# Numerical Modeling of Venous Outflow from the Cranial Cavity in the Supine Body Position
## 4 Discussion
Flow abnormalities resulting in flow separation can severely impair blood flow, and IJVs are probably one of those blood vessels where these phenomena are of clinical consequence [45]. Still, it should be emphasized that unlike cerebral and aortic aneurysms, we are very far from comprehending these flow phenomena. Our study could be seen as one of the earliest initial steps in this process. We acknowledge that there are limitations to our study. Firstly, to facilitate simulations, we built models comprising only one side of the cerebral outflow. Models with two IJVs could provide further insight into this problem, especially if only one IJV is affected. Secondly, valve leaflets in our models were stiff and immobile. In reality, they exhibit different levels of elasticity and movability. In our other in silico study, we demonstrated that there is a dynamic interplay between stenoses located upstream and valves located downstream, and that a relevant stenosis changes the valve geometry to the unfavorable one [46]. Thirdly, flow volume measurements using the \"probes\" were not very precise. Yet, even such an approximate assessment could provide useful insight into the problem studied. Besides, flow modeling using our CFD package was probably prone to developing significant errors when the flow resistances in the studied fields were high; thus, we obtained very different total flow volumes in particular models (see Table 1). Other limitations of this study are associated with the use of simplified models instead of the real morphology-based ones. Besides, the interplay between the veins and surrounding tissues should also be modeled. Also, in living subjects, the vertebral veins dilate when they become the primary outflow route from the brain (e.g., in the sitting or standing humans, but also in a case of IJV stenosis). In our simulations, the morphology of the vertebral outflow route remained unchanged, which probably accounted for decreased flow volumes in the models with stenoses (see Table 1). It should also be mentioned that in this study, the fluid was considered Newtonian. In reality, blood is a non-Newtonian fluid. Since blood is a shear-thinning fluid, slowing down its flow is associated with a higher flow resistance than that of a Newtonian fluid, which probably resulted in non-adequate flow modeling. However, the level of errors associated with non-Newtonian characteristics of blood is difficult to estimate. Of note, flow modeling of a non-Newtonian fluid cannot be conducted using such a simple CFD package, such as the one used in our study. Besides, non-Newtonian characteristics of blood is not linear, it depends on multiple factors, such as hematocrit, flow velocity, turbulences, blood plasma composition, and interactions with the endothelium. Therefore, even more powerful CFD software and computers would not solve all these variables. Yet, at least regarding the problem studied in this paper, all these shortcomings do not seem very relevant. Since this study was aimed at understanding the clinical relevance of the strictures in the IJV, despite a lack of precise flow estimations (like in the engineering studies) and alongside comparisons of these simulations to the body of current medical evidence, any resulting inaccuracies from the use of Newtonian fluid do not seem substantial. Some uncertainties regarding the interpretation of imaging studies could be explained by our CFD modeling. For example, morphological abnormalities of the IJV in the form of flattening of its upper part at the level of the first cervical vertebra were found in MR examination more frequently in multiple sclerosis patients than in healthy controls [20; 21; 47; 48; 49]. Still, in a majority of such patients, both color Doppler sonographic and catheter venography examinations were unable to reveal any hemodynamically significant stenoses at this level. Our numerical simulations provide an explanation of this discordance.