A Numerical Simulation of Pressure Variation in Arteries affected by Stenosis
Dr. H. Girija Bai
Department of Mathematics, Sathyabama University, Chennai600 119, India.
*Corresponding Author Email: girijanameprakash@gmail.com
ABSTRACT:
This paper is mainly concerned with the effect of pressure variations in different dilations of stenotic arteries. In this study, we numerically analyzed unsteady blood flow in stenotic arteries with constrictions 25%, 50%, 75% and 90%. A finite volume method is used to investigate flow in a tube with an occlusion. Using computational fluid dynamics techniques, hemodynamic factors such as velocity, pressure and streamline patterns are investigated. Wall shear stress and strain are numerically analyzed. Results are compared and validated. The study of unsteady flow reveals correlation between regions of recirculation and the lesion's location. Detection and quantiﬁcation of stenosis serve as the root for surgical interference. These techniques based on computer flow study are important for understanding the relationship between hemodynamic parameters and hazard of rupture.
KEYWORDS: Stenosis, Computational Fluid Dynamics, Hemodynamics, Numerical simulation, Finite Volume Method
INTRODUCTION:
The arteries are living organs that can adjust to and change with the varying hemodynamic conditions. Hemodynamics refers to physiological factors governing the flow of blood in the circulatory system. Blood ﬂow in arteries is dominated by unsteady ﬂow phenomena. The study of blood ﬂow under diseased circumstances is very important. The majority of deaths in developed countries is caused by cardiovascular diseases, most of which are connected with some form of irregular blood ﬂow in arteries. The deposit of cholesterol and proliferation of connective tissues in an arterial wall forms plaques which grow inside the artery and restrict the natural blood flow. The obstruction may damage the internal cells of the wall and may lead to further growth of stenosis. Stenosis refers to the abnormal narrowing of a blood vessel due to the development of atherosclerotic plaques or other types of abnormal tissue development.
This vascular disease is of frequent occurrence, particularly in mammalian arteries. If this disease takes a severe form, it may lead to fatality. The exact mechanism for the development of this vascular disease is somewhat unclear. Investigators emphasized that the formation of the intravascular plaques, the impingement of ligaments and spur on the blood vessel wall are some of the major factors for the initiation and development of this vascular disease. The hemodynamic behavior of the blood flow in arterial stenoses bears some important aspects due to engineering interest as well as feasible medical applications. Researchers have been carried out worldwide to study the blood flow in both normal and stenotic arteries. ^{15 }analyzed a finite element analysis of simple pulsatile flow in a constricted vessel. The effects of overlapping blood flow characteristics in a narrow artery were invested^{22}. Effect of NonNewtonian behavior of blood in stenotic arteries was analyzed ^{20}. ^{1,2 }made a numerical study of pulsatile flow through stenosed canine femoral arteries for lumen constrictions in the range 061%. ^{7 }Investigated flow in a tube with an occlusion by using finite difference scheme for steady and unsteady flow. ^{17 }numerically analyzed the blood flow dynamics in a stenosed, subjectspecific, carotid bifurcation using the spectral element method. ^{14} made a numerical study to investigate the capacity of the circle of Wills (CoW) to provide collateral blood supply for patients with unilateral carotid arterial stenosis. ^{16} made a study of pulsatile flow of suspension of particles in a rigid tube. ^{13} analytically discussed stenosed vessel under steady flow and considered a blood as Williamson fluid and ^{12} considering carreau fluid model of stenosed artery with heat and mass transfer effects. ^{11} studied blood flow with double stenoses in the presence of an external magnetic field and blood flow through a catheterized artery was analyzed by ^{18}. ^{21} studied a flow of micropolar fluid through catheterized artery. Pressure flow of varying width to stenosed artery was investigated by ^{19}. Numerical simulation of 2D and 3D constricted vessel was analyzed ^{5,6}. The simulation of complex dynamic processes that appear in nature or in industrial applications poses a lot of challenging mathematical problems, opening a long road from the basic problem, to the mathematical modelling, the numerical simulation and finally to the interpretation of results. Governing eqations are discretized and solved using SIMPLE algorithm^{8}. Blood pressure is represented by two numbers 120/80. The top number indicates the systolic pressure and the bottom number indicates the diastolic pressure. Systolic pressure of 140 or above or a diastolic pressure of 90 or above is considered as high blood pressure. Due to the presence of stenosis, the normal flow of blood is affected, resulting in blood recirculation and wall shear stress oscillation near the stenosis. To enforce the blood circulation in the constricted region the heart has to increase the blood pressure. Thus the main cause of the stenosis is high blood pressure. A heart attack may occur if the blood cannot flow healthy and if the heart works too rigid. The aim of the present study is to investigate the effect of normal and high blood pressure in a constricted vessel of different stenosis. The flow of pulsatile viscous fluid is considered. Using CFD analysis, the problem is numerically solved by a finite volume method which has not been thoroughly investigated so far. Limitations on the amount of the constriction are ignored. Since arterial wall is gently elastic, we neglect the wall dispensability. Change in diameter in arteries is on the order of 10% ^{10}; so error in fixed diameter is minute.
MATERIALS AND METHODS:
The geometry of the stenosed vessel is given by
, (1)
Fig.1. A stenotic arterial vessel of different constrictions The boundary conditions are u=0, v=0 on stenosed vessel,
Input pressures 12,000pa and 24,000pa give velocities u=4. 75831m/s, 6.729267m/s, v=0 on inflow segment,
f_{x}=0, f_{y}=0 on outflow segment.
Governing Differential Equations:
Equations of momentum and mass conservation of incompressible fluid can be written as:
= 0 (2)
(3)
where: ρ  density of blood, _{}velocity field, p  pressure, µ = coefficient of viscosity.
Methodology:
Computational fluid dynamics is used in many areas, such as engineering and medical field. This new field provides very detailed information about fluid characteristics. Medical science lent this new technology to study hemodynamics within the body.
A CFD solution involves the following basic steps:
· Creation of the geometry
· Choice of the models
· Apply of the boundary conditions
· Flow field computation
· Post processing
CFD helps us to understand their formation, growth, and rupture of stenosis. The purpose of our study is to show the possibility of the development of computational analyses of velocity, pressure, and patterns of streamlines. Discretization is conducted in Gambit. Governing equations were solved in FLUENT which use a finite volume method. Although blood has actually non Newtonian behavior, in the simulation, it is considered Newtonian because there were no significant differences in the distribution of wall shear stress ^{9}.
Womersley number (α) depends on: flow rate, model geometry and fluid viscosity and varies with vessel diameter.
(4)
where: r [m]  entry radius of the vessel, υ  flow rate, ρ [kg/m^{3}]  blood density, _{μ} [kg / ms]  blood viscosity.
The Womersley parameter can be interpreted as the ratio of the unsteady forces to the viscous forces. When the Womersley parameter is low, viscous forces dominate, velocity proﬁles are parabolic in shape, and the centerline velocity oscillates in phase with the driving pressure gradient. For Womersley parameters above 10, the unsteady inertial forces dominate, and the ﬂow is essentially one of pistonlike motion with a ﬂat velocity profile^{3}. The dimensionless Reynolds number (Re) gives the flow command. This number varies with the diameter of the vessel for each case.
i.e. (5)
where: ρ [kg/m^{3}]  blood density, _{} [m/s]  maximum speed of blood flow at the entrance, d [m]  diameter at the entrance of the vessel, _{μ} [Kg/ms]blood viscosity^{4}.
RESULTS:
Blood density ρ = 1060 [kg/m^{3}] and dynamic viscosity η = 0.003 [kg/ms] (Poiseuille) and inlet velocities 4.75831m/s, 6.729267m/s, governing equations were solved in Gambit and Fluent which use a finite volume method for discretization. Detailed study of velocity, pressure and streamlines are made. By considering the domain table 1, grid was generated with a pave mesh of size 0.01 was shown in fig. 2.
Table 1. Domain of the Geometry

MIN. (mm) 
MAX (mm) 
X 
0.002 
0.002 
Y 
0 
0.001 
Fig.2. Grid displays for an unsteady stenosis
Table 2. By considering Interval size = 0.01

0.25 
0.5 
0.75 
0.9 
Nodes 
37998 
35617 
33112 
31709 
Mixed wall faces 
808 
828 
858 
878 
Mixed pressoutlet 
100 
100 
100 
100 
Mixed pressinlet 
100 
100 
100 
100 
Mixed interior faces 
74482 
69690 
64635 
61799 
Quadrilateral cells 
37493 
35102 
32582 
31169 
Table 3. Volume Statistics
DIL. 
Min. Volume (m3) 
Max. Volume (m3) 
Total volume (m3) 
0.25 
1.782426e005 
2.98966e004 
3.749941 
0.50 
3.292459e005 
2.110809e004 
3.499878 
0.75 
1.435866e005 
2.437455e004 
3.249807 
0.90 
3.453424e005 
1.855253e004 
3.099763 
Unsteady flow for 10 time step size(s), 10 seconds for 1000 flow time, iterations were carried out and results are calculated.
Table 4. Area Statistics
DIL. 
Min. Face area(m2) 
Max. Face area(m2) 
0.25 
2.704820e003 
2.006518e002 
0.50 
4.997917e003 
1.733037e002 
0.75 
2.437079e003 
2.253014e002 
0.90 
5.114161e003 
1.639324e002 
The velocity magnitude of unsteady flow with inlet velocities 4.75831m/s (12000pa = 90mmHg), 6.729267m/s (24000pa = 180mmHg) for different constrictions are shown in Fig 3, and the minimum and maximum values are shown in table 5.
Fig.3. Velocity variations for different constrictions.
Table 5. Velocity And Dynamic Pressure Variation

Inlet Pressures 
Vel. mag. Min(m/s) 
Vel. mag. Max (m/s) 
Dyn. press. Min. (pa) 
Dyn. press. Max. (Pa) 
25% 
12,000 
0 
7.272042 
5147.723 
28075.43 
24,000 
0 
8.167853 
0.001276728 
35358.32 

50% 
12,000 
0 
5.010413 
0.0001470775 
13305.25 
24,000 
0 
8.213599 
0.01053289 
35755.57 

75% 
12,000 
0 
4.912563 
0.0005199683 
12790.64 
24,000 
0 
7.048975 
1.563396e05 
26334.81 

90% 
12,000 
0 
5.082479 
0.001213119 
13690.84 
24,000 
0 
7.355304 
0.007626247 
28673.29 
Dynamic pressure was calculated for different input systolic pressures 12000pa = 90mmHg, 24000pa = 180mmHg and the results are shown in fig. 4 and the values are shown in table 5.
The velocity contours from fig. 3. implicates that in the upstream region of stenosis, there is a reverse blood flow due to constriction and in the downstream region the profile shows the formation of swirls. Near the region of stenosis from 25% to 90% the swirls becomes larger and it takes much time to rejoin with the flow domain. The velocity is the peak of the stenosis region and extended to the downstream region. Totally there is no flow behind the stenosis and thus leads the formation of plaques.
Fig.4. Dynamic pressure for different constrictions.
It is clear that there is a considerable pressure variation in the direction normal to the Xaxis in the stenosed region. The axial pressure loss across the stenosis relative to the overall pressure loss across the entire stenosed vessel is of greatest physiological interest. Due to high pressure gradients across the stenosis favours the development of atheroma^{15}. The pressure gradient from the inflow segment to the peak of stenosis is much higher than the pressure gradient across the whole of the stenosis and it increases with δ, and the results are validated with previous work. Wall shear stress/strain for different input systolic (high and normal) pressures are shown in fig. 5 and the values were shown in table 6. As there is no flow near the vessel wall, shows by zero velocity, which leads much higher stress near the boundary. If the stress and strain increase it leads to burst of the vessel wall or becomes very weak.
Fig.5. Wall shear and strain with different constrictions of stenoses.
Table 6. Wall Shear Stress For Different Systolic Pressure
Wall Shear 
Inlet pressures 
Wall shear Min(pa) 
Wall shear Max(pa) 
25% 
12,000 
0 
4.441072 
24,000 
0 
2502.435 

50% 
12,000 
0 
852.6992 
24,000 
0 
2143.458 

75% 
12,000 
0 
1031.466 
24,000 
0 
1772.053 

90% 
12,000 
0 
1374.146 
24,000 
0 
2328.713 
From table 6. results on wall shear and strain rate, we observe that as the degree of stenosis increases, there is a marked increase in peak vorticity values^{15}. High stresses can damage both blood vessels and blood contents.
Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. This shows the direction of fluid element which travels at any point at any time. These streamlines are representative of the speed in a given time. Streamline patterns from fig. 6 shows the collision of blood flow and visibility of high velocities in the region of stenosis.
Fig.6. Streamline pattern of velocity magnitude.
The iterations were calculated and solution converges correct to the decimals with equation of continuity and momentum equation x  velocity, y – velocity.
Table 7. Convergence values of governing equations.
Dilations 
Inlet press. 
Converges 
Continuity 
xvelocity 
yyelocity 
25% 
12,000 
150 
2.7763e04 
5.9323e05 
6.1447e06 
24,000 
410 
7.1874e04 
5.0780e05 
1.5289e05 

50% 
12,000 
1290 
9.9412e04 
2.5209e04 
1.5758e04 
24,000 
490 
8.4035e04 
1.9714e04 
2.3638e04 

75% 
12,000 
900 
8.949e04 
7.5832e04 
5.5866e04 
24,000 
1400 
1.3405e03 
9.5489e04 
6.7213e04 

90% 
12,000 
300 
3.6289e04 
7.5636e04 
8.4883e04 
24,000 
300 
4.5121e04 
5.5669e04 
6.8779e04 
In case of nonconvergence, some parameters have to be tuned adequately. For explicit solvers the ‘cfl’ no. and for implicit solvers the under relaxation factors can be changed. The study of unsteady flow reveals several interesting new features. It appears that there is a correlation between regions of recirculation, which is a prominent feature of the unsteady flow, and the location of lesions ^{7}.
Fig7. Convergence of iterations for different systolic pressures.
CONCLUSION:
Flow analysis shows that an unsteady flow model is not similar for all stenosis. Flow characteristics are highly dependent on the geometry of the vessels and different constriction of sizes of stenosis. High velocities are obtained in the region of stenosis. Contours of velocity and pressure profiles shows the recirculation and whirlpools of blood flow near the region of stenosis and it increases with increase in constrictions and increased pressure. The results of velocity magnitude, pressure and wall shear stresses are validated with the previous results. These CFD techniques based on computer flow study are important for understanding the relationship between hemodynamic parameters and risk of rupture. The formation of a stenosis is the most dreadful biological reaction and the results show due to high blood pressure the flow gets affected totally and it leads to stroke. Hence, complete understanding of the relationship between pressure, blood ﬂow and symptoms for cardiovascular stenosis remains a critical problem. New devices to repair stenotic arteries are now being developed. In the future, the study of arterial blood ﬂow will lead to the prediction of individual hemodynamic ﬂows in any patient, the development of diagnostic tools to quantify disease, and the design of devices that mimic or alter blood ﬂow.
REFERENCES:
1. Bart J. Daly. A Numerical study of pulsatile flow through stenosed canine femoral arteries. J. Biomechanics. 9; 1976: 465475.
2. V. O’Brien and L.W. Ehrich, Simple Pulsatile flow in an artery with a constriction. J. Biomechanics. 18 (2); 1985: 117127.
3. David N. Ku. Blood flow in arteries. Fluid Mech. 29; 1997: 399–434.
4. S. Gaivas, P. Cârlescu and Ion Poeată. Computational hemodynamics in a patient  specific cerebral aneurysms models. Romanian Neurosurgery. XVIII 4; 2011: 434441.
5. H. Girija Bai, K. B. Naidu and G. Vasanth Kumar. 3DComputer Simulation of blood flow in Arteries with Multiple Stenoses. NCRTMC2013. published in the Bonfring Digital Librar., Bornfring Proceeding. 272280.
6. H. Girija Bai, K. B. Naidu and G. Vasanth Kumar. Computer Simulation of Pressure Variation in Human Blood Vessels affected by Multiple Stenoses. ICMS 2014 International Conference. Elsevier Proceedings ISSN:9789351072614: 316322.
7. H. Huang, V. J. Modi and B.R. Saymour. Fluid Mechanics of Stenosed Arteries. Int. J. Engg. Sci. 33(6); 1995: 815828.
8. Joel Guerrero. Numerical Simulation of the unsteady Aerodynamics of flapping flight. Thesis, Chapter 3: 3451.
9. Malek A.M. and Izumo S. Mechanism of endothelial cell shape change and cytoskeletal remodeling in response to ﬂuid shear stress. Journal of Cell Science. 109; 1996: 713726
10. McDonald, D. A. Blood Flow in Arteries. Camelot, Baldwin Park, CA. 1974
11. J. C. Misra, A. Sinha and G. C. Shit. Mathematical modelling of blood flow in a porous vessel having double stenoses in the presence of an external magnetic field. Int. J. Biomath. 4; 2011: 207.
12. Noreen Sher Akbar. Heat and Mass Transfer Effects on Carreau Fluid Model for Blood flow through a tapered artery with a stenosis. Int. J. Biomath. 7; 2014: 1450004
13. Noreen Sher Akbar and S. Nadeen. Blood flow analysis in tapered stenosed arteries with pseudoplastic characteristics. Int. J. Biomath. 6; 2014:1450065
14. Quan Long, Luca Luppi, Carola S. Konig, Vittoria Rinaldo, Saroj K. Das. Study of the collateral capacity of the circle of Willis of patients with severe carotid artery stenosis by 3D Computational Modeling. Journal of Biomechanics. 41; 2008: 27352742.
15. B.V. Rathish Kumar and K.B. Naidu. A transient UVP finite element analysis of a Nonlinear pulsatile flow in a stenosed vessel. IJCFD. 0, 1996: 16.
16. Sankarasubramanian.K. and Naidu. K.B. A iterative solution for pulsatile flow of suspension of particles in a rigid tube. Int. J. Pure and Appl. Math. 18; 1987: 557566.
17. Seung E. Lee, SangWook Lee, Paul F. Fischer, Hisham S. Bassiouny, Francis Loth. Direct numerical simulation of transitional flow in a stenosed carotid bifurcation. Journal of Biomechanics. 41; 2008: 25512561.
18. Shailesh Mishra, Narendra Kumar Verma and S. U. Siddiqui. A Suspension model for blood flow through a catheterized artery. Int. J. Biomath. 5; 2012: 1250033
19. A.M. Siddiqui, T. Haroon, Z. Bano and S. Islam. Pressure flow of a second grade fluid through a channel of varying width with application to stenosed artery. Int. J. Biomath. 6; 2013: 1350016.
20. Somkid Amornsamankul, Benchawan Wiwatanapataphee, Yong Hong Wu, Yongwimon Lenbury. Effect of NonNewtonian Behaviour of Blood on Pulsatile Flows in Stenotic Arteries. International Journal of Biomedical Sciences. 1(1); 2006: 4246.
21. D. Srinivasacharya and D. Srikanth . Flow of Micropolar Fluid through Catheterized Artery A Mathematical Model. Int. J. Biomath. 5; 2012: 1250016
22. V. P. Srivastava and Shailesh Mishra, Rati Rastogi. NonNewtonian Arterial Blood Flow through an Overlapping Stenosis. Applications and Applied Mathematics. 5 (1); 2010: 225238.
Received on 18.01.2017 Modified on 27.01.2017
Accepted on 14.02.2017 © RJPT All right reserved
Research J. Pharm. and Tech. 2017; 10(3): 802810.
DOI: 10.5958/0974360X.2017.00152.4