Differential pathlength factor model for near-infrared diffuse optical imaging

Continuous-wave near-infrared diffuse optical imaging (DOI) relies on accurate differential pathlength factors (DPFs) for quantitative chromophore estimation. Existing DPF definitions inherit formulation-dependent limitations that can introduce large errors in modified Beer-Lambert law (MBLL) analyses and associated inverse estimations. We propose two DPF formulations: an MBLL-based formulation derived directly from photon-transport statistics and an experimentally practical inverse-distance model that closely reproduces the results of the more rigorous MBLL-based formulation. These models are derived based on extensive Monte Carlo simulations whereby we exclusively assign the source-detector-separation dependence of optical density (OD) to the Beer-Lambert term for a given object geometry, boundary configuration, and measurement setup. The additive term G (widely referred to as the geometry-dependent term) in the MBLL is identified as the OD intercept in the limit of zero source-detector separation,thereby excluding any source-detector-separation dependence, enabling the new formulations derived here. Our absolute MBLL-based DPF model shows the relevance of logarithmic aggregation of exponentially attenuated photon contributions in the measured OD. This differs from the arithmetic mean-pathlength treatment used in conventional mean-pathlength DPF formulations, which is widely used. Our practically usable model, which we call an inverse-distance DPF model, can be used more readily for inverse solutions in imaging and is shown to be consistent with the aforementioned more rigorous model. We benchmark the two proposed DPF models against existing formulations using both the absolute and differential (where the term G cancels) forms of the MBLL. The proposed models achieve errors below 20% across broad optical conditions, whereas conventional DPFs can exceed 100% error. The theoretical predictions are further validated using controlled phantom experiments, demonstrating improved quantitative accuracy in CW-NIR imaging.