## 2 Materials and Methods  
Similarly to our previous work [22], a half-space composed of air at the top and a medium with the dielectric properties of a deflated lung at the bottom was simulated through 3D Finite Difference Time Domain (FDTD) using a recompiled version of gprMax version 3.1.5 [23; 24] which allowed double precision calculations. An FDTD solver was selected because it offers a straightforward implementation of the problem at hand. It also scales almost linearly with the problem size and it offers the possibility to create movies of the wave propagation which are important for understanding and troubleshooting. In particular gprMax was selected because it is open-source, easy to modify (we recompiled it for our purposes) and use, and geared for ground-penetrating problems which are very similar to microwave imaging of targets in biological tissues. The FDTD solves the four Maxwell's equations:  
\[\
abla\times E=-\frac{\partial B}{\partial t} \tag{1}\]  
\[\
abla\times H=J+\frac{\partial D}{\partial t} \tag{2}\]  
\[\
abla\cdot D=\rho_{v} \tag{3}\]  
\[\
abla\cdot B=0 \tag{4}\]  
together with the constitutive relations:  
\[D=\epsilon*E \tag{5}\]  
\[B=v*H \tag{6}\]  
\[J_{c}=\sigma*E \tag{7}\]  
in space on a staggered grid though the use of Yee cells [25] and electric- and magnetic-field quantities are computed at alternating time steps. The Equations (1)-(4) are also known as [PERSON]'s law (Equation (1)), [PERSON]'s circuital law (Equation (2)), [PERSON]'s law (Equation (3)), and the nonexistence of a magnetic monopole (Equation (4)) [26]. They describe how the electric field \(E\) and the magnetic field \(H\) are interconnected through the electric displacement field \(D\), the magnetic flux density \(B\), their time derivative, the volumetric charge density \(\rho_{v}\), and the current density \(J\) which is comprised of the conductive current \(J_{c}\) and the surface current \(J_{s}\). In addition, the Equations (5)-(7) represent the constitutive relations. They link \(D\), \(B\), and \(J_{c}\) to the local properties of the materials which form the computation domain. These properties are the dielectric permittivity \(\epsilon\), the magnetic permeability \(\
u\), which is usually taken as zero for biological materials, and the conductivity \(\sigma\). In general, they are time-dependent, viz. frequency dependent, and Equations (5)-(7) are convolutional relations (note the convolution sign \(*\)). The dielectric permittivity \(\epsilon_{r}\) and the conductivity \(\sigma\) for lung tissue (deflated lung taken from reference [15]) are reported in Figure 1 together with the wavelength and attenuation of the electromagnetic wave calculated for the lung tissue. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Values** \\ \hline UWB frequency & 0.5–5 GHz \\ Cancer depth & 1–6 cm \\ Antenna distribution & \(3\times 3\), \(4\times 4\), and \(5\times 5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters investigated. A schematic of the half-space used for the computations is illustrated in Figure 2 together with one antenna distribution (\(3\times 3\)) on the top of the lung tissue which is displayed in violet. Lung tissue properties were taken from the Tissue Properties Database [15] (considering deflated lung as material reference). Permittivity and conductivity were changed according to the central frequency of the electromagnetic pulse, however, no dispersive properties were actually implemented in the simulation, that is for every simulation the dielectric properties of the materials were taken as frequency independent. The excitation pulse used was the derivative of a 1 V amplitude Gaussian function (gaussiandot in gprMax). Ten central frequencies between 0.5 and 5 GHz were used as excitations and additional simulations were performed at 1 GHz to investigate extra parameters (see Supplementary Information SI).