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

One goal of photoacoustic tomography (PAT) is to estimate an initial pressure function φ 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 [J. Acoust. Soc. Am. 88, 1584 (1990)]. This equation is a correction of the thermoviscous wave equation such that its solution has a finite wave front speed and, in contrast, it can model several relaxation processes. In this sense, it is more accurate than the thermoviscous wave equation. For simplicity, we focus on the case of one relaxation process. We derive an exact formula for the time reversal image I, which depends on the relaxation time τ(1) and the compressibility κ(1) of the dissipative medium, and show I(τ(1),κ(1)) → φ for κ(1) → 0. This implies that I = φ holds in the dissipation-free case and that I is similar to φ for sufficiently small compressibility κ(1). Moreover, we show for tissue similar to water that the small wave number approximation I(0) of the time reversal image satisfies I(0) = η(0)*(x)φ with η[over ̂](0)(|k|) ≈ const. for |k| ≪ 1/c(0)τ(1), where φ denotes the initial pressure function. For such tissue, our theoretical analysis and numerical simulations show that the time reversal image I is very similar to the initial pressure function φ and that a resolution of σ≈0.036mm is feasible (for exact measurement data).