Phase Velocity Limit of High-Frequency Photon Density Waves

In frequency-domain photonmigration (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, highfrequency 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 reexamine 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 1GHzfor some biological tissue. The revised diffusion equation, including the second-order time-derivative, is often termed the PI equation. We compare the dispersion relation of the PI equation with that of the SDE. The PI phase velocity is slower than that predicted by the SDE; in fact, the SDE phase velocity is unbounded with increasing modulation frequency, while the PI phase velocity approaches c/sqrt(3) in the high frequency limit. We emphasize that the phase velocity 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