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

TL;DR

This paper develops a full-volume MRI-based framework for inferring constitutive parameters of rotator cuff tendons using variational system identification (VSI) combined with PDE-constrained optimization. By embedding voxel-wise fiber directions and testing NH, m-HGO, and a polynomial model, the authors extract parsimonious representations that capture key internal deformation patterns, including delamination-like shear bands in torn tendons. PDE refinement yields physically plausible parameters indicative of near-incompressibility and improves internal strain predictions, while VSI tends to prune high-order invariant terms, suggesting a compact, effective modeling approach. The work demonstrates the potential of full-volume inverse modeling to inform clinically relevant tendon mechanics, though it also highlights limitations in volumetric and fiber-field representations and outlines avenues for future enhancement such as phase-field fracture and region-specific modeling.

Abstract

In this work, we characterized the material properties of an animal model of the rotator cuff tendon using full volume datasets of both its intact and injured states by capturing internal strain behavior throughout the tendon. Our experimental setup, involving tension along the fiber direction, activated volumetric, tensile, and shear mechanisms due to the tendon's complex geometry. We implemented an approach to model inference that we refer to as variational system identification (VSI) to solve the weak form of the stress equilibrium equation using these full volume displacements. Three constitutive models were used for parameter inference: a neo-Hookean model, a modified Holzapfel-Gasser-Ogden (HGO) model with higher-order terms in the first and second invariants, and a reduced polynomial model consisting of terms based on the first, second, and fiber-related invariants. Inferred parameters were further refined using an adjoint-based partial differential equation (PDE)-constrained optimization framework. Our results show that the modified HGO model captures the tendon's deformation mechanisms with reasonable accuracy, while the neo-Hookean model fails to reproduce key internal features, particularly the shear behavior in the injured tendon. Surprisingly, the simplified polynomial model performed comparably to the modified HGO formulation using only three terms. These findings suggest that while current constitutive models do not fully replicate the complex internal mechanics of the tendon, they are capable of capturing key trends in both intact and damaged tissue, using a homogeneous modeling approach. Continued model development is needed to bridge this gap and enable clinical-grade, predictive simulations of tendon injury and repair.

Figures (10)

• Figure 1: Representative full volume displacement maps obtained with MRI-based strain acquisition protocol. a) Three-dimensional orientation of a sample (right shoulder), showing longitudinal (1), thickness (2), and transverse (3) directions, and anatomical axes. b) Intact and c) torn displacement maps in the longitudinal (left column), thickness (middle column), and transverse (right column) directions. *TG = tendon grip, Sup. = Superior, Inf. = Inferior, Det. = Detached tissue band.
• Figure 2: Mechanisms activated with our quasi-uniaxial tensile experimental setup. Normal strain components shown in a), b), and c), and shear strain components shown in d), e), and f) of the Lagrangian strain tensor are activated both in the intact and torn states. The torn condition activates a pronounced shear strain response. The volumetric strain component ($\mathrm{E_{vol}}=J-1$) in g) indicates that the material has a degree of compressibility. The dashed lines in the three-dimensional representation at the top left corner represent the sectioning plane for frames b) and d). *TG = tendon grip, Int. = internal, Enth. = enthesis. **Art. = articular, Burs. = bursal.
• Figure 3: Boundary conditions applied to the intact and torn states of a representative sample. Equations of the humeral head and enthesis (tendon-to-bone attachment) boundaries shown in dark red and dark orange were extracted from high-resolution images, whereas the tear region shown in light blue was found with Paraview.
• Figure 4: Procedure to find fiber directions in each 1-2 slice of high-resolution datasets. a) A representative 1-2 slice depicting the entire field of view captured by the MRI experiment. b) Binary mask isolates region of interest (tendon). c) Perimeter of mask shows bursal (cyan) and articular (green) surfaces, and tendon (magenta) and enthesis (yellow) boundaries. TG = tendon grip, Enth. = enthesis. d) Tendon and enthesis edges are removed. e) Fiber directions are found via interpolation from articular to bursal curves. Scale bar = 10 mm.
• Figure 5: Full volume displacement maps of a representative tendon (right shoulder). Each frame depicts the experimental response on the left, accompanied by forward predictions from VSI inference (first row) and adjoint refinement (second row). Displacements in the longitudinal (a, d), thickness (b, e) and transverse (c, f) direction correspond to the torn condition at the 2 mm elongation (a, b, c) and the intact condition at the 1 mm elongation (validation dataset, frames d, e, f), respectively. *TG = tendon gripper, Sup. = superior side, Inf. = inferior side. **Det. = detached tissue band.