BLOOD FLOW
Barbara Grobelnik
Advisor: dr. Igor Serša
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)
January 2008
Blood Flow
CONTENTS
Cardiovascular physiology
Physical properties of blood
 Viscosity
Steady blood flow
 Poiseuille’s equation
 Entrance effects
 Bernoulli’s equation
Oscillatory blood flow
 Windkessel model
 Wommersley equations
2
Cardiovascular Physiology
• HEART: atrium, ventricles
• BLOOD VESSELS: aorta,
arteries, arterioles,
capillaries, veinules, veins
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
right ventricle
 lungs  left
atrium
left ventricle 
aorta  organs
and tissues 
right atrium
January 2008
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
Blood Flow
3
Poiseuille flow
• Steady flow in a rigid cylindrical tube
– Pressure gradient
– Viscous force
Fp  2 r ( p1  p2 ) r
Fv  

r
(2 rL
v
) r
r
The forces are equal and opposite:
 2 v 1 v p1  p2


0
r 2 r r
L
v(r )  r 2
v(r=R)=0
v(r=0)≠∞
v(r ) 
p1  p2
4 L
p1  p2
4 L
 A ln r  B
(R2  r 2 )
L
r
R
Q   2 v(r )rdr   R 4
2r
r
0
v
p1
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p2
v
Q
 R2
 R2
Blood Flow
p1  p2
8 L
1
2
p1  p2
8 L
 v ( r  0) 
volume flow
vmax
2
average
velocity
4
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
x
– pulsatile flow in arteries
• Cylindrical shape
x
– elliptical shape (veins, pulmonary arteries), taper
• Rigid wall
x
– visco-elastic arterial walls
• Fully developed flow
x
– entrance length; branching points, curved sections
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Blood Flow
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Physical properties of blood
BLOOD =
plasma + blood cells
(55%)
(45%)
electrolyte
solution
containing
8% of
proteins
Red blood cells (95%)
White blood cells (0.13%)
Platelets (4.9%)
Reference values
RBC:
1 μm
PLASMA
density 1035 kg/m3
8 μm
January 2008
WHOLE BLOOD
1056 kg/m3
viscosity 1.3×10-3 Pa s 3.5 × 10-3 Pa s
Blood Flow
6
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 equation
In small tubes the blood
viscosity has a very low value
because of a cell-free zone near
the wall.
Fahraeus-Lindqvist
effect
   0  Kc dv / dr
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Blood Flow
7
Fahraeus-Lindqvist Effect
Cell-free marginal layer model
The Sigma effect theory
 Core region μc , vc , 0rR-
 Cell-free plasma μp , vp , R-r
R region near the wall
 velocity profile is not
continuous
 small tubes (N red blood
cells move abreast)
μ p , vp
r
μ c , vc

R
 the volume flow is
p R 3rewritten
Q
r dr
p 1 d 
dv 

 r

L
r dr 
dr 
2  L 0
 the volume flow
 N concentric laminae,
 R 4 p 1
Q
1  (1   / R) 4 (1   p / c ) 

8L  p
each of thickness ε
p N
pR 4 1 

3
Q
(n ) 

1  
2 L n1
8L  
R
1/μ
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Blood Flow
1/μ
8
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
d  dv 
viscous force
  A(r2  r1 )
dr  dr 
 - boundary layer
U
thickness at z
Fvisc  2 A(r2  r1 )

U - free stream
velocity
U2
inertial force
Fi   aV  
A(r2  r1 )
z
Fvisc  
* a=U/t=U/(z/U)
January 2008
Blood Flow
9
Entrance length
• equating the viscous and inertial force
U2
U

 k 2
z

k – proportionality constant derived from experiments, approximately 0.06
• the boundary layer thickness

z
U
• the entrance length (when =D/2 the flow
becomes fully established)
z0  kD 2
U
Pulsatile flow –
the entrance
length fluctuates

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.
January 2008
Blood Flow
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Application of Bernoulli Equation
Bernoulli
equation
p   gz  12  v 2  const.
• Flow trough stenosis
A1
p1
v1
• Flow in aneurysms
p2, v2, A2
A1v1 = A2v2
– v 2 > v1
– p2 < p1 : caving or closing
of the vessel
– decrease in v2
– reopening of the vessel
– fluttering
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Blood Flow
A1
p1
v1
p2, v2, A2
– v2 < v1
– p2 > p1 : expansion and
bursting of the vessel
– caused by the weakening
of the arterial wall
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Vacular resistance and branching
• Vascular resistance
Rv 
• Succesive branching:
– Increase in the total crosssection area
p
Q
8 L
– for Poiseuille flow Rv  4
R
– major drop in the mean
pressure in arterioles (60 mmHg)
autonomic nervous system
controls muscle tension
arterioles distend or contract
– dA1=nA2:
Mean pressure values [mmHg]:
- arteries
January 2008
p1 nR24 d 2
 4 
p2
R1
n
100
- capillaries 30-34 at arterial end,
12-15 at venous end
v1 nR22
 2 d
v2
R1
n≥2
average d=1.26
Blood Flow
velocity decreases,
pressure gradient
increases
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Turbulent Flow
• Reynolds number
v D
Re 

critical value Re > 2000
for flow in rigid straight
cylindrical pipes
• 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
January 2008
Blood Flow
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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
A typical pressure pulse curve.
Q(t ) 
p  pV dV
dp

 Di
RS
dt
dt
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
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Blood Flow
b
systole
p0
diastole
ts
T
14
Wommersley equations
• The equation for the motion of a
viscous liquid in a cylindrical tube
(general form):
 2 w 1 w 1 p  w



2
r
r r  z  t
• Arterial pulse = periodic function
p
i t

Ae

 the sum of harmonics z
The flow velocity pulse and the arterial
pressure pulse (femoral artery of a dog).
• The solution:
A * R2
w
i 2
 J 0 ( yi 3/ 2 )  it
e
1 
3/ 2 
J 0 ( i ) 

– Wommersley number  – J0(xi3/2) is a Bessel function of the
  R ( /  )
January 2008
first kind of order zero and complex
argument
– y=r/R
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The role of Wommersley number
 - unsteady inertial forces vs. viscous forces
(viscous forces dominate when   1)
The velocity profiles for
the first four harmonics
resulting from the
pressure gradient cos ωt
10-3    18
capillaries
:
3.34
4.72
5.78
aorta
6.67
 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
January 2008
Blood Flow
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The sum of harmonics
y=r/R
 Parabolic shape in
the fast systolic rush
 Phase lag between
the pressure gradient
and the movement of
the liquid
 The reversal
The time dependence of velocity
begins in the
at different distances y.
peripheral laminae
(the point of flow
reversal: 25° after gradient)
The first four harmonics summed
 The peak forward
pressure
together with a parabola (representingthe 
Back flow:
and backward
the steady forward flow).
harmonics are out of
velocities:
phase and the profile
165 cm/s at 75°
is flattened
35 cm/s at 165°
January 2008
Blood Flow
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Conclusion
• What have we learned?
• Why am I interested in
blood flow?
future experiment:
dissolving blood clots under
physiological conditions
PULSATILE FLOW
Artificial heart.
- basic equations of blood
flow
January 2008
Blood Flow
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