## 7 Conclusions  
The main goal of this research is to produce a basic model capable of describing prostate cancer cell growth subject to IAS. Such a model may be amendable to detailed and systematical mathematical and computational study aimed at revealing the near term and intermediate term growth dynamics, including cancer cells treatment resistance development. Ultimately, such models may be helpful in establishing user-friendly treatment tools for both patients and physicians. To this end, we presented two models that can accurately fit clinical PSA and androgen data simultaneously. Existing models can only fit the PSA data. While these models are simplifications of PKN, they are just as accurate in data fitting and even better at forecasting future PSA levels. Model 1 had the lowest mean MSE for data fitting of all the models, followed by PKN and Model 2--not surprisingly, due to its more biologically realistic model assumptions, Model 2 had the lowest forecast MSE, with PKN doing the worst. The unreliability of PKN's forecasts stems from its dependence on androgen data and hence lacks the ability to predict androgen dynamics. Androgen cell quota values, which are not directly measurable from data, represent a significant source of uncertainty for the PKN model. Figure 6, shows how the new models can fit clinical androgen data and reduce uncertainty. For Models 1 and 2, \(Q\) is directly computable from clinical data. Predicting the timing of resistance is a highly desirable objective of this modeling work. Mathematical models alone can not make practical predictions. However, we can use these models and apply statistical methods to produce reliable forecasts with confidence intervals [35; 36]. In [PERSON] et al. [35] and [PERSON] et al. [36], the authors have used these methods successfully in an epidemiological setting to make predictions. In a future paper, the authors plan to use the models presented in this paper to produce predictions using such statistical methods. Therefore, the main biological contribution  
Figure 5: Simulations of fittings for every model for 1.5 cycles of treatment (**left** of gray line), and one cycle of forecast (**right** of gray line). For these four patients, we can see that models fit data at comparable accuracy but Model 2 perform much better in PSA forecasting. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Min** & **Mean** & **Max** \\ \hline
**PKN Model** & 12.234 & 162.5494 & 1868.6394 \\
**Model 1** & 11.3935 & 141.9280 & 1663.0218 \\
**Model 2** & 2.2727 & 56.3478 & 278.4050 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of forecast Mean Square Error for PSA. of this work is the development of a clear and basic androgen cell-quota based model to aid our understanding of the resistance development dynamics of prostate cancer cells. The dynamics of Model 1 is characterized by globally stable steady states. The mathematical analysis of Model 2 is only partially tractable. In fact, the stability and global stability of the cancer cell coexistence steady state remain unsettled. However, with our bifurcation analysis, we observe that under ADT or IAS, \(x_{2}\) cells may drive \(x_{1}\) cells to extinction. In Figure 2, we see that with lower and realistic levels of androgen production when the patient is under ADT, \(x_{1}\) cells will eventually become extinct even at a higher level of proliferation compared to \(x_{2}\) cells. Thus, we concluded that, under continuous treatment, almost all patients will eventually become androgen resistant. However, it is still not clear if IAS delays the speed at which this occurs. With the models presented in this work, we have moved closer to the ultimate goal of modeling the androgen resistance of prostate cancer. Our work is limited by the small number of patients considered. We selected 62 patients that had at least 20 data points in the first 1.5 cycles of treatment.