Estimation of Dielectric Parameters from Ultrasound Waves in Quantitative Thermoacoustic Tomography

TL;DR

This work tackles the QTAT inverse problem of estimating dielectric parameters from thermoacoustic ultrasound by a two-stage Bayesian framework: first reconstruct the initial pressure from boundary measurements using a Gaussian acoustic posterior, then recover conductivity and relative permittivity via a Gaussian Ornstein–Uhlenbeck prior and MAP estimation using Gauss-Newton. The forward model couples Maxwell's equations for electromagnetic propagation with the acoustic wave equation, enabling p0 to be linked to the dielectric parameters through the absorbed energy density. Numerical 2D simulations show that dielectric parameters can be recovered with reasonable accuracy, but reconstruction quality strongly depends on the number of EM pulses and the sensor geometry, with limited-view configurations introducing artifacts. The study highlights practical considerations for QTAT, such as extending to 3D, incorporating finite-size bandwidth sensors, and accounting for frequency-dependent dielectric properties to bring the approach closer to realistic clinical settings.

Abstract

Thermoacoustic tomography (TAT) is an imaging modality based on the thermoacoustic effect. In TAT, a short microwave or radio wave pulse is directed to the imaged target. The energy of the electromagnetic pulse is absorbed depending on the target's dielectric parameters, resulting in a spatially varying pressure distribution via the thermoacoustic effect. This pressure, known as the initial pressure distribution, relaxes as broadband ultrasound waves that are measured on the boundary of the target. In the inverse problem of TAT, the initial pressure is estimated from the measured ultrasound waves. TAT can further be extended to quantitative TAT (QTAT), where the aim is to estimate the dielectric parameters of the imaged target by utilizing a model for electromagnetic wave propagation. In this work, we consider the QTAT problem and propose an approach for simultaneous estimation of electrical conductivity and permittivity from the ultrasound waves. The problem is approached in the framework of Bayesian inverse problems. The forward model describing electromagnetic and acoustic wave propagation is based on Maxwell's equations and the acoustic wave equation, respectively. The approach is evaluated with numerical simulations. The results show that the dielectric parameters can be accurately estimated using the proposed approach. However, the ultrasound sensor geometry and the number of electromagnetic pulses have a significant effect on the accuracy of the estimated parameters.

Figures (7)

• Figure 1: An illustration of the simulation setup. The simulated phantom (circle), and electromagnetic and acoustic domains $\Omega_\mathrm{ac}$ and $\Omega_\mathrm{el}$. The incident electric fields $E_{\mathrm{inc},m}$ are illustrated using black rectangles and propagation directions $d_m$, where $m = 1, \dots, 4$. The ultrasound sensors geometries are shown using red dots and their angular coverage is indicated using the angle $\alpha$.
• Figure 2: Simulated conductivity $\sigma_{\mathrm{true}}$ (Sm$^{-1}$, left) and relative permittivity $\epsilon_{r,\mathrm{true}}$ (right). Location of the cross section used for visualising the results is illustrated with a white dotted line.
• Figure 3: Expected values $\eta_{p_0}$ (first row), standard deviations $\tilde{\sigma}_{p_0}$ (second row), and blocks of the covariance matrices $\Gamma_{p_0}$ (third row) of the prior distribution of the initial pressure in the acoustic inverse problem using the incident electric fields $E_\mathrm{inc,1}$ to $E_\mathrm{inc,4}$ (columns 1-4). For the covariance matrices, a subsection of the covariance matrix corresponding to the pixels illustrated in Fig. \ref{['fig:phantoms']} is shown.
• Figure 4: Simulated (true) initial pressure $p_0$, (first row) and estimated initial pressures $\eta_{p_0 \vert p_t}$ (rows 2-4) for all incident electric fields $E_\mathrm{inc}$ (columns 1-4) and sensor geometries $\alpha = 90^\circ, 180^\circ$, and $360^\circ$. The sensor geometries are illustrated using red arcs.
• Figure 5: Standard deviations $\sigma_{\tilde{e}}$ and blocks of the covariance matrices $\Gamma_{\tilde{e}}$ of the noise for the electrical inverse problem for incident electric fields $E_\mathrm{inc,1}$ to $E_\mathrm{inc,4}$ (columns 1-4), and sensor geometries $\alpha = 90^\circ, 180^\circ$, and $360^\circ$ (rows 1-6). For the covariance matrices, a subsection of the covariance matrix corresponding to the pixels indicated in Fig. \ref{['fig:phantoms']} is shown. The sensor geometries are illustrated using red arcs.