Photoacoustic tomography with time-dependent damping: Theoretical and a convolutional neural network-guided numerical inversion procedure

TL;DR

This paper develops a theoretical framework and a neural network-guided numerical method for photoacoustic tomography in media with time-dependent damping, improving image reconstruction accuracy in attenuating tissues.

Contribution

It introduces a novel mathematical model for PAT with time-dependent damping and proposes a gradient-free numerical inversion method guided by neural networks.

Findings

Unique determination of initial pressure from boundary data.

Explicit series reconstruction formula for constant damping.

Robust, computationally efficient image reconstruction method.

Abstract

In photoacoustic tomography (PAT), a hybrid imaging modality that is based on the acoustic detection of optical absorption from biological tissue exposed to a pulsed laser, a short pulse laser generates an initial pressure proportional to the absorbed optical energy, which then propagates acoustically and is measured on the boundary. To account for the significant signal distortion caused by acoustic attenuation in biological tissue, we model PAT in heterogeneous media using a damped wave equation featuring spatially varying sound speed and a time-dependent damping term. Under natural assumptions, we show that the initial pressure is uniquely determined by the boundary measurements using a harmonic extension of the boundary data with energy decay. For constant damping, an expansion in Dirichlet eigenfunctions of $-c^2(\xx)\Delta$ leads to an explicit series reconstruction formula for the initial pressure. Finally, we develop a gradient free numerical method based on the Pontryagin's maximum principle to provide a robust and computationally viable approach to image reconstruction in attenuating PAT.