## Implications from Mathematical Models  
### Population-Based Models of Tumor Relapse  
In this section, we discuss the implications from mathematical models. A synthesis of our findings is presented in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Aspects of** & & \\
**mathematical Modeling** & **Summary of Findings** & **Future Exploration** \\ \hline Tumor heterogeneity & Treatment resistance appears to & Quantitative measures of the \\ and evolution & come at a competitive cost for & evolutionary cost and when it \\ Sections 3.1–3.4 & cancer, which implies that & does occur would be useful. \\  & intermittent and adaptive therapy & Furthermore, the competition \\  & would be superior to continuous & rates between sub-types of cancer \\  & therapy. & cells should be quantified. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Aspects of mathematical modeling: summary of findings and future exploration. The highly heterogeneous nature of prostate cancer, especially after it has metastasized, is one important reason why population-type models take up a large percentage of existing work. This includes simple models that describe only the interactions among sub-populations of cells, or more complex models that also account for external factors, such as environmental changes due to treatment. The population-type models, in general, focus on the interactions of the cancer sub-populations with or without external effects. On the other hand, kinetic-type models tend to view cancer as a whole and how cancer interacts within the human physiological network, often via biochemical pathways. **[PERSON] et al. (1993)**. The earliest approach to model prostate cancer progression mathematically is credited to [PERSON] et al. (2017). In their work, they developed a simple kinetic model based on Gompertzian growth that aims to explain the progression of prostate cancer, its metastasis process, and how the metastasis process affects local treatments (surgical and radiation therapy). By comparing their model simulations with the patient's survival data in [PERSON] et al. (2018), [PERSON] and colleagues suggested that local relapse is associated with more aggressive cancer phenotypes. Furthermore, due to the aggressive cancer phenotypes observed in local relapse, they suggest that local control (e.g., the complete eradication of the primary tumor and lymph node metastasis) can significantly improve the disease-free survival and metastasis-free survival time. However, after the cancer has metastasized, standard treatments show early positive results but fail to contain tumor relapse. **[PERSON] (2004)**. The relapse following positive responses to hormonal castration has long been hypothesized to be a result of prostate cancer being made up of different subpopulations of cells, each with its own level of dependency on the extracellular androgen. This concept is first demonstrated with mathematical modeling in 2001 by [PERSON] (2001). [PERSON] balances proliferation, apoptosis, and fluxes of two types of cells under radial symmetry to construct a system of partial differential equations for androgen deprivation therapy in prostate cancer, as shown in Equations (1)-(2). \[\frac{dR}{dt}=u(R(t),t). \tag{1}\]  
\[D_{p}\frac{\partial p}{\partial r}(R,t)-u(R,t)p(R,t)=0,\quad D_{q}\frac{ \partial q}{\partial r}(R,t)-u(R,t)q(R,t)=0. \tag{2}\]  
In this model, \(u(R,t)\) is the net rate of collective cellular motion, \(R(t)\) is the tumor radius and \(p(R,t),q(R,t)\) are the volume fractions of androgen dependent (AD) and androgen independent (AI) prostate cancer cells, respectively. The model assumes that a high level of androgen increases the proliferation rate of androgen dependent cells but does not affect the proliferation rate of androgen independent cells. On the other hand, a low level of androgen decreases the apoptosis rate of