why is alternans indeterminate?
DESCRIPTION
Why Is Alternans Indeterminate?. cell. tissue. Xiaopeng Zhao Mechanical, Aerospace and Biomedical Engineering University of Tennessee [email protected]. Ventricular Fibrillation. ECG. Courtesy of Flavio H. Fenton, Cornell University. stroke 167,000. sudden cardiac arrest 350,000. - PowerPoint PPT PresentationTRANSCRIPT
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Why Is Alternans Indeterminate?
Xiaopeng ZhaoMechanical, Aerospace and Biomedical EngineeringUniversity of Tennessee [email protected]
350 375 400BCLms1.4
1.6
1.8
2
aC
kaepM
350 375 400BCLms1.4
1.6
1.8
2
aC
kaepMcell tissue
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Ventricular Fibrillation
Courtesy of Flavio H. Fenton, Cornell University
ECG
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Death in the US
stroke
167,000
lung cancer
160,000http://www.wrongdiagnosis.com
sudden cardiac arrest
350,000
Ventricular Fibrillation Leads to SCA
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Basic Physiology
Calcium Contraction -- Action Potential Duration-- Diastolic Interval-- Basic Cycle Length
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Alternans: period doubling
Slow pacing1:1 response
Rapid pacing2:2 response(alternans)
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Model
Fox et al., Am J Physiol, 2002 Shiferaw et al., Biophy, 2003
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Alternans in Single Cells
Simulation of the Shiferaw-Fox Model
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Alternans in Fibers
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Alternans in Fibers
A “fiber” of Two Cells
desynchronized synchronized
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Influence of Junctional Diffusion
7 2 2voltage: 5*10 m /s, calcium: 300 m /sD D D.M. Bers, 2001
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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
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Voltage/Calcium Coupling
1
1
,
,n n n
n n n
A F D C
C G D C
n nA D B
A Single Cell
C
C
D
D
FJ
G GF
: slope of APD restitution: slope of Ca relation
D
C
FG
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Voltage/Calcium Coupling
1 1 2
1 1 1
1 2 2
1/ 2 , 1/ 2 ,
,
,
n n n n n
n n n
n n n
A F D C x F D C x
C x G D C x
C x G D C x
/ 2 / 20
0
D C C
D C
D C
F F FJ G G
G G
Two Coupled Cells
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Eigenvalues
D C
D C
F FJ
G G
/ 2 / 20
0
D C C
D C
D C
F F FJ G G
G G
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Stability Boundary0C DF G 0C DF G
Single Cell
Coupled Cells
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Spatial Synchrony
1 1 2
1 1 1
1 2 2
1/ 2 , 1/ 2 ,
,
,
n n n n n
n n n
n n n
A F D C x F D C x
C x G D C x
C x G D C x
0 1 1 and / 1 1C DG G Eigenvectors:
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Calcium Dynamics
Diaz et al., Circ Res, 2002Shiferaw et al, Biophy, 2003
CG
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Border-collision Bifurcation0C DF G 0C DF G
Single Cell
Coupled Cells
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Simulation of Mapping
350 375 400BCLms1.4
1.6
1.8
2
aC
kaepM
350 375 400BCLms1.4
1.6
1.8
2
aC
kaepM
a single cell two coupled cells
Bifurcation Diagram
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Simulation of Mapping
0 25 50 75 100cell number
240
260
280
300
DPAsm
0 25 50 75 100cell number
240
260
280
300
DPAsm
0 25 50 75 100cell number
1.4
1.6
1.8
aCM
0 25 50 75 100cell number
1.4
1.6
1.8
aCM
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Future Work
1
1
,
,
n i j n j n j
n i n i n i
A x w F D x C x
C x G D x C x
Open questions:
stability, synchronization, …experiments, prediction, control, …
Coupled Oscillators
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Take-home Message
The spatiotemporal pattern of calcium-driven alternans is indeterminate.
350 375 400BCLms1.4
1.6
1.8
2
aC
kaepM
350 375 400BCLms1.4
1.6
1.8
2
aC
kaepM
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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
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Positive Vm/Ca Coupling
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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.