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

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).