## Implications from Mathematical Models
### Population-Based Models of Tumor Relapse
\(p_{i}(a),q_{i}(a)\), and \(m(a)\) are functions that represent the effects of androgen on the proliferation, death, and mutation rates of the respective cells. The authors consider three different assumptions of how androgen affects the growth and death rates of the androgen independent population (\(p_{2}(a)\) is different for each assumption). Furthermore, the model incorporates the idea that androgen dependent cancer cells can transform into androgen independent cancer cells due to survival pressure. [PERSON] et al. used this model to compare the effects of intermittent androgen deprivation therapy verses continuous androgen deprivation therapy, making it the first mathematical modeling work to carry out such a comparison in the decade-old debate of CAS and IAS. They observed that relapse can be averted only under the assumption that the AI cell population decreases when the androgen level is normal. Their comparison concludes that if androgen dependent cancer cells have a competitive advantage over androgen independent cancer cells without treatment, then IAS is superior in prolonging the onset of treatment resistance than CAS. It is noteworthy to point out that both [PERSON] and [PERSON] make the assumption that AI cells do not have a competitive advantage over AD cells. This assumption is partially justified due to the rare occurrence of AI cells in early PCa, yet AI cells are quite common after the cancer has metastasized. While the mechanisms leading to the occurrence of AI relapse are still somewhat of a mystery, the study by [PERSON] and colleagues [20] shows strong evidence to support the assumption that AI cells are out-competed by AD cells in an androgen rich environment. In addition, [PERSON] et al. hypothesizes that in order for cancer cells to obtain treatment resistance, they must expend part of their system to develop some mechanisms to do so, which makes them less fit than the cells that do not have such mechanisms. However, [PERSON]'s and [PERSON] et al. models do not directly incorporate the competitive effect of the two populations. **[PERSON] and [PERSON] (2008), [PERSON] et al. (2016)**. To address this limitation, [PERSON] and [PERSON] [34] and [PERSON] et al. [35] proposed models that directly account for competition between the two cancer phenotypes. The competition model of [PERSON] and [PERSON] introduces a competition term with a constant rate, whereas the competition terms in [PERSON] et al. depend on the androgen level and are asymmetrical. Furthermore, [PERSON] et al. also introduce intraspecific competition, which addresses the theoretical issue of having an environment that can support an infinite amount of cancer cells in [PERSON]'s work. Numerical simulations for both models show the potential successes of IAS, which increases with higher competitive advantages for AD cells. **[PERSON] et al. (2008), [PERSON] et al. (2010)**. Aside from approaches using ordinary differential equations, [PERSON]'s model inspired spatial models by [PERSON] et al. [36] and [PERSON] et al. [37]. The work by [PERSON] et al. is a spatial extension of the [PERSON] et al. model using some assumptions from [PERSON]'s formulation. Their results give similar conclusions to [PERSON]'s. [PERSON] et al., on the other hand, analyzed the case where a mutation inhibitor is used as part of the regime for androgen deprivation therapy. They conclude that CAS can neither control nor cure PCa, even with a mutation inhibitor. However, their simulations suggest that mutation inhibition may delay the relapse of cancer. **[PERSON] and [PERSON] (2013), [PERSON] et al. (2016)** In addition to these studies, [PERSON] and [PERSON] [38] construct a spatial model of prostate cancer based on a similar framework and prove the existence and uniqueness of solutions for the model. Taking on a more computational approach, the phase-field method, which accounts for the dynamics and co-existence of healthy and cancerous cells, has also been applied in [PERSON] et al.