Bestel-Clement-Sorine model with known, spatially uniform time course S1_a(t); the parameter of interest reduces to sigma0(x).
    Section 2.1.1 introduces the rescaling S_sigma = sigma S1, which is used to decouple space and time. Regional variation in activation timing would break this reduction.
  • domain assumption The synthetic FEM solution from carpentry is an adequate proxy for clinical displacement and strain measurements.
    Used throughout Section 3 as ground truth. Clinical imaging data would involve different noise, resolution, and boundary conditions.
  • domain assumption The chosen neural network architectures (tanh MLPs with 3 hidden layers, parameter net widths 12-8-4) can represent the true displacement and parameter fields to the required accuracy.
    Assumed in Section 2.5 and 3; the paper does not provide an approximation error analysis.
  • ad hoc to paper The added boundary regularization R2(w2) with complementary weight bar-omega resolves the identifiability problem near the Dirichlet boundary without biasing the interior solution.
    Introduced in Section 2.3.2 and evaluated in Appendix S-3. The paper shows it mitigates, but does not fully remove, artifacts near the boundary.
  • domain assumption Measurement noise is additive, zero-mean Gaussian characterized by the limiting dispersion LD.
    Defined in Section 2.1.2 and used for all noisy test cases. Real imaging noise may be structured rather than white Gaussian.
  • domain assumption The apparent Pareto front analysis identifies the optimal scalarization weights for the inverse problem.
    Used in Section 2.4.1 to select loss weights for the homogeneous test case; the analysis is performed on the same test case whose errors are then reported with those weights.

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Pith. "Pith review of Physics-informed neural network estimation of active material properties in time-dependent cardiac biomechanical models." pith.science (2026). https://pith.science/paper/RBLSIIQR

@misc{pith2026250503382,
  author       = {Pith},
  title        = {Pith review of: Physics-informed neural network estimation of active material properties in time-dependent cardiac biomechanical models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RBLSIIQR}},
  note         = {Machine review of arXiv:2505.03382}
}
read the original abstract

Active stress models in cardiac biomechanics account for the mechanical deformation caused by muscle activity, thus providing a link between the electrophysiological and mechanical properties of the tissue. The accurate assessment of active stress parameters is fundamental for a precise understanding of myocardial function but remains difficult to achieve in a clinical setting, especially when only displacement and strain data from medical imaging modalities are available. This work investigates, through an in-silico study, the application of physics-informed neural networks (PINNs) for inferring active contractility parameters in time-dependent cardiac biomechanical models from these types of imaging data. In particular, by parametrising the sought state and parameter field with two neural networks, respectively, and formulating an energy minimisation problem to search for the optimal network parameters, we are able to reconstruct in various settings active stress fields in the presence of noise and with a high spatial resolution. To this end, we also advance the vanilla PINN learning algorithm with the use of adaptive weighting schemes, ad-hoc regularisation strategies, Fourier features, and suitable network architectures. In addition, we thoroughly analyse the influence of the loss weights in the reconstruction of active stress parameters. Finally, we apply the method to the characterisation of tissue inhomogeneities and detection of fibrotic scars in myocardial tissue. This approach opens a new pathway to significantly improve the diagnosis, treatment planning, and management of heart conditions associated with cardiac fibrosis.

Figures

Figures reproduced from arXiv: 2505.03382 by the authors.

Figure 1
Figure 1. Time-evolution of the Bestel-Clément-Sorine model with different values of the maximal active stiffness σ0 > 0. consider both a quasi-static approach and the time-dependent problem. In the first setting, for a fixed time point t ∗ ∈ (0, T], we denote the corresponding displacement u ∗ (X) ∈ R 3 as the solution to the quasi-static equation −∇ · P∗(u ∗ ) = f, where P∗(u) = Ppas(u) + Sa(t ∗ ) Ff0 ⊗ f0 √ Ff0 · Ff0 . Pro… view at source ↗
Figure 2
Figure 2. Ground-truth FEM solution of the displacement field. The light-shaded cube represents the reference configuration, whereas the coloured object is in deformed configuration. LD no. seeds L 2 rel. err. on u ϵSa;rel 0.00 10 3.21 × 10−2 1.24 × 10−2 0.05 10 4.84 × 10−2 4.93 × 10−2 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Comparison of training and testing losses for displacement data and PDE discrepancy and the relative error on the parameter Sa. Left: Algorithm performance considering noiseless data. Right: Performance using data corrupted with noise corresponding to LD = 0.05. The solid line depicts the geometric mean over the seeds; the shaded region is the area spanned by the trajectories. The second dashed vertical line marks t… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Analysis of apparent Pareto fronts (constant active stress parameter Sa, quasi-static approximation). The results show the different training trajectories JPDE and JBCN and the relative error in the parameter Sa, denoted by ϵSa;rel, with different weights for JOBS, JPD…
Figure 5
Figure 5. Figure 5: Time evolution of the PINN solution. Top row: ground-truth data corrupted with noise corresponding to LD = 0.05 and then sliced along the y-axis. Middle row: PINN reconstruction of the displacement field. Bottom row: absolute error between the PINN reconstruction and t…
Figure 6
Figure 6. Figure 6: One-scar test case. Ground-truth FEM solution of the displacement field. The light-shaded cube represents the reference configuration, whereas the coloured object is in deformed configuration. 4 Discussion The findings of this study suggest that our PINN-based approach…
Figure 7
Figure 7. Figure 7: Active stress field reconstruction for a case with a central scar surrounded by a grey zone (border zone). From left to right: (1) Ground-truth solution showing the spatial distribution of active stress parameter Sa with values ranging from 7.87 kPa (scar core) to 118.…
Figure 8
Figure 8. Figure 8: Two-scar test case. Ground-truth FEM solution of the displacement field. The light-shaded cube represents the reference configuration, whereas the coloured object is in deformed configuration. crucial for identifying scar boundaries. In addition, the proposed threshold…
Figure 9
Figure 9. Figure 9: Two-scar test case. The plots show the thresholded active stress field including two scars. Left: ground-truth solution. Right: PINN reconstruction of NNSa . Red areas indicate false negatives (scarred tissue incorrectly classified as healthy) and blue areas indicate f…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.