1. Why Is Alternans Indeterminate?
cell
tissue
Xiaopeng Zhao
Mechanical, Aerospace and Biomedical Engineering
University of Tennessee

2. Ventricular Fibrillation
ECG
(a) A single electrical wave produced by the heart's natural pacemaker spreads throughout the heart and induces a contraction. These waves normally occur about once every 0.8s. (b) A spiral wave of electrical activity generated in the heart with a period of about 0.2s can produce the fast oscillations characteristic of an arrhythmia called tachycardia, which often directly precedes the onset of fibrillation. (c) Multiple spiral waves produced by the breakup of a spiral wave can lead to the fast irregular oscillations characteristic of fibrillation. (Experimental movies courtesy of Richard Gray)
It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, such as the spiral waves illustrated in (c) of the above figure. These electrical waves circulate repeatedly through the tissue with higher frequency than the heart's pacemaker, thereby altering the heart's regular function and resulting in inadequate pumping. Researchers at the Center and other institutions are continuing to investigate how arrhythmias develop and how they depend on the dynamics of individual cells and cardiac tissue structure in order to develop better methods for treating and preventing arrhythmias.
Courtesy of Flavio H. Fenton, Cornell University

3. Ventricular Fibrillation Leads to SCA
Death in the US
Ventricular Fibrillation Leads to SCA
sudden cardiac arrest
350,000
stroke
167,000
lung cancer
160,000
The most severe arrhythmia is sudden cardiac arrest, which is the leading cause of death in the US. It kills about 350,000 people each year. That is more than the sum of death due to stroke and lung cancer. So, what is sudden cardiac arrest?

4. Basic Physiology Calcium Contraction
-- Action Potential Duration -- Diastolic Interval -- Basic Cycle Length

5. Alternans: period doubling
Slow pacing
1:1 response
Rapid pacing
2:2 response
(alternans)

6. Model
Fox et al., Am J Physiol, 2002
Shiferaw et al., Biophy, 2003

7. Alternans in Single Cells
Simulation of the Shiferaw-Fox Model

8. Alternans in Fibers

9. Alternans in Fibers
A “fiber” of Two Cells
desynchronized
synchronized

10. Influence of Junctional Diffusion
D.M. Bers, 2001

11. Spatial Pattern of Alternans
Existing theory:
Pattern determined by length of fiber; short fibers: spatially concordant; long fibers: spatially discordant
Shiferaw-Fox model:
Pattern determined by initial condition; spatial concordance independent of length of fiber

12. Voltage/Calcium Coupling
A Single Cell

13. Voltage/Calcium Coupling
Two Coupled Cells

14. Eigenvalues

15. Stability Boundary
Single Cell
Coupled Cells

16. Spatial Synchrony
Eigenvectors:

17. Calcium Dynamics Shiferaw et al, Biophy, 2003
Diaz et al., Circ Res, 2002

18. Border-collision Bifurcation
Single Cell
Coupled Cells

19. Simulation of Mapping Bifurcation Diagram a single cell
two coupled cells

20. Simulation of Mapping

21. Future Work Open questions:
Coupled Oscillators
Open questions:
stability, synchronization, … experiments, prediction, control, …

22. Take-home Message
The spatiotemporal pattern of calcium- driven alternans is indeterminate.

23. Acknowledgement
Code of the Shi-Fox model, courtesy of Yohannes Shiferaw
Useful discussion with Wanda Krassowska, David Schaeffer, Daniel Gauthier, Daisuke Sato, and Alain Karma

24. Positive Vm/Ca Coupling

25. Summary
In tissue, spatial coupling in APD is strong whereas the coupling in Ca is weak
Multiple alternans (period-2) solutions coexist if alternans is induced by instability in calcium dynamics
At a first approximation, the phenomenon occurs through a border-collision bifurcation.