## Implications from Mathematical Models
### Models Of Immunology  
**[PERSON] et al. (2016)**. While many of the mathematical modeling efforts for metastasized prostate cancer focus on hormonal therapy, several researchers have also taken note of the recent development of immunotherapy (vaccination) for prostate cancer. One prominent example is Provenge (or sipuleuecl-T), which has been approved by the Food and Drug Administration for treating prostate cancer. A notable example is the work by [PERSON] et al. [64] in 2016. In this work, [PERSON] and colleagues construct a system of differential equations consisting of castrate resistant and sensitive tumor cells and the immune micro-environment. After parametrizing the models using mouse data, they study the treatment efficacy of combination therapy between four different treatments (castration, vaccination, cytokine interleukin-2/IL2 neutralization and regulatory T cells/Treg depletion). The concurrent use of castration and vaccination is motivated by the review of [PERSON] et al. [65], providing evidence that ADT can increase the efficacy of immunotherapy. On the other hand, the incorporation of IL2 neutralization and Treg depletion is motivated by a study [66], showing that ADT is followed by the activation of cytotoxic T lymphocytes (killer T-cells/CTLs); however, the production of IL-2 and Treg may inhibit the activity of CTLs in the prostate lymph nodes. Their work shows the potential of using system biology type models to address complex multi-drug approaches for prostate cancer. While their study shows the potential of a system biology type approach in the modeling of prostate cancer, their limitation lies in the lack of a sufficient source of data for validation of their complex model. **[PERSON] and [PERSON] (2013)**, [PERSON] and [PERSON] (2017), [PERSON] et al. (2010)**. [PERSON] and [PERSON] [67] propose an alternative approach to modeling immunotherapy motivated by the work of [PERSON] and [PERSON] [68]. Aside from the AD and AI cells, the model also considers the number of activated T-cells, the concentration of cytokines, the concentration of androgen, and the number of dendritic cells. In computational studies, they show a small advantage of combining IAS with immunotherapy over CAS with immunotherapy. Subsequently, [PERSON] and [PERSON] [69] extend their model to study the effects that drug dosage amounts and frequencies of administration on the time to relapse. Their computational and mathematical analysis of the model show the possibility of tailoring the vaccine dosage to the patient's effective immune system to maximize the effectiveness of treatment. In another work, [PERSON] et al. [70] formulated a simple mathematical model that describes the immune response in prostate cancer patients receiving immunotherapy. [PERSON] and colleagues fit the model to individual patient data and carry out the forecasting using the estimated parameters. Their model produces robust predictions with regards to the data set of 15 patients used for validation. **[PERSON] et al. (2016), [PERSON] et al. (2016)**. The idea of predicting treatment failure time is also presented in the work of [PERSON] et al. (2017), where a simple model of cancer growth that includes three types of cancer cells with testosterone is used for hormonal therapy. The model is fitted to all patient data. Then the parameters for an individual patient are obtained using Markov Chain Monte Carlo (MCMC) from the distribution obtained from the parameter estimates of the full cohort. Using the individual parameters, 1000 simulations are carried out to predict the biochemical failure, or the onset of castrate resistance, timing for each patient. The model shows high accuracy in predicting the timing of biochemical failure for ADT. In preference of a simpler model for predicting time to cancer relapse in prostatectomy patients, [PERSON] et al. (2016) used a form of the generalized Von Bertalanffy growth law to model prostate cancer growth.