## Implications from Mathematical Models
### Population-Based Models of Tumor Relapse
\begin{table}
\begin{tabular}{c p{142.3 pt} p{142.3 pt}} \hline \hline
**Aspects of** **Mathematical Modeling** & **Summary of Findings** & **Future Exploration** \\ \hline Tracking the progression of tumor & The most commonly used biomarker for prostate cancer growth (PSA) is useful but can be unreliable. Instead, other measurements can be used in place or in concurrence with PSA to track tumor progression. & The accuracy of tracking the progression of tumors using multiple biomarkers needs to be examined. While accuracy is key, the availability of such biomarkers should also be taken into account. & \\ \hline Model types and dynamics & Ordinary differential equations build the foundation for studying prostate cancer. However, the lack of various modes of modeling implies many aspects of prostate cancer are left unexplored. & As more data becomes available, especially imaging data, spatial, memory-based, and stochastic models will become useful in capturing spatial patterns in cancer progression and interaction, specifically, the metastatic processes. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. androgen dependent cells and increases the apoptosis rate of androgen independent cells. With these assumptions, the models were able to capture the qualitative dynamics observed from experimental data of LuCaP 23 in mice [30]. Thus, [PERSON] concludes that the model supports the notion that cancer relapse occurs because AI cells have a lower apoptosis rate, rather than a higher proliferation rate, a conclusion that agrees with some experimental data [31]. This conclusion supports the hypothesis that prostate cancer cells pay a cost to evolve resistance to treatment--thus having a disadvantage against AD cells when no treatment is applied [20]. Additionally, [PERSON] shows that androgen deprivation therapy is only effective for a small region of parameters--which agrees with clinical data showing that androgen deprivation therapy is not curative, even in early stage disease (see Figure 3). As with all models, [PERSON]'s modeling effort has certain limitations. First, the assumption of spherical symmetry in the model formulation limits the application of the model to the early stage, where the cancer can be approximated with a sphere. Secondly, the assumption that increasing the level of androgen increases the apoptosis rate of androgen independent cells is arguable because androgen independent cells may still be dependent on AR, which is known as the outlaw pathway to resistance [11; 15; 32]. Regardless, [PERSON]'s work has paved the way for future modeling studies. **[PERSON] et al. (2008)**. Inspired by [PERSON]'s work, [PERSON] et al. [33] studied a system of ordinary differential equations describing the interaction of androgen dependent and independent cells, while explicitly treating the androgen level as a dynamical variable. \[\frac{da}{dt} =-\gamma a+\gamma a_{0}(1-u) \tag{3}\] \[\frac{dx_{i}}{dt} =\alpha_{i}p_{i}(a)x_{i}-\beta_{i}q_{i}(a)x_{i}+(-1)^{i}m(a)x_{1} \tag{4}\]  
Figure 3: Treatment effectiveness in [PERSON]’s model—figure reproduced from [PERSON] [29] with permission, which is distributed under a Creative Commons Attribution (CC BY) license. The vertical axis represents the proliferation rate of AD cells post treatment. The horizontal axis represents the steady state androgen concentration post treatment. The striped region indicates the tumor radius decreasing in the first and second order approximations. (**a**) The figure shows a small striped region in the parameter space where partial blockage of androgen leads to successful containment of the cancer. (**b**) Often times, successful treatment is not possible. Further analysis in the case of a complete blockage of androgen shows a small expansion of the successful region. Here, \(a\) represents the androgen level, the index \(i\) runs between 1 and 2, which represents the androgen dependent cells and androgen independent cells, respectively.