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:
![Rendered by QuickLaTeX.com I=I_0e^{(bar{-DPF}mu_ar)-G}](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-af286c7696e43e2f4a20d144a2876c3d_l3.png)
Where:
![Rendered by QuickLaTeX.com I](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png)
is the detected optical intensity.
![Rendered by QuickLaTeX.com I_0](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-d39036990f5094f2fec381165ca0dbaf_l3.png)
is the incident intensity.
![Rendered by QuickLaTeX.com bar{DPF}](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-0f5c935572d366d7a301c6b03445c2c7_l3.png)
is an average (over the absorption range 0-
![Rendered by QuickLaTeX.com mu_a](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-c46cc90b31dab60c3f5a0ef147040bd7_l3.png)
) mean pathlength factor that depends on the optical properties of tissue
![Rendered by QuickLaTeX.com bar{DPF}](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-0f5c935572d366d7a301c6b03445c2c7_l3.png)
accounts for the dependence of the mean optical pathlength on scattering and absorption.
![Rendered by QuickLaTeX.com r](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png)
is the inter-optode distance.
![Rendered by QuickLaTeX.com G](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-30a79c32f18567063fe44716929e7ced_l3.png)
is a factor that accounts for the effect of scattering.
![Rendered by QuickLaTeX.com mu_a](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-c46cc90b31dab60c3f5a0ef147040bd7_l3.png)
is the absorption coefficient.
By calculating logarithmic ratio of I and
![Rendered by QuickLaTeX.com I_0](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-d39036990f5094f2fec381165ca0dbaf_l3.png)
, we estimate the attenuation OD (optical density):
![Rendered by QuickLaTeX.com A = OD = log_{10}frac{I_0}{I} = frac{[bar{DPF} mu_a r + G]}{log_e(10)}](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-f6c2972c6eedf1a9b75e5d0373df849b_l3.png)
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
![Rendered by QuickLaTeX.com mu_{a}](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-00dc91109801f28ba97c704292f79221_l3.png)
, it can be shown that the logarithmic difference between the detected intensity at baseline and during the test condition can be written as follows:
![Rendered by QuickLaTeX.com DeltaA = DeltaOD = log_{10}(frac{I_{baseline}}{I_{test}} = DPF frac{ r Delta mu_a}{log_e(10)}](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-e30fd3b2a7ce01abfba874202c949c6a_l3.png)
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
![Rendered by QuickLaTeX.com Delta mu_a](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-7c23c80659e61832ff886983c417a2f2_l3.png)
is spatially uniform.
The change in the absorption coefficient
![Rendered by QuickLaTeX.com Delta mu_a](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-7c23c80659e61832ff886983c417a2f2_l3.png)
is the summation of the changes in the absorption contributions associated with available chromophores in the medium e.g O
![Rendered by QuickLaTeX.com _2](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-10a7e9eb9832336fb535cd82c6ea4697_l3.png)
Hb and Hb which are given by the product of their molar decadic absorption coefficients (
![Rendered by QuickLaTeX.com epsilon_{HbO_2}, epsilon_{Hb}](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-36507859701d62270859b5972d491c79_l3.png)
and their concentration changes:
![Rendered by QuickLaTeX.com Delta mu_a (lambda) = log _e(10) cdot [ epsilon_{HbO_2}(lambda) Delta c_{HbO_2} + epsilon_{Hb}(lambda) Delta c_{Hb} ]](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-6fe31c412484763a84677cccd177f9af_l3.png)
The change in absorbance can then be written for two wavelengths (
![Rendered by QuickLaTeX.com lambda_1 , lambda_2](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-1f4acbde67af3167eff120dcb81693a6_l3.png)
) and two chromophores (O
![Rendered by QuickLaTeX.com _2](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-10a7e9eb9832336fb535cd82c6ea4697_l3.png)
Hb, Hb) as below, assuming DPF to be independent of wavelength:
![Rendered by QuickLaTeX.com Delta A(lambda_2) = DPF cdot r cdot [ epsilon_{HbO_2}(lambda_2) Delta c_{HbO_2} + epsilon_{Hb}(lambda_2) Delta c_{Hb} ]](http://latamnirs.org/wp-content/ql-cache/quicklatex.com-f1f7413462779c8887ca47cb6179fb63_l3.png)
One can estimate the concentration changes of both hemoglobin species by solving this system of linear equations.
Alternative definition:
Synonym: modified Bouguer-Beer-Lambert Law
References: https://doi.org/10.1039/9781847551207
https://doi.org/10.1088/0031-9155/51/5/N02
https://doi.org/10.1088/0031-9155/33/12/008
https://doi.org/10.1088/0031-9155/49/14/N07
Related terms: Beer Lambert Law, Light Intensity, Molar Absorption