Unsteady non-Newtonian blood flow through a tapered overlapping stenosed catheterized vessel

Accepted Manuscript

Unsteady non-Newtonian blood flow through a tapered overlapping stenosed catheterized vessel N. Ali , A. Zaman , M. Sajid , J.J. Nieto , A. Torres PII: DOI: Reference:

S0025-5564(15)00172-8 10.1016/j.mbs.2015.08.018 MBS 7680

To appear in:

Mathematical Biosciences

Received date: Revised date: Accepted date:

7 April 2015 21 August 2015 28 August 2015

Please cite this article as: N. Ali , A. Zaman , M. Sajid , J.J. Nieto , A. Torres , Unsteady nonNewtonian blood flow through a tapered overlapping stenosed catheterized vessel, Mathematical Biosciences (2015), doi: 10.1016/j.mbs.2015.08.018

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ACCEPTED MANUSCRIPT

Highlights

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A mathematical study is presented for unsteady pulsatile flow of blood through a tapered overlapping stenosed catheterized vessel. Magnetic field is taken into account. The taperness of artery is considered in the present analysis. The rheology of blood is described by the constitutive equation of Carreau model. The combined effects of the non-Newtonian rheology of blood, the vessel tapering, the severity of stenosis and catheterization on blood velocity and flow rate are analyzed in detail. The flow patterns illustrating the global behavior of blood are also presented.

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Unsteady non-Newtonian blood flow through a tapered overlapping stenosed catheterized vessel N. Alia, 1, A. Zamana and M. Sajidb, J. J. Nietoc, d, A. Torrese Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

b

Theoretical Physics Division, PINSTECH, P. O. Nilore Islamabad 44000, Pakistan

c.

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela,

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a

15782, Spain

Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589 Jeddah, Saudi Arabia

e

Departamento de Psiquiatría Radiología y Salud Pública, Facultad de Medicina, Universidad de Santiago

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d

de Compostela, 15782 , Spain 1

Corresponding author: Tel :-+92519019756 (e-mail: [email protected]).

Abstract:

The unsteady flow characteristics of blood in a catheterized overlapping stenosed artery are

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analyzed in presence of body acceleration and magnetic field. The stenosed arterial segment is modeled as a rigid constricted tube. An improved shape of stenosis in the realm of the formulation of the arterial

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narrowing caused by atheroma is integrated in the present study. The catheter inside the artery is approximated by a thin rigid tube of small radius while the streaming blood in the artery is characterized by the Carreau model. Employing mild stenosis condition, the governing equation of the flow is derived

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which is then solving numerically using finite difference scheme. The variation of axial velocity, flow rate, resistance impendence and wall shear stress is shown graphically for various parameters of interest.

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The flow patterns illustrating the global behavior of blood are also presented. Keywords: Unsteady flow; Magnetic field; Carreau fluid; Body acceleration; Catheterized artery; Finite

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difference method.

1. Introduction Atherosclerosis is a cardiovascular disease caused by invasion and accumulation of white blood cells in the lumen of an artery. As a result of these accumulations, the arterial wall gets thickened and hardened. Also during this process a plaque forms. The formation of such plaque results in narrowing of arteries which is commonly known as stenosis. Stenotic sites are most commonly found in coronary arteries. It is quite apparent that the flow behavior of blood in a stenosed artery is different than that in a normal artery. 2

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The abnormal narrowing of coronary arteries is commonly diagnosed and treated by making use of catheters. The technique used for diagnosis of stenosis in coronary arteries is usually known as angiography while treatment procedure of such arteries is called as angioplasty. In diagnostic procedure, such as angiography, small catheters are inserted under X-ray guidance to the opening of coronary arteries in order to get information about location and severity of stenotic arterial segment. Once the location of the stenosis is identified, the next procedure is to treat the diseased segment through angioplasty. In

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angioplasty two types of Catheters namely, guided catheter and Doppler catheter are commonly used. Doppler catheters are used to monitor the changes in proximal and distal flow velocity during coronary angioplasty while guided catheters with a small balloon at their tips are inserted into the artery and advanced to the stenotic region of the coronary artery. The balloon is then inflated to enlarge the stenotic segment. When inserted in arteries, these catheters will significantly alter the flow patterns of blood. In

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fact it was shown by Kanai et al. [1] that that apart from other factors the error in the measurement of blood pressure by catheter-insertion may also be due to reflection of pressure wave at the tip of the catheter. Later, Back and Denton [2] provide estimates of wall shear stress during coronary angioplasty. In another paper Back [3] reported that the size as well as the eccentricity of catheter also bear the potential to increase the flow resistance. The influence of the presence and size of a catheter on mean pressure gradient across human coronary stenosed artery was also examined by Back et al. [4]. The study

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of Sarkar and Jayaraman [5] revealed that pressure drop, shear stress and impendence vary markedly in the presence of catheter. Dash et al. [6] performed the analysis of blood in a curved catheterized stenosed

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artery and found that increase in catheter size leads to a considerable increase in pressure drop, impendence and wall shear stress. Apart from the attempts reported above, some other hemodynamical studies encompassing formation and progression of plaque in coronary artery, analysis in an intracranial

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aneurysm and mathematical analysis of differential models of circulatory system can be found in refs. [7-

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9].

In above mentioned studies the rheology of blood is characterized by Newtonian fluid. However, blood exhibits remarkable non-Newtonian behavior in small arteries at low shear rates. Also non-Newtonians

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behavior of blood manifest itself in arteries with diseased conditions and in pulsatile flow case in which blood is subjected to cyclic low shear rates [10, 11]. This fact motivated various researchers to study simultaneous effects of catheter and non-Newtonian rheology on pulsatile flow of blood in a stenosed coronary artery. For example, Dash et al. [12] carried out a study to estimate the increase in flow resistance in a narrow catheterized artery using a Casson model to account for blood rheology. A mathematical model of pulsatile flow of blood in a catheterized artery using Herschel-Bulkley constitutive equation is studied analytically by Sankar and Hemalatha [13]. Reddy et al. [14] utilized the

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constitutive equation of couple stress fluid and investigated the effect of tapering angle and slip velocity on unsteady blood flow through a catheterized stenosed artery. Two-phase model of blood flow through a catheterized stenosed artery is analyzed by Srivastava and Rastogi [15]. Nadeem and Ijaz [16] investigated the effects of nanoparticles on blood flow through a catheterized artery using regular perturbation method. Steady flow of a Herschel-Bulkley fluid in a catheterized artery is analyzed by

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Sankar and Hemalatha [17]. Due to the complex rheology of blood, there is no universally agreed upon model available in the literature. The Power law, Casson and Careau models are most common non-Newtonians model in hemodynamics. Some recent attempts pertaining the usage of these models in hemodynamical studies can be found in refs. [18-21]. More recently Khandelwal et al. [22] carried out laminar flow analysis of Carreau-Yasuda model in a T- shaped channel. However, a literature review indicates that no study is

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available where pulsatile flow of blood in catheterized overlapping stenotic artery is treated using Carreau-Yasuda model.

In this paper, we propose to analyze the pulsatile flow of blood through catheterized stenosed artery considering the Carreau-Yasuda model. The geometry of the overlapping stenosis has been incorporated in the present study. Moreover, the effects of tapering angle, body acceleration and magnetic field are also

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taken into account. The present study enable us to compare various flow characteristics of blood in a catheterized vessel for Carreau, power-law and Newtonian models. The paper is structured as follow: The

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flow geometry is illustrated in Section 2. The problem is modeled with appropriate assumptions in section 3. Section 4 presents the detail of numerical procedure for solution of the developed equations. The

section 6.

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obtained results are discussed in section 5. The main conclusions of the present study are summarized in

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2. Geometry of the problem

A homogenous incompressible Carreau fluid is assumed to be flowing in a tapered stenosed artery of length L. The flow is subject to a constant applied radial magnetic field and periodic body acceleration.

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The flow analysis will be carried out in a cylindrical co-ordinate system (r ,  , z ), where r and z axes are along the radial and axial directions of the artery. It is assumed that the stenosed arterial segment is composed of two overlapping bell shaped curves whose equation is

 *  64  11 3 47 2 1 2 3 4   z  a   1    l0  z  d   l0  z  d   l0  z  d    z  d    , 48 3  R z    10  32   *z  a ,  

d  z  d  l0 , otherwise,

(1) 4

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In above expression d represents the length of non-stenotic arterial region, a the radius of undisturbed non-tapered part of the artery, l0 the length of stenotic region and  *   tan   the parameter controlling the convergence or divergence of post-stenotic region. A schematic diagram of the non-tapered artery is shown in Fig. 1.



4  al0 4

,

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The parameter  is defined as (2)

in which  denote the critical height of the stenosis appearing at two specific locations i.e.,

9l0 50

, and z  d 

3. Flow equations

41l0 50

.

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z d 

(3)

The equations that govern the flow of an incompressible fluid are

V,ii  0,

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 ai  T, ijj + G(t )  J  B, i

(4) (5)

where V are the velocity components, 𝜌 is the fluid density, T ij are the components of Cauchy stress

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tensor, J is the current density, B   B0  B1  is the total magnetic field, “,” denotes the covariant i

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derivative and a are the components of acceleration vector given as

V i t

 V rV,ri .

(6)

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ai 

The assumption of small magnetic Reynolds number enables us to neglect induced magnetic B1 in

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comparison with constant applied magnetic field B0 applied in radial direction. In view of Ohm’s law, we can write

J    E V  B ,

(7)

where  is the electrical conductivity and E is the electric field. Eq. (7) after neglecting imposed and induced electric field reduces to J  B    B0 2V i .

(8)

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The body acceleration G(t) acting on overlapping stenosed arterial can be taken of the following form [23],

G(t )  Ag cos  2bt    ,

(9)

T i j   p i j  S i j

in which 𝑝 is the pressure, 

ij

ij

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The constitutive law for Carreau fluid model is [24] (10)

is the identity tensor and S obey the following equation



S i j      0    1  Γ 2  2 

 n 1 2

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

 ij  A1 , 

(11)

In Eq. (11), 0 and  are zero-shear-rate and infinite-shear-rate viscosities, respectively,  is the time constant, n is the power law index, A1i j are the components of first Rivilin-Ericksen tensor given by (12)

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A1ij  Vi , j  V j ,i .

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and  is the second invariant of first Rivilin-Ericksen tensor defined as

=

1

2

A

pk

1

A1k p  .

(13)

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Since the flow under consideration is unsteady two-dimensional and axisymmetric, therefore we define V 1  u(r , z, t ) and V 2  0 and V 3  w(r , z, t ).

(14)

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In view of (14), the continuity equation (4) reduces to u u w    0. r r z

(15)

ij

Similarly, the momentum equation (5) and component of extra stress S take the following form:

u w  p  1   rz   u  u  w     rS rr  S , r z  r  r r z  t 





 

w w  p   w 1    u w rS rz  S zz    B02 w,      G (t )   r z  z z  t  r r 



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

 

(16)

(17)

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n 1    2 2 2 2 2         u   u   u   w   u w  rr 2 S  2     0    1  Γ  2                   ,       r   r   z    z r     r    

   S rz      0    1  Γ 2   

   u   u   w    u w      2                    r   r   z    z r     2

2

2

   u   u   w    u w     2                    r   r   z    z r    2

2

2

2

r a

Srz 

,w a U 0 0

w U0

, u

l0u

 U 0

S rz , S rr 

l0

, t 

U 0 0

n 1 2

   w   ,   z  

n 1 2

   w u     .   r z  

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The above equation can be made dimensionless by defining r 

2

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    zz S  2     0    1  Γ 2    

 z R a2 p t, z  , R  , p  , 2 l0 a U 0l0 0

S rr , S zz 

l0

U 0 0

(18)

(19)

(20)

(21)

S zz ,

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where U 0 ,  are the average velocity and angular frequency respectively. Making use of these variables Eqs. (16)-(20) after dropping bars can be casted as

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 u u  w      0,  r r  z

(22)

 u u   p   u 1    2    Re   u  w      2  rS rr  S rz  , z   r z  r  r r   t

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



 

(23)

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w w  p   w   1       Re   u   2w rS rz   2 S zz   M 2 w, (24)     B2  cos  c2 t       r z  z z  t    r r 





 

    u 2  u 2   w 2   u w 2    2                         r   r    z    z r    

    S rr   m  1  m  1  We 2    

   u 2  u 2   w  2   u w  2                               r   r    z    z r    

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    S rz   m  1  m  1  We 2   

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n 1 2

n 1 2

   w u    , z    r 

   u      ,   r  

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    S zz   m  1  m  1  We 2    

    u 2  u 2   w  2   u w  2     2                         r   r    z    z r    



n 1 2

   w    . (25)   z  



For the subsequent analysis, we assume that the stenosis is mild     a  1 , and the ratio

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  a l0  O(1). Here it is mentioned that dropping the mild stenosis assumption makes the problem more involved and extra computational effort will be required to get the solution. In fact after dropping this assumption the problem under consideration will become two-dimensional with increased number of nonlinear terms which might be difficult to handle with the present code used for simulations. Nevertheless this problem can be solved and presently it is subject of our future investigations. Moreover, the assumption of mild stenosis is not completely contrived and analysis presented here might through some

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light on various aspect of blood flow during the initial development of the stenosis. In view of above assumptions, Eqs. (22)-(25) reduce to p r

 0,

(27)

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n 1     2 2   p 1     w  w   w  2       B2  cos  c2 t      r m  1  m   1  We 2     M w,     t  z r  r  r  r           

(26)

Note that Eq. (22) does not need to be considered further because in view of the mild stenosis assumption

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the information given by it i.e. w



p z

z

 0 is already integrated in Eq. (27). Following Burton [25], we take



 A0  A1Re e

2 i p t

,

t0

(28)

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where A0 is the mean pressure gradient and A1 is the amplitude of the pulsatile component which is responsible for systolic and diastolic pressures. In dimensionless form (28) becomes



p z

 B1 1  ecos (c1t ) 

(29)

In Eqs. (22)-(29),   a 2 / 20 is the Wormsley number, Re  U 0 a / 0 is the Reynolds number, We  U 0 / a is the Weissenberg number, M  B0 a  / 0  2 is the Hartmann number and 1

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e

  Ag  A a2 A1  a2 , c1  p , c2 = b , B1  0 , B2   Ag  B1 , m   . A0   0 U 0 0 U 0 A0 0

(30)

Inserting value of  p z in axial momentum equation (27), we get

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n 1     2 2   1     w  w   w  2     B1 1  ecos (c1t )   B2  cos  c2 t      r m  1  m  1  We 2     M w,     t r  r  r  r           

(31)

The above equation is subject to following boundary and initial conditions

w(r , t ) r k  0, w(r , t ) r  R  0, w(r, 0)  0,

(32)

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The formulas of volume flow rate, wall shear stress (WSS) and resistance impendence in new variables read R



Q  wrdr , k

(33)

n 1    2 2    w   w  2  S   m  1  m   1  We ,     r   r       r  R 

M

(34)

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 p  z    Q , L

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where

(35)

 64  11 47 1 2 3 4  R ( z )  1   z  1  1   z  d    z  d    z  d    z  d    ,   z    1.5, 10 32 48 3 

CE



with

1  4 ,  





a

, 



d l0

, 

 l0 *

a

(36)

.

AC

The formula (34) actually represents the magnitude of generalized wall shear stress given in [11] under mild stenotic assumption. According to [11] 𝜏𝑠 = √𝒘 ∙ 𝒘

(37)

where 𝒘 = 𝑻𝒏 − (𝑻𝒏. 𝒏)𝒏, T is the stress tensor given through Eq. (10) and n is outward normal to the wall surface.

9

ACCEPTED MANUSCRIPT

In Eq. (37)    tan   , is called tapering parameter and  is called taper angle. The cases   0 ,   0,

  0 correspond to converging, non-tapered and diverging tapering artery respectively. These cases are shown in Fig. 2 for illustration. Employing the radial coordinate transformation [26] r R( z )

,

CR IP T

x

Eq. (32) can be transformed as

(38)

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n 1       We 2 w 2  2  w  1     w  2     B1 1  ecos (c1t )   B2  cos  c2 t      2  x m  1  m   1       M w,    t xR  x R  x  x             

(39)

and the dimensionless boundary conditions becomes

w x k  0, w x 1  0, w t  0  0.

(40)

the form

1

 

M

Similarly the volume flow rate, the shear stress at the wall and resistance impendence, respectively takes



Q  R 2  w xdx  ,



ED k

(42)

CE

PT

n 1    2 2 2   1   w   We  w  s   m  1  m   1      ,   R    R  x   x     x 1 

(41)

L  p    l z   0 Q  .

(43)

AC

Substituting the dimensionless form of pressure gradient in Eq.(43), we can write



L B1 1  ecos ( 2 t ) 

l0  1  2  w xdx  R ( z ) 0 

.

(44)

4. Solution methodology 10

ACCEPTED MANUSCRIPT

Due to nonlinear nature of Eq. (39), an exact solution is difficult to find. Therefore, a numerical solution is inevitable for further discussion. To this end, a numerical scheme, which is forward in time and central in space, is employed. If wik denotes the value of w at node xi and at time instant tk. Then, we can approximate various partial derivatives as x

2w x 2 w

and

t

wik1  wik1



2Δx





 wx ,

wik1  2wik  wik1

 Δx 

wik 1  wik Δt

2

(45)

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w

 wxx ,

.

(46)

(47)

form:

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Using the above formulas for time and spatial derivatives, Eq. (39) may be transformed in the following

    We 2 wx  Δt   k 1 k k k wi  wi    B1 1  ecos (c1t )   B2 cos  c2t     2  m  1  m  1    wx   R    xR      We 

 1    wx  R 

x  



2

n 1 2

    We 2  1   m  1  m    1    wx  R2    R     

M

R2



 

2

  

ED

1  m  wx 

2

  

n 1 2

2  

n 1 2

     

   2 k  wxx  M wi  ,     (48)

The finite difference representation of prescribed conditions is given by  0,

PT

wi1

wNk 1  0, 0

CE

w1k

at t  0, at x  1,

(49)

at x  k (cathetered radius).

The numerical solution is sought for N+1 uniformly discrete points xi ,  i  1, 2, ..N  1 with a space

AC

grid size Δx  1

N 1

at the time levels tk   k  1 Δt , where Δ𝑡 is the small increment in time. To

obtain the accuracy of the order ~ 107 , we have taken the following step sizes: Δx  0.025 and

t  0.00001.

5. Results and discussion In

this

section

graphical

results

are

displayed

for

the

following

set

of

parameters:

d  0.5, l0  1.0, L  2.7,   0.8, 0 = 0.56,  = 0.0345,   0.0,  p  2 f p ,   0.25, k  0.1.

11

ACCEPTED MANUSCRIPT

Fig. 3 demonstrates the dimensionless velocity profiles of blood at different locations of the arterial segment. Panel (a) is plotted for a specific set of parameters. Each parameter of this specific set is varied to produce plots in panels ((b)-(h)) at the same cross-sections as taken in panel (a). In this way panel (a) is considered as standard and other panels are compared with it in order to observe the effects of various parameters. In panel (b) the velocity profiles are shown by changing the radius of the catheter. This panel when compared with panel (a) indicates that the effect of increasing catheter radius is to decrease the

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magnitude of velocity significantly at each cross-section. Panel (c) illustrates the effects of increasing critical stenotic height on velocity profile. It is evident from the comparison of panel (c) and panel (a) that blood flow velocity decreases by increasing the critical height of the stenosis. The effects of power law index n on velocity profile at various location of the artery are shown in panel (d). It is seen that an increase in n decreases the flow velocity of blood. The parameter We which control the viscoelastic

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behavior of blood also bears the potential to significantly affect the velocity profile at each cross –section. It is observed through comparison of panel (a) and panel (e) that velocity profile of blood is an increasing function of Weissenberg number (We). In panel (a) the profiles are shown at the time instant t =0.3 which belongs to systolic phase. For a time instant which belong to diastolic phase i.e. t =0.45 the velocity profile are shown in panel (f). In diastolic phase pressure gradient fall down and as a result of that the magnitude of velocity at each cross-section reduces. The velocity profiles at different cross-sections of the

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artery also show decreasing behavior by increasing the strength of the magnetic field. This behavior of velocity can be readily observed by a comparison of panel (a) with panel (h). The effects of amplitude of

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body acceleration on velocity profiles at various cross-sections can be observed by comparing panels (a) and (h). The comparison shows that an increase in amplitude of body acceleration increases the magnitude of velocity. A general observation from Fig. 3 is that the rheological parameters of blood, the

PT

catheter radius, the amplitude of body acceleration, the magnetic field alter the magnitude of velocity in both stenotic and non-stenotic region. However, the geometric parameter of stenosis  effects the velocity

CE

profile only in the stenotic region. The decrease in the magnitude of velocity by inserting a catheter is of major concern to the clinicians. This may result in some complications during the treatment/diagnostic

AC

procedure. However, the non-Newtonian rheology of the blood could be exploited to tune the velocity profile in a manner so that the effects of catheter on flow velocity are minimum. Indeed, in case of Newtonian fluid, the flow velocity is only function of geometrical parameters of the stenosis. The only way to modify, for example, the flow velocity is to change the geometrical parameters of the stenosis which is of course not possible. However, for a non-Newtonian fluid the velocity in addition to the geometrical parameters of fluid is controlled by the mechanical properties of the blood. Thus the nonNewtonian feature of blood allows the flow velocity to be tuned without modification of geometrical parameters of the stenosis. It is worth mentioning that our approach basically follows the approach of 12

ACCEPTED MANUSCRIPT

Mandal [27] and that is why our results are in accordance with his results. In fact it is shown by Mandal [27] that axial velocity get reduced to a considerable extent in case of a steeper stenosis. Thus following Mandal [27], we have prescribed the pressure gradient of sinusoidal type instead of prescribing the flow rate. In this way flow rate in our study is unknown. Since pulsatile pressure gradient is the main source of flow in our case therefore no flow velocity is imposed at the inlet. On the contrary, Tzirtzilakis [28] has prescribed the flow rate at inlet and calculated the stream function (consequently the velocity). His study

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shows different flow jets and downstream recirculation zones. Similar results are reported by Lee et al [29] by prescribing the Poiseuille type flow at the inlet, where apart from different flow jets and downstream recirculation it is also shown that axial velocity increases in the case of narrowing of stenosis. From the above discussion it concluded that increase of axial velocity in the stenotic region is attributed to the prescription of flow rate at the inlet. On the other hand, the decrease of axial velocity in

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the stenotic region is attributed to the prescription of pressure gradient. It is remarked here that fixing the correct boundary conditions at inlet and outlet in term of flow rate or velocity is an active subject of research. We refer the reader to recent articles by Guerra et al. [30] and Elia et al. [31] where data assimilation techniques based on variational approach are used to address the issue. The time series of flow rate at z = 0.77 (where the maximum height of the stenosis is observed) for

M

various values of emerging parameters is shown in Fig. 4. The solid line curve corresponds to a specific set of parameters i.e. We = 1.8, n = 0.8, M = 0.5, k = 0.1, 𝛿 = 0.1, B2 =2. Each parameter from this set except B2 is varied to produce all the other curves in Fig. 4. For instance the dashed curve is generated by

ED

changing δ. It is noticed that due to pulsatile nature of flow, the flow rate also oscillates with time. For initial times its behavior is not periodic. However, it achieves periodicity after t = 2.3sec. The amplitude

PT

of oscillations shows quantitative variations by changing n, We, δ and M. In fact it is seen by examining the other curves that flow rate decreases by increasing n, M, δ and k, while it exhibits opposite trend by

CE

increasing We.

The time series of WSS at stenotic throat for some specific values of We, n, M, k and δ is shown through

AC

Fig. 5. Various curves in Fig. 5 are compared according to the rule as used in Fig. 4. It is observed from this figure that WSS is found to decrease by increasing critical height of stenosis, strength of magnetic field, radius of catheter and Weissenberg number. However, it magnitude increases by increasing amplitude of body acceleration. The time series of resistive impedance at z = 0.77 corresponding to the critical height of stenosis is calculated using Eq. (44) and shown in Fig. 6 for various parameters. A quantitative comparison of solid line curves with other curves indicates that impedance increases by increasing magnetic field strength,

13

ACCEPTED MANUSCRIPT

catheter radius, critical height of stenosis and power-law index. However, it follows opposite trend by increasing Weissenberg number. It is also observed during simulations that the magnitude of impedance at a location in overlapping region of stenosis (z = 1.26) is less than its magnitude at stenotic throat (z = 0.77). The blood flow patterns for different values of involved parameters are shown in Fig. 7. Panel (a) shows

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the flow pattern for specific values of We, n, M, δ, and t. This panel confirms the appearance of circulating bolus of fluid in the stenotic region. By increasing n from 0.8 to 0.9, the size and circulation of bolus increases as evident from panel (b). A comparison of panel (a) and panel (c) discloses the potential of applied magnetic field to reduce the strength of the circulating region. The effects of Weissenberg number on circulating bolus of fluid can be observed by comparing panel (a) and panel (d). It is noted that the strength of circulation region increases by increasing Weissenberg number. A comparison of flow

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patterns of panel (a) and panel (e) indicates that the outer most streamline of circulating region in nontapered artery splits in case of converging artery. Moreover, there is an appearance of small eddy in the downstream region for converging artery. Panel (f) shows flow pattern for a diverging artery. When compared with panel (a), this panel also indicates the splitting of the outer most streamline and small eddy in upstream region. The effects of critical height of stenosis on flow pattern can be understood from

M

a comparison of panel (a) and panel (g). It is seen that strength of circulating region increases by increasing the stenotic height. The strength of circulation bolus of fluid also varies with the passage of

ED

time. In fact the bolus size decreases in systolic phase while in diastolic its behavior is reversed panel (h). The effect of catheter radius on blood flow pattern can be well understood by comparing Figs. 7(a) and 7(i). It is interesting to note that the size and circulation of bolus increases by increasing the catheter

PT

radius.

CE

6. Concluding remarks

A mathematical model for unsteady magneto-hydrodynamics blood flow through a catheterized stenotic vessel is analyzed. The constitutive equation of Carreau model is used to represent blood rheology. A

AC

numerical solution of the governing initial-boundary value problem is obtained employing the finite difference method. The velocity and volumetric flow rate of blood, arterial wall shear stress, impedance and streamlines of flow are analyzed quantitatively for geometrical parameter of stenosis, rheological parameters of blood, catheter radius and magnetic field. It is found that blood velocity and flow rate decrease with increasing the catheter radius. On the contrary, the impedance increases with increasing the catheter radius. Moreover, the strength of recirculating zones appearing in the stenotic region is also an increasing function of catheter radius. These observations may have certain clinical implications. In fact it

14

ACCEPTED MANUSCRIPT

is desired in any clinical procedure regarding treatment/diagnosis of atherosclerosis to avoid the reduction of flow rate and to decrease the impedance (resistance to flow) due to insertion of a catheter. The present study discloses two remedies to avoid such effects. One remedy, of course, is to manufacture the catheters of very thin radius and the other remedy is to tune the rheology of the blood. The later cannot be proposed by incorporating Newtonian fluid in the flow analysis. Moreover, power-law model gives the choice to tune the velocity of blood only by exploiting the shear-thinning nature of the blood. However, the

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Carreau model gives additional solution i.e., to use the viscoelastic nature of blood to tune the velocity of blood. In the end we would like to mention that the present analysis has certain limitations. In fact a more detailed analysis, without using mild stenosis condition, which encompasses wall properties, two- or three-dimensional aspects of the flow etc. may be more helpful in bringing out the realistic results.

Acknowledgments

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The valuable suggestions of the anonymous reviewer are greatly appreciated. The second author A. Zaman is grateful to the HEC for financial assistance.

References

[1] H. Kanai, M. Iizuka, K. Sakamoto, One of the problems in the measurement of blood pressure by catheter-insertion: Wave reflection at the tip of the catheter, Med. Bio. Eng., 8 (1970) 483-496.

Bioeng., 22 (1992) 337–340.

M

[2] L. H. Back, T. A. Denton, Some arterial wall shear stress estimates in coronary angioplasty, Adv.

ED

[3] L. H. Back, Estimated mean flow resistance increase during coronary artery catheterization, J. Biomech., 27 (1994) 169–175.

[4] L. H. Back, E.Y. Kwack, M. R. Back, Flow rate–pressure drop relation in coronary angioplasty:

PT

catheter obstruction effect, J. Biomech. Eng. Trans., 118 (1996) 83–89. [5] A. Sarkar, G. Jayaraman, Correction to flow rate–pressure drop relation in coronary angioplasty:

CE

steady streaming effect, J. Biomech. 31 (1998) 781. [6] R. K. Dash, G. Jayaraman, K.N. Mehta, Flow in a catheterized curved artery with stenosis, J. Biomech. 32 (1999) 49-61.

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[7] N. Filipovic, D. Nikolic, I. Saveljic, Z. Milosevic, T. Exarchos, G. Pelosi, O. Parodi, Computer simulation of three-dimensional plaque formation and progression in the coronary artery, Compu. Fluids 88 (2013) 826-833. [8] S. Nikolov, S. Stoytchev, A. Torres, J.J. Nieto, Biomathematical modeling and analysis of blood flow in an intracranial aneurysm, Neurol. Res. 25 (2003) 497-504. [9] W. Ruan, A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model, J. Math. Anal. Appl. 343 (2008) 778-798.

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[10] C. M. Rodkiewicz, P. Sinha, J. S. Kennedy, On the application of a constitutive equation for whole human blood, J. Eng. Math. 42 (2002) 1–22. [11] A. M. Gambaruto, J. Janela, A. Moura, A. Sequeira, Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology, Math. Biosci. Eng. 42 (2011) 1–22. [12] R. K. Dash, G. Jayaraman, K. N. Metha, Estimation of increased flow resistance in a narrow catheterized artery – A theoretical model, J. Biomech., 29 (1996) 917–930.

A mathematical model, Appl Math. Mod. 31 (2007) 1497–1517.

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[13] D. S. Sankar, K. Hemalatha, Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries –

[14] J. V. R. Reddy, D. Srikanth , S.V.S.S.N.V.G. K. Murthy, Mathematical modelling of pulsatile flow of blood through catheterized unsymmetric stenosed artery—Effects of tapering angle and slip velocity, Eur. J. of Mech. B/Fluids, 48 (2014) 236–244.

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[15] V. P. Srivastava, R. Rastogi, Blood flow through a stenosed catheterized artery: Effects of hematocrit and stenosis shape, Comp. Math. Appl., 59 (2010) 1377-1385.

[16] S. Nadeem, S. Ijaz, Nanoparticles analysis on the blood flow through a tapered catheterized elastic artery with overlapping stenosis. Eur. Phys. J. Plus 129 (2014) 249.

[17] D.S. Sankar, K. Hemalatha, A non-Newtonian fluid flow model for blood flow through a catheterized artery—Steady flow. Appl. Math. Model. 31 (2007) 1847-1864.

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[18] A. Jonasova J. Vimmr, Numerical simulation of non-Newtonian blood flow in bypass models, PAMM. Proc. Appl. Math. Mech., 8 (2011) 407–423.

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[19] W. Y. Chan, Y. Ding, J. Y. Tu, Modeling of non-Newtonian blood flow through a stenosed artery incorporating fluid structure interaction, ANZIAM .J., 47 (2007) 507–523.

(2005) 398–405.

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[20] S. S. Shibeshi, W. E. Collins, The rheology of blood in a branched arterial system, Appl. Rheol., 15

[21] D. S. Sankar, U. Lee, Two-fluid non-linear model for flow in catheterized blood vessels, Int. J. Non-

CE

Linear Mech. 43 (2008) 622 – 631.

[22] V. Khandelwal, A. Dhiman, L. Baranyi, Laminar flow of non-Newtonian shear-thinning fluids in a

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T-channel. Comput. Fluids 108 (2015) 79-91. [23] M. Massoudi and T. X. Phuoc, Pulsatile Flow of a blood using a modified second grade fluid model, Compu. Math. Appl., 56 (2008) 199-211. [24] F. Yilmaz, and M. Y. Gundogdu, A critical review on blood flow in large arteries; relevance to blood rheology , viscosity models, and physiologic conditions, Korea-Australia Rheo. J., 20 (2008) 197211. [25] A. C. Burton, Physiology and Biophysics of the Circulation, Introductory Text, Year Book Medical Publisher Chicago, (1966).

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[26] S. C. Ling, H. B. Atabek, A nonlinear analysis of pulsatile flow in arteries, J Fluid Mech. 55 (1972) 493–511. [27]. P. K. Mandal, An unsteady analysis of non-Newtonian blood through tapered arteries with a stenosis, Int. J. Nonlinear. Mech. 40 (2005) 151–164. [28]. E. E. Tzirtzilakis, Biomagnetic fluid flow in a channel with stenosis, Physica D 237 66-81 (2008). [29] J.S. Lee and Y.C. Fung, Flow in locally constricted tubes at low Reynolds number, J. Appl. Mech. 37

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9-16 (1970).

[30]. T. Guerra, J. Tiago, A. Sequeira, Optimal control in blood flow simulations, Int. J. of Non-Linear Mech., 64 (2014) 57-59.

[31]. M. D' Elia, M. Perego, A. Veneziani, A variational Data Assimilation procedure for the

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incompressible Navier-Stokes equations in hemodynamics, J. Sci. Comput., 52 (2) (2011) 340-359.

Figure captions:

Fig. 1. Geometry of the overlapping stenotic artery with a catheter.

Fig. 2. Geometry of the stenotic artery with different tapering angles.

M

Fig. 3. Velocity profile at different cross sections of the artery. Fig. 4. Dimensionless flow rate profiles for: B1  2, B2  2.

ED

Fig. 5. Dimensionless shear stress profiles for: B1  2, B2  2. Fig. 6. Dimensionless resistance to flow profiles for: B1  2, B2  0.

CE

PT

Fig.7. Blood flow patterns for: B1  2, B2  2.

AC

Figures:

Fig. 1

17

ACCEPTED MANUSCRIPT

r

d = 0.5



l =1 0



a R(z) ka

L

Z

Fig. 2

0

Catheter

AC

CE

PT

ED

R(z)

M



a

Fig. 3

>0

l =1

d = 0.5 r

ka

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Catheter

18



=0

<0 L

Z

ACCEPTED MANUSCRIPT

We = 1.8, n = 0.8, M = 0.0, k = 0.1, 𝛿 = 0.1, B2 =2

We = 1.8, n = 0.8, M = 0.0, k = 0.3, 𝛿 = 0.1

(a)

1

(b)

1 0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.5

1

1.5

z

2

0

2.5

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r

r

0.8

0

0.5

(c)

1.5

2

2.5

(d)

1

0.6

0.6

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r

0.8

r

0.8

0.4

0.4

0.2 0

z

We = 1.8, n = 0.9, M = 0.0, k = 0.1, 𝛿 = 0.1

We = 1.8, n = 0.8, M = 0.0, k = 0.1, 𝜹 = 0.2 1

1

0.2

0

0.5

1

1.5

2

z

0

2.5

0

0.5

1

z1.5

2

2.5

We = 1.8, n = 0.8, M = 0.0, k = 0.1, 𝛿 = 0.1, t = 0.45

M

We = 0.5, n = 0.8, M = 0.0, k = 0.1, 𝛿 = 0.1

(e)

1

0.2 0

0.5

1

z 1.5

0.4 0.2 0

2

2.5

CE

0

PT

0.4

r

0.6

0.8 0.6

ED

r

0.8

0

(g)

r

r

0.4

0.2

0.2

0.5

1

z

1.5

2

2.5

(h)

0.6

0.4

0

z 1.5

0.8

0.6

0

1

1

AC

0.8

0.5

We = 1.8, n = 0.8, M = 0.5, k = 0.1, 𝛿 = 0.1

We = 1.8, n = 0.8, M = 0.0, k = 0.1, 𝛿 = 0.1, B2 = 0 1

(f)

1

2

0

2.5

19

0

0.5

1

z

1.5

2

2.5

ACCEPTED MANUSCRIPT

Fig. 4 1

We = 1.8, n = 0.8, M = 0.5, k = 0.1, 𝛿 = 0.1 We = 0.5 δ = 0.2 M = 0.0

0.8

𝑘 = 0.3 n = 0.9

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Q(t)

0.6

0.4

0 -0.1

0

0.5

1

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0.2

1.5

2

2.5

3

2

2.5

3

Time (t)

M

Fig. 5

We = 1.8, n = 0.8, M = 0.5, k = 0.1, 𝛿 = 0.1

1

ED

δ = 0.2 We = 0.1 M = 1.5 k = 0.3

0.8

𝜏𝑠

CE

0.4

PT

B2 = 0

0.6

AC

0.2

0 0.5

1

1.5

Time (t)

20

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Fig. 6 30

25

𝜆

20

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We = 1.8, n = 0.8, M = 0.5, k = 0.1, 𝛿 = 0.1 𝑀 = 0.0 We = 0.5 δ = 0.2 k =0.3 n = 0.9

15

5

0

0.5

1

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10

1.5

2

2.5

3

M

Time (t)

ED

Fig. 7

(b) We= 1.8, n = 0.9, M = 0.5, 𝛿 = 0.1, t = 0.3

PT

(a) We = 1.8, n = 0.8, M = 0.5, 𝛿 = 0.1, t = 0.3 1 0.9

0.8 0.7

0.5

AC

0.3

0.1 0.4

0.6

0.8

1

r

r

CE

0.7

1 0.9

0.6 0.5 0.4 0.3 0.2

1.2

1.4

1.6

1.8

0.1 0.4

2

0.6

0.8

1

1.2

1.4

1.6

1.8

z

z

(𝑐) We = 1.8, n = 0.8, M = 0.0, 𝛿 = 0.1, t = 0.3

(𝑑) We = 0.5, n = 0.8, M = 0.5, 𝛿 = 0.1, t = 0.3

21

2

ACCEPTED MANUSCRIPT

0.7

0.7

r

1 0.9

r

1 0.9

0.5

0.5

0.3

0.3

0.6

0.8

1

1.2

1.4

1.6

1.8

0.1 0.4

2

0.6

0.8

1

1.2

1.4

1.6

1.8

2

z

z

(𝑒) We = 1.8, n = 0.8, M = 0.0, 𝛿 = 0.1, 𝝃 = -0.05

CR IP T

0.1 0.4

(𝑓) We = 1.8, n = 0.8, M = 0.0, 𝛿 = 0.1, 𝝃 = 0.05

1

1

0.9

0.9

0.7

r

r

0.7

0.5

AN US

0.5

0.3

0.3

0.1 0.4

0.1

0.6

0.8

1

1.2

1.4

1.6

1.8

0.4

2

z

(𝑔) We = 1.8, n = 0.8, M = 0.0, 𝜹 = 0.2, t = 0.3

0.6

0.8

1

1.2

z

1.4

1.6

1.8

2

(ℎ) We = 0.5, n = 0.8, M = 0.5, 𝛿 = 0.1, t = 0.45

1

1

0.9

r

M

0.7

0.3

0.1 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.3

0.1 0.4

2

0.6

0.8

1

0.9

0.7

r

AC

CE

(𝑖) We = 1.8, n = 0.8, M = 0.0, 𝒌 = 0.3, t = 0.3

0.5

0.3 0.4

0.6

0.8

1

1.2

z

PT

z

0.5

ED

0.5

0.7

r

0.9

1

1.2

z

22

1.4

1.6

1.8

2

1.4

1.6

1.8

2