# Channel Modeling for In-Body Optical Wireless Communications +
## 2 Channel Model
### Absorption  
The absorption coefficient can be modeled as  
\[\mu_{a}=-\frac{1}{T}\frac{dT}{d\delta}, \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Name** \\ \hline \(B\) & Blood volume fraction \\ \(\cos(\cdot)\) & Cosine function \\ \(\delta\) & Tissue thickness \\ \(\exp(\cdot)\) & Exponential function \\ \(F\) & Fat volume fraction \\ \(f_{\text{Ray}}\) & Rayleigh scattering factor \\ \(g\) & Anisotropy factor \\ \(L\) & Pathloss \\ \(M\) & Melanin volume fraction \\ \(\mu_{a}\) & Absorption coefficient \\ \(\mu_{s}\) & Scattering coefficient \\ \(\mu_{s}\) & Reduced scattering coefficient \\ \(\sin(\cdot)\) & Sine function \\ \(W\) & Water volume fraction \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of variables. Figure 1: The structure of this paper. with \(T\) denoting the fraction of residual optical radiation at distance \(\delta\) from the origin of the radiation. Thus, the fractional change of the incident light's intensity can be written as in [23]. \[T=\exp(-\mu_{a}\delta). \tag{3}\]  
As a result, \(\mu_{a}\) can be expressed as the sum of all of the tissue's constituents, namely water, melanin, fat, and oxygenated and de-oxygenated blood. Thus, (2) can be analytically expressed as in [19]. \[\mu_{a}= BS\mu_{a(aBl)}+B(1-S)\mu_{a(dBl)}+W\mu_{a(w)}+F\mu_{a(f)}+M\mu_{a(m)}, \tag{4}\]  
where \(\mu_{a(i)}\) represents the absorption coefficient of the \(i\)-th constituent, while \(B\), \(W\), \(F\), and \(M\) represent the blood, water, fat, and melanin volume fractions, respectively. Finally, \(S\) denotes the oxygen saturation of hemoglobin. From (4), we observe that the absorption coefficients and volume fractions of each constituent are required in order to calculate the complete absorption coefficient of any tissue. At this point, it is essential to highlight the dependence of any tissue's optical properties on the type of tissue, person, and even the procedure of collecting the sample, which hinders their mathematical modeling, despite the recent advances in optical measurement techniques. Although such variations are inherent characteristics of each individual and are subject to tissue preparation protocols, it has been proven in various past publications that they are mainly dependent on the wavelength of the transmitted optical radiation. Based on the aforementioned, we utilised the non-linear regression-based machine learning algorithm presented in Algorithm 1 to derive analytical expressions for the absorption coefficients based on experimental datasets for each of the constituents [24]. In more detail, the experimental data of the absorption coefficient of water in [25, 26, 27] were used to extract a Fourier series that accurately describes the absorption coefficient as a function of the transmission wavelength, which can be written as:  
\[\mu_{a(w)}(\lambda)=a_{0(w)}+\sum_{i=1}^{7}a_{i(w)}\cos(iw\lambda)+b_{i(w)} \sin(iw\lambda). \tag{5}\]  
Moreover, the analytical expressions of the absorption coefficients of oxygenated and de-oxygenated blood were fitted on the experimental results presented in [28, 29, 30, 31] by using the sum of Gaussian functions and can be expressed as:  
\[\mu_{a(aBl)}(\lambda)=\sum_{i=1}^{5}a_{i(aBl)}\exp\left(-\left(\frac{\lambda- b_{i(aBl)}}{c_{i(aBl)}}\right)^{2}\right), \tag{6}\]  
and  
\[\mu_{a(dBl)}(\lambda)=\sum_{i=1}^{4}a_{i(dBl)}\exp\left(-\left(\frac{\lambda- b_{i(dBl)}}{c_{i(dBl)}}\right)^{2}\right). \tag{7}\]  
Furthermore, special attention is required for the appropriate preparation of fat tissue, which requires proper purification and dehydration before measuring its optical properties. As mentioned in [32], this necessary procedure can result in inconsistencies between different published works. To limit this phenomenon, we selected the measurements in [32] as they coincide with multiple published works in the visible spectrum.