## 2 Materials and Methods
### Simulation Study

Table 1: Composition and density for the components of the phantom samples (used for both simulation and experimental studies). The manufacturer for the experimental phantom was QRM GmbH, Germany [51]. Figure 2: Schematic of the simulated phantom in the XY plane (top row) and in the XZ plane (bottom row) consisting of (**a**) epoxy and HA400, (**b**) epoxy and HA800, and (**c**) epoxy and HA1200. The plane orientation is depicted in the top left corner. Figure 3: Workflow for obtaining the separate thickness of the epoxy and HA400 components using a simulated test phantom. The distance along the \(X\)-axis of the plots corresponds to the \(X\)-axis direction in Figure 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Threshold Position (keV)**} & \multicolumn{3}{c}{**Bandwidth (keV)**} \\ \cline{2-4}  & \(\Delta E\) = 3 & \(\Delta E\) = 6 & \(\Delta E\) = 9 \\ \hline \(E_{1}\) & 16 & 16 & 16 \\ \(E_{2}\) & 19 & 22 & 25 \\ \(E_{3}\) & 27 & 24 & 21 \\ \(E_{4}\) & 30 & 30 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters used for the four energies thresholds \(E_{1}\), \(E_{2}\), \(E_{3}\) and \(E_{4}\) to obtain three pairs of different energy bandwidths. The difference between the resulting retrieved thickness, \(s_{retrieved}\), and the actual thickness, \(s_{actual}\), is evaluated using:  
\[\chi^{2}=\frac{\sum_{p=1}^{n}\left(s_{p,actual}-s_{p,retrieved}\right)^{2}}{\sum_{p =1}^{n}\left(s_{p,actual}\right)^{2}} \tag{8}\]  
where \(p\) denotes the pixel number \(p=\)1, 2, \(\ldots\)\(n\). The calculated \(\chi^{2}\) error metric for all data sets is presented in Figure 5 in the absence of noise (a, c, e) and with 1% noise included (b, d, f). The noise was modelled by adding randomly generated values with a standard deviation of 1% about the mean intensity value. Figure 5 shows that in the absence of noise, the smallest bandwidth of 3 keV results in the most effective separation (smallest \(\chi^{2}\) value) of HA from epoxy for all material densities. This is due to the fact that as the bandwidth increases the assumption that Equations (4) and (5) represent quasi-monochromatic intensity images which can be represented by a single effective energy becomes increasingly invalid decreasing the effectiveness of the materials separation. Moreover, the 9 keV bandwidth was found to negatively impact the retrieval owing to the larger bandwidth and subsequent breakdown of the assumption of quasi-monochromaticity as well as the spectral overlap. Figure 5 also indicates that, under the conditions simulated here, the most effective materials separation occurs for the lowest density material (HA400) and epoxy with the highest density material (HA1200) yielding the least effective separation. However, when 1% noise is added (consistent with the typical noise level observed in the experiment) the lowest \(\chi^{2}\) value occurs for the 6 keV bandwidth. This result likely reflects the trade-off between monochromaticity (which is best for the 3 keV bandwidth) and signal-to-noise (SNR), which is highest for the 9 keV bandwidth. Figure 4: Example simulation result comparing the retrieved thickness from the matrix approach compared to the ‘actual’ value based on the known properties of the simulated sample for the (**a**) HA400 insert and (**b**) surrounding epoxy, using a 6 keV bandwidth, with (**c,d**) show their corresponding difference (actual-retrieved) in (**a,b**), respectively.