- Open Access
A 3D numerical study of the collateral capacity of the circle of Willis with anatomical variation in the posterior circulation
© Ren et al.; licensee BioMed Central Ltd. 2015
Published: 9 January 2015
The Circle of Willis (CoW) is the most important collateral pathway of the cerebral artery. The present study aims to investigate the collateral capacity of CoW with anatomical variation when unilateral internalcarotid artery (ICA) is occluded.
Basing on MRI data, we have reconstructed eight 3D models with variations in the posterior circulation of the CoW and set four different degrees of stenosis in the right ICA, namely 24%, 43%, 64% and 79%, respectively. Finally, a total of 40 models are performed with computational fluid dynamics simulations. All of the simulations share the same boundary condition with static pressure and the volume flow rate (VFR) are obtained to evaluate their collateral capacity.
As for the middle cerebral artery (MCA) and the anterior cerebral artery (ACA), the transitional-type model possesses the best collateral capacity. But for the posterior cerebral artery (PCA), unilateral stenosis of ICA has the weakest influence on the unilateral posterior communicating artery (PCoA) absent model. We also find that the full fetal-type posterior circle of Willis is an utmost dangerous variation which must be paid more attention.
The results demonstrate that different models have different collateral capacities in coping stenosis of unilateral ICA and these differences can be reflected by different outlets. The study could be used as a reference for neurosurgeon in choosing the best treatment strategy.
Stroke is a major health problem and a leading cause of adult disability in the world, and it ranks the second in the list of causes of death [1–4]. Wherein, atherosclerosis accounts for up to one-third of strokes . Atherosclerosis of supra-aortic vessels and especially at the common carotid bifurcation is a major cause of recurrent ischaemic stroke . There are two primary reasons for stroke caused by atherosclerosis: one is that the atheromatous plaque may be the source of cerebral emboli, which may result in cerebral infarction . The other is that atherosclerosis leads to vascular stenosis and the decrease of vascular lumen, which may cause inadequate blood perfusion of the downstream cerebrovascular and brain tissue .
Carotid occlusive disease amenable to re-vascularization accounts for 5-12% of new strokes [9–11]. As for patients with severe internal carotid artery (ICA) stenosis, carotid endarterectomy (CEA) has been recognized as an effective therapy for re-vascularization [12, 13]. However, in fact, it is often compared with carotid artery stenting (CAS), another therapy employed, when selecting the best treatment strategy for an individual patient. Thus, as an important index, the collateral capacity of the cerebral artery is very necessary to be pre-known .
The circle of Willis (CoW) is a ring-like arterial structure, and it links the two main cerebral artery systems, namely the internal carotid artery system and the vertebrobasilar system, and it is also the primary collateral pathway locating in the base of brain. When the blood flow of an unilateral artery declines caused by stenosis, it can be compensated from the contralateral side by the CoW. The collateral capacity of the CoW improves cerebral perfusion in ischemic areas and may diminish the effect of ischemic events [15, 16]. Unfortunately, based on the radiological and anatomical studies [17, 18], no more than 50% of the general population have a complete CoW. The possible variations include hypoplasia or completely absent blood vessels, which may occur in the anterior circulation or the posterior circulation. The variations may influence the collateral capacity of the CoW and the risk of ischaemic stroke. Therefore, evaluating the collateral capacity of different configurable CoWs in patients is very important .
There have been some studies performed on haemodynamics of CoW with different anatomical variations in recent years. Some researchers have treated the cerebral vasculature as 1D structure, however, basing on Poiseuille flow, it cannot capture the effects of the complex arterial geometry, especially the effects of blood vessel junctions [20, 21]. Although 2D models can improve the accuracy of simulation , it's still necessary to establish 3D models for obtaining the more realistic hemodynamic data. There are some existing researches to investigate the collateral capacity of the CoW, based on 3D model [23, 24]. However, these studies just considered two or three general anatomical variations, in order to understand the influences of other variations, more variational cases should also be paid attention. Moreover, according to the recent CTA study of variations of the CoW, the population of China has a higher prevalence of compromised posterior collateral . Therefore, the present work aims to investigate the collateral capacity of the comprehensive CoW with anatomical variations in the posterior circulation.
Materials and methods
Basic model reconstruction
Diameters of each artery branch . Diameter1 and Diameter2 represent the shrunken and enlarged diameters of the corresponding vessels, respectively.
Normal type circle (NT): all of the arterial segments are normal, and PCA-P1 has a larger diameter than the PCoA.
Fetal-type posterior (FTP): the diameter of PCA-P1 is smaller than the unilateral PCoA, and the ICA is the main blood supplier to the PCA.
Transitional-type (TransT): the PCoA and PCA-P1 share an equal diameter.
Full FTP (fuFTP): the PCA-P1 is absent, and the PCA arises from the ICA.
Bilateral FTP (bFTP): the variation of FTP occurs on both the two sides.
Bilateral PCoA hypoplasia (bhypPCoA): both the left and right PCoA are hypoplastic, namely, the diameters of bilateral PCoA are less than 1 mm.
Unilateral PCoA hypoplasia (uhypPCoA): only one side of PCoA is hypoplastic, namely, the diameter of unilateral PCoA is less than 1 mm.
Unilateral PCoA absent (uabPCoA): one side of PCoA is absent.
Stenosed model reconstruction
There are now two major definitions about the stenosis in the ICA. In the North American Symptomatic Carotid Endarterectomy Trial (NASCET), ICA stenosis was classified angiographically: Degree of stenosis (Std) = (1--[narrowest ICA diameter/diameter normal distal cervical ICA]) × 100% . While basing on the same NASCET data, Moneta et al.  defined the degree of stenosis in symptomatic patients as the ratio of an internal carotid artery to common carotid artery peak systolic velocity (ICA/CCA PSV). Here, we adopt the first definition of stenosis.
Dimensions of different sketches and the change of MCARR with different stenosis.
Unstructured meshes were created with ICEM CFD (ANSYS, Version 15.0). Element sizes ranged between 0.3 mm and 0.35 mm, with denser elements in high curvature regions. On average, meshes of each model consisted of 0.47 million nodes and 2.5 million elements. Mesh independence were confirmed by performing a denser mesh than those mentioned above. By comparing the VFR difference between different meshes, the relative errors were less than 3%.
where P is the blood pressure, v is the velocity vector, ρ is the blood density (1056 kg/m3) and τ represents the shear stress term. The fluid viscosity is set as 0.004Pa.s.
CFD software Fluent 6.3 (ANSYS) was used to perform the simulations. Blood pressure in the realistic physiological conditions is pulsatile, but it's only change the mean cerebral blood flow , so we chose the steady pressure as the inlet and outlet boundary condition. As for the choice of pressure, rather than velocity or mass flow, it's because the change of volume flow rate, which reflects the influence of different models, is the indicator to get. Therefore, six outlets were set as a pressure of 75 mmHg, and the four inlets were set as 82.14 mmHg (ICAs) and 80.17 mmHg (VAs). The reason for the different pressure between ICA and VA is that the pressure drops between the two arteries are different. When we choose the real blood flow measured by PC-MRI as the boundary condition , the results verify this difference.
In the simulation, the convergence criterion was satisfied when the residual of continuity was less than 10-4 and the velocity component is less than 5.0e-5.
Here, we only considered the influence of the anatomical variations in the posterior circulation and introduced different severities stenosis in the ICAR, so the other conditions were consistently kept.
Influence of the degree of stenosis
Flow rate variations of MCAs&ACAs with different CoW models
Because the anatomical variations of the CoW are mainly located in the posterior circulation in this study, namely PCA-P1 and PCoA, the PCAs have different flow variations compared with the ACAs and the MCAs. Moreover, the ACAs and the MCAs are the direct branches of the ICAs, so the stenosis in the ICAR will firstly influence the flow rate of the ACAR and MCAR. Based on the above understanding, we will discuss the PCAs from the ACAs and the MCAs separately.
The blood reduction percentage of ACA and MCA compared with their respective no-stenosis model when ICAR stenosed by 79%.
Flow rate variations of PCAs with different CoW models
The blood reduction percentage of PCAL and PCAR compared with no-stenosis model when ICAR stenosed by 79%.
Discussion and conclusions
The CoW, served as the primary collateral pathway of the cerebral artery, has a major role in maintaining sufficient blood perfusion for brain tissue. A good collateral capacity is beneficial to relieve symptom caused by unilateral artery stenosis or occlusion. The anatomical variations of the CoW lead to the differences for their collateral capacity, and understanding these differences is utmost important for clinicians.
In recent years, the collateral capacity of different configured CoW has been studied. Long et al.  have studied the collateral capacity of the CoW with severe carotid artery based on 3D models, and this literature mainly concerned about the change of pressure. Alastruey et al.  have assessed the effects of anatomical variations and occlusions on cerebral flows with 1D computing models, and they have found that ACoA is a more critical collateral pathway than PCoAs if ICA is occluded, and the worst case is that a CoW lacks of the first segment of ACA, and the contralateral ICA is occluded. Fleur et al.  have studied the influence of FTP on cerebral collateral circulation, and considered that patients with an FTP may be more prone to suffer vascular insufficiency, especially for the case of full FTP.
In this paper, we have reconstructed eight 3D models with anatomical variations in the posterior circulation, and assessed their collateral capacity in term of the flow rate changes. Compared with other studies, the main contribution of this paper is that the anatomical variations in the posterior circulation were investigated and we have considered more variations. The major concern of this study are blood flow rates of various cerebral arteries, however, there is not enough data based on all of the cases in this study, therefore, we just comparing the distribution of flow of afferent and efferent arteries with published PC-MRI measurements [27, 28, 31] for some special models, such as the NT and fuFTP models. In the case of NT model with no-stenosis, the VFR of unilateral ICA, MCA and ACA with this measurement are about 210-280 ml/min, 130-155 ml/min and 75-90 ml/min, respectively. While in our simulation, the corresponding values are 245 ml/min, 143 ml/min and 89 ml/min, respectively. In the case of fuFTP model with no-stenosis, relative contribution by VFR of the ipsilateral ICA, contralateral ICA and BA is 46.6:41.6:11.7, while we find the specific value is 46.5:43:10.5. The present results agree well with the in vivo data in the literature. In fact, because of the auto-regulation of the cerebral arteries, also including the vast collateral anastomoses and individual difference, it is hard to evaluate the collateral capacity of different configured CoWs quantitatively. However, the tendencies reflected by the anatomical variations are helpful in assessing the collateral capacity of the CoW qualitatively.
In conclusion, with the help of CFD simulation, we have found that the collateral capacities of CoW with anatomical variations in the posterior circulation are different in dealing with unilateral stenosis, and these differences can be reflected by the blood flow rates of efferent arteries. As for the ACAs and MCAs, the TransT model possesses the best collateral capacity. But for the PCAs, unilateral stenosis of ICA has the weakest influence on the uabPCoA model. Meanwhile, we must pay more attention to the fuFTP type for all of the efferent arteries, which have a notable blood flow reduction.
For patients suffering from the ischemic stroke caused by atherosclerosis, it's important to assess the collateral capacity of the cerebral artery, which serves as an important index for selecting the best treatment strategy, especially for the CoW. The study may help neurosurgeons with the risk stratification for patients with cerebral arterial diseases.
Finally, we have to point out the limitations of the work: (1) the results and conclusions are based on ideal models, not the patient-specific models; (2) pulsatile boundary condition has been ignored.
YR: M.E., Biomechanics Laboratory, School of Biological Science and Medical Engineering, Southeast University, Nanjing, China; QC: Ph.D, Associate Professor, Biomechanics Laboratory, School of Biological Science and Medical Engineering, Southeast University, Nanjing, China; ZL: Ph.D, Full Professor, Principal Investigator, Biomechanics Laboratory, School of Biological Science and Medical Engineering, Southeast University, Nanjing, China;
This study was partially supported by the National 973 Basic Research Program of China [No.2013CB733800] and the National Natural Science Foundation of China (NSFC) (No. 11272091, 11422222, 31470043, 31300780).
Publication of this article was paid with funding from the National Natural Science Foundation of China (NSFC, No 11272091).
This article has been published as part of BioMedical Engineering OnLine Volume 14 Supplement 1, 2015: Cardiovascular Disease and Vulnerable Plaque Biomechanics. The full contents of the supplement are available online at http://www.biomedical-engineering-online.com/supplements/14/S1
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