Analytic function for predicting light fluence rate of circular fields on a semi-infinite turbid medium

Accurate determination of in-vivo light fluence rate is critical for preclinical and clinical studies involving photodynamic therapy (PDT). The light fluence distribution in tissue depends on both the tissue optical properties and the incident field size. This study compares the longitudinal light fluence distribution inside biological tissue in the central axis of circular uniform light field with different radii for a range of in-vivo tissue optical properties (absorption coefficients (µ_a) between 0.01 and 1 cm^-1 and reduced scattering coefficients (µ_s') between 2 and 40 cm^-1). This was done using Monte-Carlo simulations for a semi-infinite turbid medium in an air-tissue interface. The end goal is to develop simple analytical expressions that would fit the results from the Monte Carlo simulation for circular beams with different radii. A 6-parameter model (ϕ/ϕ_air=(1-b⋅e^-λ)(C₂e^-λ+C₃e^-λ)) can be used to fit MC simulation. Each of these parameters (b, C₂, C₃, λ₁, λ₂, and λ₃) is expressed as a function of tissue optical properties and beam radius. These results can then be compared against the existing expressions in the literature for broad beam for analysis in both accuracy and applicable range. The analytical function can be used as rapid guide in PDT to calculate in vivo light fluence distribution for known tissue optical properties.