## 2 Models
### Statistical Methods  
The various mathematical models described in the previous section have been fitted to the volume vs. time data set obtained with the on-lattice Monte Carlo ABM for each one of the simulated MTS. These fits have been carried out by using the Levenberg-Marquardt method [62], in which the model parameters are adequately changed until the minimum of the function  
\[\chi^{2}\,=\,\sum_{i=1}^{N}\frac{\left(V_{i}\,-\,V(t_{i})\right)^{2}}{\sigma_{i }^{2}} \tag{7}\]  
is reached. In the previous equation, \(t_{i}\) indicates the time values at which the MTS volumes \(V_{i}\) are obtained in the simulation, \(\sigma_{i}\) labels the corresponding volume uncertainty provided by the simulation, and \(V(t_{i})\) is the estimation of the MTS volume obtained with the specific mathematical model. The results of the fitting procedure include the fitting parameters that provide the best fit of the model to simulated data and their uncertainties that are calculated following standard procedures [62]. To measure the goodness of the fit, the figure-of-merit \(\chi\) has been used; it is defined as:  
\[\chi\,\equiv\,\frac{\chi^{2}}{\
u}\,. \tag{8}\]  
Here, \(\
u\,=\,N\,-\,n_{\text{par}}\) indicates the number of degrees of freedom in the fit that is calculated as the difference between the number of data in the sample to which the model is fitted, \(N\), and the number of model parameters, \(n_{\text{par}}\). In this case, \(n_{\text{par}}=2\) for the E-model and \(n_{\text{par}}=3\) for the other models. The quantity \(\chi\) represents the \(\chi^{2}\) per degree of freedom, and it is worth noting that good fits can be assumed if \(\chi\,=\,1\)[62]. A first analysis consisted in evaluating the capabilities of the mathematical models to reproduce the behavior of the volume of the simulated MTS in the whole growth period. However, the spheroids showed significant volume changes in the first growth days, and the data were too noisy. To avoid numerical problems in the fitting procedure, the mathematical models were fitted to \(V_{i}\) data in the interval \(i\,=\,[6-60]\). In this case, the figure-of-merit measuring the goodness of the fits carried out was labeled \(\chi_{\text{all}}\). In addition, the predictive capabilities of these models were analyzed. First, the \"retrospective\" predictive power was studied by fitting the models to volumes \(V_{i}\), \(i\,=\,[40-60]\)  
Figure 1: Volume vs. time of growth (in days) corresponding to the two samples generated with the Monte Carlo ABM. The whole regions covered by samples #1 and #2 are indicated in gray and yellow, respectively. Solid lines show some of the simulated MTS growth curves. Figure 2: Best fits of the potential (dashed blue lines), Bertalanffy (dotted-dashed red lines), logistic (solid green lines), and Gompertz ( dotted black lines) models to three simulated MTS belonging to sample #1. Black solid circles show the volumes obtained in the ABM simulation. The MTS #138 (**a**) is the spheroid, among all in sample #1, whose data best fit the potential and Bertalanffy models; the MTS #179 (**b**) is the spheroid whose data best describe the logistic model, and the MTS #248 (**c**) is the spheroid whose data best fit the Gompertz model. The values of \(\chi_{\rm all}\) obtained in each case are also shown. Figure 4 shows the distributions of the \(\chi_{\rm all}\) values obtained in the first analysis carried out. In this case, the mathematical models have been fitted to the whole sets of volume data obtained in the simulations. The fits to data in the sample #1 (those without variability in the ABM parameters) produced the distributions shown with dark colors, while those plotted with light colors (and outlined in black) correspond to the sample #2 in which a variability in the values of the ABM parameters was included in the simulations.