## 3 Results
### Sensitivity Analysis and Uncertainty Propagation  
We finally perform a sensitivity analysis on model parameters fit to the experimental data (listed in Table 3), to identify which have the most influence on the therapeutic potential of ADT and Provenge administered in combination. Please note that of the 23 fitted parameters, the dimensionless parameters \(e_{M}\) and \(\omega_{M}\) were held fixed at 1 and not included in this analysis, which is equivalent to assuming that ADT does not significantly impact castration-resistant cells. We begin with a local sensitivity analysis wherein each parameter is varied independently about its best fit value. The percentage change in tumor volume relative to the baseline case induced by the optimal ADT and Provenge treatment schedule identified in the previous subsection, is recorded. A parameter is considered sensitive if a 50% change in its value induces at least a 50% change in output, that is, the parameter has a greater  
Figure 6: Predicted time courses of cell numbers after concurrent administration of ADT and Provenge, starting on day 21. (**a**) Number of tumor cells (\(N+M+D_{A}+D_{N}\)) with ADT alone (dashed blue curve) or combination therapy (solid blue curve). The horizontal black line corresponds to the treatment failure threshold (tumor volume of 420 mm\({}^{3}\)). (**b**) Number of mature DCs in the lymphoid compartment after combination therapy. (**c**) Number of activated CTLs in the lymphoid compartment after combination therapy. (**d**) Number of activated CTLs after combination therapy. (**e**) Number of activated Th cells in the lymphoid compartment after combination therapy. Black arrows indicate days on which Provenge was administered. than linear effect. Figure 8a,b reveals that model predictions of tumor xenograft response to combination treatment are most sensitive to: \(\delta_{K}\), the kill rate of tumor cells by CTLs; \(\alpha_{N}\) and \(\alpha_{M}\), the proliferation rates of \(N\) and \(M\) cells, respectively; \(\delta_{NA}\) and \(\delta_{MA}\), the rates of \(N\) and \(M\) cell apoptosis, respectively; and \(K_{T}\), the half-saturation constant for the rate of CTL-induced tumor cell death. We finally investigate how uncertainty in these six key parameters might affect our predictions on xenograft response to optimal combination treatment. For this, we select ranges for the significant parameters, that induce at most a 50% variance in tumor volume relative to the baseline case. We recognize that this threshold can be chosen differently, but do not believe that our results will be qualitatively different. The resulting multidimensional parameter space is sampled uniformly, using Latin Hypercube Sampling--a standard sampling technique in performing sensitivity and uncertainty analyses [36]. A total of 100,000 samples were drawn, to ensure uniform coverage of parameter space. Each parameter vector may be thought of as a virtual mouse. Having generated a population of heterogeneous test mice, we then simulate the treatment of each mouse's xenograft with combination ADT and Provenge per the optimal protocol. We record the time to treatment failure (\(t_{fail}\)), defined here as the time interval from ADT initiation to when the tumor volume increases over its value at the start of ADT, following any treatment-induced decrease. If treatment does not induce a decrease in tumor volume, \(t_{fail}\) is taken as 0. Figure 7: Optimizing ADT and Provenge scheduling when given in combination. (**a**) Predicted time to treatment failure as we vary the time of administration of the first dose of Provenge. In all cases, ADT was administered starting on day 21. (**b**) Predicted time courses of cell numbers after optimized administration of ADT and Provenge with ADT starting on day 21 and Provenge starting on day 15 (solid blue curve). (**c**) Predicted time courses of cell numbers after administration of ADT and Provenge per current clinical protocol with ADT starting on day 21 and Provenge starting on day 48, when ADT has failed (solid blue curve).