## 2 Materials and Method
### Experimental Verification of Simulations  
This experimental verification was aimed at detecting possible errors in the simulations, i.e., errors in defining the materials, shapes, and sizes of the simulated objects or in the original spectrum of X-rays and other errors at various stages of the simulations. The simulations were verified through a comparison of thicknesses of the half-value layer _HVL_ obtained from the simulations and experiments. Using a Siemens Novation mammography  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Thickness of Breast (mm)** & **Glandularity (\%)** \\ \hline
32 & 67 \\
45 & 41 \\
53 & 29 \\
60 & 20 \\
75 & 9 \\
90 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Typical values of glandularity corresponding to breasts of a given thickness after compression according to the _European Guidelines for Quality Assurance in Breast Cancer Screening and Diagnosis (4 th Edition)_[33].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & \multicolumn{2}{c}{**Adipose Tissue**} & **Glandular Tissue** & **Cancerous Tissue** \\  & \(\rho_{a}\) = 0.93 g/cm\({}^{3}\) & \(\rho_{g}\) = 1.04 g/cm\({}^{3}\) & \(\rho_{t}\) = 1.058 g/cm\({}^{3}\) \\ \hline H & 0.112 & 0.102 & 0.102 \\ C & 0.619 & 0.184 & 0.184 \\ N & 0.017 & 0.032  
machine (Siemens, Munich, Germany) (serial number 2654, mobile installation at the Maria Sklodowska-Curie National Oncology Institute in Gliwice), equipped with a two-path X-ray tube (Mo and W anode), the thickness of the half-value layer _HVL_ was measured. The experiment was carried out for 3 voltages (24 kV, 28 kV, and 32 kV) and 3 anode-filter combinations (Mo-Mo, Mo-Rh, and W-Rh). Measurement in the narrow beam conditions was made with a Marcus ionization chamber with a volume of 0.055 cm\({}^{3}\) (PTW 23343, serial number 2810, PTW, Freiburg, Germany) and a PTW Unidos electrometer (type 10001, serial number 10612, PTW, Freiburg, Germany). The radiation emitted from the X-ray tube was limited by a system of two lead collimators, each 3 mm thick. The first one was 33 cm from the lamp's focal point, while the second one was 43 cm away. Round holes were placed in collimators with a diameter of 3 cm. Aluminum filters (Gammex 115H, serial number 800051-1809, purity 0.999) were placed between the collimators. An ionization chamber was located below the collimators, 53 cm from the focus. The geometry used in measurements is schematically presented in Figure 5. The thickness of the half-value layer was determined by the logarithmic interpolation method [30] using the following formula:  
\[HVL=\frac{X_{1}\Big{(}\frac{2R_{2}}{R_{0}}\Big{)}-X_{2}\Big{(}\frac{2R_{1}}{R_ {0}}\Big{)}}{\ln\Big{(}\frac{R_{2}}{R_{1}}\Big{)}}, \tag{8}\]  
where \(X_{1}\) and \(X_{2}\) indicate the thicknesses of the filters (\(X_{1}>X_{2}\)); \(R_{0}\) is the electrometer reading for exposures without Al filters; \(R_{1}\) and \(R_{2}\) are the electrometer readings for exposures with an additional filter with a thickness of \(X_{1}\) and \(X_{2}\), respectively.  
Figure 5: Scheme of the system for the _HVL_ measurement. An analogous system was simulated to verify the simulation through comparison of the _HVL_ values measured and those determined in simulations.