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