# The Impact of Intermittent Androgen Suppression Therapy in Prostate Cancer Modeling
## 5 Discussion  
Mathematical models for prostate cancer have made significant progress since the work of [PERSON] (1975). Consequently, recent approaches have been focused on the application aspects in clinical settings. However, there are still obvious questions without an easy answer; for example, how one would define hormonal resistance in a rigorous mathematical sense and yet still meaningful in applications, or the identifiability of the model parameters. These difficult questions, that link theoretical work and application, ask for mechanistic models that can test theories against data. To this end, we aim to build upon a series of previous works to create a model with direct links to clinical data. We study the model and explore various aspects relating to using it. In this section, we summarize our findings and point out the limitations of our work. (A1) From Table 3, both the BK model and the new model achieved similar accuracy for fitting. Since we used a two-part fitting scheme with exponential weight, the fitting was good when PSA quickly relapsed after the patient was put off treatment; see patients 39, 77, and 79 in Figure 1. On the other hand, when PSA took a longer time to reach the peak, the accuracy of fitting was not as good as in the previous cases; for instance, in patient 28 in Figure 1. In other words, because we chose to use a fixed exponential weight for fitting, the earlier the data points, the less weight that those data points contributed in the parameter estimation process. Thus, for patients with an abnormally long first cycle, most of the earlier data were essentially not accounted for during fitting; see patient 66 in Figure 3. To resolve this problem, a possible scheme is to use a linear weight. Specifically, we would first normalize the weight contribution from the peak PSA level to be 1. Then, the weight contribution of the points before the peak is the relative height with respect to the peak. This would ensure that earlier points have a more significant contribution to the fitting. While it is well-suggested that individual patients should have their unique parameter set, we further suggest that the process of studying the patient data should be unique, in order to capture the patient's characteristics; for example, a patient-specific weight for fitting in this case. (A2) From Table 4, the new model slightly outperformed the BK model, in terms of prediction, which is noteworthy as the new model used fewer parameters for fitting. An interesting observation is that the new model did not overshoot forecasting, perhaps due to the parameter ranges established in Table 2. This can become a problem, if there is a stable equilibrium below the upper threshold PSA level required for putting the patient on treatment. In such a case, the model would be unrealistic for the parameter values which influence the equilibrium. While this problem can be mediated by forcing hormonal resistance within five years, as done in ([PERSON] et al., 2018), it is perhaps better done by optimization of parameter ranges via mathematical analysis. We further note that when the data had good regularity, in terms of the similarity in the shape of data for different cycles (see patients 28, 77, and 79), the forecasting was quite accurate. However, when unforeseen changes occur in the data (patient 39), the model may fail to capture this change. Hence, the use of a stochastic model or uncertainty quantification is necessary for such applications. (A3) [PERSON] et al. (2018) noted the problem of the sudden jump in PSA between the transition of the on- and off-treatment intervals in the fitting and forecasting of the BK model. Our result suggests that the main cause was the abnormally large value of the rate of primary androgen production (\(\gamma_{1}\)). By adjusting this value in accordance to [6], we resolved this problem, as seen for patients 28, 39, 77, and 79 in Figure 1. [PERSON] et al. [18] also mentioned a delay that sometimes occurs during the relapsing period of PSA; for example, in patient 77.