## 2 Materials and Methods
### Parameter Estimation  
Where possible, parameter values were taken from the literature or fit to available experimental data. The parameters taken from the literature are summarized in Table 2. The remaining model parameters were estimated by fitting model output to data from experiments reported in [28; 29], in a least squares sense. This was implemented in Matlab using the curve-fitting tool 'fminsearchbnd' coupled with the built-in ordinary differential equation solver 'ode23s'. These data are described below. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Dimension** & **Reference** \\ \hline \(\delta_{NN}\) & 0.1066 & per day & [16] \\ \(\delta_{MN}\) & 0.1066 & per day & [16] \\ \(\delta_{AI}\) & 0.3151 & per day & [30] \\ \(\delta_{AM}\) & 0.4 & per day & [31] \\ \(\mu_{T8}\) & 0.012 & per day & [13] \\ \(s_{T8}\) & 0.0038 & \(10^{7}\) cells per day & [14; 32; 33] \\ \(s_{T4}\) & 0.0072 & \(10^{7}\) of cells per day & [14; 32; 33] \\ \(s_{TR}\) & 0.0005 & \(10^{7}\) of cells per day & [14; 32; 33] \\ \(\delta_{T}\) & 0.462 & per day & [13] \\ \(\delta_{TL}\) & 0.1199 & per day & [13] \\ \(\alpha_{T8}\) & 16 & per day & [14] \\ \(\alpha_{T4}\) & 1.9 & per day & [14] \\ \(\alpha_{TR}\) & 2.1 & per day & [14] \\ \(\lambda_{R}\) & 0.022 & per day & [14] \\ \(\alpha_{G}\) & \(1.1\times 10^{-7}\) & ng/mL per cell per day & [14] \\ \(\alpha_{GT}\) & \(1.8\times 10^{-8}\) & ng/mL per cell per day & [14] \\ \(\delta_{G}\) & 14.2857 & per day & [14] \\ \(\alpha_{I}\) & 1.7 \(\times\) 10\({}^{-5}\) & ng/mL per cell per day & [14] \\ \(\delta_{I}\) & 12.5 & per day & [14] \\ \(k_{I}\) & 0.3 & ng/mL & [14] \\ \(TR0\) & 2 & \(10^{7}\) cells & [14] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values obtained from the literature. **Mouse xenograft experiments:** In mouse xenograft experiments described in [28], eight to ten week old male FVB/NJ mice were injected subcutaneously with Myc-CaP cells. Myc-CaP cells are androgen-sensitive; however after treatment with ADT, castration resistance emerges spontaneously. ADT with degarelix acetate, a GnRH (gonadotropin releasing hormone) receptor antagonist that inhibits androgen production, was initiated on day 21. Tumor volume was recorded periodically (see Figure 2a, blue circles). Additionally, the numbers of dendritic cells, Th cells, CTLs, and Tregs were measured before treatment, post-treatment, and once castration resistance had emerged (see Figure 3, blue and red circles). The emergence of castration resistance was defined as a re-growth of tumor volume to \(\sim\)420 mm\({}^{3}\) after an initial decline post-treatment. Best fits to this data-set are shown in Figures 2a and 3 (solid lines). Figure 3: Model fits to time-course data of: (**a**) total number of dendritic cells; (**b**) number of activated CTLs; (**c**) number of activated Th cells; (**d**) number of Tregs. Black arrows indicate when treatment with ADT was initiated, blue/red circles together with error bars represent experimental data and solid lines show best fit model simulations. Data taken from [28]. Red circles represent data reserved for model validation, and were not used in the fitting process. Figure 2: Model fits to tumor xenograft time-course data of: (**a**) total tumor cell number; (**b**) number of apoptotic tumor cells pre-ADT. Black arrows indicate when treatment with ADT was initiated, blue circles together with error bars represent experimental data and solid lines show best fit model simulations. Data taken from [28; 29]. **Cell death measurements under pre-treatment:** In [29], experiments were performed to quantify the degree of apoptosis in PCa xenografts. Briefly, 4-5 week old nude/athymmic mice were implanted subcutaneously with CWR22 tumors. CWR22 is an androgen-dependent human PCa cell line. The percentage of apoptotic cells was recorded periodically. These data (blue circles), together with best fit (solid line) are shown in Figure 2b.