Boundary conditions for the diffusion equation in radiative transfer

Abstract

Using the method of images, we examine the three boundary conditions commonly applied to the surface of a semi-infinite turbid medium. We find that the image-charge configurations of the partial-current and extrapolated-boundary conditions have the same dipole and quadrupole moments and that the two corresponding solutions to the diffusion equation are approximately equal. In the application of diffusion theory to frequency-domain photon-migration (FDPM) data, these two approaches yield values for the scattering and absorption coefficients that are equal to within 3%. Moreover, the two boundary conditions can be combined to yield a remarkably simple, accurate, and computationally fast method for extracting values for optical parameters from FDPM data. FDPM data were taken both at the surface and deep inside tissue phantoms, and the difference in data between the two geometries is striking. If one analyzes the surface data without accounting for the boundary, values deduced for the optical coefficients are in error by 50% or more. As expected, when aluminum foil was placed on the surface of a tissue phantom, phase and modulation data were closer to the results for an infinite-medium geometry. Raising the reflectivity of a tissue surface can, in principle, eliminate the effect of the boundary. However, we find that phase and modulation data are highly sensitive to the reflectivity in the range of 80-100%, and a minimum value of 98% is needed to mimic an infinite-medium geometry reliably. We conclude that noninvasive measurements of optically thick tissue require a rigorous treatment of the tissue boundary, boundary approach.

Measuring tissue hemodynamics and oxygenation by continuous-wave functional near-infrared spectroscopy—how robust are the different calculation methods against movement artifacts?

“…MD method1: spatially resolved spectroscopy. A more fundamental description of the light transport through tissue than the MBLL is the radiation transfer equation (Haskell et al 1994, Lindkvist et al 2013 1 c…”

Tissue parameters determining the visual appearance of normal skin and port-wine stains

“…The reason for this is that the reflected light has contributions not only from very superficial regions but also from deeper dermal layers. The diffusion approximation assumes an almost isotropic light distribution and radiance L can be expressed by a series expansion of the form (4,16) (A1) where 9 and j are, respectively, the fluence rate and the diffuse photon flux vector. The first term in this series expansion corresponds to an isotropic radiance and the second term represents the deviation from isotropy in the direction given by the unit directional vector 1 The irradiance on a surface normal to the flux then becomes,…”

“…The diffuse photon flux vector is given by, A very useful boundary condition at the skin-air interface is obtained by relating the reflected part of the irradiation at the inside of the interface to the irradition propagating back into the skin (16) …”

“…The basic criterion of optical diffusion theory that the flux shall be much smaller than the fluence rate is thus only satisfied to a limited degree at the surface where A=0.2 -0.17 for n=1.33 -1.4 (16). However, the ratio is small enough to justify the use of the diffusion theory as a reasonably good approximation.…”

Monitoring Spatially Heterogeneous Dynamics in a Drying Colloidal Thin Film

“…A detailed theoretical description of the static scattering process can be found in references [14,[29][30][31]. Briefly, the radial distribution of scattered light is approximately given by I(r x,y , s) (3/4πsl * ) exp −3r 2 x,y /4sl * , where l * is a transport mean free path.…”