Bio-Medical Materials and Engineering 24 (2014) 459–466 DOI 10.3233/BME-130831 IOS Press
459
Hemodynamic Numerical Simulation and Analysis of Oscillatory Blood Flow in growing Aneurysms Lifang Wanga, Xiaohua Zhou b, Mingxiu Shena, Yanping Suna , Guifang Sun c,* a
Department of Automatic Control and mechanical engineering, Kunming University, Kunming , China. b School of Electrical and Information Engineering, Guangxi University of Science and Technology, Liuzhou , China. c Clinical skill center,Kunming Medical University,Kunming ,China.
Abstract. Hemodynamics plays a crucial role in the formation, progression and rupture of intracranial aneurysms. Understanding these mechanisms is important to improve current diagnosis and treatment of intracranial aneurysms. In this study we simulate and analyze the pressure gradients and the blood flow fields in growing intracranial aneurysms. Firstly, the pressure gradients are obtained according to the blood velocity waveform at the axis of the inlet to the artery, which can be acquired by transcranial Doppler technology. Then, blood flow fields are calculated by solving the linearized Navier-Stokes equations and continuity equation using the Fourier series method. Results show that the higher the aneurysm dilatation degree is, the lower the maximum oscillatory velocity will be. Therefore, the oscillatory velocity may be used to analyze the characteristics of blood flow signals from aneurysm and to forecast the size of aneurysm. This sensitive parameter can be utilized for the detection of vessel diseases, which is promising to provide a useful reference in clinical application. Keywords: hemodynamics; aneurysms; numerical simulation; oscillatory blood flow, pressure gradients
1. Introduction Intracranial aneurysms are pathological dilatations of the vessel wall which usually occur near arterial bifurcations in the circle of Willis [1-2]. The most serious consequences of intracranial aneurysms result from their rupture and the resulting intracranial hemorrhage, which have an incidence rate of about 1/10000 and a mortality and morbidity rate of 25%~50% [3-4]. The formation of intracranial aneurysm can be attributed to various factors, but hemodynamics is thought to be an important factor in the formation, progression and rupture of intracranial aneurysms [5-6]. Understanding the flow dynamics is important to improve current diagnosis and treatment of intracranial aneurysms. Nowadays, image-based computational Fluid Dynamics (CFD) software was This work was supported by the grant of $pplied Basic Research Projects of Yunnan Province (2010ZC161), the Key Scientific Research Project of Office of Education in Yunnan Province (2010Z027). * Corresponding author. E-mail: [email protected]. 0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
460
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
applied to simulate the blood flow to extract patient-specific hemodynamics information [7–9]. However, individual blood flow simulation is subject to the limitations of time and facilities. Numerical blood flow simulation provides the visualization and quantification of the hemodynamics in temporal and spatial detail. In this study we simulate and analyze the pressure gradients and the blood flow fields in growing intracranial aneurysms using the Fourier series method to evaluate the role of blood flow fields in the development, which is promising for the future clinical application in intracranial aneurysms. 2. Methods 2.1. Aneurysm model The simulation model of aneurysm is depicted in Fig. 1 and the geometry of the aneurysmal wall is given by
ª § π x · ·º δ § R ( x ) = R0 «1 + ¨¨ 1 + cos ¨ ¸ ¸¸ » , x ∈ [ − x0 , x0 ] «¬ 2 R0 © © x0 ¹ ¹ »¼
(1)
where R0 is the radius of the normal vessel, R ( x ) denotes the radius of aneurysmal dilatation segment. 2x0 is the axial length of the aneurysm and δ is the measure of the degree of the dilatation.
Fig.1. The geometric model of the aneurysm
Fig.2. The mean inlet axial velocity by applying TCD technology
2.2. Governing equations and boundary conditions Blood flow was considered as a pulsatile, axisymmetric, incompressible Newtonian fluid with blood density ρ = 1050 kg / m 3 , blood viscosity η = 3 . 5 × 10 − 3 Pa .s and modeled by the unsteady NavierStokes equations with cylindrical polar coordinates [10]:
461
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
∂u 1 ∂ ( rv ) + = 0; ∂x r ∂r
∂u ∂p η § ∂ 2u 1 ∂u · =− + ¨ 2 + ¸; ∂t ∂x ρ © ∂r r ∂r ¹
∂p ∂p = =0 ∂r ∂θ
(2)
where u and v are symbols for the axial and radial flow velocities, respectively, p represents the pressure, ρ is blood density, η is blood viscosity, and t is time, x and r represent coordinates of axial and radial directions, respectively. Wall condition: Assuming rigid and no slip boundary conditions were applied at the walls, which can be expressed as:
v r=R = 0 ; u r=R = 0
(3)
Symmetry condition: Owing to symmetrical characteristic of blood fluid on the tube axis, the boundary condition on the symmetry axis is:
∂u ∂r
r =0
=0
(4)
2.3. Solutions for the Equations Blood flow fields are calculated by solving the Navier-Stokes equations and continuity equation using the Fourier series method. The blood flow in heart circulation system generated by periodic pumping action of the heart can be reasonably considered to be periodic, so some flow parameters, ∂p and velocity (axial velocity u and radial velocity v ), can be such as pressure p , pressure gradient ∂x expressed as the Fourier series in the following form [10]:
u=
V § N U0 § N · jω t · + Re ¨ ¦ U i e jω i t ¸ ; v = 0 + Re ¨ ¦ Vi e i ¸ ; 2 2 © i =1 ¹ © i =1 ¹
C ∂p § N = 0 + Re ¨ ¦ C i e 2 ∂x © i =1
jω i t
· ¸ ¹
(5)
2π represents angular frequency, T denotes the cardiac cycle, Re is the symbol for T the real part, and j = − 1 ,C is Fourier series coefficient of the pressure gradient. Because the pressure gradient, axial velocity and radial velocity are all the body's physiological parameters, the number of the Fourier series terms N is finite and usually taken as 14 to get a better approximation. By inserting (5) in (1) (2), (3) (4) and solving the differential equations, the oscillatory blood flow velocities at the axis in the aneurysms with various dilatation degrees are written as [11] where ω i = iω =
u ( y, t ) =
2
(
R 2 y 2 − y1 4η
° N R2 C 0 + Re ® ¦ 2 ° i =1 jηα i ¯
)
½ ª J § j 3 2α y · º i ¸ » « 0 ¨© ¹ C e jω i t ° = u + u ¾ 1 2 « » i 3 ° « J 0 §¨ j 2 α i y1 ·¸ » ¹¼ ¬ © ¿
(6)
ρω i r R( x) ′ dy ( x ) , J and J denote the , y1 = , α i = R0 , i = 1, 2,3, , y1 = 1 0 1 dx R0 R0 η Bessel function of the zeroth and first order respectively; u1 denotes steady velocity, u2 denotes
where y =
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L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
pulsatile velocity. From above we know that the axial velocity components can be totally determined by given pressure gradient. The pressure gradients are obtained according to the blood velocity waveform at the axis of the inlet to the artery (as shown in Fig. 2), which can be acquired through Doppler technology. In order to obtain pressure gradient, V 0 and V i are inserted into the corresponding velocity boundary conditions to yield the following equations [11]: 3 2 J1 §¨ j 2αi y1 ·¸ dCi 2 y1 © ¹ Ci = 0 + dx y1 J §¨ j 32α y ·¸J §¨ j 32α y ·¸ i 1 i 1 0 © ¹ 2© ¹
′ dC 0 4 y 1 + C0 = 0 ; dx y1
′
(7)
where J 2 is the Bessel function of the second order. Corresponding boundary conditions are needed for solving equations (7). Assume that u ( x * ,0 , t ) is the velocity waveform at the axis ( y = 0 ) at a certain position ( x = x * ) in the upstream even segment of the aneurysmal artery, its Fourier series can be written as [12]
(
)
u x * ,0, t =
U 0* § N · + Re ¨ ¦ U i* e jω i t ¸ 2 © i =1 ¹
(8)
Comparing U 0* and U i* with (6) and letting y = 0 and y 1 = 1 , one can obtain the coefficients of the pressure gradient at the site x = x * of the artery: C0 = − *
4η U R0
* 0 2
;
C i = j ηα i U i *
2
*
ª º½ ° 2« »° 1 / ®R0 « − 1» ¾ 3 ° « J 0 §¨ j 2 α i ·¸ »° ¹ ¬ © ¼¿ ¯
(9)
In accordance with boundary conditions (9), by solving the differential equations (7) we can get the following C 0 and Ci [13]. C0 = C0
*
1 4 y1
3 § ( j 32α )2 J § j 32α · − 2 j 32α J § j 32α · · J0 §¨ j 2αi y1 ·¸ ¨ i i¸ i 1¨ i ¸¸ 0¨ * © ¹ © ¹ ¸. © ¹ Ci = Ci .¨ 2 3 3 3 ¨¨ ¸¸ § j 2α y · J § j 2α y · − 2 j 32α yJ § j 32α y · J0 §¨ j 2αi ·¸ ¨ i 1 ¸ 0¨ i 1¸ i 1¨ i 1¸ © ¹ © ¹© ¹ © ¹ ¹ ©
(10)
(11)
3. Experiments and results In the experimental studies, the pressure gradients and the oscillatory blood flow fields in intracranial aneurysms with dilatation degrees of 30% and 70% at the position x = R 0 and x = 3R 0 are simulated and analyzed, respectively. The following parameters are used: normal vessel radius R 0 = 0.5 cm, aneurysm length 2 x 0 = 8 R 0 , cardiac cycle T = 1s . According to the inflow
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
463
boundary conditions, the pressure gradients and the oscillatory blood flow velocities in aneurysms with various dilatation degrees at different positions are simulated using previously mentioned methods and listed as follows. 3.1. Pressure gradient
Fig.3. Pressure gradient with dilatation degrees of (a) 30%, (b) 70%
Fig. 3 shows that the influence of local dilatation on pressure gradient in dilatational arterial segment is not very obvious. 3.2. Blood flow field 1). Steady velocity
Fig.4. Steady velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=3R0
Fig.5. Steady velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=R0
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L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
Fig.4 and Fig.5 show that the steady velocity is parabolic distribution, which shows the characteristics of Poiseuille flow. Namely, it does not change along with the time constant. 7he more the vessel dilatation is, the lower the steady velocity in aneurysms will be and the maximal steady velocity with dilatation degrees of 70% at the position x=R0 is the smallest. Furthermore, under the condition of the same dilatation degree, the steady velocity in aneurysms will increase with the increase of the axial distance. In order to further compare the effect of aneurysm dilatation degree on the maximum steady velocity, the maximum steady velocity with different dilatation degrees at different position are listed in table 1. Table 1 The maximum steady velocity in aneurysm with different dilatation degrees at different positionV (m/s) dilatation degree position x=3R0 x=R0
30%
70%
0.2223 0.1535
0.1993 0.0949
2). Pulsatile velocity
Fig.6. Pulsatile velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=3R0
Fig.7. Pulsatile velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=R0
Fig.6 and Fig.7 show that the pulsatile blood flow field distributions have no longer been monotonous profile and change with time and radial position of the vessel. Within the quite big radial region, the pulsatile velocity along vessel radius changes flatly. The velocity in aneurysm will decrease, and the more the vessel dilatation is, the lower the pulsatile velocity will be. In addition, under the condition of the same dilatation degree, it will increase with the increase of the axial distance. The maximal pulsatile velocity with dilatation degrees of 70% at the position x=R0 is the
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
465
smallest. In order to further compare the effect of aneurysm dilatation degree on the maximum pulsatile velocity, the maximum pulsatile velocity with different dilatation degrees at different position are listed in table 2. Table2 The maximum pulsatile velocity in aneurysm with different dilatation degrees at different position (m/s) dilatation degree position x=3R0 x=R0
30%
70%
0.2504 0.1697
0.2232 0.0961
3). Oscillatory velocity
Fig.8. Oscillatory velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=3R0
Fig.9. Oscillatory velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=R0
Fig.8 and Fig.9 show that the oscillatory velocity also changes with time and radial position of the vessel. The oscillatory velocity reaches its maximum at the centerline. With the increase of the dilatation degree, the maximal axial oscillatory velocity will also decrease, and the maximal oscillatory velocity with dilatation degrees of 70% at the position x=R0 is the smallest. Furthermore, the oscillatory velocity near the vessel wall drops quickly in normal vessel and in aneurysm with small dilatation degrees at the downstream outlet. As a comparison, the oscillatory velocity near the vessel wall with the biggest dilatation will not drop quickly. In order to further compare the effect of aneurysm dilatation degree on the maximum oscillatory velocity, the maximum oscillatory velocity with different dilatation degrees at different position are listed in table 3.
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Table 3 The maximum oscillatory velocity in aneurysm with different dilatation degrees at different position (m/s) dilatation degree position x=3R0 x=R0
30%
70%
0.4465 0.3073
0.3997 0.1892
4. Conclusion In this study we simulate and analyze the pressure gradients and the blood flow fields in growing intracranial aneurysms under periodically oscillatory flow. Results show that the higher the aneurysm dilatation degree is, the lower the maximum oscillatory velocity will be. Therefore, the oscillatory velocity may be used to analyze the characteristics of blood flow signals from aneurysm and to forecast the size of aneurysm, which is promising to provide a useful reference in clinical application from this sensitive parameter. References [1] [2] [3] [4] [5] [6] [7] [8]
[9] [10] [11] [12] [13]
Lasheras J. C. The biomechanics of arterial aneurysms [J], Annual review of fluid mechanic, 2007 (39), pp. 293-319. Wiebers DO, Piepgras DG, Meyer FB, etc. Pathogenesis, natural history, and treatment of unruptured intracranial aneurysms [J], Mayo clinic proceedings, 2004(79), pp.1572–1583. Winn HR, Jane JA, Taylor J, Kaiser D, Britz GW. Prevalence of asymptomatic incidental aneurysms: review of 4568 arteriograms. Journal of neurosurgery, 2002(96), pp. 43–49. Hop JW, Rinkel GJ, Algra A, etc, Case-fatality rates and functional outcome after subarachnoid hemorrhage: a systematic review [J]. Stroke, 1997 (28), pp. 660-664. Tamura T, Jamous MA, Kitazato KT, etc. Endothelial damage due to impaired nitric oxide bioavailability triggers cerebral aneurysm formation in female rats [J]. Journal of Hypertension, 2009(27), pp. 1284-1292. Sforza D, Putman CM, Cebral JR. Hemodynamics of cerebral aneurysms [J]. Annual review of fluid mechanic, 2009 (41), pp. 91–107. Steinman DA, etc. Image-based computational simulation of flow dynamics in a giant intracranial aneurysm [J]. AJNR American Journal of Neuroradiology, 2003(24), pp.559–566. Valencia A, Morales H, Rivera R, Bravo E, Galvez M. Blood flow dynamics in patient-specific cerebral aneurysm models: the relationship between wall shear stress and aneurysm area index [J]. Medical engineering & physics. 2008(30), pp. 329–340. Shojima M, Oshima M, Takagi K, etc. Magnitude and role of wall shear stress on cerebral aneurysm: computational fluid dynamic study of 20 middle cerebral artery aneurysms [J]. Stroke, 2004(35), pp. 2500–2505. Liu Z., Cardiovascular Hydrodynamics [M], Fudan university Presses, Shanghai, 1986. Gong K., Liu Z. Shear stress distribution of oscillatory blood flow in local expansion artery [J]. Chinese quarterly of mechanics, Vol.24, No. 1, 2003 (3). Liu B., Guo B. and Liu Z., Velocity profile of periodic oscillatory blood flow in circle tube[J], 2002 (3) , pp.15-22. Gao L., An Analytical simulation model of Doppler ultrasound blood flow signals under the condition of the bilateral symmetrical stenosis vascular (Master Thesis), 2012.
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459
Hemodynamic Numerical Simulation and Analysis of Oscillatory Blood Flow in growing Aneurysms Lifang Wanga, Xiaohua Zhou b, Mingxiu Shena, Yanping Suna , Guifang Sun c,* a
Department of Automatic Control and mechanical engineering, Kunming University, Kunming , China. b School of Electrical and Information Engineering, Guangxi University of Science and Technology, Liuzhou , China. c Clinical skill center,Kunming Medical University,Kunming ,China.
Abstract. Hemodynamics plays a crucial role in the formation, progression and rupture of intracranial aneurysms. Understanding these mechanisms is important to improve current diagnosis and treatment of intracranial aneurysms. In this study we simulate and analyze the pressure gradients and the blood flow fields in growing intracranial aneurysms. Firstly, the pressure gradients are obtained according to the blood velocity waveform at the axis of the inlet to the artery, which can be acquired by transcranial Doppler technology. Then, blood flow fields are calculated by solving the linearized Navier-Stokes equations and continuity equation using the Fourier series method. Results show that the higher the aneurysm dilatation degree is, the lower the maximum oscillatory velocity will be. Therefore, the oscillatory velocity may be used to analyze the characteristics of blood flow signals from aneurysm and to forecast the size of aneurysm. This sensitive parameter can be utilized for the detection of vessel diseases, which is promising to provide a useful reference in clinical application. Keywords: hemodynamics; aneurysms; numerical simulation; oscillatory blood flow, pressure gradients
1. Introduction Intracranial aneurysms are pathological dilatations of the vessel wall which usually occur near arterial bifurcations in the circle of Willis [1-2]. The most serious consequences of intracranial aneurysms result from their rupture and the resulting intracranial hemorrhage, which have an incidence rate of about 1/10000 and a mortality and morbidity rate of 25%~50% [3-4]. The formation of intracranial aneurysm can be attributed to various factors, but hemodynamics is thought to be an important factor in the formation, progression and rupture of intracranial aneurysms [5-6]. Understanding the flow dynamics is important to improve current diagnosis and treatment of intracranial aneurysms. Nowadays, image-based computational Fluid Dynamics (CFD) software was This work was supported by the grant of $pplied Basic Research Projects of Yunnan Province (2010ZC161), the Key Scientific Research Project of Office of Education in Yunnan Province (2010Z027). * Corresponding author. E-mail: [email protected]. 0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
460
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
applied to simulate the blood flow to extract patient-specific hemodynamics information [7–9]. However, individual blood flow simulation is subject to the limitations of time and facilities. Numerical blood flow simulation provides the visualization and quantification of the hemodynamics in temporal and spatial detail. In this study we simulate and analyze the pressure gradients and the blood flow fields in growing intracranial aneurysms using the Fourier series method to evaluate the role of blood flow fields in the development, which is promising for the future clinical application in intracranial aneurysms. 2. Methods 2.1. Aneurysm model The simulation model of aneurysm is depicted in Fig. 1 and the geometry of the aneurysmal wall is given by
ª § π x · ·º δ § R ( x ) = R0 «1 + ¨¨ 1 + cos ¨ ¸ ¸¸ » , x ∈ [ − x0 , x0 ] «¬ 2 R0 © © x0 ¹ ¹ »¼
(1)
where R0 is the radius of the normal vessel, R ( x ) denotes the radius of aneurysmal dilatation segment. 2x0 is the axial length of the aneurysm and δ is the measure of the degree of the dilatation.
Fig.1. The geometric model of the aneurysm
Fig.2. The mean inlet axial velocity by applying TCD technology
2.2. Governing equations and boundary conditions Blood flow was considered as a pulsatile, axisymmetric, incompressible Newtonian fluid with blood density ρ = 1050 kg / m 3 , blood viscosity η = 3 . 5 × 10 − 3 Pa .s and modeled by the unsteady NavierStokes equations with cylindrical polar coordinates [10]:
461
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
∂u 1 ∂ ( rv ) + = 0; ∂x r ∂r
∂u ∂p η § ∂ 2u 1 ∂u · =− + ¨ 2 + ¸; ∂t ∂x ρ © ∂r r ∂r ¹
∂p ∂p = =0 ∂r ∂θ
(2)
where u and v are symbols for the axial and radial flow velocities, respectively, p represents the pressure, ρ is blood density, η is blood viscosity, and t is time, x and r represent coordinates of axial and radial directions, respectively. Wall condition: Assuming rigid and no slip boundary conditions were applied at the walls, which can be expressed as:
v r=R = 0 ; u r=R = 0
(3)
Symmetry condition: Owing to symmetrical characteristic of blood fluid on the tube axis, the boundary condition on the symmetry axis is:
∂u ∂r
r =0
=0
(4)
2.3. Solutions for the Equations Blood flow fields are calculated by solving the Navier-Stokes equations and continuity equation using the Fourier series method. The blood flow in heart circulation system generated by periodic pumping action of the heart can be reasonably considered to be periodic, so some flow parameters, ∂p and velocity (axial velocity u and radial velocity v ), can be such as pressure p , pressure gradient ∂x expressed as the Fourier series in the following form [10]:
u=
V § N U0 § N · jω t · + Re ¨ ¦ U i e jω i t ¸ ; v = 0 + Re ¨ ¦ Vi e i ¸ ; 2 2 © i =1 ¹ © i =1 ¹
C ∂p § N = 0 + Re ¨ ¦ C i e 2 ∂x © i =1
jω i t
· ¸ ¹
(5)
2π represents angular frequency, T denotes the cardiac cycle, Re is the symbol for T the real part, and j = − 1 ,C is Fourier series coefficient of the pressure gradient. Because the pressure gradient, axial velocity and radial velocity are all the body's physiological parameters, the number of the Fourier series terms N is finite and usually taken as 14 to get a better approximation. By inserting (5) in (1) (2), (3) (4) and solving the differential equations, the oscillatory blood flow velocities at the axis in the aneurysms with various dilatation degrees are written as [11] where ω i = iω =
u ( y, t ) =
2
(
R 2 y 2 − y1 4η
° N R2 C 0 + Re ® ¦ 2 ° i =1 jηα i ¯
)
½ ª J § j 3 2α y · º i ¸ » « 0 ¨© ¹ C e jω i t ° = u + u ¾ 1 2 « » i 3 ° « J 0 §¨ j 2 α i y1 ·¸ » ¹¼ ¬ © ¿
(6)
ρω i r R( x) ′ dy ( x ) , J and J denote the , y1 = , α i = R0 , i = 1, 2,3, , y1 = 1 0 1 dx R0 R0 η Bessel function of the zeroth and first order respectively; u1 denotes steady velocity, u2 denotes
where y =
462
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
pulsatile velocity. From above we know that the axial velocity components can be totally determined by given pressure gradient. The pressure gradients are obtained according to the blood velocity waveform at the axis of the inlet to the artery (as shown in Fig. 2), which can be acquired through Doppler technology. In order to obtain pressure gradient, V 0 and V i are inserted into the corresponding velocity boundary conditions to yield the following equations [11]: 3 2 J1 §¨ j 2αi y1 ·¸ dCi 2 y1 © ¹ Ci = 0 + dx y1 J §¨ j 32α y ·¸J §¨ j 32α y ·¸ i 1 i 1 0 © ¹ 2© ¹
′ dC 0 4 y 1 + C0 = 0 ; dx y1
′
(7)
where J 2 is the Bessel function of the second order. Corresponding boundary conditions are needed for solving equations (7). Assume that u ( x * ,0 , t ) is the velocity waveform at the axis ( y = 0 ) at a certain position ( x = x * ) in the upstream even segment of the aneurysmal artery, its Fourier series can be written as [12]
(
)
u x * ,0, t =
U 0* § N · + Re ¨ ¦ U i* e jω i t ¸ 2 © i =1 ¹
(8)
Comparing U 0* and U i* with (6) and letting y = 0 and y 1 = 1 , one can obtain the coefficients of the pressure gradient at the site x = x * of the artery: C0 = − *
4η U R0
* 0 2
;
C i = j ηα i U i *
2
*
ª º½ ° 2« »° 1 / ®R0 « − 1» ¾ 3 ° « J 0 §¨ j 2 α i ·¸ »° ¹ ¬ © ¼¿ ¯
(9)
In accordance with boundary conditions (9), by solving the differential equations (7) we can get the following C 0 and Ci [13]. C0 = C0
*
1 4 y1
3 § ( j 32α )2 J § j 32α · − 2 j 32α J § j 32α · · J0 §¨ j 2αi y1 ·¸ ¨ i i¸ i 1¨ i ¸¸ 0¨ * © ¹ © ¹ ¸. © ¹ Ci = Ci .¨ 2 3 3 3 ¨¨ ¸¸ § j 2α y · J § j 2α y · − 2 j 32α yJ § j 32α y · J0 §¨ j 2αi ·¸ ¨ i 1 ¸ 0¨ i 1¸ i 1¨ i 1¸ © ¹ © ¹© ¹ © ¹ ¹ ©
(10)
(11)
3. Experiments and results In the experimental studies, the pressure gradients and the oscillatory blood flow fields in intracranial aneurysms with dilatation degrees of 30% and 70% at the position x = R 0 and x = 3R 0 are simulated and analyzed, respectively. The following parameters are used: normal vessel radius R 0 = 0.5 cm, aneurysm length 2 x 0 = 8 R 0 , cardiac cycle T = 1s . According to the inflow
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
463
boundary conditions, the pressure gradients and the oscillatory blood flow velocities in aneurysms with various dilatation degrees at different positions are simulated using previously mentioned methods and listed as follows. 3.1. Pressure gradient
Fig.3. Pressure gradient with dilatation degrees of (a) 30%, (b) 70%
Fig. 3 shows that the influence of local dilatation on pressure gradient in dilatational arterial segment is not very obvious. 3.2. Blood flow field 1). Steady velocity
Fig.4. Steady velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=3R0
Fig.5. Steady velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=R0
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L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
Fig.4 and Fig.5 show that the steady velocity is parabolic distribution, which shows the characteristics of Poiseuille flow. Namely, it does not change along with the time constant. 7he more the vessel dilatation is, the lower the steady velocity in aneurysms will be and the maximal steady velocity with dilatation degrees of 70% at the position x=R0 is the smallest. Furthermore, under the condition of the same dilatation degree, the steady velocity in aneurysms will increase with the increase of the axial distance. In order to further compare the effect of aneurysm dilatation degree on the maximum steady velocity, the maximum steady velocity with different dilatation degrees at different position are listed in table 1. Table 1 The maximum steady velocity in aneurysm with different dilatation degrees at different positionV (m/s) dilatation degree position x=3R0 x=R0
30%
70%
0.2223 0.1535
0.1993 0.0949
2). Pulsatile velocity
Fig.6. Pulsatile velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=3R0
Fig.7. Pulsatile velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=R0
Fig.6 and Fig.7 show that the pulsatile blood flow field distributions have no longer been monotonous profile and change with time and radial position of the vessel. Within the quite big radial region, the pulsatile velocity along vessel radius changes flatly. The velocity in aneurysm will decrease, and the more the vessel dilatation is, the lower the pulsatile velocity will be. In addition, under the condition of the same dilatation degree, it will increase with the increase of the axial distance. The maximal pulsatile velocity with dilatation degrees of 70% at the position x=R0 is the
L. Wang et al. / Hemodynamic numerical simulation and analysis of oscillatory blood flow
465
smallest. In order to further compare the effect of aneurysm dilatation degree on the maximum pulsatile velocity, the maximum pulsatile velocity with different dilatation degrees at different position are listed in table 2. Table2 The maximum pulsatile velocity in aneurysm with different dilatation degrees at different position (m/s) dilatation degree position x=3R0 x=R0
30%
70%
0.2504 0.1697
0.2232 0.0961
3). Oscillatory velocity
Fig.8. Oscillatory velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=3R0
Fig.9. Oscillatory velocity with dilatation degrees of (a) 30%, (b) 70% at the position x=R0
Fig.8 and Fig.9 show that the oscillatory velocity also changes with time and radial position of the vessel. The oscillatory velocity reaches its maximum at the centerline. With the increase of the dilatation degree, the maximal axial oscillatory velocity will also decrease, and the maximal oscillatory velocity with dilatation degrees of 70% at the position x=R0 is the smallest. Furthermore, the oscillatory velocity near the vessel wall drops quickly in normal vessel and in aneurysm with small dilatation degrees at the downstream outlet. As a comparison, the oscillatory velocity near the vessel wall with the biggest dilatation will not drop quickly. In order to further compare the effect of aneurysm dilatation degree on the maximum oscillatory velocity, the maximum oscillatory velocity with different dilatation degrees at different position are listed in table 3.
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Table 3 The maximum oscillatory velocity in aneurysm with different dilatation degrees at different position (m/s) dilatation degree position x=3R0 x=R0
30%
70%
0.4465 0.3073
0.3997 0.1892
4. Conclusion In this study we simulate and analyze the pressure gradients and the blood flow fields in growing intracranial aneurysms under periodically oscillatory flow. Results show that the higher the aneurysm dilatation degree is, the lower the maximum oscillatory velocity will be. Therefore, the oscillatory velocity may be used to analyze the characteristics of blood flow signals from aneurysm and to forecast the size of aneurysm, which is promising to provide a useful reference in clinical application from this sensitive parameter. References [1] [2] [3] [4] [5] [6] [7] [8]
[9] [10] [11] [12] [13]
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