## 3 Data-Constrained Modelling for the Samples
### Calculation Results of Data-Constrained Modelling  
Table 5 shows the volume fractions obtained by DCM and error analysis results of each composition for samples 1-6. The value differences in Table 5 are the differences between the volume fractions of the compositions computed by DCM and the corresponding actual volume fractions (Table 2). The relative errors in Table 5 are the ratios between the absolute values of the difference values and the actual values, as shown in Table 2, of each \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sample Number**} & \multirow{2}{*}{**Equivalent Energy (keV)**} & \multicolumn{3}{c}{\(\text{n}^{(\text{m})}\) **(1/cm)**} & \multirow{2}{*}{**S**(\(\text{n}^{\text{th}}\))} \\ \cline{3-3} \cline{5-6}  & & **Pore** & & \(\text{SiO}_{\text{2}}\) & \multicolumn{1}{c}{**Fe**} \\ \hline
1 & 33.630 & 0.0000 & 1.5170 & — & 0.0000000 \\
2 & 79.510 & 0.0000 & 0.4534 & 4.6860 & 0.0001920 \\
3 & 75.600 & 0.0000 & 0.4716 & 5.2520 & 0.0009310 \\
4 & 80.145 & 0.0000 & 0.4506 & 4.6040 & 0.0021175 \\
5 & 82.030 & 0.0000 & 0.4429 & 4.3750 & 0.0017580 \\
6 & 80.060 & 0.0000 & 0.4510 & 4.6150 & 0.0013900 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computing parameters of DCM for each sample. \(\text{S}^{(\text{1})}\) represents the self-energy parameter of the silica composition. composition. From Table 5, it can be seen that for sample 1, DCM accurately obtained the volume fractions of silica and pores in the sample with the default value of the self-energy parameters as 0. The value difference of the computed volume fractions was only 0.6% compared to the actual values. For samples 2-6, adjustments were made to the self-energy parameter of SiO\({}_{2}\) in order to align the calculated volume fractions of SiO\({}_{2}\) as closely as possible with the actual values. In these cases, the calculated volume fractions of pores and iron are also reasonably consistent with the actual results. Comparatively, samples 3 and 4 exhibit a higher calculation accuracy than other samples containing iron. The reason for this is the low iron content in sample 2 (only 0.9%), where even small calculation errors can result in large relative errors. For samples 5 and 6, with actual iron volume fractions of 10.24% and 16.90%, respectively, the two samples exhibit more pronounced partial volume effects and metal artifacts on their CT slices due to the significantly higher absorption coefficient of iron compared to silica. This led to increased calculation errors in DCM. As shown in Figure 6a, most areas are mixed colors of blue and green, indicating that pores and silica mostly exist in the form of partially occupied pixels. In Figure 6b-d, the color of red corresponding to iron composition is less visible, primarily due to its lower volume fractions in these three samples, where its color is masked by others. Figure 6d shows some yellow areas, suggesting the coexistence of iron and pores in those regions. A large number of light red regions are clearly visible in Figure 6e,f, corresponding to the high volume fractions of iron in samples 5 and 6. However, areas displaying as 100% red are relatively scarce, indicating that iron in these samples also occupies voxels partially. Furthermore, an increase in iron particle volume fractions may lead to an increase in X-ray linear absorption coefficients of voxels around them due to metal artifacts and point spread functions, potentially resulting in an overestimation of iron content in the calculations. In Figure 6a-c, some compositions exhibit a ring-shaped distribution form, which is more likely caused by the ring artifacts present in the CT slices of the samples, as shown in Figure 2. Despite our efforts to improve CT imaging quality through methods such as background subtraction and the use of filters, there may still be unavoidable image artifacts present in the CT slices, such as ring artifacts, motion artifacts, and beam hardening artifacts.