Quasi-static in vivo elastography from internal displacement information only

TL;DR

This work addresses the frequency-dependent limitations of dynamic elastography by introducing a noise-robust quasi-static elastography framework that reconstructs relative stiffness from in vivo displacement fields without boundary data or displacement-derivative amplification. Building on the Virtual Fields Method, it reformulates the forward problem into a robust weak form and applies Galerkin discretization, enabling region-wise stiffness estimation under a linear isotropic model with known ν. The approach is validated in silico with phantom-like geometries and in vivo on the thigh using cuff-induced deformations, demonstrating repeatability across seven sessions and physiologically consistent stiffness changes during isometric knee activation. The method has potential for broader in vivo biomechanics applications and offers a path toward boundary-free, high-spidelity stiffness mapping in soft tissues, with extensions to 3D, anisotropy, and absolute scaling discussed for future work.

Abstract

As disease often alters the structural properties of soft tissue, noninvasive elastography techniques have emerged to quantitatively assess in vivo mechanical properties. Magnetic Resonance Elastography (MRE) based on dynamic deformations is the standard technique for imaging mechanical properties, but the viscoelastic nature of soft tissue makes the results dependent on the actuation frequency, which can be limiting. In this proof-of-principle study we propose a noise robust framework for reconstructing relative stiffness properties from quasi-static in vivo displacement fields captured on a physiological time scale. The acquisition is performed using a pneumatic pressure cuff to induce tissue deformation in a controlled manner. The reconstruction does not require boundary information which is generally hard to access in vivo nor spatial derivatives of displacement fields that are known to amplify noise. The validity of our framework is corroborated with in silico experiments on a numerical phantom. In vivo experiments on the thigh of a volunteer demonstrate the repeatability of the method. As an application, the quantitative change in muscle stiffness during isometric knee flexion is investigated which yielded physiologically meaningful results.

Figures (8)

• Figure 1: (Left) Schematic depiction of the coaxial domain mimicking muscle surrounding a bone discretized using quadrilaterals. The mesh is chosen extremely coarse for illustrative purposes. (Middle) The bilinear shape function $N_{\tilde{\imath}}(\vec{x})$ in the neighborhood of $\vec{x}_{\tilde{\imath}}$. The family of bilinear shape functions $N(\vec{x})$ around all nodes form a basis for the solution $\boldsymbol{C}(\vec{x})$. (Right) The special shape function $\mathcal{N}_{\tilde{\imath}}(\vec{x})$ in the neighborhood of $\vec{x}_{\tilde{\imath}}$. The family of special shape functions $\mathcal{N}(\vec{x})$ around all internal nodes form a basis for the test functions $\vec{\eta}(\vec{x})$.
• Figure 2: (Top) Schematic representation of the set-up with the pneumatic pressure cuff and coil placement. (Bottom) Volunteer in supine position on the scanner table (anterior RF coil not shown).
• Figure 3: Displacement fields (green arrows) at different time points during inflation and deflation of the pressure cuff. The full video with a spatial resolution of $3.5$mm isotropic and a temporal resolution of $345.6$ms can be found in the Supplementary materials.
• Figure 4: In silico Experiment 1.1. On the domain, two materials types are present ($n_{seg}=2$) with different Young's moduli (inner ring: $30$kPa, outer ring: $15$kPa). The relative Young's moduli are reconstructed for each spatial location such that the sum of all nodes equals $\alpha=n_{node}$.
• Figure 5: In silico Experiment 1.2. The dashed lines in the box plot are the ground truth relative Young's moduli. The green arrows in the domain represent one displacement field (out of $8$ used for reconstruction) and the black circle in the middle the bone structure. On the domain, two material types are present ($n_{seg}=2$) with different Young's moduli (material A: $15$kPa, material B: $30$kPa). The relative Young's moduli are reconstructed for the two segments such that their sum equals $\alpha=1$.