## 4 Mathematical Models in Clinical Settings
### Optimal Schedule and Patient Classification  
Due to the lack of a gold standard when it comes to determining a treatment schedule for IAS, its full benefits may not be realized [20]. Furthermore, in some instances, patients may benefit more from CAS. To tackle this issue, [PERSON] and colleagues studied patient classification based on whether a patient would benefit more or less from IAS as compared to CAS [40]. Another classification system was introduced by [PERSON] et al. [87], where the type of treatment resistance is studied based on cell death rate analysis. In treating prostate cancer, mathematical models have been used to study potential optimal treatment schedules that may give the best chance for patients. While various studies have been done on this topic, the goal of an optimal schedule has varied among them. For instance, [PERSON] and [PERSON] [94] choose the objective to be the minimization of the amount of time a patient is on treatment while still keeping IAS effective in controlling the cancer progression. [PERSON] et al. [42] instead focus on delaying the relapse of cancer as long as possible, and [PERSON] et al. [95] study three potential objectives: minimizing average tumor volume, tumor mass variance, or average density of androgen independent cells. While these are obvious objectives to be minimized for prostate cancer(or cancer in general), other creative alternatives such as minimizing cancer activity at any given time, or PSA doubling rate, may provide better control of cancer.  
The study of drug combinations can be considered as a subset of the optimal schedule. Various studies have been done on finding the best combinations of drugs for certain cases [52; 61]. While these  
Figure 8: Example of varying sensitivity for parameter during IAS-figure reproduced from [PERSON] et al. [92] with permission, which is distributed under a Creative Commons Attribution (CC BY) license. (**a**) The sensitivity index of the mutation parameter \(m_{1}\) with respect to androgen independent cancer cells \(x_{2}\) for intermittent treatment. (**b**) The sensitivity index of the mutation parameter \(m_{1}\) with respect to androgen dependent cancer cells \(x_{1}\) for intermittent treatment. Note that while the dynamical behaviors of the sensitivity indexes can be non-trivial, they should be explainable using biological understanding of the system.  
Figure 7: (**a**) Figure reproduced from [PERSON] et al. [41], with permission, which is distributed under a Creative Commons Attribution (CC BY) license. The spread denotes the 80% confidence interval using the bootstrap method, where the red crosses represent a patient’s data. (**b**) Figure reproduced from [PERSON] et al. [49] with permission, which is distributed under a Creative Commons Attribution (CC BY) license. In the fitting portion, the fittings using five different sets of parameters are nearly indistinguishable. However, in the forecasting portion, only one set of parameters provides accurate forecasting.  
studies hold promises, their implications hinge on how well the mathematical models represent the underlying biological system, which means drawing definitive conclusions is difficult due to the aforementioned problems with model validation. Furthermore, finding an optimal time (free terminal time) for a treatment may aid the design of a treatment, especially in the case of adaptive therapy. A comprehensive collection of tutorials for optimal control application in bio-sciences are presented by [PERSON] and [PERSON] [96].