# Numerical Analysis of Human Cancer Therapy Using Microwave Ablation
## 2 Description and Formulation of the Problem  
### Thermal Problem  
The biological tissue investigated is the liver. A micro-coaxial antenna is used to generate an electric field. The heat transfer problem in the biological medium is described by the bio-heat equation [28]  
\[\rho C_{p}\frac{\partial T}{\partial t}\ =\ \
abla(KVT)+\rho_{b}C_{b}\omega_{b}(T_{b }-T)+Q_{ext}+Q_{meta} \tag{1}\]  
where \(T\) is the tissue's temperature, \(\rho\) is its density, \(C_{p}\) is its specific heat capacity, \(k\) is its thermal conductivity, \(\rho_{b}=1060\) kg\(\cdot\)m\({}^{-3}\) is the blood density, \(C_{b}=3600\) Jkg\({}^{-1}\) denotes the specific heat capacity of the blood, \(\omega_{b}\) represents the blood perfusion rate, and \(T_{b}\) is the blood temperature. In this study, we assume \(T_{b}=37\)\({}^{\circ}\)C.  
The first term in Equation (1) represents the transient term whereas the second term denotes the heat conduction. The third term represents the heat dissipation by the blood flow. \(Q_{ext}\) denotes the external heat source and \(Q_{meta}\) is the heat source from the metabolism. The value of \(Q_{meta}\) is usually neglected compared to the other heat terms.  
The external source term \(Q_{ext}\) results from the heat generated by the electromagnetic field, i.e.,  
\[Q_{ext}=\frac{\sigma}{2}\|E_{0}\|^{2} \tag{2}\]  
where \(\sigma\) denotes the electric conductivity of the biological tissue (S/m) and \(E_{0}\) is the amplitude of the electric field generated by the micro-coaxial antenna.  
The thermal conductivity, density, and specific heat of the liver are dependent on temperature. Their expressions can be extracted from the literature [29]. For the normal tissue, we have  
\[k_{liter}(T)=4190[0.133+1.36W_{a}(T)] \tag{3}\]  
\[\rho_{liter}(T)=1300-300W_{a}(T) \tag{4}\]  
\[C_{p,liter}(T)=0.419[0.37+0.63W_{a}(T)] \tag{5}\]  
In the above equations, the temperature is expressed in degrees Celsius. The parameter \(W_{a}(T)\) is the measured remaining tissue water content versus temperature of the liver tissue [30]. It is given by  
\[W_{a}(T)=\left\{\begin{array}{ll}0.778-0.779\times\exp\bigl{(}\frac{T-106}{ 3.42}\bigr{)}&T\leq 103\)\({}^{\circ}\)C \\ -0.03924(T-103)+0.454392&103\)\({}^{\circ}\)C \(<T\leq 104\)\({}^{\circ}\)C \\ 0.778\times\exp\bigl  
Regarding the properties of the tumor, we used the following values: \(k_{tumor}=0.57\;\mathrm{W/m}\;^{\circ}\mathrm{C}\), \(\rho_{tumor}=1040\;\mathrm{kg\;m^{-3}}\), and \(C_{p,tumor}=3960\;\mathrm{J\;kg^{-1}\;K^{-1}}\). These values are independent of the temperature [7].  
The blood flow induces a convective heat flux between the blood flow and the tissue. This heat flux depends on the temperature difference \((T_{b}-T)\) and the blood flow rate. Hence, it is influenced by blood perfusion [31]. It is represented by the term \(\rho_{b}C_{b}\omega_{b}(T_{b}-T)\). The rate of blood perfusion versus temperature is given by  
\[\omega_{b}(T)=\omega_{0}+a\;T \tag{7}\]  
In the above equation, \(T\) is also expressed in degrees Celsius, whereas the unit of the rate \(\omega_{b}\) is s\({}^{-1}\). The constants involved in Equation (7) are given by \(\omega_{0}=6.08\times 10^{-3}\) s\({}^{-1}\) and \(a=2.1\times 10^{-5}\) s\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\).  
The source term in Equation (1) depends on the electric field amplitude. The next paragraph will explain how this quantity is computed.