## 3 Thermal Heat Modeling  
The biothermal heating effect by MMW/sub-THz waves on the skin is a complex process. The relevant parameters include frequency, power density, irradiation time, heat conduction, blood flow velocity in skin tissue, convection between skin surface and air, sweat evaporation and metabolic rate, etc. [48]. When electromagnetic powers are focused onto the skin surface, they will be absorbed and converted into heat, which is equivalent to establishing a heat source inside the skin tissue. The illumination spot inside the skin tissue acts as a heat source and the energy it generates can be expressed by the equation \(Q_{r}(x)=(I_{0}\,\Lambda/L)\cdot e^{-\pi/L}\)[48], with \(x\) as the depth of the source, \(I_{0}\) as the illumination power density, and \(L\) as the penetration depth in the skin. The parameter \(\Lambda=\mid\)\(\tau\)\(\mid^{2}\) refers to the power transfer coefficient and can be calculated by [PERSON]'s principle with the transmission coefficient as \(\tau=2\eta_{2}\text{cos}\theta_{1}/(\eta_{2}\text{cos}\theta_{1}+\eta_{1} \text{cos}\theta_{2})\). Here parameters \(\eta_{1}\) and \(\eta_{2}\) are the wave impedance of electromagnetic waves in air and skin, respectively. Parameters \(\theta_{1}\) refers to the angle of incidence and \(\theta_{2}\) to the angle of reflection. The classical Bioheat Transfer Equation (BHTE) models, such as Pennes' BHTE, Weinbaum-Jiji (W-J) BHTE, and the thermal wave model of bioheat transfer (TWMBT), can be adapted to express the heat transfer mechanism inside the tissue. The Pennes' BHTE model lays a solid and favorable foundation for establishing and studying the distribution of tissue temperature, due to its superiorities in simple structure, strong practicality, and wide range of applications. The W-J BHTE model is based on the fine anatomy of skin tissue [49] and takes into account the changes in number density, size, blood flow velocity, and direction of blood vessels in the tissue's microstructure along with the penetration direction of electromagnetic waves. Its solution is more applicable to micro-vascular tissue due to its complexity but is not as efficient for a single skin layer. The TWMBT model considers the finite speed of heat transmission and can reflect the fluctuation characteristics of heat flow [50], which is not necessary or required here. In this work, we employ [PERSON] BHTE model, which is considered to be more reasonable in describing the microstructure and temperature distribution of subcutaneous areas caused by external heat sources [51]. It is also regarded as a common method to study the dependence of the biothermal heating effect on specific parameters, such as temperature, frequency, illumination time, and power density. There are several assumptions for establishing the model: (1) the convective heat transfer process of blood occurs in the dermis with capillaries, (2) the tissue and blood in capillaries have the same temperature and are equal to the tissue temperature, and (3) the blood perfusion rate in local capillaries is isotropic. We also consider the skin tissue as a semi-infinite, homogeneous, and isotropic medium, with the heat exchange as a one-dimensional process, which is available to describe the heat transfer due to the thermal heating effect [52]. Thus, [PERSON] BHTE model can be simplified to be a one-dimensional form as  
\[\rho c\frac{\partial T}{\partial t}=k\frac{\partial^{2}T}{\partial x^{2}}+W_{ b}c_{b}(T_{b}-T)+Q_{m}+Q_{r} \tag{6}\]  
where parameters \(\rho\), \(c\), and \(k\) are the density, specific heat, and thermal conductivity of the skin, respectively. Parameter \(W_{b}\) refers to the blood perfusion rate per unit volume with \(c_{b}\) as the blood-specific heat and \(T_{b}\) as the arterial blood temperature. Parameter \(T\) is the skin temperature, \(Q_{m}\) is metabolic heat production, and \(Q_{r}\) is the electromagnetic wave heat source term.