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

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 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

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

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),,(2

1

fddf

dddfL

00

z

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Filament finding result

FHN Model: time=2

Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

time=999

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

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

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