Adding scattering for photon migration: exact identities and experimental validation

Exact identities for radiative transfer are derived in scattering and absorbing media that relate emitted, received, and absorbed power (in the full domain or in arbitrary sub-volumes) to two trajectory statistics: the mean optical pathlength ⟨ℓ⟩ traveled by photons and the mean number of scattering events ⟨k⟩. For the received signal, the logarithmic derivatives with respect to absorption and scattering take particularly simple forms. In the continuous-wave case, ∂ ln (PR) /∂ μa = −⟨ℓ⟩R and ∂ ln (PR) /∂ μs = ⟨k⟩R/μs − ⟨ℓ⟩R, with analogous relations for time-resolved measurements. The second identity implies a counter-intuitive and yet highly consequential fact: the difference ⟨k⟩R/μs − ⟨ℓ⟩R is generally nonzero, which is required to ensure that the detected signal is sensitive to scattering variations. Motivated by this observation, we formalize an “adding scattering” protocol, analogous to established adding-absorption approaches, to estimate scattering sensitivity via finite-difference changes in μs in both Monte Carlo simulations and experiments. The predicted relations are validated via independent Monte Carlo implementations and with measurements in calibrated liquid phantoms. Finally, the role of these identities as exact benchmarks for forward solvers in photon migration studies is discussed.