The variability of the circulus arteriosus (Willisii): order or anarchy?

The variation of the circulus arteriosus is studied using multivariate methods. The data which form the basis of this study are 19 measurements of half the circumference of the arteries that form the circle of Willis and its afferent and efferent branches; 100 circles of Willis were measured for this purpose. Since the number of variables per individual is large, multivariate statistical techniques are the most appropriate method to gain insight in the relations of vessel sizes that exist within the circle of Willis. So a principal component analysis was performed on the data. The results clearly show a number of relations between vessel sizes. In general, inverse relationships were found of vessels that have (at least partially) an identical irrigation area: both internal carotid arteries and the ipsilateral posterior communicating artery show an intimate relationship and are together inversely related to the basilar artery and the precommunicating part of the posterior cerebral artery. Inverse relationships are also found for both vertebral arteries and both precommunicating parts of the anterior cerebral arteries. The homonymous efferent arteries appear to be closely related and show an independent variation. Together the first six principal components explain 69% of the variance. These results support a haemodynamical hypothesis on the explanation of the variability of the circle of Willis. Moreover, the differential growth in the head-neck region during the first two decades of life is postulated to be the origin of a part of the variation.     

Blood vessel size of circulus arteriosus cerebri (circle of Willis): a statistical research on 100 human subjects

The brain weight of 100 fresh cadavers of Italian subjects (60 males and 40 females), aged between 17 and 84 years, was obtained and the corrected circumference of the following blood vessels was measured: basilar artery, internal carotid arteries, anterior and posterior cerebral arteries, and anterior and posterior communicating arteries. The cerebral 'potential flow' was expressed in each case by adding the circumference of the basilar artery to that of the internal carotid arteries. Moreover, the sides and the perimeter of the circle of Willis as well as the length of the basilar artery were calculated. The statistical analysis of the obtained data yielded the following main results: (1) the brain weight decreases with aging, is lower in females than in males and and is statistically correlated neither with the circumferences of the considered arteries and the 'potential flow' nor with the perimeter of the arterial polygon (circle of Willis); (2) the arteries of the left side appear to be larger than those of the right one; (3) no significant difference exists in the circumference and length of the arteries between males and females; (4) the increase of the perimeter of the arterial polygon is achieved by means of a harmonious increase of all its sides; (5) the anterior and posterior communicating arteries have an anarchic pattern, because of the relatively frequent anomalies and the lack of a correlation between their circumference and that of the vessel of origin or of outlet.     

A mathematical model of the flow in the circle of Willis

A mathematical model of the flow in the circle of Willis has been designed and the effects of (a) the large anatomical variation of the communicating arteries and (b) physiological changes of the resistances of the vertebral arteries have been studied. The influence of the posterior perforating arteries on the flow in the posterior communicating arteries has been investigated as well, with special attention being paid to the possible occurrence of a 'dead point'. In the model, the influence of diameters of the communicating arteries on the flow in the afferent vessels and the segments of the circle turns out to be considerable, especially in the range of the anatomical variation of the diameters. Within this range flow reductions due to an increased resistance of the vertebral artery will be compensated for by the system. Assuming that the values and ratios of the peripheral resistances are within the physiological range, a dead point is not to be expected in the flow in the posterior communicating arteries.     

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.     

One-dimensional and three-dimensional models of cerebrovascular flow

The Circle of Willis is a ring-like structure of blood vessels found beneath the hypothalamus at the base of the brain. Its main function is to distribute oxygen-rich arterial blood to the cerebral mass. One-dimensional (1D) and three-dimensional (3D) computational fluid dynamics (CFD) models of the Circle of Willis have been created to provide a simulation tool which can potentially be used to identify at-risk cerebral arterial geometries and conditions and replicate clinical scenarios, such as occlusions in afferent arteries and absent circulus vessels. Both models capture cerebral haemodynamic autoregulation using a proportional-integral (PI) controller to modify efferent artery resistances to maintain optimal efferent flow rates for a given circle geometry and afferent blood pressure. The models can be used to identify at-risk cerebral arterial geometries and conditions prior to surgery or other clinical procedures. The 1D model is particularly relevant in this instance, with its fast solution time suitable for real-time clinical decisions. Results show the excellent correlation between models for the transient efferent flux profile. The assumption of strictly Poiseuille flow in the 1D model allows more flow through the geometrically extreme communicating arteries than the 3D model. This discrepancy was overcome by increasing the resistance to flow in the anterior communicating artery in the 1D model to better match the resistance seen in the 3D results.     

Experimental study of hemodynamics in the circle of willis

Background

The Circle of Willis (CoW) is an important collateral pathway of the cerebral blood flow. An experimental study of the cerebral blood flow (CBF) distribution in different anatomical variations may help to a better understanding of the collateral mechanism of the CoW.

Methods

An in-vitro test rig was developed to simulate the physiological cerebral blood flow in the CoW. Ten anatomical variations were considered in this study, include a set of different degrees of stenosis in L-ICA and L-ICA occlusion coexist with common anatomical variations. Volume flow rates of efferent arteries and pressure signals at the end of communicating arteries of each case were recorded. Physiological pressure waveforms were applied as inlet boundary condition.

Results

In the development of L-ICA stenosis, the total CBF decreases with the increase of stenosis degree. The blood supply of ipsilateral middle cerebral artery (MCA) was affected most by the stenosis of L-ICA. Anterior communicating artery (ACoA) and ipsilateral posterior communicating artery (PCoA) function as important collateral pathways of cerebral collateral circulation when unilateral stenosis occurred. The blood supply of anterior cerebral circulation was compensated by the posterior cerebral circulation through ipsilateral PCoA when L-ICA stenosis degree is greater than 40% and the affected side was compensated immediately by the unaffected side through ACoA. Blood flow of the anterior circulation and the total CBF reached the minimum among all cases studied when L-ICA occlusion coexist with the absence of PCoA.

Conclusion

The results demonstrated the flow distribution patterns of the CoW under anatomical variations and clarified the collateral mechanism of the CoW. The flow ACoA is the most sensitive indexes to the morphology change of ipsilateral ICA. The relative independence of the circulation in anterior and posterior sections of the CoW is not broken and the function of ipsilateral PCoA is not activated until a severe stenosis of unilateral ICA occurs. PCoA is the most important collateral pathway of the collateral circulation and the missing of PCoA has the highest risk of stroke when the ipsilateral ICA has severe stenosis. These findings may provide the basis for future therapeutic and diagnosis applications.

     

Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows

Blood flow in the circle of Willis (CoW) is modelled using the 1-D equations of pressure and flow wave propagation in compliant vessels. The model starts at the left ventricle and includes the largest arteries that supply the CoW. Based on published physiological data, it is able to capture the main features of pulse wave propagation along the aorta, at the brachiocephalic bifurcation and throughout the cerebral arteries. The collateral ability of the complete CoW and its most frequent anatomical variations is studied in normal conditions and after occlusion of a carotid or vertebral artery (VA). Our results suggest that the system does not require collateral pathways through the communicating arteries to adequately perfuse the brain of normal subjects. The communicating arteries become important in cases of missing or occluded vessels, the anterior communicating artery (ACoA) being a more critical collateral pathway than the posterior communicating arteries (PCoAs) if an internal carotid artery (ICA) is occluded. Occlusions of the VAs proved to be far less critical than occlusions of the ICAs. The worst scenario in terms of reduction in the mean cerebral outflows is a CoW without the first segment of an anterior cerebral artery combined with an occlusion of the contralateral ICA. Furthermore, in patients without any severe occlusion of a carotid or VA, the direction of flow measured at the communicating arteries corresponds to the side of the CoW with an absent or occluded artery. Finally, we study the effect of partial occlusions of the communicating arteries on the cerebral flows, which again confirms that the ACoA is a more important collateral pathway than the PCoAs if an ICA is occluded.     

Migraine with Aura Is Associated with an Incomplete Circle of Willis: Results of a Prospective Observational Study

Objective

To compare the prevalence of an incomplete circle of Willis in patients with migraine with aura, migraine without aura, and control subjects, and correlate circle of Willis variations with alterations in cerebral perfusion.

Methods

Migraine with aura, migraine without aura, and control subjects were prospectively enrolled in a 1∶1∶1 ratio. Magnetic resonance angiography was performed to examine circle of Willis anatomy and arterial spin labeled perfusion magnetic resonance imaging to measure cerebral blood flow. A standardized template rating system was used to categorize circle of Willis variants. The primary pre-specified outcome measure was the frequency of an incomplete circle of Willis. The association between circle of Willis variations and cerebral blood flow was also analyzed.

Results

170 subjects were enrolled (56 migraine with aura, 61 migraine without aura, 53 controls). An incomplete circle of Willis was significantly more common in the migraine with aura compared to control group (73% vs. 51%, p = 0.02), with a similar trend for the migraine without aura group (67% vs. 51%, p = 0.08). Using a quantitative score of the burden of circle of Willis variants, migraine with aura subjects had a higher burden of variants than controls (p = 0.02). Compared to those with a complete circle, subjects with an incomplete circle had greater asymmetry in hemispheric cerebral blood flow (p = 0.05). Specific posterior cerebral artery variants were associated with greater asymmetries of blood flow in the posterior cerebral artery territory.

Conclusions

An incomplete circle of Willis is more common in migraine with aura subjects than controls, and is associated with alterations in cerebral blood flow.     

Anatomy of the arteries of the head in the domestic fowl

The anatomy of the cephalic arterial system in the fowl was studied in 24 specimens by means of latex-injected preparations and by dissection. Branches of the external carotid artery supply the extracranial regions. The vertebral arteries unite with the occipitals and have no major communications with the encephalic system. Blood can reach the brain directly from the internal carotid artery and indirectly by way of the extensive cerebral-extracranial anastomoses; especially prominent are those to the palatine and sphenomaxillary arteries from the maxillary and facial branches of the external carotid artery. A large external ophthalmic artery supplies the temporal rete and eye musculature, and an internal ophthalmic artery links the rete and the cerebral vessels. The circle of Willis is incomplete both anteriorly and posteriorly; the anterior cerebral arteries do not unite and the basilar artery is generally asymmetrical in origin. The basilar artery tapers caudally and is continued as the ventral spinal artery.     

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.     

Hemodynamic role of the circle of Willis in stenoses of internal carotid arteries. An analytical solution of a linear model

A mathematical model of blood flow through the circle of Willis was developed, within a linear framework. Comprehensive analytical solutions, including a remarkably small number of parameters, were derived in the cases of obstructive lesions of extracranial carotid arteries. The influence of these lesions and the role of anterior and posterior communicating arteries on the blood pressure at the entry of the cerebral territories were quantified and analyzed emphasizing that the responses of the system of Willis to obstructive carotid lesions are extremely varied, depending on the communicating artery anatomy. Comparison with numerical results obtained by using a non-linear model showed no physiologically significant differences. Such a model might be an essential tool for an accurate assessment of the cerebral hemodynamics in carotid diseases.     

High-Resolution Postcontrast Time-of-Flight MR Angiography of Intracranial Perforators at 7.0 Tesla

Background and Purpose

Different studies already demonstrated the benefits of 7T for precontrast TOF-MRA in the visualization of intracranial small vessels. The aim of this study was to assess the performance of high-resolution 7T TOF-MRA after the administration of a gadolinium-based contrast agent in visualizing intracranial perforating arteries.

Materials and Methods

Ten consecutive patients (7 male; mean age, 50.4 ± 9.9 years) who received TOF-MRA at 7T after contrast administration were retrospectively included in this study. Intracranial perforating arteries, branching from the parent arteries of the circle of Willis, were identified on all TOF-MRA images. Provided a TOF-MRA before contrast administration was present, a direct comparison between pre- and postcontrast TOF-MRA was made.

Results

It was possible to visualize intracranial perforating arteries branching off from the entire circle of Willis, and their proximal branches. The posterior cerebral artery (P1 and proximal segment of P2) appeared to have the largest number of visible perforating branches (mean of 5.1 in each patient, with a range of 2–7). The basilar artery and middle cerebral artery (M1 and proximal segment M2) followed with a mean number of 5.0 and 3.5 visible perforating branches (range of 1–9 and 1–8, respectively). Venous contamination in the postcontrast scans sometimes made it difficult to discern the arterial or venous nature of a vessel.

Conclusion

High-resolution postcontrast TOF-MRA at 7T was able to visualize multiple intracranial perforators branching off from various parts of the circle of Willis and proximal intracranial arteries. Although confirmation in a larger study is needed, the administration of a contrast agent for high-resolution TOF-MRA at 7T seems to enable a better visualization of the distal segment of certain intracranial perforators.     

Study of the collateral capacity of the circle of Willis of patients with severe carotid artery stenosis by 3D computational modeling

This numerical study aims to investigate the capacity of the circle of Wills (CoW) to provide collateral blood supply for patients with unilateral carotid arterial stenosis. The basic 3D geometry of the CoW was reconstructed based on a magnetic resonance angiogram of a normal human subject. A total of 52 computational fluid dynamics simulations were performed for four geometry configurations of the CoW with an artificially inserted axisymmetric stenosis of different luminal area reductions in an internal carotid artery (ICA) under a variety of boundary conditions. The CoW geometric configurations included (a) a normal CoW with all communicating arteries; (b) as model (a) but with enlarged communicating arterial diameters; (c) as (a) but with the ipsilateral posterior communicating artery missing, and (d) as (c) but with enlarged communicating arteries. It is found that the blood perfusion pressure drop between the ipsilateral ICA and the middle cerebral artery (MCA) only becomes significant when the degree of stenosis is greater than 86%. The cerebral autoregulation range varied significantly between the different CoW configurations for the severe stenosis cases. Without causing the flow rates to decrease at the efferent arterial ends, the mean perfusion pressure in the ipsilateral ICA can drop from 100 to 73, 67, 92 and 84mmHg for the CoW models (a)-(d) with 96% luminal area reduction stenosis, respectively. The additional pathways are able to raise the ipsilateral MCA pressure significantly without reducing the total flow perfusion. Cerebral autoregulation effects were not directly included in the study. Therefore, the findings in the study should be interpreted with cautions when comes to the biological and clinical significance.     

Cross-flow at the anterior communicating artery and its implication in cerebral aneurysm formation

The anterior communicating artery (ACoA) is an important element of the circle of Willis. While the artery itself is short and small, a large number of intracranial aneurysms can be found at the ACoA. Four subject-specific ACoA models are constructed from 3D rotational angiographic images. The ACoA of these models ranged from 1.7 to 2.7 mm in diameter and 1.5 to 5.7 mm in length. Pulsatile flows through these four ACoA models are studied numerically. Blood is found to move in two opposite directions simultaneously within the ACoA, giving a much higher wall shear at the ACoA. These two opposite flow streams produce a cross-flow that is dependent on the flow rates at the anterior cerebral arteries and internal carotid arteries (ICAs). A larger and shorter ACoA allows flow through the ACoA easily, leading to a greater cross-flow and higher hemodynamic forces on the artery. This cross-flow may disappear when there is a sufficient net flow for a smaller and longer ACoA. Wall shear stress can be as high as 185 Pa at smaller ACoAs, but it can be lowered by asymmetric waveforms at the ICAs. A functional circle of Willis also promotes cross-flow at both the ACoA and posterior communicating arteries.     

On the subclavian steal syndrome in vitro studies.

An elastic model of the arterial system has been used in which a specially designed pumping unit simulated the heart action. Physiological pressures and normal geometry, size, and flow distribution together with the normal cardiac output and use of prosthetic heart valves are the features of the system. Atherosclerosis was simulated by introducing blockages of known cross-section at specific sites of predilection. It has been shown that, for some specific occlusion magnitude in the left or right subclavian, or in the brachycephalic arteries, the stagnant no blood flow condition will appear in the left vertebral, or the right vertebral, or right common carotid, or the right internal carotid arteries. For larger occlusions the blood flow in these arteries reverses its direction, i.e., the "steal syndrome" appears. It is shown that besides the known single steal syndrome there exists also a double steal syndrome, i.e., blood reverses its flow direction simultaneously in two arteries, both on the right side of the arterial system. This blood is taken from the circle of Willis, which at the same time is significantly supplemented by the increased blood flow through the other arteries leading into the circle of Willis.     

Three-dimensional hemodynamics analysis of the circle of Willis in the patient-specific nonintegral arterial structures

The hemodynamic alteration in the cerebral circulation caused by the geometric variations in the cerebral circulation arterial network of the circle of Wills (CoW) can lead to fatal ischemic attacks in the brain. The geometric variations due to impairment in the arterial network result in incomplete cerebral arterial structure of CoW and inadequate blood supply to the brain. Therefore, it is of great importance to understand the hemodynamics of the CoW, for efficiently and precisely evaluating the status of blood supply to the brain. In this paper, three-dimensional computational fluid dynamics of the main CoW vasculature coupled with zero-dimensional lumped parameter model boundary condition for the CoW outflow boundaries is developed for analysis of the blood flow distribution in the incomplete CoW cerebral arterial structures. The geometric models in our study cover the arterial segments from the aorta to the cerebral arteries, which can allow us to take into account the innate patient-specific resistance of the arterial trees. Numerical simulations of the governing fluid mechanics are performed to determine the CoW arterial structural hemodynamics, for illustrating the redistribution of the blood flow in CoW due to the structural variations. We have evaluated our coupling methodology in five patient-specific cases that were diagnosed with the absence of efferent vessels or impairment in the connective arteries in their CoWs. The velocity profiles calculated by our approach in the segments of the patient-specific arterial structures are found to be very close to the Doppler ultrasound measurements. The accuracy and consistency of our hemodynamic results have been improved (to \(16.1 \pm 18.5\) %) compared to that of the pure-resistance boundary conditions (of 43.5 \(\pm \) 28 %). Based on our grouping of the five cases according to the occurrence of unilateral occlusion in vertebral arteries, the inter-comparison has shown that (i) the flow reduction in posterior cerebral arteries is the consequence of the unilateral vertebral arterial occlusion, and (ii) the flow rate in the anterior cerebral arteries is correlated with the posterior structural variations. This study shows that our coupling approach is capable of providing comprehensive information of the hemodynamic alterations in the pathological CoW arterial structures. The information generated by our methodology can enable evaluation of both the functional and structural status of the clinically significant symptoms, for assisting the treatment decision-making.     

Thalamic arterial pattern: an endocast and scanning electron microscopic study in normotensive male rats

Rat cerebral vasculature serves as a model for study of the pathophysiology of stroke in humans. Human thalamic arteries show a high incidence of stroke. The objective is to describe the thalamic arterial vascular pattern in normotensive male rats as the initial step for quantitative histochemical studies of enzyme activities in the walls of these vessels. Intracardiac injections of methyl methacrylate monomer provide detailed vascular endocasts. The thalamic vascular bed defined by in situ dissection, serial reconstruction, and light and scanning electron microscopy of endocasts contained four groups of vessel: ventral medial thalamic arteries, thalamic branches from the posterior cerebral artery, and ventral lateral and ventral anterior thalamic arteries. Thalamic vessels are muscular arterioles that, after three to four bipinnate branches, feed into a continuous capillary bed (no loops). The parent vessels and their subsequent branches have been evaluated in terms of their mean internal diameters, mean interbranch intervals, and branch angles. The arterial patterns to rat and human thalami are very similar, with the exception of the anterior choroidal artery which is missing in the rat. The branches supplying the thalamus in both the rat and human are closely associated with the circle of Willis; however, the constituent parts of the circle in rat vary from the pattern in human brain. The rat thalamic arteries show morphological features similar to those seen in the stroke-prone ganglionic arteries in the human basal ganglia.     

The vascularization of the human tela choroidea of the lateral ventricle

The human tela choroidea of the lateral ventricle is vascularized by arteries arising from the two systems which form the arterial circle of the base, i.e. the internal carotid system and the vertebral basilar system. This blood supply is given by one anterior choroidal artery and by several posterior choroidal arteries. These arteries anastomose to form multiple indirect and remote links between the carotid and vertebral basilar systems. The capillary networks of the tela choroidea of the lateral ventricle consists of a velar network and of a choroidal network. This duality is constantly observed in the choroid formations of the human brain. The venous vascularization of the tela is tributary of the venous circle of the base of the brain through choroidal veins that drain either into the internal cerebral veins or into the basal veins.     

1D simulation of blood flow characteristics in the circle of Willis using THINkS

One-dimensional (1D) simulation of the complete vascular network, so called THINkS (Total Human Intravascular Network Simulation) is developed to investigate changes of blood flow characteristics caused by the variation of CoW. THINkS contains 158 major veins, 85 major arteries, and 77 venous and 43 arterial junctions. THINkS is validated with available in vivo blood flow waveform data. The overall trends of flow rates in variations of the CoW, such as the missing anterior cerebral artery (missing-A1) or missing posterior cerebral artery (missing-P1), are confirmed by in vivo experimental data. It is demonstrated that the CoW has the ability to shunt blood flow to different areas in the brain. Flow rates in efferent arteries remain unaffected under the variation of CoW, while the flow rates in afferent vessels can be subject to substantial changes. The redistribution of blood flow can cause particular vessels to undergo extra flow rate and hemodynamic stresses.     

川金丝猴脑的动脉供应

为了探讨川金丝猴脑动脉供应的形态学特征,为脑生物学研究提供结构基础,用血管铸型和组织透明方法追踪观察了川金丝猴幼体脑动脉的来源和分支分布。结果表明川金丝猴与人脑的动脉供应基本相同,也由颈内动脉和椎动脉供应。上述动脉的分支于垂体周围形成大脑动脉环。颈内动脉通过大脑前动脉和大脑中动脉主要供应大脑半球前部的血液,椎动脉参与形成基底动脉、小脑动脉系和大脑后动脉,供应脑干、小脑和大脑后部的血液。另外,川金丝猴幼体左、右大脑前动脉间缺少前交通动脉。