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Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle
Xianfeng Song, Sima Setayeshgar
Department of Physics, Indiana University, Bloomington
Model Construction
Motivation
Ventricular fibrillation (VF) is the main cause of sudden cardiac
death in industrialized nations, accounting for 1 out of 10
deaths. Experimental evidence strongly suggests that selfsustained waves of electrical wave activity in cardiac tissue are
related to fatal arrhythmias.
Parallelization
Fiber paths as:
geodesics on fiber surfaces
CPUs
Speed up
2
1.42 ± 0.10
circumferential at midwall
4
3.58 ± 0.16
8
7.61 ±0.46
16
14.95 ±0.46
32
28.04 ± 0.85
2
L   f ( ,
1
d
,  )d
d
Scaling of Ventricular
Turbulence
Cross-section along azimuthal direction
f
d  f 



 d   ' 
z
subject to:
 0  0

Fiber trajectory:
L  0 
Mechanisms that generate and sustain
VF are poorly understood. One
conjectured mechanism is:

  1
 1 
  1  a 2 sec 1 
Breakdown of a single spiral (scroll)
wave into a disordered state, resulting
from various mechanisms of spiral
wave instability.
inner surface
Patch size: 5 cm x 5 cm
Time spacing: 5 msec
Fiber trajectories on nested pair of conical surfaces
From Idealized to FullyRealistic Geometrical modeling
Anatomical canine ventricular model
Transmembrane potential propagation
u
Cm
   ( Du )  I m
t
Cm: capacitance per unit area
of membrane
D: diffusion tensor
u: transmembrane potential
Im: transmembrane current
Transmembrane current, Im, described by simplified FitzHughNagumo type dynamics
I m  ku(u  a)(u  1)  uv
v 
1v 
 v  ku(u  a  1
   
t 
2  u 
Log(total filament length) and Log(filament
number) versus Log(heart size)
The average filament length, normalized by
average heart thickness, versus heart size
outer surface
Governing Equations
W.F. Witkowksi, et al., Nature 392, 78 (1998)
Rectangular slab
The communication can be minimized when parallelized along
azimuthal direction. Computational results show the model has a
very good scalability.
These results are in agreement with those obtained with the fully
realistic canine anatomical model*, using the same electrophysiology.
Phase Singularities
[*] A. V. Panfilov, Phys. Rev. E 59, R6251 (1999)
Tips and filaments are phase singularities that act as organizing
centers for spiral (2D) and scroll (3D) dynamics, respectively,
offering a way to quantify and simplify the full spatiotemporal
dynamics.
Finding all tips
Choose an unmarked tip as current tip
Add current tip into a new filament,
marked as the head of this filament
set reversed=0
Add current tip into
current filament
Find the closest unmarked tip
Is the
distance smaller than a certain
threshold?
Set reversed=1
Mark the current tip
Yes
Set the closest tip as current tip
No
v: gate variable
Parameters: a=0.1, m1=0.07,
m2=0.3, k=8, e=0.01, Cm=1
Definition: Distance between two tips
Set the head of current Yes
filament as current tip
Is revered=0?
No
Yes
Are there any unmarked tips?
The phase transition between
fibrillation state to stable state
(1) If two tips are not on a same fiber
surface or on adjacent surfaces,
the distance is defined to be
infinity
(2) Otherwise, the distance is the
distance along the fiber surface
3D simulation tends to be more unstable than 2D
simulation
On the boundary between instable and stable
parameters, the decrease in domain size could
increase the instability
No
End
J.P. Keener, et al., in Cardiac Electrophysiology, eds.
D. P. Zipes et al. (1995)
Courtesy of A. V. Panfilov, in Physics Today,
Part 1, August 1996
Filament-finding Algorithm
Diffusion Tensor
Construct minimally realistic model of LV for studying electrical wave
propagation in three dimensional anisotropic myocardium that adequately
addresses the role of geometry and fiber architecture and is:
t=2
Transformation matrix R
 Simpler and computationally more tractable than fully realistic models
 Easily parallelizable and with good scalability
 More feasible for incorporating realistic electrophysiology,
electromechanical coupling
t = 999
Local Coordinate
Dlocal
 D//

 0
 0

0
D p1
0
0 

0 
D p 2 
The left images show the
simulation at time t=2 and
t=999 units. The right
images show the filament
finding results,
corresponding to the scroll
waves.
Lab Coordinate
Dlab  R 1 Dlocal R
Peskin asymptotic model: first
principles derivation of toroidal fiber
surfaces and fiber trajectories as
approximate geodesics.
Fibers on a nested pair of surfaces in the LV,
from C. E. Thomas, Am. J. Anatomy (1957).
Fibers on a nested pair of surfaces in the LV, from
C. E. Thomas, Am. J. Anatomy (1957).
Fiber angle profile through LV thickness:
Comparison of Peskin asymptotic model and dissection results,
from C. S. Peskin, Comm. in Pure and Appl. Math. (1989).
Numerical Implementation
Numerical Convergence
The results for filament number agree
to within error bars for spatial mesh
size dr=0.7 and dr=0.5. The result for
dr=1.1 is slightly off, which could be
due to the filament finding algorithm.
Working in spherical coordinates, with the
boundaries of the computational domain
described by two nested cones, is equivalent
to computing in a box.
Standard centered finite difference scheme
is used to treat the spatial derivatives, along
with first-order explicit Euler time-stepping.
We have constructed and implemented a minimally realistic fiber
architecture model of the left ventricle for studying electrical
wave propagation in the three dimensional myocardium.
Our model adequately addresses the geometry and fiber
architecture of the LV, as indicated by the agreement of filament
dynamics with that from fully realistic geometrical models.
LV Fiber Architecture
Early dissection results revealed
nested ventricular fiber surfaces,
with fibers given approximately
by geodesics on these surfaces.
Conclusions and Future Work
Filament Number and Filament Length
versus Heart size
The computation time for dr=0.7 for
one wave period in a normal heart size
is less than 1 hour of CPU time using
FHN-like electrophysiological model.
Our model is computationally more tractable, allowing reliable
numerical studies. It is easily parallelizable and has good
scalability.
As such, it is more feasible for incorporating
Realistic electrophysiology
Biodomain description of tissue
Electromechanical coupling