Investigating a
Period-Doubling Bifurcation
in Cardiac Tissue Using
Alternate Pacing
Carolyn Bergera , Hana Dobrovolnya , Soma
Kalbc , Salim Idrissce , David Schaefferb , Daniel
Gauthierad and Wanda Krassowskacd
Duke University
Supported by NSF PHY-0243584 and
NIH 1 R01-HL-72831
a
Department of Physics
b
Department of Math
c
Department of Biomedical Engineering
d
Center for Nonlinear and Complex Systems
e
Pediatric Cardiology
Introduction to Cardiac
Dynamics
Experimental Setup
Single Cell Cardiac Dynamics
1:1⇒2:2 Transition
1:1
2:2 (alternans)
Increase Pacing Rate⇒bifurcation to 2:2
2:2 behavior linked to ventricular
arrythmias and sudden cardiac death
We are interested in the 1:1⇒2:2 transition
New Technique:
Alternate Pacing Protocol(APP)
alternate pacing induces alternate
response
Measure the gain, defined as
Gain
=
APDl −APDs
2δ
Purpose of the APP
Small beat-to-beat changes in the BCL, 2δ ,
will induce a large difference between
APDl and APDs
Hypothesis: APD variations will be more
prominent in tissue susceptible to
alternans
Goal: identify the tissue’s propensity
towards alternans before the actual onset
Simple Mathematical
Explanation
AP Dn+1 = f (DIn )
where DIn = BCL − AP Dn
APPLY Alternate Pacing Protocol
Linear approximation to 1-D map
Gain
=
f 0 (DIn )|DI ∗
1−f 0 (DIn )|DI ∗
f 0 (DIn )|DI ∗ = 1 condition for bifurcation
DIVERGENCE f 0 (DIn )|DI ∗ ⇒ 1
1-D Map Simulation
AP Dn+1 = f (DIn ) = Amax − Ae−DIn /τ
(Nolasco, Dahlen. J. Appl. Physio. (1968))
Bifurcation Diagram for the 1−D map
650
δ=5 ms
600
APDl
550
APD
500
APDs
450
400
Steady State Pacing
Alternate Pacing
350
300
600
650
700
750
800
850
BCL
Gain
=
APDl −APDs
at BCL =
2δ
700ms
⇒ Gain ∼ 4.78
The Theoretical Gain
for AP Dn+1 = f (DIn )
Gain for 1−D Map
40
35
linear approximation
30
Gain
25
20
15
numerical
10
5
0
640
bifurcation pt
650
660
670
BCL
680
690
700
For linear approximation ⇒ gain diverges
generic result for other mapping models
AP Dn+1 = f (AP Dn , DIn )
AP Dn+1 = f (AP Dn , DIn , DIn−1 )
APP in Experiment
Experimental Setup
Number of Animals: 13
Number of Trials: 50
Data from One Trial
δ = 20ms
Experimental Bifurcation Diagram
660
640
APD (ms)
620
600
580
560
steady state pacing
540
alternate pacing
520
700
alternans at 650 ms
750
800
BCL (ms)
850
900
Experimental Gain
1
0.8
Gain
0.6
0.4
0.2
0
700
alternans at 650 ms
750
800
BCL
850
900
Summary
Mapping models show gain becoming
large near the bifurcation
Fox Ionic Model (implemented by Dr. Kalb)
also shows large gain near the bifurcation
theoretically we expect the gain to become
large near the bifurcation point;
experimentally we find the gain does not
exceed one
APP serves as another protocol that can
validate new map and ionic models
Study calcium and voltage dynamics
together