# Numerical Modeling of Venous Outflow from the Cranial Cavity in the Supine Body Position
## 2 Materials and Methods
Figure 6: Model of the cerebral venous outflow comprising the internal jugular vein (**above**) and the vertebral venous plexus (**below**). Magenta squares represent the probes—in these areas flow characteristics was measured. Flow is from the left to the right. In this model, there are no stenoses in the internal jugular vein; consequently, most of the outflow utilizes the jugular pathway. The flow was simulated using 2500 consecutive steps, each of them lasting 0.28 ms, which in total was equivalent to 0.7 s of the real-time flow. Using our computer, it took around 24 h of computational time for each case. The parameters of the fluid were set up to be similar to those observed during blood flow in the IJV. The velocity component along the long axis of the model for the initial field, of which the area was 2.08 cm\({}^{2}\), was set up at 8 cm/s, which enabled the inflow of fluid into the model without strictures in the jugular route at the level of 400 mL/min, which was equal to approximately 50% of the physiological cerebral blood flow [22]. The velocity profile in the initial field was flat and constant. The walls of all models were rigid. The initial pressure was 750 Pa (equivalent to 7.65 cm H\({}_{2}\)O; a physiological pressure in the IJV in the supine body position). The dynamic viscosity of the fluid was set to 2.78 \(\times\) 10\({}^{-3}\) kg/m/s (which is the dynamic viscosity of whole blood at 37 \({}^{\circ}\)C). The density of the modeled fluid was constant and set at 1055 kg/m\({}^{3}\), which is the density of blood at room temperature [23; 24]. We augmented the model of cerebral venous outflow in the supine body position, which has been validated in our previous study [22], with two strictures (Figure 7). The first one was positioned at the beginning of the IJV, mimicking the nozzle-like stenosis evoked by a narrow jugular foramen, an enlarged transverse process of the atlas, or an elongated styloid process of the temporal bone. Such pathological strictures can be seen in some patients, particularly those presenting with neurological symptoms [12]. The second stricture was positioned at the end of the IJV, mimicking a stenotic jugular valve. Such aberrant jugular valves are also found in some patients [14]. Figure 7: Model of the cerebral venous outflow comprising the internal jugular vein (**above**) and the vertebral venous plexus (**below**); magenta squares represent the probes. Flow is from the left to the right. There are 2 strictures in the internal jugular vein: in the beginning (white arrow) and at its end (black arrow). In the model with stenoses, part of the flow is shifted towards the vertebral pathway. We constructed models with following types of upper JIV structures:  
* No stenosis in the upper part of the JV (cross-sectional area of 0.94 cm\({}^{2}\));
* Minor elliptically shaped stenosis in the upper part of the JIV (cross-sectional area of 0.42 cm\({}^{2}\));
* Minor irregularly shaped stenosis in the upper part of the JIV (cross-sectional area of 0.60 cm\({}^{2}\));
* Major elliptically shaped stenosis in the upper part of the JIV (cross-sectional area of 0.24 cm\({}^{2}\));
* Major elliptically shaped stenosis in the upper part of the JIV (cross-sectional area of 0.24 cm\({}^{2}\)) with opening positioned eccentrically;
* Major elliptically shaped stenosis in the upper part of the JIV (cross-sectional area of 0.24 cm\({}^{2}\)) with a longer narrowed segment (12 mm instead of 2.5 mm);
* Very narrow elliptically shaped stenosis in the upper part of the JIV (cross-sectional area of 0.07 cm\({}^{2}\));
* Very narrow elliptically shaped stenosis in the upper part of the JIV (cross-sectional area of 0.07 cm\({}^{2}\)) with opening positioned eccentrically;
* Major stenosis in the upper part of the JIV comprising 2 eccentrically positioned openings (cross-sectional area of 0.48 cm\({}^{2}\));
* Major stenosis in the upper part of the JIV comprising eccentrically positioned short flattening (cross-sectional area of 0.48 cm\({}^{2}\)).