Hybrid Solutions of the Radiative Transfer Equation
DOI: 10.1117/3.824746.ch5
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Excerpt

In order to solve the RTE without simplifying assumptions, we have to resort to numerical methods. These methods convert the integral form of the transport equation into a system of algebraic equations that can be treated numerically by using a computer. The finite element method, the discrete ordinates method, or the spherical harmonics method are examples of numerical procedures for reconstructing solutions of the RTE. Among the numerical procedures, there are also statistical methods like the MC method, which offers a way of simulating the physical propagation of photons in scattering media. With the MC method, the physical quantities that describe photon migration, such as the flux and the fluence rate, are calculated after running a large number of photon trajectories. The MC can be considered a numerical method to solve the RTE even though the equation itself is not implemented in the code as it is usually done in other numerical methods. Besides the approximate solutions obtained making use of the DE, very few analytical solutions are available for studying light propagation through highly scattering media.

Approximate analytical solutions subjected to the same limitations of those of the DE are also obtained with a random walk method.

Solutions based on the DE are affected by the intrinsic approximations of this theory that show different consequences for steady-state and time-dependent sources. For steady-state sources, solutions based on the DE cannot describe photons detected at short distances. For time-dependent sources, DE solutions provide a poor description of early received photons since the DE is well established only for photons undergoing many scattering events.

The limitations of the DE can present a serious problem in modeling photon migration, especially when small volumes of diffusive media are considered. To overcome these limitations, several improved solutions of the DE have been proposed. For this reason, the availability of analytical solutions of the RTE not subjected to the intrinsic limitations of the DE is desirable.

© 2010 Society of Photo-Optical Instrumentation Engineers

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