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 self-sustained 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 cmTime spacing: 5 msec

From idealized to fully realistic geometrical modeling

Rectangular slab Anatomical canine ventricular model

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

J.P. Keener, et al., in Cardiac Electrophysiology, eds.D. P. Zipes et al., 1995

Courtesy of A. V. Panfilov, in Physics Today, Part 1, August 1996

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: 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

Fibers on a nested pair of surfaces in the LV,from C. E. Thomas, Am. J. Anatomy (1957).

Model construction (cont’d)

Nested 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

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Governing Equations

Transmembrane potential propagation

Transmembrane current, Im, described by simplified FitzHugh-Nagumo type dynamics*

mm IuDt

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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

* R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996)

Cm: capacitance per unit area of membrane

D: diffusion tensoru: transmembrane potential

Im: transmembrane current

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

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plocal

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Local Coordinate Lab Coordinate

Transformation matrix R

RDRD locallab1

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

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

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.

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 all tips are consumed

“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: t = 2

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

t = 999

Numerical Convergence

Filament Number and Filament Length versus Heart size

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.

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

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

Log(total filament length) and Log(filament number) versus Log(heart size)

The average filament length, normalized by average heart thickness, versus heart size

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