Time reversal for photoacoustic tomography based on the wave equation of Nachman, Smith and Waag

Abstract

The goal of \emph{photoacoustic tomography} (PAT) is to estimate an \emph{initial pressure function} \(\varphi\) from pressure data measured at a boundary surrounding the object of interest. This paper is concerned with a time reversal method for PAT that is based on the dissipative wave equation of Nachman, Smith and Waag\cite{NaSmWa90}. This equation has the advantage that it is more accurate than the \emph{thermo-viscous} wave equation. For simplicity, we focus on the case of one \emph{relaxation process}. We derive an exact formula for the \emph{time reversal image} \(\I\), which depends on the \emph{relaxation time} \(\tau_1\) and the \emph{compressibility} \(\kappa_1\) of the dissipative medium, and show \(\I(\tau_1,\kappa_1)\to\varphi\) for \(\kappa_1\to 0\). This implies that \(\I=\varphi\) holds in the dissipation-free case and that \(\I\) is similar to \(\varphi\) for sufficiently small compressibility \(\kappa_1\). Moreover, we show for tissue similar to water that the \emph{small wave number approximation} \(\I_0\) of the time reversal image satisfies \(\I_0 = \eta_0 *_\x \varphi\) with \(\hat \eta_0(|\k|)\approx const.\) for \(|\k|<< \frac{1}{c_0\,\tau_1}\). For such tissue, our theoretical analysis and numerical simulations show that the time reversal image \(\I\) is very similar to the initial pressure function \(\varphi\) and that a resolution of \(\sigma\approx 0.036\cdot mm\) is feasible (in the noise-free case).