What are arteries for?
• Conduits
– To conduct blood to the organs and periphery
• Impedance matching
– Minimise cardiac work
– Minimise pulse pressure
– Control flow according to demand
Conduit arteries: large
arteries near the heart and
their main branches
Questions
• Why are conduit arteries distensible?
• What are arteries made of?
• Why do large arteries become stiffer with age (and
disease)?
• Why are some people affected more than others?
Arteries are distensible because:
• The wheel has yet to evolve in the animal
kingdom (bacteria have propellers)
• Therefore(?) the heart is a pulsatile pump.
• Its output consists of a pulse wave
superimposed on a steady component.
120
100
80
1 sec
Aortic pulse wave
Pressure [mmHg]
Systolic pressure
120
100
80
Average pressure
1 second
Diastolic pressure
Pulse pressure = systolic pressure - diastolic pressure
Pressure [mmHg]
Systolic pressure
120
100
80
Average pressure
1 second
Diastolic pressure
Pulse pressure = systolic pressure - diastolic pressure
• Average pressure determined by resistance of peripheral arteries
• Pulse pressure determined by elasticity of large arteries
The pulse is a wave of dilatation
With thanks to Chris Martyn
Similar to a surface wave
A heavenly wave
Speed of the wave is related to
the stiffness of the artery it is
traveling in
The stiffer the artery;
the higher the wave speed
Wave speed is proportional to the square
root of arterial stiffness
Blood vessel elasticity
•
•
•
•
•
Inelastic
Non linear
Large strains
Anisotropic
Viscoelastic
}
Pseudo elasticity
Strain energy function/incremental approach
Uni-axial expts./circumferential direction
Quasi static experiments
Stress, strain and elastic modulus
A reminder.
• Stress (, sigma)
– Force per unit area
= (F/A)
• Strain (, epsilon)
– Change in length per unit length
= (L/L0)
• Elastic (Young’s) modulus (E)
– stress/strain
= /
• Poisson’s ratio (, nu)
– transverse strain /longitudinal strain= -x/y
– for incompressible materials

F L0
=
A L
2.0
Relative Radius
R
R
P
P
1.5
R
R
P
P
1.0
0
100
Pressure (mmHg)
200
(s)

-
Cir cumf erential strain
(e)
=
Cir cumf erential elastic modulus
(E)
Relative Radius
2.0
R
-
PR
h
C R

C
R
R
-
R
Incremental strain
(2)
1  
2
0.75
1.5
R
(1)
R
inc 
R
P
PPR
(3)
2
P
Rh
(4)
2.0
RR
Mean circumf erential stress
1.5
R
P
P
1.0
0
P
1.0
Incremental
stress
0

 inc 
100
100
Pressure (mmHg)
PR
h
Pressure (mmHg)
Incremental elastic modulus

(structural stiffness)
R
200
 inc
PR 2
E inc 
 0.75
inc
Rh
200
Variation of Einc with stretch
Einc [Nm-2 x 105]
15
10
5
0
1.0
1.2
1.4
1.6
1.8
R/Ro
2.0
2.2
2.4
2.6
Structural & functional stiffness
Functional
stiffness
Structural
stiffness

PR
Ep 
R
R
E inc  0.75E p
h
PR 2
E inc  0.75
Rh


Geometry

h
E p  1.5E inc
R
Structure

 0.75
PR
R
x
R
h
Some haemodynamics
P
W
Q
= k1
Steady flow resistance
µl
R
µ:
l:
R:
k1:
4
ˆ
P
Zc 
Qˆ
= k2
viscosity
length
inner radius
constant
Characteristic impedance
(pulsatile flow “resistance”)
Ep
R
2
Just a touch more
Ep
Zc = k2 2
R
c k3 Ep
rc
 2
pR
c:
r:
pulse wave velocity
tissue & blood density
Measure pulse wave velocity non
invasively to estimate functional stiffness
Summary
The relationship between vessel dimensions,
elasticity and blood flow
Structural stiffness
PR 2
E inc  1.5
R h
Functional stiffness
Ep 
Characteristic impedance
Pˆ
Zc 
Qˆ 
PR
 R
(a measure of all the factors which
combine to limit pulsatile flow due to a
pulsatile pressure gradient)


h
 1.5E inc
R
k
Ep
R2
Structure &
geometry
Functional
stiffness &
diameter
Electrical analogue
R1
R3
L1
C1
R2
R1 : resistance of large vessels
L1 : inertia of blood
C1 : compliance of large vessels
R2 : peripheral resistance
R3 : source resistance of heart
Reflections
In the arterial system reflections of pressure and flow
waves occur wherever there is a change in the local
fluid impedance
• Decrease in diameter or increase in stiffness ->
positive reflection of pressure
negative reflection of flow
• If no reflections: pressure and flow waves are the same shape
• Arterial disease usually associated with increased reflections (except
aneurysms)
• Energy is lost so cardiac output must increase to maintain a given flow
Wave reflection
∆t
Pressure
Flow
Time
P
Q
resistance
W
characteristic impedance
Pˆ
Zc 
Qˆ

mean pressure/mean flow
pulsatile pressure/pulsatile flow
Fourier analysis
H1
H2
H1 + H2 + H3
H4
2
1
0
-1
-2
90
180
270
360
H1 + H2
H3
H1+H2+H3+H4
Mean
Measured
Q(t) = q0
+ q1Cos(t - 1)
+ q2Cos(t - 2)
+ q3Cos(t - 3)
+ ...
P(t) = p0
+ p1Cos(t - 1)
+ p2Cos(t -  2)
+ p3Cos(t -  3)
+ ...
|Z| = |pn|/|qn|Pressure/Flow
F = n - n Pressure - Flow