Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging

TL;DR

The paper develops a rigorous mathematical framework for estimating the effective speed of sound in tissue using speckle-based analysis within a reflection-matrix ultrasound setting. By combining stochastic homogenization of the Helmholtz equation with paraxial asymptotics, it derives a representation for the scattered field in a random multi-scale medium and analyzes how mis-matching the backpropagation speed $c$ shifts and attenuates the focal spot. The key contributions include a confocal-imaging functional for unknown $c$, a PSF-based focusing criterion, and an ensemble-averaging (then spatial-averaging) approach that yields a practical estimator for $c^\star$ from a single realization of the medium, complemented by numerical illustrations. The work advances quantitative ultrasound by enabling in situ, model-based speed-of-sound estimation from speckle data, with potential to improve image fidelity and provide diagnostic markers of tissue state.

Abstract

In this paper, we present a mathematical model and analysis for a new experimental method [Bureau and al., arXiv:2409.13901, 2024] for effective sound velocity estimation in medical ultrasound imaging. We perform a detailed analysis of the point spread function of a medical ultrasound imaging system when there is a mismatch between the effective sound speed in the medium and the one used in the backpropagation imaging functional. Based on this analysis, an estimator for the speed of sound error is introduced. Using recent results on stochastic homogenization of the Helmholtz equation, we provide a representation formula for the field scattered by a random multi-scale medium (whose acoustic behavior is similar to a biological tissue) in the time-harmonic regime. We then prove that statistical moments of the imaging function can be accessed from data collected with only one realization of the medium. We show that it is possible to locally extract the point spread function from an image constituted only of speckle and build an estimator for the effective sound velocity in the micro-structured medium. Some numerical illustrations are presented at the end of the paper.

Figures (8)

• Figure 1: Illustration of the geometry for an ultrasound imaging experiment.
• Figure 2: Plot of $(\xi_1,\xi_2) \mapsto \vert \mathcal{G}(\xi_1,\xi_2)\vert$.
• Figure 3: Normalized plot of $c \mapsto \vert\int_\mathcal{B} \omega^2 \mathcal{G}^2\left(0,\frac{\omega \ell^2}{c^\star \vert y_0\vert} \left( \frac{c^\star}{c}\right)^2-1\right) \mathrm{d} \omega\vert$ for realistic values of the parameters $\ell, \vert y_0\vert, \mathcal{B},\ldots$ (see section \ref{['sec:numerical']}).
• Figure 4: Shape of the point spread function $z\mapsto \left\vert F^c(z,y_0)\right\vert$ with three different backpropagation speed. Here $y_0=(0.01,0.06)$ for all three graphs. The quantity $\frac{y_0^\shortparallel}{\ell}$ is the numerical aperture at $y_0$ where $\ell$ is the length of the linear probe $\mathcal{P}$. The focal spot at threshold $0.1$ is the area in yellow/light blue.
• Figure 5: Source term used in the simulation.

Theorems & Definitions (49)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Remark 3.1
• Definition 3.1: Virtual Domain
• Definition 3.2