## 2 Methods and Materials
### Arterial Input Function  
Although in the proposed approach, the trained neural network enables the determination of GFR without annotation of a patient-specific arterial input function, information about the time course of the contrast agent's concentration in a tissue-feeding artery is required at the training stage. As further described, we use a subset of image-derived AIFs to construct a training data set. Therefore, in this section, we present our custom method for automatic AIF determination. Most frequently, AIF is determined based on signal averaged over a manually delineated region of interest near to the perfused organ [18; 19]. Specifically, in the case of renography it is recommended to locate the AIF region-of-interest (ROI) near the place where the aorta bifurcates into common iliac arteries. This location is in relative proximity to the kidney, and is simultaneously sufficiently far from the upper aorta section, where the signal coming from the blood that enters the imaging slab at high velocity is high, because of the inflow artifact and not because of the contrast agent [19]. Moreover, [PERSON] et al. in [18] confirmed the significant dependence of renal perfusion and the filtration coefficients on the size of the chosen AIF ROI. In [20], AIF was determined from voxels chosen semi-automatically on the maximum signal-enhancement maps as the brightest locations in the abdominal aorta, directly below the inlets of the renal arteries. Note that the manual or semi-automatic approaches suffer from poor reproducibility and subjectivism, and they can be time-consuming and can require the availability of well-trained staff. In addition, it may be necessary in such cases to obtain patient-specific hemodynamic parameters in order to adjust the DCE-MRI protocol to the arterial pulse wave, and thus reduce the inflow artifact. Therefore, there were several attempts made to automate determination of the AIF. A common approach employs clustering techniques [21]. On the other hand, in [22], the authors used Kendall's coefficient of concordance. These attempts, however, were either applied in preclinical studies, or to different organs such as the head or neck, but not to the kidney. We postulate a fully automatic procedure for determining a patient-specific AIF based on the clustering of abdominal aorta voxels' time-series. Our approach consists of three stages, as illustrated in Figure 2. The algorithm starts with delineation of the abdominal aorta region. This task, further decomposed into four steps, is accomplished through the vessel segmentation of one time frame of the DCE-MR image series. The appropriate time frame is selected for maximum signal enhancement in the bottom-central region of the image, near the aorta's bifurcation into common iliac arteries. The peak intensity value in this region corresponds to the time point when the CA fills the whole aorta lumen, thus ensuring the highest contrast between the background tissue and the aorta at its entire length. Next, the image is blurred by the low-pass Gaussian filter (\(\sigma\ =\ 3\) voxels) to suppress fine structures that could be misinterpreted as vessels. Such a smoothed image is then submitted to the vessel enhancement routine. In this step, we computed a multi-scale vesselness function in the form proposed by [PERSON] [23], using 5 scales of radius standard deviation ranging from 0.5 to 2.5 mm. As a result, the filtered image contains only the elongated bright structures. Apart from the aorta, it may contain smaller vessels, like renal arteries and some other elongated structures. In order to remove them, the flood fill operation is executed, with the lower threshold set to an empirically found value equal to 12.5% of the maximum image intensity. The upper bound is set to maximum, as we found the aorta to be the brightest structure visible in the time frames selected for segmentation. Similarly,the seed point of the flood fill operation was determined as the brightest point in the central part of the image along the coronal direction.