The Radiative Transfer Equation in Photoacoustic Imaging

Abstract

Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, photoacoustic tomography is divided into two steps. In the first step, one solves an inverse problem for the wave equation to determine how tissue absorbs light as a result of a boundary illumination. The second step is generally modeled by either diffusion or transport equations, and involves recovering the optical properties of the region being imaged. In this paper we, we provide an overview of mathematical progress in photoacoustic and thermoacoustic tomography, as well as inverse problems for the radiative transfer equation. We then address quantitative photoacoustic tomography modeled by the radiative transfer equation, and in particular, we show that the absorption coefficient in the stationary transport equation can be recovered given certain internal information about the solution. Our new result will consider the variable index of refraction case, which will correspond to an inverse transport problem on a Riemannian manifold with internal data and a known metric. We will prove a stability estimate for a functional of the absorption coefficient of the medium by finding a singular decomposition for the distribution kernel of the measurement operator. Finally, we will use this estimate to recover the desired absorption properties.