A Solution of the Junction Riemann Problem for 1D Hyperbolic Balance Laws in Networks including Supersonic Flow Conditions on Elastic Collapsible Tubes

The numerical modeling of one-dimensional (1D) domains joined by symmetric or asymmetric bifurcations or arbitrary junctions is still a challenge in the context of hyperbolic balance laws with application to flow in pipes, open channels or blood vessels, among others. The formulation of the Junction Riemann Problem (JRP) under subsonic conditions in 1D flow is clearly defined and solved by current methods, but they fail when sonic or supersonic conditions appear. Formulations coupling the 1D model for the vessels or pipes with other container-like formulations for junctions have been presented, requiring extra information such as assumed bulk mechanical properties and geometrical properties or the extension to more dimensions. To the best of our knowledge, in this work, the JRP is solved for the first time allowing solutions for all types of transitions and for any number of vessels, without requiring the definition of any extra information. The resulting JRP solver is theoretically well-founded, robust and simple, and returns the evolving state for the conserved variables in all vessels, allowing the use of any numerical method in the resolution of the inner cells used for the space-discretization of the vessels. The methodology of the proposed solver is presented in detail. The JRP solver is directly applicable if energy losses at the junctions are defined. Straightforward extension to other 1D hyperbolic flows can be performed.

Cerebral venous steal equation for intracranial segmental perfusion pressure predicts and quantifies reversible intracranial to extracranial flow diversion

Cerebral perfusion is determined by segmental perfusion pressure for the intracranial compartment (SPP), which is lower than cerebral perfusion pressure (CPP) because of extracranial stenosis. We used the Thevenin model of Starling resistors to represent the intra-extra-cranial compartments, with outflow pressures ICP and Pe, to express SPP = Pd–ICP = FFR*CPP–Ge(1 − FFR)(ICP–Pe). Here Pd is intracranial inflow pressure in the circle of Willis, ICP—intracranial pressure; FFR = Pd/Pa is fractional flow reserve (Pd scaled to the systemic pressure Pa), Ge—relative extracranial conductance. The second term (cerebral venous steal) decreases SPP when FFR < 1 and ICP > Pe. We verified the SPP equation in a bench of fluid flow through the collapsible tubes. We estimated Pd, measuring pressure in the intra-extracranial collateral (supraorbital artery) in a volunteer. To manipulate extracranial outflow pressure Pe, we inflated the infraorbital cuff, which led to the Pd increase and directional Doppler flow signal reversal in the supraorbital artery. SPP equation accounts for the hemodynamic effect of inflow stenosis and intra-extracranial flow diversion, and is a more precise perfusion pressure target than CPP for the intracranial compartment. Manipulation of intra-extracranial pressure gradient ICP–Pe can augment intracranial inflow pressure (Pd) and reverse intra-extracranial steal.

Shock wave propagation along the central retinal blood vessels

Retinal haemorrhage is often observed following brain injury. The retinal circulation is supplied (drained) by the central retinal artery (vein) which enters (leaves) the eye through the optic nerve at the optic disc; these vessels penetrate the nerve immediately after passing through a region of cerebrospinal fluid (CSF). We consider a theoretical model for the blood flow in the central retinal vessels, treating each as multi-region collapsible tubes, where we examine how a sudden change in CSF pressure (mimicking an injury) drives a large amplitude pressure perturbation towards the eye. In some cases, this wave can steepen to form a shock. We show that the region immediately proximal to the eye (within the optic nerve where the vessels are strongly confined by the nerve fibres) can significantly reduce the amplitude of the pressure wave transmitted into the eye. When the length of this region is consistent with clinical measurements, the CSF pressure perturbation generates a wave of significantly lower amplitude than the input, protecting the eye from damage. We construct an analytical framework to explain this observation, showing that repeated rapid propagation and reflection of waves along the confined section of the vessel distributes the perturbation over a longer lengthscale.