Iteratioal Joural of Emergig Treds & Techology i Computer Sciece (IJETTCS Volume 6, Issue 3, May- Jue 27 ISSN 2278-6856 A mathematical model of blood flow through diseased blood vessel Sapa Rata Shah, Aamika School of Computatioal ad Itegrative Scieces, Jawaharlal Nehru Uiversity, New Delhi-67, (Idia Abstract The flow of blood i two-dimesioal through the costricted radially o-symmetric multiple steosed artery is ivestigated i this paper. Blood is cosidered a Herschek-Bulkley fluid ad it is characterized by the geeralized form of Navier-Stokes equatio. The equatios of the proposed model are solved ad closed from expressios for the blood flow characteristics, resistace to flow, wall shear stress at maximum depressio for differet poits of a sigle loop steosis. It is foud that resistace to flow is icreased as the height of steosis icreases. Ad wall shear stress is icreased with axial velocity for icreasig values of steosis shape parameter. Keywords: Steosis, Herschek-Bulkley fluid, Resistace to flow, Wall shear stress, Radially o-symmetric.. INTRODUCTION Atherosclerosis is the major cause of heart attack ad stroke. This disease causes a ogoig steosis of the lume ad hardeig of the artery wall because of accumulatio of lipids i the itima [2]. The successive build-up of deposits i the arterial wall may form a plaque that protrudes i to the lume ad restricts the blood flow. If the carotid artery is affected the it may cause stroke, whe the coroary artery. There are also correspodig chages i the forces (shear ad ormal stresses exerted by the flowig blood o the plaque surface. The abormal arrowig i a blood vessel is caused by steosis Fig..(a. Ivestigators [5, 8] have emphasized that the formatio of itravascular plaques ad the impigemet of ligamets ad spurs o the blood vessel wall are some of the major factors for the iitiatio ad developmet of this vascular disease. The fruitful study of [3,6] has poited out that the variatio of resistace to flow ad the wall shear stress with the axial distace are physiologically importat quatities. [,7] have show theoretical results of for the velocity profiles, pressure drop, wall shearig stress ad separatio pheomea for special geometries for Newtoia model of blood. I the series of the papers [7] the effects o the cardiovascular system ca be uderstood by studyig the blood flow i its viciity. I these studies the behavior of the blood has bee cosidered as a Newtoia fluid. However, it may be oted that the blood does ot behave as a Newtoia fluid uder certai coditios. It is geerally accepted that the blood, beig a suspesio of cells, behaves as a o-newtoia fluid at low shear rate [9]. It has bee poited out by [] that the flow behaviour of blood i a tube of small diameter (less tha.2 mm ad at less tha 2sec - shear rate, ca be represeted by a power-low fluid model. I these discussed models, the ivestigators have ot dealt with the radially o-symmetric steosis. I this preset aalysis mathematical model for the blood flow through a radially o-symmetric steosis has bee formulated for improved geeralized geometry of multiple steosis located at equispaced poits. For simplicity the graphical aalysis is performed for a sigle loop of steosis havig maximum depressio at differet poits. Fig.. (a Artery with multiple steosis 2. FORMULATION OF THE PROBLEM I the preset aalysis, it is assumed that the steosis develops i the arterial wall ad symmetrical about the axis but o-symmetrical with respect to radial co-ordiates. I such a case the radius of artery, R(z ca be writte as: Fig..(b Volume 6, Issue 3, May Jue 27 Page 282
Iteratioal Joural of Emergig Treds & Techology i Computer Sciece (IJETTCS Volume 6, Issue 3, May- Jue 27 ISSN 2278-6856 Fig..(b.Geometr y of Steosed artery R(z ad R is the radius of the artery with ad without steosis, respectively. L is the legth of artery ad L is the steosis legth, d idicates the distace betwee equispaced poits, k is umber of steosis that appears i arterial lume, α.is a positive iteger, m is parameter determiig the shape of steosis i artery ad δ deotes the maximum height of steosis at Z. / (m kd k L L / m z. α (2 Coservatio Equatio ad boudary coditios: The equatio of motio for lamiar ad icompressible, steady, fully-developed, oe-dimesioal flow of blood whose viscosity varies alog the radial directio i a artery reduces to [4]: P (r τ, r r z P, r (3 where (z, r are co-ordiates with z measured alog the axis ad r measured ormal to the axis of the artery. Followig boudary coditios are itroduced to solve the above equatios, u / r = at r = u = at r = R (z. is fiite at r = P = P o at z = P = P L at z = L (4 Aalysis of the problem: Herschel-Bulkley fluid model- The stress-strai relatio of Herschel-Bulkley fluid is give as: f ( f ( where du dr du dr, dp r, 2, dp, 2 ad µ deotes Herschel-Bulkley viscosity coefficiet, o is yield stress, is shear stress, R c is the radius of the plug-flow regio, u is the axial velocity alog the z directio ad is the flow behavior idex. The relatio correspod to the vaishig of the velocity gradiets i regios, i which the shear stress τ is less tha the yield stress τ o this implies a plug flow wherever τ τ o whe the shear rates i the fluid are very high, τ τ o, the power-law fluid behavior is idicated. By equatio (2 ad (3 we get, / p / = - r - R c, du dr 2 μ (6 the flow of flux, Q, is defied as, d u Q= 2 p u r dr = p r - dr, d r R R 2 substitutig the value of f ( from equatio (4 i equatio (6, / (3 + π P R Q = f (y, 2 2μ ( + (8 ((/ 4 ((/ 2 f ( y [2( ( R ((/ 2 R 4 (( ((/ 2((/ 3 R (( ((/ 3 ( ], R R c y =. R Usig equatio (7 we have, (( / 3 (5 dp 2 μ 2 Q P = - = ( + 3 ( + R π f (y (9 to determie λ, we itegrate equatio (8 for the pressure P L ad P o are the pressure at z = ad z = L, respectively, where L is the legth of the tube. 2μ ΔP=P -P = 2Q + L π R L +3 (+3 R(z R f(y ( (7 Volume 6, Issue 3, May Jue 27 Page 283
Iteratioal Joural of Emergig Treds & Techology i Computer Sciece (IJETTCS Volume 6, Issue 3, May- Jue 27 ISSN 2278-6856 The resistace to flow is give by the coefficiet λ is defie as follows: P - P L Q ( o usig equatio (9 ad ( gives, 2 μ 2Q ( + +3 (M R π (2 d d + L L d z d z d z M = + + 3 + f d R (z d + L (f f(y R 4 ( = [2( y (/ + 2 y f (/ 4 (3 (3 + (( y (( y ] (2 (3 Whe there is o steosis i artery the R = R o, the resistace to flow, grows or radius of artery decreases (this referred to as Fahraeus-Lidquist effect i very thi tubes. I Fig.4 the variatio of wall shear stress with steosis shape parameter (m has bee show. This figure depicts that wall shear stress decreases as steosis shape parameter (m icreases. These results are similar with the results of []. Fig.5 describes the variatio of wall shear stress (τ with steosis size. This figure depicts that wall shear stress (τ icreases as steosis size icreases. These results are cosistet to the observatio of [3,6]. Resistace to flow 5 45 4 35 3 25 2 5 5 δ/r =., α=. 2 4 6 8 Steosis Shape Parameter δ/r =., α=. δ/r =., α=.2 N 2 μ 2Q ( + L R +3 π (f (3 Fig.2.Variatio of Resistace to flow with Steosis shape parameter from equatio ( ad ( the ratio of ( / N is give as: d + L L L N d R (z L (f d z = - + R o + 3 f(y (4 3.Result ad Discussio I order to have estimate of the quatitative effects of steosis shape parameter (m= 2..., ad steosis size o resistace to flow ad wall shear stress, computer codes were developed ad to evaluate the aalytical results obtaied for resistace to blood flow ad wall shear stress for diseased system associated with steosis due to the local depositio of lipids have bee determie. The results are show i Fig 2-5 by usig the values of parameter based o experimetal data i steosed artery. Fig.2 reveals the variatio of resistace to flow ( with steosis shape parameter (m. It is observed that the resistace to flow ( decreases as steosis shape parameter (m icreases, maximum resistace to flow ( occurs at (m = 2, i. e. i case of symmetric steosis. The result is cosistig with the result of [2,9]. Fig.3 cosists the variatio of resistace to flow ( with steosis size (δ/r. It is evidet that resistace to flow icreases as steosis size icreases. Resistace to flow icrease as steosis Resistace to flow 45 4 35 3 25 2 5 5.2.4.6.8 Steosis Size δ/r =., α=. Fig.3. Variatio of Resistace to flow with steosis size δ/r =., α=. δ/r =., α=.2 Volume 6, Issue 3, May Jue 27 Page 284
Iteratioal Joural of Emergig Treds & Techology i Computer Sciece (IJETTCS Volume 6, Issue 3, May- Jue 27 ISSN 2278-6856 Wall Shear Stress 8 6 4 2 8 6 4 2 Wall Shear Stress 4 35 3 25 2 5 5 2 4 6 8 δ/r =., α=. Steosis Shape Parameter Fig.4. Variatio of Wall Shear Stress with Steosis shape parameter 4.CONCLUSION A theoretical ad aalytical study of blood flow through multi-shape steosed artery has bee carried out. The umerical experimet is helpful for biologist ad medical practitioers to aalyze the effect of blood flow i presece of radially o-symmetric multiple steosed artery. The radially o-symmetric multiple steosis have importat effect o flow tha sigle-regular shape steosis. REFERENCES []. Bhatagar, A., Srivastava, R.K., Sigh, A.K., A umerical aalysis for the effect of slip velocity ad steosis shape o oewtoia flow of blood, Iteratioal Joural of Egieerig, (25, Vol. 28, Issue 3, pp. 44-446. [2]. Bhatagar, A., Srivastava, R.K., Sigh, A.K., A umerical aalysis for the effect of slip velocity ad steosis shape o oewtoia flow of blood, δ/r =., α=. δ/r =., α=.2 δ/r =., α=..2.4.6.8 Steosis Size Fig.5. Variatio of wall shear stress with steosis size δ/r =., α=. δ/r =., α=.2 Iteratioal Joural of Egieerig, (25, Vol. 28, Issue 3, pp. 44-446. [3]. Biswas, D. ad Ali, M., Two-layered mathematical model for blood flow iside a asymmetric steosed artery with velocity slip at iterface, Iteratioal Joural of Mathematical Archive, (24, Vol. 5, Issue 2,pp. 293-3. [4]. Jai, N., Sigh, S., Steady flow of blood through a atherosclerotic artery: A o-newtoia model, Iteratioal Joural of Applied Mathematics ad Mechaics, (22,8, pp. 52-63. [5]. Joshi, P., Pathak, A. ad Joshi, B.K. Two-Layered Model of Blood Flow through Composite Steosed Artery. Applicatio ad Applied Mathematics, Joural of Bio-Sciece ad Bio-Techology, (29, Vol. 3(, pp. 27-38. [6]. Kumar, S., Diwakar, C., Blood flow resistace for a small artery with the effect of multiple steoses ad post steotic dilatatio, Iteratioal Joural of Egieerig Scieces & Emergig Techologies, (23, Vol. 6, Issue, pp. 57-64. [7]. Moha, V., Prashad, V. ad Varshey, N.K., Effect of icliatio of a steosed artery o casso fluid flow with periodic body acceleratio, Iteratioal Joural of Advaced Scietific ad Techical Research, (23, Vol. 4, Issue 3, pp. 365-37. [8]. Sakar, A.R., Guakala, S.R. ad Comissiog, D.M.G., Two-layered Blood Flow through a Composite Steosis i the Presece of a Magetic Field, Iteratioal Joural of Applicatio or Iovatio i Egieerig & Maagemet, (23, Vol. 2, Issue 2, pp. 3-4. [9]. Sakar, A.R., Guakala, S.R. ad Comissiog, D.M.G., Two-layered Blood Flow through a Composite Steosis i the Presece of a Magetic Field, Iteratioal Joural of Applicatio or Iovatio i Egieerig & Maagemet, (23, Vol. 2, Issue 2, pp. 3-4. []. Srivastava V.P. ad Mishra S., "No-ewtoia arterial blood flow through a overlappig steosis", Appl. ad Appl. Math.: A It. J. (AAM, (2, Vol. 5, Issue, pp. 225 238. AUTHOR Dr. Sapa Rata Shah, Associate Professor, School of Computatioal ad Itegrative Scieces, Jawaharlal Nehru, New Delhi, received M.Sc. ad Pd.D degree i Mathematics from Christ Church P. G. College, ad Harcourt Butler Techical Uiversity, Kapur respectively. She has bee doig research work i Applied Mathematics, Biomechaics, Biomathematics sice 22. She has published more tha sevety research papers i various reputed iteratioal jourals. Volume 6, Issue 3, May Jue 27 Page 285
Iteratioal Joural of Emergig Treds & Techology i Computer Sciece (IJETTCS Volume 6, Issue 3, May- Jue 27 ISSN 2278-6856 Mrs. Aamika, Systems Aalyst, School of Computatioal ad Itegrative Scieces, Jawaharlal Nehru, New Delhi, received B.E. degree i Computer Scieces from M.D.U Rohtak,, Post graduatio diploma i advace desig ad developmet from CDAC Noida ad M.S. degrees i Software Systems from BITS Pilai, Rajestha. She is perusig her research work i applied Mathematics Volume 6, Issue 3, May Jue 27 Page 286