## Data, Parameter Ranges, and a Framework for Clinical Application
### Parameter Ranges  
One of the main issues that is common to existing work is determining the values of model parameters. Here, we also provide references with regard to some important parameters.  
_Cancer maximum proliferation rates_: by making the assumption that measurements of cell growth and death for hormonally untreated patients to be that of AD cells, the study of tumor doubling time at various stages in [PERSON] et al. [101] can be used to suggest that the range for the proliferation rates of AD cells is (0.004-0.081)\([day]^{-1}\). Similarly, we can find a range of (0.001-0.046)\([day]^{-1}\) for AI cells. Note that these ranges fall within the expectation that AD cells out-compete AI cells in an androgen-rich environment. Additionally, growth rates for specific cell types can be obtained in vitro [20; 61]. However, appropriate scaling or fitting is necessary to account for differences in tumor environment. Since the environment in a laboratory is ideal for the growth of cancer cells, their growth rate should be lower in practice. Thus, we consider these ranges to be for maximum proliferation rates.  
_Cancer cell death rates_: similar to cancer growth rates, we estimate the ranges of cancer death rates for AD and AI cells to be (0.001-0.0525)\([day]^{-1}\) and (0.015-0.0775)\([day]^{-1}\), respectively.  
_Cancer cell maximal transformation rate:_ Robust estimate of cancer transformation rates from experimental data is lacking. However, numerical experiments and fitting show that (\(10^{-5}\)-\(10^{-4}\))\([day]^{-1}\) is an appropriate range for the maximal transformation rate [24; 33]. The transformation rate encompasses the mutation rate. While there are several estimates of the PCa cell mutation rate, mathematical models often use transformation rates instead of the more specific mutation rate.  
PSA clearance rate and production rates by healthy and cancerous cells: initial estimates [73] were for human-derived mouse xenograft sublines LuCaP 23.1, 23.8, and 23.12. These values were later extrapolated for human prostate cancer in Section 5.3.1 [15] using the study of [PERSON] et al. [101] and [PERSON] et al. [102]. The PSA production rate from healthy cells is \((2.870\times 10^{-5}\)-\(1.354\times 10^{-4})[ng][ml]^{-1}[mm]^{-3}[day]^{-1}\). On the other hand, estimating PSA production rate is perhaps best carried out using specific cell sublines. For LuCaP 23.1, 23.8, and 23.12, this rate is estimated to be 1.7210, 2.1841, and 6.9722 \([ng][ml]^{-1}[mm]^{-3}[day]^{-1}\), respectively [73]. Additionally, the PSA clearance rate is within the range of \((0.1754\)-\(0.4030)[day]^{-1}\)[103]. We summarize these in Table 3.  
Some parameters are difficult to estimate robustly, for instance, the competition rates of cancer cells. In the case of difficulty in estimating certain parameters from laboratory data, researchers also rely on experts' opinions or fix the parameters in an ad hoc way [20; 43]. Alternatively, the model can also be derived from physical laws (e.g., conservation laws), which introduce competition rates without adding additional explicit parameters for competition between cells [47; 48; 50]. For readers interested in how some parameters are estimated in existing models, we refer to studies [16; 51], where binding rates are estimated from multi-level processes.