# The Impact of Intermittent Androgen Suppression Therapy in Prostate Cancer Modeling
## 2 Model and Method
### Parameter Estimations  
For parameter fitting, we use function _fmincon_ from MATLAB (version 2018b), which utilizes an _interior point algorithm_ to minimize an objective error function within the range provided in Table 2. Previously, the objective error function was calculated as the sum of the mean squared error (MSE) fitting error for PSA and androgen, because MSE punishes particularly large errors (which is undesirable in clinical applications). This emphasis on good fit can be problematic for forecasting, as the peak PSA level (during the off-treatment interval) is much larger than the PSA level during the on-treatment interval. Specifically, MSE will prioritize large errors, thus _fmincon_ will estimate parameter values to achieve high PSA level to match the data at peak level, which leads to overestimated forecasts. This problem is escalated, as the time for on- and off-treatment intervals are taken from data. Furthermore, the problem is not resolved using other error measures. For instance, using the relative error metric causes the fitting to skew toward smaller data points due to their number, and so forecasting is often significantly underestimated ([PERSON] et al., 2018). Instead, taking into account the characteristic of the treatment schedule and its effect on PSA level, we employ a weighted least square \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Param** & **Description** & **Range** & **Unit** \\ \hline \(\mu\) & max proliferation rate & 0.001–0.09 & [day]\({}^{-1}\) \\ \(\mu_{1}\) & max proliferation rate (AD cells) & 0.001–0.09 & [day]\({}^{-1}\) \\ \(\mu_{2}\) & max proliferation rate (AI cells) & 0.001–0.09 & [day]\({}^{-1}\) \\ \(q_{1}\) & min AD cell quota & 0.41–1.73 ** & [nmol][day]\({}^{-1}\) \\ \(q_{2}\) & min AI cell quota & 0.01–0.41 ** & [nmol][day]\({}^{-1}\) \\ \(b\) & baseline PSA production rate & 0.0001–0.1 & [\(\mu\)g][nmol]\({}^{-1}\)[day]\({}^{-1}\) \\ \(\sigma\) & tumor PSA production rate & 0.001–1 & [[\(\mu\)g][nmol]\({}^{-1}\)[L]\({}^{-1}\)[day]\({}^{-1}\) \\ \(\epsilon\) & PSA clearance rate & 0.0001–0.1 & [day]\({}^{-1}\) \\ \(d_{1}\) & max AD cell death rate & 0.001–0.09 & [day]\({}^{-1}\) \\ \(d_{2}\) & max AI cell death rate & 0.01–0.001 & [day]\({}^{-1}\) \\ \(\delta_{1}\) & density death rate & 1–90 * & [L]\({}^{-1}\)[day]\({}^{-1}\) \\ \(\delta_{2}\) & density death rate & 1–90 * & [L]\({}^{-1}\)[day]\({}^{-1}\) \\ \(R_{1}\) & AD death rate half-saturation & 0–3 & [nmol][L]\({}^{-1}\) \\ \(R_{2}\) & AI death rate half-saturation & 1–6 & [nmol][L]\({}^{-1}\) \\ \(c\) & maximum mutation rate & 0.00001–0.0001 * & [day]\({}^{-1}\) \\ \(K\) & mutation rate half-saturation level & 0.8–1.7 * & [nmol][day]\({}^{-1}\) \\ \(\gamma_{1}\) & primary androgen production rate & 0.008–0.8 & [day]\({}^{-1}\) \\ \(\gamma_{2}\) & secondary androgen production rate & 0.001–0.1 * & [day]\({}^{-1}\) \\ \(m\) & diffusion rate from \(A\) to \(Q\) & 0.01–0.9 & [day]\({}^{-1}\) \\ \(A_{0}\) & maximum serum androgen level & 27–35 ** & [nmol][day]\({}^{-1}\) \\ \(x_{1}(0)\) & Initial subpopulation of AD cells & 0.009–0.02 & [L] \\ \(x_{2}(0)\) & Initial subpopulation of AI cells & 0.0001–0.001 & [L] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter definition and range, established via clinical data and previous computational work. Note that the parameters with an asterisk next to their fitting range are fixed in the new model, e.g., \(c\), \(K\), \(\delta_{1}\), \(\delta_{2}\), and \(\gamma_{2}\) are fixed at 0.00015, 1, 5, and 0, respectively. The ranges of parameters with a double asterisk are dependent on the data from the first 1.5 cycles for each patient. error that will allow emphasis on the recent data, and a controlled weight on the error contribution from the peak level.