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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
Goal
Model Construction
Results
Discussion and future plan
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Minimally Realistic Model: Goal
 Construct a minimally realistic model of the left ventricle for studying
electrical wave propagation in the three dimensional anisotropic
myocardium.
 Adequately addresses the role of geometry and fiber architecture on
electrical activity in the heart
 Simpler and computationally more tractable than fully realistic
models
 More feasible to incorporate contraction into such a model
 Easy to be parallelized and has a good scalability
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Nested Cone Approximation
• A simple nested cone geometry, represents the left
ventricle which does not incorporate the valves.
fi=8
fe=16
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Fiber construction
 Construction principles
 Peskin Asymptotic Model (Derived by Peskin in 1996)
The fiber paths are approximate geodesics on the fiber surfaces.
 Requiring the fibers to be circumferential where the double sheets meet at midwall
 Euler-Lagrange equations (f: fiber trajectory):
2
L   f ( ,
1

 Result
d
,  ) d
d
f
d  f 



 d   ' 
z

  0
0

  1
 1 
  1  a 2 sec 1 
Fiber
paths on
the inner
sheet
Fiber
paths on
the outer
sheet
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Governing equations
 Governing equation (a conventional parabolic partial differential
equation)
Cm
u
   ( Du )  I m
t
 Transmembrane current Im was described using a simplified excitable
dynamics equations of the FitzHugh-Nagumo type (R. R. Aliev and A.
V. Panfilov, 1996)
I m  ku(u  a)(u  1)  uv
v 
v 
    1  v  ku(u  a  1
t 
2  u 
Parameters: a=0.1, 1=0.07,2=0.3,k=8,=0.01, Cm=1
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 finite difference scheme is
used to treat the spatial derivatives,
along with explicit Euler timestepping
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
Parallelize the code
The communication can be minimized when parallelized along the theta
direction
Computational results show the model has a very good scalability
CPUs
Speed
up
2
1.40
4
3.65
8
7.80
16
15.50
32
29.20
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Finding the filament
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
Set reversed=1
Is the
distance smaller than a certain
threshold?
Mark the current tip
Yes
Set the closest tip as current tip
No
Set the head of current Yes
filament as current tip
Definition: Distance between two tips
Is revered=0?
No
Yes
Are there any unmarked tips?
No
End
(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
Result - Simulation
FHN Model:
Filament initially time=2
Color denotes the u variable in FHN model. The movie
shows the spread of excitation in the cone shaped model
from time=0-30.
The filament after breaking up
time=999
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Result - Convergence
The results of filament length agree
within error bar for three different mesh
sizes
The results of filament number agree
within error bar between dr=0.7 and
dr=0.5. The result for dr=0.5 is slightly
off, which could be due to the filament
finding algorithm
The computation time for dr=0.7 for
one wave period in normal heart size is
approximately 3 hours of cpu time
Filament number and Filament length vs Heart size using our electro-physiological model
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Result - Filaments
Both filament length
Scaling of ventricular turbulence. The log of the
total length and the log of the number of filaments
both have linear relationship with log of heart size,
but with different scale factor.
The average filament length/avearge
heart thickness versus the heart size.
It clearly show that the this average
tends to be a constant
Discussion and 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