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Uniaxial poroelastic tendon model with crimped fibre recruitment

This paper presents a one-dimensional poroelastic model of tendon mechanics that incorporates crimped fibre recruitment, demonstrating how the straightening and re-crimping of fibrils create a protective softening mechanism during loading and induce asymmetric, load-dependent hysteresis during unloading compared to a purely neo-Hookean matrix.

Original authors: Zoe C. Godard, Sarah L. Waters, Derek E. Moulton

Published 2026-06-24

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CC BY 4.0

Original authors: Zoe C. Godard, Sarah L. Waters, Derek E. Moulton

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tendon not just as a simple rope, but as a sponge filled with water and thousands of tiny, crinkly springs. This is the core idea of the research paper by Godard, Waters, and Moulton. They created a mathematical model to understand how tendons (the tissues connecting muscle to bone) behave when you pull on them and let go.

Here is a breakdown of their findings using simple analogies:

1. The Two Models: The "Rigid Sponge" vs. The "Crimped Spring Sponge"

To understand what makes a tendon special, the researchers compared two imaginary versions of it:

The "NH Model" (The Rigid Sponge): Imagine a sponge made of a single, uniform material. It's soft, but it stretches evenly. If you pull it, it gets harder to stretch the more you pull. This is a standard way scientists usually model soft tissues.
The "FIB Model" (The Crimped Spring Sponge): This is the new, more realistic model. Imagine the same sponge, but this time, inside it are thousands of tiny, wavy springs (collagen fibers) that are initially crinkled up like a slinky. These springs are embedded in a very soft gel (the non-collagenous matrix).

The Key Difference: In the real tendon (the FIB model), those tiny springs are wavy. They don't do any work until you pull the sponge enough to straighten them out. Once they are straight, they become very stiff and take over the heavy lifting.

2. What Happens When You Pull (Loading)

When you apply a constant pull to the tendon:

The "Toe Region": At first, the tendon stretches easily. Why? Because you are just uncrinkling the wavy springs. They are like a tangled headphone cord; you have to pull the slack out before the cord gets tight.
The "Soft" Stretch: Because the springs are busy uncrinkling, the whole tendon feels softer and stretches further than the "Rigid Sponge" model would predict.
The Protection Mechanism: This is a clever biological trick. Because the springs uncrinkle gradually, the tendon can stretch a lot without snapping the springs or tearing the soft gel. It acts like a natural shock absorber, protecting the structural parts from being overstressed.

3. The "Sponge" Effect: Water and Time

Tendons are mostly water (55–70%). When you pull on them, the water has to move around inside the sponge. This movement isn't instant; it's like trying to squeeze water out of a wet sponge—it takes time.

Diffusion: The speed at which the tendon reacts to your pull depends on how stiff it is.
The Result: Because the "Crimped Spring" tendon is softer at the start, the water moves slower, and the tendon takes longer to reach its final stretched position compared to the "Rigid Sponge."

4. What Happens When You Let Go (Unloading)

This is where the model reveals a surprising asymmetry (a one-way street):

The Rigid Sponge: When you let go, it snaps back quickly and symmetrically. The path it takes to relax is the same as the path it took to stretch.
The Crimped Spring Sponge: When you let go, it is much slower to relax.
- Why? As the tension drops, the straight springs start to get wavy again (re-crimping). As they get wavy, the whole tendon becomes very soft.
- The Feedback Loop: Because the tendon gets softer, the water inside moves even slower. It's like trying to push a car that is slowly turning into a puddle of jelly. The relaxation drags on.

5. The "Hysteresis" Loop (The Energy Loss)

If you plot the stretching and the letting-go on a graph, the two lines don't match up. They form a loop.

The Paper's Claim: This loop (called hysteresis) happens because of the interaction between the water moving and the springs uncrinkling/uncrimping.
The Load Matters: If you pull very hard, the springs stay straight longer, and the loop gets smaller (the tendon behaves more like the rigid sponge). If you pull gently, the springs crinkle up quickly, the tendon gets very soft, and the loop gets huge. The tendon "remembers" the difference between being pulled and being let go.

Summary of the Discovery

The paper argues that the "wavy" nature of collagen fibers isn't just a detail; it's a crucial feature.

It protects the tendon: It allows the tissue to stretch far without breaking the fibers.
It creates a time delay: The tendon reacts slowly to being pulled and even slower to relaxing.
It explains the "loop": The difference between pulling and letting go is caused by the fibers getting wavy again while the water struggles to move through the softening tissue.

The researchers conclude that by including these "crimped springs" in their math, they can better explain real-world experiments and understand how tendons handle the complex job of connecting muscle to bone.

Technical Summary: Uniaxial Poroelastic Tendon Model with Crimped Fibre Recruitment

Problem Statement
Tendons are complex soft tissues composed primarily of collagen fibrils, a non-collagenous matrix (NCM), and water (55–70%). While the mechanical role of "crimped" collagen fibrils—where fibrils straighten and recruit load only after being uncrimped—is well-documented experimentally, existing mathematical models often fail to integrate this recruitment behavior within a poroelastic framework. Previous poroelastic models have either ignored fibrils entirely, treating the solid skeleton as a homogeneous neo-Hookean material, or included fibers without accounting for the progressive recruitment due to crimp. This study addresses the gap in understanding how fibre recruitment within a non-linear poroelastic framework influences the tissue's response to loading and unloading, specifically regarding diffusion rates, stress-strain asymmetry, and hysteresis.

Methodology
The authors developed a one-dimensional, non-linear poroelastic model (the FIB model) representing a tendon bounded by muscle and bone. The solid skeleton is modeled as a composite of:

Crimped Fibrils: Stiff, initially crimped collagen fibrils embedded in a matrix. They bear no load until straightened (uncrimped). The recruitment is governed by a distribution of crimp angles $\theta(r)$ , where fibrils straighten progressively as the tissue stretches.
Non-Collagenous Matrix (NCM): A soft, neo-Hookean background matrix that bears load continuously.

The model assumes incompressible solid grains and fluid, with fluid flow governed by Darcy's law and porosity-dependent permeability (Kozeny-Carman law). The governing equation is a non-linear diffusion equation for porosity, where the diffusion coefficient $D$ depends on the local stiffness ( $\partial s' / \partial \hat{\Phi}$ ).

The FIB model is compared against a control model (NH model) where the solid skeleton is a purely neo-Hookean material with no fibril recruitment, using equivalent macroscopic Young's moduli. The study investigates two boundary conditions at the muscle interface:

Loading: Instantaneous application of a constant tensile stress to a relaxed tendon.
Unloading: Instantaneous release of the load from a pre-stretched tendon.

Simulations were performed using a finite volume method in space and an implicit variable-step size method in time.

Key Contributions

Integration of Recruitment into Poroelasticity: The study extends a previous poroelastic framework to explicitly include a fibre-recruitment rule based on fibril crimp, allowing for a direct comparison between a fibre-reinforced tissue and a homogeneous neo-Hookean tissue.
Derivation of a Bespoke Stress Law: The authors derived a specific stress law for the FIB model that accounts for the transition from crimped (zero load) to taut (linear load-bearing) states, distinguishing the stress contributions of the fibrils and the NCM.
Identification of Diffusion-Stiffness Coupling: The work highlights that the diffusion coefficient is not constant but varies dynamically with fibril recruitment. As fibrils recruit, stiffness increases, increasing the diffusion rate; conversely, as fibrils re-crimp, stiffness drops, reducing diffusion.

Results

Loading Phase:
- Softer Response: The FIB model exhibits a softer stress-strain response than the NH model. Consequently, under the same applied load, the FIB tendon stretches further (higher steady-state strain/porosity).
- Slower Approach to Steady State: Despite stretching more, the FIB model approaches steady state more slowly than the NH model. This is because the initial recruitment phase involves low stiffness (near-zero diffusion), slowing the propagation of the load response through the tissue.
- Spatial Heterogeneity: The FIB model exhibits sharp turning points in stress, porosity, and flux profiles at the location where fibrils are just becoming taut ( $Z_c$ ), a feature absent in the smooth profiles of the NH model.
Unloading Phase:
- Asymmetry and Slow Relaxation: The FIB model displays significant asymmetry between loading and unloading. During unloading, as the tissue relaxes, fibrils re-crimp, causing the tissue to soften. This softening reduces the diffusion coefficient, creating a feedback loop that drastically slows the relaxation process. The tissue often fails to reach a fully relaxed state within the simulation time, leaving a non-negligible fraction of fibrils still taut.
- Hysteresis: This asymmetry manifests as a hysteresis loop in the average stress-strain curve. The hysteresis is most pronounced at lower applied loads where fibrils re-crimp early in the unloading phase. At higher loads, fibrils remain taut longer, allowing for faster initial relaxation and reduced hysteresis, though the system still does not fully relax within the imposed timeframe.
Protective Mechanism: The model demonstrates that the presence of crimped fibrils allows the tendon to undergo larger overall extensions without excessively straining either the fibrils or the NCM, acting as a natural protection mechanism for structural components.

Significance and Claims
The paper claims that incorporating crimped fibre recruitment into a poroelastic framework provides a more biologically realistic interpretation of tendon mechanics than homogeneous neo-Hookean models. The primary significance lies in explaining the origin of hysteresis and loading/unloading asymmetry not solely through viscoelasticity, but through the interaction between the diffusive nature of fluid flow and the stiffness-dependent recruitment of fibres.

The authors note that the FIB model parameters (crimp angle, fibril fraction, fibril modulus) are more directly measurable and biologically interpretable than the phenomenological constants required for neo-Hookean laws. The model successfully reproduces experimental observations of toe regions and hysteresis. However, the authors modestly acknowledge limitations, including the assumption that crimped fibrils bear zero load (which may not be strictly true), the exclusion of fibre sliding, and the simplification to a one-dimensional geometry. The framework is presented as a foundation for future work on tendon damage (e.g., fibril breakage) and higher-dimensional modeling.