# Monte-Carlo-Based Estimation of the X-ray Energy Spectrum for CT Artifact Reduction
## 3 Results and Discussion  
The normalized X-ray energy spectra obtained both from Monte Carlo simulation and MLEM methods for different tube voltages (in the range of 60-200 kV) are shown in Figure 8, from which good agreement between both methods can be observed. Figure 6: X-ray energy spectrum of a tungsten X-ray tube at 100 kV reported by the Institute of Physics and Engineering in Medicine (IPEM) (report no. 78). Figure 7: Normalized energy absorption response function of GadOx (1–230 keV). In order to better evaluate the agreement between the obtained results, the root mean square error (RMSE) between the spectra obtained from Monte Carlo simulation and MLEM method was calculated for each tube voltage using the following equation:  
\[RMSE=\ \left[\frac{\sum_{i=1}^{N}\left(S_{i}(Monte\ Carlo)-S_{i}(MLEM)\right)^{2}}{N} \right]^{0.5} \tag{7}\]  
where \(S_{i}(Monte\ Carlo)\) and \(S_{i}(MLEM)\) represent the intensity in each 1 keV energy bin in the spectrum obtained by Monte Carlo simulation and the MLEM method, respectively. N represents the total number of 1 keV energy bins in each spectrum, for example, N = 160 represents the spectrum obtained for a tube voltage of 160 kV. The RMSE versus tube voltage is provided in Figure 9. As can be observed from Figure 9, by increasing the tube voltage, the RMSE decreases up to a tube voltage of 100 kV and remains almost constant up to 160 kV. After 160 kV, the RMSE increases again. The reason for the higher RMSE for lower tube voltages is that the majority of low energy photons are absorbed in the initial, thicker steps (mainly steps no. 1-3), and only a small portion of high energy photons passes through the remaining steps. A lower number of transmitted photons leads to a lower signal-to-noise ratio for pixel values belonging to thick steps, which consequently causes a higher error in MLEM calculations. The reason the RMSE for higher tube voltages is relatively high is that when the tube voltage is increased, the ratio of high energy photons is increased as well. These high energy photons, in turn, cause lower contrast in pixel values belonging to the thin steps (especially for step no. 11 and step no. 12), which consequently leads to a higher RMSE for MLEM calculations. Figure 8: \(Cont.\)  
Figure 8: Comparison of obtained X-ray energy spectra from both Monte Carlo and Maximum Likelihood Expectation Maximization (MLEM) methods for tube voltages in the range of (**a**) 60 kV, (**b**) 80 kV, (**c**) 100 kV, (**d**) 120 kV, (**e**) 140 kV, (**f**) 160 kV, (**g**) 180 kV and (**h**) 200 kV. Figure 9: The results of root mean square error (RMSE) between the spectra obtained from Monte Carlo simulation and the MLEM method versus tube voltage of the X-ray tube. As described in Section 2.1.2, after validating the simulation geometry of the FleXCT system, the required X-ray energy spectrum (in front of the X-ray tube and independent of the flat panel detector response) for beam hardening and scattering CT artifact reduction applications was calculated. As an example, the energy spectrum of exiting X-ray radiations from the FleXCT X-ray tube was calculated for tube voltage of 230 kV for two different cases: (1) no additional filter, (2) 1 mm copper filter (the results are shown in Figure 10). The intensity values in Y-axis of Figure 10 are per one particle source (in fact, MCNP code calculates scores per one source particle). A copper filter is typically used to pre-harden the X-ray emission spectrum to prevent cupping effects in CT scans. By comparing both spectra, we see that a 1 mm copper filter completely absorbs photons below 40 keV, whereas the intensity of photons in the energy range of 40-130 keV is heavily reduced. The emission spectra above 130 keV remain almost unchanged for both cases.