# The Impact of Intermittent Androgen Suppression Therapy in Prostate Cancer Modeling
## 2 Model and Method
### Parameter Range
* \(K\) is the half-saturation level of the androgen-dependent mutation rate from AD cells to AI cells. Its unit is [nmol][L]\({}^{-1}\). [PERSON] and [PERSON] used the range (0-1), while [PERSON] et al. [7] used (0.08-1.7). Since no experimental validation is available, and as \(K\) is one of the least sensitive parameters, we fix \(K\) to be 1 [nmol][L]\({}^{-1}\). * \(R_{i}\) are the half-saturation level of androgen-dependent death rates for AD and AI cells. Its unit is nmol/L. While no experimental information is available, computation results in [8,23] show that (0-3) and (1-6) are reasonable ranges for \(R_{1}\) and \(R_{2}\), respectively. Thus, we employ these ranges. * \(c\) is the maximum mutation rate from AD cells to AI cells. Its unit is [day]\({}^{-1}\). While there is no experimental ranges for \(c\), numerical experimentation using a mathematical prostate model from [PERSON] et al. [5] established a reasonable range for \(c\) to be (\(10^{-5}\)-\(10^{-4}\)). This is often fixed at about 0.00015 in the literature [7,23]. Since \(c\) is one of the most insensitive parameters, we choose to fix it at 0.00015 [day]\({}^{-1}\). * \(b\) is the baseline PSA production rate. Its unit is [\(\mathrm{\SIUnitSymbolMicro}\)][nmol]\({}^{-1}\)[day]\({}^{-1}\). [PERSON] and [PERSON] [7,8] used two different ranges for \(b\). Taking the lower and upper limits of both, we will use (0.0001-0.1) as the range for \(b\). We note that, intuitively, the baseline production of PSA is produced by normal prostate cells; however, to simplify the model, we assume that this production is proportional to the intracellular androgen level, because the lack or abundance of androgen directly affects the product from prostate cells. * \(\sigma\) is the androgen-dependent PSA production rate of the cancer cells. Its unit is [\(\mathrm{\SIUnitSymbolMicro}\)][nmol]\({}^{-1}\)[L]\({}^{-1}\)[day]\({}^{-1}\). [PERSON] et al. [23] used several ranges within (0.001-1), so we employ (0.001-1) as the range for \(\sigma\). * \(\epsilon\) is the PSA clearance rate. Its unit is [day]\({}^{-1}\). While no clinical experiment supports a range for \(\epsilon\), [PERSON] et al. [7] used 0.08, and [PERSON] and [PERSON] [8] extended this to (0.001-0.01). We further extend the range to (0.0001-0.1) for our parameter estimation. * \(\gamma_{1}\) is the androgen production rate associated with the testes, the primary source of androgen production. Its unit is [day]\({}^{-1}\). Previously, the value used in [PERSON] et al. [8] seemed large, which resulted in an instantaneous increase in androgen level after the on-treatment interval stopped [18]. On the contrary, [PERSON] et al. [5] set \(\gamma_{1}\) to be 0.08. Thus, for our purpose, we use (0.008-0.8) as a possible range for \(\gamma_{1}\). * \(\gamma_{2}\) is the secondary androgen production rate, produced from sources other than the testes, such as the adrenal gland. The range established from [8] is (0.001-0.1) for \(\gamma_{2}\), where their \(\gamma_{1}=20\). However, the value of \(\gamma_{2}\) should only be a small fraction of \(\gamma_{1}\). Thus, in our case, the value of \(\gamma_{2}\) is negligible. Hence, we fix it at 0 for the fitting process. * \(m\) is the diffusion rate that links serum androgen level with intracellular androgen level. It is unitless. This rate is taken at 1 implicitly in [7,8]. Here, We allow \(m\) to vary from 0.01 to 0.9, to incorporate the level of synergy between \(A\) and \(Q\). * \(A_{0}\) is the maximum serum androgen level. Clinical data shows that typical serum androgen range is below 27 [nmol][L]\({}^{-1}\)[19]. Furthermore, computational work often assumed \(A_{0}<35\) [nmol][L]\({}^{-1}\)[5,8]. Since this range is patient-specific, for parameter fitting we find the maximum serum androgen (\(max_{A}\)) in the first 1.5 cycles (from data) and use the range (\(max_{A}-5,max_{A}+5\)) for \(A_{0}\). Aside from the ranges for parameters, we also discuss the initial conditions for the model.