# The Impact of Intermittent Androgen Suppression Therapy in Prostate Cancer Modeling
## 2 Model and Method
### Parameter Range  
Previous work employed various parameter ranges without a detailed discussion of how the parameter range is obtained. This caused an unnecessary inconsistency across the literature. Here, we present our method of approximation of the parameter range used for fitting in Table 2. This is possible, partially because our model has clear biological links to clinical data and previous work. In doing so, we hope to build upon this, in future work, for coherency across models. First, we note that the units of \(x_{i}\), \(Q\), \(A\), and \(P\) are respectively L, nmol/L, nmol/L, and ug/L. From this, the following ranges and unit of parameters are established:  
* \(\mu_{1}\) is the maximum proliferation rate for the AD cancer subpopulation. Its unit is \([\text{day}]^{-1}\). [PERSON] et al. [21] studied the proliferation and death rates of prostate cancer cells. We interpret their result for metastatic prostate cancer cells from hormonally untreated patients to be suitable for AD cells because, before hormonal treatment, most cancer cells should be androgen-dependent. This gives us a range of (0.004-0.081) for \(\mu_{1}\). Similarly, for \(\mu_{2}\) (the maximum proliferation rates for AI cancer subpopulation), its range (0.001-0.046) is derived from the result for _hormonally failing patients_. This is because we expect AI cells to take over, once the patients become hormonal refractory. Taking into account experimental and survey errors (especially in this case, where the cancer population size is small), we use (0.001-0.09) as the range for both \(\mu_{1}\) and \(\mu_{2}\) in the parameter fitting process. * \(d_{1}\) is the maximum death rate for AD cells. Its unit is \([\text{day}]^{-1}\). Its range is also derived from [PERSON] et al. [21], under similar assumptions in the case of the proliferation rates. The ranges for \(d_{1}\) and \(d_{2}\) are (0.001, 0.0525) and (0.015, 0.0775), respectively. For parameter fitting, we use the ranges (0.001, 0.09) and (0.01, 0.09) for \(d_{1}\) and \(d_{2}\), respectively. * \(q_{1}\) is the minimum androgen cell quota for AD cells. It has the same unit as \(Q\). \([\text{nmol/L}]\). [PERSON] [22] showed in his review paper that hormonal suppression therapy aims to reduce serum androgen to below 0.69 nmol/L, which is considered the castration level (this threshold varies in practice, but no more than 1.73 nmol/L). The lowest threshold recorded in [PERSON] [22] is 0.41 nmol/L. This gives us an approximate bound for \(q_{1}\), (0.41-1.73) because, once serum androgen drops below this range, the growth of AD cells is repressed. \(q_{2}\), the minimum androgen cell quota for AI cells, should be smaller than 0.41 nmol/L, to allow AI cells some growth during castration. Since no information is available on the lower bound of \(q_{2}\), we approximate the range to be (0.01-0.41). Since the value of \(q_{i}\) is patient-specific, a fixed range will not suffice for every patient. Thus, we find the minimum serum androgen \((min_{A})\) in the first 1.5 cycles (from the data) and use the ranges \((min_{A}+0.2,min_{A}+0.5)\) and (0.01, \(min_{A}+0.2)\), respectively, for \(q_{1}\) and \(q_{2}\) to reflect the condition \(q_{1}>q_{2}\) and to represent the effect of cell quota on growth. * The \(\delta_{i}\)'s are the density death rates. Their unit is \([\text{L}]^{-1}[\text{day}]^{-1}\). No experimental information is available for the density death rate of cancer cells. Additionally, our sensitivity results in  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline \hline
**Param** & \(\mathbf{\mu_{1}}\) & \(\mathbf{\mu_{2}}\) & \(\mathbf{q_{1}}\) & \(\mathbf{q_{2}}\) & \(\mathbf{b}\) & \(\mathbf{\sigma}\) & \(\mathbf{e  
Table 1 indicate that their importance is negligible, thus we fix them at \(10^{-5}\) [mm\({}^{3}\)]\({}^{-1}\)[day] \({}^{-1}\), or 10 [L]\({}^{-1}\)[day]\({}^{-1}\) using the value taken from [PERSON] and [PERSON] [8]. We keep \(\delta_{i}\) in our model to avoid unrealistic overgrowth of the cancer cell population.