We deal with light diffusion in mismatched N-layered turbid media having a finite or an infinitely thick N'th layer. We focus on time-resolved light propagation in both the frequency and time domains. Based on our results for the steady-state domain, solutions of the N-layered diffusion equations in the frequency and time domains are obtained by applying the Fourier transform technique. Different methods for calculation of the inverse Fourier transform are studied to validate the solutions, showing relative differences typically smaller than 10(-6). The solutions are compared to Monte Carlo simulations, revealing good agreement. Finally, by applying the Laplace and Fourier transforms we derive a fast ( approximately 1 ms) and accurate analytical solution for the time domain reflectance from a two-layered turbid medium having equal reduced scattering coefficients and refractive indices in both layers.
"In order to solve the forward model, the time-resolved reflectance R(t) at 30.5 mm from the incident point was calculated by the analytical solution of the diffusion equation (Liemert and Kienle 2010). This distance is suitable since modifications of the optical properties of the brain can be seen at larger distances, whereas modifications of the optical properties of the upper layers can be seen at smaller distances. "
[Show abstract][Hide abstract] ABSTRACT: We investigated the performance of a neural network for derivation of the absorption coefficient of the brain from simulated non-invasive time-resolved reflectance measurements on the head. A five-layered geometry was considered assuming that the optical properties (except the absorption coefficient of the brain) and the thickness of all layers were known with an uncertainty. A solution of the layered diffusion equation was used to train the neural network. We determined the absorption coefficient of the brain with an RMS error of <6% from reflectance data at a single distance calculated by diffusion theory. By applying the neural network to reflectance curves obtained from Monte Carlo simulations, similar errors were found.
Physics in Medicine and Biology 06/2011; 56(11):N139-44. DOI:10.1088/0031-9155/56/11/N02 · 2.76 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We deal with light diffusion in N-layered turbid media. The steady-state diffusion equation is solved for N-layered turbid media having a finite or an infinitely thick N'th layer. Different refractive indices are considered in the layers. The Fourier transform formalism is applied to derive analytical solutions of the fluence rate in Fourier space. The inverse Fourier transform is calculated using four different methods to test their performance and accuracy. Further, to avoid numerical errors, approximate formulas in Fourier space are derived. Fast solutions for calculation of the spatially resolved reflectance and transmittance from the N-layered turbid media ( approximately 10 ms) with small relative differences (<10(-7)) are found. Additionally, the solutions of the diffusion equation are compared to Monte Carlo simulations for turbid media having up to 20 layers.
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