EXTREME THEORETICAL PRESSURE OSCILLATIONS IN CORONARY BYPASS Ana Pejović-Milić, Ryerson...
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Transcript of EXTREME THEORETICAL PRESSURE OSCILLATIONS IN CORONARY BYPASS Ana Pejović-Milić, Ryerson...
EXTREME THEORETICAL PRESSURE
OSCILLATIONS IN CORONARY BYPASS
Ana Pejović-Milić, Ryerson University, CA
Stanislav Pejović, University of Toronto, CA
Bryan Karney, University of Toronto, CA
The pulsatile hemodynamics of a coronary bass
Fluid dynamics of the bypass loop is influenced by:wave reflection from junctionsnarrowing of coronary vessels (Duan et al., 1995)mechanical stiffness of the coronary bypass (Alderson et al., 2001)complexity of the vascular networkbypass lengthexact position of the narrowing in the coronary vessel and its stiffnessaging of vessels
This work: a mathematical/hydraulic method to analyse the resonance/stability of localized bypass loop.
The pulsatile hemodynamics of a coronary bass
Investigated pulsatile conditions of a simplified human bypass implant
Approach utilises transfer matrix and graph theory, with computer simulation
Modelling includes wave reflections and the elasticity of the blood vessels
Chosen dimensions corresponds to typical vessel lengths and diameters in human coronary circulation
Coronary bypass loop modelCoronary bypass loop model
Elements of model:Blood vessel segments (1 – 6):Aorta - 1, 2, and 3 Coronary bypass - 4 (between junctions B and C)Coronary artery – 5 (diseased) and 6 (healthy)Junctions - A, B, and C
For narrowed segment 5, diameter was varied (0.4 - 0.01 cm), thus simulating different stages of aorta stenosis.
Steady oscillatory
condition calculated
in the simplified
bypass loop along
with local vascular
network.
A
B
C
1
2
3
1
4
1 5 6
aort
a
bypass
coronary artery
Coronary bypass loop modelCoronary bypass loop modelThe hemodynamic model focuses on pressure distribution in each segment, computed for the input amplitude pressure at the heart of 1.
Modelling assumptions:vessel wall is thin
slightly elastic
blood vessel wall is free to move under the forces of the flow field
A
B
C
1
2
3
1
4
1 5 6
Theory of Oscillatory Flow
Equations of motion and continuity:
Wave velocity a is given by
0
t
c
x
cc
x
hg
h
tc
h
x
a
g
c
x
2
0
dEea
Notation:
h - piezometric head,
c - flow velocity
x - distance along vessel axis, t – time
g - gravitational acceleration
d - arterial diameter
e - wall thickness
ρ - blood density
E - Young’s modulus of wall material
a - wave velocity
hydraulic inertance:
L = 1/(gA) hydraulic capacitance:
C = gA/a2,
Theory of Oscillatory FlowRearranging equations reveals wave form:
Solution has general form:
2
2
2
2
t
qCL
x
q
2
2
2
2
t
hCL
x
h
h H x est
Amplitudes of pressure (H) and flow (Q) oscillations at downstream (D) and upstream (U) ends of pipes related by
UC
C
DQ
H
lZ
l
lZl
Q
H
cosh
sinh
sinh- cosh
CLs
where
gA
a
CsZC
Extreme Pressures Without BypassResults for 3 different diameters of coronary artery
0.4 cm - corresponds to a healthy artery
0.2 cm – narrowed coronary artery
0.6 cm - enlarged coronary artery
0
1
2
3
4
0 50 100 150
Frequency (Hz)
No
rmalized
pre
ssu
re
d = 4mm
d = 2mm
d = 6mm
Normal Artery (4 mm)
first natural frequencies for 6 coronary diameters
from 0.2 cm (narrowed) to 0.6 cm (enlarged)
Mode Shape
0.5
1
1.5
2
2.5
3
3.5
4
1 11 21
Distance from the heart (mm)
No
rma
lize
d p
res
su
re a
mp
litu
de
s
d = 0.6 cmm, f = 35Hz
d = 0.5 cm, f = 35Hz
d = 0.4 cm, f = 17Hz
d = 0.35 cm, f = 71Hz
d = 0.3 cm, f = 71Hz
d = 0.2 cm, f = 71Hz
Effects of wave reflections on pressure amplitudes in the bypass loop along section 1 – 2 – 4 – 6
2.5 % Stenosis (coronary artery 1 mm radius)
0.6
0.8
1
1.2
1.4
1.6
1.8
25 % Stenosis (coronary artery 1 mm radius)
0.6
0.8
1
1.2
1.4
1.6
1.8
50 % Stenosis (coronary artery 2 mm radius)
0.6
0.8
1
1.2
1.4
1.6
1.8
No stenosis (coronary artery 4 mm radius)
0.60.8
11.21.41.61.8
0 10 20 30 40
Distance from the heart (cm)
1 Hz
5 Hz
10 Hz
Resonance
A
B
C
1
2
3
1
4
1 5 6
No
rmal
ized
pre
ssu
re a
mp
litu
de
Effects of wave reflections on pressure amplitudes in coronary bypass loop along the section 1 – 5 – 6:
A
B
C
1
2
3
1
4
1 5 6
2.5 % Stenosis (coronary artery 0.1 mm radius)
0.5
1
1.5
2
2.5
3
3.5
25 % Stenosis(coronary artery 1 mm)
0.5
1
1.5
2
2.5
50 % Stenosis(coronary artery 2 mm radius)
0.5
1
1.5
2
2.5
No stenosis (coronary artery 4 mm radius)
0.5
1
1.5
2
2.5
1 10 19
Distance from the heart (cm)
1 Hz
5 Hz
10 Hz
Resonance
No
rmal
ized
pre
ssu
re a
mp
litu
de
Conclusion: Either narrowed or enlarged coronary artery amplifies the amplitude of the pressure fluctuation, due to the reflection of waves.
Reporting for the first time the steady oscillatory condition for the coronary tree as well as coronary bypass loop following its surgical implantation.
Results and conclusions relate to the effects of wave reflections only.
Results are complementary to other studies, and must be viewed in conjunction with other associated effects on diameter changes of coronary artery.
Future modeling:
When friction in vessels is simulated a continuous wave reflection occurs.Coronary arteries, which enter the heart, might be contracting continuously, thus producing additional excitations propagating with the wave speed though the surrounding arteries upstream to the left ventricle.Heart is a type of a reciprocating pump pushing the blood into aorta and the pressure is caused by this pulsatile flow. Thus, it would ultimately be more appropriate to assume an excitation of flow (not pressure) at the entrance of aorta.
Pressure wave excited at the heartTime 0.04 s
A
B
C
1
2
3
1
4
1 5 6
Direction of the waves
Measured
from th
e hear
t
heart
bypass
Pressure Excitation
More Information?
Talk at The Serbian Academy of Science and Arts, Mechanics Department of Mathematical Institute tomorrow at 6:00 pm
Heart dynamics with time domain (Ana Pejović-Milić, Stanislav Pejović, Bryan Karney)
Talk at University tomorrowMulti-faceted role of transients (Karney)
Analysis in time domainFigure 3.13;2. Simultaneous blood pressure records made at a series of sites along the aorta in the dog, with distance measured from the beginning of the descending aorta. From Olson, R.M. (1968) Aortic blood pressure and velocity as a function of time and position.
Increase in amplitude of systolic
pressure with distance from the heart
is the phenomena of wave reflection
(elastic brunching system).
Slamming of aortic valve
High frequency excitation
Figure 2.2:1. Blood flow through the heart. The arrows show the direction of blood flow.
SVC = superior vena cava; IVC - inferior vena cava; RA = right atrium; RV = right ventricle; PA = pulmonary artery; LV = left ventricle; T = tricuspid; P = pulmonary; AO = aortic; M = mitral.
FromFolkow and Neil (1971) Circulation, Oxford Univ. Press, New York,
aortic valve is slamming
Figure 2.3:1. The electric system of the heart and the action potentials at various locations in the heart. From Frank Netter (1969).
Pressure wave excited at the heartTime 0.01 s
A
B
C
1
2
3
1
4
1 5 6
x axis
Pressure wave excited at the heartTime 0.03 s
Pressure wave excited at the heartTime 0.03 s
A
B
C
1
2
3
1
4
1 5 6
Direction of the waves
Pressure wave excited at the heartTime 0.04 s
A
B
C
1
2
3
1
4
1 5 6
Direction of the waves
Pressure wave excited at the heartTime 0.53 s
Maximum pressure
Minimum pressure
Measured
from th
e hear
t
heart
bypass
Pressure Excitation