The 3D inverse optoacoustic source problem on the beam axis

Today, optoacoustics is widely used in the life sciences, e.g. for imaging of biological tissue. While the direct problem of absorption of light in biological media consists of solving the optoacoustic wave equation for an initial pressure distribution p0(r), the inverse problem, i.e. the reconstruction of optical properties from observed signals is not sufficiently understood, yet. For the particular case of a Gaussian transverse beam profile, the signal p(z,t) at a point z along the beam axis, at the retarded time t, is given by an integral equation, which is linear in the initial pressure profile p0(t) on the boundary of the absorbing medium. This integral equation resembles a Volterra equation of the second kind with known kernel, where p(z,t) is given and p0(t) is an unknown function to be solved for. For this, existing inversion schemes need to be adapted. Here, we study the inversion of synthetic signals that correspond to different initial pressure distributions, compare the inversion in the far-field to an approximate method based on the solution of a simple differential equation and consider the effect of noise on the quality of the reconstructed profile.