## 5 Materials and Methods
### Mathematical Modeling  
The Pennes equation is a mathematical equation that describes the transfer of heat in biological tissue. The equation was first proposed by [PERSON], a Norwegian physicist, in 1948 [24]. It is used to model the heat transfer between blood vessels and surrounding tissue in the body. It takes into account various factors, such as blood flow, metabolic heat generation, and tissue perfusion. The equation states that the rate of heat transfer is proportional to the temperature difference between the blood and surrounding tissue, as well as the blood flow rate, tissue perfusion rate, and metabolic heat generation. The equation also includes a term for thermal conductivity, which describes the ability of tissue to conduct heat. The [PERSON] equation is often used in medical applications, such as hyperthermia treatments for cancer. It can help predict the temperature distribution in tissue during treatment, which can help optimize treatment parameters and avoid overheating or underheating of the tissue [25]. Overall, the Pennes equation provides a mathematical framework for understanding and modelling the heat transfer in biological tissue, which has important implications for medical treatments and other applications. There are four layers in the breast and tumor, so there are five values for \(I\), which are as follows: muscle, \(i=1\); gland, \(i=2\); fat, \(i=3\); skin, \(i=4\); and tumor, \(i=5\). In addition, there are two other variables: blood perfusion rate, \(\omega_{b,i}\), and thermal conductivity of the tissues, \(k_{i}\), as well as the specific mass, \(\rho_{b}\), the specific heat, \(C_{b}\), and the arterial temperature of the blood, \(T_{b}\). Furthermore, in each tissue layer, there are metabolic heat generation, \(Q_{i}\), and temperature, \(T_{i}\). \[k_{i}\
abla^{2}T_{i}+\rho_{b}C_{b}\omega_{b,i}(T_{b}+T_{i})+Q_{i}=0 \tag{1}\]  
The equation is a form of the heat transfer equation, which describes how energy is transferred through a medium. Here is an explanation of each term in the equation:  
\(k_{i}\) is the thermal conductivity of the material in which the temperature field, \(T_{i}\), is defined. \(\
abla^{2}T_{i}\) is the Laplacian operator of the temperature field, \(T_{i}\), which describes how the temperature varies spatially. \(P_{b}\) is the density of the surrounding fluid or medium, and \(C_{b}\) is its specific heat capacity. The term \(P_{b}\)\(C_{b}\) represents the thermal capacity of the medium. \(\omega_{b,i}\) is the volume fraction of the fluid or medium in which the temperature field, \(T_{i}\), is defined. \(T_{b}\) is the temperature of the surrounding fluid or medium, and \(Q_{i}\) is any heat source within the breast in which the temperature field, \(T_{i}\), is defined. Together, these terms describe how energy is transferred through a material and its surroundings. The first term on the left-hand side represents heat conduction within the material, while the second term represents the exchange of energy between the material and its surroundings due to convection. The term \(Q_{i}\) represents any internal heat sources within the breast. By solving this equation, one can determine the temperature distribution within a breast as it interacts with its surroundings. Equation (1) can be solved by suitable boundary conditions on the breast skin, temperature, and heat flux continuity conditions at each tissue layer. Core body temperature was determined in the lower layer of the muscle, and the heat flux at the lower layer of the skin was taken as zero. In addition, the skin surface is subjected to convection.