Unsupervised full-field Bayesian inference of orthotropic hyperelasticity from a single biaxial test: a myocardial case study

TL;DR

This work develops a physics‑informed, unsupervised Bayesian framework (EUCLID with stochastic variational inference) to identify the full set of nonlinear, cross‑correlated orthotropic Holzapfel–Ogden parameters from a single heterogeneous biaxial test on myocardial tissue. By embedding the 3D full‑field displacement data into a weak‑form momentum balance and using hierarchical priors with an inferred noise variance, the method quantifies parameter uncertainty while solving a high‑dimensional inverse problem. The study demonstrates that deliberate geometric and microstructural heterogeneity markedly improves identifiability of shear‑related parameters, even under measurement noise, and that single‑shot inferences can predict multimodal loading responses with high fidelity (R^2 values approaching 1.0 in many cases). These results offer a significant reduction in tissue manipulation and exemplify how experimental design can be tuned to maximize observability in full‑field inverse constitutive characterization, with potential applicability to other complex biological tissues. The approach provides a practical path toward uncertainty‑aware, localized tissue characterization in scenarios with limited sample availability.

Abstract

Fully capturing this behavior in traditional homogenized tissue testing requires the excitation of multiple deformation modes, i.e. combined triaxial shear tests and biaxial stretch tests. Inherently, such multimodal experimental protocols necessitate multiple tissue samples and extensive sample manipulations. Intrinsic inter-sample variability and manipulation-induced tissue damage might have an adverse effect on the inversely identified tissue behavior. In this work, we aim to overcome this gap by focusing our attention to the use of heterogeneous deformation profiles in a parameter estimation problem. More specifically, we adapt EUCLID, an unsupervised method for the automated discovery of constitutive models, towards the purpose of parameter identification for highly nonlinear, orthotropic constitutive models using a Bayesian inference approach and three-dimensional continuum elements. We showcase its strength to quantitatively infer, with varying noise levels, the material model parameters of synthetic myocardial tissue slabs from a single heterogeneous biaxial stretch test. This method shows good agreement with the ground-truth simulations and with corresponding credibility intervals. Our work highlights the potential for characterizing highly nonlinear and orthotropic material models from a single biaxial stretch test with uncertainty quantification.

Figures (12)

• Figure 1: Schematic overview of our unsupervised full-field stochastic variational inference framework for orthotropic hyperelastic tissue behavior. Quantification of the tissue's microstructural organization, such as spatially varying fiber orientations, provides essential architectural information across the sample domain. In parallel, point-wise measurements of displacements $\hat{\bm{u}}$ and reaction forces $\hat{\bm{R}}^{\beta}$ are acquired under quasi-static biaxial loading using digital volume correlation. The sample geometry is discretized into a finite element mesh, enabling reconstruction of continuous deformation fields $\bm{F}$ from the measured data. A nonlinear orthotropic constitutive model $\psi$, parameterized by a set of cross correlated material parameters $\bm{\theta}$, serves as the mechanistic basis for inference. Given the reconstructed deformation and microstructural organization fields, the model predicts stress responses at the element level, which are used to compute internal and external nodal forces. Residuals are formulated through the weak form of the conservation of linear momentum, minimized pointwise for unconstrained degrees of freedom and in aggregate at boundaries with known reaction forces. The inverse problem is cast as a stochastic variational inference task, where a variational guide distribution $q(\bm{\theta})$ approximates the true posterior $\pi \left( \bm{\theta} \,|\, \hat{\bm{u}},\hat{\bm{R}}^{\beta} \right)$ by minimizing the Kullback-Leibler divergence $\mathrm{KL} \left( \pi \left( \bm{\theta} \,|\, \hat{\bm{u}},\hat{\bm{R}}^{\beta} \right) \, || \, q \left( \bm{\theta} \right) \right)$. Our physics-informed constitutive inference framework enables efficient and scalable inference from complex high dimensional full-field deformation datasets, accommodating nonlinear material behaviors and cross-correlated parameter effects.
• Figure 1: Synthetic data generation: training sample dimensions and loading conditions Tissue slab geometry and applied boundary conditions for the training samples. We subject all training samples undergo to a biaxial loading protocol $\lambda_1 : \lambda_ 2 - (1:2), (1:1), (2:1)$ up to $15\%$ of the initial length of the sample. All samples are of size 10 $\times$ 10 $\times$ 1 mm$^3$. The geometrically heterogeneous tissue samples have a circular occlusion in the center with a radius of $R=1$ mm.
• Figure 2: Schematic of the hierarchical inference model. This hierarchical structure of the Bayesian inference model allows for simulteneous placement of prior distributions on the noise in the likelihood and the constitutive model parameters. The red and circular distributions represent hyperpriors for $V_s$ and $\sigma^2$, the lightblue circular distribution represent the prior on the material model parameters and the black and rectangular distribution represents the likelihood of the linear momentum balance. The black arrows denote the hierarchical structure dependencies. The hyperpriors are distributed according to inverse-gamma distributions. The model parameters are distributed according to a truncated multivariate normal distribution. Truncation occurs at numerical zero to enforce polyconvexity in the material model. The likelihood is assumed to be normally distributed.
• Figure 3: Specimen slicing and microstructural organization. Top: Schematic of the left ventricle and three considered slicing orientations to extract myocardial tissue slabs: circumferential–longitudinal (circ-long), circumferential–radial (circ-rad), and a rotated variant of the latter (rad-heli). Bottom: Each slicing orientation leads to a distinct microstructural configuration within the specimen. In circ-long slabs, the fiber direction lies in-plane and the sheet direction spans the through-thickness. Circ-rad slabs display a transmural fiber rotation from epicardium to endocardium, with the sheet direction aligned in the radial plane. The rad-heli orientation rotates the slab around the radial axis, aligning all three microstructural directions approximately within the plane of the specimen. Fiber directions ($\bm{f}$) are shown in red, sheet directions ($\bm{s}$) in pink, and normal directions ($\bm{n}$) in brown.
• Figure 4: Deformation profiles of geometrically homogeneous tissue slabs with varying microstructural heterogeneity. Maximum principal stretches are shown for circ-long-hom (top row), circ-rad-hom (middle row), and rad-heli-hom (bottom row) specimens subjected to biaxial loading protocols ($\lambda_1 : \lambda_2 = (1:2), (1:1), (2:1)$) up to $\lambda^{\text{max}} = 1.15$. Rightmost columns display distributions of invariants $I_1-3$, $I_{4ff}-1$, $I_{4nn}-1$, and $I_{4fs}$ for each specimen type respectively using approximated kernel density estimations denoted as $\hat{\pi}(I_*)$. The kernel density estimation is truncated at an arbitrary level of 20 to show the spread of the invariants more clearly.