## 4 Discussion and Conclusions  
In this work, we developed two mathematical models (Model 1 and Model 2) to study how ZA can induce IPP accumulation in tumor cells and predict how to modulate the immune system to kill tumor cells. Increased levels of PAgs accumulate in metabolically stressed, transformed, and infected cells, which are thus sensed by V\(\gamma\)9V\(\delta\)2 T cells. ZA can enhance V\(\gamma\)9V\(\delta\)2 T cell's anti-tumor functions thanks to the increasing IPP in target cells. In this regard, butylrophilin 3A1 (BTN3A1), expressed by both immune cells and tumor cells, is considered TCR-mediated sensing of phosphoAntigens (IPP). Indeed, IPP binding the intracellular domain of the protein determines a conformational change useful to be recognized by V\(\gamma\)9V\(\delta\)2 TCR. Thus, BTN3A1, together with BTN2A1, are critical determinants in the recognition of human tumors by V\(\gamma\)9V\(\delta\)2 T cells and trigger a cytotoxic activity. Finally, IPP is measured in the medium of cancer cell lines as a released product of treated target cells [47], and it has been proposed to bind the extracellular immunoglobulin-like domain of BTN3A1 [48] through mechanisms that are not completely clear. The generation of anti-BTN3A 20.1 mAb strongly boosts V\(\gamma\)9V\(\delta\)2 T cells' cytolytic function [49], and the newly acquired information about activation mechanisms of butylrophilin opens new perspectives in T cell-based immunotherapies. To this end, we show two models formulated by time-course measurements of ZA uptake and IPP accumulation into tumor cells from the observations of two papers, published independently and in different years [39; 40]. From a mathematical point of view, both models are simple. Although Model 1 can follow the trend of the experimental data, it neglects the biologically important interaction ZA + Enzyme + IPP, which is instead considered in Model 2 without a major increase in complexity. The inhibition of FPPS by ZA results in the accumulation of the early metabolites of the mevalonate pathway, IPP [21] and its isomer DMAPP [22]. IPP becomes conjugated to adenosine-5-monophosphate (AMP) to form a novel ATP analog ApppI, while DMAPP appears to lead to the formation of ApppD. The detection of these molecules is not contemplated in the present models. There are no observations available on their concentration in time-course assays, but in the future,it would be relatively straightforward to expand these models, filling this gap. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter [Unit]** & **SD** & **CV** & **LLC** & **ULC** \\ \hline \(k_{ZY}\) [/h] & \(0.001\) & \(19.30\) & \(0.004\) & \(0.0092\) \\ \hline \(\rho_{ZY}\) [pmol/mgprot/\(\mu\)M] & \(4.922\) & \(19.30\) & \(15.81\) & \(35.11\) \\ \hline \(k_{XG}^{M\alpha 1}\) [/h] & \(0.064\) & \(62.80\) & \(-0.023\) & \(0.226\) \\ \hline \(\lambda_{GZ}\) [/pmol/mgprot] & \(0.258\) & \(146.3\) & \(-0.330\) & \(0.683\) \\ \hline \(k_{XG}^{M\alpha 2}\) [/h] & \(0.051\) & \(63.10\) & \(-0.019\) & \(0.181\) \\ \hline \(k_{XGU}\) [/h/\(\%\)] & \(0.525\) & \(1029\) & \(-0.977\) & \(1.080\) \\ \hline \(k_{RULZ}\) [/h/(pmol/mgprot)] & \(0.051\) & \(459.7\) & \(-0.791\) & \(0.988\) \\ \hline \end{tabular}
\end{table} 
Table 5: Summary of the standard deviation (SD), percent coefficient of variation (CV), and lower and upper confidence limits for the zoledronate kinetic model and IPP antigen dynamics sub-model (Model 1 and Model 2) parameters. Because the parameter \(k_{XG}\) is present in both models, the notation \(k_{XG}^{M\alpha 1}\) and \(k_{XG}^{M\alpha 2}\) was used in the Table to differentiate them. Focusing the attention on ZA and IPP, the proposed models reproduce very well observations already established in several studies to describe the uptake of ZA and the following accumulation of IPP, as summarized below:  
1. ZA is taken up by macrophages and osteoclast cells in endocytic vacuoles; acidification of vacuoles seems to be required to release it into the cytoplasm. 2.