Blood flow through arteries in a pathological state: A theoretical study

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1 International Journal of Engineering Science 44 (6) Blood flow through arteries in a pathological state: A theoretical study J.C. Misra *, G.C. Shit Department of Mathematics, Indian Institute of Technology, Kharagpur 71, India Received 14 July 5; received in revised form 5 October 5; accepted 1 December 5 (Communicated by G. LUKASZEWICZ) Abstract The aim of this paper is to develop a mathematical model for studying the nonnewtonian flow of blood through a stenosed arterial segment. Herschel Bulkley equation has been taken to represent the nonnewtonian character of blood. The problem is investigated by a combined use of analytical and numerical techniques. It is noticed that the resistance of flow and the skinfriction increase as the stenosis height increases. The results are compared with data available in the existing literature presented by previous researchers. Ó 6 Elsevier Ltd. All rights reserved. Keywords: Herschel Bulkley fluid; Stenosis; NonNewtonian fluid; Skinfriction 1. Introduction It is known that stenosis is a dangerous disease and is caused due to the abnormal growth in the lumen of the arterial wall. While the exact mechanism of the formation of stenosis in a conclusive manner remains somewhat unclear from the standpoint of pathology/physiology, deposition of various substances like cholesterol on the endothelium of the arterial wall and proliferation of the connective tissues are believed to be the factors that accelerate the growth of the disease. It has been observed that whole blood, being predominantly a suspension of erythrocytes in plasma, behave as a nonnewtonian fluid at low shear rates in microvessels (cf. [1 6]). Existing literature in the area also reveals that the shear rate of blood is low in the stenosed region. These two experimental observations emphasize that flow of blood in the stenosed portion of an artery must exhibit nonnewtonian characteristics. Some researchers studied the power law fluid model of blood arguing that under certain conditions, blood behaves like a power law fluid. Casson [7] examined the validity of Casson model in studies pertaining to the * Corresponding author. Fax: address: jcm@maths.iitkgp.ernet.in (J.C. Misra). 75/$ see front matter Ó 6 Elsevier Ltd. All rights reserved. doi:1.116/j.ijengsci

2 J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) Nomenclature s s H s R s d k k Q r z u R R(z) L L p k n shear stress yield stress skinfriction nondimensional skinfriction stenosis height flow resistance nondimensional flow resistance volumetric flow rate radial coordinate axial coordinate axial velocity radius of the artery radius of the artery at stenosed portion halflength of the artery halflength of the stenosis pressure viscosity coefficient fluid index flow characteristics of blood and reported that at low shear rates the yield stress for blood is nonzero. Blair and Spanner [8] reported that blood behaves like a Casson fluid in the case of moderate shear rate flows and that there is not much of difference between Casson s and Herschel Bulkley s plots. It is worthwhile to note that the Herschel Bulkley model is of general type and the results obtained by this model enables one to derive the corresponding Newtonian fluid, Bingham fluid and also power law fluid by putting specific results for a values for the parameters. A survey of the existing literature on the experimental and theoretical studies of blood flow various parts of the arterial tree under normal as well as pathological conditions further indicates that in certain situations the Bingham plastic fluid model (a particular case of Herschel Bulkley model) gives a better description of the rheological properties of blood [9,1]. The physiological importance of studies pertaining to the variations of the resistance to flow and the wall shear stress with the axial distance and the depth of the stenosis were discussed by Shukla et al. [11,1] and Chaturani and Ponnalagorsamy [1] by considering the fluid to be nonnewtonian. Halder [14] dealt with a problem of blood flow through a stenosed artery by considering blood by a power law fluid and observed that the maximum resistance to flow is attained at the throat of the stenosis, in the case of a symmetrical stenosis. A mathematical analysis for the flow of blood in a stenosed arterial segment was studied by Misra and Chakravarty [15] by treating blood as a Newtonian viscous fluid. Misra et al. [6] subsequently developed a mathematical model for investigating the behaviour of blood flow in a stenosed segment of an artery by taking into account the viscoelastic behaviour of blood. In view of the above the present investigation has been devoted to the problem of blood flow through a stenosed segment of an artery where the rheology of blood is described by Herschel Bulkley model. Computational results for the variation of skinfriction with axial distance in the region of the stenosis are presented graphically. The qualitative and quantitative changes in the skinfriction, the flow resistance and the volumetric flow rate at different stages of the growth of the stenosis have also been reported.. The problem and its solution Let us consider steady laminar fully developed onedimensional flow of blood obeying the constitutive equation given by Herschel Bulkley model through a stenosed artery.

3 664 J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) The equation governing the flow of blood is taken in the form dp ¼ 1 dðrsþ ð1þ r dr in which s represents the shear stress of blood considered as Herschel Bulkley fluid and p the pressure at any point. The constitutive equation may be put as du dr ¼ f ðsþ ¼1 k ðs s HÞ n ; s P s H ¼ ; s < s H ðþ where u stands for the axial velocity of blood and s H is the yield shear stress and k, n are parameters which represent nonnewtonian effects. Let us consider a bellshaped stenosis geometry given by RðzÞ ¼R 1 d exp m z ðþ R R where R stands for the radius of the arterial segment outside the stenosis, R(z) is the radius of the stenosed portion of the arterial segment under consideration at a longitudinal distance z from the leftend of the segment; d is the depth of the stenosis at the throat and m is a parametric constant; characterises the relative length of the constriction, defined as the ratio of the radius to half length of stenosis, i.e., ¼ R L. Considering the stenosis geometry to be of the form (cf. Fig. 1) RðzÞ ¼ 1 ae bz R with a ¼ d R and b ¼ m. R Eqs. (1) and () are to be solved subject to the boundary conditions u ¼ at r ¼ RðzÞ ðnoslip conditionþ ð5þ s is finite at r ¼ ðregularity conditionþ ð6þ ð4þ Integrating Eq. (1) and using (6), we get s ¼ r dp The skinfriction s R is given by s R ¼ R dp where R = R(z). ð7þ ð8þ r L L R R δ o Z L L Fig. 1. Geometry of the arterial segment with stenosis.

4 The volumetric flow rate Q is given by the Rabinowitsch equation Z sr Q ¼ pr s f ðsþds s R where s and s R are given by Eqs. (7) and (8) respectively. Substituting the value of f(s) from Eq. () into Eq. (9), we obtain Q ¼ pr s n R kðn þ Þ 1 s nþ1 H 1 þ s R " # sh s H þ n þ s R ðn þ 1Þðn þ Þ s R When s H sr 1, Eq. (1) reduces to Q ¼ pr kðn þ Þ s R n þ n s H ð11þ n þ use of (8) in Eq. (11) yields dp ¼ 1=n n kqðn þ Þ þ pr þn pr þn ðn þ Þ n þ s H s R Integrating (1) along the length of the artery and using the conditions that P = P 1 at z = L and P = P at z = L, we obtain 1=n Z P 1 P ¼ n kqðn þ Þ z¼l þ n þ s Z z¼l H ð1þ z¼ L n þ1 n þ s R z¼ L R R where R R can be obtained by using Eq. (4). Thus the resistance to flow k defined by k ¼ P 1 P Q has the expression " # 1=n k ¼ 4n kðn þ Þ 6 Q n 1 4ðL L pr þn Þþ J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) Z L R n þ1 R R R 7 5 þ 4 n þ s H 4ðL L Þþ n þ s R Z L ð9þ ð1þ ð1þ ð14þ 5 ð15þ In the absence of any constriction, the resistance to flow k N is given by! 4ðn þ ÞL ðn þ Þ 1 n 1=n kq k N ¼ 4 þ s H 5 ð16þ QR pr n þ In dimensionless form, the flow resistance may be expressed as R R k ¼ k k N ¼ 1 L L þ 1 L ðn þ Þ 1 n kq pr ðn þ Þ 1 n kq pr! 1=n I 1 þ s H n þ I! 1=n þ s H n þ ð17þ where I 1 ¼ R L and I ðr R Þ n þ1 ¼ R L Substituting the expression for R R I 1 ¼ Z L where a ¼ d R and b ¼ m. R ð R R Þ. ð1 ae bz Þ n þ1 and I ¼ from (4), the integrals I 1 and I reduce to Z L ð1 ae bz Þ ð18þ

5 666 J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) For zero yield stress, Eq. (17) reduces to k ¼ 1 L þ 1 Z L L L ð19þ ð1 ae bz Þ n þ1 Integrals I 1 and I calculated by using Gauss twopoint Quadrature formula, now take the forms I 1 ¼ L ae ffiffi þ 1 ðbl =1Þð4þ p Þ n þ1 1 ae ffiffi 7 4 ðbl =1Þð4 p 5 ðþ Þ n þ1 and τ Present study (HerschelBulkley model) Power law model z Fig.. Variation of skinfriction along the length of the arterial segment. (Comparison between Herschel Bulkley fluid model and power law fluid model [1].).. n=. 1.8 τ % 1. 5% 1% z Fig.. Variation of skinfriction along the vessel length for different n (s H =.5, k = 4.). n = 1 represents the Bingham plastic fluid model.

6 J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) n=. τ % 1. 5% 1% z Fig. 4. Variation of skinfriction along the vessel length for different n (s H =.1, k = 4.). I ¼ L ae ffiffi 1 ðbl =1Þð4þ p þ Þ 1 ae ffiffi ðbl =1Þð4 p 5 ð1þ Þ The skinfriction obtained from (7) and (1) is given by s R ¼ R dp 1=n ¼ kqðn þ Þ þ n þ pr n þ s H In the absence of any constriction i.e., when R(z) =R, the expression for the skinfriction reads 1=n kqðn þ Þ s N ¼ þ n þ pr n þ s H representing the skinfriction for a normal arterial segment. ðþ ðþ. 1.8 =. =.5 =.1 τ n= z Fig. 5. Variation of skinfriction along the vessel length for different value of n and s H (k = 4. and 5% stenosis).

7 668 J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) A nondimensional expression for skinfriction may be put as 1=n s ¼ s ðkqðn þ ÞÞ 1=n þ nþ ðpr R nþ Þ1=n R R sh ¼ s =n h i ð4þ N ðkqðn þ ÞÞ 1=n þ nþ ðpr Þ1=n s H R R nþ. Numerical computation and discussion of results The analytical expressions (17) and (4) derived in the preceding section have been computed. The computational results for the skinfriction obtained from the present study are displayed in Fig. (solid line). In. 1.8 =. =.5 =.1 τ n= z Fig. 6. Variation of skinfriction along the vessel length for different values of n and s H (k =. and 9% stenosis) n= 1 8 τ a= δ/r Fig. 7. Variation of skinfriction with stenosis height (s H =.1, k = 7. and 5% stenosis).

8 J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) order to compare our results based on the Herschel Bulkley model for blood flow, the results reported by Chaturani and Ponnalagorsamy [1] (who considered power law fluid model) are also presented in the same figure by dotted line. In order to examine the effects of the various parameters involved, on the skinfriction and the flow resistance, computation has been carried out for different values of n, s H and k. In the present study the values of the rheological parameters (n, k, s H ) are taken as n = 1., 1.5,.; k =., 4., 7. (CP) n /sec n 1 ; s H =.,.5,.1,.5 dyne/cm as per the experimental study of Young and Tsai [16]. Figs. 6 illustrate the individual effects of the said parameters in the case of 1%, 5% and 9% stenosis. It may be noted that the skinfriction increases with the size of the stenosis and that as the value of n diminishes, the magnitude of the skinfriction increases. It is interesting to observe (cf. Figs. 5 and 6) that the effect of yield stress s H on the magnitude of the skinfriction is only marginal. This implies that it is the viscosity that brings about the major changes in the skinfriction. A comparison between Figs. 5 and 6 reveals that the rate n= 1 1 λ 8 9% 5% 6 4 1% a=δ/r Fig. 8. Variation of flow resistance with stenosis height for different values of n (s H =.5, k = 4.). λ n= a=δ/r Fig. 9. Variation of flow resistance with stenosis height for different values of n (s H =.1, k = 7.). 9% 5%

9 67 J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) of change in skinfriction diminishes as the stenosis size increases. All the figures reveal that the nonnewtonian behaviour of blood has a reducing effect on the skinfriction of the arterial wall. The results presented in these figures further indicate that as the nonnewtonian behaviour is more and more prominent, there will occur a gradual reduction in the skinfriction. Fig. 7 depicts the variation of the skinfriction with the height of the stenosis when the blockage of the vessel is 5%. It is observed that increase in the stenosis height up to a certain level, does not have any bearing on skinfriction, beyond which the skinfriction increases with the height of the stenosis. One may further note that for n = 1, the rate of increase is maximum, the rate decreases when n assumes greater values. The nature of variation is similar in the case of stenosis of any height; however, the skinfriction does depend on the degree of blockage quantitatively. (For the sake of brevity, the results for other degrees are omitted from presentation.) cm /sec.9.8 Q.7.6 =. =.5 =.1 = z cm Fig. 1. Variation of the rate of flow with axial distance for different s H (n =1,k = 4. and 5% stenosis) cm /sec.4 Q.5..5 n= Fig. 11. Variation of the rate of flow with axial distance for different n (s H =.1, k = 7. and 9% stenosis). z cm

10 The variation of the flow resistance with the height of the stenosis is shown in Figs. 8 and 9 for different values of n, s H and k. The results presented in these figures indicate that the flow resistance increases with the increase in the stenosis size and further that as the fluid index diminishes, the flow resistance increases. Fig. 1 shows that the quantum of the volumetric flow rate decreases with an increase in yield stress, while Fig. 11 reveals that with n increasing, the volumetric flow rate diminishes. 4. Summary and conclusions The present study deals with a theoretical investigation of the characteristics of blood flow through a stenosed segment of an artery. Blood is represented by Herschel Bulkley (nonnewtonian) fluid model. Using appropriate boundary conditions, analytical expressions of the skinfriction and the flow resistance have been obtained. The computed results are presented graphically for different values of the parameters n, k and s H that stand respectively for fluid index, viscosity coefficient and the yield stress. The study shows that the skinfriction and the resistance to flow are maximum at the throat of the stenosis and minimum at the ends of the stenosis, i.e., at the onset of the normal portion of the arterial segment. It also ensures that the volumetric flow rate of blood is maximum at the onset of the stenosis and minimum at the throat of the stenosis. A increase in the size of the stenosis renders resistance to blood flow through arteries in the brain, heart and other organs of the body. This may lead to stroke, heart attack and various cardiovascular diseases. This analysis bears the potential to examine under the purview of a single study the results for several models proposed earlier, viz. the Bingham plastic fluid model, the power law fluid model and the Newtonian viscous model. The results for these different models have been compared. So this study is more useful for the purpose of simulation and validation of different models in different conditions of arteriosclerosis. This study also provides a scope for estimating the influence of the various parameters mentioned above on different flow characteristics and to ascertain which of the parameters has the most dominating role. Acknowledgements The authors are thankful to the referees for their valuable comments, basing upon which the manuscript has been revised to the present form. References J.C. Misra, G.C. Shit / International Journal of Engineering Science 44 (6) [1] S.E. Charm, G.S. Kurland, Nature 6 (1965) [] C.E. Huckaba, A.N. Hahn, Bull. Math. Biophys. (1968) [] R.L. Whitemore, Rheology of the Circulation, Pergamon Press, New York, [4] M.M. Lih, Transport Phenomena in Medicine and Biology, Wiley, New York, [5] J.C. Misra, B.K. Kar, Math. Comput. Modell. 15 (1991) [6] J.C. Misra, M.K. Patra, S.C. Misra, J. Biomech. 6 (199) [7] N. Casson, in: C.C. Mill (Ed.), Rheology of Disperse Systems, Pergamon Press, London, 1959, pp [8] G.W. Scott Blair, D.C. Spanner, An Introduction to Biorheology, Elsevier Scientific Publishing Company, Amsterdam, Oxford and New York, [9] F.R. Eirich, RheologyTheory and Applications, vol., Academic press Inc. Publishers, New York, S.F., London, 196. [1] Y.C. Fung, Biomechanics, SpringerVerlag, New York, Heidelberg, Berlin, [11] J.B. Shukla, R.S. Parihar, B.R.P. Rao, Bull. Math. Biol. 4 (198) [1] J.B. Shukla, R.S. Parihar, S.P. Gupta, Bull. Math. Biol. 4 (198) [1] P. Chaturani, R. Ponnalagorsamy, Biorheology (1986) [14] K. Halder, Bull. Math. Biol. 47 (1985) [15] J.C. Misra, S. Chakravarty, J. Biomech. 19 (1986) [16] D.F. Young, F.Y. Tsai, J. Biomech. 6 (197) 95 4.