Frequency-domain optical tomographic image reconstruction algorithm with the simplified spherical harmonics (SP3) light propagation model.

We introduce here the finite volume formulation of the frequency-domain simplified spherical harmonics model with n-th order absorption coefficients (FD-SPN) that approximates the frequency-domain equation of radiative transfer (FD-ERT). We then present the FD-SPN based reconstruction algorithm that recovers absorption and scattering coefficients in biological tissue. The FD-SPN model with 3rd order absorption coefficient (i.e., FD-SP3) is used as a forward model to solve the inverse problem. The FD-SP3 is discretized with a node-centered finite volume scheme and solved with a restarted generalized minimum residual (GMRES) algorithm. The absorption and scattering coefficients are retrieved using a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. Finally, the forward and inverse algorithms are evaluated using numerical phantoms with optical properties and size that mimic small-volume tissue such as finger joints and small animals. The forward results show that the FD-SP3 model approximates the FD-ERT (S12) solution within relatively high accuracy; the average error in the phase (<3.7%) and the amplitude (<7.1%) of the partial current at the boundary are reported. From the inverse results we find that the absorption and scattering coefficient maps are more accurately reconstructed with the SP3 model than those with the SP1 model. Therefore, this work shows that the FD-SP3 is an efficient model for optical tomographic imaging of small-volume media with non-diffuse properties both in terms of computational time and accuracy as it requires significantly lower CPU time than the FD-ERT (S12) and also it is more accurate than the FD-SP1.