## 2 Models
### On-Lattice Monte Carlo Agent-Based Model  
In order to investigate their predictive capabilities, the classical mathematical models described in the previous section have been fitted to data obtained with an on-lattice ABM recently developed and tested [53], which has been used as a reference. In this ABM, cells are situated at the vertices of a cubic grid and are organized in layers forming regions characterized by one of the three possible states considered: proliferative (\(k=\mathrm{p}\)), quiescent or hypoxic (\(k=\mathrm{h}\)), or necrotic (\(k=\mathrm{n}\)). Proliferative cells may divide according to a probability binomial distribution, \(p_{\mathrm{div}}=p_{\mathrm{max}}\,g(d;d_{0},\kappa)\), where the nutrient gradient inside the MTS, \(g\), depends on the depth of the cell inside the MTS, \(d\), and the two parameters \(d_{0}\) and \(\kappa\). After each iteration, which corresponds to a complete cellular cycle, simple evolution rules, linked to the distance of cells to nutrients and \(\mathrm{O}_{2}\), are used to simulate how these cells change from one state to another, updating their status in the aggregate. Exfoliation of the peripheral cells is also considered according to a binomial distribution with probability \(p_{\mathrm{exfol}}\). Region thicknesses are calculated as \(T_{k}=T\,\bar{n}_{k}\), where \(T\) is a scaling factor that takes into account the cell size and \(\bar{n}_{k}\) is the average of the cell number in each of the three possible states, in the three orthogonal directions. With these thicknesses, MTS volumes are calculated as:  
\[V\;=\;\frac{4\,\pi}{3}\left(\sum_{k=\text{p,h,n}}\frac{T_{\text{k}}}{f_{\text{k}}^ {1/3}}\right)^{3}\,. \tag{6}\]  
Here, the \(f_{\text{k}}\) are packing factors that take into account the intercellular space and the cell compression in each MTS region. The feasibility of the model was tested by comparing simulated growth curves with the experimental data measured for an MTS sample of the MCF-7 cell line. The parameters \(p_{\text{max}}=0.37\), \(p_{\text{exfol}}=0.01\), and \(T=15\,\upmu\text{m}\), as well as the number of proliferative and hypoxic layers, \(L_{\text{p}}=6\) and \(L_{\text{h}}=3\), were chosen from the experiment [22; 26; 60]. The values of the packing factors, \(f_{\text{p}}=0.9\), \(f_{\text{h}}=1.0\), and \(f_{\text{n}}=1.1\) were selected according to experimental evidence [61]. Finally, \(d_{0}=6\), \(\kappa=0.8\), were fine-tuned to reproduce the experimental results obtained in [53] for the MCF-7 line. The shapes of the experimental and simulated MTS are very similar, with the necrotic nucleus and the hypoxic and proliferating cell regions well described. The ABM also reproduces the linear relation experimentally found between the radius of the necrotic nucleus and that of the whole MTS: both the slope (\(b_{\text{exp}}=1.078\pm 0.096\) and \(b_{\text{sim}}=1.006\pm 0.002\)) and the independent term (\(a_{\text{exp}}=-151\pm 44\,\upmu\text{m}\) and \(a_{\text{sim}}=-142.5\pm 1.6\,\upmu\text{m}\)) of the respective linear regressions coincide within the uncertainties (given with a coverage factor 2). Furthermore, the dependence of the number of hypoxic plus proliferative cells with the total volume of the MTS shows the same behavior as the experimental data [60], agreeing within uncertainties with them (see reference [53]). Two samples with 1000 virtual MTS each have been generated with the ABM. These are the data to which the mathematical models have been fitted in order to test their predictive capabilities. In sample #1, the model parameters are left fixed at the values indicated above for all simulated tumors. In this way, only the variability linked to the Monte Carlo randomness of the different processes involved in the simulation has been taken into account. The variability observed in the MTS volumes measured in the experiment performed in [53] is within the statistical variability found in this sample #1 of simulated MTS.