The modified Beer–Lambert law revisited

TLDR

The modified Beer–Lambert law underpins continuous‑wave near‑infrared spectroscopy and assumes homogeneous absorption changes and constant scattering loss. The study aims to quantify the error introduced by assuming constant scattering loss in the differential Beer–Lambert law. The authors use the differential Beer–Lambert law, measuring attenuation at multiple wavelengths, and analyze a semi‑infinite homogeneous cerebral cortex model to estimate the magnitude of scattering‑related errors. The study finds that changes in scattering and absorption equally affect attenuation, with a 1 % increase in scattering raising estimated concentration changes by ~0.5 µM, highlighting the need to account for scattering cross‑talk.

Abstract

The modified Beer–Lambert law (MBLL) is the basis of continuous-wave near-infrared tissue spectroscopy (cwNIRS). The differential form of MBLL (dMBLL) states that the change in light attenuation is proportional to the changes in the concentrations of tissue chromophores, mainly oxy- and deoxyhaemoglobin. If attenuation changes are measured at two or more wavelengths, concentration changes can be calculated. The dMBLL is based on two assumptions: (1) the absorption of the tissue changes homogeneously, and (2) the scattering loss is constant. It is known that absorption changes are usually inhomogeneous, and therefore dMBLL underestimates the changes in concentrations (partial volume effect) and every calculated value is influenced by the change in the concentration of other chromophores (cross-talk between chromophores). However, the error introduced by the second assumption (cross-talk of scattering changes) has not been assessed previously. An analytically treatable special case (semi-infinite, homogeneous medium, with optical properties of the cerebral cortex) is utilized here to estimate its order of magnitude. We show that the per cent change of the transport scattering coefficient and that of the absorption coefficient have an approximately equal effect on the changes of attenuation, and a 1% increase in scattering increases the estimated concentration changes by about 0.5 µM.