Study of blood flow through a catheterized artery

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1 Available online at.pelagiaresearchlibrary.com Advances in Applied Science Research, 11, (6): 11-1 Study of blood flo through a catheterized artery ISSN: CODEN (USA): AASRFC Narendra Kumar Verma a, Shailesh Mishra b, Shafi Ullah Siddiqui, Ram Saran Gupta * Department of Mathematics Harcourt Butler Technological Institute, Kanpur, India * Department of Mathematics Kamla Nehru Institute of Technology, Sultanpur, India _ ABSTRACT The problem of blood flo through a symmetric stenosis during artery catheterization assuming blood to behave like a Netonian fluid is discussed in the present analysis. The analytical expressions for the blood flo characteristics, namely, the impedance, the all shear stress in the stenotic region and the shear stress at stenosis throat are derived. The impedance increases ith the increasing size of the catheter and assumes considerable higher magnitude in a catheterized artery than its corresponding magnitude in uncatheterized for any given set of parameters fixed. Also, for any given catheter size the impedance increases ith the stenosis size (height and length). The all shear stress distribution in the stenotic region possesses almost an opposite characteristics in catheterized artery in comparison to its variations in an uncatheterized artery. The variations in the magnitude of the shear stress at stenosis throat are observed having opposite characteristics in comparison to the variations in the magnitude of impedance (flo resistance). Keyords: Catheter, stenosis, all shear stress, throat, impedance. _ INTRODUCTION Circulatory disorders are mostly responsible for over seventy five percent of all deaths and stenosis or arteriosclerosis is one of the frequently occurring diseases. The generic medical term Stenosis means a narroing of any body passage, tube or orifice, and is a frequently occurring cardiovascular disease in mammalian arteries. Stenosis or arteriosclerosis is the abnormal and unnatural groth in the arterial all thickness that develops at various locations of the cardiovascular system under diseased conditions hich occasionally results into serious consequences (myocardial infraction, cerebral strokes, argina pitoris, etc.). It is believed that stenoses are caused by the impingement of extravascular masses or due to intravascular atherosclerotic plaques hich develop at the all of the artery and protrude into the lumen. Regardless of the cause, it is ell established that once an obstruction has developed, it results 11

2 Ram Saran Gupta et al Adv. Appl. Sci. Res., 11, (6):11-1 into significant changes in blood flo, pressure distribution, all shear stress and the impedance (flo resistance). In the region of narroing arterial constriction, the flo accelerates and consequently the velocity gradient near the all region is steeper due to the increased core velocity resulting in relatively large shear stress on the all even for a mild stenosis. An account of most of the theoretical and experimental investigations on the subject may be had from Young [16], Srivastava et al. [1, 13], Sarkar and Jayraman [11], Mekheimer and El-Kot [7], Rathod and Asha [8], Reddy and Venkataramana [9]. Arterial stenosis is associated ith significant changes in the flo of blood, pressure distribution, all shear stress and the flo resistance (impedance). The flo accelerates and consequently the velocity gradient near the all region is steeper due to the increased core velocity resulting in relatively large shear stress on the all even for a mild stenosis, in the region of narroing arterial constriction. The flo rate and the stenosis geometry are the reasons for large pressure loss across the stenosis. The use of catheters is of immense importance and has become a standard tool for diagnosis and treatment in modern medicine. Transducers attached to catheters are of great use in clinical orks and the technique is used for measuring blood pressure and other mechanical properties in arteries. A catheter is composed of polyester based thermoplastic polyurethane, medical grade polyvinyl chloride, etc. When a catheter is inserted into the stenosed artery, the further increased impedance or frictional resistance to flo ill alter the velocity distribution. To treat arteriosclerosis in balloon angioplasty, a catheter ith a tiny balloon attached at the end is inserted into the artery. The catheter is carefully guided to the location at hich stenosis occurs and the balloon is then inflated to fracture the fatty deposits and iden the narroed portion of the artery. Kanai et al. [5] established analytically that for each experiment, a catheter of an appropriate size is required in order to reduce the error due to the ave reflection at the tip of the catheter. Gunj et al. [], Anderson et al. [1] and Wilson et al. [1] have studied the measurement of translational pressure gradient during angioplasty. eimgraber et al. [6] have reported high mean pressure gradient across the stenosis during balloon angioplasty. The mean flo resistance increase during coronary artery catheterization in normal as ell as stenosed arteries has been studied by Back et al. [], Sarkar and Jayaraman [11] discussed the changed flo patterns of pulsatile blood flo in a catheterized stenosed artery. Dash et al. [3] further studied the problem in a stenosed curved artery. Most recently, Sankar and Hemlatha [1] studied the flo of Herschel Bulkley fluid in a catheterized blood vessel. An effort is made in the present ork to estimate the increased impedance and other flo characteristics during artery catheterization in an artery ith a symmetric stenosis assuming that the floing blood to behave like a Netonian fluid. The all in the vicinity of the stenosis is usually relatively solid hen stenoses develop in the living vasculature. The artery length is considered large enough as compared to its radius so that the entrance, end and special all effects can be neglected. Formulation of the problem Consider the axisymmetric flo of blood through a catheterized artery ith a symmetric stenosis. The artery is assumed to be a rigid circular tube of radius R and the catheter as a coaxial rigid tube of radius R c. The artery length is considered large enough as compared to its radius so that the entrance, end and special all effects can be neglected. The geometry of the stenosis, assumed to be manifested in the arterial segment is described in Fig. 1 as 115

3 Ram Saran Gupta et al Adv. Appl. Sci. Res., 11, (6):11-1 R(z) R δ = cos R = 1 otherise, π z d ; d z d +, (1) Fig. 1 Geometry of a symmetric stenosis in a catheterized artery. here R(z) is the radius of the tube ith stenosis, is the length of the stenosis and d indicates its location, δ is the maximum projection of the stenosis at = d + /. z Blood is assumed to be represented by a Netonian fluid and folloing the report of Young [15] and considering the axisymmetric, laminar, steady, one-dimensional flo of blood in an artery, the general constitutive equation in a mild stenosis case, under the conditions: δ/r << 1, e (δ / ) << 1and R / ~ O(1), may be stated as R dp µ = r u, () dz r r r dp =, (3) dr here (r,z) are cylindrical polar coordinates system ith 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 u = at r = R(z), () u = at r = R, (5) Conditions () and (5) are the standard no slip conditions at the artery and catheter alls, respectively. c 116

4 Ram Saran Gupta et al Adv. Appl. Sci. Res., 11, (6):11-1 Analysis The expression for the velocity obtained as the solution of equation () subject to the boundary conditions () and (5), is given as R dp u = (R/R ) (r/r ) µ dz The flo flux, Q is thus calculated as + ) (R /R ) } log(r/r c c ) log r R. (6) here ε = R /R. c Q = π R Rc πr = r u dr ) ε } dp ) ε } 8µ (R/R ) dz + ε log )/ε}, (7) From equation (7), one no obtains dp 8µ Q = φ(z), (8) dz πr ith φ(z) = 1/F(z), F(z) = ) ε } ) ε } { } (R/R ) + ε. log (R/R ) / ε The pressure drop, p ( = p at z =, p at z = ) across the stenosis in the tube of length, is obtained as here d dp p = dz dz 8µ Q = ψ, (9) πr d+ [ φ(z) ] R/R = 1dz + [ φ(z) ] R/R (1) dz + [ φ(z) ] ψ = from d d+ R/R = 1 The first and the third integrals in the expression for ψ obtained above are straight forard hereas the analytical evaluation of second integral is almost a formidable task and therefore shall be evaluated numerically. The expressions for the impedance (flo resistance), λ, the all shear stress distribution in the stenotic region, τ and shear stress at the stenosis throat, τ s in their non-dimensional form as 1 / λ = + + η π π ` ( θ ε ) dα dz [ θ + ε ( θ ε ) /log( θ / ε )], (1) 117

5 Ram Saran Gupta et al Adv. Appl. Sci. Res., 11, (6):11-1 τ = R / R (11) {( / ) }[ ( / ) {( / ) }/ log(( / ) / )], R R ε R R + ε R R ε R R ε = b (1) { }[ { }/ log( / )], b ε b + ε b ε b ε τs here θ θ(α) = a + b cosα, α = π ( π/ )(z d / ), a a( z) = 1 δ / R, b = δ / R, 1 { 1+ ε + (1 ε ) / logε} η = (1 ε ), λ/λ λ =, (, τs ) ( τ, τs )/τ τ =, λ 8 / = µ πr, τ = µq πr are the flo resistance and shear stress, respectively for a / 3 Netonian fluid in a normal artery (no stenosis), and λ, τ and τ s are the impedance, all shear stress and shear stress at stenosis throat, respectively in their dimensional form obtained from the definitions: λ = p/q, τ ( R/)dp/dz τs τ. = =, ( ) R / R = b Numerical results and discussion To discuss the results of the study quantitatively, computer codes are no developed for the numerical evaluations of the analytical results obtained in equations (1)-(1). The various parameter values are selected from Young [15] as: (cm) = 1; (cm) = 1,, 5; ε (nondimensional catheter radius) =,.1,.,.3,.,.5,.6; δ/r (non-dimensional stenosis height) =,.5,.1,.15,.. It is to note here that the present study corresponds to the flo in uncatheterized artery for parameter value ε =. 6 5 / =1 Numbers ε.3 λ δ/r Fig. Variation of flo resistance,λ ith δ/r for different ε. 118

6 Ram Saran Gupta et al Adv. Appl. Sci. Res., 11, (6):11-1 The impedance, λ increases ith the catheter size, ε for any given stenosis height, δ/r and also increases ith stenosis height, δ/r for any given catheter size, ε (Fig.). One notices that for any given stenosis height, a significant increase in the magnitude of the flo resistance, λoccurs for any small increase in the catheter size, ε (Fig.). The flo resistance, λ steeply increases ith the catheter size, ε (.3) and depending on the height of the stenosis, attains a very high asymptotic magnitude ith increasing the catheter size,ε (Fig.3). The all shear stress distribution, τ in an uncatheterized artery increases from its approached magnitude (i.e., at z = ) in the upstream of the throat ith the axial distance and achieves its maximal at the stenosis throat (i.e., at z = d + o/) and then decreases in the donstream and attains its approached magnitude at the end of the constriction profile (i.e. at z/o = 1). Interestingly, hoever, the shear stress distribution, τ possesses an opposite characteristics in a catheterized artery. 16 = =1 Numbers δ/r. 1 λ ε Fig.3 Variation of impedance,λ ith ε for different δ/r. The flo characteristic, τ decreases from its approached magnitude in the upstream, achieves its minimal at the throat and then increases in the donstream and attains its approached magnitude at the end of the constriction profile. 119

7 Ram Saran Gupta et al Adv. Appl. Sci. Res., 11, (6): Numbers ε ε= 1. τ ε=.1. ε= z/ o Fig. Wall shear stress distribution, τ in stenotic region for different ε Numbers ε τ s δ/r.15. Fig.5 Variation of shear stress at stenosis throat,τ s ith δ/r for different ε. One further notice that τ decreases ith increasing catheter size, ε for other parameters fixed (Fig. ). Shearing stress at the stenosis throat, τ s increases ith the stenosis size (height and length) for any given catheter size, ε (Figure 5). The all shear stress at the maximum height of the stenosis, τ s decreases ith catheter size,ε. 1

8 Ram Saran Gupta et al Adv. Appl. Sci. Res., 11, (6): Numbers δ/r τ s ε Fig.6 Variation of shear stress at the stenosis throat, τ s for different δ/r. The flo characteristic, τs assumes higher magnitude for higher stenosis height for small catheter size, ε [beteen ε = and 1.3 (approximately)] but the property reverses for large values of ε (Fig. 6). One notices that τ s achieves an asymptotic magnitude hen the catheter size becomes approximately fifty percent of the artery size. CONCUSION To estimate for the increased impedance and shear stress during artery catheterization, flo through a symmetric stenosis has been analyzed assuming that the floing blood is represented by a Netonian fluid. The impedance increases ith increasing catheter size and strongly depends on the stenosis height is an important information. Thus the size of the catheter must be chosen keeping in vie of stenosis height during the medical treatment. The impedance increases ith the increasing size of the catheter and assumes considerable higher magnitude in a catheterized artery than its corresponding magnitude in uncatheterized for any given set of other parameters fixed. Also, for any given catheter size the impedance increases ith the stenosis size (height and length). The all shear stress distribution in the stenotic region possesses almost an opposite characteristics in catheterized artery in comparison to its variations in an uncatheterized artery. The variations in the magnitude of the shear stress at stenosis throat are observed having opposite characteristics in comparison to the variations in the magnitude of impedance (flo resistance). REFERENCES [1] Anderson HV, Roubin GS, eimgruber PP, Cox WR, Douglas Jr. JS, King III SB, Gruentzig AR, Circulation, 1986, 73, 13. [] Back H, Kack EY, Back MR, J. Biomed. Eng., 1996, 118, 83. [3] Dash RK, Jayaraman G, Mehta KN, J. Biomech., 1999, 9. 11

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