## 4 Discussion  
The aim of this study was to compare the global and regional CVR between patients with PCS\({}_{\text{fmc}}\), patients with PCS\({}_{\text{mn}}\), and healthy controls using bh-fMRI. To date, the diagnosis of PCS remains a diagnosis of exclusion [2], making it challenging for clinicians and unsatisfactory for patients. In this study, we aimed to investigate whether CVR measured with bh-fMRI could be a complementary marker for the diagnosis of PCS, which could influence patient management and therapy. The CVR can change over time in patients with different neurological manifestations and may, therefore, be a biomarker for disease progression and treatment response [6,19]. Furthermore, the exact pathomechanism behind PCS is an ongoing topic of research [5,6]. Therefore, an additional motivation for this study was to investigate whether vascular damage is the most important pathomechanism, which may also influence long-term treatment options. Treatment options that affect endothelial function, such as beta-blockers, angiotensin-converting enzyme inhibitors, angiotensin receptor blockers, and statins, have already been discussed [11]. In line with the findings of [PERSON] et al. [15], patients with PCS in this pilot study showed a slight trend toward lower CVR. However, no significant global or regional differences in the percentage signal change were observed after adjusting for time since infection, age, sex, and white matter lesions. There were no significant global or regional differences in TTP after adjustment for time since infection, age, sex, and white matter lesions. It was noticeable that the patients with PCS tended to show a reduced TTP (approximately 3 s). A possible explanation could be the pulmonary impairment of PCS patients as part of the SARS-CoV-2 infection. Pulmonary diseases have an impact on the change in PaCO\({}_{2}\)[16]. A limited pulmonary reserve in patients with PCS could result in a different baseline PaCO\({}_{2}\) and a faster increase in the PaCO\({}_{2}\), and, therefore, a lower TTP. In this study, there was no significant correlation between the CVR and white matter changes. One reason might be that it is unclear whether the observed white matter lesions were due to SARS-CoV-2 infection, although we controlled for age to account for age-related microvascular lesions. However, it is likely that most of the observed lesions were nonspecific. Another possible explanation is that structural vascular changes that induce anatomical white matter changes are expected to occur later than functional changes [6] and may not yet be evident in PCS patients. According to previous studies, neurological symptoms are not associated with white matter lesions [34,35]. Several previous studies investigating the CVR with TCCD in patients with acute SARS-CoV-2 infection described significantly lower CVR in PCS patients than in control groups [19, 20]. [PERSON] et al. reported contrary results and found no significant differences between PCS patients and healthy controls [21]. Compared with the study of [PERSON] et al. and most of the other studies using TCCD, the present study involved a longer time period between the SARS-CoV-2 infection and the acquisition of MRI data [6, 15]. This might be one reason why no significant CVR differences were reproduced. [PERSON] et al., who compared the CVR between neurological PCS patients with a relatively long time period since infection (approximately 10 months) and controls, found significant differences in the CVR between the two groups [18]. However, in this study, the time period between SARS-CoV-2 infection and MRI was even longer (approximately 20 months). This study has limitations. One limitation was the small sample size, which may account for the lack of significance of the results. To overcome this limitation, we aimed to provide more robust estimates of the CVR using propensity score matching to reduce bias and using bootstrapping resampling to provide confidence intervals.