## 2 Models
### Statistical Methods
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**All Data**} & \multicolumn{2}{c}{**Retrospective**} & \multicolumn{2}{c}{**Prospective**} \\ \cline{3-7}  & & \(\chi_{\text{all}}\) & \(\chi_{\text{ret}}^{\text{fit}}\) & \(\chi_{\text{ret}}^{\text{pr}}\) & \(\chi_{\text{pro}}^{\text{fit}}\) & \(\chi_{\text{pro}}^{\text{pr}}\) \\ \hline \multirow{2}{*}{\#1} & G & 3.23 & 1.09 & 29.08 & 1.04 & 50.49 \\  & & [1.89:6.29] & [0.51:1.82] & [5.47:68.67] & [0.49:1.87] & [3.78:126.70] \\ \cline{2-7}  & L & 171.49 & 1.77 & 478.86 & 3.58 & 1050.98 \\  & & [139.06:258.91] & [0.81:2.95] & [360.00:613.40] & [1.97:5.51] & [956.80:1140.00] \\ \cline{2-7}  & P & 22.39 & 2.56 & 64.59 & 2.35 & 340.05 \\  & & [17.50:27.41] & [1.34:4.18] & [44.49:87.20] & [1.12:3.91] & [212.30:501.50] \\ \cline{2-7}  & B & 30.94 & 1.08 & 22.99 & 1.40 & 304.76 \\  & & [21.74:46.02] & [0.50:1.78] & [4.74:48.63] & [0.64:2.52] & [92.49:694.50] \\ \hline \multirow{2}{*}{\#2} & G & 3.79 & 1.11 & 29.76 & 1.11 & 58.79 \\  & & [1.29:9.83] & [0.45:2.19] & [1.46:94.18] & [0.44:2.12] & [1.35:183.90] \\ \cline{2-7}  & L & 179.48 & 1.80 & 480.19 & 3.75 & 1061.00 \\  & & [56.72:378.29] & [0.74:3.55] & [147.10:959.40] & [1.14:8.28] & [467.80:1833.00] \\ \cline{2-7}  & P & 24.17 & 2.60 & 65.14 & 2.65 & 340.71 \\  & & [6.22:57.37] & [1.12:4.85] & [15.11:139.60] & [0.71:6.36] & [54.31:835.20] \\ \cline{2-7}  & B & 31.02 & 1.08 & 23.37 & 1.44 & 315.34 \\  & & [11.46:64.49] & [0.46:2.17] & [4.18:52.42] & [0.57:2.76] & [67.76:836.80] \\ \hline \end{tabular}
\end{table} 
Table 1: Average values and 95% CI of the \(\chi\) distributions obtained in the various fits carried out in the present study. Figure 5: Distribution histograms of the values of \(\chi_{\rm ret}\) obtained in the retrospective fits of the four mathematical models considered to the simulated MTS in samples #1 (dark colors) and #2 (light colors). Left panels correspond to the fits to data in the time interval [40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60] day and right panels to the retrospective fits to data in the time interval [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39] day. The results obtained for the Gompertz (**a**,**b**), logistic (**c**,**d**), potential (**e**,**f**) and Bertalanffy (**g**,**h**) models are shown. Solid circles and uncertainty bars correspond to sample #1, whereas solid curves and light colored areas stand for sample #2. The following points must be noted:  
1. Similar results were found for G- and B-models that showed average values tending to 1 with increasing \(t_{k}\). The former appeared to be slightly robust in this sense, showing a less noising behavior for \(t_{k}<20\) day. 2. For L- and P-models, a non-negligible dependence on the data region fitted occurred. It is worth pointing out the fluctuating behavior with \(t_{k}\) observed in both cases. 3. This also results in the differences between the two samples studied being larger for L- and P-models than for G- and B-models. The analysis that has been done may help to understand how tumors, particularly small tumors that could remain as surgical residues or micro-metastases, evolve before or during radiotherapy, or to investigate the effects provoked by the delays and interruptions that often occur throughout the treatment. In the case of delays in starting radiotherapy, the prospective predictive power of the models is of great interest. For interruptions, both the capacity for retrospection, if the trend of the growth is preserved, and prospection are relevant. These problems can be analyzed with the ABM model and, afterwards, it would be possible to see if the classical models are able to describe what happens in these cases. The E-model deserves a particular comment. As it has been shown, this model can reproduce the growth curves only when the data considered for the fit are limited to a short time period. On the other hand, its predictive capabilities are very poor.