Mathematical models of the flow in the basilar artery

The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. This study deals with the question whether a parabolic (Poiseuille) profile will have been established before the basilar artery divides into both posterior cerebral arteries. The inlet length (that is, the downstream distance needed for the flow to become approximately equal to the limiting Poiseuille flow) and velocity profiles have been computed from two- and three-dimensional mathematical models in which flow pulsatility and vessel wall distensibility have been neglected and the complex geometry of the junction has been taken into account in a simplified form. The results show that the flow at the end of the basilar artery is far from being parabolic and that an asymmetry in the entrance flow will be carried along towards the end of the basilar artery, thus affecting flows in the circle of Willis.     

Numerical simulations of the blood flow through vertebral arteries

Vertebral arteries are two arteries whose structure and location in human body result in development of special flow conditions. For some of the arteries, one can observe a significant difference between flow rates in the left and the right arteries during ultrasonography diagnosis. Usually the reason of such a difference was connected with pathology of the artery in which a smaller flow rate was detected. Simulations of the flow through the selected type of the vertebral artery geometry for twenty five cases of artery diameters have been carried out. The main aim of the presented experiment was to visualize the flow in the region of vertebral arteries junction in the origin of the basilar artery. It is extremely difficult to examine this part of human circulation system, thus numerical experiments may be helpful in understanding the phenomena occurring when two relatively large arteries join together to form one vessel. The obtained results have shown that an individual configuration and diameters of particular arteries can exert an influence on the flow in them and affect a significant difference between flow rates for vertebral arteries. It has been assumed in the investigations that modelled arteries were absolutely normal, without any pathology. In the numerical experiment, the non-Newtonian model of blood was employed.     

Blood flow dynamics in saccular aneurysm models of the basilar artery

Blood flow dynamics under physiologically realistic pulsatile conditions plays an important role in the growth, rupture, and surgical treatment of intracranial aneurysms. The temporal and spatial variations of wall pressure and wall shear stress in the aneurysm are hypothesized to be correlated with its continuous expansion and eventual rupture. In addition, the assessment of the velocity field in the aneurysm dome and neck is important for the correct placement of endovascular coils. This paper describes the flow dynamics in two representative models of a terminal aneurysm of the basilar artery under Newtonian and non-Newtonian fluid assumptions, and compares their hemodynamics with that of a healthy basilar artery. Virtual aneurysm models are investigated numerically, with geometric features defined by beta = 0 deg and beta = 23.2 deg, where beta is the tilt angle of the aneurysm dome with respect to the basilar artery. The intra-aneurysmal pulsatile flow shows complex ring vortex structures for beta = 0 deg and single recirculation regions for beta = 23.2 deg during both systole and diastole. The pressure and shear stress on the aneurysm wall exhibit large temporal and spatial variations for both models. When compared to a non-Newtonian fluid, the symmetric aneurysm model (beta = 0 deg) exhibits a more unstable Newtonian flow dynamics, although with a lower peak wall shear stress than the asymmetric model (beta = 23.2 deg). The non-Newtonian fluid assumption yields more stable flows than a Newtonian fluid, for the same inlet flow rate. Both fluid modeling assumptions, however, lead to asymmetric oscillatory flows inside the aneurysm dome.     

An immunohistochemical study on the innervation of SP-IR nerves in the cerebral arteries of the bent-winged bat

Summary The overall distribution of substance P (SP) immunoreactive (IR) nerves surrounding the cerebral arteries of the bent-winged bat were investigated immunohistochemically. In this microchiropteran species, the walls from the vertebral artery to the caudal part of the basilar artery have considerably well-developed plexuses of SP-IR nerves, whereas no demonstrable SP-IR fibers were found in the crostral part of the basilar artery, and in more rostrally located arteries the nerve supply was very sparse or occasionally lacking. This innervation pattern has not yet been established for the cerebral arterial systems of other mammals that have been studied under normal conditions, but it is very similar to the pattern of SP-IR innervation observed in the guinea pig and cat of which the trigeminal ganglia have been destroyed. From the combination of this and other immunohistochemical findings, it is suggested that SP-IR nerves innervating the vertebral and basilar arteries of the bent-winged bat originate from the upper cervical dorsal root ganglia (DRG) and enter the cranial cavity along the vertebral artery and through the meninges.Abbreviations BA basilar artery - CSN cervical spinal nerves - ICS internal carotid system - SCG superior cervical ganglion - SNB sympathetic nerve bundle - VA vertebral artery - VBS vertebro-basilar system     

An immunohistochemical study on the innervation of SP-IR nerves in the cerebral arteries of the bent-winged bat

The overall distribution of substance P (SP) immunoreactive (IR) nerves surrounding the cerebral arteries of the bent-winged bat were investigated immunohistochemically. In this microchiropteran species, the walls from the vertebral artery to the caudal part of the basilar artery have considerably well-developed plexuses of SP-IR nerves, whereas no demonstrable SP-IR fibers were found in the rostral part of the basilar artery, and in more rostrally located arteries the nerve supply was very sparse or occasionally lacking. This innervation pattern has not yet been established for the cerebral arterial systems of other mammals that have been studied under normal conditions, but it is very similar to the pattern of SP-IR innervation observed in the guinea pig and cat of which the trigeminal ganglia have been destroyed. From the combination of this and other immunohistochemical findings, it is suggested that SP-IR nerves innervating the vertebral and basilar arteries of the bent-winged bat originate from the upper cervical dorsal root ganglia (DRG) and enter the cranial cavity along the vertebral artery and through the meninges.     

Selective vasodilatation effect of sargahydroquinoic acid, an active constituent of Sargassum micracanthum, on the basilar arteries of rabbits

Sargahydroquinoic acid (2), a major active constituent of Sargassum micracanthum collected from the coast of the East Sea in Korea, showed a selective vasodilatation effect on the basilar arteries of rabbits. Therefore, treatment with sargahydroquinoic acid may selectively accelerate cerebral blood flow through dilatation of the basilar artery without lowering systemic blood pressure.     

Computational modelling of emboli travel trajectories in cerebral arteries: influence of microembolic particle size and density

Ischaemic stroke is responsible for up to 80 % of stroke cases. Prevention of the reoccurrence of ischaemic attack or stroke for patients who survived the first symptoms is the major treatment target. Accurate diagnosis of the emboli source for a specific infarction lesion is very important for a better treatment for the patient. However, due to the complex blood flow patterns in the cerebral arterial network, little is known so far of the embolic particle flow trajectory and its behaviour in such a complex flow field. The present study aims to study the trajectories of embolic particles released from carotid arteries and basilar artery in a cerebral arterial network and the influence of particle size, mass and release location to the particle distributions, by computational modelling. The cerebral arterial network model, which includes major arteries in the circle of Willis and several generations of branches from them, was generated from MRI images. Particles with diameters of 200, 500 and 800  $\upmu \hbox {m}$ and densities of 800, 1,030 and 1,300 $\hbox {kg/m}^{3}$ were released in the vessel’s central and near-wall regions. A fully coupled scheme of particle and blood flow in a computational fluid dynamics software ANASYS CFX 13 was used in the simulations. The results show that heavy particles (density large than blood or a diameter larger than 500  $\upmu \hbox {m}$ ) normally have small travel speeds in arteries; larger or lighter embolic particles are more likely to travel to large branches in cerebral arteries. In certain cases, all large particles go to the middle cerebral arteries; large particles with higher travel speeds in large arteries are likely to travel at more complex and tortuous trajectories; emboli raised from the basilar artery will only exit the model from branches of basilar artery and posterior cerebral arteries. A modified Circle of Willis configuration can have significant influence on particle distributions. The local branch patterns of internal carotid artery to middle cerebral artery and anterior communicating artery can have large impact on such distributions.     

Influence of inlet boundary conditions on the local haemodynamics of intracranial aneurysms

Haemodynamics is believed to play an important role in the initiation, growth and rupture of intracranial aneurysms. In this context, computational haemodynamics has been extensively used in an effort to establish correlations between flow variables and clinical outcome. It is common practice in the application of Dirichlet boundary conditions at domain inlets to specify transient velocities as either a flat (plug) profile or a spatially developed profile based on Womersley's analytical solution. This paper provides comparative haemodynamics measures for three typical cerebral aneurysms.

Three dimentional rotational angiography images of aneurysms at three common locations, viz. basilar artery tip, internal carotid artery and middle cerebral artery were obtained. The computational tools being developed in the European project @neurIST were used to reconstruct the fluid domains and solve the unsteady Navier–Stokes equations, using in turn Womersley and plug-flow inlet velocity profiles. The effects of these assumptions were analysed and compared in terms of relevant haemodynamic variables within the aneurismal sac. For the aneurysm at the basilar tip geometries with different extensions of the afferent vasculature were considered to study the plausibility of a fully-developed axial flow at the inlet boundaries.

The study shows that assumptions made on the velocity profile while specifying inlet boundary conditions have little influence on the local haemodynamics in the aneurysm, provided that a sufficient extension of the afferent vasculature is considered and that geometry is the primary determinant of the flow field within the aneurismal sac. For real geometries the Womersley profile is at best an unnecessary over-complication, and may even be worse than the plug profile in some anatomical locations (e.g. basilar confluence).     

On the effect of parent-aneurysm angle on flow patterns in basilar tip aneurysms: towards a surrogate geometric marker of intra-aneurismal hemodynamics

As a part of previous computational fluid dynamic (CFD) validation studies, particle image velocimetry (PIV) of two anatomically realistic basilar artery tip aneurysm models revealed two distinct types of flow (one of which has yet to be reported in the literature), characterized by the location and strength of the intra-aneurismal vortex. We hypothesized that these distinct "hemodynamic phenotypes" could be anticipated by a simple geometric parameter: the angle of the aneurysm bulb relative to the parent artery. An idealized basilar tip aneurysm model was constructed to allow independent control of this angle, and CFD simulations were carried out for angles ranging from 2 degrees to 30 degrees , these extremes corresponding to the angles measured from the two anatomically realistic models. The gross hemodynamics predicted by the idealized model for 2 degrees and 30 degrees were consistent with those seen in the corresponding anatomically realistic models. For the idealized model, the flow type switched at an angle between 8 degrees and 12 degrees . Sensitivity studies suggested that, near these angles, the hemodynamic phenotype was sensitive to inflow momentum. Outside this range, however, the parent-bulb angle appeared to be a robust predictor of hemodynamic phenotype. Our findings suggest that blood flow dynamics in basilar artery tip aneurysms fall into one of the two broad phenotypes, each subject to distinct hemodynamic forces. That the general features of these flow types may be anticipated by a relatively simple-to-measure geometric parameter could help ease the introduction of hemodynamic information into routine clinical decision-making.     

Effects of interrupted and uninterrupted occlusion of the basilar artery on cerebral blood flow, and on neurological and histological outcome in rats with subarachnoid hemorrhage.

Most neurosurgeons consider temporary vessel occlusion for aneurysmal clipping an effective technique that facilitates dissection between the aneurysm and the parent vessel. It is generally believed that repeated short periods of cerebral ischemia are safer for the brain than a single long episode. The aim of this study was to identify whether interrupted and uninterrupted vessel occlusion differs with regard to changes in brain tissue and cerebral hemodynamics after subarachnoid hemorrhage (SAH). Fifty Spraque Dawley rats (300-350 g) were placed under general anaesthesia and ventilated. The basilar artery was exposed through a transclival approach. Baseline local cerebral blood flow (LCBF) values was measured, and then the basilar artery was punctured, causing subarachnoid hemorrhage (SAH). Group I (n = 24) was subjected to 60 min of interrupted basilar artery occlusion, defined as 5 min of reperfusion after every 10 min of occlusion, group II (n = 26) 60 min of uninterrupted artery occlusion. Three days after completion of the experiment, each rat was neurologically evaluated and decapitated. Coronal brain slices were obtained and stained to assess infarct volume. Immediately after SAH, LCBF fell by 58% in group I, and by 52% in group II. In group I, each ischemic insult brought a similar reduction in LCBF, and after each release of the occlusion there was a rapid rise in flow. In group II, the LCBF values dropped initially and remained at low levels until the end of the study. The 2,3,5 triphenyltetrazolium chloride stained sections showed similar volumes of brainstem infarction in both groups (38.3 +/- 9.2 mm3 vs. 34.3 +/- 8.7 mm3, respectively; p > 0.05). The results suggest that there is no neuroprotective advantage to either interrupted or uninterrupted temporary blockage of blood flow during neurovascular procedures after SAH in the basilar artery region.     

Causes of variations in the pathway of the basilar and vertebral arteries

Artery loops at the root exit zones of cerebral nerves are regarded as causes of certain diseases, e.g. trigeminal neuralgia or hemifacial spasm. The factors, which may cause such loops and displacements of arteries, however, are still not known sufficiently. In order to find out more about such causes, 60 corpses were examined. We recorded the variations in the positions of vertebral and basilar arteries and correlated them with the respective age at the time of death. We found that those showing atypical artery positions and loops were generally of older age. We further examined possible influences of blood flow factors on variations of artery positions. Our sample indicated such influence of flow factors on displacements of basilar artery, but they seemed to be of lesser importance than the effect of ageing.     

Amplification of sumatriptan-induced contraction in rabbit saphenous vein but not in basilar artery

The modulation of serotonin (5-HT(1B/1D)) receptor-induced vascular contractility by histamine and U-46619 was compared in the rabbit basilar artery and saphenous vein. In the saphenous vein, histamine (5 x 10(-7) M) significantly increased the potency (from pEC(50) of 6.0 to 6.8) and efficacy (from 52.3% to 88.2%) of sumatriptan. Likewise, U-46619 (5 x 10(-9) M) also increased the potency (from pEC(50) of 5.9 to 6.6) and efficacy (from 51.8% to 92.1%) of sumatriptan in the saphenous vein. In contrast, equieffective concentrations of histamine (10(-7) M) and U-46619 (3 x 10(-9) M) failed to amplify contraction to sumatriptan in the basilar artery. Contraction to sumatriptan was inhibited by nitrendipine (10(-7) M) in the basilar artery but not in the saphenous vein, suggesting that different contractile signaling mechanisms are operating in these tissues. Furthermore, U-46619- and thrombin-induced contractility in the basilar artery were also not amplified by histamine, suggesting that lack of amplification of contraction in the basilar artery was not restricted to sumatriptan but was rather a characteristic of this cerebral vessel. These data suggest that in the in vivo plasma milieu sumatriptan will more markedly contract the peripheral saphenous vein than the basilar artery, a cerebral blood vessel.     

Participation of Lipoxygenase Products from Arachidonic Acid in the Pathogenesis of Cerebral Vasospasm

To examine the possible involvement of lipoxygenase products from arachidonic acid in the pathogenesis of delayed vasospasm after subarachnoid hemorrhage (SAH), we measured the contents of hydroxyeicosatetraenoic acids (HETEs) in the subarachnoid clot, the cerebrospinal fluid, and the basilar artery, using the canine "two-hemorrhage" model. Lipoxygenase activity in the subarachnoid clot and the basilar artery was measured, ex vivo, using samples obtained 7 days after SAH. For a quantitative analysis of HETEs, each sample was homogenized with either ice-cold saline or methanol. The lipid extract was then submitted to reverse-phase HPLC. The identity of each HETE was further confirmed using straight-phase HPLC and gas chromatography-mass spectrometry. When the basilar artery was homogenized with ice-cold saline, a significant increase in the 5-HETE content was observed on SAH day 8. However, when the artery was homogenized with methanol, HETEs were not detected. In the case of incubation in the presence of arachidonic acid and calcium ionophore A23187, the 5-lipoxygenase activity was remarkably increased in the basilar artery exposed to SAH, compared to that of normal dogs. The subarachnoid clot contained a significant amount of 12-HETE (average 1.8 nmol/g wet weight) from day 2 to day 8. The administration of 1,2-bis(nicotinamido)propane significantly ameliorated vasospasm in the two-hemorrhage model, simultaneously inhibiting the 5-lipoxygenase activity of the basilar artery. Our observations show that the activities of 12- and 5-lipoxygenases are significantly increased after SAH in the subarachnoid clot and the basilar artery, respectively.(ABSTRACT TRUNCATED AT 250 WORDS)     

Vasodilatation effect of farnesylacetones,active constituents of Sargassum siliquastrum,on the basilar and carotid arteries of rabbits

Two farnesylacetones, 311 and 312, major active constituents of Sargassum siliquastrum collected from the coast of the East Sea in Korea, showed a moderate vasodilatation effect on the basilar arteries of rabbits. Therefore, treatment with farnesylacetones 311 and 312 may selectively accelerate cerebral blood flow through dilatation of the basilar artery.     

Influence of exercise on dilatation of the basilar artery during diabetes mellitus.

Our goal was to examine whether exercise training alleviates impaired nitric oxide synthase (NOS)-dependent dilatation of the basilar artery in Type 1 diabetic rats. To test this hypothesis, we measured in vivo diameter of the basilar artery in sedentary and exercised nondiabetic and diabetic rats in response to NOS-dependent (acetylcholine) and -independent (nitroglycerin) agonists. To determine the potential role for nitric oxide in vasodilatation in sedentary and exercised nondiabetic and diabetic rats, we examined responses after NG-monomethyl-l-arginine (l-NMMA). We found that acetylcholine produced dilatation of the basilar artery that was similar in sedentary and exercised nondiabetic rats. Acetylcholine produced only minimal vasodilatation in sedentary diabetic rats. However, exercise alleviated impaired acetylcholine-induced vasodilatation in diabetic rats. Nitroglycerin produced dilatation of the basilar artery that was similar in sedentary and exercised nondiabetic and diabetic rats. l-NMMA produced similar inhibition of acetylcholine-induced dilatation of the basilar artery in sedentary and exercised nondiabetic and diabetic rats. Finally, we found that endothelial NOS (eNOS) protein in the basilar artery was higher in diabetic compared with nondiabetic rats and that exercise increased eNOS protein in the basilar artery of nondiabetic and diabetic rats. We conclude that 1) exercise can alleviate impaired NOS-dependent dilatation of the basilar artery during diabetes mellitus, 2) the synthesis and release of nitric oxide accounts for dilatation of the basilar artery to acetylcholine in sedentary and exercised nondiabetic and diabetic rats, and 3) exercise may exert its affect on cerebrovascular reactivity during diabetes by altering levels of eNOS protein in the basilar artery.     

Numerical simulation of pre- and postsurgical flow in a giant basilar aneurysm

Computational modeling of the flow in cerebral aneurysms is an evolving technique that may play an important role in surgical planning. In this study, we simulated the flow in a giant basilar aneurysm before and after surgical takedown of one vertebral artery. Patient-specific geometry and flowrates obtained from magnetic resonance (MR) angiography and velocimetry were used to simulate the flow prior to and after the surgery. Numerical solutions for steady and pulsatile flows were obtained. Highly three-dimensional flows, with strong secondary flows, were computed in the aneurysm in the presurgical and postsurgical conditions. The computational results predicted that occlusion of a vertebral artery would result in a significant increase of the slow flow region formed in the bulge of the aneurysm, where increased particle residence time and velocities lower than 2.5 cms were computed. The region of slow flow was found to have filled with thrombus following surgery. Predictions of numerical simulation methods are consistent with the observed outcome following surgical treatment of an aneurysm. The study demonstrates that computational models may provide hypotheses to test in future studies, and might offer guidance for the interventional treatment of cerebral aneurysms.     

Numerical simulations of the blood flow in the patient-specific arterial cerebral circle region

The Cerebral Circle Region, also known as the Circle of Willis (CoW), is a loop of arteries that form arterial connections between supply arteries to distribute blood throughout the cerebral mass. Among the population, only 25% to 50% have a complete system of arteries forming the CoW. 3D time-varying simulations for three different patient-specific artery anatomies of CoW were performed in order to gain a better insight into the phenomena existing in the cerebral blood flow. The models reconstructed on the basis of computer tomography images start from the aorta and include the largest arteries that supply the CoW and the arteries of CoW. Velocity values measured during the ultrasound examination have been compared with the results of simulations. It is shown that the flow in the right anterior artery in some cases may be supplied from the left internal carotid artery via the anterior communicating artery. The investigations conducted show that the computational fluid dynamic tool, which provides high resolution in both time and space domains, can be used to support physicians in diagnosing patients of different ages and various anatomical arterial structures.     

The variability of the circle of Willis: univariate and bivariate analysis

The results are presented of a statistical analysis of the variability of the circle of Willis using univariate and bivariate methods. For this purpose 100 circles of Willis were available. From each circle 19 indexes of arterial size were determined, the basilar artery was measured in two places. Half the circumference was measured. This data yielded no evidence of differences between left- and right-sided vessels in the sample. An important source of variation is the general size of all vessels considered. When the data are cleared from this general size variation, correlation coefficients reveal interesting relations between the vessels. The posterior communicating arteries are strongly related to the ipsilateral carotid artery, whereas a strong inverse relationship exists with the basilar artery and the precommunicating part of the ipsilateral posterior cerebral artery. These relationships can be understood from the expected patterns of the blood flow in these vessels. Similar relationships can be found in the anterior part of the circle of Willis and in the vertebro-basilar junction. In a different manner, based on previous haemodynamic studies, the relation between blood flow and vessel size within the circle of Willis can be demonstrated by relating the ratios of the sizes of afferent and efferent arteries to the sizes of the posterior communicating arteries, an "intuitive" model. The supposed correlations of the outcome of this "intuitive" model with the size of the communicating arteries appeared to by highly significant. It is concluded that the variations of the circle of Willis are related to the individual variations of the blood flow in this arterial network.     

Gender differences in myogenic regulation along the vascular tree of the gerbil cochlea

Regulation of cochlear blood flow is critical for hearing due to its exquisite sensitivity to ischemia and oxidative stress. Many forms of hearing loss such as sensorineural hearing loss and presbyacusis may involve or be aggravated by blood flow disorders. Animal experiments and clinical outcomes further suggest that there is a gender preference in hearing loss, with males being more susceptible. Autoregulation of cochlear blood flow has been demonstrated in some animal models in vivo, suggesting that similar to the brain, blood vessels supplying the cochlea have the ability to control flow within normal limits, despite variations in systemic blood pressure. Here, we investigated myogenic regulation in the cochlear blood supply of the Mongolian gerbil, a widely used animal model in hearing research. The cochlear blood supply originates at the basilar artery, followed by the anterior inferior cerebellar artery, and inside the inner ear, by the spiral modiolar artery and the radiating arterioles that supply the capillary beds of the spiral ligament and stria vascularis. Arteries from male and female gerbils were isolated and pressurized using a concentric pipette system. Diameter changes in response to increasing luminal pressures were recorded by laser scanning microscopy. Our results show that cochlear vessels from male and female gerbils exhibit myogenic regulation but with important differences. Whereas in male gerbils, both spiral modiolar arteries and radiating arterioles exhibited pressure-dependent tone, in females, only radiating arterioles had this property. Male spiral modiolar arteries responded more to L-NNA than female spiral modiolar arteries, suggesting that NO-dependent mechanisms play a bigger role in the myogenic regulation of male than female gerbil cochlear vessels.     

Hemodynamics analysis of a stenosed carotid bifurcation and its plaque-mitigating design

Considering transient two-dimensional laminar flow in a diseased carotid artery segment with realistic inlet and outflow conditions, detailed velocity profiles, pressure fields, wall shear stress distributions and coupled, localized plaque formations have been simulated. The type of outflow boundary condition influences to a certain degree the extent of plaque build-up, which in turn reduces "disturbed flow" phenomena such as flow separations, recirculation zones, and wavy flow patterns in the artery branches during portions of the pulse. Based on computer experiments varying key geometric factors, a plaque-mitigating design of a carotid artery bifurcation has been proposed. Elimination of the carotid bulb, a smaller bifurcation angle, lower area ratios, and smooth wall curvatures generated a design with favorable hemodynamics parameters, leading to reduced plaque build-up by factors of 10 and 2 in the internal carotid and in the external carotid, respectively.