diffusion equation (de)

Functional near-infrared spectroscopy (fNIRS) is an imaging method in which a light source and detector are installed on the head; consequently, the re-emission of light from human skin contains information about cerebral hemodynamic alteration. The spatial probability distribution profile of photons penetrating tissue at a source spot, scattering into the tissue, and being released at an appropriate detector position, represents the spatial sensitivity. 

Modeling light propagation in a human head is essential for quantitative near-infrared spectroscopy and optical imaging. The specific form of the distribution of light is obtained using the theory of perturbation. An analytical solution of the perturbative diffusion equation (DE) and finite element method (FEM) in a Slab media (similar to the human head) makes it possible to study light propagation due to absorption and scattering of brain tissue. 

The simulation result indicates that sensitivity is slowly decreasing in the deep area, and the sensitivity below the source and detector is the highest. The depth sensitivity and computation time of both analytical and FEM methods are compared. The simulation time of the analytical approach is four times larger than the FEM. 

In this paper, an analytical solution and the performance of FEM methods when applied to the diffusion equation for heterogeneous media with a single spherical defect are compared. The depth sensitivity along with the computation time of simulation has been investigated for both methods. For simple and Slab modes of the human brain, the analytical solution is the right candidate. Whenever the brain model is sophisticated, it is possible to use FEM methods, but it costs a higher computation time.