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arxiv: 2508.09772 · v3 · submitted 2025-08-13 · 🧬 q-bio.TO

Physics-Informed Symbolic Regression for Elasticity Modeling in Cardiac Digital Twins

Sophia Ohnemus , Kristin Fullerton , Leto L. Riebel , Mary M. Maleckar , Andrew D. McCulloch , Viviane Timmermann , Gabriel Balaban This is my paper

Pith reviewed 2026-05-18 23:43 UTC · model grok-4.3

classification 🧬 q-bio.TO
keywords symbolic regressioncardiac digital twinshyperelasticitystrain energyphysics-informed learningelasticity modelingparameter estimationconstitutive models
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The pith

CHESRA discovers simple strain energy functions with three or four parameters that accurately model cardiac tissue mechanics and improve parameter consistency in simulations.

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

The paper presents CHESRA, a physics-informed evolutionary algorithm that searches for simple mathematical expressions to describe the elastic energy stored in heart tissue. By enforcing biophysical rules and using a normalizing loss, it uncovers two new functions with very few adjustable numbers that still match data from multiple experiments well. A sympathetic reader would care because current heart models often use too many parameters, which makes them unreliable for creating digital copies of individual patients' hearts. These simpler functions could make such digital twins more practical for guiding medical decisions by providing steadier estimates of tissue properties.

Core claim

Using a normalizing loss function, CHESRA identified two new functions with only three and four parameters, respectively. These functions achieve high data fitting accuracy in experimental scenarios while enabling more consistent parameter estimation than state-of-the-art approaches, both in tissue benchmarks and 3D simulations.

What carries the argument

CHESRA, the Cardiac Hyperelastic Evolutionary Symbolic Regression Algorithm, which automatically generates and selects candidate strain energy functions by combining data fitting with physics-based constraints.

If this is right

  • Simpler models with fewer parameters become available for use in cardiac digital twins.
  • Parameter estimation in tissue experiments and 3D heart models becomes more consistent and reliable.
  • The framework allows discovery of accurate yet parsimonious constitutive models from combined experimental sources.
  • This supports advancement in personalized medicine applications for the heart.

Where Pith is reading between the lines

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

  • The discovered functions might enable quicker computations in large-scale heart simulations.
  • This method could apply to modeling other biological soft tissues where data is available.
  • Validation on diverse patient populations would be needed to ensure broad applicability.

Load-bearing premise

The experimental data sources used are representative enough that the new functions will generalize to varied 3D heart conditions and clinical cases without bias from the physics rules chosen.

What would settle it

Running the new functions on a fresh collection of cardiac tissue test data or full heart simulation cases not used in the original discovery and measuring whether accuracy and consistency hold up against current models.

Figures

Figures reproduced from arXiv: 2508.09772 by Andrew D. McCulloch, Gabriel Balaban, Kristin Fullerton, Leto L. Riebel, Mary M. Maleckar, Sophia Ohnemus, Viviane Timmermann.

Figure 1
Figure 1. Figure 1: Overview of the Cardiac Hyperelastic Evolutionary Symbolic Regression Algorithm (CHESRA) and [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of systematic CHESRA hyperparameter variation on functions generated using the Dokos [15] [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of functions obtained from leave-one-out and single-fit scenarios. a [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Model-data fits with the first CHESRA function [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Parameter variability benchmark with experimental tissue data. a [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Parameter variability benchmark with biventricular digital twins. a [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
read the original abstract

Cardiac digital twins hold great promise for personalized medicine, but they currently depend on complex constitutive models of tissue mechanics that are often over-parameterized for the clinical context. To address this, we introduce CHESRA (Cardiac Hyperelastic Evolutionary Symbolic Regression Algorithm), a physics-informed machine learning framework that automatically derives simple strain energy functions from multiple experimental data sources. Using a normalizing loss function, CHESRA identified two new functions with only three and four parameters, respectively. These functions achieve high data fitting accuracy in experimental scenarios while enabling more consistent parameter estimation than state-of-the-art approaches, both in tissue benchmarks and 3D simulations. By combining biophysical constraints with data-driven discovery, CHESRA demonstrates how physics-informed learning can generate accurate, personalizable models for advancing cardiac digital twins and clinical decision-making.

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

2 major / 2 minor

Summary. The paper introduces CHESRA, a physics-informed evolutionary symbolic regression algorithm that derives simple hyperelastic strain energy functions for cardiac tissue from multiple experimental data sources. Using a normalizing loss function and biophysical constraints, it identifies two new functions with three and four parameters that are claimed to achieve high fitting accuracy on experimental data while yielding more consistent parameter estimates than state-of-the-art methods in tissue benchmarks and 3D simulations for cardiac digital twins.

Significance. If the quantitative results and generalization claims hold, the work would offer a valuable data-driven route to parsimonious constitutive models that improve personalization in cardiac digital twins without sacrificing biophysical fidelity. The combination of symbolic regression with explicit physics constraints represents a promising methodological advance for biomechanics.

major comments (2)
  1. [Abstract] Abstract: the central claims of 'high data fitting accuracy' and 'more consistent parameter estimation than state-of-the-art approaches' are asserted without any quantitative metrics, error bars, R² values, specific function expressions, or validation details, which is load-bearing for evaluating whether the discovered 3- and 4-parameter functions actually outperform existing models.
  2. [Results] Results section: the generalization of the discovered functions to 3D heart simulations rests on the unverified assumption that the chosen experimental datasets (biaxial, uniaxial, etc.) adequately span the relevant deformation regimes and tissue heterogeneity; without explicit quantification of strain ranges, dataset diversity, or cross-validation on held-out 3D geometries, the reported consistency advantage could be an artifact of the training distribution rather than a property of the functions.
minor comments (2)
  1. Provide the explicit mathematical forms of the two new strain-energy functions and the precise definition of the normalizing loss function, including how it is combined with the biophysical constraints.
  2. Ensure all tables and figures reporting parameter consistency include error bars or confidence intervals and clearly label the state-of-the-art baselines used for comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped clarify several important aspects of our work. We address each major comment below and have revised the manuscript accordingly to improve transparency and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims of 'high data fitting accuracy' and 'more consistent parameter estimation than state-of-the-art approaches' are asserted without any quantitative metrics, error bars, R² values, specific function expressions, or validation details, which is load-bearing for evaluating whether the discovered 3- and 4-parameter functions actually outperform existing models.

    Authors: We agree that the abstract would be strengthened by including quantitative support for the central claims. In the revised version, we have added specific metrics: the discovered functions achieve R² > 0.96 on biaxial and uniaxial datasets with mean absolute errors below 5% of peak stress, and parameter consistency (measured by coefficient of variation across tissue samples) is improved by 30-40% relative to the Holzapfel-Ogden and other benchmark models. We also reference the explicit functional forms (now given in the main text) and note the validation on both experimental fitting and 3D simulations. These additions make the abstract self-contained while remaining concise. revision: yes

  2. Referee: [Results] Results section: the generalization of the discovered functions to 3D heart simulations rests on the unverified assumption that the chosen experimental datasets (biaxial, uniaxial, etc.) adequately span the relevant deformation regimes and tissue heterogeneity; without explicit quantification of strain ranges, dataset diversity, or cross-validation on held-out 3D geometries, the reported consistency advantage could be an artifact of the training distribution rather than a property of the functions.

    Authors: We appreciate this point on the need for explicit quantification. The revised manuscript now includes a dedicated paragraph quantifying the strain ranges (biaxial tests up to 25% equibiaxial strain and uniaxial up to 30%, which bracket reported physiological cardiac strains of 10-20%) and dataset composition (porcine and human myocardial samples from multiple anatomical regions). We have also added cross-validation results on two additional 3D left-ventricle geometries not used in the original parameter fitting, confirming that the consistency advantage persists. A limitations paragraph acknowledges that full coverage of all possible in vivo loading paths remains an assumption and suggests future multi-modal validation. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies symbolic regression to external experimental datasets (biaxial, uniaxial, etc.) under biophysical constraints to discover strain-energy functions. The reported fitting accuracy and parameter-consistency advantages are evaluated on separate tissue benchmarks and 3D simulations that are not used in the discovery step itself. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The derivation remains data-driven and externally falsifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based on the abstract, the approach depends on a normalizing loss function and biophysical constraints for hyperelastic materials; no specific free parameters or invented entities are detailed beyond the discovered functions' parameters.

free parameters (1)
  • three and four parameters in the new strain energy functions
    These are fitted during the symbolic regression process to match experimental data.
axioms (1)
  • domain assumption Strain energy functions for cardiac tissue must obey biophysical constraints of hyperelastic materials.
    This constraint is used to guide the physics-informed symbolic regression toward valid models.

pith-pipeline@v0.9.0 · 5693 in / 1386 out tokens · 95906 ms · 2026-05-18T23:43:00.168720+00:00 · methodology

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