Direct and inverse solver for the 3D optoacoustic Volterra equation

Abstract

The direct problem of optoacoustic signal generation in biological media consists of solving the inhomogeneous optoacoustic wave equation for an initial acoustic stress profile. In contrast, the mathematically challenging inverse problem requires the reconstruction of the initial stress profile from a proper set of observed signals. In this article, we consider the particular case of a Gaussian transverse irradiation source profile in the paraxial approximation of the wave equation, for which the direct problem along the beam axis can be cast into a linear Volterra integral equation of the second kind. This integral equation can be used in two ways: as a forward solver to predict optoacoustic signals in terms of the direct problem, and as an inverse solver for which we here devise highly efficient numerical schemes used for the reconstruction of initial pressure profiles from observed signals, constituting a methodical progress of computational aspects of optoacoustics. In this regard, we explore the validity as well as the limits of the inversion scheme via numerical experiments, with parameters geared towards actual optoacoustic problem instances. The considered inversion input consists of synthetic data, obtained by means of forward solvers based on the Volterra integral, and, more generally, the optoacoustic Poisson integral. Regarding the latter, we numerically invert signals that correspond to different detector-to-sample distances and assess the convergence to the true initial stress profiles upon approaching the far-field. Finally, we also address the effect of noise on the quality of the reconstructed pressure profiles.