## Implications from Mathematical Models
### Models of Cellular Kinetics  
**[PERSON] and [PERSON] (1998)**, [PERSON] et al. (2006)**, [PERSON] et al. (2020)**, [PERSON] et al. (2020)**. As hormonal therapy is the standard of care for metastatic prostate cancer, it is crucial to obtain a good understanding of how the drugs used in hormonal therapy affects the androgen level and consequently the prostate cancer cells. To this end, the work by [PERSON] and [PERSON] in 1998 [9] has paved the way for an initial framework of androgen regulation of prostate growth. Its extension and validation using rat data was later carried out in [PERSON] et al. [51]. Recently, [PERSON] et al. [52] also formulated a pharmacokinetics model that incorporates the specific properties of a drug on androgen production in a prostate cancer model and tests it with clinical patient data. While [PERSON] et al. focus on the effects of combination hormonal drugs (Cyproterone acetate and Leuprolide acetate) on a macroscopic level using phenomenological functional responses, the stochastic differential equation pharmacokinetics model by [PERSON] et al. [53] attempts to capture the molecular interactions (oxygen consumption and protons production of cancer cells) between the cells and a newer hormonal drug (enzalutamide) using mice data. In essence, these works are successful in their attempts to replicate the drug effects on the dynamics of androgen and prostate growth; however, the complexity of these models often make them mathematically untractable and highly difficult to fit with high certainty. To address this issue, in both pharmacokinetics attempts, the parameter estimations are carried out in a multi-level process. **[PERSON] et al. (2010)** Utilizing the intracellular kinetics framework of the androgen receptor, testosterone, and \(5\alpha\)-dihydrotestosterone in the work of [PERSON] et al., [PERSON] et al. [54] built a multi-scale model to study the effect of androgen in the evolutionary process from the benign to treatment stage of prostate cancer. The model couples the AR kinetics model with a state-transition model, where 100 states included with each state representing a different strain of prostate cells with varying AR expression. This reflects the observation that AR up-regulation is one of the most important pathways in which cancer cells becomes AI. Their results demonstrate how a heterogeneous population of prostate cancer cells can be skewed to select for androgen independent phenotypes in a low androgen environment. This suggests that while low levels of androgen may delay the appearance of malignant cancer cells, it may increase the chance of more aggressive cancer phenotypes. This result offers a reasonable explanation to the observation that finasteride, a 5-\(\alpha\) reductase inhibitor, can reduce the overall rate of prostate cancer but may increase the rate of high-grade prostate cancer [55]. **[PERSON] et al. (2011), [PERSON] and [PERSON] (2013)**. While [PERSON] et al. considered the role of androgen in the evolutionary dynamics of prostate cells using hundred of cell types, [PERSON] et al. [16] applied the framework into a two cell types model under androgen deprivation therapy (both intermittent and continuous). Since their model is rather complex, they carry out multi-level fitting to obtain the estimates for the parameter values separately. By fitting average patient data reported in [PERSON] et al. [56], they show that their model not only contains patient specific parameters but is also capable of reproducing a variety of clinically observed dynamics of cancer progression. The model predicts similar conclusions as previous work: intermittent androgen suppression therapy is superior compared to continuous therapy when the AD cells have a competitive advantage over the AI cells and vice versa. Due to the complexity of the model, not much analytical work can be done. Furthermore, the large number of parameters presented in the model means clinical parameter estimation for each patient is difficult.