From Idealized to Fully- Realistic Geometrical modeling Scaling of Ventricular Turbulence Phase...
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From Idealized to Fully-Realistic Geometrical modeling
Scaling of Ventricular Turbulence
Phase Singularities
Numerical Implementation
Model Construction
Conclusions and Future Work
We 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 ConvergenceMotivationVentricular fibrillation (VF) is the main cause of sudden cardiac death in industrialized nations, accounting for 1 out of 10 deaths. Experimental evidence strongly 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, Bloomington
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.
Rectangular 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
Construct 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).
LV Fiber ArchitectureEarly dissection results revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces.
inner surface outer surface1
1
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 spatial mesh size 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.
Fiber trajectories on nested pair of conical surfaces
Fiber paths as:
geodesics on fiber surfaces
circumferential at midwall
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 left images show the simulation at time t=2 and t=999 units. The right images show the filament finding results, corresponding to the scroll waves.
Peskin asymptotic model: first principles derivation of toroidal fiber surfaces and fiber trajectories as approximate geodesics.
Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957).
Fiber angle profile through LV thickness: Comparison of Peskin asymptotic model and dissection results,
from C. S. Peskin, Comm. in Pure and Appl. Math. (1989).
Cross-section along azimuthal direction