Xianfeng Song, Department of Physics, Indiana University

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 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 scalable

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

Anatomical Heart

A nested layered geometry for the left ventricleA single macroscopic fiber bundle starting at the

basal plane outside the midwall traverses down toward the apex on an outer surface, and at some point before reaching the apex, changes direction, traverses back along an inner surface reinserting at the basal plane inside the midwall.

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.

i=8

e=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):

Result

'

),,(2

1

f

d

df

dd

dfL

00

z

11

12 sec1

a

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)

Transmembrane current Im was described using a simplified excitable dynamics equations of the FitzHugh-Nagumo type (R. R. Aliev and A. V. Panfilov, 1996)

mm IuDt

uC

)(

1(2

1

aukuvu

v

t

v

uvuaukuIm )1)((

Parameters: a=0.1, 1=0.07,2=0.3,k=8,=0.01, Cm=1

Diffusion Tensor

2

1

//

00

00

00

p

plocal

D

D

D

D

Local Coordinate Lab Coordinate

Transformation matrix R

RDRD locallab1

Numerical Implementation

Working in spherical polar coordinates, with the boundaries of the computational domain described by two nested cones, reduces the numerics to computing in a box.

Standard finite differencing is used to treat the spatial derivatives, along with explicit Euler time-stepping

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

CPUsSpeed 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 filamentFinding all tips

Add current tip into a new filament, marked as the head of this filament

Find the closest unmarked tip

End

Choose an unmarked tip as current tip

Is the distance smaller than a certain

threshold?

Set the closest tip as current tip

Mark the current tip

set reversed=0

Add current tip into current filament

Set the head of current filament as current tip

Is revered=0?

Are there any unmarked tips?

set reversed=1

Definition: Distance between two tips

(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

Finding the filament

Finding all tips

Add current tip into a new filament, marked as the head of this filament

Find the closest unmarked tip

End

Choose an unmarked tip as current tip

Is the distance smaller than a certain

threshold?Set the closest tip as current tip

Mark the current tip

set reversed=0Add current tip into

current filament

Set the head of current filament as current tip

Is revered=0?

Are there any unmarked tips?

Set reversed=1

Definition: Distance between two tips

(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

Yes

No

Yes

Yes

No

No

Result - Simulation

FHN Model:

Color denotes the u variable in FHN model. The movie shows the spread of excitation in the cone shaped model.

Filament initially

The filament after break up

Result - Convergence

Filament number and Filament length vs Heart size

As the mesh size decreases, the quantitive behavior convergent to a certain value

The result for dr=0.7 agree with the result for dr=0.5 within the error

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.

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 very good scalability

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