Scaling of Ventricular Turbulence

Phase Singularities

Numerical Implementation

Model Construction (cont.)

Conclusions and Future WorkWe have constructed and implemented a minimally realistic fiber architecture model of the left ventricle for studying electrical wave propagation in the three dimensional myocardium.

Our model adequately addresses the geometry and fiber architecture of the LV, as indicated by the agreement of filament dynamics with that from fully realistic geometrical models.

Our model is computationally more tractable, allowing reliable numerical studies. It is easily parallelizable and has good scalability.

As such, it is more feasible for incorporating Realistic electrophysiologyBiodomain description of tissueElectromechanical coupling

Parallelization Numerical Convergence

Model Construction

MotivationVentricular fibrillation (VF) is the main cause of sudden cardiac death in industrialized nations, accounting for 1 out of 10 deaths.Strong experimental evidence suggests that self-sustained waves of electrical wave activity in cardiac tissue are related to fatal arrhythmias.

Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle Xianfeng Song, Sima Setayeshgar

Department of Physics, Indiana University

W.F. Witkowksi, et al., Nature 392, 78 (1998)

Patch size: 5 cm x 5 cmTime spacing: 5 msec

Mechanisms that generate and sustain VF are poorly understood. One conjectured mechanism is: Breakdown of a single spiral (scroll) wave into a disordered state, resulting from various mechanisms of spiral wave instability.

From idealized to fully realistic geometrical modelingRectangular slab Anatomical canine ventricular model

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

Minimally realistic model of LV for studying electrical wave propagation in 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 realistic electrophysiology, electromechanical coupling,

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

Early dissection results revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces.

Our modelAdopted Nested cone geometry fiber surfaces the fiber paths are both geodesics on fiber surfaces and circumferential at midwall.

inner surface outer surface

11

12 sec1

a

'0

),,(2

1

f

d

dfL

dd

dfL

00

zsubject to:

Fiber trajectory:

Transmembrane potential propagation

mm IuDt

uC

)(

1(2

1

aukuvu

v

t

v

uvuaukuIm )1)((

Cm: capacitance per unit area of membraneD: diffusion tensoru: transmembrane potentialIm: transmembrane current

v: gate variableParameters: a=0.1, m1=0.07, m2=0.3, k=8, e=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

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

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.

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

t = 2

t = 999

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.

Crossection along azimuthal direction

Fiber trajectories on nested pair of conical surfacesFiber path equation

Governing Equations

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

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.

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

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

These 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)

Filament finding algorithm

The filament finding results. The left pictures show the simulation at time=2 and time=999. The right pictures show the filament finding results, corresponding to the scroll waves.