## 6 Comparison of Models  
We use data from the Vancouver Prostate Center (Vancouver, BC, Canada) to validate and compare the accuracy of each model. From the 109 patients registered, 103 were eligible for interruption of treatment, with a PSA response rate of 95% [9]. Using the criteria of having at least 20 data points for both androgen and PSA in the initial 1.5 cycles, we select 62 from those 109 patients. The individual PSA and androgen mean square error (MSE) are provided in Table 2 from these 62 selected patients. Notice that the PKN model did not include an androgen equation, and thus we cannot compare the fittings of androgen with the PKN model. For the PKN model, we interpolated androgen serum data using a cubic spline interpolation between every androgen data point. This created a function in terms of time that was utilized as \(A\) in  
Figure 4: Normalized sensitivities of Model 2. Figure 3: Normalized sensitivities of Model 1. (13) and (14). We implemented the method used by [PERSON] et al. [13] for generating future androgen levels by generating a rectangular function based on the average off and on-treatment serum androgen values. Parameter ranges were taken from PKN [13] and [PERSON] et al. [28], and the reader is referred to these papers for more details on forecasting serum PSA levels and parameter values of the PKN model. For every patient selected, we fitted 1.5 cycles of treatment and performed parameter estimation. Then, to measure the forecasting ability of every model, we ran the models for one more cycle of data using the parameters estimated from the initial 1.5 cycles. To compare models, we conduct simulations with MATLAB's (MATLAB 9.1, The MathWorks, Inc., Natick, MA, USA) built in function fmincon, which uses the Interior Point Algorithm, to find the optimum parameters for each patient. The algorithm searches for a minimum value in a range of pre-specified parameter ranges, which we estimated from various literature sources. We use this algorithm to minimize the MSE for PSA and androgen data. The MSE is calculated with the following equations:  
\[P_{error}=\frac{\sum_{i=1}^{N}(P_{i}-\hat{P_{i}})^{2}}{N},\]  
\[Q_{error}=\frac{\sum_{i=1}^{N}(Q_{i}-\hat{Q_{i}})^{2}}{N},\]  
where \(N\) represents the total number of data points, \(P_{i}\) represents the PSA data value, and \(\hat{P_{i}}\) the value from the model. Likewise, \(Q_{i}\) represents the androgen data value, and \(\hat{Q_{i}}\) the value from the model. We then use an equally weighted combination of both errors  
\[error=P_{error}+Q_{error},\]  
as our objective function, which is then minimized with fmincon. Figure 5 shows PSA fitting and forecasting simulations for patients 1, 15, 17, and 63. We selected these patients to display the typical behavior shown in all 62 patients. Patient 1 shows that Models 1, 2, and PKN fit data with about the same accuracy. However, PKN overshoots in forecasting and Model 2 outperforms Model 1 in forecasting. Patient 17 shows that PKN underestimates future PSA levels, but Models 1 and 2 both perform well. Patients 15 and 63 provide the cases where PKN does a better forecast while Models 1 and 2 still do better. The rest of the patients can be classified similarly. Table 2 documents the error of fitting 1.5 cycles of treatment and Table 3 displays the errors in forecasting one more cycle of treatment. On average, PKN and Model 1 perform prediction at the same level of accuracy. However, Model 2 performs prediction on average about three times better than the PKN model and Model 1.