## 3 Results  
### MATLAB Calculations  
Figure 3 depicts the normalized empirical data obtained via the standard 9C model for twelve patients and the theoretical estimates from the MATLAB program. The former are illustrated by dots with error bars, and the optimal estimates are plotted with six continuous curves. In contrast, the numeric calculation was finalized in each specific case to ensure that only a minimal ATtot was reached, and then each curve was plotted. Thus, the lines corresponded to stable and converged solutions derived via multiple iterations of the Gaussian elimination according to the inverse problem algorithm [14]. Tables 2 and 3 summarize the derived biological half-lives and branching ratios for the nine first-order simultaneous differential Equations (1)-(9) adopted in this work. In addition, calculations via the 4C model according to the ICRP-56 report and the simplified 9C model were integrated for comparison. As shown in Tables 2 and 3, no significant discrepancies between the standard and simplified 9C models were observed. Thus, it was acceptable to ignore the feed paths from body fluid to oral, i31, and kidney to the whole body, i87, which simplified the computation. Figure 2: The obtained results from (**A**) cases 2 and (**B**) case 8 with the zero initial values for kidney and bladder; the outcome differs from the experimental data from in vivo gamma camera scanning for comparison. The six plots were consecutively downloaded from the facility after the gamma scanning after 1, 4, 24, 48, 72, and 168 elapsed hours. The marked ROIs were assigned as St (stomach), WB (body fluid + whole body), Th (thyroid), BI (bladder), and BKG (background) to derive the count \(\times\) pixel\({}^{-1}\)\(\times\) s\({}^{-1}\) for further analysis. Figure 3: The normalized empirical data for patients and theoretical estimates via MATLAB. The former are illustrated by dots with error bars, and the optimal estimates are plotted with three continuous lines. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Case No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8 *** & **9** & **10** & **11** & **12** & **Average** \\ \hline
**4C model** & & & & & & & & & & & & & & & \\ Stomach & 0.5 & 0.3 & 0.3 & 0.33 & 0.35 & 1 & 0.45 & 0.25 & 0.7 & 1.3 & 0.8 & 0.5 & 0.57 \(\pm\) 0.33 \\ BF\&WB & 15 & 14 & 10 & 11 & 12 & 15 & 13 & 10 & 12.5 & 11 & 12 & 12 & **12.29 \(\pm\) 1.71** \\ Thyroid & 35 & 50 & 40 & 40 & 50 & 40 & 45 & 2 & 40 & 45 & 15 & 42 & **37.00 \(\pm\) 14.27** \\ \hline
**Standard 9C** & & & & & & & & & & & & & \\ Oral & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.4 & 0.1 & 0.4 & 0.1 & 0.1 & 0.1 & 0.15 \(\pm\) 0.12 \\ Stomach & 0.8 & 0.5 & 0.3 & 0.5 & 0.7 & 1 & 0.35 & 0.25 & 0.7 & 1.3 & 0.8 & 0.35 & 0.63 \(\pm\) 0.32 \\ BF\&WB & 17 & 30 & 17 & 12 & 12 & 20 & 13.5 & 8 & 12.5 & 13 & 12 & 13.5 & **15.04 \(\pm\) 5.63** \\ Liver & 7 & 7 & 2.5 & 8 & 7.2 & 7 & 5 & 4 & 4 & 3.5 & 4.6 & 5 & 5.40 \(\pm\) 1.77 \\ Thyroid & 32 & 25 & 35 & 50 & 48 & 40 & 50 & 2 & 35 & 50 & 15 & 50 & **36.00 \(\pm\) 15.61** \\ Kidney & 3 & 2.8 & 3.5 & 6 & 1.5 & 1.5 & 4 & 1.5 & 5.5 & 6.5 & 2.5 & 4 & 3.53 \(\pm\) 1.74 \\ Bladder & 5 & 2.5 & 1.3 & 2 & 2 & 1 & 3 & 1 & 3 & 5 & 3 & 3 & 2.65 \(\pm\) 1.34 \\ \hline
**Simplified 9C** & & & & & & & & & & & & & \\
**model** & & & & & & & & & & & & & \\ Oral & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.4 & 0.1 & 0.4 & 0.1 & 0.1 & 0.1 & 0.15 \(\pm\) 0.12 \\ Stomach & 1.12 & 0.5 & 0.3 & 0.5 & 0.7 & 1 & 0.35 & 0.25 & 0.7 & 1.3 & 0.8 & 0.5 & 0.67 \(\pm\) 0.33 \\ BF\&WB & 20 & 30 & 15 & 12 & 12 & 20 & 13.5 & 8 & 12.5 & 11 & 12 & 12 & **14.83 \(\pm\) 5.91** \\ Liver & 8 & 7 & 2.5 & 8 & 7.2 & 7 & 7 & 4 & 4 & 3.5 & 4.6 & 4.8 & 5.63 \(\pm\) 1.92 \\ Thyroid & 35 & 25 & 22 & 50 & 48 & 40 & 50 & 2 & 35 & 48 & 15 & 42 & **34.33 \(\pm\) 15.42** \\ Kidney & 10 & 2.8 & 3 & 6 & 1.5 & 1.5 & 1.5 & 1.5 & 5.5 & 6.5 & 2.5 & 7 & 4.11 \(\pm\) 2.81 \\ Bladder & 5 & 2.5 & 1.3 & 2 & 2 & 1 & 1 & 1 & 3 & 5 & 3 & 5.5 & 2.69 \(\pm\) 1.66 \\ \hline \hline \end{tabular}
\end{table}