A STEADY FLOW OF BLOOD THROUGH A STENOSED ARTERY WITH A BALLOON-CATHETER TECHNIQUE

Rupesh K. Srivastav 1 , Manisha Gupta 2 and Poonam Sinha 3 . 1. Department of Mathematics, AIMT, Lucknow-226026, India. 2. Math Section-IT Department, Higher College of Technology, Sultanate of Oman. 3. Department of Mathematics, S.M.S.Govt. Science College, Gwalior-474001, India. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

In the present paper, an analysis of blood flow through a stenosed artery with a balloon based catheter technique has been carried out where the blood is treated as Newtonian fluid. The governing equation of motion is solved and closed form expressions for dimensionless resistance to flow, the wall shear stress and the shearing stress on the wall at maximum height of the stenosis are obtained. The behavior of these flow characteristics in this constricted annular region has been discussed. Their variations with different flow parameters are plotted in figures.

…………………………………………………………………………………………………….... Introduction:-
Blood is a marvelous fluid which nurtures the life. Blood flow in artery has some important aspects due to engineering as well as medical application point of view. For over a couple of centuries, the theoretical and experimental studies of blood flow through the circulatory system of living mammals, has been the subject of scientific research. Mathematical modeling of blood flow has been subject subject of modifications in order to account for the new evidences uncovered through the improved initial experimental observation.
The frequently occurring cardiovascular disease, stenosis, is defined as a partial occlusion of the blood vessels due to accumulation of cholesterol, fats and the abnormal growth of tissue. Once the constriction is formed, the blood flow is significantly altered and fluid dynamical factor play important roles as stenosis continues to enlarge leading to development of diseases such as heart attack and stroke. Young and Tsai (1973) discussed some characteristics of flow of blood in stenosed arteries. Quite a good number of analytical studies pertaining to the blood flow through stenosed arteries have been carried out by Young and Tsai (1973); By assuming the artery to be circulatory cylindrical in shape, Mishra and Panda (2005) Srivastav RK (2014, 2015, 2017; and many others; in order to analyze the arterial constriction on the flow characteristics of blood. In large vessels, which include large cavities such as the ventricles and atria inside the myocardium as well as the large arteries and veins, the blood essentially behaves as a Newtonian fluid (Caro et al., 1978) and non-Newtonian effects which are basically induced at low shear rates die out (Perktold et al., 1999). No single model, Newtonian or non-Newtonian, can capture all the features of the blood complexities (Yilmaz et al., 2008) and hence different models are used to represent different characteristics of the blood rheology.

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Catheters are of extensive use in modern medicine. Catheters are also being used in diagnostic technique, e.g. X-ray angiography as well as in treatment procedures of various arterial diseases. An inserted catheter in an artery will increase the impedance and will modify the pressure distribution and alter the flow field. A review of most of the experimental and theoretical investigation on artery catheterization has been presented by Srivastava at al. (2008). Dash et al. (1996) considered the steady and pulsatile flow of Casson fluid in a narrow artery when catheter is inserted into it and estimated the increase in frictional resistance in the artery due to catheterization. In ballooncatheter technique, a catheter with an inflatable balloon at its tip which is used during a catheterization procedure to enlarge a narrow opening or passage within the artery. The catheter is carefully guided to the location at which stenosis occurs in coronary arteries and then balloon is then inflated to fracture the fatty deposits and widen the narrowed portion, McMahon et al. (1971), Gabe (1972). Gunj et al. (1985); Anderson et al. (1986) and Wilson et al. (1988) have studied the measurement of translational pressure gradient during angioplasty.
With the above discussion in mind, the goal of this investigation is to study the effect of a balloon-catheter technique on various physiologically important flow characteristics i.e. impedance, wall shear stress distribution in stenotic region and shear stress at stenosis throat, as on flow patterns in artery with overlapping stenosis through a mathematical model.

Formulation Of The Problem:-
Consider the axisymmetric flow of blood through the gap between an artery of circular cross section with an overlapping stenosis specified at the position as shown in Figure-1 and a catheter having a balloon. The geometry of arterial wall with the overlapping stenosis and balloon model are defined by functions R(z) and (z) R c respectively and described as where R 0 are the radius of the tube without stenosis, L 0 is the stenosis length, d indicates stenosis location, δ is the critical height of the stenosis into the lumen, appears at the locations: z = d+L 0 /6 and d+5L 0 /6, z being the axial coordinate. The maximum height attained by the balloon at z=d+L 0 /2 is 0 δ and kR 0 is the radius of the inner tube which keeps the balloon in position, k<<1, and d z represents the axial displacement of balloon.
where (r, z) are cylindrical polar coordinates system with z measured along the tube axis and r measured normal to the axis of the tube, R e is the tube Reynolds number, p is the pressure and (u, µ) is the fluid (velocity, viscosity). The boundary conditions are R, (4) and (5) are the standard no slip conditions at the artery and balloon-catheter walls, respectively.

Analysis:-
The expression for the velocity obtained as the solution of equation (3) subject to the boundary conditions (4) and (5), is given as The flow flux, Q is thus calculated as From equation (7), one now obtains across the stenosis in the tube of length, L is obtained as , dz dz dp Δp where   Shear stress affects endothelial cell (Glagov et al. 1988). The balloon helps to widen blood vessels and improve blood flow. To inflate and deflate the balloon several times to push the blockage aside to restore blood flow in blood vessel. The wall shear stress distribution, w  decreases with increasing value of z d for some fixed value of 0 R δ and k in stenotic region at any axial distance, For the analysis of blood flow characteristics through stenosed artery, wall shear stress near the throat of the stenosis deserves special attention. It has significant role in the development of arterial diseases. The peak of the wall shear stress is believed to cause severe damage to arterial lumen. The variation of shearing stress at stenosis throat with stenosis size, 0 R δ is shown in Figure-8 and 9. Figure-8    , balloon height, limits the usefulness of present study which allows the use of fully developed flow equations and leads to a local poiseuille-like flow. Balloon based catheter technique system that applies mechanical pressure to blood vessel walls has dramatically advanced the standard of care for patients with vascular disease. The insertion of balloon with catheter into artery leads to considerable increase in the magnitudes of fluid characteristics. Balloon inflation opens blocked arteries i.e. treatment of stenosis and restores normal blood flow. Balloon based catheter technique system has emerged as the standard tool of care for coronary arterial disease. The present analysis may be useful for medical purpose.