This study focuses on the behavior of blood flow in the stenosed vessels. Blood is modelled as an incompressible non-Newtonian fluid which is based on the power law viscosity model. A numerical technique based on the finite difference method is developed to simulate the blood flow taking into account the transient periodic behaviour of the blood flow in cardiac cycles. Also, pulsatile blood flow in the stenosed vessel is based on the Womersley model, and fluid flow in the lumen region is governed by the continuity equation and the Navier-Stokes equations. In this study, the stenosis shape is cosine by using Tu and Devil model. Comparing the results obtained from three stenosed vessels with 30%, 50%, and 75% area severity, we find that higher percent-area severity of stenosis leads to higher extrapressure jumps and higher blood speeds around the stenosis site. Also, we observe that the size of the stenosis in stenosed vessels does influence the blood flow. A little change on the cross-sectional value makes vast change on the blood flow rate. This simulation helps the people working in the field of physiological fluid dynamics as well as the medical practitioners.
Cardiovascular disease is one of the major causes of deaths in developed countries. Most cases are associated with some form of abnormal flow of blood in stenotic arteries. Therefore, blood flow analysis through stenosed vessels has been identified as one of the important area of research in the recent few decades. One of the motivations to study the blood flow was to understand the conditions that may contribute to high blood pressure. Past studies indicated that one of the reasons a person having hypertension is when the blood vessel becomes narrow. Thus, many researches have been done for analysis of blood flow in stenosed vessels. Some recent investigations in these areas are cited in [
The simulation model of the stenosed vessel is depicted in Figure
The geometric model of the stenosis.
The 3D simulation model of the stenosed vessel.
In order to model the pulsatile blood flow, Womersley-Evans theory is used to obtain the spatial and temporal distribution of velocity profile [
Axial centerline velocity waveform obtained from Womersley model (at
Blood is non-Newtonian fluid, because of presence of various cells. This means that when shear stress is plotted against the shear rate at a given temperature, the plot shows a nonstraight line with a nonconstant slope as shown in Figure
Classification of non-Newtonian fluids.
In this study, power law fluid model is used to simulate the behavior of blood fluid. A power law fluid, or the Ostwald-de Waele relationship, is a type of generalized Newtonian fluid for which the shear stress,
Different states of flow behaviour index.
|
<1 | 1 | >1 |
---|---|---|---|
Type of fluid | Pseudoplastic | Newtonian fluid | Dilatant |
The viscous, incompressible flow in a long tube with stenosis at the specified position is considered. Let (
There is no shear along the axis of the tube which may be stated mathematically as
Finite-difference discretization of the equations has been carried out in the present work in staggered grid, popularly known as MAC cell. In this type of grid alignment, the velocities and the pressure are evaluated at different locations of the control volume. The time derivative terms are differenced according to the first order accurate two-level forward time differencing formula. The convective terms in the momentum equations are differenced with a hybrid formula consisting of central differencing and second order upwinding. The diffusive terms are differenced using the three-point central difference formula. The source terms are centrally differenced keeping the position of the respective fluxes at the centers of the control volumes. The pressure derivatives are represented by forward difference formulae. Discretization of the continuity equation at (
Here
The finite difference equation approximating the momentum equation in the
Here
In this study we have simulated and analyzed pulsatile flow of a power law fluid as a model for blood flow in the cardiovascular system. The model is used to study the critical flow in stenotic arteries with three different severities of 30%, 50%, and 70% (shown in Figures
Upstream from the stenosis, the velocity profile in the
The velocity
A rapid fall in pressure is observed as the occlusion is approached. Higher percentage area severity leads to greater pressure drops around the stenosis (shown in Figures
Pulsatile flow velocity, pulse pressure, and variation of shear rate with respect to time at an upstream point (
Wall shear stress at the end of diastole
Blood flow velocity distribution at peak systole obtained from simulation of stenosed vessel with severities of 30%.
Pressure along a longitudinal line at peak systole related to stenosed vessel with severities of 30%.
Flow velocity of blood along a longitudinal line at peak systole related to stenosed vessel with severities of 30%.
Blood flow velocity distribution at peak systole obtained from simulation of stenosed vessel with severities of 50%.
Pressure along longitudinal line at peak systole related to stenosed vessel with severities of 50%.
Flow velocity of blood along a longitudinal line at peak systole related to stenosed vessel with severities of 50%.
Blood flow velocity distribution at peak systole obtained from simulation of stenosed vessel with severities of 70%.
Pressure along a longitudinal line at peak systole related to stenosed vessel with severities of 70%.
Flow velocity of blood along a longitudinal line at peak systole related to stenosed vessel with severities of 70%.
The results also show a similar pattern in the pulsatile velocity, in the pulse pressure, and in the variation of shear rate in cardiac cycles. These confirm the features of the characteristic of the periodic motion. Therefore, in the presence of a narrowing vessel lumen with different area severity, the flow experiences resistance, which causes an increase in the shear stress and in the pressure drop. Higher percent-area severity of stenosis produces a higher pressure drop, a higher blood speed, a higher shear rate, and a higher wall shear stress.
Comparing the results obtained from three stenotic tubes with 30%, 50%, and 70% area severity, we find that higher percent area severity of stenosis leads to higher extrapressure jumps, higher blood speeds around the stenosis site, higher shear rate, and higher wall shear stress.
In this paper, we have derived a simple mathematical model that can represent the blood flow in the arteries. We observe that the size of the stenosis in stenosed vessels does influence the blood flow. A little change in the cross-sectional value makes vast change in the blood flow rate.
It should be noted that blood flow in a small stenotic artery is an extremely complex phenomenon. There are many unresolved modeling problems such as the flow in the arterial wall which is deformed during the cardiac period. The presented work only focuses on blood flow in the stenosed vessels. Also, this simulation helps the people working in the field of physiological fluid dynamics as well as the medical practitioners.