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Transition from Ventricular Tachycardia to Ventricular Fibrillation as Function of Tissue Characteristics in a Computer Model
Transition from Ventricular Tachycardia to Ventricular
Fibrillation as Function of Tissue Characteristics in a
Computer Model
Flavio H. Fenton
1
, Alain Karma
2
, Harold M. Hastings
3
, and Steven J. Evans
4
A simplified quantitative ionic model of cardiac action
potential, which reproduces accurate restitution curves, is
used in conjunction with global tissue characteristics such
as rotational cell anisotropy and periodic boundaries to
study the transition from ventricular tachychardia (VT) to
ventricular fibrillation (VF). We give an explanation for
the experimental observation that there is a minimum
tissue mass required for this transition to occur.
Key words Computer simulations, Reentry, VF and VT
I. I
NTRODUCTION
Spiral waves of electrical activity in cardiac tissue are life
threatening because they act as high frequency sources of
waves that induce tachycardia, an abnormal rapid heart beat
that is not controlled by the natural pace maker of the heart.
Once initiated in the ventricle, ventricular tachycardia (VT)
usually decays within a few seconds into ventricular
fibrillation (VF), a more spatiotemporally disorganized
electrical activity leading to sudden cardiac death.
Experiments in the heart using simultaneous multi-site
electrode mappings as well as voltage sensitive dyes, have
shown that in many cases VF is a consequence of several
wandering spiral waves [1,2]. Therefore, it has been suggested
that the rapid transition from VT to VF can be associated with
the breakup of a spiral wave into multiple offspring [3,4] (see
Fig. 1).
To understand this transition, there have been multiple and
extensive studies of two-dimensional spiral waves and three-
dimensional scroll waves in isotropic excitable media using
various models, ranging from simple generic, to detailed ionic
models with explicit membrane processes which accurately
reproduce the cardiac action potential at the single cell level.
[5]
Consequently a fair amount is known about their behavior, and
several mechanisms for breakup have been found [3,4].
Nevertheless, very little is known about their behavior in bulk
ventricular muscle (formed by cardiac fibers in which wave
propagation is faster parallel to their axis than perpendicular to
them and the fiber axis rotates transmurally across the
ventricular walls).
Furthermore, none of the previously found breakup
mechanisms occurring in isotropic excitable media can explain
the results obtained in 1914 by Garrey [6], and multiple
experiments thereafter, corroborating that a minimum mass of
cardiac tissue seems to be necessary for the development of
VT into VF.
Fig. 1 Schematic representation of the transition from normal heart beat to
ventricular tachycardia (VT) and then to ventricular fibrillation (VF) on an
ECG recording.
The initiation of spiral waves (or reentrant waves) is relatively
well understood. They can occur when the propagating
electrical waves produce by the sino-atrial node are blocked by
pre-existing fixed obstacles in the tissue [7] which have very
little or no conduction at all such as scars, calcified plaques, or
orifices.
Blocking of the wave can also be induce by residual obstacles
of necrosis or poorly excitable myocardial regions produced
by prior myocardial infractions.
In any case, the waves can then break and pin to this obstacles,
propagating around them, in a topologically analogous way to
plane wave propagation, with a period determined by the size
of the obstacle and the conduction velocity [7].
Functional reentry is an alternate mechanism for the initiation
of spiral waves, which in contrast to anatomical reentry, can
occur in healthy tissue. In healthy tissue, reentrant spiral
waves can be created and maintained, without obstacles or
local inhomogeneities by functional conduction blocks caused
by: (i) a non-uniform dispersion of repolarization produced by
abrupt changes in stimulation rates and, (ii) unidirectional
blocks created by stimuli induced at the same or other site.
Fig. 2 shows the initiation of spiral waves by a premature
stimulation. Fig. 2 Initiation of spiral waves by a premature stimulus. Dark areas represent
unexcited tissue while yellow-red represents different levels of voltage
excitation. Case 1 (panels A-B): shows a point premature stimulus that
occurs too soon following a plane wave. The tissue behind the plane wave is
refractory and the stimulus dies out. Case 2 (panels A-B): The premature
stimulus occurs too late. The tissue behind the plane wave is readily excitable
and the stimulus produces a bulls eye pattern wave. Case 3 (panels A-L) the
premature point stimulus occurs successfully during the window of
vulnerability producing two mirror image spiral waves of AP that rotate
indefinitely. (See [8] )
When a spiral wave is formed, the point where the front and
the back of the wave meet is called the spiral wave tip. In 3D
the spiral wave becomes a scroll wave and the tip-point
becomes a line or vortex filament as shown in the results
section. One characteristic of spiral wave dynamics is the
trajectory of the spiral wave tip. Depending on the dynamical
tissue characteristics, spiral wave tips can follow different
types of paths [9], ranging from circular to linear trajectories
with very sharp turns. Between these alternatives there is a
gamut of trajectories that follow winding series of loop-like
bends which turn upon themselves, and are known as
meandering trajectories (see Fig. 3).
Fig 3 Color contour plots of action potential spiral waves in different regimes
(from high to low excitability). Black regions represent polarized tissue and
light regions depolarized. Linear, meander and circular trajectories have been
noticed experimentally in cardiac tissue see for example [9].
II R
ESTITUTION
C
URVES
In cardiac tissue, it is possible to define two characteristic
curves that describe how the duration and velocity of a wave
depend on the time interval since the previous activation,
during which the medium recovers its resting properties.
One is the APD restitution, which relates the duration the AP
at a given point in the tissue, with the previous diastolic
interval (DI) at the same point. This DI, or recovery time,
measures the time between successive repolarization and
depolarization of the cell membrane, measured at the same
membrane potential threshold as the APD. The other is the
conduction velocity (CV) restitution, which relates in similar
way, the instantaneous conduction velocity of an excitation
wavefront to the previous DI at the point where the velocity is
measured. These two curves thus incorporate, in a functional
form, the ionic complexity of the tissue as well as its
electrophysiological characteristics.
Fig. 4 A) APD restitution curve and B) CV restitution curve. In cardiac tissue
there is a minimum DI for which there is no propagation of AP. It is the
minimum time the cells need to recover before another excitation can be
induced. These restitutions correspond to the BR model [5].
Numerical simulations with different models have shown that
when the APD restitution curves is steep with slope > 1,
breakup of spiral waves can occur by slow recovery fronts [3]
and by APD oscillations [4]. Nevertheless, most experimental
studies in normal tissue have reported APD restitution curves
with slope < 1. Therefore it remains unclear whether steep
restitution can occur only in diseased tissue, different protocols
are needed to measure APD restitution, or whether other
mechanisms are responsible for spiral wave breakup.
III T
HE
I
ONIC
M
ODEL
To model the propagation of electrical activity and the
dynamics of spiral waves in cardiac tissue. We use a simple
ionic model [10], which is able to reproduce arbitrary
monophasic generic and experimental APD and CV restitution
curves. The total membrane current of the model is given by
the sum of three independent phenomenological ionic currents:
I
t
= I
fi
(V;v) + I
so
(V) +I
si
(V;w).
Where
(i)

I
fi
(V;v) is a fast inward inactivation current used to
depolarize the membrane when an excitation greater
than threshold is induced. This current controls the
CV restitution. It depends on the voltage V, an
inactivation gate variable v and an activation gate that
is extremely fast so its replaced by a Heaviside step
function. I
fi
(V;v) = -v(V-0.13)(1-V)p/ d
. This current
is directly analogous to the sodium current in more
complex models such as the Beeler-Reuter (BR) and
Luo-Rudy (LR) models [5].
(i)

I
so
(V) is a slow time independent rectifying outward
current used to repolarize the membrane back to the
resting potential. This current is analogous to the time independent potassium current in the BR and LR
models [5]. I
so
(V)=V(1-p)/ o
+ p/ r.
(iii)

I
si
(V;w) is a slow inward inactivation current used to
balance I
so
(V) and produce a plateau in the action
potential. It depends on an inactivation gate variable
of an activation