Transcript fiber-v3 - Department of Physics

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
Numerical Convergence
Parallelization
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.
Fiber paths as:
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.
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
L
2
1
d
f ( ,
,  ) d
d
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  a sec    1
 1 
2
Breakdown of a single spiral (scroll)
wave into a disordered state, resulting
from various mechanisms of spiral
wave instability.
inner surface
1
outer surface
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)
Patch size: 5 cm x 5 cm
Time spacing: 5 msec
u
Cm
   ( Du )  I m
t
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.
Scaling of Ventricular
Turbulence
Finding all tips
From Idealized to FullyRealistic Geometrical modeling
Rectangular slab
Anatomical canine ventricular model
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
I m  ku(u  a)(u  1)  uv
Set reversed=1
Is the
distance smaller than a certain
threshold?
Mark the current tip
Yes
Yes
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
Are there any unmarked tips?
(2) Otherwise, the distance is the
distance along the fiber surface
Filament-finding Algorithm
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:
LV Fiber Architecture
Early dissection results revealed
nested ventricular fiber surfaces, with
fibers given approximately by
geodesics on these surfaces.
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).
[*] A. V. Panfilov, Phys. Rev. E 59, R6251 (1999)
End
Transformation matrix R
 More feasible for incorporating realistic electrophysiology,
electromechanical coupling
These results are in agreement with those obtained with the fully
realistic canine anatomical model*, using the same electrophysiology.
No
Courtesy of A. V. Panfilov, in Physics Today,
Part 1, August 1996
 Easily parallelizable and with good scalability
The average filament length, normalized by
average heart thickness, versus heart size
Set the closest tip as current tip
Definition: Distance between two tips
No
Diffusion Tensor
 Simpler and computationally more tractable than fully realistic models
Log(total filament length) and Log(filament number)
versus Log(heart size)
No
Set the head of current Yes
filament as current tip
J.P. Keener, et al., in Cardiac Electrophysiology, eds.
D. P. Zipes et al. (1995)
Add current tip into
current filament
Find the closest unmarked tip
Conclusions and Future Work
Local Coordinate
Dlocal
 D//

 0
 0

0
D p1
0
0 

0 
D p 2 
Lab Coordinate
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.
1
Dlab  R Dlocal R
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
t = 999
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.
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.
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