New paper on "The Illusion of Fit: Spatially Resolved Assessment of Constitutive Model Validity in Elastography and Physics-Based Inverse Problems"

Abstract:
Inferring the mechanical properties of soft tissues from measured deformations is a fundamental challenge in elastography and, more broadly, in physics-model-based inverse problems. A critical but rarely examined assumption underlying all existing approaches, i.e. direct, indirect, and learning-based alike, is that the assumed constitutive law correctly describes the material being imaged. When this assumption fails, inversion still produces parameter estimates that appear plausible, creating an illusion of fit: practitioners obtain seemingly reasonable material property distributions with no indication that the constitutive model may be invalid in parts of the domain, a situation that can actively mislead clinical interpretation.

We propose a probabilistic framework that transforms constitutive model validity from an implicit assumption into an explicit, spatially resolved inference target. The key architectural departure from standard formulations is to treat the stress field as an independent latent variable rather than deriving it from displacements and material properties through the constitutive law. This enables a pointwise comparison between the stress required by mechanical equilibrium and the stress predicted by the assumed constitutive model. Both sets of governing equations are incorporated into the probabilistic learning objective as virtual observables with separate precision hyperparameters: the conservation law precision is set a priori to a small value reflecting its undisputed validity, while the constitutive precision is inferred from the data under a sparsity-promoting prior. The resulting spatial distribution of constitutive precisions provides an uncertainty-aware map of where the assumed model is supported by the data and where it is not. Inference is carried out via stochastic variational inference, making the approach forward-model-free.

We validate the framework on synthetic harmonic elastography experiments using a brain slice geometry with a localized anisotropic inclusion embedded in an otherwise linear elastic domain. The inferred precision field correctly identifies the inclusion with a contrast of five orders of magnitude relative to the valid surrounding domain, robust across noise levels from 35 to 25 dB and under a four-fold reduction in observation density. A phantom experiment using real ultrasound-based displacement measurements from a material known to follow linear elastic behavior confirms that the method produces no false positive constitutive violations, while recovering the true stiffness contrast within the 99% credibility interval. The framework is appllicable to any PDE-constrained inverse problem in which conservation laws can be trusted but closure relations cannot.

Full paper: www.sciencedirect.com/science/article/pii/S0045782526004810 

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