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-SP_N) that approximates the frequency-domain equation of radiative transfer (FD-ERT). We then present the FD-SP_N based reconstruction algorithm that recovers absorption and scattering coefficients in biological tissue. The FD-SP_N 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 high accuracy; the average errors 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.