## 1 Introduction
Upon IPP activation, V\(\gamma\)9V\(\delta\)2 T cells produce and release pro-inflammatory cytokines (such as tumor necrosis factor-alpha (TNF-a), chemokines, interferon-\(\gamma\) (IFN-\(\gamma\))) and, with the addition of IL2, V\(\gamma\)9V\(\delta\)2 proliferates and also acquires cytotoxic functions. As mentioned before, considering the immuno-modulatory function of ZA, extensive evidence from preclinical studies [27] showed that ZA exerts its anticancer actions in different ways: (1) by a direct effect on tumor cells, inhibiting proliferation and inducing apoptosis in vitro (with sensitivity to this effect largely depending on the ability of tumor cells to internalise sufficient amounts of N-BP to inhibit FPPS); (2) indirectly by affecting bone resorption, thereby reducing tumor cell migration to the bone [28]; (3) inducing IPP production by tumor cells and tumor-associated macrophages (TAM) that activate cytotoxic V\(\gamma\)9V\(\delta\)2 T cells against the tumor; (4) as very recently demonstrated [29], by causing the shift of TAM in the M1 phenotype; (5) by inhibiting the adhesion of tumor cells to the extracellular matrix proteins and, thus, by impairing the process of tumor cell invasion and metastasis [30]. Several theoretical models have been developed in recent years to further refine treatment regimens for cancer patients [31]. Typically, cellular response to a drug is often evaluated by a variety of in vitro assays and is generally interpreted using dose-response curves. In these assays, the drug is typically applied to a cell population over a wide range of concentrations and evaluated over time (usually 72 h) to test the effects on cell death (often indirectly). Based on in vitro experiments with ZA, several clinical trials for patients with different types of tumors have been performed [32; 33]. Small changes in experimental timing or pharmaceutical concentrations appear to significantly impact the outcomes [34]. Several methods have been used to quantify ZA and IPP based on radioactive tracing, gas chromatography-mass spectrometry (GC-MS), or by liquid chromatography-tandem MS (LC-MS/ MS) [35]. An in vitro approach could be used to support the formulation of mathematical models, with the main aim of providing quantitative tools describing the pharmacokinetics and pharmacodynamics (PKPD) of the compound under investigation [31]. [PERSON] et al. [36] proposed an in vitro model of primary human tubular kidney cells to mimic the most important physiological characteristics of molecular uptake/transport by the tubular epithelium _in vivo_. At present, the pharmacokinetics of ZA have been assessed on clinical data of patients with bone metastases from a variety of primary cancers. The relationship between the dose and drug safety is also studied [17; 37]. However, while in the studies mentioned above, the temporal relationship between ZA-treatment and its effects has received great attention from different viewpoints, to date, a model describing the PKPD of ZA and IPP is missing. Such a model would be very useful due to recent evidence supporting the generation of optimised protocol to expand efficient V\(\gamma\)9V\(\delta\)2 T cells [38], with enhanced activation and differentiation of human V\(\gamma\)9V\(\delta\)2 T cells upon the stimulation of short-term-expanded \(\gamma\delta\) T cell lines with L ascorbic acid 2 phosphate (pVC). Attempts to find novel approaches to expand V\(\gamma\)9V\(\delta\)2 T cells, approaches that could be transferred to the clinical care of cancer patients, would benefit from the quantitative indications that such a model could provide.