## 4 Discussion and Conclusions  
The main goal of this study was to evaluate the possibility of estimating the parameters of the two-compartment filtration model using an artificial neural network. As shown in the Results section, especially in Table 4, overall the task was accomplished successfully. The estimated single- and two-kidney estimates of glomerular filtration rate fall into the physiologically feasible range, and are close to the reference values calculated with the Trust Region Reflective algorithm (a non-linear least-squares curve-fitting method), as well as those measured by blood tests. Comparison with the latter method, which serves as a gold standard in clinics, appears particularly optimistic. The mean difference between the corresponding measurements is to \(\mu_{d}~{}=~{}2.35\) mL/min/1.73 m\({}^{2}\), with the agreement interval \(\kappa~{}=~{}\mu_{d}\pm 36.16\) mL/min/1.73 m\({}^{2}\). This interval is even narrower than the one obtained for the NLLS method (\(\kappa~{}=~{}\mu_{d}\pm 63.97\) mL/min/1.73 m\({}^{2}\)), which signals the greater precision of the designed MLP structures. This effect was achieved thanks to the usage of dropout layers between the hidden ones, and the training of the network with data vectors purposely corrupted with noise. Although the trained MLP network performs relatively well in predicting \(K^{trans}\), \(v_{p}\) and \(\Delta\) parameters, it actually fails in estimating accurate values for the VIRF decay time constant \(T_{g}\). The corresponding \(R^{2}\) determination coefficients calculated for the test sets do not exceed the value of 0.64, which is remarkably less than for the other three parameters (0.85-0.92). \(T_{g}\), however, does not affect the calculation of GFR, which depends only on \(K^{trans}\). In contrast with the NLLS procedure, a neural network does not actually fit a modeled curve to the input data. A failure in predicting a given parameter value does not necessarily cause an error in another parameter's estimation. Figure 12: (a–d): Box plots showing distribution of 2 CFM parameter values estimated by MLP and NLLS methods. Provided \(p\)-values were obtained for the \(t\)-test under the null hypothesis that mean estimated values are the same for both methods. It is instructive to compare the results obtained in this study to similar measurements presented in [16], partly conducted on the same subjects. The scores presented therein were obtained using a conventional curve-fitting approach, and were divided with respect to examination session. The mean differences and limits of agreement reported in their study were \(\mu_{d}\ =\ 1.5\) mL/min/1.73 m\({}^{2}\) and \(\kappa\ =\ \mu_{d}\pm 43.2\) mL/min/1.73 m\({}^{2}\) (Session 1), and \(\mu_{d}\ =\ 6.1\) mL/min/1.73 m\({}^{2}\) and \(\kappa\ =\ \mu_{d}\pm 31.9\) mL/min/1.73 m\({}^{2}\) (Session 2). Hence, our MLP-based method estimates true GFR values at a comparable level, demonstrating either slightly better (in the case of Session 1) or poorer (Session 2) precision. Apart from different parameter estimation methods, the observed discrepancies may be caused by other factors, including various post-processing algorithms, e.g., image registration and segmentation. Above all, a different pharmacokinetic model has been utilized, namely the [PERSON]'s two-compartment separable model [7]. Regarding the PK models themselves, the most common of them are often claimed to be too simple to properly represent patient-specific biophysiological processes. Then, even the most precise parameter estimation may not lead to accurate GFR calculation. There have been several attempts to develop more advanced models, including multi-compartmental [32] or patient-specific models [33], as well as those with a modified functional form of CA retention in the renal tissue compartments [34]. As such, the reliability of our proposed MLP-based method for DCE-MRI processing should be further examined with respect to various mathematical models of kidney perfusion.