In this talk, we introduce an elastic imaging method where elastic properties (i.e. mass density and Lamé parameters) of a weakly scatterer are reconstructed from full-field data of scattered waves. We linearise the inverse scattering problem under consideration using Born's, Rytov's or Kirchhoff's approximation. Primarily, one appeal to the Fourier diffraction theorem developed in our previous work [1] for the pressure-pressure mode (i.e. generating Pressure incident plane waves and measuring the Pressure part of the scattered data). Then, we reconstruct the inverse Fourier transform of the pressure-pressure scattering potential using the inverse $\textit{nonequispaced discrete Fourier transform}$ for 2D transmission acquisition experiments. Finally, we quantify the elastic parameter distributions with different plane wave excitations.

[1] B. Mejri, O. Scherzer. A new inversion scheme for elastic diffraction tomography. arXiv:2212.02798, 2022.

New high-resolution, three-dimensional imaging techniques are being developed that probe the breast without delivering harmful radiation. In particular, photoacoustic tomography (PAT) and ultrasound tomography (UST) promise to give access to high-quality images of tissue parameters with important diagnostic value. However, the involved inverse problems are very challenging from an experimental, mathematical and computational perspective. In this talk, we want to give an overview of these challenges and illustrate them with data from an ongoing clinical feasibility study that uses a prototype scanner for combined PAT and UST.

In quantitative photoacoustic tomography, information about a target tissue is obtained by estimating its optical parameters. In this work, we propose a one-step methodology for estimating spectral optical parameters directly from photoacoustic time-series data. This is carried out by representing the optical parameters with their spectral models and by combining the models of light and ultrasound propagation. The inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores, two scattering related parameters, and the Grüneisen parameter are estimated simultaneously. The methodology is evaluated using numerical simulations.

We study the issues of stability and reconstruction of the anisotropic conductivity $\sigma$ of a biological medium $\Omega\subset\mathbb{R}^3$ by the hybrid inverse problem of Magneto-Acoustic Tomography with Magnetic Induction (MAT-MI). More specifically, we consider a class of anisotropic conductivities given by the ​symmetric and uniformly positive definite matrix-valued functions $A(x,\gamma(x))$, $x\in\Omega$,​ where the one-parameter family $t\mapsto A(x, t)$, $t\in[\lambda^{-1}, \lambda]$, ​is assumed to be $\textit{a-priori}$ known. Under suitable conditions that include $A(\cdot, \gamma(\cdot))\in C^{1,\beta}(\Omega)$, with $0<\beta\leq 1$, we obtain a Lipschitz type stability estimate of the scalar function $\gamma$ in the $L^2(\Omega)$ norm in terms of an internal functional that can be physically measured in the MAT-MI experiment. We demonstrate the effectiveness of our theoretical framework in several numerical experiments, where $\gamma$ is reconstructed in terms of the internal functional. Our result extends previous results in MAT-MI where the conductivity $\sigma$ was either isotropic or of the simpler anisotropic form $\gamma D$, with $D$ an $\textit{a priori}$ known matrix-valued function in $\Omega$. In particular, the more general type of anisotropic conductivity considered here allows for the anisotropic structure to depend non-linearly on the unknown scalar parameter $\gamma$ to be reconstructed. This is joint work with Romina Gaburro, Shari Moskow and Isaac Woods