Numerical and 1-D modeling of systemic circulation ...

Numerical and 1-D modeling of systemic circulation along with cerebral vasculature Seyedeh Sarah Salehi [email protected]

Bahar Firoozabadi [email protected]

Mohamad Said Saidi [email protected]

Mechanical Engineering Department Sharif University of Technology Tehran, Iran.

Abstract— The brain is one of the vital organs in the body. The main cerebral distribution center of blood flow in the brain is the circle of Willis (CoW). In more than 50% of healthy brains and in more than 80% of dysfunctional ones, at least one artery of the circle of Willis is absent or underdeveloped . These variations reduce the collateral flow availability and increase the risk of stroke and transient ischemic attack in patients with atherosclerosis. Thus it is essential to simulate the circle of Willis and investigate the effects of stenosis. In this work the systemic arteries along with the circle of Willis are simulated using the finite volume method and one- dimensional equations of conservation of mass and momentum. The structured tree is considered as the outlet boundary condition. The results show that blood flow rate, pressure and velocity through vessels are consistent with previous results. Meanwhile, the internal Carotid artery (ICA) stenosis is simulated and the collateral capacity of the circle of Willis is shown. As the area of the ICA decreases, the velocity of blood through communicating arteries is increased but this ability is reliable to a special extent of area reduction in the ICA.

In this method the arterioles are considered as the arterial tree which is based on a particular and optimized model. In fact, an asymmetric structured tree is used and the impedance of it is computed in a certain way. The circle of Willis is made up of 16 vessels. “Fig 1.b” shows the circle of Willis in which the basilar artery (artery no. 22 ) and internal carotid arteries (arteries no. 11 & 12) are connected to it. The vertebral arteries join intracranially to form the basilar artery (artery no.22) and each of internal carotid arteries branches and enters the CoW. The 1-D incompressible Navier-Stokes equations are the governing equations. Finite volume method is used to solve these equations.

Keywords- circle of Willis, stenosis, structured tree, systemic circulation.

I.

INTRODUCTION

So far there have been several models to simulate the vascular system. To simulate three- dimensional flow, considerable computational costs are required. The onedimensional simulations can be applied successfully where the flow does not exhibit significant three dimensional effects [1]. To calculate the blood flow and pressure in vessels it is necessary to solve the continuity and momentum equations and there should be a state equation that relates the area and pressure. In order to gain this aim, the boundary conditions should be known. In this study the systemic circulation with 48 main arteries and the CoW are the geometry at hand. “Fig1.a” shows the main arteries in the body. The cardiac output is the inlet boundary condition for CoW. Blood through aorta reaches the ICAs and then enters the CoW. The outlet boundary condition is chosen based on the structured tree concept [2].

Figure1. a) 48 arteries in the body(left), b) one dimensi onal model of the CoW connected to arteries(right)

The circle of Willis is made up of right and left posterior cerebral arteries (rPCA & lPCA (artreis no.32 & 33)), right and left posterior communicating arteries(rPCoA & lPCoA(arteries no.19 & 20)), right and left anterior cerebral arteries(rACA & lACA(arteries no.29 & 30)) and anterior communicating artery(ACoA(artery no.31)). Blood is delivered to the brain through the two internal carotid arteries and the two vertebral arteries. The two vertebral arteries communicate and reach the basilar artery (artery no.22). It then bifurcates and the posterior part of the brain receives blood. Each ICA bifurcates and middle cerebral arteries (arteries no.23 &24) and anterior cerebral arteries(arteries no. 29 &30) deliver blood to middle and anterior part of the brain. This collateral pathway has two advantages. First, in case of missing or occluded arteries, it provides blood through this collateral

blood supply to the outflow arteries and second, it makes regular blood supply to the brain. II.

PHYSIOLOGICAL DATA

Geometrical data of cerebral arteries are based on results of experimental measurements. Elastic parameters of the vessels are estimated with the relation between module of elasticity and radius and experimental data. Total blood flow rate in the brain is about 12 ml/s. Physiological data of the vessels of the circle of Willis is presented in table 1 [3].

and is the cross sectional area, h is the vessel wall thickness at ( ) and it is assumed that are zero. E is the module of elasticity, σ is poison ratio which is considered as ½ [4]. β is calculated from the equation below and is related to pulse wave propagation velocity, c. 

c2 

L(cm)

(cm)

(cm)

R/L. vertebral R/L. int. carotid I(ICA) Basilar R/L. PCA I R/L. PCA II R/L. PoCA R/L. int. carotid II(ICA) R/L. MCA R/L. ACA I R/L. ACA II ACoA

13.5 17.6 2.90 0.50 8.60 1.50 0.50 11.9 1.20 10.30 0.30

0.150 0.250 0.162 0.107 0.105 0.073 0.200 0.143 0.117 0.120 0.100

0.136 0.200 0.162 0.107 0.105 0.073 0.200 0.143 0.117 0.120 0.100

III.

SIMULATION METHOD

The mathematical model is based on solution of nonlinear and one dimensional equations of flow and pressure pulse wave propagation. Governing equations are the continuity and momentum equations for a Newtonian fluid. These equations are [4]: (1) A ( AU )  0 t x  (2) U ( AU ) 1 p f U   t x  x A  where, x is the longitudinal length along the vessel, t is the time, A(x,t) is the cross sectional area of the vessel, U(x,t) is the average longitudinal velocity, p(x,t) is the average internal pressure along the cross section, ρ is blood viscosity(1050 kg/m3) and f(x,t) is the friction force. The relation between these two equations is completed with an equation that relates the cross sectional area and pressure (state equation). It is assumed that the vessel wall is thin, homogenous and elastic. This equation is (3)  p  p0  ( A  A0 ) A0  Where



 hE (1   2 ) 



 

Then, the state equation in vessels will be as follows: p( x, t )  p0 

TABLE I. TABLE 1. PHYSIOLOGICAL DATA OF THE CIRCLE OF WILLIS, L REPRESENTS THE LENGTH OF THE VESSEL, ، IS THE INLET RADIUS OF THE VESSEL AND ، IS THE OUTLET RADIUS Arterial segment

 1 / 2 A 2 A0

4 Eh (1  3 r0

A0 ) A

  



The elastic module is a function of the vessel radius in diastole and is calculated from the following equation [5].

Eh  k1 exp( k2 r0 )  k3 r0

(7)

Where, = 2*10^7 g/(s2cm) = -22.53(1/cm) = 8.65*10^5 g/(s2cm) A. Assumptions In the simulation process, the following assumptions were considered:  The blood is a Newtonian and incompressible fluid,  Blood flow in arteries is one dimensional,  The equations are expressed in terms of variables averaged on cross sections.  The vessel walls are elastic and cross sectional area of the vessel is not only related to internal pressure and radius but also is dependent on module of elasticity and vessel wall thickness,  Blood flow is laminar and velocity profile is parabolic,  The vessels are tapered in their longitudinal direction,  The vessel walls are incompressible but blood leakage is modeled through small arteries,  Gravity is not considered and body is modeled as horizontal. B. Inlet boundary condition The circle of Willis has three inlets, left and right internal carotid arteries and the basilar artery. In this study, the circle of Willis is coupled to the 48 arterial of the systemic arteries and thus the inlet boundary condition is

the heart outflow or blood enters aorta that is measured experimentally. In this way the blood flow rate into the first artery is known at any time. Blood flow rate from the heart is varying with different conditions. Age, gender and transient states like exercise are variables that change the blood flow rate from the heart. In this work this profile is considered as constant.” Fig.2” shows the inlet blood flow profile to the aorta that is the inlet boundary condition. The heart pulse period is 1 second.

Figure2. Inlet flow rate to the aorta as initial condition[2]

C. Boundary condition in bifurcations “Fig.3” shows a bifurcation. If it is assumed that bifurcation is at one point, we require three equations to complete the system of equations. One of them is the continuity equation. Since at the bifurcation we have no leakage, blood flow out of the parent artery is equal to the blood flow that enters the daughter arteries. This equation is as follows: q parent ( L, t )  qd1(0, t )  qd 2 (0, t )

In one dimensional modeling, the outlet boundary condition is critically important because it should be able to simulate the flow in arterioles and even capillaries. In this work, the structured tree model is used as the outlet boundary condition. This outlet boundary condition has some advantages and diminishes some disadvantages of other outlet boundary conditions. Indeed, in this study some assumptions are added, such as that governing equations are linearized when blood flow passes through this tree. So we can take an equation dependent on time that relates pressure and flow rate. This outlet boundary condition is physiological and has less computational cost [6]. “Fig.4” shows a schematic of the structured tree which is a representation of small vessels. The impedance of the structured tree is calculated with the expansion of the Womersely equation. Therefore with the impedance of the structured tree, we can develop an equation that relates pressure and flow. There are two algorithms to extract pressure from impedance and flow rate. In the first one, the Furrier transform is used and thus pressure is the multiplication of impedance and flow in frequency domain. Pressure is transferred to time domain by using the inverse Furrier transform. In the second algorithm, the impedance is transfered to time domain with the inverse Furrier transform and then pressure is calculated in time domain by using of convolution integral. However, the result of both algorithms is the same. The geometry of the structured tree is defined with two factors α and β.







D. Outlet boundary condition

 Figure3. The branching point Figure4. The structured tree

The continuity of static pressure relates the outlet pressure of parent vessel with the inlet pressure of daughter vessels. However, it is assumed there is no pressure loss. Since the flow is laminar, this assumption is reliable. This assumption is expressed in following equation p parent ( L, t )  pd1(0, t )  pd 2 (0, t )

(9)

The power law is used to relate the parent vessel to daughter vessels in a bifurcation. Zamir [7] developed this equation.

rparent  rd1  rd 2

(10)

The value of ξ is dependent on flow, whether it is laminar or turbulent. Experimental results show that this value can be between 2 and 3. Area fraction η is defined as (11) 

velocity(m/s)



rd12  rd22 2 rparent

0.5 0.4 0.3 0.2 0.1 0

and asymmetric ratio is

 

(12)

rd1 rd 2

is the length to radius ratio and is calculated from the equation below. L lrr  (13) r  (14) rd1  rroot  (15) rd 2  rroot  i j i (16) ri, j    rroot  In the systemic arteries the value of is variable between 10 and 50. By combination the power law and area fraction, the parameters α and β are

  (1    / 2 ) 1 /

(17)

  

(18)

For systemic arteries, α and β are 0.9 and 0.6 respectively. IV.

RESULTS

Blood flow rate, pressure and velocity through vessels of the circle of Willis are obtained and are compared with previous results. One of these is showed in the “Fig.5”. In this figure the velocity through middle cerebral artery (MCA) is compared with the results of Alastruey et al. [4].

0 present work Alastruey

0.5

1

1.5

time(s)

Figure5. Verification of the velocity profile through MCA with the results of Alastruey et al. [4].

As can be seen, the results of this model are consistent with the results of Alastruey et al. [4]. Thus using these boundary conditions are good for modeling the systemic arteries along with cerebral arteries. Moreover, in this work, the internal carotid artery (ICA) stenosis and its effects on blood flow distribution are simulated. It is found that when the cross sectional area of the ICA decreases, blood velocity through communicating arteries increases, but as area reduction reaches 84% of the initial cross sectional area of the ICA, the communicating arteries cannot deliver sufficient blood. It means that the circle of Willis has a collateral pathway and in case of stenosis in one side, the other side compensates the blood flow rate. Long et al. [8] investigated the capacity of the circle of Willis to provide collateral blood supply for patients with unilateral carotid arterial stenosis. He found that the ability to deliver sufficient blood flow in stenosis is lower than 86% of the area reduction. However, in his work the circle of Willis was considered as the simulation geometry, since using 3D computational fluid dynamics for all the vessels of the cerebral had a great computational cost. Variations in the velocity profile through communicating artery with area reduction in ICA are shown in “Fig.6”. When the stenosis in ICA is greater than 84%, the direction of flow in the communicating artery would be reversed and it is in consistent with the results of Long et al. [8].

0.4

R=0.00275 R=0.0015 R=0.001

velocity(m/s)

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(s)

Figure 6. Variations in the velocity profile through communicating artery with decreasing the cross sectional area of the ICA,”R” represents the radius of the ICA

Pressure difference between middle cerebral artery (MCA) and ICA is calculated and it is found that when the degree of stenosis in ICA is greater than 84%, this pressure drops significantly as can be seen in “Fig.7”.

8000 R=0.00275 R=0.002 R=0.0015 R=0.001 R=0.0005

7000

pressure(pa)

6000 5000 4000 3000 2000 1000 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(s)

Figure7. Variations in blood pressure drop through MCA with decreasing ICA cross section.

V.

CONCLUSION

In this study, a model of 48 arteries in systemic circulation along with cerebral vasculature was considered and 1-D governing equations are solved numerically using finite volume method. The innovation is using the structured tree as the outlet boundary condition. Blood flow rate, pressure and velocity through vessels were calculated and compared with previous results and good agreement was taken. ICA stenosis was also studied and the collateral pathway of the circle of Willis was shown. VI.

REFERENCES

[1] L. Grinberg, T. Anor, E. Cheever, J. Madsen and G. Karniadakis, “ Simulation of the human intracranial arterial tree,” Phil. Trans. R. Soc, vol. 367, pp.2371-2386, 2011.

[2] S.M. Olufsen, S.P. Charles, W.Y. Kim, E. Pederson and A. Larsen , “Numerical simulation and experimental validation of blood flow in arteries with structured tree outflow conditions”. Annals of Biomedical Engineering, vol. 28 , pp.1281-1299, 2000. [3] F. Liang and K. Fukasaku, “A computational model study of the influence of the anatomy of the circle of willis on cerebral hyperperfusion following carotid artery surgery”, BioMedical Engineering OnLine, 2011. [4] J. Alastruey , ” Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows”, J. Biomechanics, vol. 40, pp.1794– 1805, July 2006. [5] N. Stergiopulos, D. Young, T. Rogge, “Computer simulation of arterial flow with applications to arterial and aortic stenosis”, J. Biomechanics, vol. 25, pp. 14771488, 1992. [6] W. Cousins and P.A. Gremaud, ” Boundary conditions for hemodynamics: The structured tree revisited”, J. computational physics, vol. 231, pp.60866096, 2012. [7] M. Zamir, ”Optimality principles in arterial branching”, J. Theor Biol, vol. 62, pp. 227-251, 1976. [8] Q. Long , L. Luppi , S. Koing , C. , Rinaldo and V. , K. Das , ” Study of collateral capacity of the circle of Willis of patients with severe carotid artery stenosis by 3D computational modeling”, J. Biomechanics , vol. 41 ,pp. 2735-2742, June 2008.