Saurabh Minor

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Minor Project ReportModelling of Flow in Blood Vessels

Project Supervisor: Dr. Gaurav Sharma

Submitted by:

Saurabh Kr. Verma 08112033

Jubin Shah - 08112012


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CHECKLIST

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Signature of student:

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4. Whether the report includes the following items? Title page Yes / No Checklist Yes / No Certificate Yes / No Abstract Yes / No Table of contents Yes / No List of Figures Yes / No List of Tables Yes / No Nomenclature Yes / No Introduction Yes / No Literature Survey/Review Yes / No Description of Experimental Setup / Software / Procedure Yes / No Results & Discussion Yes / No Conclusion Yes / No Scope for future work Yes / No References Yes / No

5. Whether the references are written in standard acceptable format? Yes / No6. Overall rating of the quality of report: (a) Excellent (b) Good (c) Average (d) Poor (e)

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Project Supervisor: Dr. Gaurav Sharma


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CERTIFICATE

This is to certify that the work presented in this Project Report is solely carried out by us

under the guidance of Dr. Gaurav Sharma and has not been submitted by me in part or full

form for any other degree/diploma/certificate of any other Institute/ Organization/University.

_______________________

Saurabh Kr. Verma Jubin Shah

This is to certify that the project report titled Modeling of Flow in Blood Vesselsbeing

submitted by Saurabh Kr. Verma and Jubin Shah of Department of Chemical Engineering, IIT

Roorkee, is a bonafide work carried out by them under my guidance and supervision.

________________________

Dr. Gaurav Sharma

Assistant Professor

Indian Institute of Technology, Roorkee

Date: ____________


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Abstract

In this text, a Hagen-Poiseuille model is presented for the flow of blood in blood vessels. A

circular Poiseuille model is used to simulate the flow. The flow of blood is assumed to be

non-Newtonian. Considering the nature of blood and blood vessels various effects are

demonstrated using the model. Following model based on certain assumptions regarding the

flow and behaviour of blood vessels. The model is used to determine the bifurcation of blood

vessels with the application of Murrays law. The Hagen -Poiseuille model is used for

understand the reflection of pressure waves at Arterial Junctions.


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TABLE OF CONTENTS

Section Title Page No.

Abstract 4

Table of Contents 5

Nomenclature 6

1 Introduction 7

2 Literature Review 8

2.1 Physical properties of Blood 8

2.1.1 Viscosity of blood 8

2.1.2 Fahraeus-Lindqvist Effect 10

2.2 Total Peripheral Resistance 11

3

3.1

Modelling of Flow in Blood Vessels

Hagen-Poiseuille flow Model

13

13

3.2

3.3

Accounting Fahraeus-Lindqvist Effect in the model

Effect of Tube Wall Elasticity on Hagen Poiseuille Model

16

18

3.4 Pulsatile Blood Flow Theory 19

4 Conclusions 21

5 References 21


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NOMENCLATURE

u

R

p

Velocity

Viscosity

Frequency of pulse wave

Kinematic viscosity

Resistance in Peripheral Resistance Unit(PRU)

Pressure in vessel

Plasma layer thickness

Shear Stress

Shear Rate

y Yield Stress

CF Fibrinogen content(in gm. per 100 ml)

H Hematocrit Fraction

p Plasma viscosity

Kc Casson viscosity

Q Flow Rate

C Subscript C, Core

P Subscript P, Plasma

T Subscript T, Tube

a Radius of capillary


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1. IntroductionBlood shows anomalous viscous properties. The anomalous behaviour of blood is principally

due to the suspension of particles in plasma. The plasma solution in the blood obeys the

linear Newtonian model for viscosity. However, blood as a whole is often considered as non-

Newtonian fluid, particularly when the characteristic dimension of the flow is close to the cell

dimension. Due to its complexity and anomalous behaviour, it is very difficult to analyse it.

The two types of anomaly are due to low shear and high shear effects. When blood flows

through larger diameter arteries at high shear rates, it behaves like a Newtonian fluid. The

apparent viscosity of blood decreases with decreasing blood vessel diameter, when

measurements are made in capillaries of diameter less than 300 m. Pries et al. studied the

effect of the tube diameter and the hematocrit ratio on the blood viscosity and found that for

tube diameters greater than 1 mm, the blood viscosity is independent of the diameter while

for tube diameter less than 1 mm, the blood viscosity is strongly dependent on the tube

diameter. They also reported the viscosity increases non-linearly with the hematocrit.

The study of the behaviour of blood flow in the blood vessels provides understanding on

connection between flow and the development of diseases such as atherosclerosis,

thrombosis, aneurysms etc. and how the flow dynamics is changed under these conditions.

The understanding of the flow dynamics past prosthetic devices such as heart valves, vascular

grafts and artificial hearts will help improving the design of the implants. The functioning of

several extra-corporeal flow devices such as blood oxygenators and dialysis machines, which

are commonly used in modern medicine, can be improved if blood flow behaviour through

the devices is well understood.

The Hagen-Poiseuille model which describes the laminar flow in a rigid straight circular tube

is applied to explain the entrance effects and branching in the blood flow in vessels.


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2. Literature Review

2.1 Physical Properties of Blood

The whole blood consists of formed elements that are suspended in plasma. The plasma is a

dilute electrolyte solution containing about 8% by weight of proteins. About 45% by volume

of whole blood consist of formed elements and about 55% of plasma in the normal human

blood. The formed elements of blood are red blood cells (95%), white blood cells (0.13%)

and platelets (4.9%). The diameter of red blood cell is about 8.5 m at the thickest portion

and about 1 m at the thinnest portion. Its membrane is flexible and the cell can pass through

capillaries of diameter as small as 5 m assuming a bent shape.

2.1.1 Viscosity of BloodViscosity is an important property of a flowing fluid. The viscosity of blood depends on the

viscosity of the plasma and its protein content, the hematocrit, the temperature, the shear rate

and the narrowness of the blood vessel or capillary. The white blood cells and platelets have

not much significant effect on viscosity owing to their lower concentration. The effect of

narrowness of vessel diameter on viscosity of blood is known as fahraeus-lindqvist effect.

The viscosity of plasma depends on its protein composition and ranges 1.1-1.6 centipoise.

The viscosity of blood is about 3.2 cP at 45% physiological hematocrit. At a 60% hematocrit

level, the relative viscosity of blood is 8 with respect to water (viscosity of water at room

temperature is about 0.8 cP). The dependency of viscosity on flow rate in vessels is quite

complicated. The blood viscosity increases with decreasing temperature which is

approximately 2% for each C.

The shear stress and shear rate has significant effect on blood viscosity. Relation between

and is complicated over the whole blood due to following reasons. In a blood volume at

rest, above a minimum hematocrit of about 5-8%, blood cells form a continuous structure.

Yield stress y (finite stress) is required to break this continuous structure into a suspension of


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aggregates in the plasma. This yield stress depends also on the concentration of plasma

proteins, in particular fibrinogen. An empirical correlation is given by the following

expression:

y = (H - 0.1) (CF + 0.5) (Eqn 2.1)

Where H>0.1 and 0.21< CF < 0.46. The yield stress (y) is in the range 0.01 < y < 0.06

dynes/cm2

for 45% H. For whole blood at lower shear rates, < 200 sec-1, variation of with

is nonlinear. This behaviour at low is non-Newtonian. At higher shear rates, < 200 sec-1

,

The relationship is linear and the viscosity approaches an asymptotic value of about 3.5 cP.

The equations describing relationship between and are described by Casson model and it

is expressed as follows:

p) =kc + y/p) (Eqn 2.2)Another expression based on least square fit is given by Whitmore is as follows:

p) =1.53 + 2.0 (Eqn 2.3)

In blood vessels smaller than 500 m in diameter, apparent viscosity starts to effect by

inhomogeneous nature of blood.

Fig 2.1: A least square fit of apparent viscosity as a

function of shear rate in Casson Model


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2.1.2 Fahraeus-Lindqvist Effect

It was observed that in very small diameter tubes (15 m to 500 m) the apparent viscosity of

blood has a very low value. The viscosity increases with the increase in tube diameter and

approaches an asymptotic value at tube diameters larger than about 0.5 mm. This

phenomenon is referred to as the Fahraeus-Lindqvist effect this is because when blood of

constant haematocrit flows from a large vessel into a small vessel, the haematocrit in the

small vessel decreases as the tube diameter decreases. As the blood flows through a tube, the

blood cells tend to rotate and move towards the centre of a tube. Hence, a cell-free layer

exists near the wall. In tubes with small diameter, the area of the cell-free zone is comparable

to the central core. The net effect of the cell-free zone with a lower viscosity (viscosity of

plasma alone) is to reduce the apparent viscosity of flow through the tube. As the tube

diameter increases, the effect of the cell-free zone reduces and the viscosity coefficient

approaches the asymptotic value.

We consider laminar blood flow in straight, horizontal, circular, feed and capillary tubes.

Now based on law of conservation of blood cells we establish a relationship between QF, QC,QP, HF, HT, HC, and a.

Fig 2.2: The Fahraeus Effect

Fig 2.3: The variation of the blood

viscosity with the tube diameter,

illustrating the Fahraeus-Lindqvist

effect.


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QFHF = QCHC, QC + QP = QF, and HTa2

= HC (a-)2

(Eqn 2.4)

2.2 Total Peripheral Resistance ConceptArterioles are the primary site of vascular resistance, and blood flow distribution to variousregions is controlled by changes in resistances offered by various arterioles. To quantify the

resistance of the arterioles in an average sense total peripheral resistance concept is

introduced. The vascular resistance is given as the pressure difference over the volume flow.

R=p/Q (Eqn 2.5)

Both pressure difference and flow rate are time averaged. Now, pressure difference is the

difference between aortic valve and right atrium time averaged pressures.

p = pA - pV (Eqn 2.6)

With pV= 0, p = pA , time averaged arterial pressure. Then pA is estimated as:

PA = QR (Eqn 2.7)

PA = (1/3) pS + (2/3) pD = pD + (1/3) (pSpD) (Eqn 2.8)

Where, S and D subscripts are systolic and diastolic pressures respectively. A small change in

the radius of the vessel will affect the resistance to flow considerably. The mean arterial

Fig 2.4: Vessel diameter, total cross-

section area, and velocity of flow.


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pressure is normally about 100 mmHg and has fallen very little in the smallest arteries. In the

capillaries it is generally agreed to be about 30 mmHg at the arterial end and about 15 mmHg

at the venous end. The pressure in large veins will only be a few mmHg. Most of the fall (up

to 60 mmHg) will occur in arterioles less than 200 m in diameter. As the resistance is

proportional to the drop in mean pressure, it is apparent that the resistance of the arterioles

constitutes the largest proportion of whole. With the alteration of muscle tension, which is

controlled by the autonomic nervous system, the arterioles can be distended or contracted

selectively to vary the amount of flow into the various segments of the body.


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3. Modelling of Blood FlowIn the human circulatory system, blood flow is general unsteady. The systolic and diastolic

pumping makes it pulsatile in most regions. In pulsatile flow, the flow has a periodic

behaviour and a net directional motion over the cycle. Pressure and velocity profiles vary

periodically with time, over the duration of a cardiac cycle. A dimensionless parameter called

the Womersley number, , is used to characterize the pulsatile nature of blood flow, and it is

defined by:

= a (Eqn 3.1)

The Womersley number is a composite parameter of the Reynolds number (Re=2au/, and

the Strouhal number (St=2a /u). The square of the Womersley number is called the Stokes

number. The Womersley number is the ratio of unsteady inertial forces to viscous forces in

the flow. With a normal heart rate of 72 beats per minute, = (2 72/60) =8 rad/s.

3.1. Hagen-Poiseuille Model of FlowIn this model, blood flow is modelled as a laminar, steady, incompressible, fully developed

flow of a Newtonian fluid through a straight, rigid, cylindrical, horizontal tube of constant

circular cross section. In the normal body, blood flow in vessels is generally laminar.

However, at high flow rates, particularly in the ascending aorta, the flow may become

turbulent at or near peak systole. Disturbed flow may occur during the deceleration phase of

the cardiac cycle. Turbulent flow may also occur in large arteries at branch points. However,

under normal conditions, the critical Reynolds number, Rec, for transition of blood flow in

long, straight, smooth blood vessels is relatively high, and the blood flow remains laminar.

Let us make some approximate analysis. The aorta is about 40 cm long and the average

velocity u of flow in it is about 40 cm/s. The lumen diameter at the root of the aorta is d=25

mm, and the corresponding Re= u d/ is 3000. The maximum Reynolds number may be as

high as 9000. The average value for Re in the vena cava is also about 3000. Arteries have

varying sizes and the maximum Re is about 1000. For Newtonian fluid flow in a straight

cylindrical rigid tube, Rec is about 3000. However, aorta and arteries are distensible tubes,

and this Re criterion does not apply. In the case of blood flow, laminar flow conditions


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generally prevail even at Reynolds numbers as high as 10,000. In summary, the laminar flow

assumption is reasonable in many cases.

Several assumptions in deriving the Poiseuille law were made and the validity of these

assumptions in models describing blood flow should be critically examined. The conditions

under which Poiseuilles equation applies are the following:

1. The liquid is homogeneous with constant viscosity. Blood is a suspension of particlesbut, in tubes in which the internal diameter is large compared with the size of the red

blood cells, it behaves as a Newtonian liquid.

2. The liquid does not slip at the wall. This is the assumption that velocity is zero whenr = R.

3. The flow is laminar (the liquid is moving parallel to the walls of the tube). There isno experimental evidence of sustained turbulence in the human circulation in the

absence of diseased states.

4. The rate of flow is steady. As the flow in all large arteries is markedly pulsatile, it isclear that Poiseuilles equation cannot be applied in these vessels.

5. The tube is long compared with the region being studied. Close to the inlet of a tube,flow has not yet become established with the parabolic velocity profile. Similarly, the

Fig 3.1: Poiseuille Flow


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flow passes through branching points and curved sections, where the flow is

appropriately altered. Clearly, the assumption of fully developed flow is not valid.

6. The tube is cylindrical in shape. Most arteries of the systemic circulation are circularin cross-section, but many veins and the pulmonary arteries tend to be elliptical. The

requirement of parallel walls is probably never exactly met in blood vessels because

individual arteries taper as they progress toward the periphery.

7. The tube is rigid; the diameter does not vary with the internal pressure. Blood vesselsare viscoelastic structures, and their diameter is a function of pressure. The

interaction between the distensible arterial wall and the flowing blood is an important

factor in the description of the flow dynamics.

All arteries are not circular tubes but may have tapering cross sections, while the veins and

pulmonary arteries have elliptical cross-section. However, there remain many situations

where the Hagen-Poiseuille model is reasonably applicable. Considering the vessel in

cylindrical coordinates (r, , x), the axial flow velocity, u (which is a function of radius), in a

pipe of radius a, is given by:

u(r) = {(r2a

2)/4} (dp/dx) (Eqn 3.2)

In a fully developed flow, the pressure gradient (dp/dx) is considered as constant, and it is

expressed in terms of the overall pressure difference:

(dp/dx) = -p/L = - (p1p2)/L (Eqn 3.3)

And now after substituting this expression in velocity expression we get:

u(r) = (a2p/4L) (1r

2/a

2) (Eqn 3.4)

Maximum velocity occurs at the centre of vessel and is given by:

umax = {a2p/4L} (Eqn 3.5)

The volumetric flow rate can be calculated as:

Q =

= - {a4/8} (dp/dx) = {a4/8} (p/L) = (umax/2) a

2(Eqn 3.6)

The above equation is hagen-poiseuille equation of model. The average velocity over the

circular conduit calculated:


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V = Q/A = Q/a2 = umax/2 (Eqn 3.7)

The shear stress at the tube wall may be calculated using the expression:

xr|r=a= = - (du/dx)|r=a = -(a/2)(dp/dx) = (a/2)(p/L) (Eqn 3.8)

The Hagen-Poiseuille equation and its derivatives are most applicable to flow in the muscular

arteries. Modifications are likely to be required outside this range. With the results for the

Hagen-Poiseuille flow, we have from:

Total Peripheral Resistance = R = p/Q = 8L/a4 (Eqn 3.9)

Following equation shows that peripheral resistance to the flow of blood is inversely

proportional to the fourth power of vessel diameter.

3.2. Accounting Fahraeus-Lindqvist effect in the modelThe flow of blood in vessels is divided into two differentiated regions:

a) A central core containing RBCs with axial flow velocity ucand viscosity cb) A cell free plasma layer of thickness with axial flow velocity u pand viscosity p

The shear stress is then estimated by the following expression:

xr = - c (duc/dr) = - (r/2) (p/L) (Eqn 3.10)

Now applying the boundary conditions;

duc/dr = 0, r = 0

Fig 3.2: Fahraeus-Lindqvist Effect


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xr|c = xr|p , r = (a)

Shear stress distribution in the plasma region is;

xr

= - p (dup/dr) = - (r/2) (p/L) (Eqn 3.11)

And boundary conditions are,

uc

= up

, at r = (a)

up

= 0 , at r = a

Now, uc

and up

are determined in core and plasma region by integrating equation (3.10) and

(3.11) using boundary conditions respectively.

uc

= (a2/4p) (p/L) [1 {(a)/a}

2(p/c) (r/a)

2+ (p/c) {(a)/a}

2] , for 0 a

(Eqn 3.12)

up

= (a2/4p) (p/L) [1 (r/a)

2] , for (a) a (Eqn 3.13)

The volumetric flows in plasma and core region are obtained by following integration:

Qp = 2

u

pdr = (p/8pL) [a

2(a)

2]

2(Eqn 3.14)

and,

Qc = 2

u

cdr = (a2p/4pL) [a

2{1(p/2c)} {(a)

4/a

2}] (Eqn 3.15)

Total flow rate in the vessel is obtained by summing the two flow rates.

Q = Qc + Qp = (a2p/4pL) [1(1/a)

4(1 - p/c)] (Eqn 3.16)

Now, comparing this expression with the equation (3.6) we get apparent viscosity which

satisfies the Hagen-Poiseuille model for the flow in blood vessels.

app = p [1(1 - /a)4

(1 - p/c)]-1

(Eqn 3.17)

Since, the thickness of plasma layer is very small as compared to radius of vessel so in the

expression (1/a)

4

is expanded using binomial expansion, i.e.


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/a


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On integrating we get following stress,

= {(p(x)pe) a(x)} /h (Eqn 3.22)

Hooks law gives circumferential strain as,

e = /E = (a(x)a0)/a0 = {a(x)/a0}1 (Eqn 3.23)

The radial stress rr is neglected because it is too small as compared to in thin-walled

vessel. The wall is considered thin i.e. (h/a)


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immediately goes into systemic circulation. A part of the blood is used to distend the aorta

and a part of the blood is sent to peripheral vessels. The distended aorta acts as an elastic

reservoir, the rate of outflow from which is determined by the total peripheral resistance of

the system. As the distended aorta contracts, the pressure diminishes in the aorta. The rate of

pressure decrease in the aorta is much slower compared to that in the heart chamber. In other

words, during the systole part of the heart pumping cycle, the large fluctuation of blood

pressure in the left ventricle is converted to a pressure wave with a high mean value and a

smaller fluctuation in the distended aorta. In the Windkessel theory, blood flow at a rate Q(t)

from the left ventricle enters an elastic chamber (the aorta) and a part of this flows out into a

single rigid tube representative of all of the peripheral vessels. The rigid tube offers constant

resistance R equal to the total peripheral resistance that was evaluated in the Hagen-Poiseuille

model. From the law of conservation of mass, assuming blood is incompressible:

Rate of Inflow into Aorta = Rate of change of volume of elastic chamber

+ Rate of outflow into rigid tube (Eqn 3.26)

Let the instantaneous blood pressure in the elastic chamber be p(t), and its volume be v(t).

The pressure on the outside of the aorta is taken to be zero. The rate of change of volume of

an elastic chamber is given by:

dv/dt = (dv/dp) (dp/dt) (Eqn 3.27)

In (3.27), the quantity (dv/dp) is the compliance, K, of the vessel and is a measure of the

distensibility. Compliance at a given pressure is the change in volume for a change in

pressure. Here pressures are always understood to be transmural pressure differences.

Compliance essentially represents the distensibility of the vascular walls in response to a

certain pressure. Also, from (16.9), the rate of flow into peripherals is given by (p(t)/R),

where we have assumed pv = 0. Therefore, (3.27) becomes

Q(t) = K(dp/dt) + {p(t)/R} (Eqn 3.28)

Above equation is in linear form and can be solved as:

p(t) = (1/K)e-t/RK

e/RKd + p0e

-t/RK (Eqn 3.29)

In (3.29), p0 would be the aortic pressure at the end of diastolic phase. A fundamental

assumption in the Windkessel theory is that the pressure pulse wave generated by the heart is


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transmitted instantaneously throughout the arterial system and disappears before the next

cardiac cycle. In reality, pressure waves require finite but small transmission times, and are

modified by reflection at bifurcations, bends, tapers, and at the end of short tubes of finite

length, and so on. We will now account for some of these features.

4. ConclusionBlood flow phenomena are often too complex that it would be possible to describe them

entirely analytically, although simple models, such as Poiseuille model, can still provide

some insight into blood flow. The understanding of governing laws that apply in the pulsatile

blood flow is crucial for my future work. For the planned experiment of dissolving blood

clots under physiological conditions of pulsatile flow and for building a model to describe

such flow through the clot channel the pulsatile flow dynamics and entrance effects should be

studied thoroughly.

8. References

[1] Pijush K.Kundu. Fluid Mechanics, Fifth Edition

[2] W.W. Nichols, M.F. ORourke. McDonalds Blood Flow in Arteries. Theoretical

experimental and clinical principles. Fifth Edition. Hodder & Arnold, London, 2005

[3] D. O. Cooney. Biomedical Engineering Principles. An Introduction to Fluid, Heat and

Mass Transport Processes. Marcel Dekker, New York, 1976

[4] R. L. Whitmore. Rheology of Circulation. Pergamon Press, New York, 1968

[5] C. G. Caro, T. J. Pedley, R. C. Scroter, W. A. Seed. The Mechanics of Circulation.

Oxford Medical Publications, Oxford, 1978

[6] Y. C. Fung. Biodynamics: Circulation. Springer-Verlag, New York, 1984

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