## Implications from Mathematical Models
### Models of Cellular Kinetics
Therefore, in subsequent work, [PERSON] and [PERSON] [57] simplify their model and carry out mathematical analysis on the simplified version. By defining rigorously the definition of treatment viability and failure, they were able to compare the efficacy of continuous vs. intermittent therapy, which yields a similar conclusion to previous work. Additionally, they discover that even if it is possible to control a tumor with an optimal schedule of intermittent therapy, a sub-optimal one may still lead to the emergence of treatment resistance. **[PERSON] et al. (2017)**. In the same spirit of using a simple phenomenological model, but in contrast to [PERSON] et al. where the focus is to examine the mutation process that governs the dynamics of the cancer progression, [PERSON] et al. [20] utilized a three population Lotka-Volterra type model to investigate solely the competition aspect of cancer sub-populations. The model takes the form:  
\[\frac{dx_{i}}{dt}=r_{i}x_{i}\left(1-\frac{\sum_{j=1}^{3}a_{ij}x_{j}}{K_{i}}\right) \tag{10}\]  
and  
\[\frac{dP}{dt}=\sum_{i=1}^{3}x_{i}-0.5P. \tag{11}\]  
Here, \(a_{ij}\) is the competition matrix, where \(x_{1},x_{2},\) and \(x_{3}\) represent AD cells, treatment-sensitive AI cells, and treatment-resistant AI cells, respectively. The PSA level is given by the variable \(P\). The model is used in concurrence with their clinical study. Their work comes at a time where there is no conclusive evidence of whether IAS is better than CAS in terms of prolonging the time to the relapse of cancer. They argue that the result of previous clinical comparisons of intermittent and continuous treatments that study whether intermittent therapy gives any benefit in terms of delaying the onset of treatment resistance [58; 59] are not well supported. [PERSON] and colleagues were able to offer clinical evidence that the previous protocol for intermittent therapy, in essence, has the same effect as continuous therapy. Furthermore, they show that by changing the standard protocol, they were able to obtain significant improvement in the delay of treatment resistance for intermittent therapy with abiraterone over previous results. Their results show that AD cells are likely to have a significant competitive advantage over AI cells when treatment is not applied. In the same work, they support an alternative form of therapy, namely adaptive therapy. The main idea behind adaptive therapy is not to cure cancer but to obtain a stable tumor burden and maintain it indefinitely [60]. Compared to standard treatment where a stable tolerable dose is used, adaptive therapy uses varying drug dosages that change, depending on the response of the tumor to the drug, resulting in a lower overall drug usage than the standard treatment over the same treatment period. By using a smaller dosage, the treatment does not wipe out the drug-sensitive population. Hence, the treatment can be said to rely on the sensitive cancer population to control the resistance population due to their competitive advantage. **[PERSON] et al. (2018, 2019)**. Subsequently, the work by [PERSON] et al. was extended further in [PERSON] et al. [61], where an additional compartment was added in order to implement multi-drug therapy. By noting the limitation of parameter identification, [PERSON] et al. put forward an assumption that aids the parameter fitting process: parameters corresponding to cancer cells are similar across different patients. What distinguishes the cancer progression between different patients is the initial relative size of the cancer subpopulations. Using this assumption, [PERSON] and colleagues parametrized the parameters and fixed them across patients. They then carry out the data fitting process only on the initial size of each cancer subpopulation. They argue that the identifiability of their parameter estimation process is supported by the theorem in [PERSON]'s paper [62]. Their work shows that the incorporation of an additional drug (adding of docetaxel to abiraterone) may further delay the onset of treatment failure.