Transcript Electrical Wave Propagation in a Minimally Realistic Fiber Architecture

Electrical Wave Propagation in a
Minimally Realistic Fiber Architecture
Model of the Left Ventricle
Xianfeng Song, Department of Physics, Indiana University
Sima Setayeshgar, Department of Physics, Indiana University
March 17, 2006
This Talk: Outline
Motivation
Model Construction
Numerical Results
Conclusions and Future Work
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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 that selfsustained waves of electrical wave activity in
cardiac tissue are related to fatal arrhythmias.

Mechanisms that generate and sustain VF are
poorly understood.

Conjectured mechanism for understanding VF:
Breakdown of a single spiral (scroll) wave into
a disordered state, resulting from various
mechanisms of spiral wave instability.
W.F. Witkowksi, et al., Nature 392, 78 (1998)
Patch size: 5 cm x 5 cm
Time spacing: 5 msec
From idealized to fully realistic
geometrical modeling
Rectangular slab
J.P. Keener, et al., in Cardiac Electrophysiology, eds.
D. P. Zipes et al., 1995
Anatomical canine ventricular model
Courtesy of A. V. Panfilov, in Physics Today,
Part 1, August 1996
Construct a minimally realistic model of left ventricle for studying electrical wave propagation in
the 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
 Easily parallelizable and with good scalability
 More feasible for incorporating contraction
Model Construction
 Early dissection revealed nested
ventricular fiber surfaces, with fibers
given approximately by geodesics
on these surfaces.
 Peskin Asymptotic Model
C. S. Peskin, Comm. on Pure and Appl. Math. 42, 79 (1989)
Conclusions:
Fibers on a nested pair of surfaces in the LV,
from C. E. Thomas, Am. J. Anatomy (1957).
 The fiber paths are approximate
geodesics on the fiber surfaces
 When heart thickness goes to zero,
all fiber surfaces collapse onto the
mid wall and all fibers are exact
geodesics
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Model construction (cont’d)
Nested cone
geometry
and fiber
surfaces
Fiber paths
To be geodesics
To be circumferential at the mid wall
2
L   f ( ,
1
d
,  ) d
d
f
d  f 




 d   ' 
z

  0
0

  1
 1 
Fiber
paths on
the inner
sheet
  1  a 2 sec 1 
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Fiber
paths on
the outer
sheet
Governing Equations
 Transmembrane potential propagation
Cm
u
   ( Du )  I m
t
Cm: capacitance per unit ar
D: diffusion tensor
u: transmembrane potentia
Im: transmembrane current
 Transmembrane current, Im, described by simplified
FitzHugh-Nagumo type dynamics*
I m  ku(u  a)(u  1)  uv
v 
v 
    1  v  ku(u  a  1
t 
2  u 
v: gate variable
Parameters: a=0.1, 1=0.07, 2=0.3,
k=8, =0.01, Cm=1
* R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996)
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Numerical Implementation
 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.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Diffusion Tensor
Transformation matrix R
Local Coordinate
Dlocal
 D//

 0
 0

0
D p1
0
0 

0 
D p 2 
Lab Coordinate
Dlab  R 1 Dlocal R
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Parallelization
The communication can be minimized when parallelized along
azimuthal direction.
Computational results show the model has a very good scalability.
CPUs
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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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.
Color denotes the transmembrane potential.
Movie shows the spread of excitation for 0 < t <
30, characterized by a single filament.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same fiber
surface or on adjacent surfaces, the
distance is defined to be infinity.
Otherwise, the distance is the
distance along the fiber surface
Find all tips
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Random choose a tip
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Search for the closest tip
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Make connection
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue doing search
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
The closest tip is too far
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Reverse the search direction
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Complete the filament
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Start a new filament
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Repeat until all tips are consumed
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding result
t=2
FHN Model:
t = 999
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Numerical Convergence
The results for filament length agree
to within error bars for three different
mesh sizes.
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.
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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Scaling of Ventricular Turbulence
The average filament length, normalized by
Log(total filament length) and Log(filament number)
average heart thickness, versus heart size
versus Log(heart size)
Both filament length
The 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)
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Conclusion

We constructed a minimally realistic model of the left ventricle for studying
electrical wave propagation in the three dimensional myocardium and
developed a stable filament finding algorithm based on this model

The model can adequately address the role of geometry and fiber
architecture on electrical activity in the heart, which qualitatively agree with
fully realistic model

The model is more computational tractable and easily to show the
convergence

The model adopts simple difference scheme, which makes it more feasible
to incorporate contraction into such a model

The model can be easily parallelized, and has a good scalability
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore