# The Impact of Intermittent Androgen Suppression Therapy in Prostate Cancer Modeling
## 2 Model and Method
### Sensitivity Analysis for the New Model  
Sensitivity analysis of the parameters plays an important part in evaluating and improving the robustness of a model. Previous studies focus on the local sensitivity at a fixed time point [8; 18]. While this can provide valuable information, the dynamics of IAS therapy are broken down into on- and off-treatment intervals, with distinct characteristics in both parts. Additionally, understanding the effect each parameter has on the dynamics of the model can lead to the fixing of certain parameters, during the parameter estimation process. If some of the less sensitive parameters can be fixed, then we essentially reduce the uncertainty in the remaining parameters, which increases their identifiability. Here, we employ the normalized sensitivity equation from section 5.5 in [20], to carry out the sensitivity analysis of the new model:  
\[S_{p}=\frac{\partial x}{\partial p}\frac{p}{x},\]  
where \(p\) is the parameter, and \(x\) is the variable of interest; for example, \(\mu\) and PSA level, respectively. Note that, since we will be varying each parameter by \(1\%\) with respect to their value, we can use the following approximation of the sensitivity equation:  
\[S_{p}=\frac{\partial x}{\partial p}\frac{p}{x}\approx\frac{x_{varied}-x}{0.01p }\frac{p}{x}=\frac{x_{varied}-x}{0.01x}, \tag{13}\]  
where \(x_{varied}\) is the value of \(x\) after increasing a parameter by \(1\%\). The result of the sensitivity analysis is summarized in Table 1.  
While the sensitivity process is perhaps more relevant when carried out on patient data, there is a large variation in the data between patients, and so finding a consensus among all patients can be difficult. Instead, we: (1) Synthesize a set of parameters within the range in Table 2, and a treatment schedule that produces the expected dynamics observed in clinical data. (2) Using these parameters, we carry out the sensitivity process by increasing one parameter at a time by \(1\%\) of its value. (3) We approximate \(S_{p}\) at each time point for 1 treatment cycle using Equation (13). The process is repeated for all parameters. Finally, we record the maximum order of sensitivity in Table 1.  
Table 1 shows the maximum effect that changing a parameter by \(1\%\) has on the variables. For instance, by looking at the intersection between \(c\) and \(A\), we see that changing \(c\) by \(1\%\) can cause \(A\) to change by a maximum order of \(-8\). To compare the sensitivity of these parameters, we add up their sensitivity coefficients. Doing so, we find that the parameters with combined order less than \(-10\) (least sensitive) are \(c\), \(K\), \(\delta_{1}\), \(\delta_{2}\), and \(d_{2}\). Since \(d_{1}\) and \(d_{2}\) serve similar purposes and \(d_{1}\) is a sensitive parameter, we choose to not to fix \(d_{2}\). In summary, for the fitting process of the new model, we fix \(c\), \(K\), \(\delta_{1}\), \(\delta_{2}\), and \(\gamma_{2}\). Note that we do not carry out the sensitivity analysis for \(\gamma_{2}\), due to its small value; in fact, we fix \(\gamma_{2}\) to 0 for the new model (although a small value would suffice). The values that we fix for \(c\), \(K\), \(\delta_{1}\), and \(\delta_{2}\) are 0.00015, 1, 5, and 5, respectively.