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.     

Effect of natriuretic peptides on cerebral artery blood flow in healthy volunteers

The natriuretic peptides (NPs), atrial natriuretic peptide (ANP), brain natriuretic peptide (BNP) and C-type natriuretic peptide (CNP), have vasoactive functions that concern humans and most animals, but their specific effects on cerebral circulation are poorly understood. We therefore examined the responsiveness of cerebral arteries to different doses of the natriuretic peptides in animals and humans. We conducted a dose-response experiment in guinea pigs (in vitro) and a double-blind, three-way cross-over study in healthy volunteers (in vivo). In the animal experiment, we administered cumulative doses of NPs to pre-contracted segments of cerebral arteries. In the main study, six healthy volunteers were randomly allocated to receive two intravenous doses of ANP, BNP or CNP, respectively, over 20 min on three separate study days. We recorded blood flow velocity in the middle cerebral artery (V_MCA) by transcranial Doppler. In addition, we measured temporal and radial artery diameters, headache response and plasma concentrations of the NPs. In guinea pigs, ANP and BNP but not CNP showed significant dose-dependent relaxation of cerebral arteries. In healthy humans, NP infusion had no effect on mean V_MCA, and we found no difference in hemodynamic responses between the NPs. Furthermore, natriuretic peptides did not affect temporal and radial artery diameters or induce headache. In conclusion, natriuretic peptides in physiological and pharmacological doses do not affect blood flow velocity in the middle cerebral artery or dilate extracerebral arteries in healthy volunteers.     

Abnormalities of the Vertebrobasilar Circulation due to Subclavian Artery Disease

The studies of Hutchinson and Yates on caroticovertebral stenosis have stimulated further interest in cerebrovascular disease. As a consequence, investigation of the four vessels, supplying the brain is now routine in the assessment of patients being considered for surgery for occlusive disease of the vessels. This has led to some interesting and unexpected findings.The effect of subclavian artery disease on the vertebrobasilar circulation was studied in two patients. Angiography showed the vertebral arteries to function as a collateral pathway when an occlusion of the subclavian artery was present proximal to the origin of the vertebral arteries. In one case both the carotid and the vertebral arteries were implicated in the collateral supply. A normal circulation was restored by subclavian endarterectomy.Studies of other workers have shown that such a circulation may reduce the cerebral blood flow by about 40%, but neither patient described in this report had signs or symptoms of cerebrovascular disease. It is evident that our understanding of the effects of cerebrovascular disease is far from complete.     

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。     

Blood flow velocity--a constantly acting factor in dilatation of the large arteries

Acute experiments on cats have demonstrated the presence of renal artery sensitivity to the bloodstream velocity; the artery is enlarged with the blood flow increase and is narrowed with its decrease. The dilatation of the artery caused by the double increase of the blood flow (from 20 to 40 ml/min) is 23 +/- 8%. It has been established experimentally that carotid, femoral and renal arteries of cats respond to minor changes (of the order of 1 ml/min) in the volumetric velocity of the bloodstream by changing their diameters. Therefore, the arteries continuously follow the bloodstream velocity changes by changing their diameters. It may thus be concluded that the bloodstream itself is the artery dilatation factor.     

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.     

Value of color-Doppler imaging in diagnosis of intrabulbar and intraorbital tumors

Patients with intrabulbar and intraorbital tumours have been investigated by colour Doppler imaging. In the first group there were 20 patients with malignant melanoma of the uvea and in the second group there were 19 patients with cavernous hemangioma. The third group was a control group represented by 20 healthy individuals. In the group of patients with uveal melanoma it has been established that every patient, except one, has a blood flow inside of tumour tissue. Internal blood flow of intraorbital hemangioma was slower than inside the melanoma and the resistance index was lower. Comparison of blood flow in all patients has shown that there is no statistically significant difference between ophthalmic artery, central retinal artery and posterior ciliary arteries except in maximal blood flow in central retinal artery and posterior ciliary arteries.     

Computation of steady three-dimensional flow in a model of the basilar artery

The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. To study the flow phenomena in the basilar artery, computations have been performed using a finite element (FE) method. We consider steady flow in a geometrically symmetric confluence. For simplicity, channels with a rectangular cross-section have been used. Both symmetric and asymmetric flow cases have been considered. The results show that for the Reynolds number of interest the flow downstream of the junction is highly three-dimensional, and that the flow at the end of the basilar artery, where it splits again, will not be fully developed. The computed phenomena have been confirmed by laser Doppler velocity measurements.     

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,     

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.     

Hemodynamic Characteristics of the Vertebrobasilar System Analyzed Using MRI-Based Models

The vertebrobasilar system (VBS) is unique in human anatomy in that two arteries merge into a single vessel, and it is especially important because it supplies the posterior circulation of the brain. Atherosclerosis develops in this region, and atherosclerotic plaques in the vertebrobasilar confluence can progress with catastrophic consequences, including artery occlusion. Quantitative assessments of the flow characteristics in the VBS could elucidate the factors that influence flow patterns in this confluence, and deviations from normal patterns might then be used to predict locations to monitor for potential pathological changes, to detect early signs of disease, and to evaluate treatment options and efficacy. In this study, high-field MRI was used in conjunction with computational fluid dynamics (CFD) modeling to investigate the hemodynamics of subject-specific confluence models (n = 5) and to identify different geometrical classes of vertebrobasilar systems (n = 12) of healthy adult subjects. The curvature of the vessels and their mutual orientation significantly affected flow parameters in the VBS. The basilar artery geometry strongly influenced both skewing of the velocity profiles and the wall shear stress distributions in the VBS. All five subjects modeled possessed varying degrees of vertebral asymmetry, and helical flow was observed in four cases, suggesting that factors other than vertebral asymmetry influence mixing of the vertebral artery flow contributions. These preliminary studies verify that quantitative, MR imaging techniques in conjunction with subject-specific CFD models of healthy adult subjects may be used to characterize VBS hemodynamics and to predict flow features that have been related to the initiation and development of atherosclerosis in large arteries. This work represents an important first step towards applying this approach to study disease initiation and progression in the VBS.     

Hemodynamic analysis of Hyrtl anastomosis in human placenta

The Hyrtl anastomosis is a common connection between the umbilical arteries near the cord insertion in most human placentas. It has been speculated that it equalizes the blood pressure between the territories supplied by the umbilical arteries. However, its functional role in the regulation and distribution of fetal blood flow to the placenta has not yet been explored. A computational model has been developed for quantitative analysis of hemodynamic characteristic of the Hyrtl anastomosis in cases of discordant blood flow in the umbilical arteries. Simulations were performed for cases of either increased placental resistance at the downstream end or reduced arterial blood flow due to some pathologies upstream of one of the arteries. The results indicate that when placental territories of one artery impose increased resistance to fetal blood flow, the Hyrtl anastomosis redistributes the blood flow into the second artery to reduce the large pressure gradients that are developed in the affected artery. When one of the arteries conducts a smaller blood flow into the placenta and a relatively smaller pressure gradient is developed, the Hyrtl anastomosis rebuilds the pressure gradients in the affected artery and redistributes blood flow from the unaffected artery to the affected one to improve placental perfusion. In conclusion, the Hyrtl anastomosis plays the role of either a safety valve or a pressure stabilizer between the umbilical arteries at the placental insertion.     

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.     

The dissections of craniocervical arteries

Dissection of craniocervical arteries internal carotid artery (ICA), or vertebral artery (VA) is an increasingly recognized entity and infrequent cause of stroke. We investigated 8 patients (4 women and 4 men) with dissections of the craniocervical arteries. Diagnostic procedures for detection of craniocervical dissection included: extracranial ultrasound-color Doppler flow imaging (CDFI) of carotid and vertebral arteries, transcranial Doppler sonography (TCD) and radiological computed tomography (CT) and digital subtractive angiography (DSA) examinations. Ultrasound findings (CDFI of carotid and vertebral arteries) were positive for vessel dissection in seven patients (or 87.5 per cent) and negative in one patient. DSA was consistent with dissection in five patients (or 62.5 per cent), negative in one, while in two patients the examination was not performed due to known allergy to contrast media. Five patients (62.5 per cent) were treated with anticoagulants, one with suppressors of platelet aggregation, and two patients were operated. Six patients (75 per cent) after the treatment showed partial recovery of neurological defects, and an improvement of ultrasound finding of dissected arteries. In one patient, following operation, stroke developed with deterioration of motor deficit, and one patient was readmitted three months later due to a newly developed stroke and soon died. The diagnosis should be suspected in any young or middle-age patient with new onset of otherwise unexplained unremitting headache or neck ache, especially in association with transient or permanent focal neurological deficits.     

Effect of alterations in femoral artery flow on abdominal vessel hemodynamics in swine

In support of an in vivo investigation in swine of the influence of changes in fluid dynamic wall shear on arterial macromolecular permeability, a procedure has been developed to alter the flows in the porcine posterior arterial vasculature by opening and closing a reversible arteriovenous shunt placed on one of the femoral arteries. Laparoscopic techniques were used to place appropriately modified Transonic Systems ultrasonic flow probes on both external and circumflex iliac arteries, and on the terminal aorta. Flow measurements were made prior to shunt placement, and with the shunt open and closed, to measure the influence of altered external iliac artery flow on the distribution to the infrarenal abdominal vessels. Similar experiments were carried out to relate the flow rates in the external iliac arteries to those in the femoral arteries, which are more accessible. Based on the relationships among the measured flow rates, rules have been developed to estimate the major infrarenal flows in the pig, at baseline and with the shunt opened and closed, from only the flow rates measured at the two femoral arteries.     

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.     

Errors in power-law estimations of inflow rates for intracranial aneurysm CFD

Patient-specific inflow rates are rarely available for computational fluid dynamics (CFD) studies of intracranial aneurysms. Instead, inflow rates are often estimated from parent artery diameters via power laws, i.e. Q ∝ Dⁿ, reflecting adaptation of conduit arteries to demanded flow. The present study aimed to validate the accuracy of these power laws. Internal carotid artery (ICA) flow rates were measured from 25 ICA aneurysm patients via 2D phase contrast MRI. ICA diameters, derived from 3D segmentation of rotational angiograms, were used to estimate inflow rates via power laws from the aneurysm CFD literature assuming the same inlet wall shear stress (WSS) (n = 3), velocity (n = 2) or flow rate (n = 0) for all cases. To illustrate the potential impact of errors in flow rate estimates, pulsatile CFD was carried out for four cases having large errors for at least one power law. Flow rates estimated by n = 3 and n = 0 power laws had significant (p < 0.01) mean biases of −22% to +32%, respectively, but with individual errors ranging from −78% to +120%. The n = 2 power law had no significant bias, but had non-negligible individual errors of −58% to +71%. CFD showed similarly large errors for time-averaged sac WSS; however, these were reduced after normalizing by parent artery WSS. High frequency WSS fluctuations, evident in 2/4 aneurysms, were also sensitive to inflow rate errors. Care should therefore be exercised in the interpretation of aneurysm CFD studies that rely on power law estimates of inflow rates, especially if absolute (vs. normalized) WSS, or WSS instabilities, are of interest.     

Nonquasi-steady character of pulsatile flow in human coronary arteries

This experiment was conducted to determine if the pulsatile flow through the proximal portion of the left coronary artery system in man exhibits quasi-steady characteristics. Steady and pulsatile flows were passed through an idealized model whose dimensions were based on a vascular cast. The mean Reynolds number was 180 and the unsteadiness number was 2.7. Velocity profiles were measured by laser Doppler anemometry at several locations along diameters in the parent and both daughter channels in the neighborhood of the "left main" bifurcation. Analysis of the results along one diameter in the "left main" channel shows that unsteady flow in the larger coronary arteries may not be simulated by a series of steady flow experiments.     

Role of the vascular laboratory in the diagnosis of aortic arch occlusive disease: Screening of 2000 patients

A significant difference in bilateral systolic brachial artery pressures and the presence of systolic bruit over a subclavian artery suggests occlusive disease in the innominate or subclavian arteries. Two thousand patients with these signs underwent ultrasonography of the carotid, subclavian, and brachial arteries in our laboratory. Doppler recordings of a vertebral artery, with hyperemia testing of the ipsilateral arm, identified those with a subclavian steal syndrome. In each case, angiography confirmed the laboratory diagnosis. The effectiveness of corrective surgery was evaluated noninvasively. Illustrative cases are presented and discussed.     

An investigation of an autonomic innervation of the vertebral artery using monoamine histofluorescence

Blood flow to the hindbrain, via the paired vertebral arteries, must be uncompromised for adequate neurological functioning of its vital centres. Therefore, it would seem unlikely that the intracranial vertebral artery would need to vasoconstrict, thus reducing its blood flow. In order to investigate the existence and location of a noradrenaline-mediated constrictor mechanism in the wall of the intracranial vertebral artery, transverse sections of ten baboon and ten monkey vessels were stained with sucrose-potassium phosphate-glyoxylic acid (counterstained with malachite-green). This method allows the visualisation of catecholaminergic nerves when the sections are exposed to ultraviolet light. In this study of primate vascular tissue, however, none of the monkey or baboon vertebral artery sections showed the presence of noradrenergic nerves in the tunica media - tunica adventitia junction or penetrating the tunica media of the arteries. These findings indicate that the intracranial vertebral artery does not have a neurogenic vasomotor function in primates.