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Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle
Xianfeng Song, Department of Physics, Indiana UniversitySima Setayeshgar, Department of Physics, Indiana University
March 17, 2006
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This Talk: Outline
Goal
Model Construction
Results
Conclusions
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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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
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Model Construction - Background
Anatomical structurePicture goes here
Peskin Asymptotic ModelC. 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
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Model construction –Nested Cone ApproximationNested cone
geometry and fiber surfaces
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Fiber paths on the inner sheet
Fiber paths on the outer sheet
Fiber pathsTo be geodesicsTo be circumferential at the mid wall
11
12 sec1
a
'
),,(2
1
fddf
dddfL
00
z
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Governing equations Governing equation
Cm: capacitance per unit area of membraneD: diffusion tensoru: 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))
mm IuDtuC
)(
1(2
1
aukuv
uv
tv
uvuaukuIm )1)(( 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
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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
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Diffusion Tensor
2
1
//
000000
p
plocal
DD
DD
Local Coordinate Lab Coordinate
Transformation matrix R
RDRD locallab1
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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ParallelizationThe 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
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Tips, Filaments
Tip: The point around which the spiral wave (in 2 dimensions) are generated
Filament: The core around which that the scroll wave (in 3 dimensions) rotates
Color denotes the transmembrane potential. The movie shows the spread of excitation in the
cone shaped model from time=0-30.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Find all tips
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
“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
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Filament finding algorithm
Random choose a tip
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Search for the closest tip
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Make connection
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
“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
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Filament finding algorithm
Continue doing search
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Continue
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Continue
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Continue
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
The closest tip is too far
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Reverse the search direction
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Continue
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Complete the filament
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Start a new filament
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding algorithm
Repeat until consuming all tips
“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
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
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Filament finding result
FHN Model: time=2
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
time=999
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Numerical Convergence
Filament number and Filament length vs Heart size
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
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
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Agreement with fully realistic model
Both filament length
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
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 normalized by average heart thickness versus the heart size. It clearly show that the this average
tends to be a constant
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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