Photoacoustic imaging taking into account thermodynamic attenuation

In this paper we consider a mathematical model for photoacoustic imaging which takes into account attenuation due to thermodynamic dissipation. The propagation of acoustic (compressional) waves is governed by a scalar wave equation coupled to the heat equation for the excess temperature. We seek to recover the initial acoustic profile from knowledge of acoustic measurements at the boundary. We recognize that this inverse problem is a special case of boundary observability for a thermoelastic system. This leads to the use of control/observability tools to prove the unique and stable recovery of the initial acoustic profile in the weak thermoelastic coupling regime. This approach is constructive, yielding a solvable equation for the unknown acoustic profile. Moreover, the solution to this reconstruction equation can be approximated numerically using the conjugate gradient method. If certain geometrical conditions for the wave speed are satisfied, this approach is well--suited for variable media and for measurements on a subset of the boundary. We also present a numerical implementation of the proposed reconstruction algorithm.