On the Intensity-based Inversion Method for Quantitative Quasi-Static Elastography

TL;DR

The paper addresses quantitative parameter recovery in quasi-static elastography from two internal images before and after deformation. It introduces the intensity-based inversion method (IIM), a one-step regularized framework that directly minimizes $\mathcal{T}_\alpha(\mathbf{a}) = \|\mathcal{F}(\mathbf{a}) - \mathcal{I}_1\|^2_{L_2(\Omega)} + \alpha \|\mathbf{a}-\mathbf{a}_0\|^2_X$ with $\mathcal{F}(\mathbf{a}) = \mathcal{I}_2\circ G(\mathbf{a})$ and $\mathcal{R}(\mathbf{a}) = \|\mathbf{a}-\mathbf{a}_0\|^2_X$. The authors develop a convergence theory for noisy data, covering restricted and full noise scenarios, establishing stability and order-optimal rates under standard source-type conditions; they show the theory applies to linear elasticity with explicit differentiability and Lipschitz properties. Numerical experiments, including simulated optical coherence elastography data, demonstrate the IIM’s robustness to noise, segmentation imperfections, and complex inclusion geometry, and show it can outperform a representative two-step approach in recovering Lamé parameters. The work also discusses practical extensions to general material models, multiple measurements, and varied minimization strategies, underscoring the method’s potential for reliable, physics-informed elastography in clinical and research settings.

Abstract

In this paper, we consider the intensity-based inversion method (IIM) for quantitative material parameter estimation in quasi-static elastography. In particular, we consider the problem of estimating the material parameters of a given sample from two internal measurements, one obtained before and one after applying some form of deformation. These internal measurements can be obtained via any imaging modality of choice, for example ultrasound, optical coherence or photo-acoustic tomography. Compared to two-step approaches to elastography, which first estimate internal displacement fields or strains and then reconstruct the material parameters from them, the IIM is a one-step approach which computes the material parameters directly from the internal measurements. To do so, the IIM combines image registration together with a model-based, regularized parameter reconstruction approach. This combination has the advantage of avoiding some approximations and derivative computations typically found in two-step approaches, and results in the IIM being generally more stable to measurement noise. In the paper, we provide a full convergence analysis of the IIM within the framework of inverse problems, and detail its application to linear elastography. Furthermore, we discuss the numerical implementation of the IIM and provide numerical examples simulating an optical coherence elastography (OCE) experiment.

Figures (11)

• Figure 1.1: Schematic depiction of the workflow in quasi-static quantitative elastography. Imaging of the sample before/after deformation and inverse problems IP1-IP3 for obtaining material parameters.
• Figure 2.1: Illustration of our mathematical setting for quasi-static elastography: initial, non-deformed object geometry $\Omega_1$ (left) and deformed object geometry $\Omega_2$ (right); here $\mathbf{g}$, is a fixed applied downward displacement. The shaded domain in both images corresponds to one possible choice of $\Omega$ in \ref{['domain_condition']}.
• Figure 6.1: Schematic drawing of the object domain $\Omega_1$ with displacement-traction boundaries $\Gamma_D$ and $\Gamma_T$. Here, $\mathbf{g}_D$, is a fixed applied downward displacement, which does not have to be constant.
• Figure 7.1: Schematic depiction of simulated material sample structure (left) and synthetic OCT images before the application of artificial speckle (right). Inclusion numbering corresponding to Table \ref{['tab_ground_truth']}.
• Figure 7.2: Synthetic OCT images (including speckle) before (left) and after compression (right), corresponding to $\mathcal{I}_1$ and $\mathcal{I}_2$.

Theorems & Definitions (38)

• Example 2.1
• Lemma 3.1
• proof
• Remark 4.1
• Theorem 4.1
• proof
• Proposition 4.2
• proof
• Theorem 4.3
• proof