## 6 Conclusions  
Mathematical models play a vital role in the study of prostate cancer dynamics. Over the past two decades, various models have been developed to examine different aspects of prostate cancer, including treatment options and schedules. In this review, we collect and synthesize the results of existing studies to share conclusions that are well agreed upon and raise questions that are in need of further investigation. The study of prostate cancer at the multi-cell level gives insights on the treatment efficacy with respect to the competition among the cells. Specifically, the consensus among existing work is that IAS is superior to CAS at delaying the relapse of metastasized cancer, if AD cells have some competitive advantages over AI cells in an androgen-rich environment. While some studies have shown that AD cells can indeed out-compete AI cells, there is no extensive evidence that this is true in general. Instead of suggesting that the competition advantage is an intrinsic property of AD cells, this property needs to be examined on a case-by-case basis because many factors such as the physiology of the cancer or the specific cancer phenotype can affect the relative competition between cancer cells. Additionally, we find that lowering the androgen level as a preventive measure of prostate cancer may result in selection for aggressive androgen independent phenotypes, if the tumor forms. On the other hand, studies that focus on tools used to track prostate cancer progression, such as PSA, show that while PSA is an overall good biomarker for cancer growth, it can display poor correlation in many cases due to the patient's specific PSA production rates by cancer cells and PSA degradation rates. Instead, the relative velocity of PSA appears to be a better alternative for monitoring tumor progression. Furthermore, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Range** & **Unit** & **Source** \\ \hline Max proliferation rate (AD) & 4.00\(\times 10^{-3}\) - 8.10\(\times 10^{-2}\) & \([day]^{-1}\) & [101] \\ Max proliferation rate (AI) & 1.00\(\times 10^{-3}\) - 4.60\(\times 10^{-2}\) & \([day]^{-1}\) & [101] \\ Death rate (AD) & 1.00\(\times 10^{-3}\) - 5.25\(\times 10^{-2}\) & \([day]^{-1}\) & [101] \\ Death rate (AI) & 1.50\(\times 10^{-2}\) - 7.75\(\times 10^{-2}\) & \([day]^{-1}\) & [101] \\ Max transformation rate & 1.00 \(\times 10^{-5}\) - 1.00 \(\times 10^{-4}\) & \([day]^{-1}\) & [24] \\ PSA clearance rate & 1.75\(\times 10^{-1}\) - 4.03\(\times 10^{-1}\) & \([day]^{-1}\) & [103] \\ PSA production rate (healthy) & 2.87 \(\times 10^{-5}\) - 1.35 \(\times 10^{-4}\) & \([ng][ml]^{-1}[mm]^{-3}[day]^{-1}\) & [15] \\ PSA production rate (cancer) & 1.72\(\times 10^{0}\) - 6.97\(\times 10^{0}\) & \([ng][ml]^{-1}[mm]^{-3}[day]^{-1}\) & [73] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Ranges for some commonly used parameter values in mathematical models for prostate cancers. Note that for more accurate ranges, the specificity of the situation needs to be taken into account, for example, sublines of cancer cells. to tackle the problem of identifiability of models, multiple biomarkers should be used to enhance the accuracy of models. Thus far, there exists mathematical models for almost all clinical stages of prostate cancer. Some models are even flexible enough to be used across many stages. However, the application of each model is still limited to certain stages. Having a comprehensive model that can describe tumor development as a whole may be useful for clinical practice. Treatment effectiveness is one of the most examined issues using mathematical models with a focus on hormonal therapy. This is often coupled with the study of optimal schedule, especially for IAS and adaptive therapies. While the studies so far show potential, they lack key components for use in clinical settings. First, the drugs being considered in a mathematical model are often generalized--meaning the specific properties of the drug are not accounted for.