## 2 Materials and Methods
### Mathematical Model  
In the present section, two different IPP-ZA pharmacokinetic models (PKM), Model 1 and Model 2, are described. The first model (Model 1) is an ordinary differential equation (ODE) system with 3 equations and 11 parameters; the second model (Model 2) is instead a differential algebraic equation (DAE) system with 4 differential equations, 1 algebraic equation and 13 parameters. Both models aim to describe the two different experimental situations (continuous and pulse experiments) with the same ZA kinetics (Equations (1) and (2)) but different formulations for IPP dynamics. For greater clarity, the biological variables considered (Table 1) will be indicated in the mathematical formulation as follows: \(Y\) (uM) is the zoledronate concentration in the medium; \(Z\) (pmol/mg prot) is the zoledronic acid (ZA) intracellular concentration; \(G\) (pmol/mg prot) is the antigen (IPP) concentration; \(B\) ((%)) is the percentage of bound enzyme and \(U\left((\%)\right)\) is the percentage of unbound enzyme. Each model variable is represented with a compartment (Figure 1), with transfer rates from one compartment to another being indicated with \(k_{ij}\), where \(j\) represents the compartment of origin and \(i\) represents the arrival compartment. Zoledronate kinetics are described with the same equations (Equations (1) and (2)) in both models (Model 1 and Model 2). The supernatant (medium) ZA concentration \(Y\) [uM] variation over time is:  
\[\frac{dY(t)}{dt}=\ -(k_{ZY}+k_{XY})\ Y(t)\ -\ \chi_{pulse}\ \delta(t-t^{*})[Y(t)-Y^{*+}]\ \ +\ \frac{k_{YZ}}{\rho_{ZY}}\ Z(t),\quad Y(t_{0})\ =\ 25\ [\upmu\text{M}] \tag{1}\]  
where \(k_{ZY}\) (/h) is the ZA transfer rate from medium to cells; \(k_{XY}\) (/h) is the medium ZA loss (degradation) rate; \(k_{YZ}\) (/h) is the transfer rate from cell ZA to medium ZA; and \(\rho_{ZY}\) (pmol/mgprot/uM) is the parameter converting uM to pmol/mgprot. \(\chi_{pulse}\) is an indicator variable that distinguishes the type of experiment:  
\[\chi_{pulse}=\left\{ \begin{array}{ll}0&\ continuous\ experiment\\ 1&\ pulse\ experiment.\end{array} \right.\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Variable** & **Unit** & **Meaning** \\ \hline \(Y\) & uM & Pre-zoledronate concentration (zoledronate in medium) \\ \(Z\) & pmol/mg prot & Zoledronate acid concentration (in adhered cells) \\ \(G\) & pmol/mg prot & Isopentenyl pyrophosphate antigen concentration \\ \(B\) & \% & Percentage of bound enzyme \\ \(U\) & \% & Percentage of unbound enzyme \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Biological meanings of the variables used in Model 1 and Model 2. \(Y^{*+}\) (\(\mu\)M) is the ZA concentration in the medium after the removal occurring at time instant \(t^{*}\) (after one hour). \(Z\) (pmol/mgprot) is the ZA concentration in the cells, and its variation over time is:  
\[\frac{dZ(t)}{dt}=\ \rho_{ZY}\ k_{ZY}\ Y(t)\ \ -\ \ (k_{XZ}+k_{YZ})\ Z(t),\ \ \ Z(t_{0})\ \ =\ 0 \tag{2}\]  
where \(k_{XZ}\) (/h) is the within-cell ZA loss or degradation rate. The pharmacodynamics of ZA, i.e., the effect of ZA on within-cell antigen concentrations (IPP), is formalized differently in the two models. The first model does not consider the formation of a complex made up of ZA + FPPS + IPP (Equation (3)), which is conversely taken into account in the second model in Equations (4)-(6).