Up to 50% cases of Subarachnoid Hemorrhage (SAH) result in fatality and 10%–15% lead to death before reaching a hospital [
An efficient and costeffective way to modeling natural phenomena is numerical simulation. Increasing power of computing causes numerical simulation that plays a key role in recognition of various problems in engineering and medicine. The important point is the reliability of obtained results and the meaningful results of a numerical simulation that depend on variety of aspects.
The classification of related subjects to numerical simulation is a valuable step. Because of the complexity of blood supply system in the brain and also longtime history of related research, it is necessary to review these studies. Two main groups are considered here: (1) fluid and structure and (2) flow and simulation. The tree of this classification is shown in Figure
Classification of related subjects to the numerical simulation of blood flow in CoW.
In fluid and structure section, there are three subcategories: blood properties, vessel properties, and cerebrovascular features. On the other side, the flow and simulation is divided to three subcategories: geometry and computational domain, numerical approaches, and flow regime features. The CoW is a collection of major vessels connected together and located at the bottom of the cerebral cavity and it is a common place for aneurysm [
Physiological properties of CoW’s vessels, such as length, radius, thickness, and elastic modulus [
Arteries  Length (cm)  Initial diameter (cm)  Thickness (cm)  Elastic modulus (Pa) × 10^{6}  

1  Ascending aorta  4  2.4  0.163  0.4 
2  Aortic arch I  2  2.24  0.126  0.4 
3  Brachiocephalic  3.4  1.24  0.08  0.4 
4  Aortic arch II  3.9  2.14  0.115  0.4 
5  L. common carotid  20.8  0.5  0.063  0.4 
6  R. common carotid  17.7  0.5  0.063  0.4 
7  R. subclavian  3.4  0.846  0.067  0.4 
8  Thoracic aorta  15.6  2  0.11  0.4 
9  L. subclavian  3.4  0.846  0.067  0.4 
10  L. ext. carotid  17.7  0.3  0.038  0.8 
11  L. int. carotid I  17.7  0.4  0.05  0.8 
12  R. int. carotid I  17.7  0.4  0.05  0.8 
13  R. ext. carotid  17.7  0.3  0.038  0.8 
14  R. vertebral  14.8  0.272  0.034  0.8 
15  R. brachial  42.2  0.806  0.067  0.4 
16  L. brachial  42.2  0.806  0.067  0.4 
17  L. vertebral  14.8  0.272  0.034  0.8 
18  L. int. carotid II  0.5  0.4  0.05  1.6 
19  L. PCoA  1.5  0.146  0.018  1.6 
20  R. PCoA  1.5  0.146  0.018  1.6 
21  R. int. carotid II  0.5  0.4  0.05  1.6 
22  Basilar  2.9  0.324  0.04  1.6 
23  L. MCA  11.9  0.286  0.036  1.6 
24  R. MCA  11.9  0.286  0.036  1.6 
25  L. ACA, A1  1.2  0.234  0.029  1.6 
26  R. ACA, A1  1.2  0.234  0.029  1.6 
27  L. PCA, P1  0.5  0.214  0.027  1.6 
28  R. PCA, P1  0.5  0.214  0.027  1.6 
29  L. ACA, A2  10.3  0.24  0.030  1.6 
30  R. ACA, A2  10.3  0.24  0.030  1.6 
31  ACoA  0.3  0.148  0.019  1.6 
32  L. PCA, P2  8.6  0.21  0.026  1.6 
33  R. PCA, P2  8.6  0.21  0.026  1.6 
It was evident that the risk of transient ischemic attack is lower in the patients with healthy collateral circulations [
The variations of CoW usually affect more than one segment. Papantchev et al. studied six types of common variations and their effects [
The vessels of human are classified in three groups by size: arteries, veins, and capillaries. The artery and vein consist of three and two layers, respectively. The three layers are Tunica media, Tunica intima, and Tunica adventitia. Furthermore, the muscle layer in vein is considerably thinner than the artery’s muscle layer [
Entire circulatory system is coated by endothelial cells. Low wall shear stress (less than 0.4 Pa) causes the proliferation of endothelial cells and also atherosclerosis. On the other side, high wall shear stress (more than 1.5 Pa) causes the reorientation of endothelial cells in the flow direction [
The properties of blood vessels.
Vessel  Number  Diameter (cm) 


Aorta  1  2.5  3400 
Arteries  159  0.4  500 
Arterioles  400  0.005  0.7 
Capillaries  4500  0.0008  0.002 
Venules  4000  0.002  0.01 
Veins  40  0.5  140 
Vena cava  18  3  3300 
Obviously, the blood properties are required to simulate flow in arteries. The average adult has about 4 to 5 liters of blood. The red blood cells constitute about 45% of the volume of blood, and the remaining cells (white blood cells and platelets) less than 1%. The fluid part of blood is called plasma and is about 55% [
The Reynolds number is an important flow criterion to identify the flow regime, that is, laminar or turbulent. Generally, the Reynolds number is the ratio of inertial forces to viscous forces. The Reynolds number is a dimensionless parameter
If the Reynolds number is lower than the critical Reynolds, the flow is simulated as a laminar flow. The problem is finding the critical Reynolds number. In some cases the flow condition is on the border of turbulence and proper turbulent model is required [
The other main feature of fluid is viscosity. In the nonNewtonian fluid, there is a nonlinear relation between shear stress and shear rate. The nonNewtonian fluids are classified into the shear thinning and shear thickening. In shear thinning, increasing shear rate reduces the viscosity, and in shear thickening it is vice versa.
The human blood is considered as Newtonian fluid for all rates of shear when hematocrits are up to 12 percent. When the size of particles is significantly large in comparison with the channel’s dimension, the fluid behaves like nonNewtonian fluid [
Consequently, in the large vessels such as arteries the blood is Newtonian, but in small vessels such as capillary, it is considered as nonNewtonian fluid. The effect of vessel’s diameter on the viscosity is called FahraeusLindqvist. As the diameter decreases (between 10 and 300 micrometers), the blood viscosity is reduced, because of the motion of erythrocytes in the center of the vessel and leaving plasma near the wall. A comprehensive study on different viscosity models of blood was performed by Yilmaz and Gundogdu. The models are classified into two groups including time independent and time dependent flow behavior models [
The shear forces due to the viscosity rises the pressure drop in network of vessels. Generally, the HagenPoiseuille equation is used to compute pressure drop of a laminar and incompressible flow in long cylindrical pipe:
If an oscillating pressure gradient exists in the flow field, the velocity profile is a parabolic and proportional to the instantaneous flow rate and also the flow field is unsteady. Actually, in in vivo blood, the flow is affected by a pulsatile source. The inertia of the fluid in the central core prevents the core from following the applied gradient of pressure [
Another important property of blood is density. Same as viscosity, not only the amount of density is not fixed for all cases, but also it is function of different variables, such as species, gender, and body posture. The density of human blood is approximately 1056~1060 (kg/m^{3}). The density of blood plasma is approximately 1025 and 1125 (kg/m^{3}) [
The pressure gradient affording blood flow to the brain is called the Cerebral Perfusion Pressure (CPP). The variation of CPP is limited, because it can cause brain tissue to become ischemic, or rising Intracranial Pressure (ICP) [
The autoregulation is a significant ability of human body to regulate the proper perfusion pressure for sensitive and vital organs. The brain arteries can respond to the perfusion pressure change by vasodilate or vasoconstrict, which is called autoregulation.
The autoregulation mechanism consists of two features, the autoregulation curve and the autoregulation dynamics [
The upper and lower limits of the permeability are related to the arterial pressures limits of the autoregulated range. Therefore, the autoregulation dynamics are modeled using permeability (
The cerebral blood flow is a complex system, and CoW is just one of the important parts. Therefore, the systematic analysis is necessary for recognizing the functionality of CoW in this dynamic system. From the numerical modeling view point, there are three models of arterial blood flow [
The lumped element model describes the behavior of a distributed system into discrete entities under certain assumptions. The Windkessel model of the arterial network reveals a relation between pressure and flow at a specific arterial condition, without wave propagation consideration.
The simplified equations of fluid motion are used to study the flow field. The wavetransmission characteristics are formulated by Womersley’s oscillatory flow theory because of the hyperbolic feature of equations. A comprehensive literature review on the 1D models was performed by Reymond et al. [
In order to simulate accurately a segment of arterial tree, more comprehensive 3D model is required. Computational Fluid Dynamic (CFD) is a powerful tool to simulate complex flow field. The 0D and 1D modeling can reveal some useful and practical aspects about a real complex system. In order to investigate a phenomenon in detail, more accurate and applicable methods are required. If whole system of cerebral vessels is simulated by an accurate threedimensional approach, the computational cost is beyond the power of current computing devices. The studies based on CFD are classified into three: the development, progression, and rupture of cerebral aneurysms [
Combination of above methods can help improve prediction of complex problems. Quarteroni and Veneziani coupled 3D modeling of blood flow with a systemic, zerodimensional, lumped model of circulation [
One of the earliest studies about the mathematical model of saccular aneurysm in CoW was performed by Austin in 1971 [
Cassot et al. developed a mathematical model to simulate the CoW flow and predict the effectiveness of anterior and posterior communicating arteries on the brain blood pressure [
The equations of 1D models are based on the governing equations of flow motion. The laminar flow in arteries is same as steady and fully developed Poiseuille flow, therefore, the axial pressure gradient is computed as [
In order to make conservation form, the equations are coupled in a system of equations as
There are three variables in this system of equations, and an additional equation is needed to solve them. The third equation describes the relation between variation of artery’s area and pressure (i.e., fluidstructure interaction). Therefore, the Laplace law for thin and homogenous elastic artery’s wall is used to make this equation as follows:
The onedimensional model can be deduced from the wave propagation relation. The average speed of blood in vessels is 0.5 m/s, but the wave speed is 5 m/s. The blood pressure and flow pulsations are considered as the wave propagation in the arterial network, which consist of information about the cardiovascular system [
An important factor in the instantaneous vessel’s area is pressure difference between two sides of a vessel’s wall, which is called transmural pressure (
Generally, the characteristic lines are defined in hyperbolic equations and they determine the behavior of solutions using invariant quantities along certain trajectories. The conservative form of hyperbolic system of equations appears as [
The speed of waves from the heart (forward) is
In order to solve this system, the boundary conditions are required. The boundary conditions are reflecting and nonreflecting. In nonreflecting boundary conditions, the forward and backward waves are used at the inlet and outlet boundaries, respectively. Thus, the information from inside and outside of the domain are combined.
It should be noted that there are some difficulties in solution, such as discontinuity. A discontinuity is considered as a sudden jump; for example, stent or in a more prevalent case it is a vessel branching.
In the brain, aneurysms are often located at the lateral sides of curved vessels and at the bifurcations. The ruptured aneurysm is a common reason of death and also the surgery of aneurysm is high risk. Therefore, the recognition of different related subjects can improve the prediction of its behaviors. Different scientists studied the subjects, such as the reasons of aneurysm, the growth trend, and the effective factors. The effect of hemodynamic factors on the initiation, growth, and rupture of aneurysms is shown in different studies [
Furthermore, the interaction between blood and vessel’s wall play crucial role to simulate semireal pattern of fluid flow. The pressure and also blood velocity in large arteries are affected by the vessel deformability. The fluidsolid interaction (FSI) methods are used to predict this interaction. In simulation of a real situation, if the wall tension exceeds the wall tissue strength, the rupture of tissue should occur [
Generally, there are some common points in FSI methods:
generated loads on the structure by fluid,
response of structure and impacts on the flow field,
computational domain and grid adaptation according to new conditions.
Based on significant development in computational fluid approaches and FSI methods and also advanced software to generate complex computational domain from in vivo medical imaging, the prediction of aneurysm behavior become more accurate and reliable [
One of the first successful attempts to use FSI in predicting the abdominal aortic aneurysm was performed by Di Martino et al. [
The functionalities and anomalies of CoW are studied using CFD tools [
Kim simulated the flow condition of CoW in two cases, including absence of left PCoA and absence of ACoA. It was shown that the autoregulation mechanism is strongly affected by the communicating arteries, PCoA and ACoA [
In spite of these significant abilities, there are some difficulties to reach accurate and applicable predictions:
geometry generation according to the real situation,
generation of proper and efficient computational grid,
appropriate and efficient solver selection,
accurate initial and boundary conditions,
trace the result during the run carefully to improve the simulation,
present the final results in a userfriendly and inferable manner.
The most important subject in geometry generation is similarity with the physical domain. The simplest numerical form of aneurysm is a spherical and symmetric sac. But the actual geometry of aneurysms is irregular. Some software can generate real geometry from Xray scans. For example, the mimiced software can produce actual geometry of organs from MRI of CTscan files [
After the geometry and grid generation, the initial and boundary conditions are required. Generally, the boundary value problem is a differential equation with boundary conditions. The boundary conditions cause unique solution for problem, which is called wellposed problem. In addition, the relation between computational domain and outer regions is established with boundary condition. The boundary condition types are inlet, outlet, wall, symmetry, and periodic. Another important factor is initial condition, which specifies the condition of domain at the beginning.
The main step is selecting an appropriate numerical solver. As mentioned before, there are two regimes of flow, that is, laminar and turbulent. In laminar flow, the layers are parallel without any disruption, lateral mixing, and eddies [
On the other hand, the turbulent flow is completely a complex regime with disruption, lateral mixing, and eddies. The implementation of discretized governing equation in turbulent flow needs very fine grid, and so unacceptable computational cost. This method is called Direct Numerical Simulation (DNS). DNS is not an applicable method in real situation. Alternatively, some simplified methods were developed in order to model the turbulent flow and reduce considerably the computational cost, such as ReynoldsAveraged NavierStokes (RANS). The comprehensive description of different methods of turbulent flow simulation is presented by Andersson et al. [
The critical point in virtual simulation is the accuracy of prediction and consistency with the reality. The accuracy depends on the variety of factors. In the current paper, the important related subjects are reviewed in a classified manner. Although, different 1D and 3D simulations of CoW and cerebral aneurysm were carried out, many questions have been left. Some of important objectives of future studies are listed here.
(1) The cerebral perfusion is a vital subject and it is a function of flow rate and pressure. The amount of flow rate is known in normal condition and also the effect of some anomalies on flow rate was studied by 1D simulation [
(2) The 0D or 1D modeling can help generate boundary conditions for 3D simulation. In addition, the 3D simulation can help improve the 1D modeling. As a result, 0D or 1D models and 3D simulation can have an effect on each other, simultaneously. It means that the local 3D simulation, that is, CoW, and global modeling, that is, cerebral blood system, should perform in a coupled condition. Regarding anomalies influence on the functionalities of CoW and also whole cerebral blood supply system, what is the relation between anomalies and cerebral blood supply?
There is no financial interest related to the material in the paper.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Study concept and design was made by S. R. Ghodsi, V. Esfahanian, and S. M. Ghodsi. Drafting of the paper was made by S. R. Ghodsi. The sponsor had no role in the design and conduct of the study; collection, management, and analysis of the data; or preparation, review, and approval of the paper.
This study was carried out in VFE Research Institute in University of Tehran and based on a Project entitled as “3D Numerical Simulation of Flow in the Circle of Willis in Normal and Common Anomalies.” In addition, many thanks to Sina Trauma Research Center, because of spiritual and financial sponsorship of the project.