## 4 Mathematical Models in Clinical Settings
### Uncertainty, Identifiability, And Sensitivity  
Bootstrapping and the ensemble Kalman filter are two standard methods that have been used to study the uncertainty in prostate cancer models [41; 49]. As shown in those works, existing models often contain large uncertainty in their prediction, especially when longer forecasts are needed to make a decision on optimal schedule, see Figure 7a. Other variance-based uncertainty quantification methods can also be applied for similar purposes, such as the work by [PERSON] et al. [71]. In the case of sensitivity analysis, a standard approach is to vary the parameters one by one by some percentage and evaluate the corresponding changes at a fixed time point [47]. This method can also be extended to study the sensitivity over the entire treatment [50; 92]. The advantage of studying the sensitivity of a parameter over time is evident in intermittent and adaptive therapy. Since there are orders of magnitudes difference for certain variables (such as PSA level) for different phases of the treatment, the sensitivity of the parameter is also affected, see Figure 8. Thus, having a better understanding of how the sensitivity of each parameter changes as the treatment progresses can provide a better tool to optimize treatment. Additionally, sampling-based methods to account for simultaneous effects of all parameters is also possible; however, this should be done in a relatively small range after the parameters have been identified for a specific patient.  
There are a variety of methods to estimate the parameters of a model. However, recently, when the identifiability of some prostate cancer models were examined, [PERSON] et al. [49] show that these models are not practically identifiable. They further show that for an unidentifiable model, different sets of parameters may yield an indistinguishable fitting but result in vastly different forecasting, see Figure 7b. Such a result is troublesome as it could undermine the applicability of mathematical models in a clinical setting. To take into account these issues, some researchers rely on the 2n + 1 law by [PERSON] [62]. However, this law requires models to be structurally identifiable--a condition that is often not examined when applying the law. Alternatively, if one carries out the sensitivity analysis of a model before hand, then information revealing the least sensitive parameters could be used to enhance the identifiability of the model. The issue of identifiability ties back to the complexity of the model and the limited data in clinical settings. It essentially requires a sufficient number of data points and data types, to uniquely determine the parameters of a model. In [PERSON] et al. [49], the authors suggest the use of an observer experiment to quantify the minimum required data that would allow the model to be identifiable. The suggested technique is to use the Fisher information matrix because of its ability to test different combinations of data sets easily. While the process is tedious, such information can provide insight into the types of data needed for a model to be useful. Additionally, if the functional forms of some parameters are available (either through study of parameter evolution or study of the nonlinear relationships between parameters), it may directly address the issue of model identifiability [93].