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In frequency-domain photon migration (FDPM), two factors make high modulation frequencies desirable. First, with frequencies as high as a few GHz, the phase lag versus frequency plot has sufficient curvature to yield both the scattering and absorption coefficients of the tissue under examination. Second, because of increased attenuation, high frequency photon density waves probe smaller volumes, an asset in small volume in vivo or in vitro studies. This trend toward higher modulation frequencies has led us to re-examine the derivation of the standard diffusion equation (SDE) from the Boltzman transport equation. We find that a second-order time-derivative term, ordinarily neglected in the derivation, can be significant above 1 GHz for some biological tissue. The revised diffusion equation, including the second-order time-derivative, is often termed the P1 equation. We compare the dispersion relation of the P1 equation with that of the SDE. The P1 phase velocity is slower than that predicted by the SDE; in fact, the SDE phase velocity is unbounded with increasing modulation frequency, while the P1 phase velocity approaches c/sqrt(3) is attained only at modulation frequencies with periods shorter than the mean time between scatterings of a photon, a frequency regime that probes the medium beyond the applicability of diffusion theory. Finally we caution that values for optical properties deduced from FDPM data at high frequencies using the SDE can be in error by 30% or more.
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