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Comparative Mathematical Study of Blood Flow Through Stenotic and Aneurysmatic Artery with the Presence and Absence of Blood Clots

Numerical predictions of blood flow and hemodynamic properties through stenosis and aneurysm artery have been studied in the presence of blood clots at the constricted area. The finite element method has been used to solve the partial differential equations of continuity, momentum, Oldroyd-B, and bioheat transport in cartesian coordinates systems. The present investigation carries the potential to compute blood velocity, pressure, and drag coefficients with significance at the throat of stenosis and aneurysm. The models have also been employed to study simulation, blood clots, and hemodynamic characteristics for all modifications. The impact of shearthinning on blood flow is intensified compared to the viscoelastic properties. It is found that the maximum effect of the drag coefficient is visible at the hub of stenotic for nonclotting models. The highest pressure and the lowest velocity are gained for the presence of blood clots at the constraint area. The impact of stenosis and aneurysm artery, drag coefficient, and blood clots on blood flow is the main physical outcome that may be reported in medical science to identify atherosclerosis diseases. The quantitative analysis has been completed numerically with the physiological significance of hemodynamic factors of blood flow which shows the validity of the present model.


Introduction
Cardiovascular system of human body is affected due to irregular cell increases or decreases in blood vessel which leads to arterial diseases in various organs. Although it is not clearly identified of exact mechanisms for invent of this matter, but is found that blood flow disorder, initiation of arterial diseases and irregularities have been created from a mild stenosis [1,2]. A good number of experimental, theoretical and numerical study has been continued for understanding of stenotic blood flow. Smith [3] has been studied details of steady flow through an axisymmetric stenosis and show that blood flow strongly depends on the stenotic geometry condition. The real understanding that the blood pulsatile flow cannot be ignored, many laboratory work, theoretical study and computation analyses on the blood flow have been completed at stenotic conditions for Newtonian fluid [4][5][6][7][8]. It is very important that the blood flow will obey Newtonian law for high shear rate flow. In case of low shear rate, the fluid behavior will be non-Newtonian specially if the blood flow is in smaller arteries and in the down steam of the stenosis. It has been pointed out that blood shows non-Newtonian performances (Thixotropy, Viscoelasticity and Shear thinning) for cardiovascular diseases.
Recently researchers [9] has been given more concentration for non-Newtonian blood flow analysis with various stenosis conditions and still have a lot of scope to work in this area.
In this paper, we also consider aneurysm condition of blood vessel, it is another severe situation that about 75 percent of all patients have died before reaching the hospital and it is extreme risk for human body [10]. Ingoldby et al. [11] have shown these prehospital deaths are counted, the overall mortality rate may exceed 90 percent. Ernst [12] has studied that the increasing median age of the population contributes to an increasing incidence. It is important to identifying the risk features that may have key role in aneurysm growth and rupture has become a unified multidisciplinary task to clear understanding on the pathogenesis and growth of aneurysm. Numerical and experimental study has performed to interpret of blood flow through anuerysmatic artery during last and present decades [13][14][15] and biomechanics has been receiving the attention of researchers on this area and providing numerical solution of human blood flow system in various conditions [16]. In most of the inquiries, the flow is considered in cylindrical pipes with uniform cross-section area. But, is well known that blood vessels deformed at the change of bone shape for various reasons in human body. Hence the idea of blood flow in a various cross-section forms the prime basis of a large class of problems in understanding blood flow pattern. Most of the study have considered long, narrow and tapering vessels for blood flow analysis [17][18][19] of Newtonian and non-Newtonian case. Thus, the effects of stenotic and aneurysmatic vessel with the non-Newtonian performance of the flowing blood seem to be identical significant and hence definitely deserve special consideration. Therefore, we are motivated for mathematical study of blood flow through stenotic and aneurysmatic artery with blood clot. Thus, an effort is made in the present theoretical study to develop a mathematical model and numerical analysis in order to investigate the significant characteristics of the blood flow through a rigid stenotic and aneurysmatic vessel. The generalized cross model is used for non-Newtonian behavior of flowing blood flow. Malek et al. [20] widely studied the existence and uniqueness as well as the stability characteristics of streaming flow problems. The present problem is one of the major physiological significance, due attention is also paid to comparable study in the presence of blood clot and without blood clot for considered model.
We have used the following notations which are available in mathematical structures. The dimensionless number Weissenberg number, Reynold number and Prandtl number are denoted by Wi, Re and Pr respectively,  denotes stress tensor, P is the pressure, and stress tensor is denoted by . The velocity components are U and V along X and Y respectively. The perfusion coefficient is f, v is viscoelasticity components and heat source is denoted by Q.

Preliminaries
This section briefly reviews some fundamental concepts from the previous literatures. The goal is to establish some notations and results that will be used in the later section of this paper.
From the invent of stenosis or aneurysm effects on blood flow in our human arteries have created serious disorders in circulatory system that leads various cardiovascular diseases such as arteriosclerosis, bleeding, stroke, kidney damage etc. [21,22]. At present time medical researchers, bioengineers and numerical scientists join efforts with the purpose of providing numerical simulations of human blood flow system in different conditions. The exchange of knowledge and data information is main purpose of this kind of collaboration that can be used in simulations. A relation has been established by Thurston [23] between blood shear rate and viscoelasticity of blood flow. At first, Wille [24] has described numerical study of pulsatile blood flow through aneurysms vessel and shown existence of vortex with various size during the cardiac cycle. Oka [25] has studied a taper analysis of blood flow to identify arterial disease and shown the pressure development is a vital factor. Kumar et al. [26] have explained the pulsatile suspension flow in a dilated blood vessel. They have determined that the additional complicate of hemodynamics in diseased areterial vessels having aneurysm and stenosis artery. During blood circulation system, the hemodynamical features of blood can plays a vital role to develop sever fatal cardiovascular diseases. Anand et at. [27] has described a model to blood flow simulation with Oldroyd-B fluid properties. They have [28] also exposed a model with blood clots for the formation and analysis. Shih et al. [29] have studied on skin surface with sinusoidal heat flux condition. A thermodynamic framework has studied by Rajagopal and Srinivasa [21] to show the blood viscoelastic response with various configurations. With the presence of hematocrit in a local aneurysm, the systematic analysis of flow features through a tube and artery have studied by Mukhopadhyay and Layek [10]. Prokop and Kozel [30] have shown a numerical simulation of generalized Newtonian and Oldroyd-B fluid with an extended computation domain. Nadeem et al. [2] have shown the reduce of the wall shear stress and resistance impedance to blood flow with the influence of metallic nanoparticles. The pulsatile blood flow has explicated by Achab et al. [31] to evaluate the blood flow characteristics and the wall shear stress under physiological conditions through a stenosis artery. Recently, Nasir and Alim [32] have described the blood flow through stenotic artery with various flow rates numerically. Later they [33] have also demonstrated the numerical investigation of blood flow through double stenotic artery.
A large scale numerical analysis is carried out by performing mathematical computations of the desired quantities having more physiological significance to explore the effects of stenotic vessel, aneurysmatic artery, the blood clot, the drag coefficient, dimensionless numbers and the non-Newtonian behaviour of the streaming blood on the physiological flow phenomena which are extensively quantified through their graphical representations presented at result section with appropriate scientific discussions. A comparative study is exposed in the presence of blood clot and without blood clot to know the significances of coagulation of blood. A code validation results are also shown with Prokop et al [30] to authenticate the applicability of the present study.

Modeling and analysis of blood flow
This section contains main results of this paper. At first, we specify the physical model. Then the formulate the mathematical model using partial differential equations (PDE) and its boundary conditions. Finally, numerical technique is discussed to solve the mathematical model efficiently.

Model Specifications
The treated model is a two-dimensional rigid artery with stenosed and aneurysm vessel wall in cartesian system. The considered model in the present study is displayed in figure 1a. The figures 1b and 1c are main motivation for considering present model. The model is assumed with stenosed and aneurysmal hight hs and ha respectively where hs=R and ha=3R. The upper and bottom stenosed vessel walls are cool (Tc) and heated (Th) while the rest of the walls are adiabatic and impermeable with no slip conditions. The velocity profile is prescribed at entrance and pressure is fixed to constant at outlet. The blood flow acceleration is expected at stenotic and aneurysmatic cross-sectional area. A mathematical structure has used by Achab et al [31] in cosine shaped for stenosis model. The modified mathematical model of physical construction in the presence of stenosis and aneurysm is in cosine shaped as follows

Mathematical Model
The functioning fluid is assumed to be laminar blood flow and incompressible with shearthinning and viscoelasticity properties. The generalized Oldroyd-B model is used to capture non-Newtonian properties of the blood flow. Using non-dimensional scale variables, the partial differential equations (1), (2), (3) and (4) are obtained in vector form for 2D stenosis and aneurysmal vessel in domain  as follows: Continuity Equation: Momentum Equations: Where, U and V are the velocity components along the X and Y axes respectively, P is the pressure,  is stress tensor, f is perfusion coefficient, and Weissenberg number (Wi), Reynold number (Re), Prandtl number (Pr) are dimensionless numbers. To non-dimensionalized the above equations we incorporating the dimensionless variables given below: The boundary conditions are required to solve the governing equations and there are four disjoint parts of the boundary wall: rigid wall, outlet, blood clot and inlet. The axial velocity profile u is considered along x-axis and set zero velocity along y-axis at inlet artery. The extra stress components Te and pressure Pi are prescribed at the inlet. A constant pressure Po is imposed on the pressure at the outlet. On rigid walls, all velocity components are set to zero.
On blood clot, the wall is heated and cooled at stenotic artery.

Numerical Solutions
The discretized momentum and Oldroyd-B equations subjected to the boundary conditions simultaneously will be solved using the Matlab programming [36] & Mathematical programming package COMSOLMULTIPHYSICS [37] or the dependent variables (velocity, pressure and stress tensor). The numerical procedure [38,39] have been used in this work is based on the Galerkin weighted residual method of finite element formulation and details are given here below. After the appling the weighted residual method then To generate the boundary integral terms associated with the surface tractions, extra stress tensor and temperature the Equations (5.2-5.5) become after appling Gauss's theorem.
Here Equations (5.2-5.3) specify surface tractions (Sx, Sy) along outflow boundary S0, Equations (5.4-5.5) specify velocity components, stress tensor and fluid temperature that can be applied force from domain along wall boundary Sw. The six-node triangular element is used for the development of the finite element equations. All six nodes are associated with velocities, temperature as well as stress tensor; only the corner nodes are associated with pressure. This means that a lower order polynomial is chosen for pressure and which is satisfied through continuity equation. The basic unknowns for the above differential equations are the velocity components U, V the stress tensor,  and the pressure, P. The velocity component and the stress tensor distributions and linear interpolation for the pressure distribution according to their highest derivative orders in the differential Equations (1)(2)(3)(4)(5) Where, α = 1, 2, … …, 6; λ= 1, 2, 3; Nα are the element interpolation functions for the velocity components and the stress tensor, and Hλ are the element interpolation functions for the pressure. Substituting the element velocity component distributions, the stress tensor distribution, and the pressure distribution from Equations (1-5) the finite element equations can be written in the form, Where, the coefficients in element matrices are in the form of the integrals over the element area and along the element edges S0 and Sw as, This leads to a set of algebraic equations with the incremental unknowns of the element nodal velocity components, temperatures, and pressures in the form, If the percentage of the overall change compared to the previous iteration is less than the specified value, then the iteration process is terminated.

Numerical experiments
To evaluate the accuracy and confirmation of this work we compute a validation test with Prokop et al [30] for Newtonian (N), generalized Newtonian (GN), Oldroyd-B (OD) and generalized Oldroyd-B (GD) cases. We calculate velocity, pressure and drag coefficient with various dimensionless numbers for considered model. In the present numerical study, the following thermal properties and tissue [29] are considered: 0  Wi  1, 0 < Re  3000,

Stenotic and Aneurysmatic Effects on blood flow
The blood flow simulation is shown in figs. 4

Stenotic and Anuerysmatic effects on pressure distribution
The pressure distribution of blood flow is exhibited for all cases in fig. 7 having blood clot and without blood clot. In fig. 7(left), the steep contour plots display the pressure gained a minimum value at the separation point and decline gradually along vessel axis. Due to formation of blood clots, the pressure peaked at reattachment point at stenosis region and the pressure gradient is high for all events which is agree with Muraki [41] test. The

Drag coefficient effects on Blood flow
In fig. 9, the drag coefficient (resistance) of blood flow increases at blood clot model and nonblood clot model. It is found that in the absence of blood clots at stenotic area, the drag coefficient is high compare to having blood clots in stenosis artery for Newtonian, Generalized

Conclusion
In this paper we have studied the blood flow patterns through stenotic and anuerysmatic artery having or absence of blood clot for four cases numerically. We have shown that the effect of stenotic vessel, anuerysmatic artery, blood clot, dimensionless number and drag coefficients are very momentous in blood flow. The above factors are correlated to blood viscoelasticity and blood shear thinning behavior. It is found that the blood velocity and pressure have intensive change at the throat of stenosis (no blood clot) compare to blood clot stenosis model. The blood flow patterns have more affected by high Reynold numbers (turbulence flow) at generalized Oldroyd-B case on the contrary insignificant changed occur for different Weissenberg numbers in present problem. Effects of drag coefficient on blood flow at blood clot regimes are highlighted to explore their impacts on blood flow structure and its characteristics. The efficiency and competence of the theoretical result have been discussed by numerical experiment using finite element technique. Based on the above computational results we conclude that: a. The shear-thinning effects are related to the blood velocity and pressure and it is more pronounced than the viscoelastic ones.
b. The blood flow parameters are predominant in the recirculation zones compare to blood clot model.
c. The effect of drag coefficient on blood flow is more extreme in stenotic (without blood clot) artery for all models.
d. Due to presence of blood clot at the throat of stenosis the recirculation zones are more elliptic for all models and minimum velocity and maximum blood pressure are observed.
e. The blood velocity has peaked maximum value at narrow cross-sectional area (center of stenotic artery) for absence of blood clot in the model and gained minimum value at wide cross-sectional area with presence of blood clot in the model which is theoretically true.
Besides, we have computed blood pressure for present model and the result is quiet agree with theory for all cases.