We developed a mathematical model of the cerebral venous outflow for the simulation of the average blood flows and pressures in the main drainage vessels of the brain. The main features of the model are that it includes a validated model for the simulation of the intracranial circulation and it accounts for the dependence of the hydraulic properties of the jugular veins with respect to the gravity field, which makes it an useful tool for the study of the correlations between extracranial blood redistributions and changes in the intracranial environment. The model is able to simulate the average pressures and flows in different points of the jugular ducts, taking into account the amount of blood coming from the anastomotic connections; simulate how the blood redistribution due to change of posture affects flows and pressures in specific points of the system; and simulate redistributions due to stenotic patterns. Sensitivity analysis to check the robustness of the model was performed. The model reproduces average physiologic behavior of the jugular, vertebral, and cerebral ducts in terms of pressures and flows. In fact, jugular flow drops from ϳ11.7 to ϳ1.4 ml/s in the passage from supine to standing. At the same time, vertebral flow increases from 0.8 to 3.4 ml/s, while cerebral blood flow, venous sinuses pressure, and intracranial pressure are constant around the average value of 12.5 ml/s, 6 mmHg, and 10 mmHg, respectively. All these values are in agreement with literature data. mathematical modeling; cerebral outflow; posture dependence; jugular veins collapse; collateral routes THE EXTRACRANIAL VENOUS SYSTEM represents an important determinant of the brain circulation, but its role in the pathology of the central nervous system is not fully understood yet (4,26). It is recognized that, in supine position, the jugular veins represent the main outflow route for the cerebral circulation (2,6,25,36,38), being able to carry most of blood flow from the brain and from other extracerebral territories (ϳ700 -720 ml/ min; Refs. 27, 33) with respect to a cerebral blood inflow of ϳ750 ml/min (45). However, the jugular venous system exhibits important flow limitation during upright posture changes, because the jugular veins tend to collapse as a consequence of the decrease of transmural pressure due to the gravitational field, causing a significant increase in resistance (3,7,12,14,26). In the absence of other routes for extracranial outflow, this flow limitation would have dramatic effects on the cerebral circulation, since, apart from brief transient time intervals, the average cranial arterial inflow is expected to be equal to the cranial venous outflow for the mass preservation.As a consequence, large attention has been devoted to the venous circulation in the upright state, in an effort to understand which alternative routes can carry the brain venous outflow. It has long been postulated that the vertebral venous system may provide an important alternative route for venous outflow when standing or sitting; this was f...