## 2 Elements of Mathematical Models for Clinical Applications
### Population Structure and Dynamics  
To have a complete understanding of the structure of prostate cancer and how the subpopulations of cells interact, with or without treatment, would be similar to having solved the problem of cancer from a theoretical perspective. A system-biology approach treats cancer as a unified population of cells that is part of a larger ecosystem. On the other hand, cancer can also be looked at like a society with many interacting sub-types of cancer cells. While it would perhaps be ideal to understand the details of each aspect of prostate cancer, it is impossible to do so. Instead, each model has to make certain simplifying assumptions that aim to answer a specific set of questions. Thus, we will examine the underlying assumptions of the population structure and dynamics on which each model was built and analyzed. Specifically, we aim to explore the following features of mathematical models in more detail.  
_Tumor heterogeneity and evolution:_ treatment failure due to cancer cells developing resistance is an unsolved problem in treating advanced prostate cancer. The most likely explanation is that over the course of treatment, selective pressure allows selection for resistant cell types. Thus, all existing population-type prostate cancer models incorporate some mechanisms that describe this heterogeneity and the evolution of the cancer population during treatment. For this feature, we study the existing literature to obtain conclusions that explore the heterogeneity and evolution of a tumor and its effects.  
_Tracking the progression of tumor size:_ having accurate and consistent measurements for the growth of the tumor that can be incorporated into mathematical models is crucial for clinical applicability and model assessment. While imaging data is perhaps best suited when it comes to measuring the changes in the tumor, the challenges of collecting high-quality spatial data hinder this possibility. Instead, time-serial PSA data has often been used as a proxy for the progression of prostate cancer. In this regard, we synthesize suggestions from existing work to provide an assessment of PSA as a biomarker for prostate cancer and explore other alternatives.  
_Model types and dynamics:_ some of the most successful prostate cancer models make use of the characteristics of the cancer progression or the observed data, while others construct more mechanistic models that make use of either fundamental or physical laws, like conservation laws, and/or the biochemical networks of cancers. In essence, mathematical modeling of prostate cancer is at the interface between simple classical mathematical models and complex system-biology type models, which offers rich opportunities to study interesting properties of these models from a mathematical, biological, or computational perspective. Additionally, while modeling with ordinary differential equations still dominates existing studies, the potential for using other types of dynamics has also been explored. For this reason, we compare and contrast different types of models and how they may offer insights for treating prostate cancer.