Transcript “Distance” between two tips - Department of Physics | Indiana

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
Conclusions
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
Model Construction - Background
Anatomical structure
Picture goes here
Peskin Asymptotic Model
C. S. Peskin, Communications on Pure and Applied
Mathematics 42, 79 (1989)
Conclusions:
 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 –
Nested Cone Approximation
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
 Governing equation
Cm
u
   ( Du )  I m
t
Cm: capacitance per unit area of membrane
D: diffusion tensor
u: transmembrane potential
 Transmembrane current Im was described using a simplified excitable
dynamics equations of the FitzHugh-Nagumo type (R. R. Aliev and A. V.
Panfilov, Chaos Solitons Fractals 7, 293(1996))
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
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
Tips, Filaments
 Tip: The point around
which the spiral wave
(in 2 dimensions) are
generated
Color denotes the transmembrane potential. The
movie shows the spread of excitation in the
cone shaped model from time=0-30.
 Filament: The core
around which that the
scroll wave (in 3
dimensions) rotates
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 consuming all tips
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament finding result
time=2
FHN Model:
time=999
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Numerical 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=1.1 is slightly
off, which could be due to the filament
finding algorithm
Filament number and Filament length vs Heart size
The computation time for dr=0.7 for
one wave period in normal heart size is
less than 1 hours of cpu time using our
electro-physiological model
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Agreement with fully realistic model
The average filament length normalized
Scaling of ventricular turbulence. The log of the
by average heart thickness versus the
total length and the log of the number of filaments
Both filament length
heart size. It clearly show that the this
both have linear relationship with log of heart size,
average tends to be a constant
but with different scale factor.
The results agree with the simulation on the fully realistic model using the same
electro-physiological model (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