Modified Beer-Lambert Law

Definition: A common approach to measuring changes in the tissue absorption using continuous waves methods is based on a generalization of Beer-Lambert law in conjunction with the assumption that the scattering properties of tissue do not vary with time. The modified Beer-Lambert law (mBLL) can be expressed as follows:

$I=I_0e^{(-\overline{DPF}\mu_ar)-G}$

Where: $I$ is the detected optical intensity. $I_{0}$ $I_0$ is the incident intensity. $\overline{DPF}$ is an average (over the absorption range 0- $\mu_a$ ) mean pathlength factor that depends on the optical properties of tissue $\overline{DPF}$ accounts for the dependence of the mean optical pathlength on scattering and absorption and $r$ is the inter-optode distance. $G$ is a factor that accounts for the effect of scattering and $\mu_{a}$ $\mu_a$ is the absorption coefficient. By calculating logarithmic ratio of I and $I_0$ , we estimate the attenuation OD (optical density):

$A = OD = log_{10}\frac{I_0}{I} = \frac{[\bar{DPF} \mu_a r + G]}{log_e(10)}$

which is the integral form of the mBLL. It is valid for an arbitrary medium in terms of its geometry and spatial distribution of the scattering coefficient. If we assume that G is constant and that the absorption change in the medium between a baseline condition and a test condition is small compared to $\mu_{a}$ , it can be shown that the logarithmic difference between the detected intensity at baseline and during the test condition can be written as follows:

$\Delta A = \Delta {OD} = log_{10}(\frac{I_{baseline}}{I_{test}}) = DPF \frac{ r \Delta \mu_a}{log_e(10)}$

where: Differential pathlength factor (DPF) is defined as the ratio of the mean pathlength of detected photons at baseline to the inter-optode distance. The last equation is the differential form of the mBLL and it is valid for an arbitrary medium in terms of its geometry and spatial distribution of the optical properties at baseline. The only requirement is that $\Delta \mu_a$ is spatially uniform. The change in the absorption coefficient $\Delta \mu_a$ is the summation of the changes in the absorption contributions associated with available chromophores in the medium e.g. HbO $_2$ and Hb which are given by the product of their molar decadic absorption coefficients ( $\epsilon_{HbO_2}, \epsilon_{Hb}$ ) and their concentration changes:

$\Delta \mu_a (\lambda) = log _e(10) \cdot [ \epsilon_{HbO_2}(\lambda) \Delta c_{HbO_2} + \epsilon_{Hb}(\lambda) \Delta c_{Hb} ]$

The change in absorbance can then be written for two wavelengths ( $\lambda_1 , \lambda_2$ ) and two chromophores (HbO $_2$ , Hb) as below, assuming DPF to be independent of wavelength:

$\Delta A(\lambda_1) = DPF \cdot r \cdot [ \epsilon_{HbO_2}(\lambda_1) \Delta c_{HbO_2} + \epsilon_{Hb}(\lambda_1) \Delta c_{Hb} ]$