## 1 Introduction  
Perivascular spaces (PVSs), also known as Virchow-Robin spaces, are the pial-lined and fluid-filled cavities surrounding the walls of arteries, arterioles, veins, and venules in the brain [1]. PVSs have a significant role in the glyphmatic system and waste clearance process [2, 3] and are key to regulating immunological responses [4]. Several studies on enlarged perivascular spaces (EPVS) in the brain indicate that their presence is an indication of neuronal dysfunctions and that they could serve as a biomarker to diagnose brain diseases such as dementia [5, 6, 7], multiple sclerosis [8], stroke [9, 10, 11, 12, 13], Parkinson's [14], and small vessel diseases [15, 16, 17, 18, 19]. Due to its soft tissue contrast and high-resolution features, magnetic resonance imaging (MRI) is the most often utilized imaging tool in the assessment of EPVS. In MRI scans, PVSs are observed as well-defined oval/rounded/tubular structures with smooth margins. However, quantification of EPVS is challenging due to the difficulty of discriminating them from lacunes (both lacunes and EPVS have a cerebrospinal fluid (CSF)-like intensity on T1-weighted and T2-weighted MRI and are observed as hyperintensities on T2-weighted MRI and hypo-intensities on T1-weighted MRI scans [20, 21]). In general, clinical studies manually count the number of EPVS presented in a region of interest [20, 22, 23, 24] (basal ganglia, hippocampus, midbrain, and centrum semiovale) and rate the presence of EPVS on a five-point scale [20]. However, there is an intra- and inter-reader variability of detecting EPVS due to their tiny structures and morphological ambiguity [20]. In brain disease diagnosis, the manual assessment of EPVS is the key step but is challenging and time consuming due to (i) the difficulty in discriminating the EPVS from brain lesions with similar appearances and (ii) the existence of multiple EPVS in single scans. Several studies have developed automatic and semi-automatic methods for EPVS segmentation. For instance: an automated EPVS segmentation method based on Haar-like features was proposed in [25]. The authors of [26] proposed an interactive segmentation method based on intensity thresholding, while those of [27] used a Frangi filter to enhance EPVS area and performed segmentation of individual EPVS. Recent studies have begun to apply deep learning to EPVS quantification as the popularity of deep learning algorithms have grown due to their superior performance. For example, [28] proposed a multi-channel and multi-scale strategy approach for segmenting EPVS using 7 Tesla MRI scans, while [29] proposed a GP-Unet model for segmenting EPVS using 1.5 Tesla MRI scans. However, most of these methods focus on the segmentation and the detection of the EPVS, and as a result they often lead to inaccurate EPVS rating scores due to their low segmentation performance (existing segmentation methods do not satisfy the performance parameters). Further, the production of the training dataset for deep learning-based segmentation requires the medical experts to manually label EPVS as masks--a time-consuming and laborious process. This is because as many as 20-50 EPVSs can be contained in a single slice. In contrast, counting the EPVS as a rating score is significantly simpler. The development of methods for directly rating the EPVS could overcome the mentioned drawbacks in existing segmentation methods. In particular, since clinical decisions are made based on the rating score of EPVS, the direct measure of the EPVS rating score is a more convenient way--it bypasses the labor-intensive segmentation practice. Therefore, to deal with the above problems, we propose the direct estimation model of EPVS rating without segmentation or detection. Currently, only one study reports a successful segmentation-free quantification of EPVS using a 3D regressive convolutional neural network (CNN) [30]. However, this study used 7.0 Tesla MRI scans for its evaluation.