# An Energy Balance Clustering Routing Protocol for Intra-Body Wireless Nanosensor Networks
## 2 System Model
### Intra-Body Terahertz Channel Model  
The path loss of the terahertz channel consists of three parts: the loss experienced by the terahertz wave propagating in the medium, the loss caused by the molecular absorption and the scattering loss during signal propagation [18] Therefore, the path loss can be expressed as:  
\[PL(f,d)=PL_{spr}(f,d)\times PL_{abs}(f,d)\times PL_{sca}(f,d) \tag{1}\]  
where \(PL(f,d)\) is the total path loss, \(PL_{spr}(f,d)\), \(PL_{abs}(f,d)\) and \(PL_{sca}(f,d)\) represent the propagation loss, molecular absorption loss and scattering loss respectively. Propagation loss represents the transmission loss of electromagnetic waves in a physical medium and can be modeled by a path loss formula _Friis_, which is defined as:  
\[PL_{spr}(f,d)=\left(\frac{4\pi dn_{m}}{\lambda_{0}}\right)^{2} \tag{2}\]  
where \(d\) is the propagation distance, \(\lambda_{0}\) is the free space wavelength and \(n_{m}\) represents the refractive index of human tissue. At the frequency of 1 THz (\(\lambda_{0}=300\ \mu m\)), the propagation loss is quite high, severely limiting the transmission range of the nanonode. Therefore, in the human body scenario, the communication distance between the nanonode and NC is drastically reduced. Molecular absorption loss refers to the attenuation caused by molecular absorption. Usually, when electromagnetic waves propagate through the material, a portion of the electromagnetic energy radiated by the nano-antenna is converted to the internal kinetic energy of the molecule. This process depends on the operating frequency and the biological tissue considered. For example, the absorption loss caused by propagation in the skin or blood is different. Thus, the molecular absorption of each material  
Figure 1: Layered clustering structure of intra-body Wireless NanoSensor (iWNNNs). can be characterized by an absorption loss factor \(\alpha(f)\), which varies with frequency. Therefore, given \(\alpha(f)\), the molecular absorption loss can be obtained by the following formula:  
\[PL_{abs}(f,d)=e^{\alpha(f)d} \tag{3}\]  
Human tissue is composed of different types of substances. The deflection of the beam caused by this microscopic inhomogeneity will cause scattering of particles and affect the propagation of electromagnetic waves. The path loss due to scattering can be expressed as:  
\[PL_{sca}(f,d)=e^{\mu d} \tag{4}\]  
where \(d\) is the propagation distance, and \(\mu\) is the scattering loss factor. It can be seen from [19] that the scattering loss factor is too small compared to the molecular absorption loss factor, which is almost negligible. Therefore, the path loss can be expressed as:  
\[PL(f,d)=\left(\frac{4\pi dn_{m}}{\lambda_{0}}\right)^{2}e^{\alpha(f)d} \tag{5}\]  
The phenomenon of molecular absorption from the medium not only affects the characteristic of the channel in terms of attenuation, but also introduces noise. This is the main noise in the terahertz band. The power spectral density of molecular absorption noise can be expressed as:  
\[N(f,d)=K_{B}T_{0}\left(1-e^{-\alpha(f)d}\right) \tag{6}\]  
where \(K_{B}\) represents the Boltzmann constant and \(T_{0}\) represents the reference temperature. According to Shannon theory, the channel capacity can be calculated by the following formula:  
\[C=Blog_{2}\left(1+\frac{S}{N}\right) \tag{7}\]  
where \(B\) represents the total bandwidth of the system and \(S\)/\(N\) represents the signal-to-noise ratio. However, the terahertz channel is frequency selective and the noise is non-white noise, so the entire bandwidth is divided into many narrow sub-bands. These sub-bands are very narrow, so the sub-band channels are non-selective and the noise therein is white noise.