blood flow

BLOOD FLOWBarbara GrobelnikAdvisor: dr. Igor Serša

January 2008 Blood Flow 2

Introduction

The study of blood flow behavior:• Improving the design of implants (heart valves, artificial heart) and extra-corporeal flow devices (blood oxygenators, dialysis machines)• Understanding the connection between flow characteristics and the development of cardiovascular diseases (atherosclerosis, thrombosis)

CONTENTSCardiovascular physiologyPhysical properties of blood

Viscosity

Steady blood flow Poiseuille’s equation Entrance effects Bernoulli’s equation

Oscillatory blood flow Windkessel model Wommersley

equations


Cardiovascular Physiology

MAIN FUNCTIONS:• to deliver oxygen and nutrients to the cells• to remove cellular wastes and carbon dioxide• to maintain organs at a constant temperature and pH

• HEART: atrium, ventricles• BLOOD VESSELS: aorta, arteries, arterioles, capillaries, veinules, veins

left ventricle aorta organs and tissues right atrium

right ventricle lungs left atrium

mean diameter

[mm]

number of vessels

aorta 19 - 4.5 1

arteries 4 – 0.15 110.000

arterioles 0.05 2.7 ∙106

capillaries 0.008 2.8 ∙109


Poiseuille flow• Steady flow in a rigid cylindrical tube

– Pressure gradient– Viscous force

p 1 2

v

2 ( )

(2 )vr r

F r p p r

F rL r

21 2

2

2 1 2

2 21 2

4

4

10

( ) ln

( ) ( )

p p

L

p p

L

p pv v

r r r L

v r r A r B

v r R r

R4 1 2

0

2 1 2 max

8

102 8 2 2

2 ( )

( )

p p

L

Q p p vr

LR

Q v r rdr R

v R v

r

L

r

p1 p2

2 r

v

The forces are equal and opposite:

volume flow

average velocity

v(r=R)=0

v(r=0)≠∞


Poiseuille flow - assumptions• Newtonian fluid

– in large blood vessels (at high shear rates)

• Laminar flow– Reynold’s numbers below the critical value of about 2000

• No slip at the vascular wall– endothelial cells

• Steady flow– pulsatile flow in arteries

• Cylindrical shape– elliptical shape (veins, pulmonary arteries), taper

• Rigid wall– visco-elastic arterial walls

• Fully developed flow– entrance length; branching points, curved sections

x

x

x

x


Physical properties of bloodBLOOD =plasma + blood cells (55%) (45%)

PLASMA WHOLE BLOOD

density 1035 kg/m3 1056 kg/m3

viscosity

1.3×10-3 Pa s

3.5 × 10-3 Pa s

Reference values

electrolyte solution containing 8% of proteins

Red blood cells (95%)

White blood cells (0.13%)

Platelets (4.9%)

RBC:

8 μm

1 μm


Viscosity• Viscosity varies with samples

– variations in species– variations in proteins and RBC

• Temperature dependent– decrease with increasing T

• Blood– a non-Newtonian fluid at low shear rates (the agreggates of RBC)– a Newtonian fluid above shear rates of 50 s-1

– Casson’s equation0 /cK dv dr

In small tubes the blood viscosity has a very low

value because of a cell-free zone near the wall.

Fahraeus-Lindqvist effect


Fahraeus-Lindqvist EffectCell-free marginal layer model Core region μc , vc , 0rR- Cell-free plasma μp , vp , R-r R

The Sigma effect theory velocity profile is not

continuous small tubes (N red blood

cells move abreast)

the volume flow is rewritten

region near the wall

1p d dvr

L r dr dr

4

411 (1 / ) (1 / )

8 p cp

R pQ R

L

the volume flow

1/μ

rμc , vc

μp , vp

R

3

02

RpQ r dr

L

43

1

1( ) 1

2 8

N

n

p pRQ n

L L R

N concentric laminae, each of thickness ε

1/μ


Entrance length• The flow of fluid from a reservoir to a pipe

– flat velocity profile at the entrance point– the fluid in contact with the wall has zero velocity (‘no slip’)– retardation due to shearing adjacent to the wall– boundary layer (where the viscous effects are present)– acceleration in the core region to maintain the same volume

of flow– parabolic velocity profile FULLY DEVELOPED FLOW

2 1

2 12

( )

( )

visc

visc

d dvF A r r

dr dr

UF A r r

viscous force

- boundary layer thickness at zU - free stream velocity

2

2 1( )i

UF aV A r r

z inertial force

* a=U/t=U/(z/U)


Entrance length• equating the viscous and inertial force

k – proportionality constant derived from experiments, approximately 0.06

• the boundary layer thickness

• the entrance length (when =D/2 the flow becomes fully established)

2

2

U Uk

z

z

U

20

Uz kD

The above derivation is valid only for the flow originating from a very large reservoir, where the velocity profile at the entrance point is relatively flat. In other cases, the entrance length is shorter.

Pulsatile flow – the entrance length fluctuates


Application of Bernoulli Equation

• Flow trough stenosis

– v2 > v1

– p2 < p1 : caving or closing of the vessel

– decrease in v2

– reopening of the vessel– fluttering

• Flow in aneurysms

– v2 < v1

– p2 > p1 : expansion and bursting of the vessel

– caused by the weakening of the arterial wall

212 .p gz v const Bernoulli

equation

p2, v2, A2

p1

v1A1 p2, v2, A2

A1

p1

v1A1v1 = A2v2


Vacular resistance and branching• Vascular resistance

– for Poiseuille flow– major drop in the mean pressure in arterioles (60 mmHg)

autonomic nervous system controls muscle tension arterioles distend or contract

• Succesive branching:– Increase in the total cross-section area

– dA1=nA2:

v

pR

Q

4

8v

LR

R

21 2

22 1

4 21 2

42 1

v nRd

v R

p nR d

p R n

Mean pressure values [mmHg]:

- arteries 100

- capillaries 30-34 at arterial end, 12-15 at venous end

n ≥ 2 average d=1.26

velocity decreases, pressure gradient increases


Turbulent Flow• Reynolds number

• Flow in the circulatory system is normally laminar• Flow in the aorta can destabilize during the deceleration phase of late systole

– too short time period for the flow to become fully turbulent

• Diseased conditions can result in turbulent blood flow

– vessel narrowing at atherosclerosis, defective heart valves– weakening of the wall, progression of the disease

R ev D

critical value Re > 2000for flow in rigid straight cylindrical pipes


Unsteady flow models• The pressure pulse:

– generated by the contraction of the left ventricle– travels with a finite speed through the arterial wall– change in a shape due to interaction with reflected waves

• Windkessel model– the arteries: a system of interconnected tubes with

a storage capacity

– distensibility Di = dV/dp

– Inflow – Outflow = Rate of Storage

SYSTOLE DIASTOLE

Q=Q0, 0 t ts Q=0, ts t T

p(t) b-(b-p0)e-t/a p(t) e(T-t)/a

A typical pressure pulse curve.

p0

ts T

bd d

( )d d

Vi

S

p p V pQ t D

R t t

diastolesystole


Wommersley equations• The equation for the motion of a viscous liquid in a cylindrical tube (general form):

• Arterial pulse = periodic function

the sum of harmonics

• The solution:

– J0(xi3/2) is a Bessel function of the first kind of order zero and complex argument– y=r/R

2

2

1 1w w p w

r r r z t

i tp

zAe

3/ 220

2 3/ 20

( )*1

( )i tJ yiA R

w ei J i

( / )R

The flow velocity pulse and the arterial pressure pulse (femoral artery of a dog).

– Wommersley number


The role of Wommersley number - unsteady inertial forces vs. viscous forces

(viscous forces dominate when 1) 10-3 18

capillaries aorta

: 3.34 4.72 5.78 6.67

The velocity profiles for the first four

harmonics resulting from the pressure

gradient cos ωt Parabolic profile is not formed

The laminae near the wall move first

Solid mass in the centre

Increase in : flattening of the central region, reduction of amplitude and reversal of flow at the wall


The sum of harmonics

y=r/R

The first four harmonics summed together with a parabola (representing the steady forward flow).

The time dependence of velocity at different

distances y.

Parabolic shape in the fast systolic rush

Phase lag between the pressure gradient and the movement of the liquid

The reversal begins in the peripheral laminae (the point of flow reversal: 25° after the pressure

Back flow: harmonics are out of phase and the profile is flattened

gradient) The peak

forward and backward velocities: 165 cm/s at 75° 35 cm/s at 165°


Conclusion• What have we learned?

- basic equations of blood flow

• Why am I interested in blood flow?

future experiment: dissolving blood clots under physiological conditions

PULSATILE FLOW

Artificial heart.

blood flow

Technology

blood flow wommersley

poiseuille flow steady

flow characteristics

flow trough stenosis

blood density

blood bloodviscosity

blood flow7

blood flow10