## 4 Discussion  
Advanced PCa is typically treated with ADT due to its dependence on testosterone for survival [2; 3]. However, resistance to ADT frequently emerges and Provenge is the first vaccine approved to treat this late stage cancer [6]. Provenge is currently approved for administration once ADT has failed, but has displayed limited clinical success in terms of improving median survival or time of disease progression [7]. Here, our aim was to maximize the therapeutic potential of Provenge when given in combination with ADT. Toward this goal, we developed a mechanistic mathematical model of tumor and immune system interactions, taking care to incorporate biological detail relevant to the mechanism of action of both therapies. We calibrated and validated our model versus data from mouse xenograft experiments and used it to investigate the therapeutic potential of Provenge. Model simulations of xenograft treatment with Provenge alone show that while it inhibits the rate of tumor growth, it is unable to induce any tumor regression. This perhaps explains the findings of the first phase III clinical trial of Provenge, which reported only a modest median overall survival benefit of 4.1 months (25.8 versus 21.7 months over the placebo group) [38]. In general, higher doses of the vaccine resulted in greater tumor growth rate inhibition, with a saturating effect. Our simulations further suggested that targeting the anti-inflammatory Tregs in combination with Provenge might enhance its therapeutic efficacy, colorredan idea that is currently under experimental evaluation [39]. We then varied the timing of Provenge administration relative to ADT onset, in order to maximize the time to treatment failure, defined as the time interval between ADT onset and when the tumor xenografts reach a target volume. The optimal protocol was identified as administering one dose of Provenge _prior_ to ADT, and giving the other doses later. This mimics the protocol identified in [10]. Essentially, Provenge targets androgen-dependent and castration-resistant cells within the tumor. However, if the tumor is too large when Provenge is first given, it has limited efficacy. An optimal protocol is achieved when the CTL-to-target cell ratio within the xenografts is as high as possible. We finally performed sensitivity and uncertainty analyses on fitted model parameters, to determine which parameters are critical determinants of response to combination ADT and Provenge. Surprisingly, the proliferation and death rates of castration-resistant cells were found to be non-significant indicators of response. Rather, a high degree of androgen sensitivity, coupled with a high degree of tumor cell susceptibility to targeted death by anti-tumor CTLs, characterized a positive response to treatment. Moreover, the majority of a heterogeneous (virtual) mouse population did not respond to ADT combined with Provenge. This mimics the results of the clinical trial of Provenge in human subjects, which reported no significant difference in time to progression compared with control [38]. Even though only a small fraction of the mice responded to combination therapy, their response was remarkable, with a median time to failure of almost 63 days. This suggests that a rare sub-population of prostate cancer patients may benefit strongly from ADT administered in combination with Provenge, highlighting the need for continued experimental and modeling efforts in this direction. Explicating the results of clinical trials of novel anti-cancer therapeutics, and optimizing treatment schedules when these therapies are given in combination with current the standard-of-care requires tremendous investments of time, money, and human resources. Mathematical modeling such as that proposed here can aid in this drug development process by conducting any number of virtual experiments at a fraction of the cost and time.