## 4 Discussion  
### Comparative Analysis against Other Models  
Several researchers used the two-compartmental (2C) model, which contained only thyroidal and extrathyroid (remainder) compartments, for solving the radioactive I-131 concentration evolution in the human body and predicting its residual values [7,15]. Alternatively, [PERSON] adopted the three-compartmental model (thyroid, transfer, and other organs) to predict the same mechanism in the human body [16], although the predicted results were quite inaccurate. Therefore, other researchers have extended the [PERSON] model to four compartments (stomach, thyroid body fluid, and the whole body) with one excretion compartment to maintain the close realm of the simultaneous differential equations and easier convergence of the numeric results [5,9,17]. In this work, the number of compartments was increased to nine, the additional five being oral, liver, remainder, kidney, and bladder. The radioactive I-131 could be easily detected in the oral and liver compartments, whereas the kidney and bladder were the major excreting organs reducing the liquid I-131 concentration. Thus, it was essential to adopt the revised 9C model to interpret the real metabolic mechanism of radioactive I-131 in the human body. The numeric analysis was performed via the self-developed program run in MATLAB. The low AT\({}_{\text{tot}}\) of 0.048 \(\pm\) 0.019 proved the feasibility of the proposed approach in finding the optimal solution (ref. Table 4).  
A more detailed investigation of the I-131 concentration evolution in each compartment revealed that oral and ST compartments exhibited simple decay with their effective half-lives (1/T\({}_{\text{eff}}\) = 1/T\({}_{\text{bio}}\) + 1/T\({}_{\text{rad}}\)). The biological half-lives of oral and ST were 0.15 \(\pm\) 0.12 and 0.63 \(\pm\) 0.32, h (ref. Table 2). This implied that the contributions of both the oral and the ST compartments were negligible in the group. In contrast, the biological half-lives of liver, kidney, and bladder, 5.40 \(\pm\) 1.77, 3.55 \(\pm\) 1.74, and 2.65 \(\pm\) 1.34 h, shifted more toward transient equilibrium in the radionuclides' concentration change with BF + WB, whereas the BF + WB in reality had a biological half-life 15.04 \(\pm\) 5.63 h (ref. Figure 3). Noteworthy is that the larger the biological half-life of the receiving compartment (i.e., liver, kidney, or bladder in this work) the longer it takes to reach its maximal concentration. However, each compartment possessed its unique time-dependent I-131 concentration curve, which could not be simply interpreted as transient equilibrium in radionuclide chain decay. The real metabolic mechanism was quite complicated and could be assessed by solving simultaneous differential equations (ref. Equations (1)-(9)). It is executable to further explore the excretion from the kidney or bladder by collecting time-dependent concentration of I-131 in urine since the quantified biological half-life and branching ratios are instrumental in the correlated analysis. Table 5 shows further the biological half-lives in the major compartments, whole body (BF + WB), and thyroid calculated in this and previous studies to solidify the comparison among different approaches. The results obtained via different practical methods and theoretical simulations are quite controversial. Thus, the internal doses for thyroidectomy patients have to be specifically considered for accurate estimation in clinical surveys.