Am J Physiol Heart Circ Physiol 307: H1024–H1035, 2014.
First published August 8, 2014; doi:10.1152/ajpheart.00593.2013.
Turbulent electrical activity at sharp-edged inexcitable obstacles in a model
for human cardiac tissue
Rupamanjari Majumder,1,2 Rahul Pandit,1,3 and A. V. Panfilov4,5
1
Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, India; 2Laboratory of
Experimental Cardiology, Department of Cardiology, Leiden University Medical Center, Leiden, The Netherlands;
3
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India; 4Department of Physics and
Astronomy, Gent University, Ghent, Belgium; and 5Moscow Institute of Physics and Technology (State University),
Dolgoprudny, Moscow Region, Russia
Submitted 8 August 2013; accepted in final form 26 July 2014
Majumder R, Pandit R, Panfilov AV. Turbulent electrical activity at sharp-edged inexcitable obstacles in a model for human cardiac
tissue. Am J Physiol Heart Circ Physiol 307: H1024 –H1035, 2014.
First published August 8, 2014; doi:10.1152/ajpheart.00593.2013.—
Wave propagation around various geometric expansions, structures,
and obstacles in cardiac tissue may result in the formation of unidirectional block of wave propagation and the onset of reentrant arrhythmias in the heart. Therefore, we investigated the conditions
under which reentrant spiral waves can be generated by high-frequency stimulation at sharp-edged obstacles in the ten TusscherNoble-Noble-Panfilov (TNNP) ionic model for human cardiac tissue.
We show that, in a large range of parameters that account for the
conductance of major inward and outward ionic currents of the model
[fast inward Na⫹ current (INa), L—type slow inward Ca2⫹ current
(ICaL), slow delayed-rectifier current (IKs), rapid delayed-rectifier
current (IKr), inward rectifier K⫹ current (IK1)], the critical period
necessary for spiral formation is close to the period of a spiral wave
rotating in the same tissue. We also show that there is a minimal size
of the obstacle for which formation of spirals is possible; this size is
⬃2.5 cm and decreases with a decrease in the excitability of cardiac
tissue. We show that other factors, such as the obstacle thickness and
direction of wave propagation in relation to the obstacle, are of
secondary importance and affect the conditions for spiral wave initiation only slightly. We also perform studies for obstacle shapes
derived from experimental measurements of infarction scars and show
that the formation of spiral waves there is facilitated by tissue
remodeling around it. Overall, we demonstrate that the formation of
reentrant sources around inexcitable obstacles is a potential mechanism for the onset of cardiac arrhythmias in the presence of a fast
heart rate.
cardiac arrhythmias; computer modeling; reentry; spiral waves; pacing
through the heart encounters multiple functionally and anatomically nonconducting regions at different spatial scales (3, 28, 34, 44, 62– 64, 72). An
electrical activation wave interacts with these structural discontinuities, which exist in the heart in the form of thin planes
of tissues, sheets interconnected by small trabeculae (34), or
fibrotic tissue septa, formed as a result of aging, infarction, or
hypertrophy. The influence of inhomogeneities on electrical
conduction in the heart and their possible roles in the onset of
cardiac arrhythmias have been a subject of extensive research
over the years (11, 29, 37, 38, 44, 59 – 61, 63, 64, 66, 68, 69).
THE ELECTRICAL SIGNAL PROPAGATING
Address for reprint requests and other correspondence: A. V. Panfilov, Dept.
of Physics and Astronomy, Gent Univ., Krijgslaan 281, S9, 9000 Gent,
Belgium (e-mail: Alexander.Panfilov@ugent.be).
H1024
From the point of view of the genesis of cardiac arrhythmias,
the most important effect of conduction inhomogeneities is the
development of functional blocks in the path of a propagating
wave and the subsequent formation of reentrant sources of
arrhythmias, i.e., spiral waves. In most cases, such propagation
blocks result from source-sink mismatch, i.e., the condition
that arises when the local current generated by the wave front
has to be spread to a large area because of tissue geometry and
the wave cannot sufficiently excite the tissue in front of it. This
situation has been studied by Rohr et al. (56), who have shown
that, at the point of abrupt expansion of cardiac tissue, wave
propagation can be slowed down or blocked. Similar effects in
two-dimensional (2D) geometry, for the interaction of the
wave front of a plane wave stimulus with a narrow isthmus,
have been investigated both experimentally and numerically by
Cabo et al. (9). In an experimental preparation of sheep
ventricular epicardial tissue, they created isthmuses, both parallel and perpendicular to the fiber orientation, and observed
that the planar wave front of the electrical stimulus is diffracted
at the isthmus to an elliptical front, with curvature similar to
that produced by point stimulation. The curvature of the front
varies as a function of distance from the isthmus. The velocity
of the wave front is also affected. Their studies reveal that
changes in curvature may lead to slow conduction or propagation block. However, in the situations studied by Cabo et al. (9)
and Rohr et al. (56) wave breaks lead to the total blockage of
wave propagation; they do not result in the formation of
reentrant sources of arrhythmias.
Situations in which the conduction heterogeneities may
result in the formation of new arrhythmia sources were studied
in a 2D active medium, described by the simplified Fitz-HughNagumo equations, by Panfilov and Pertsov (47). They showed
that when an electrical wave bends around an inexcitable strip,
the wave detaches from the boundary with the formation of a
tip; this tip eventually evolves into a rotating spiral wave. This
effect was then studied theoretically and experimentally by
Cabo et al. (10) in sheep epicardial tissue, by using voltagesensitive dyes and video imaging techniques, under the influence of high-frequency stimulations or with partial blockage of
the Na⫹ channels of the myocyte cell membrane. Such destabilization led to the formation of self-sustained vortices of
electrical activity in the heart, in a manner similar to vortex
shedding in hydrodynamic flows past obstacles (73).
Sometimes the location of an inexcitable obstacle (e.g., an
infarction scar) is intramural, with a preserved endocardial/
epicardial rim (5, 17); in such situations, during normal sinus
activation or during ectopic activation occurring at some dis-
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
tance away from a scar, this scar faces the incoming waves
directly. In such a case, wave breaks and vortices can also be
formed under the conditions of high-frequency stimulation,
which has been shown in generic models of excitable media
(48, 50, 65) and in experiments with the Belousov-Zhabotinsky
(BZ) reaction (1).
The aim of this paper was to study a possible process of
spiral wave formation at inexcitable heterogeneity in the ionically realistic ten Tusscher-Noble-Noble-Panfilov (TNNP)
model for human ventricular cells (67). We show that, under
the influence of high-frequency stimulation, a train of plane
waves impinging on an inexcitable obstacle can result in the
formation of spiral waves. Furthermore, in the cardiac model
that we study, the critical frequency at which such formation
can take place is related to the frequency of rotation of a spiral
wave in such tissue. We illustrate how the process of spiral
wave formation depends on the size of the obstacle, its shape,
and the excitability of the medium. We also discuss the
importance of this mechanism for the onset of cardiac arrhythmias.
METHODS
In a continuum model, electrical wave propagation in cardiac tissue
can be described with the monodomain formulation (31), which in two
dimensions (2D) is
⭸V
⭸t
⫽ div共DgradV兲 ⫺
Iion
Cm
(1)
where V is the transmembrane potential, Cm is the membrane capacitance per unit area, div and grad indicate the divergence and the
gradient operators, and D is a symmetrical 2 ⫻ 2 matrix responsible
for the anisotropy of cardiac tissue. For human cardiomyocytes, we
have described the total ionic current Iion by using the TNNP model
(67). It is expressed as a sum of the following ionic currents:
Iion ⫽I1 ⫹ I2;
I1
⫽INa ⫹ ICa ⫹ Ito ⫹ IKS ⫹ IKr ⫹ IK1;
I2
⫽INaCa ⫹ INaK ⫹ IpCa ⫹ IpK ⫹ IbNa ⫹ IbCa .
(2)
INa is the fast inward Na⫹ current, ICa the L—type slow inward Ca2⫹
current, Ito the transient outward current, IKs the slow delayed-rectifier
current, IKr the rapid delayed-rectifier current, IK1 the inward rectifier
K⫹ current, INaCa the Na⫹/Ca2⫹ exchanger current, INaK the Na⫹-K⫹
pump current, IpCa the plateau Ca2⫹ current, IpK the plateau K⫹
currents, IbNa the background Na⫹ current, and IbCa the background
Ca2⫹ current. Time t is measured in milliseconds, voltage V in
millivolts, conductances (GX) in nanosiemens per picofarad, the intracellular and extracellular ionic concentrations (Xi, Xo) in millimoles
per liter, and current densities (IX) in microamperes per unit area in
square centimeters, as used in second-generation models (see, e.g.,
Refs. 7, 35, 36, 67). For a detailed list of the parameters of this model
and the equations that govern the spatiotemporal behaviors of the
transmembrane potential and currents see, e.g., References 61 and 67.
The diffusion constant D is a tensor, whose elements are D11 ⫽
D11
0.00154 cm2/ms, D22 ⫽
, and D12 ⫽ D21 ⫽ 0. We assume that the
4
x-axis is directed along the direction of the cardiac muscle fiber,
whereas the y-axis is directed transverse to the fiber. Cm measures the
membrane capacitance per unit area (␮F/cm2).
To solve Eq. 1 for the temporal evolution of the transmembrane
potential V, we use the forward-Euler method, with a time step ␦t ⫽
0.02 ms; we compute the spatial part of Eq. 1 by using a centered,
second-order finite-difference scheme. The diffusion constant is dif-
ferent along the longitudinal and transverse directions (D11 ⫽ 4D22),
so, to ensure dynamical stability of our numerical scheme, we use
different spatial step sizes for the longitudinal (x) and transverse (y)
directions. We use ␦x ⫽ 0.04 cm and ␦y ⫽ 0.02 cm, such that
D11
D22
⫽ 2 . To model the inexcitable obstacle, we set D ⫽ 0 at the
␦x2
␦y
location of the obstacle and apply zero-flux Neumann boundary
conditions at its edges.
For our studies with the 2D TNNP model, we use a large tissue
containing 512 ⫻ 512 grid points to reduce boundary effects. Thus the
size of the tissue is 20.48 cm ⫻ 10.24 cm. We pace the medium
periodically by applying an external, high-frequency stimulation
along the left boundary of the tissue (x ⫽ 1). We choose the amplitude
of the applied voltage stimulus to be roughly three times the threshold
value, i.e., 150 pA/pF. We start periodical pacing at a high frequency,
which results, however, in a 1:1 response at the stimulation site. This
frequency, which is different for different values of the ionic conductances studied in this article, ranges from 2.5 Hz to 4.0 Hz. At a given
frequency, we keep pacing until the spatial wavelengths of the plane
waves in the wave train stabilize, i.e., the difference between two
successive wavelengths is ⬍5%.
We then increment the frequency by 0.01 Hz and repeat the pacing
procedure. We continue incrementing the frequency until a further
increment leads to the disappearance of a pulse at the stimulation site.
If we need to obtain waves with frequencies close to the maximal
frequency, we repeat simulations from the last 1:1 point, with a
smaller value of the increment (half of the previous value) up to 0.001
Hz. We then stop the external stimulation and allow the system to
evolve.
For the configuration shown in Fig. 9, we use a tissue containing
624 ⫻ 1,248 grid points, with dx ⫽ 0.04 cm, dy ⫽ 0.02 cm, and dt ⫽
0.02 ms. The physical size of the tissue used is 24.96 cm ⫻ 24.96 cm.
To generate a spiral, we use the S1-S2 protocol, with the S1 stimulus
applied at the left boundary (x ⫽ 1) of the tissue and the S2 stimulus
applied, for 2 ms at t ⫽ 440 ms after the application of the first
stimulus, when the wave back of the S1 wave crosses one-fourth of
the extent of the tissue in the x-direction. The S2 stimulus, with
amplitude 150 pA/pF, is applied over the region y ⱕ 312. The core of
the spiral wave forms at the point of intersection of the S2 stimulus
and the wave back of the S1 wave.
In most of our studies, we use rectangular, conduction-type obstacles of different sizes; in addition, we also study two representative
cases of r-shaped obstacles. Such obstacles are constructed from
rectangles of different dimensions. Figure 1 shows schematic diagrams of the different kinds of obstacles that we use in our studies;
Fig. 1B shows the r-shaped obstacle, Fig. 1C shows a laterally
inverted r-shaped obstacle, and Fig. 1D shows the distribution of
obstacles in a large tissue, which we use in our simulations, along with
the positions and sizes of each of the obstacles in units of grid points.
For simulations involving many obstacles (Fig. 9, B and C) and the
simulation shown in Fig. 10D, we surround each inexcitable obstacle
by a localized heterogeneity with conductance GX. To do this, we use
a phase-field-type formulation (see, e.g., Ref. 23); we define a parameter ␸ whose values are, initially, ␸ ⫽ 0 inside the obstacles and ␸ ⫽
1 outside. Then we evolve ␸ spatiotemporally for a short time by
using the following relaxation-type equations:
⭸␾
⭸t
⫽ ␨2 ⵜ ␾ ⫺
G共 ␾兲 ⫽
⭸G
⭸␾
;
⫺ 共2␾ ⫺ 1兲2
4
⫹
共2␾ ⫺ 1兲4
(3)
8
This causes the sharp boundary between the ␸ ⫽ 1 and ␸ ⫽ 0 regions
to smoothen out. We then use this distribution of ␸ for the next part
of the calculation. The parameter ␨ controls the spatial range over
which the heterogeneity exists. We choose ␨ ⫽ 0.025. Next, to
AJP-Heart Circ Physiol • doi:10.1152/ajpheart.00593.2013 • www.ajpheart.org
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
GIMP 2.6 to coarse-grain the resulting image, in order to remove
noise and filter out a well-defined boundary for the scar. This is
important because in numerical simulations the treatment of boundaries often becomes a crucial determinant of the results of the study.
The original image was obtained from a slice of a rat heart; a human
heart is substantially larger than a rat heart, so we have rescaled this
image as follows. We have taken mirror images of the scar along the
x-axis and the y-axis (to make it symmetrical and thus circumvent
boundary effects arising from asymmetry of the scar) and have scaled
it up 50 times; the total area of the rescaled scar we use is ⬃17 cm2,
which is close to that reported in a human heart (19.3 ⫾ 13.4 cm2)
(17). Finally, we have used MATLAB 7.5 to read the resulting image
and convert it into a binary input file that stores the configuration of
the scar.
RESULTS
Fig. 1. Schematic diagrams of the types of obstacles used in our study. A: a
rectangular obstacle of height h and width d; B: the r-shaped obstacle with
vertical segments of height h1 and h2 and horizontal segments with widths d1
and d2. C: the laterally inverted “r-shaped” obstacle. D: multiple obstacles.
Dimensions of the 6 obstacles and their exact positions in the tissue as depicted
on left are listed in table on right. Here n denotes the obstacle number
(indicated on left above each obstacle); px and py denote, respectively, the xand y-grid point locations of the left bottom corner of each obstacle. Three
arrows on left of A indicate the direction of the incident waves relative to
obstacles A, B, and C (see text).
introduce the ionic heterogeneity around the obstacles, we relate the
ionic conductance GX for ion X to ␸ as follows:
GX ⫽ fGXnormal ⫹ ␾GXnormal共1 ⫺ f 兲
GXnormal
for ␾ ⫽ 0,
otherwise
(4)
Here, GXnormal is the normal value of GX from the original parameter
1
set of the TNNP model. In Fig. 9B, we use f ⫽ . To check the
3
validity of our results in more realistic conditions (Figs. 9C and 10D),
we modify the properties of cardiac tissue around each obstacle, as
prescribed by Arevalo et al. (4), to mimic gray zone (GZ) cells;
specifically, we reduce the peak Na⫹ current to 38% of its normal
value (f ⫽ 0.38), the peak L—type Ca2⫹ current to 31% of its normal
value (f ⫽ 0.31), and the peak K⫹ currents IKr and IKs to 30% and
20% of their respective normal values, i.e., f ⫽ 0.30 and 0.20,
respectively.
In the case studied in Fig. 9A, no localized heterogeneity is
assumed around the obstacles; thus the value of GX remains constant
over the whole tissue; in the case studied in Fig. 9B, all conductances
except GKr remain spatiotemporally constant; GKr obeys relation
1
shown in Eqs. 4 and 5 with f ⫽ . This is similar to the work of
3
Campbell et al. (12), who also introduce a gradient in IKr to induce a
heterogeneity around the obstacles.
Finally, to model realistic scar tissue, we apply the following
protocol: We use GIMP 2.6 (a Linux-based open-source imaging
software package) to read the image from Fig. 2B of Reference 57.
We then desaturate its color code and create two object groups. In the
first group, which represents the scar, we include all the different
heterogeneities shown in the figure; in the second group, which
represents the cardiomyocytes, we include everything outside the scar
from Fig. 2B of Reference 57. We then use the thresholding tool of
When a plane wave, generated from a line stimulus that is
applied along the left edge of the tissue, impinges on an
obstacle, it splits at the front and develops a gap between the
upper and lower portions of the wave. We find that, at pacing
periods (Tpacing) higher than a critical value Tcritical, this gap
shrinks and disappears as the wave front of the interacting
wave leaves the right edge of the obstacle. Figure 2 illustrates
the process described above for a rectangular obstacle of size
0.8 cm ⫻ 3.6 cm when paced with a period Tpacing ⬎ Tcritical.
The incident plane wave stimulus hits the obstacle (Fig. 2, A
and D) and splits into two parts (Fig. 2, B and E), which go
around the obstacle and rejoin (Fig. 2, C and F); no spiral
formation takes place in this case.
However, when the period of pacing becomes equal to the
critical period (i.e., Tpacing ⫽ Tcritical), we find that the wave
cannot follow the boundary of the obstacle, as the wave front
and wave back of the wave leave the right edge of the obstacle.
The split waves thus produced then curl about their respective
broken tips, and spiral formation takes place within the tissue.
Figure 3 illustrates the process described above with reference
to a rectangular obstacle of size 0.8 cm ⫻ 3.6 cm when paced
with a period that is equal to the critical pacing period. The
incident plane wave of electrical activation hits the obstacle (as
shown in Fig. 3, A and D) and wave breaks appear; they detach
from the obstacle, and the gap between the breaks increases
(Fig. 3, B and E); finally, these wave breaks give rise to two
fully developed spiral waves (Fig. 3, C and F).
Previous studies of chemical waves interacting with inexcitable obstacles (1, 48) have shown that Tcritical is close to the
period of spiral waves rotating in the same medium. Therefore,
in order to study the dependence of Tcritical on parameters of the
model, as well as on the time period (Tspiral) of a spiral wave in
the absence of obstacles, we explored the variations of both
Tcritical and Tspiral for the same parameter values. In our study
we varied the values of the conductances GNa, GCa, GK1, GKr,
and GKs, which control the main inward ionic currents, INa and
ICa, and the main outward ionic currents, IK1, IKr, and IKs,
respectively. These currents are major determinants of action
potential shape and are targets of antiarrhythmic drugs of class
I (Na⫹ channel blockers), class III (K⫹ channel blockers), and
class IV (Ca2⫹ channel blockers) (75). These currents are also
changed during various cardiac pathologies, e.g., long and
short QT syndromes (30, 41, 52). Figure 4, A–E, illustrate the
dependence of Tcritical and Tspiral on the values of GNa, GCa,
GK1, GKr, and GKs, respectively, expressed as percentages of
their original values. We observe that, in a wide range of
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
H1027
Fig. 2. Interaction of plane waves with a period Tpacing ⫽ 310 ms with a rectangular obstacle of size 0.8 cm ⫻ 3.6 cm for original values of the parameters in
the ten Tusscher-Noble-Noble-Panfilov (TNNP) model (A–C) and for conductance GNa reduced to 10% of the original value, keeping other parameters unchanged
(D–F). Here, Tpacing is greater than the critical pacing period Tcritical. The incident plane wave stimulus hits the obstacle (A and D) and splits into 2 parts (B and
E) (indicated by arrow), which go round the obstacle and rejoin (C and F); no spiral formation takes place. The approaching of wave breaks toward each other
is most clearly evident in D–F, where the action potential duration is shorter than in A–C.
parameters, Tspiral is close to Tcritical, and only for few values
of the parameters do we see significant differences between
Tspiral and Tcritical (see Fig. 4).
For spiral formation, another important parameter is the
height of the obstacle h. For Tpacing ⯝ Tcritical, spiral formation
takes place only when h is above a critical value. If the obstacle
does not meet this critical size requirement, then, no matter
what the pacing period, spiral formation does not occur;
instead, we observe a situation similar to that illustrated in Fig.
2. To study the role of h in the formation of spirals, we perform
studies similar to those of Fig. 3 for progressively decreasing
obstacle sizes [decreasing h, with width (d) held fixed at 0.8
cm], until we find a minimum h below which spiral waves do
not form. We record this value of h as hcritical and repeat the
entire procedure for other values of GNa, GCa, GK1, GKr, and
GKs; Fig. 5, A–E, respectively, illustrate our results. We observe that hcritical is around a few centimeters. In addition, we
see that the critical h is substantially affected by the parameters
of the model. When we decrease GNa, the obstacle size decreases more than twofold. A decrease of GCa results in a
fivefold decrease in the value of hcritical. The trend is just the
opposite for conductances GK1, GKr, and GKs. We note that as
GK1 increases the critical obstacle size reduces to half of its
value, whereas an increase in GKr leads to a fivefold reduction
in the obstacle size; the variation with GKs follows the same
trend as that for GK1.
Another characteristic spatial constant of the system is the
spatial distance between the successive waves in the wave
train, which we refer to as the spatial wavelength (␭). Note that
this spatial wavelength is a product of the velocity and the
period (cycle length) and it differs from the wavelength widely
used in physiological literature, which is the product of the
1
Supplemental Material for this article is available online at the Journal
website.
Fig. 3. Interaction of plane waves with a period Tpacing ⫽ Tcritical with a rectangular obstacle of size 0.8 cm ⫻ 3.6 cm for the original parameter values of the
TNNP model (A–C; Tcritical ⫽ 260 ms) and for GNa reduced to 10% of the original value, keeping all other parameters unchanged (D–F; Tcritical ⫽ 464 ms). The
incident plane wave stimulus hits the obstacle (A and D), and wave breaks appear (B and E) (indicated by arrow); when the stimulation is stopped, these breaks
evolve into spirals (C and F). Again, as in Fig. 2, the separation of wave breaks, away from each other in the case of spiral formation (as indicated by arrow),
is more evident in D–F, where the spatial size of the propagating wave is shorter than in A–C because the velocity of wave propagation is slower at GNa ⫽ 10%.
The mechanism for the formation of secondary spirals is demonstrated in Supplemental Video S1.1
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
Fig. 4. Dependence of Tcritical and Tspiral on
the conductances of the main ionic currents.
A–E show the dependence of Tcritical and
Tspiral on GNa, GCa, GK1, GKr, and GKs,
respectively, expressed as % of their original
values. Tspiral denotes the time period of a
spiral produced in a homogeneous tissue, in
the absence of obstacles. Tcritical denotes the
critical period of pacing required to trigger
the formation of spirals in the same tissue
but in the presence of a single obstacle of
size 0.8 cm ⫻ 3.6 cm, located at the center of
the tissue. We observe that Tspiral and Tcritical
for most parameter values are close to each
other. The only exceptions to this statement
are the cases of GNa ⫽ 5%, where Tcritical is
significantly shorter than Tspiral (P ⬍ 10⫺10;
Tspiral ⫽ 1,662.15 ⫾ 64.53 ms, Tcritical ⫽
436.74 ⫾ 3.23 ms). Also, Tspiral is significantly shorter than Tcritical for 25% GK1 (P ⫽
0.01; Tspiral ⫽ 324 ⫾ 5 ms, Tcritical ⫽ 350 ⫾
2 ms), for GKs ⫽ 25% (P ⫽ 0.02; Tspiral ⫽
290 ⫾ 1.1 ms, Tcritical ⫽ 300 ⫾ 2.1 ms), and
for GKr ⫽ 175% (P ⫽ 0.04; Tspiral ⫽ 237 ⫾
1.1 ms, Tcritical ⫽ 249 ⫾ 2.5 ms).
conduction velocity and the action potential duration (APD).
The wavelength changes when we change the conductivities of
the ionic currents; therefore, we study whether the change in
hcritical can be explained just by the change in ␭. In Fig. 6, we
hcritical
plot the dimensionless size of the obstacle given by
vs.
␭
different ionic conductances (cf. Fig. 5 for hcritical). We observe
hcritical
that the plots for hcritical vs. GX and
vs. GX follow similar
␭
trends. If we change GNa from 100% to 10%, hcritical reduces by
hcritical
a factor of 3.5, whereas
reduces by a factor of 1.75.
␭
Similarly, if we reduce GCa from 100% to 10%, hcritical reduces
hcritical
by a factor of 7, whereas
reduces by a factor of 4.23.
␭
Again, if we reduce GK1 from 200% to 25%, hcritical increases
hcritical
by a factor of 2.5, whereas
increases by a factor of 1.94.
␭
Similarly, if we change GKr from 200% to 25%, hcritical
increases by a factor of 4.25, whereas
hcritical
increases by a
␭
factor of 3.16. Finally, if we reduce GKs from 200% to 25%,
hcritical
hcritical increases by a factor of 2.25, whereas
increases
␭
by a factor of 1.8. All these data indicate that the wavelength
has a significant contribution in determining hcritical: ⬃50% of
the observed changes can be attributed to the change of ␭.
However, there is also a wavelength-independent component,
and additional factors underlying the change in hcritical require
further investigation.
Next, we study how the geometry of the obstacle affects
Tcritical. We obtain Tcritical for different values of GNa and GCa
from plots similar to those in Fig. 4. In these studies, we set
h ⫽ 3.6 cm (the critical h for GNa ⫽ 100% and GCa ⫽ 100%)
and vary d (see Fig. 1A). We find that, for a rectangular
obstacle, if h meets the critical requirement then Tcritical is
independent of the value of d. This is illustrated in Fig. 7
(dashed line). However, Tcritical is influenced slightly by the
shape of the obstacle for nonrectangular obstacles. To demon-
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
H1029
Fig. 5. Dependence of hcritical on the conductances of the major ionic currents in the
model. A–E show the dependence of hcritical
on GNa, GCa, GK1, GKr, and GKs, respectively, expressed as % of their original values. We observe that hcritical is of the order of
a few centimeters for d ⫽ 0.8 cm.
strate this influence, we conduct studies on obstacles of two
representative shapes. In one case we use an r-shaped obstacle
(see Fig. 1B), and in the other case we use a laterally inverted
r-shaped obstacle (see Fig. 1C).
In both of these cases, we use h1 ⫽ 3.6 cm, h2 ⫽ d2 ⫽ 0.8
cm, and d2 ⫽ d. The dependence of Tcritical on d for an r-shaped
obstacle is shown in Fig. 7 (dash-dot line); the dependence of
Tcritical on d for a laterally inverted r-shaped obstacle is also
shown in Fig. 7 (solid line with circles). We observe that
Tcritical increases very slightly with an increase in d in the case
of the r-shaped obstacle; by contrast, Tcritical decreases very
slightly with an increase in d in the case of the laterally
inverted r-shaped obstacle.
In addition to its dependence on the shape and size of the
obstacle, Tcritical depends on factors such as the angle of incidence
of the external stimulation on the boundary of the obstacle. To
verify this, we position the obstacle at the center of the tissue, as
shown in the schematic diagram of Fig. 8A. The position of the
obstacle is marked by a filled rectangle; we then pace the medium
from a square region of area 0.8 cm ⫻ 0.8 cm, which is shown in
Fig. 8A by an open square. We then shift the geometric center of
the pacing site along the perimeter of an ellipse, and, for each
location of the pacing site, we measure Tcritical. We measure the
angle ␪ with respect to the horizontal. The size of the obstacle
used for these studies is 0.8 cm ⫻ 3.6 cm. Figure 8B shows
the angular dependence of Tcritical for the original values of the
parameters GNa and GCa. We observe that Tcritical at ␪ ⫽ 0° agrees
with our findings for pacing by a line stimulus. As ␪ increases,
Tcritical decreases slightly. At large values of ␪ (⬎60°), spiral
formation is no longer observed, because the pacing site is then
close to the short edge of the obstacle, the length of which is less
than the critical size of the obstacle required for the formation of
spirals.
From our results we can conclude that formation of breaks at
obstacles is not very likely to be a primary mechanism of spiral
wave formation in cardiac tissue from the normal sinus rhythm.
This is because such break formation requires that the frequency of the upcoming waves is faster than that of a spiral
wave, and thus the heart rate should already be as fast as that
during cardiac arrhythmia (3– 4 Hz). However, it can be a
potential secondary mechanism for spiral wave formation if the
arrhythmia has appeared by a different mechanism, although,
even in that case, the frequency of the arrhythmia should be
faster than that of rotating spiral waves. This means that in a
homogeneous cardiac tissue a rotating spiral wave cannot
produce a secondary reentrant source, as the frequency of the
spiral wave must be lower than the critical frequency necessary
for break formation. It is reasonable to assume that properties
AJP-Heart Circ Physiol • doi:10.1152/ajpheart.00593.2013 • www.ajpheart.org
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
hcritical
on the conduc␭
tances of the major ionic currents in the
hcritical
model. A–E show the dependence of
␭
on GNa, GCa, GK1, GKr, and GKs, respectively, expressed as % of their original values.
Fig. 6. Dependence of
of tissue around obstacles can differ from those in the rest of
the cardiac tissue. In such cases, we can observe the formation
of secondary spiral waves via the mechanism we have elucidated above. To check this in direct numerical simulations, we
study spiral wave dynamics in a large, homogeneous, anisotropic tissue, in the presence of many obstacles of varying
lengths, which are located at different angles with respect to
the core of the parent spiral wave. The results of this study are
illustrated in Fig. 9. In Fig. 9A, we use the original parameter
values of the TNNP model throughout the tissue. We find that
the presence of multiple obstacles does not affect spiral wave
dynamics significantly, except for a local influence around
each obstacle; this influence is confined in space and does not
spread as the system evolves. Such wave patterns, reminiscent
of the excitation during ventricular tachycardia (VT), are
characterized by a single stable reentrant source (25). In Fig.
9B, we illustrate a state of ventricular fibrillation (VF), characterized by multiple wavelet reentrant patterns (25, 40),
brought on by the presence of a heterogeneity in GKr localized
in the region surrounding each obstacle. This is in consonance
with the work of Campbell et al. (12).
We have also performed simulations in which the local
heterogeneity, around the obstacles, is based on the measured
properties of cardiac tissue around an infarction scar region. In
particular, we use a GZ-type heterogeneity (see METHODS). The
results for this simulation are illustrated in Fig. 9C. We see
secondary spiral formation around the obstacles; Fig. 9A shows
a state that is the analog of VT, whereas Fig. 9, B and C, show
analogs of VF.
Finally, we conduct three representative simulation studies
of high-frequency pacing and spiral wave dynamics in the
presence of experimentally obtained and designed infarct tissue. In these studies, we simulate a sheet of myocardial cells
containing a patch of inexcitable obstacles; the shape of this
patch is obtained from Reference 57. By using the GIMP 2.6
imaging software package, we read the image from Fig. 2B of
Reference 57 (for details, see METHODS). Because the original
image was captured from a slice of a rat heart and the size of
the human heart is substantially bigger than that of the rat
heart, we had to rescale the image 50 times; this resulted in a
total area of the scar of ⬃17 cm2, which is close to that
reported in the human heart (19.3 ⫾ 13.4 cm2) (17).
We apply the pacing protocols that we have used to obtain Figs.
2 and 9; we then study wave propagation around such realistic
obstacles under periodical forcing and also in the presence of a
preexisting spiral wave (in both homogeneous and heterogeneous
cases). The results of these calculations are illustrated in Fig. 10.
We find that periodic forcing causes the formation of secondary
spirals (Fig. 10A) and the critical period for spiral formation in this
case is 254 ms, which is slightly shorter than the critical period of
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
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primary spiral wave forms secondary spiral waves. This result
is illustrated in Fig. 10D (and its spatiotemporal evolution is
demonstrated in Supplemental Video S6).
DISCUSSION
Fig. 7. Dependence of Tcritical on the horizontal length d of the obstacle.
Dash-dot line is for an r-shaped obstacle; solid line with circles indicates the
case of a laterally inverted r-shaped obstacle; dashed line is for a rectangular
obstacle. We observe that, in the case of the rectangular obstacle, Tcritical is
independent of d for the values of d used. Tcritical increases very slightly with
increase in d in the case of the r-shaped obstacle, whereas Tcritical decreases
very slightly with increase in d in the case of the laterally inverted r-shaped
obstacle.
spiral formation around a rectangular obstacle, for original parameters of the TNNP model (260 ms). This may be in part because
of the angular dependence of Tcritical (see Fig. 8) and in part
because of differences in the interaction of waves with a boundary
of complex shape. Spiral wave formation here takes place at
sharp, distinct regions in the obstacle, where the curvature
changes abruptly: We obtain four secondary spirals (see Fig. 10A),
which, after the cessation of the forcing, evolve in time into fully
developed, self-sustained spiral waves (Fig. 10B and Supplemental Video S5).
Furthermore, to study the effect of such an obstacle on
preexisting spiral waves of electrical activation in the medium,
we use a spiral wave as the initial condition and place the
obstacle at the center of the tissue. The primary spiral wave is
produced by an S1-S2 cross-field protocol, such that its core is
located away from the center of the tissue. We find that, as the
system is allowed to evolve, the primary spiral wave remains
detached from the obstacle; secondary spiral formation does
not take place (Fig. 10C); however, if we introduce a GZ-type
ionic heterogeneity around the obstacle, then the tip of the
In this report we show that high-frequency simulation can result
in the formation of spiral waves around inexcitable obstacles. We
carried out studies in the TNNP ionic model for human ventricular
tissue and show that, in a large range of parameters, the critical
time period of pacing necessary for spiral formation is related to
the period of a spiral wave rotating in the same tissue. We also
show that there is a minimal size of the obstacle for which the
formation of spirals is possible, and this size is fairly large: In
normal conditions it is ⬃3 cm and it decreases with a decrease of
the excitability of cardiac tissue, and at low tissue excitabilities it
can be as low as 0.7 cm. These sizes are comparable to typical scar
sizes reported in clinical studies (17). We also show that other
factors, such as the obstacle shape and the direction of stimulation
in relation to the obstacle, are of secondary importance and affect
the conditions for spiral wave initiation only slightly.
Effects of major ionic currents. In the present study we have
changed the conductances GNa, GCa, GK1, GKr, and GKs that
control the main inward and outward ionic currents. For inward
currents we have considered blocks up to 5% of the normal
value, and for the outward currents we have considered blocks
up to 20%, as well as gains up to 200% of the original value.
The blocking of these currents mimics the action of various
antiarrhythmic drugs. In many cases, a drug can affect multiple
currents. For example, it has been reported that one of the most
widely used antiarrhythmic preparations, namely, amiodarone,
blocks the inward Na⫹ and Ca2⫹ channels as well as the
outward K⫹ channels (24, 27, 33, 43, 53, 74, 78, 80, 82, 84);
its effects on a single cardiomyocyte include a slight elevation
in the normal resting membrane potential of healthy and
ischemic cells and prolongation of the APD, although in some
cases it can lead to APD shortening as well, and it leads to a
reduction of the action potential amplitude. Amiodarone is also
found to decrease the conduction velocity and the wavelength
of signals in healthy cardiac tissue, whereas in ischemic tissue
the same drug leads to an increase of the wavelength. Cardiac
arrhythmias can also occur as a result of adverse side effects of
noncardiological drugs, e.g., cisapride (13, 19, 76, 80), which
works mainly by inhibiting IKr. In a cardiomyocyte, cisapride
Fig. 8. Dependence of critical frequency of spiral wave on orientation of the obstacle. A: schematic representation of the tissue indicating the position of the obstacle
(filled rectangle) and the various pacing sites (open squares). The size of the obstacle is 0.8 cm ⫻ 3.6 cm, and the pacing site is a region enclosed within a square of
side 0.8 cm. B: angular dependence of Tcritical for the original values of the parameters GNa, GCa, GK1, GKr, and GKs. We observe that Tcritical at ␪ ⫽ 0° agrees with our
findings for pacing by a line stimulus. As ␪ increases, Tcritical decreases slightly. At large values of ␪ (⬎60°) spiral formation no longer takes place because the pacing
site is then close to the short edge of the obstacle, the length of which is less than the critical size of the obstacle required for the formation of spirals.
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
Fig. 9. Formation of wave breaks in the presence of multiple obstacles in a medium with preexisting spiral wave activity. A–C show spiral wave dynamics in
a homogeneous, anisotropic tissue, with many obstacles. In all 3 cases, we use the original parameter values of the TNNP model. In B, around each obstacle we
decrease GKr smoothly from its original value to up to 1/3 of the original value, at the boundaries of the obstacles. This regional slowing down of the electrical activation
of the medium facilitates the formation of secondary spirals in B; such secondary spirals are completely absent in A. In C, we further modify properties of cardiac tissue
around the obstacles, to mimic changes around an infarction scar (see METHODS for details). All 3 images are recorded at t ⫽ 5 s after the formation of the primary spiral
wave. The spatiotemporal evolution of transmembrane potential V in cases A–C is shown in Supplemental Videos S2–S4, respectively.
does not affect the resting membrane potential or the action
potential amplitude significantly but prolongs the APD in both
healthy and ischemic cases; furthermore, it does not affect the
signal conduction velocity but increases the wavelength in
cardiac tissue. Therefore, to study the effects of drugs on the
mechanisms studied in this report one needs not only to
analyze the change caused by the individual ion channels but
also to study specific combinations reproducing specific drug
Fig. 10. Formation of wave breaks around a realistic infarction scar, by
high-frequency pacing. A shows the formation of secondary wave breaks
around the 4 potential sources of this scar (image recorded at t ⫽ 7.62 s after
the first line stimulus was applied). B shows the development of these wave
breaks into self-sustained spiral waves, 6.08 s after the first appearance of
sustained secondary spirals. C illustrates the effect of the presence of the scar
tissue on the dynamics of preexisting spiral waves in the medium but in the
absence of gray zone (GZ)-type heterogeneity around the scar (image recorded
at t ⫽ 4.78 s after the formation of the primary spiral wave). D shows the
dynamics of the primary spiral wave, in the presence of a realistic scar and
GZ-type heterogeneity at the infarction scar boundary (image recorded at t ⫽
4.78 s after the formation of the primary spiral wave).
effects. Such research can be done in the future with modeling
approaches as in Reference 80.
Similar questions arise in various forms of genetic diseases,
such as in the Brugada syndrome, in which there is a 50% INa loss
of function localized in the right ventricular outflow tract; this is
considered severe (8, 79). However, in addition to the change in
INa, the Brugada syndrome is also associated with several other
changes, e.g., an increase in the fibrosis of cardiac tissue (15).
Therefore, it is interesting to study the combination of changes
in ionic currents with other factors (e.g., fibrosis); such issues
can also be addressed by using modeling approaches (45, 69).
Critical pacing period and critical obstacle size. We have
shown that the critical obstacle height hcritical and pacing rate
Tcritical depend on the conductances of inward (GNa, GCa) and
outward (GK1, GKr, GKs) currents. The critical pacing rate is
determined in a wide range of parameters by the period of a
spiral wave rotating in the same tissue. Such dependence can
be qualitatively explained in the following way: When a plane
wave, from a train of pulses paced at the critical pacing period,
impinges upon an obstacle, it breaks at the edges of the
obstacle that are parallel to the direction of propagation of the
incident waves. After breaking at the obstacle, the wave forms
two free ends. Under normal conditions of tissue excitability,
these free ends grow toward each other and rejoin to form an
intact wave; spiral formation does not take place. However, the
velocity of motion of the broken wave’s free ends, toward each
other, decreases with a decrease in the excitability of the
medium (49, 65). Tissue excitability is known to decrease in
the presence of high-frequency pacing of the medium (48, 65).
Furthermore, rotating spiral waves can be considered to exhibit
periodic (e.g., circular) motion of a wave break. Thus the
period of a spiral wave can be considered as the pacing period,
at which the free ends of the broken wave cease to move
toward each other, thus enabling the formation of spiral waves.
This explains why Tcritical is comparable to the time period of
rotation of the spiral wave. A more accurate estimate of Tcritical
for low-dimensional models (i.e., models with fewer variables
than the present model) can be obtained by using the analytical
methods developed in Reference 65. It would be interesting to
develop a more quantitative description of the observed pro-
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
cesses. Interesting results in that direction can potentially be
derived from the study of Starobin et al. (65), who have studied
the formation of vortices around single and multiple L-shaped
obstacles in generic models of excitable media and have
derived analytical estimates for wave front-obstacle separation
and vortex formation by using singular perturbation theory.
They have shown that any decrease in tissue excitability favors
vortex formation; also, the formation of secondary vortices
takes place in the presence of multiple obstacles. It would be
interesting to study whether their approach can explain the
large difference between Tspiral and Tcritical at low excitability
(as shown in Fig. 4A) and the differences for points where
Tcritical ⬎ Tspiral or Tcritical ⬍ Tspiral shown in Fig. 4A.
Our results show that a decrease of the inward currents or an
increase of the outward currents decreases the critical obstacle
size and also the APD. Both these factors decrease the APD
and some (GNa) velocity of the wave. As the formation of two
counterrotating spiral waves requires two action potentials to
enter the gap separating the free ends of the waves, we can
assume that a smaller spatial break separation is necessary at
the obstacle for a shorter spatial APD. This can explain the
observed decrease of the critical obstacle size shown in Fig. 5.
Critical period and refractoriness. We relate the critical
period for spiral wave formation via pacing to the period of the
spiral wave rotating in the same tissue. However, the period of
spiral wave rotation, by itself, is determined by the refractory
properties of cardiac tissue and an excitable gap, which is again
determined by the curvature (source-sink mismatch) effects
(14, 46, 51). In highly excitable tissue, the refractoriness is
expected to be the main determinant of the spiral wave period
(46). This may explain the findings of References 10 and 50,
where the refractory period has been shown to be a potential
determinant of the critical pacing frequency for spiral wave
formation. The value of the refractory period, however, depends on the frequency of excitation of cardiac cells and other
experimental conditions, such as the threshold of stimulation,
stimulation protocol, etc. (6, 86). It would be interesting to
perform a detailed study of the relation between the spiral wave
period and the refractory period of cardiac tissue in various
conditions and possibly relate the results observed in our study
to other measurable characteristics of cardiac tissue.
Dependence on shapes, geometries, and types of obstacles.
In this article, we study the formation of spiral waves, with
high-frequency stimulation, around an inexcitable obstacle
with no-flux conditions at its boundaries. However, similar
processes can also occur in the presence of ionic heterogeneities in cardiac tissue. Xu and Guevara (83) studied the effects
of high-frequency stimulation, in a heterogeneous medium
with heterogeneity induced by regional myocardial ischemia,
by using the modified Luo-Rudy ionic (LRI) model for cardiac
tissue. They observed that at a high pacing frequency the wave
front separates increasingly from the boundary of heterogeneity, which eventually results in the formation of a spiral wave;
the core of this spiral lies outside the heterogeneity. It would be
interesting to study a similar process in the TNNP model and
relate the frequency of onset of spiral waves in that situation
with the frequency of spiral waves rotating in the same tissue.
Our results are similar to the vortex shedding results in Reference 10. As in Reference 10, we show that high-frequency pacing
and low excitability favor the formation of spiral waves at sharp
H1033
edges. The main mechanism of spiral wave formation is the
detachment of the wave from the obstacle boundary.
Although Cabo et al. (10) show that the main mechanism for
the formation of spiral waves is the detachment of the wave from
the obstacle boundary, in our simulations we find different patterns of interaction between the wave and the inexcitable obstacle:
In particular, we must account for both the upcoming wave and
the wave following the boundary of the medium. Furthermore, we
extend this line of research by providing a generic estimate of the
critical pacing period Tcritical necessary for spiral wave formation,
and we relate it to the period of a spiral wave. We estimate the
critical size hcritical of the obstacle necessary for spiral formation at
different levels of tissue excitabilities and account for its dependence on GNa, GCa, GK1, GKr, and GKs.
The relation of the critical period of stimulation Tcritical and the
period of spiral wave rotation has not been studied so far in direct
experiments in cardiac tissue. However, Cabo et al. (10) found
that spiral formation at an inexcitable barrier in sheep epicardial
tissue occurred only if the interval between stimuli did not exceed
⬎10% of the refractory period of the tissue. A similar condition
has also been proposed theoretically by Pertsov et al. (50). As in
cardiac tissue, the spiral wave period is close to the refractory
period; the finding by Cabo et al. indicates that, indeed, the spiral
wave period can be a good estimate for Tcritical.
We show in Figs. 9 and 10 that, in the presence of heterogeneity, a rotating spiral wave can induce a fibrillatory excitation pattern about an obstacle. This is similar to the mechanism of mother rotor fibrillation studied in References 12 and
58. It would be interesting to study the dynamics of the
observed fibrillatory patterns obtained by this mechanism for a
long time interval and characterize them in terms of regularity
and their power spectra. However, if obstacles are associated
with tissue remodeling, as, e.g., around an infarction scar, the
formation of secondary spiral waves around the obstacles
becomes possible and may potentially result in the onset of
ventricular fibrillation by the mother rotor mechanism (58). We
defer a detailed study of this issue to future work.
We have performed simulations for large-sized tissues in
order to separate the effects of 1) the interaction of waves with
the inexcitable obstacle and 2) boundary effects. Because wave
interactions with an obstacle or scar are local, all our results
regarding the formation of wave breaks and their initial dynamics should be unchanged for tissues whose size is smaller
than those we consider. The formation of sustained spiral
waves from wave breaks requires some space, so such spiral
formation might well depend on the size of the tissues. We
propose to study these issues in detail in future research in
which we plan to extend our work here by using anatomically
accurate ventricular models and boundaries, three-dimensional
tissue slabs with muscle fiber orientation.
Limitations. The principal limitations of our study are the
following: We do not use a detailed representation of the
anisotropy of cardiac tissue; in many important situations wave
propagation is three dimensional, and thus it would be interesting to study similar processes in three dimensions, especially in an anatomically realistic heart with various shapes of
realistic scars; we do not study in detail the effects of possible
tissue remodeling around the obstacles, except in one simulation (Fig. 10); our study can also benefit from using a bidomain
description for the cardiac tissue, especially if we study defibrillation of the observed patterns.
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OBSTACLE-INDUCED TURBULENCE IN HUMAN CARDIAC TISSUE
ACKNOWLEDGMENTS
R. Majumder thanks Soling Zimik for his help during the revision of the
manuscript.
GRANTS
We thank the Department of Science and Technology (DST), India, the
University Grants Commission (UGC), India, the Council for Scientific and
Industrial Research (CSIR), India, and the Robert Bosch Centre for Cyber
Physical Systems (IISc) for support and the Supercomputer Education and
Research Centre (SERC, IISc) for computational resources. The research of
A. V. Panfilov is supported by the Research-Foundation Flanders (FWO
Vlaanderen).
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
Author contributions: R.M. performed experiments; R.M. and A.V.P. analyzed data; R.M. and A.V.P. interpreted results of experiments; R.M. prepared
figures; R.M. and A.V.P. drafted manuscript; R.M., R.P., and A.V.P. edited
and revised manuscript; R.M., R.P., and A.V.P. approved final version of
manuscript; A.V.P. conception and design of research.
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