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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
Model Construction (cont.)
Motivation
Ventricular fibrillation (VF) is the main cause of sudden cardiac
death in industrialized nations, accounting for 1 out of 10
deaths.Strong experimental evidence suggests that selfsustained waves of electrical wave activity in cardiac tissue are
related to fatal arrhythmias.
2
L   f ( ,
1
d
,  )d
d
f
d  f 
L  0 



 d   ' 
z
subject to:
 0  0

Fiber trajectory:

  1  a sec    1
 1 
2
Mechanisms that generate and sustain
VF are poorly understood. One
conjectured mechanism is:
Breakdown of a single spiral
(scroll) wave into a disordered
state, resulting from various
mechanisms of spiral wave
instability.
1
inner surface
Speed up
2
1.42 ± 0.10
4
3.58 ± 0.16
8
7.61 ±0.46
16
14.95 ±0.46
32
28.04 ± 0.85
Fiber trajectories on nested pair of
conical surfaces
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.
The communication can be minimized when parallelized along
azimuthal direction. Computational results show the model has a
very good scalability.
Governing Equations
Transmembrane potential propagation
W.F. Witkowksi, et al., Nature 392, 78 (1998)
u
Cm
   ( Du )  I m
t
From idealized to fully realistic geometrical modeling
Anatomical canine ventricular model
Cm: capacitance per unit area of
membrane
D: diffusion tensor
u: transmembrane potential
Im: transmembrane current
Phase Singularities
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.
Courtesy of A. V. Panfilov, in Physics Today,
Part 1, August 1996
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:
 Simpler and computationally more tractable than fully realistic models
Scaling of Ventricular
Turbulence
Finding all tips
Transmembrane current, Im, described by simplified
FitzHugh-Nagumo type dynamics
v: gate variable
Parameters: a=0.1, m1=0.07,
v 
v 
    1  v  ku(u  a  1 m2=0.3, k=8, e=0.01, Cm=1
t 
2  u 
Choose an unmarked tip as current tip
Add current tip into a new filament,
marked as the head of this filament
set reversed=0
Set reversed=1
Is the
distance smaller than a certain
threshold?
Yes
The average filament length, normalized by
average heart thickness, versus heart size
Set the closest tip as current tip
Definition: Distance between two tips
Is revered=0?
(1) If two tips are not on a same fiber
surface or on adjacent surfaces,
the distance is defined to be
infinity
No
Diffusion Tensor
Yes
 More feasible for incorporating realistic electrophysiology, electromechanical coupling,
Mark the current tip
Log(total filament length) and Log(filament number)
versus Log(heart size)
No
Set the head of current Yes
filament as current tip
 Easily parallelizable and with good scalability
Add current tip into
current filament
Find the closest unmarked tip
I m  ku(u  a)(u  1)  uv
J.P. Keener, et al., in Cardiac Electrophysiology, eds.
D. P. Zipes et al. (1995)
CPUs
The results for filament number agree
to within error bars for 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.
outer surface
Fiber path equation
Patch size: 5 cm x 5 cm
Time spacing: 5 msec
Rectangular slab
Numerical Convergence
Parallelization
Are there any unmarked tips?
These results are in agreement with those obtained with the fully
realistic canine anatomical model, using the same electrophysiology.
A. V. Panfilov, Phys. Rev. E 59, R6251 (1999)
(2) Otherwise, the distance is the
distance along the fiber surface
No
End
Transformation matrix R
Filament finding algorithm
Conclusions and Future Work
Model Construction
Local Coordinate
Dlocal
 D//

 0
 0

0
D p1
0
0 

0 
D p 2 
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.
Lab Coordinate
Dlab  R 1 Dlocal R
Early dissection results revealed nested
ventricular fiber surfaces, with fibers
given approximately by geodesics on
these surfaces.
t=2
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.
Numerical Implementation
Fibers on a nested pair of surfaces in the LV,
from C. E. Thomas, Am. J. Anatomy (1957).
Working in spherical coordinates, with
the boundaries of the computational
domain described by two nested cones, is
equivalent to computing in a box.
Our model
Adopted Nested cone geometry fiber
surfaces
the fiber paths are both geodesics on
fiber surfaces and circumferential at
midwall.
Crossection along azimuthal direction
Standard centered finite difference
scheme is used to treat the spatial
derivatives, along with first-order explicit
Euler time-stepping.
t = 999
The filament finding results. The left pictures show the
simulation at time=2 and time=999. The right pictures show
the filament finding results, corresponding to the scroll waves.
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