# Veins Depth Estimation Using Diffused Reflectance Parameter
## 4 Mathematical Model Derivation  
A mathematical model to calculate the blood vessel's depth with the measured diffused reflectance from the surface of skin has been derived. When light is made incident on the medium, a part of it gets reflected, absorbed and transmitted through the medium. The total thickness of medium is infinite in accordance to the light intensity and wavelength. With the considered wavelength and intensity, the light rays cannot get transmitted through the four-layered skin model, hence ignoring the transmission part. The following equation (Equation (3)) expresses the relationship of incident light to absorbed and diffused (scattered) light ([19], pg 62, eq 2.52).  
\[\mathrm{I_{0}=Ic+Id}, \tag{3}\]  
where \(\mathrm{I_{0}}\) is the incident light launched in the direction normal to the surface. \(\mathrm{I_{d}}\) and \(\mathrm{I_{c}}\) represent the diffused and absorbed component of light, respectively.  
Intensity of light absorbed while propagating through the Scattering-Dominant medium ([17], pg 70, eq. 2.572) ([19], pg 62, eq. 2.57) can be expressed as:  
\[\mathrm{I_{c}=I_{0}\;(1-Rd^{2})\;exp\;[-(\mu_{a}+\mu_{s})\;z]}, \tag{4}\]  
where z is total depth, \(\mu_{a}\) is absorption coefficient, \(\mu_{s}\) is the scattering coefficient and Rd is diffused reflectance parameter. Substitution of Ic from Equation (4) into Equation (3) results in the following equation:  
\[\mathrm{I_{0}=I_{0}\;(1-Rd^{2})\;exp\;[-(\mu_{a}+\mu_{s})z+I_{d}]}, \tag{5}\]  
Finally, with a little math's we get the relationship between vessels depth and diffused reflectance parameter \(Rd\).  
\[z\;\;=\;-\frac{\ln\left(\frac{1}{(1+R_{d})}\right)}{(\mu_{a}+\ \mu_{s})}, \tag{6}\]  
The summation of the absorption coefficient \(\mu_{a}\) and scattering coefficient \(\mu_{s}\) in Equation (6) represents the attenuation of light intensity while propagating through the medium. The parameter 'z' in Equation (6) is the total thickness of layered medium. In this model, veins are supposed to be at the bottom of fourth layer, hence 'z' can be referred as the depth of veins from the top surface of the layered medium. In the proposed four-layered model of skin, the absorption and scattering properties of each layer is different, depending upon the constituents of each layer. Therefore, the total attenuation is the  
Figure 2: Block diagram of proposed method for blood vessel’s depth estimation.  
sum of \(\mu_{a}\) and \(\mu_{s}\) parameters of all layers of the skin model. For the vessel depth estimation in layered medium like skin, the Equation (6) is modified as follows:  
\[z = -\frac{\ln\left(\frac{1}{\left(1+R_{d}\right)}\right)}{\left(\mu_{a}+ \ \mu_{s}\right)}\times\text{C} \tag{7}\]  
where C = \(\frac{Z_{ref}}{Z_{ini}}\).  
The parameter 'C' is the ratio between the reference depth \(Z_{ref}\) to the initial depth \(Z_{init}\). It is empirically derived to compensate the initial reflectance of light from the very top layer with high scattering coefficient. In this work, \(Z_{ref}\) is the total thickness of layered medium in each iteration of simulation. \(Z_{init}\) is the thickness value of the first layer with high scattering coefficient. In the four-layered model, the first layer is the stratum corneum with scattering coefficient '\(\mu_{s}\) = 1050' with a thickness '\(t\) = 0.002 cm'. Therefore, the value of \(Z_{init}\) for this work is set to 0.002 as reference to the thickness of first layer. The reference depth \(Z_{ref}\) value can be estimated from experimentation on phantoms, where depth of vessel will be known against the similar value of measured diffused reflectance. These experiments will be performed in future work in order to validate the proposed model.