# Safety Assessment of Microwave Breast Imaging: Heating Analysis on Digital Breast Phantoms
## 2 Materials and Methods
### Mathematical Modeling  
The EM problem was simulated in Sim4 Life (version 7.2, [https://sim4 life.swiss/](https://sim4 life.swiss/) accessed on 30 September 2024), which implements a finite-difference time-domain solver with absorbing boundary conditions implemented with perfectly matched layers (PML). The anatomical models were discretized in voxels, whose dimensions range from 7 mm in the background to 1 mm in the antenna and breast region. The size of the computational domain is 1000 mm \(\times\) 362 mm \(\times\) 1000 mm, with an additional padding of 300 mm \(\times\) 20 mm \(\times\) 300 mm. This allows for neglecting possible spurious reflection effects in the region of interest that may arise from imperfect absorption conditions on the boundaries. To avoid missing any detail or symmetry, it was necessary to use a finer mesh of 0.4 mm in the power supply, represented by a coaxial cable connected to the horn antenna. In the breast, a mesh with a resolution not larger than 1 mm ensured good accuracy in calculating the SAR distribution because the penetration depth at the highest considered frequency is equal to 2.4 mm. Results were scaled to an input power of 1 mW, equal to that used by the MammoWave system. Heat transfer in biological tissues was modeled mathematically by [PERSON] equation [24], which, with the temperature increase \(\theta\) (K) as unknown, is given by  
\[\rho\mathrm{C}\frac{\partial\theta}{\partial t}=\
abla\cdot(k\
abla\theta)\ -h_{b}\theta+Q_{em} \tag{1}\]  
where C is the specific heat capacity (\(\mathrm{J}\cdot\mathrm{kg}^{-1}\cdot\mathrm{K}^{-1}\)), \(\rho\) is the mass density (kg\(\cdot\mathrm{m}^{-3}\)), t is time (s), k is the thermal conductivity (W\(\cdot\mathrm{m}^{-1}\cdot\mathrm{K}^{-1}\)), h\({}_{b}\) is the perfusion coefficient (W\(\cdot\mathrm{m}^{-3}\cdot\mathrm{K}^{-1}\)), and Q\({}_{em}\) is representing the EM heat-source density (W\(\cdot\mathrm{m}^{-3}\)). The temperature increase as unknown allows us to write [PERSON] equation without explicitly including the metabolic heat and the blood temperature, which are highly subject-dependent parameters [24]. EM heat source density is expressed as:  
\[Q_{em}=\frac{\sigma|E|^{2}}{2}=\rho\ \mathrm{SAR} \tag{2}\]  
where \(|E|\) is the peak value of the electric field generated in the considered point of tissue (V\(\cdot\mathrm{m}^{-1}\)), \(\sigma\) the electric conductivity of the tissue (S\(\cdot\mathrm{m}^{-1}\)), and SAR the specific absorption rate of the radiation in the tissue (W\(\cdot\mathrm{kg}^{-1}\)). Because of the short duration of the exposure with respect to the time scale of the thermal conduction in biological tissues, the precautionary hypothesis of adiabatic heating was assumed, namely both the diffusion of heat towards neighbor voxels (described in the equation by the term with thermal conductivity) and the diffusion towards the blood (described by the term with the perfusion coefficient) were neglected. Thus, SAR and temperature increase became linearly related as follows:  
\[\theta=\frac{t_{e}}{C}\mathrm{SAR} \tag{3}\]  
Figure 4: (**a**) Configuration of horn and Vivaldi antennas with respect to the breast model; (**b**) three positions assumed by the Vivaldi antenna in the cases simulated labeled as C, D, and E, including the transmitting antenna. where \(\rm t_{e}\) is the duration of exposure (s). Henceforth, the duration of exposure is set to \(\rm t_{e}=15\) s. Over the entire duration of a single breast diagnostic exam, the active time of the VNA was estimated as the product of 10 ms (VNA ignition time for every signal transmission) times 1601 (number of steps of frequency sampling, 5 MHz each, in the range from 1 GHz to 9 GHz) times 800 (number of combinations of 10 transmitting, 5 positions each one with 2 transmissions, and 80 receiving positions scanned with a two-port VNA) [11]. Therefore, 15 s is an overestimation of the actual duration of exposure.