🧬 biology

Constitutive parameter inference using physics-based data-driven modeling in full volume datasets of intact and torn rotator cuff tendons

This study employs variational system identification and PDE-constrained optimization on full-volume datasets to infer constitutive parameters for intact and torn rotator cuff tendons, demonstrating that while simplified models can capture key deformation trends, current constitutive formulations still require refinement to fully replicate complex internal mechanics for clinical predictive simulations.

Original authors: Carla Nathaly Villacís Núñez, Siddhartha Srivastava, Ulrich Scheven, Asheesh Bedi, Krishna Garikipati, Ellen M. Arruda

Published 2026-05-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Carla Nathaly Villacís Núñez, Siddhartha Srivastava, Ulrich Scheven, Asheesh Bedi, Krishna Garikipati, Ellen M. Arruda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a rotator cuff tendon not just as a rope, but as a complex, layered, 3D sponge made of tiny fibers. When you pull on this sponge, it doesn't just stretch straight; it twists, shears, and squishes in complicated ways, especially if it's torn.

This paper is like a high-tech detective story where the researchers tried to figure out the "secret recipe" (the mathematical rules) that governs how this sponge behaves, using a special kind of MRI camera and some very smart math.

Here is the breakdown of their adventure:

1. The Problem: Looking at the Surface vs. Seeing the Whole Picture

Usually, when scientists study how tendons break, they look at the outside surface, like looking at a car's paint job to guess how the engine works. They use cameras to watch the surface stretch. But the authors realized this is like trying to understand a tornado by only watching the leaves on the ground. You miss the swirling winds inside.

They wanted to see the entire 3D volume of the tendon, including the deep, hidden layers, to understand how it actually moves and tears.

2. The Experiment: The "Stretch and Snap" MRI

The team took sheep shoulders (which are very similar to human shoulders) and put them inside a special MRI machine.

The Setup: They held the tendon in a custom 3D-printed holder and pulled it gently, then pulled it harder.
The Twist: They did this on two types of samples: one that was healthy (intact) and one where they had surgically created a tear (75% detached).
The Magic Camera: Instead of just taking a picture, the MRI acted like a "motion tracker" for every tiny pixel (voxel) inside the tendon. It mapped exactly how every single point inside the 3D block moved, twisted, and sheared.

3. The Detective Work: Variational System Identification (VSI)

Now they had a massive amount of data showing how the tendon moved, but they didn't know the "rules" (the math formulas) that caused that movement.

They used a method called Variational System Identification (VSI). Think of this like a game of "Guess the Rules."

They had a library of three different mathematical "recipes" (models) that could describe how soft tissue behaves.
- Recipe A (Neo-Hookean): A very simple, basic recipe. Like trying to describe a complex cake using only flour and water.
- Recipe B (Modified HGO): A fancy, complex recipe with many ingredients, including specific terms for fiber direction and high-order math.
- Recipe C (Polynomial): A middle-ground recipe that uses a mix of terms.
They fed the MRI movement data into a computer program to see which recipe could "replay" the tendon's movement most accurately.

4. The Big Findings

Here is what the math told them:

The Simple Recipe Failed: The basic "Neo-Hookean" recipe (Recipe A) was too simple. It could guess the general stretch, but it completely failed to predict the twisting and shearing that happened inside the torn tendon. It was like trying to predict a car crash using only a toy car model; it missed the real physics.
The Fancy vs. The Simple Surprise: The researchers expected the most complex recipe (Recipe B) to win. Surprisingly, a simplified version of the polynomial recipe (Recipe C) performed just as well as the fancy one, but with far fewer ingredients. It turned out that you didn't need the most complicated math to get the job done; you just needed the right ingredients.
The "Fiber" Ingredient is Key: The most important thing they learned was that you must include the direction of the fibers in your math. If you ignore which way the fibers are pointing, the model breaks. When they added a map of the fiber directions (like a GPS for the tiny threads inside the tendon), the models suddenly became physically realistic.
Refining the Guess: The first guess (VSI) was good, but they used a second step (PDE-constrained optimization) to "fine-tune" the numbers. This was like taking a rough sketch and polishing it until the lines were perfect. This step fixed a major issue where the computer thought the tendon was squishy (compressible) when it should have been stiff and solid (incompressible).

5. The "Aha!" Moment: Seeing the Invisible

The most exciting part was seeing the shear bands.

In the torn tendon, the MRI showed that the layers of the tendon were sliding against each other (delamination) right at the edge of the tear.
The simple model couldn't see this.
The models with the fiber directions and the right math could see this sliding. They successfully reproduced the "internal friction" that happens when a tendon tears, which is crucial for understanding why tears get worse.

6. The Conclusion

The paper concludes that:

You don't need to cut the tendon into tiny pieces to study it; you can study the whole thing in one piece using full-volume MRI.
You don't need the most complicated math possible; a simpler model works if you include the fiber direction.
Current models are getting better at predicting how tendons tear, but they still need work to be perfect.

In short: They built a 3D digital twin of a torn tendon, figured out the math rules that make it move, and proved that looking at the inside of the tendon is the only way to truly understand how it breaks.

Technical Summary: Constitutive Parameter Inference in Rotator Cuff Tendons

Problem Statement
Rotator cuff tendon tears are a leading cause of shoulder disorders, yet the clinical management of partial-thickness tears remains a subject of debate due to a lack of mechanistic frameworks to predict tear progression. Traditional biomechanical characterization relies on surface-level strain measurements (e.g., Digital Image Correlation) or stress-strain curves from excised samples. These approaches fail to capture the complex, three-dimensional internal mechanics, such as shear strains and delamination mechanisms, which are critical for understanding tear growth. Furthermore, existing finite element models often rely on simplified boundary conditions or regional subdivision that may not reflect the native, continuous mechanical behavior of the tendon. There is a need for a modeling framework that utilizes full-volume, non-destructive data to infer constitutive parameters that accurately represent both intact and injured tissue states.

Methodology
The authors developed a physics-based data-driven modeling framework integrating full-volume Magnetic Resonance Imaging (MRI) data with Variational System Identification (VSI) and Partial Differential Equation (PDE)-constrained optimization.

Experimental Data Acquisition: Experiments were conducted on ovine infraspinatus tendons (validated as a proxy for human supraspinatus tendons). Samples were tested in both intact and torn states (75% bursal footprint detachment) under quasi-uniaxial tension at two displacement levels (1 mm and 2 mm). A custom displacement-encoded MRI sequence (APGSTEi) captured full-volume displacement fields, revealing internal strain behaviors including volumetric, tensile, and shear mechanisms.
Mathematical Framework: The study employed VSI, which utilizes the weak form of the stress equilibrium equation. This approach relates basis operators derived from candidate strain energy density functions ( $W$ ) to the observed displacement fields. The strain energy was decomposed into volumetric, isochoric, and anisotropic contributions dependent on the invariants of the right Cauchy-Green tensor ( $I_1, I_2, I_3, I_4$ ).
Constitutive Models: Three candidate models were evaluated for parameter inference:
- Neo-Hookean (NH): A simplified isotropic model.
- Modified Holzapfel-Gasser-Ogden (m-HGO): An anisotropic model incorporating fiber dispersion and higher-order terms in invariants $I_1$ and $I_2$ .
- Polynomial (Pol): A reduced polynomial model based on invariants $I_1, I_2,$ and fiber-related $I_4$ .
Inference and Optimization:
- VSI with Step-wise Regression: Initial parameters were inferred by minimizing a loss function based on the equilibrium residual. A step-wise regression strategy with an F-test was used to suppress inactive deformation mechanisms, yielding a parsimonious representation of the strain energy function.
- PDE-Constrained Optimization: To address the ill-posedness of the inverse problem (particularly regarding bulk modulus in near-incompressible materials) and refine parameters, an adjoint-based optimization framework was applied. This minimized the error between forward finite element solutions and experimental displacement fields, subject to the equilibrium constraints.
Model Setup: The framework incorporated voxel-wise fiber direction maps derived from high-resolution anatomical images, accounting for the tendon's curvature around the humeral head. Boundary conditions were prescribed directly from experimental measurements to avoid contact simulation complexities.

Key Results

Activation of Multiple Mechanisms: A single quasi-uniaxial tensile test activated volumetric, tensile, and shear mechanisms simultaneously due to the tendon's complex geometry and boundary conditions. Torn tendons exhibited distinct high shear strain bands at the cut-uncut interface, indicative of delamination.
Model Performance:
- The Neo-Hookean (NH) model failed to reproduce key internal features, particularly the shear behavior in injured tendons, and predicted unrealistic strain distributions (e.g., compressive regions not observed experimentally).
- The Modified HGO (m-HGO) model successfully captured the tendon's deformation mechanisms with reasonable accuracy. However, VSI indicated that higher-order terms in $I_1$ and $I_2$ were negligible, suggesting a reduced representation was sufficient.
- The Polynomial (Pol) model, reduced to only three terms (neo-Hookean form plus a quadratic fiber term), performed comparably to the m-HGO model. This suggests that a simplified anisotropic formulation can capture the dominant mechanics.
Parameter Refinement: PDE-constrained optimization significantly improved the physical plausibility of the inferred parameters. While VSI often predicted bulk moduli at the lower boundary (implying poor identifiability), the refined models yielded parameters consistent with near-incompressible behavior. The refinement also introduced fiber dispersion ( $\kappa > 0$ ) in most tendons, whereas VSI often predicted perfect alignment.
Error Reduction: Forward predictions using refined parameters showed reduced displacement and strain errors, particularly in the longitudinal direction and in reproducing the sliding behavior of detached tissue. The m-HGO and Pol models significantly outperformed the NH model in capturing internal shear strain localization.

Significance and Claims
The authors claim this work represents the first tendon-specific modeling framework trained and validated using full-volume data, incorporating native architectural features into a high-fidelity inference scheme.

Unified Modeling Approach: The results suggest that rotator cuff tendons can be modeled as continuous structures with homogeneous material properties, challenging the trend of regional subdivision. The internal mechanical heterogeneity and strain trends were effectively captured by traditional constitutive models when informed by full-volume data.
Parsimony: The study demonstrates that VSI can extract a minimal yet sufficient representation of the strain energy density function. A three-term polynomial model was found to be as effective as the more complex m-HGO formulation, offering a computationally efficient alternative for preliminary investigations.
Clinical Relevance: By avoiding tissue excision and utilizing native attachments, this methodology enables the characterization of material properties in situ. The authors posit that this framework lays the foundation for individualized, predictive simulations of tendon injury and repair, potentially guiding treatment selection for partial-thickness tears.
Limitations: The authors modestly acknowledge that current models do not fully replicate all complex internal mechanics. Limitations include potential local minima in optimization, the assumption of unchanged material properties post-injury, and the reliance on interpolated fiber direction fields. Future work is proposed to incorporate phase-field fracture modeling and time-dependent behavior to further bridge the gap between simulation and clinical reality.