## 3 Results
### Selecting Network Configuration  
In the first phase of experiments, we aimed to select an architecture for the neural network suitable to solving the problem of fitting the 2 CFM to the DCE-MRI data. We began with deciding upon  
Figure 4: Examples of automatically determined arterial input function region-of-interests (ROIs) (**a**,**b**). For comparison, manually delineated ROIs placed directly above aorta bifurcation into iliac arteries are shown as red rectangles. The corresponding fitted model curves are shown in (**c**,**d**) for manual ROIs and in (**e**,**f**) for the automatic ones. Figure 5: Bland–Altman plots of agreement for manually (**a**) and automatically (**b**) determined AIF ROIs. Measurements were evaluated against normality using Shapiro–Wilk test. The obtained \(p\)-values were equal to 0.52 (iohexol-based glomerular filtration rate (GFRs)), 0.29 (image-derived GFRs with manual ROIs) and 0.12 (for image-derived GFRs with automatically determined ROIs), thus disallowing us to reject the null hypothesis of the data with a normal distribution. the number of hidden layers and their activation units. Figure 6 presents the learning curves for three configurations of two-layer architectures, whereas the plots shown in Figure 7 correspond to the three-layer network configurations listed in Table 2. Although the selected learning curves were obtained only for one of the leave-one-out repetitions, they are representative for all the subjects in the study. It can be seen that in all cases, the loss function initially decreases rapidly, and converges to a value in the range 0.01-0.13 depending on the network architecture and the existence of noise in the training data. However, for configurations without the dropout layer, it is observable that the MSE value in a validation subset heavily fluctuates until the very end of the training process. It must be noted that for the single-hidden-layer configurations (Figure 6), the problem with the convergence of the loss function also persists in the largest tested network, with 50 hidden units, despite even inserting a dropout layer before the output nodes. This observation, combined with potential overfitting (which manifests in larger errors in a validation sample, with respect to a training set), substantiates the use of two hidden layers. In the latter case, the loss function stabilizes after around 40 iterations for 'dropout' configurations with higher numbers of neurons (architectures number 6, 8, 10 and 12). It is especially pronounced if the training data was corrupted by noise. Apparently, increased data variability due to the presence of noise, along with the inclusion of a dropout layer, ensures better loss function convergence and helps prevent the network from overfitting. As can be observed in Figure 7 (second column), the loss function scores on the validation set start to surpass the values obtained for the training set quite early in the learning process. This overfitting dynamic is further confirmed by the MAPE scores obtained after 100 learning epochs (see Figure 8). Without the dropout mechanism, the error remains higher for the validation set than for the training one, for almost every network configuration. Figure 6: Learning curves for various two-layer MLP configurations—training set scores in blue, validation set scores in orange. The plots in the first two and second two columns correspond to architectures without and with the dropout layer, respectively. The curves in columns 1 and 3 were obtained for the noise-free training data, whereas in columns 2 and 4 for the data corrupted with Gaussian noise. Figure 7: Learning curves for various three-layer MLP configurations—training set scores in blue, validation set scores in orange. The plots in the first two and two second two columns correspond to architectures without and with the dropout layer, respectively. The curves in columns 1 and 3 were obtained for the noise-free training data, whereas in columns 2 and 4 for the data corrupted with Gaussian noise.