# Assessment of Electromagnetic Exposure to a Child and a
## 3 Bioheat Equation and Boundary Conditions  
The steady-state distribution of temperature in tissues is determined using the following Pennes bioheat equation [63]:  
\[\rho c\frac{\partial T}{\partial t}=\
abla\cdot(k\
abla T)+\rho Q+\rho S-\rho_{b}c _{b}\rho\omega(T-T_{b}) \tag{3}\]  
where \(\rho\left(kg/m^{3}\right)\) represents the medium's density, \(c\) (\(J/\left(kg\cdot^{\circ}C\right)\)) is the specific heat capacity, \(T\left({}^{\circ}C\right)\) denotes the temperature of the tissue, \(t\) (s) refers to time, \(k\) (\(W/(m^{\circ}C)\)) indicates the efficiency of heat conduction within the tissue, \(Q\) (\(W/kg\)) refers to the rate of metabolic heat generation, \(S\) (\(W/kg\)) corresponds to specific absorption rate (SAR), \(\omega\left(m^{3}/\left(kg\cdot s\right)\right)\) represents the rate of blood flow through tissue, \(\rho_{b}\left(kg/m^{3}\right)\) is the density of the mass, \(c_{b}\) (\(J/\left(kg\cdot^{\circ}C\right)\)) indicates the thermal capacity of the material, and \(T_{b}\) (\({}^{\circ}C\)) is the heat level of blood [64]. When the bioheat equation is used in a simulation that aims to determine the temperature rise (Tincr) from SAR exposure, and SAR-induced temperature rises are expected to remain below 1 \({}^{\circ}C\), all tissue and boundary parameters (\(\rho\), \(c\), \(k\), \(\omega\), \(Q\), \(T_{n}\), \(h\), and \(T_{b}\)) can be treated as temperature and time independent. We therefore write the total temperature as follows:  
\[T=T_{base}+T_{incr}\]  
Substitute into the following standard Pennes equation:  
\[\rho c\frac{\partial T_{incr}}{\partial t}=\
abla\cdot(k\
abla T_{incr})+\rho S -\rho_{b}c_{b}\rho\omega T_{incr} \tag{4}\]  
The following boundary conditions were applied in this study to determine the temperature distribution [65; 66]:  
Mixed boundary condition:  
\[k\frac{\partial T_{incr}}{\partial n}+hT_{incr}=0 \tag{5}\]  
where \(h\) denotes the coefficient of the heat transfer, \(T_{incr}\) represents the rise in temperature, and \(n\) is the unit normal vector directed outward from the surface interface. Mixed boundary conditions were implemented at the tissue-air interfaces. The heat transfer coefficient (\(h\)) between the skin and the surrounding air was defined as \(h_{skin-air}=8\left(W/\left(m^{2}\cdot K\right)\right)\) and between the cornea and the external air \(h\) was set as \(h_{ cornea-air}=20\left(W/\left(m^{2}\cdot K\right)\right)\)[65]. The boundary condition of Neuman:  
\[k\frac{\partial T_{incr}}{\partial n}=0 \tag{6}\]  
The boundary condition of Neumann was enforced at the interfaces of biological tissues and air-filled cavities, covering the internal passages of the pharynx, bronchi, trachea, and esophagus. The boundary condition of Dirichlet:  
\[T_{incr}=0 \tag{7}\]  
The Dirichlet boundary conditions were defined at the interface adjoining the circulatory system that includes the veins, arteries, the umbilical cord and all the tissues that are surrounded by blood-perfused, excluding air-filled tissues, maternal and fetal cerebrospinal fluid (CSF), maternal and fetal vitreous humor, amniotic fluid, teeth, stomach lumen, andthe small intestine lumen. Additionally, the ambient external temperature was steady set to T\({}_{\text{external}}\) = 24 \({}^{\circ}\)C, and the human's temperature fixed at T\({}_{\text{human}}\) = 37 \({}^{\circ}\)C.  
A non-uniform grid mesh was utilized, and the minimum grid resolution of 1 mm was applied in proximity to the source and within specific tissues of the anatomical phantoms of humans for both electromagnetic and thermal simulations. In the surrounding environment, representing free-space air, the maximum grid resolution was set at 30 mm for both considered frequencies. This resolution was achieved by adjusting the maximum step at the two frequencies, corresponding to the distinct wavelengths (\(\lambda\)) in free space, which are \(\lambda\) = 299.79 mm at 1000 MHz and \(\lambda\) = 166.67 mm at 1800 MHz.