## 3 Results
### Model 1, Continuous Experiment  
In the case of the continuous experiment, only the data by [PERSON] et al. [39] are available. The cells were treated with 25 uM ZA initially and left untouched for 48 h. The intracellular concentration of ZA was determined by comparing the radioactivity of the medium, washes and cell extracts relative to the amount of protein in the cell extract. In this case, the amount of IPP increased gradually with the increasing concentration of ZA \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**Kinetic Model**} \\ \cline{3-4}  & & **Continuous** & **Pulse** \\ \hline \(k_{ZY}\) & /h & 0.007 & 0.607 \\ \hline \(k_{XY}\) & /h & 0.004 & 0.029 \\ \hline \(\gamma^{*+}\) & \(\mu\)M & — & 0 \\ \hline \(k_{YZ}\) & /h & 0.001 & 18.06 \\ \hline \(\rho_{ZY}\) & pmol/mgprot/\(\mu\)M & 25.46 & 14.91 \\ \hline \(k_{XZ}\) & /h & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Zoledronate kinetic model parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**Model 2**} \\ \cline{3-4}  & & **Continuous** & **Pulse** \\ \hline \(k_{G}\) & pmol/mgprot/h & 141.7 & 22.18 \\ \hline \(k_{XG}\) & /h & 0.081 & 0 \\ \hline \(k_{XClU}\) & /h/\% & 0.051 & 0.224 \\ \hline \(k_{XG}^{QQ}\) & /h & 5.181 & 22.44 \\ \hline \(G_{0}\) & pmol/mgprot & 27.32 & 0.988 \\ \hline \(\lambda_{UBG}\) & /(pmol/mgprot) & 1.806 & 0.017 \\ \hline \(k_{BULZ}\) & /h/(pmol/mgprot) & 0.099 & 0.150 \\ \hline \(k_{UBG}\) & /h & 0.168 & 0.057 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model 2: IPP antigen dynamics sub-model parameters. in the cells over time. Therefore, the highest intracellular concentrations were achieved at 48 h of ZA exposure and corresponded to 1624 pmol/mgprot of IPP in cells. Notice that the indicator variable \(\chi_{pulse}=0\) (Equation (1)) in this experimental situation, as mentioned in the MM section above. Figure 2 shows the time course of the ZA (panel (a)) and IPP (panel (b)) starting from \(t=0\) to \(t=48\) h. The continuous light blue curve represents the predicted time courses as derived by Model 1 Equations (2) and (3), whereas red asterisks indicate observations [39]. The model fits the observed data very well. In response to the administration of ZA, there is an increase in the level of IPP, described by a sigmoid-like curve with a slow initial increase and near steady-state at the end. The parameter values estimated from the continuous experiment are reported in Tables 2 and 4. In Figure 2(panel (a) and panel (b)), it is possible to note that between 24 h and 48 h there is a variation of intracellular ZA concentration of about 80 pmol/mgprot. There is a more modest increase between 12 h and 24 h (about 30 pmol/mgprot), contrary to what we observe for IPP levels, which increase more between 12 h and 24 h than between 24 h and 48 h.  
Based on the evidence of in vivo experimental data [36], it seems reasonable to assume that a value of transfer rate from medium to tumor cells of ZA (\(k_{ZY}=0.007\) /h) eight-fold larger than the transfer from the cell to the medium (\(k_{YZ}=0.001\) /h) is due to a preferential transport of the molecule through fluid-phase endocytosis into the cytosol [20]. After 24 h, the ZA concentration increased by 33 times compared to its first hour value (\(Z(t_{24})=33Z(t_{1})\), with \(t\) expressed in hours), and at 48 h, the ZA concentration was twice the value at 24 h (\(Z(t_{48})=2Z(t_{24})\), with \(t\) expressed in hours). This behavior is described by the experimental data and correctly reproduced by Model 1.