Sensitivity factors for diffuse optics - derivation of equations for finite changes

We present analytical equations for time-resolved signals in diffuse optics and their sensitivity factors in homogeneous media, derived from perturbation-based equations valid for infinitesimal changes and then extended to finite changes in optical properties. Time-resolved optical measurements are becoming increasingly available due to technological advancements, expanding their use in various techniques such as near-infrared spectroscopy (NIRS). These measurements acquire distributions of times of flight (DTOF) of photons, and the measurands considered in this study are attenuation (A), mean time of flight (m ₁), and variance (V) of the DTOF. Established methods for recovering changes in the absorption coefficient (Δμ _a) from ΔA, Δm ₁, or ΔV use perturbation-based sensitivity factors, which are valid for infinitesimal changes. We derived sensitivity factor equations that relate ΔA, Δm ₁, and ΔV to finite (including large) changes in absorption (Δμ _a), reduced scattering coefficient $\left(\Delta {\mu }_s^{\text{'}}\right)$ , and source-detector distance (Δr). The derivations rely on solutions of the diffusion equation (DE) for homogeneous infinite (IM) and semi-infinite (SM) media within the diffusion approximation $({\mu }_s^{\text{'}}\gg {\mu }_a)$ , without introducing additional assumptions. We also present analytical equations for the modified Beer-Lambert law (MBLL) that are valid for finite Δμ _a in IM and SM, which can be directly applied in continuous-wave NIRS data analysis. Sensitivity factors and the MBLL require knowledge of the baseline optical properties and the differential pathlength factor (DPF), and we assessed how errors in these parameters affect the recovered Δμ _a. We also assessed cross-talk, in which scattering changes $\Delta {\mu }_s^{\text{'}}$ lead to spuriously recovered absorption changes Δμ _a, and vice versa. The proposed framework can be used to improve the accuracy of methods for estimating changes in optical properties.