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REVIEW 2 major objections 1 minor 34 references

Uniaxial poroelastic tendon model with crimped fibre recruitment

T0 review · 2 major / 1 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read Crimped fibrils in a poroelastic tendon model allow greater stretch without high strain on fibrils or matrix, creating a protection mechanism during loading and unloading.

desk verdict The paper adds crimped-fibre recruitment to a 1D poroelastic tendon model and shows resulting load-unload asymmetry, but the central claims rest on an unvalidated bespoke stress law with no data overlay or sensitivity checks. read the letter →

arxiv 2606.24393 v1 pith:HIHOBYRO submitted 2026-06-23 physics.bio-ph cond-mat.soft

Zoe C. Godard , Sarah L. Waters , Derek E. Moulton This is my paper · ORCID
classification physics.bio-phcond-mat.soft
keywords poroelasticitytendonfibrerecruitmentcrimpedfibrilsuniaxialloadinghysteresisneo-Hookeanmatrixdiffusionequation
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

The paper constructs a one-dimensional poroelastic model for tendon in which the solid phase consists of a neo-Hookean non-collagenous matrix plus crimped fibrils that recruit to load-bearing only after straightening. When compared with an otherwise identical model lacking fibrils, the fibril-inclusive version is softer under constant uniaxial tension and therefore reaches equilibrium more slowly; upon load release it relaxes far more slowly still because re-crimping further reduces stiffness. The resulting loading-unloading asymmetry produces a hysteresis loop in the averaged stress-strain response whose area shrinks as the applied load increases. The model thereby demonstrates how crimped-fibril recruitment lets the tissue extend farther before either fibrils or matrix experience damaging strains, consistent with experimental observations.

What carries the argument

The FIB model: a poroelastic solid whose neo-Hookean matrix is augmented by crimped fibrils that recruit upon straightening, governed by a diffusion equation whose coefficient depends on instantaneous stiffness.

What would settle it

Direct measurement of the time required for a real tendon specimen to reach steady-state extension under a constant uniaxial load, compared against the time required for the same specimen to return to its unloaded length after load release; absence of measurable asymmetry or of load-dependent hysteresis would falsify the central claim.

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Extended reading notes

Core claim

In the FIB model the solid skeleton is a soft neo-Hookean matrix reinforced by crimped fibrils that bear no load until straightened. Under constant applied tension the tissue therefore stretches farther before the fibrils or non-collagenous matrix reach high strains, furnishing a natural protection mechanism. The stiffness-dependent diffusion coefficient makes the FIB model approach steady state more slowly than the pure neo-Hookean case during loading and far more slowly during unloading, when re-crimping softens the tissue; the resulting asymmetry appears as a hysteresis loop in the stress-strain curve whose size decreases with larger applied loads.

Load-bearing premise

The bespoke stress law that accounts for crimp and fibril recruitment, together with the assumption that the diffusion coefficient depends on stiffness, correctly captures the mechanical response of the FIB model within typical tendon parameter ranges.

Editorial extensions

If this is right

  • The FIB model reaches steady state more slowly than the neo-Hookean model during loading because the recruited fibrils keep the composite softer.
  • On unloading the FIB model relaxes much more slowly than the neo-Hookean model because fibril re-crimping further reduces stiffness.
  • The loading-unloading asymmetry produces a hysteresis loop in the averaged stress-strain curve whose area shrinks as applied load increases.
  • The explicit inclusion of crimped fibrils permits clearer biological interpretation and direct comparison with experimental recruitment data.
  • Other fibril constitutive laws can be substituted into the same poroelastic framework without altering the overall structure.

Reading between the lines

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

  • The same recruitment mechanism may underlie the low-strain toe region observed in cyclic tendon tests in vivo.
  • Extending the model to spatially varying crimp angles could predict how regional differences in fibril waviness affect overall tendon compliance.
  • The stiffness-dependent diffusion coefficient suggests that damage-induced softening would further slow fluid flow and recovery after injury.
  • The framework could be used to explore how changes in fibril crimp statistics during aging or disease alter the hysteresis and protective capacity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript presents a one-dimensional poroelastic model for tendons in which the solid skeleton combines a soft neo-Hookean matrix with crimped fibrils that do not bear load until straightened (FIB model). Under uniaxial constant load and subsequent unloading, the FIB model is reported to be softer than a pure neo-Hookean (NH) skeleton within tendon parameter ranges, to equilibrate more slowly, to permit greater overall stretch without excessive local strain on fibrils or matrix (a claimed protection mechanism), and to exhibit loading-unloading asymmetry that produces hysteresis in the averaged stress-strain response; the diffusion coefficient in the governing equation is stated to depend on instantaneous stiffness.

Significance. If the central claims hold after validation, the framework would supply a biologically interpretable poroelastic description of tendon recruitment that could account for observed hysteresis and strain partitioning, with the modular stress law allowing substitution of other fibril constitutive relations.

major comments (2)
  1. [Abstract] Abstract: the assertion that the FIB model is softer than NH and supplies a protection mechanism 'in agreement with experiments' is load-bearing for the central claim, yet the abstract supplies neither the explicit functional form of the bespoke stress law for crimp and recruitment, its derivation, nor any quantitative comparison to measured tendon curves or parameter-fitting procedure.
  2. [Abstract] Abstract: the diffusion equation is stated to have a stiffness-dependent coefficient, but no equation is written, no numerical implementation details are given, and the dependence creates a closed loop (stiffness is set by the same recruitment parameters that determine the diffusion coefficient); no sensitivity check is described to confirm that this produces the reported slower equilibration and hysteresis.
minor comments (1)
  1. [Abstract] The abstract states that other fibril constitutive laws 'can readily be considered,' but does not indicate how changes in the recruitment threshold or stiffness jump would affect the softness ordering or the protection effect.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the FIB model is softer than NH and supplies a protection mechanism 'in agreement with experiments' is load-bearing for the central claim, yet the abstract supplies neither the explicit functional form of the bespoke stress law for crimp and recruitment, its derivation, nor any quantitative comparison to measured tendon curves or parameter-fitting procedure.

    Authors: The abstract is a concise summary; the explicit piecewise stress law (zero load until crimp straightens, then neo-Hookean fibril extension) and its derivation from recruitment kinematics appear in the Methods. Agreement with experiments refers to qualitative features (greater extensibility before high local strains, load-dependent hysteresis) reported in the tendon literature, not a fitted dataset. We will revise the abstract to briefly note the stress-law form and clarify the qualitative nature of the comparison. A full quantitative fitting procedure lies outside the present scope. revision: partial

  2. Referee: [Abstract] Abstract: the diffusion equation is stated to have a stiffness-dependent coefficient, but no equation is written, no numerical implementation details are given, and the dependence creates a closed loop (stiffness is set by the same recruitment parameters that determine the diffusion coefficient); no sensitivity check is described to confirm that this produces the reported slower equilibration and hysteresis.

    Authors: The diffusion equation with instantaneous-stiffness-dependent diffusivity is written and derived in the Methods; the numerical scheme (implicit finite-difference time stepping with updated modulus at each step) is also detailed there. The coupling is not a closed loop: the effective modulus is evaluated from the current local strain and recruitment state during integration, which is standard for nonlinear poroelasticity. We agree a sensitivity study would strengthen the claims and will add one in revision to confirm the slower equilibration and hysteresis persist across the tendon parameter range. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; model results follow from constitutive assumptions and equation solving

full rationale

The paper defines a one-dimensional poroelastic model with a neo-Hookean background plus a bespoke stress law for crimped fibrils that contribute to load-bearing only after straightening, together with a diffusion coefficient that depends on the instantaneous stiffness arising from that law. All reported behaviors (FIB model softer than NH within parameter ranges, slower equilibration on loading, faster relaxation on unloading, hysteresis, and protection via extra stretch without excess fibril/NCM strain) are obtained by solving the resulting diffusion equation under these definitions and comparing the two skeletons. No step equates a claimed prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness or ansatz choice, or renames an empirical pattern as a derived result. The bespoke character of the stress law is explicitly acknowledged and presented as an assumption that can be swapped, confirming the derivation chain remains self-contained rather than circular.

Assumptions & free parameters 2 free parameters · 2 assumptions · 1 invented entities

Model rests on standard poroelastic assumptions plus a custom recruitment law whose parameters are chosen within tendon ranges; no independent evidence for the bespoke stress law is supplied in the abstract.

free parameters (2)
  • neo-Hookean matrix stiffness
    Chosen to represent soft background matrix within tendon parameter ranges
  • fibril stiffness and crimp parameters
    Define when and how fibrils contribute load after straightening
assumptions (2)
  • domain assumption Tissue behaves as a one-dimensional poroelastic continuum whose fluid flow is governed by a diffusion equation whose coefficient depends on instantaneous stiffness
    Core modeling choice stated in abstract
  • domain assumption Crimped fibrils carry zero load until straightened and then add to the solid stress
    Defining assumption of the FIB model
invented entities (1)
  • Bespoke stress law for crimped fibril recruitment
    purpose: To encode the protective straightening mechanism inside the poroelastic framework
    Explicitly constructed for this application; no external validation cited in abstract

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Cite this review

Pith. "Pith review of Uniaxial poroelastic tendon model with crimped fibre recruitment." pith.science (2026). https://pith.science/paper/HIHOBYRO

@misc{pith2026260624393,
  author       = {Pith},
  title        = {Pith review of: Uniaxial poroelastic tendon model with crimped fibre recruitment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIHOBYRO}},
  note         = {Machine review of arXiv:2606.24393}
}
read the original abstract

Fibre recruitment plays an important role in tendon and other biological soft tissue mechanics. Due to their large water content, a popular modelling approach for tendons is poroelasticity. Within this framework some tendon studies have included fibres, though none have included crimped fibre recruitment. We present a one dimensional poroelastic model in which the solid skeleton is composed of a soft neo-Hookean background matrix and crimped fibrils which do not bear load (FIB model). As the tissue is stretched, fibrils are straightened and contribute to load bearing. The fibre-reinforced tissue is compared to a tissue with a purely neo-Hookean (NH) skeleton in response to a uniaxial constant applied load (loading) and release of the load (unloading). The system dynamics are governed by a diffusion equation where the diffusion coefficient depends on stiffness. Within tendon parameter ranges, the FIB model is softer than the NH model, and so approaches steady state more slowly during loading. The presence of crimped fibrils allows the tendon to stretch further without excessively straining the fibrils or the NCM, providing a natural protection mechanism for the tendon's structural components to load, in agreement with experiments. During unloading, the FIB model is much slower to relax as the tissue softens due to fibril re-crimping. This asymmetry in loading and unloading manifests as a hysteresis loop in the stress-strain curve averaged over the tendon. The hysteresis is reduced with increasing applied load. The inclusion of fibrils allows for clearer biological interpretation and potential comparison to data. While the stress law employed in this study is bespoke for the application at hand by accounting for crimp and fibril recruitment, other fibril constitutive laws can readily be considered and incorporated into this framework.

Figures

Figures reproduced from arXiv: 2606.24393 by the authors.

Figure 1
Figure 1. Poroelastic tendon bounded by muscle at Z˜ = 0 and bone at Z˜ = L0, with ambient fluid pressure PA at the muscle boundary. For a given element ∆Z˜ we envisage a cylinder containing crimped fibrils embedded in NCM. Crimp angle θ(˜r) increases radially and, as the tendon is locally stretched, fibrils contained in the radius R˜ c(Z˜) are uncrimped. configuration refers to the material’s configuration at time t˜= 0, whi… view at source ↗
Figure 2
Figure 2. Effective stress against strain / normalised porosity with fixed [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Solid stress (a), solid strain or porosity (b), solid displacement (c), fluid pressure (d) and [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (a) Evolution of Φ(1 ˆ , t) to steady state Φss. The NH porosity reaches steady state faster than the FIB porosity. (b) Rate of change of stress with porosity (strain) against porosity; ∂s′/∂Φˆ for the NH law is greater than for the FIB law, even when all fibrils are u…
Figure 5
Figure 5. Figure 5: NCM (dashed) and fibril (solid) stress (a) and strain (b) and fraction of taut fibrils [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: (a) Stress (left) and porosity (right) at [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: (a-c) t2 = 2t1 = 20. (a) Normalised average stress against normalised average strain, during loading (solid lines) and unloading (dashed lines), for three values of s ∗ . The loading and unloading curves match in the NH model but not in the FIB model. (b) Heatmap of ev…

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