Stability in the linearized problem of quantitative elastography

TL;DR

This paper analyzes the stability of linearized inverse problems in quantitative elastography, providing conditions for stable parameter recovery and highlighting the importance of avoiding singular strain fields.

Contribution

It introduces a stability analysis framework based on PDE theory for linearized elastography problems, characterizing kernels and injectivity of forward operators.

Findings

Stability established for shear modulus, pressure, and density.

Singular strain fields can compromise stability.

Additional measurements improve stability.

Abstract

The goal of quantitative elastography is to identify biomechanical parameters from interior displacement data, which are provided by other modalities, such as ultrasound or magnetic resonance imaging. In this paper, we analyze the stability of several linearized problems in quantitative elastography. Our method is based on the theory of redundant systems of linear partial differential equations. We analyze the ellipticity properties of the corresponding PDE systems augmented with the interior displacement data; we explicitly characterize the kernel of the forward operators and show injectivity for particular linearizations. Stability criteria can then be deduced. Our results show stability of shear modulus, pressure and density; they indicate that singular strain fields should be avoided, and show how additional measurements can help in ensuring stability.