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.     

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.     

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.     

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.     

Chemical activation of thin-fiber phrenic afferents. 2. Cardiovascular responses

To assess the effects of groups III and IV (thin-fiber) phrenic afferents on arterial pressure, heart rate, and distribution of cardiac output, we injected capsaicin into phrenic arteries of in situ isolated and innervated left diaphragms of dogs anesthetized with chloralose, vagotomized, and mechanically ventilated. Blood flow in the ascending aorta, common carotid, renal, superior mesenteric, and femoral arteries was measured by electromagnetic and Doppler flow probes. Injection of 1 mg capsaicin into the left phrenic artery produced congruent to 15% increase in mean arterial pressure and congruent to 7% increase in heart rate with no change in aortic flow. Phrenic arterial flow decreased by 64%, renal arterial flow by 16%, and superior mesenteric arterial flow by 10%, whereas carotid flow increased by 13% and flow to the right gastrocnemius muscle did not change. Mean arterial pressure, heart rate, and blood flow distribution (with the exception of the decline in phrenic blood flow) returned to baseline within 60 s of the injection. Injection of 1.5 mg capsaicin into the right isolated and innervated gastrocnemius produced congruent to 35% increase in mean arterial pressure, 17% rise in heart rate, and no change in aortic blood flow. Phrenic and carotid arterial flow rose by 240 and 41%, respectively, whereas renal and superior mesenteric flow declined by 50 and 20%, respectively. In conclusion, thin-fiber phrenic afferents have an excitatory effect on arterial pressure and heart rate. They redistribute blood flow away from the renal and intestinal vascular beds and toward the carotid vascular bed. On the other hand, the cardiovascular reflex from thin-fiber phrenic afferents seems less potent than that from limb muscle afferents.     

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.     

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.     

凝闭双侧椎动脉预处理对大鼠全脑缺血损伤的防护作用

目的：观察凝闭双侧椎动脉与夹闭双侧颈总动脉之间的不同时间间隔对Pulsinelli四血管闭塞法全脑缺血模型的影响、以及在凝闭单侧椎动脉的基础上夹闭双侧颈总动脉后的脑缺血的特点。方法：84只Wistar大鼠．随机分为以下4组：对照组、双侧椎动脉凝闭组、全脑缺血组、单侧椎动脉凝闭＋双侧颈总动脉夹闭组。全脑缺血组中,根据凝闭双侧椎动脉与夹闭双侧颈总动脉之间的时间间隔不同,又分为24h间隔、48h间隔和72h间隔3个亚组。观察大鼠脑缺血过程中的反应包括瞳孔散大、对光反射等情况,脑缺血后恢复翻正反射所需要的时间、以及动物的一般状况,并应用硫堇染色法观察海马CA1区锥体神经元迟发性死亡的情况：结果：全脑缺血72h间隔亚组的大鼠,脑缺血过程中的反应、脑缺血后的一般状况和锥体神经元迟发性死亡程度均明显重于全脑缺血24h间隔亚组及48h间隔亚组,但24h间隔亚组与48h间隔亚组之间无显著差异一单侧椎动脉凝闭＋双侧颈总动脉夹闭组大鼠的凝闭侧瞳孔散大、对光反射消失、海马CA1区神经元大量死亡;而未凝闭侧未见上述相关变化。结论：凝闭双侧椎动脉本身也具有脑缺血预处理样作用,对其后48h内夹闭双侧颈总动脉所致的严重脑缺血具有一定程度的保护作用;大鼠椎动脉对脑干及海马的血液供应均存在明显的同侧优势效应,     

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.     

Computational models of blood flow in the circle of Willis

A two-dimensional, steady state model of the circle of Willis has been developed. To simulate the peripheral resistance of the cerebrovascular tree, blocks of porous media were used. Their effective resistance was kept constant, disregarding the effects of arterial auto-regulation. The model was then used to simulate different common abnormalities of the circle of Willis while a range of varying boundary conditions was imposed to the right internal carotid artery (ICA). The total flux was tabulated and compared favourably with both clinical measurements and other models of the circle of Willis. Relevant fluid dynamics effects were also observed and analysed. The present model demonstrates that the use of CFD can produce physiological results if the appropriate boundary conditions are used. We can provide clinicians with a priority list of the severity of the flux reduction for the considered abnormalities for different degrees of stenosis of the right ICA. From this study it is apparent that the redistribution of blood via the circle of Willis is mainly driven by changes in the vascular resistance of the brain rather than in the local arterial geometry. The use of valid peripheral resistances allows for a more realistic model of the circle of Willis but also highlights the need for more accurate means to estimate the vascular resistance of a patient.     

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.     

Contribution of the vertebral artery to cerebral circulation in the rat snake Elaphe obsoleta

Blood supplying the brain in vertebrates is carried primarily by the carotid vasculature. In most mammals, cerebral blood flow is supplemented by the vertebral arteries, which anastomose with the carotids at the base of the brain. In other tetrapods, cerebral blood is generally believed to be supplied exclusively by the carotid vasculature, and the vertebral arteries are usually described as disappearing into the dorsal musculature between the heart and head. There have been several reports of a vertebral artery connection with the cephalic vasculature in snakes. We measured regional blood flows using fluorescently labeled microspheres and demonstrated that the vertebral artery contributes a small but significant fraction of cerebral blood flow (∼13% of total) in the rat snake Elaphe obsoleta. Vascular casts of the anterior vessels revealed that the vertebral artery connection is indirect, through multiple anastomoses with the inferior spinal artery, which connects with the carotid vasculature near the base of the skull. Using digital subtraction angiography, fluoroscopy, and direct observations of flow in isolated vessels, we confirmed that blood in the inferior spinal artery flows craniad from a point anterior to the vertebral artery connections. Such collateral blood supply could potentially contribute to the maintenance of cerebral circulation during circumstances when craniad blood flow is compromised, e.g., during the gravitational stress of climbing. J. Morphol. 238:39–51, 1998. © 1998 Wiley-Liss, Inc.     

Analysis of mechanical parameters on the thromboembolism using a patient-specific computational model

Ischemic stroke is a major cause of death and long-term disabilities worldwide. In this paper, we aim to represent a comprehensive simulation of the motion of emboli through cerebrovascular network within patient-specific computational model. The model consists of major arteries of the circle of Willis reconstructed from magnetic resonance angiography images, pulsatile flow and emboli with different sizes and material properties. Here, the fluid–structure interactions method was used to simulate the motion of deformable and rigid emboli through cerebral arteries. Hemodynamic changes in the circle of Willis due to the entrance of embolus are observed. The effect of material properties on the distribution ratio and dynamics of motion of the emboli in the cerebral arterial network is also analyzed. Our results reveal that as the rigidity of emboli increases, higher proportion of them tend to enter to the larger arteries (e.g., middle cerebral artery). Scrutinizing the amount of stress acting on the emboli represented in this paper can broaden our understanding of the rheological phenomenon (e.g., lysis or growth of emboli during embolism). The approach of considering different material properties of the thrombus in a patient-specific computational model not only enable us to better understand the roll of biomechanical parameters causing the embolism, but also lead to a better clinical decision making to manage patients with stroke.     

颈性眩晕中医证型与经颅超声脑动脉血流检测结果的相关性分析

摘要 目的：分析颈性眩晕中医证型与经颅超声脑动脉血流检测结果的相关性。方法：选取2021年5月-2023年5月收治颈性眩晕患者作为研究对象,根据不同中医证型分为痰湿中阻组、肝阳上亢组、肝肾阴虚组和气血亏虚组,每组各纳入20例;并另选取健康体检患者30例作为对照组,均给予多普勒超声检查。分析不同中医证型者与对照组者多普勒超声检查特征与脑动脉血流变化[左右椎动脉、基底动脉及大脑中动脉收缩期峰值血流速度（VS）、平均血流速度(Vm)、舒张期峰值血流速度(Vd)及搏动指数(PI)]。结果：痰湿中阻组、肝肾阴虚组和气血亏虚组左右椎动脉、基底动脉及大脑中动脉VS均低于对照组（P＜0.05）;肝阳上亢组左右椎动脉、基底动脉及大脑中动脉VS均高于对照组（P＜0.05）;不同中医证型组间VS比较,肝阳上亢组＞痰湿中阻组＞肝肾阴虚组＞气血亏虚组;痰湿中阻组左右椎动脉、基底动脉均高于对照组（P＜0.05）,大脑中动脉Vm均低于对照组（P＜0.05）;肝阳上亢组左右椎动脉、基底动脉及大脑中动脉Vm均高于对照组（P＜0.05）;肝肾阴虚组左右椎动脉、基底动脉及大脑中动脉Vm均低于对照组（P＜0.05）;气血亏虚组大脑中动脉Vm均低于对照组（P＜0.05）,左右椎动脉、基底动脉Vm和对照组无显著性差异（P>0.05）;不同中医证型组间Vm比较,肝阳上亢组＞痰湿中阻组＞气血亏虚组＞肝肾阴虚组;痰湿中阻组左右椎动脉、基底动脉Vd均低于对照组（P＜0.05）,大脑中动脉Vd均高于对照组（P＜0.05）;肝阳上亢组左右椎动脉、基底动脉及大脑中动脉Vd均低于对照组（P＜0.05）;肝肾阴虚组左右椎动脉及大脑中动脉Vd均低于对照组（P＜0.05）,基底动脉Vd和对照组无差异（P＞0.05）;气血亏虚组左右椎动脉Vd均低于对照组（P＜0.05）,基底动脉Vd和对照组无差异（P＞0.05）,大脑中动脉Vd均高于对照组（P＜0.05）;不同中医证型组间Vd比较,气血亏虚组＞痰湿中阻组＞肝肾阴虚组＞肝阳上亢组;痰湿中阻组和气血亏虚组左右椎动脉、基底动脉及大脑中动脉PI均低于对照组（P＜0.05）;肝阳上亢组和肝肾阴虚组左右椎动脉、基底动脉及大脑中动脉PI均高于对照组（P＜0.05）;不同中医证型组间PI比较,肝阳上亢组＞肝肾阴虚组＞痰湿中阻组＞气血亏虚组。结论：不同中医证型的眩晕患者会出现不同程度脑动脉血流动力学异常,且不同组间存在差异,通过经颅多普勒超声检查,可以对眩晕中医证型提供参考价值。     

树鼩(Tupaia belangeri chinensis)的脑底动脉环

用25只树鼩,从升主动脉灌注带色的橡胶乳液,在解剖显微镜下进行解剖观察,用目测微尺进行测量。大多数树鼩(22只)有完整的脑底动脉环。由左、右大脑前动脉向内侧各发一前交通动脉组成大脑前总动脉。前交通动脉口径为大脑前动脉的75～85％。后交通动脉口径与大脑后动脉相近,连于颈内动脉与大脑后动脉(基底动脉的分支)之间。测量了组成脑底动脉环有关动脉的口径。由于后交通动脉足够粗大,只有中断左、右颈总动脉和左、右椎动脉,才能造成全脑缺血。     

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.     

Investigations of flow and pressure distributions in physical model of the circle of Willis

The paper presents the results of experiments concerning flow in the model of cerebral supplying arteries and the circle of Willis (CW). Vascular phantom was prepared on the basis of anatomical specimens. The most typical artery shapes and dimensions were considered. Pressure distribution in six characteristic points is provided, and so are the average flow rates in the anterior, middle and posterior section of the brain. Tests were run in the conditions replicating the physiological state (i.e. when the supplying arteries were fully patent) and in pathological conditions, in which the internal carotid and vertebral arteries were occluded on one or both sides. Thus obtained results were compared with the results of computer simulations based on linear and non-linear flow models. To estimate the non-linear resistance of vascular segment two phenomenological formulae were proposed. High degree of correlation between the values obtained from experiments and those registered in non-linear computer model proves usefulness of proposed formulae. It verifies the hypothesis that non-linearity of flow characteristics of the vessel segments to a great extent is caused by their tortuousity and small length in relation to diameter. Non-linear effects are particularly pronounced in conditions of pathological occlusion of supplying vessels.     

Capacities of angiography in the imaging of abnormal changes in the cerebral arteries

The study was based on the angiographic examination of 233 patients with prior subarachnoidal hemorrhage. Angiographic study was performed using the Seldinger technique by contrasting both carotid and vertebral arteries. Twenty-three patients in whom arterial aneurysm had been detected by digital subtraction angiography underwent 3D angiography. The authors improved a procedure during which a contrast agent was manually injected into the internal carotid or vertebral artery, by using a 20-ml disposal syringe with controlled maximum developed pressure and flow increase rate up to 2.0 ml/sec for 4-5 sec during rotary scanning and the administration of the radiocontrast medium was stopped when an image appeared on the monitor at 190 degrees (190.0, 200.0) C-arm rotation. This procedure could decrease significantly the volume of the administered contrast agent from 18 to 8 (8.0, 10.0) ml and reduce the time of radiation exposure from 6 to 4 (4.0, 5.0) sec. The improved angiographic modes for the right vertebral and right carotid artery could visualize pathological changes in these arteries and establish a relationship, namely: due to degenerative dystrophic processes of the cervical spine there is a tendency for higher pathological changes in the vertebral arteries with an increased stage of osteochondrosis in the cervical spine (R = 0.95; p = 0.014).     

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.