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Virtual Sources and Sinks During Extracellular Field
Shocks in Cardiac Cell Cultures
Effects of Source-Sink Interactions Between Adjacent Tissue Boundaries
Aleksandar A. Kondratyev, MD, PhD; Jean-Philippe Didon, PhD; Helene Hinnen-Oberer;
Mathieu Lemay, PhD; Jan P. Kucera, MD; Andre G. Kleber, MD
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Background—One mechanism by which extracellular field shocks (ECFSs) defibrillate the heart is by producing changes
in membrane potential (Vm) at tissue discontinuities. Such virtual electrodes may produce new excitation waves or affect
locally propagating action potentials. The rise time of Vm determines the required duration of a single defibrillation pulse
to reach a critical threshold for activation or for the modification of ion channel function, and depends on the electric
and microstructural characteristics of the tissue.
Methods and Results—We used optical mapping of Vm in patterned cultures of neonatal rat ventricular myocytes to assess
the relationship between cardiac structure and the early time course of Vm during ECFSs. At monolayer boundaries, the
time course of Vm showed a close fit to the theoretical change predicted by theory, with a membrane time constant of
2.65⫾0.19 ms (n⫽13) and a length constant of 159⫾6 ␮m (n⫽10). Experiments in patterned strands, mimicking
the resistive boundaries that occur naturally in the heart, explained the observation that the rate of rise and the
maximal amplitudes of the Vm changes are inversely related because of electrotonic interactions between structural
boundaries. Interrupting ECFSs by very short intervals diminished Vm, but did not cause major changes in its
overall time course.
Conclusions—Interaction between virtual sinks and sources decreases the magnitude of the changes in Vm but accelerates
its time course. For efficient defibrillation, short ECFSs are needed, with an amplitude adapted to match the boundary
interaction. (Circ Arrhythm Electrophysiol. 2012;5:391-399.)
Key Words: electrophysiology 䡲 defibrillation 䡲 electric mapping
C
ardiac cells form a network that allows rapid propagation
of the electric impulse and coordinated mechanical
contraction. This process is determined by the electric properties of the cells, the extracellular space, and the architecture
of the cellular network.1 The same factors also determine how
the tissue reacts to an electric field shock applied between
extracellular electrodes. Such shocks are used to defibrillate
the heart and to measure the passive electric properties of the
tissue itself.2– 4
ulation artifacts. Early work using this technique has shown
that an ECFS can produce either membrane depolarization or
hyperpolarization. Thus, changes of opposite polarity have
been observed within very short distances (⬇1 mm) in
perfused rabbit hearts.7 Recent studies on the effect of short
defibrillation pulses have demonstrated the importance of
intramyocardial virtual electrodes caused by the coronary
vasculature.9
While membrane potential sources close to the extracellular electrodes are mostly produced by the macroscopic
boundaries of the heart itself,10 “far-field” virtual sources11,12
most likely are caused by the bidomain nature5,13 and the
structural organization of myocardial tissue.14,15 Theoretically, structural discontinuities may occur in both intra- and
extracellular domains. In a tissue culture model, we have
shown that such histological boundaries form major sources
for transmembrane current flow and that they represent
preferential sites for the initiation of propagated excitation,16
Clinical Perspective on p 399
The effect of extracellular field shocks (ECFSs) on cardiac
tissue is complex and depends on the field strength of the
shock, the tissue structure, the bidomain nature of cardiac
tissue, and the electric properties of the cardiac cells.2,5 The
introduction of optical mapping of transmembrane potential
(Vm) has enabled the direct measurement of Vm in whole
hearts6,7 and tissue cultures,8 without interference from stim-
Received October 6, 2011; accepted February 22, 2012.
From the Department of Physiology, University of Bern, Bern, Switzerland (A.A.K., H.H-O., M.L., J.P.K., A.G.K.); Schiller Incorporated Laboratory,
Wissembourg, France (J-P.D.).
The online-only Data Supplement is available at http://circep.ahajournals.org/lookup/suppl/doi:10.1161/CIRCEP.111.968180/-/DC1.
Correspondence to Dr André G. Kléber, Department of Pathology, Beth Israel Deaconess Medical Center, DANA 752, Harvard Medical School, 330
Brookline Avenue, Boston, MA 02215, E-mail [email protected]
© 2012 American Heart Association, Inc.
Circ Arrhythm Electrophysiol is available at http://circep.ahajournals.org
391
DOI: 10.1161/CIRCEP.111.968180.
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whereas resistive barriers formed by individual cell borders
play only minor roles.8
The time course of Vm in response to an ECFS is important
because it will determine the time needed by an ECFS to
produce biologically relevant changes of membrane potential.
These changes include the initiation, prolongation, or shortening of action potentials, which may interrupt fibrillation.
Work in cell cultures using high resolution optical mapping
of Vm has shown that, depending on pulse strength, the
responses of Vm to ECFS exhibit 3 typical phases.17 The very
early phase produces symmetrical hyper- and depolarization
at the respective anodal and cathodal boundaries. This symmetry suggests that it can be attributed to the passive linear
electric properties of the myocyte network. The second phase,
which is superimposed on the first phase at intermediate
levels of shock strength, most likely is caused by a change in
the time- and voltage-dependent properties of ion channels.
The third phase, observed at high field strengths, is characterized by a decrease in the Vm response and probably is
associated with electroporation.8,18 –20 A relatively fast initial
change in membrane potential may increase the efficiency of
short defibrillation pulses to excite tissue at the sites of
intramyocardial virtual sources, and it has been suggested
recently that repetitive short pulses (requiring less energy)
can be efficient in defibrillation.9,21
In the present study we used the experimental model of
patterned cardiac cell cultures to assess the time course of the
first, very early phase of Vm changes after application of an
ECFS. Whereas the tissue culture model is limited and cannot
assess the full spectrum of variables determining the effect of
ECFSs attributable to the bidomain nature of cardiac tissue,
such as anisotropy, fiber curvature, heterogeneous extracellular resistance, and heterogeneous extracellular fields, it is
nevertheless ideally suited for the reproducible control of the
architecture of the cellular network.22 The time dependence of
Vm during an ECFS reflects the charge and discharge of the
myocyte membrane capacitance, because the extracellular
and intracellular spaces have no major capacitive elements.
Therefore, use of the cell culture model, devoid of the
features playing a role in a bidomain model (as mentioned
above), seems appropriate to test the specific effect of local
tissue boundaries, which are present in the myocardium in
vivo,14,23 on the time dependence of ECFS. In order to
provide a mechanistic explanation for the observed interrelation between Vm and structure, we compared the experimental with theoretical results derived using a 1-dimensional
model of cardiac tissue.4,24
Methods
Production of Patterned Cell Cultures
The production of patterned cell cultures from neonatal rat heart cells
has been described in detail.8,16,22 In brief, hearts from neonatal rats
were excised, enzymatically digested to form a cell suspension,
preplated to eliminate fibroblasts, and seeded on coverslips at a
density of 0.5x106 cells per mL. Before seeding, defined fibronectin
patterns were produced by microphotolithography to determine cell
attachment and thus to produce well defined cell culture boundaries.
After seeding, the cultures were kept in an incubator for 6 to 8 days
at 35°C.
Figure 1. Schematic illustration of the experimental setup. A,
Schematic of a cell culture dish showing a continuous layer of
cultured neonatal rat heart cells patterned in the shape of half a
disk (black). A bipolar electrode was used to stimulate the culture at an S1–S1 interval of 500 ms. Two platinum electrodes
(gray) were arranged in such a way to produce a homogenous
electric field (arrows) oriented perpendicular to the linear border
of the cell culture. B, Schematic of a cell culture dish showing
the culture pattern used to assess the effects of boundaries on
the time course of Vm during an extracellular field shock.
Strands of 200 ␮m in width emerge from a bulk monolayer of
cells (black). The bulk is stimulated at an S1–S1 interval of 500
ms. The electric field (arrows) produced by the platinum electrodes (gray) is oriented perpendicular to the strands. C, Electric
equivalent circuit of a passive cable of length L. L indicates the
distance between resistive borders; ri, resistance of the intracellular space per unit length; rm, membrane resistance per unit
length; cm, membrane capacitance per unit length (see onlineonly supplement for further explanation).
For experiments, 2 types of patterns were used, as illustrated in
Figure 1. For the determination of passive cable properties, patterns
were constructed in the shape of a half disk with a sharp rectilinear
boundary. For the investigation of the interactions between adjacent
boundaries, linear strands were patterned with a width of 200 ␮m
(interboundary distance). In all cultures the arrangement and shape
of the myocytes was isotropic.8
Stimulation and Optical Mapping of
Transmembrane Potential
Transmembrane potential was recorded by multisite optical mapping
of transmembrane potential, as described in detail previously8,16 (see
online-only supplement). The change in transmembrane potential
was expressed in percentage of action potential amplitude
(%APA).25
Application of Extracellular Field Shocks
Propagated action potentials were elicited via electric stimulation
using a bipolar electrode at a basic (S1-S1) interval of 500 ms.
ECFSs were applied via 2 platinum electrodes placed in the tissue
bath (Figure 1). The electric field produced in this way was
homogeneous, as previously shown.8 ECFSs were produced by a
custom-built device. This device produced single or repetitive
Kondratyev et al
Membrane Potential and Extracellular Shocks
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393
Vm⫽V0⫹Vamplitude*erf共公共t/␶兲兲
Mathematically, changes in amplitude and shape of Vm during an
ECFS due to interactions between boundaries can be derived from
the 1-dimensional cable equation corresponding to the equivalent
circuit described in Figure 1C, and the application of the so-called
superimposition and reflexion principles (see online-only Data Supplement Equation IS).24
Computations and fitting algorithms were implemented in MATLAB,
and the exponential fit (equation 1) needed for the determination of
␭ was programmed in LabView (National Instruments).
Statistics
Experimental values were compared using the nonpaired t test where
appropriate. Differences were considered significant at P⬍0.05.
Unless specified otherwise, values are expressed as mean⫾ SD.
Results
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Early Time Course of the Change in Vm During
Application of an Extracellular Field Shock
Figure 2. Changes of Vm after application of a hyperpolarizing
extracellular field shock during phase 4. A, Schematic representation of the cell culture with 5 measuring sites (circles) close
to the culture border. The white arrow represents the extracellular field (6.36 V/cm). Superimposed signals from the 5 locations
show hyperpolarization followed by the upstroke of the action
potential. B, Hyperpolarizing portion of the change in Vm from
the location indicated by the filled circle on Panel A. The red
curve corresponds to the fit with equation 2.
rectangular voltage pulses that could be varied with respect to field
strength, number, interpulse interval, polarity, duration, and latency
after the basic stimulus (S1-S2 interval).
Computation of Time Dependent Changes in
Transmembrane Potential, Determination of the
Space Constant ␭ and the Membrane
Time Constant ␶
In our experiments we consistently applied ECFS exactly perpendicular to the tissue boundaries. We have previously shown that such
an arrangement produces changes in Vm with isopotential lines
parallel to the boundaries (see online-only Data Supplement Figure
IS). This relationship between the patterned tissue boundary and the
extracellular field facilitates the computation of the change in Vm,
because a 1-dimensional model can be used. The linear equivalent
circuit illustrated in Figure 1C comprises the intercellular resistance
per unit length (ri), the resistance of the cell membrane (rm), and
membrane capacitance (cm). The membrane time constant (␶) and
length constant (␭) are defined in their usual way (␭2⫽rm/ri; ␶⫽rm *
cm).24 For L⬎⬎␭, which corresponds to the single boundary case in
the cell culture (Figure 2A), ␭ can be obtained from the exponential
decay of Vm along the cable at the end of a long ECFS (t 3 ⬁):2– 4,24
(1)
Vm⫽V0⫹Vamplitude*exp(⫺x/␭)
and the ␶ is obtained from the change in Vm close to the boundary by:
The time course of Vm during application of an ECFS during
phase 4 and the plateau phase of the action potential was
determined at the border of a dense culture of myocytes
(Figure 2A). Selection of this single boundary geometry
represents the “extreme” case, where interaction between
boundaries is excluded (L3⬁). Figure 2A shows 5 signals
from measuring sites close to the culture border during a
hyperpolarizing ECFS applied 30 ms before an S1 stimulus.
Superimposition of the signals reflects the homogeneity of
the electric field and Vm, and at the culture border. The
change in Vm produced by the field pulse is depicted in
Figure 2B. The red line in Figure 2B corresponds to the fit
with equation 2 and shows that Vm during the ECFS follows
closely the time course predicted by equation 2 with a
membrane time constant of 3.08 ms. Approximately 90% of
maximal hyperpolarization was achieved after about 4 ms. In
each experiment, the membrane time constant ␶ was obtained
as the average from the fit of equation 2 to 5 signals located
at the culture boundary. In 13 different cultures, the mean
value of ␶ amounted to 2.70⫾0.19 ms. During phase 4,
determination of ␶ following depolarizing ECFS was not
possible, because action potentials were elicited close to the
boundary, which precluded an appropriate fit with equation 2
in most experiments.
Vm elicited by an ECFS of 6 to 8 ms during the plateau
phase of the action potential is depicted in Figure 3A.
Similarly to a hyperpolarizing pulse during phase 4, there was
an accurate fit to equation 2 that yielded a membrane time
constant of 3.24 ms. At the boundary, 90% hyperpolarization
was reached after approximately 5 ms. The mean ␶ was
3.57⫾0.47 ms (n⫽8). As shown previously17 and as illustrated in Figure 3A, the time course of hyperpolarizing pulses
during the plateau was different from depolarizing pulses.
Hyperpolarization produced an initial change in Vm symmetrical to depolarization, and after the initial pulse segment of
approximately 0.5 ms duration, a subsequent rapid repolarization was observed, probably corresponding to so called
“all-or-nothing” repolarization,17,26 caused by activation of
repolarizing ion currents. This precluded the determination of
␶ for hyperpolarizing pulses.
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Figure 3.Changes of Vm after application of an extracellular
field shock during the plateau phase of the action potential. A,
Schematic representation of the cell culture with 5 measuring
sites (circles) close to the culture border. The white arrow
represents the extracellular field (6.36 V/cm). Superimposed
signals from the 5 locations show changes in Vm following
hyperpolarizing and depolarizing shocks. The black arrow
marks the onset of “all-or-nothing” repolarization. B, Depolarizing portion of the change in Vm from the location indicated
by the filled circle on Panel A. The red curve corresponds to
the fit with equation 2.
The Effect of Adjacent Tissue Boundaries on the
Time Course of Extracellular Field Shocks
To assess the effect of interaction of virtual electrodes at
tissue boundaries on the time course and amplitude of Vm
during an ECFS (phase 4 of the action potential), we
applied ECFSs perpendicular to 200 ␮m wide strands
(Figure 1B). Figure 4A illustrates the peak deflection of
Vm (measured at the end of the pulse) in response to a
single ECFS of 4 ms, as a function of position across the
strand (from the border adjacent to the anode to the border
adjacent to the cathode; mean⫾ SE, n⫽5, field strength
6.36V/cm). The interaction of the virtual electrodes at
tissue boundaries is reflected in the shape of the peak Vm
deflection profile. The linear aspect of the profile is in
agreement with previous observations in cell cultures.8 The
maximal amplitude measured at the cathodal boundary of
a strand of 200 ␮m in width amounted to 23⫾2%APA
(n⫽5). In contrast, at a single boundary, the same field
strength produced a significantly larger depolarization
(filled square with asterisk), of 38⫾4%APA (n⫽7,
P⬍0.05, nonpaired t test), demonstrating experimentally
that the proximity of tissue boundaries decreases the
maximal deflection of Vm induced by an ECFS.
Figure 4. A, Effect of virtual source-sink interaction on the maximal deflection of Vm across a strand of 200 ␮m in width during
an extracellular field shock (ECFS; 6.36 V/cm) applied during the
plateau phase of the action potential (filled circles). For comparison, the peak change in Vm at the border of a large monolayer
(absence of virtual source-sink interaction) is given for the same
field strength (filled square). See text for details. B, Time course
of Vm during an ECFS. Each data point represents the mean
value from 5 experiments. The changes in Vm are depicted for
recording sites located at different distances from the median
axis of the strand (labels). The interrupted curve (open symbols)
corresponds to the theoretical result (see online-only Data Supplement Equation I), with ␶⫽3.5 ms and ␭⫽180 ␮m. Note that
the maximal Vm change is already reached at t ⬍␶.
Figure 4B illustrates Vm at various distances from the
middle axis of the strand (⫺103 ␮m to 103 ␮m) in the same
experiments shown in Figure 4A. The ECFS resulted in an
initial rapid change of Vm reaching a plateau level at
approximately 2 ms, that is, earlier than in the presence of a
single boundary and in the absence of virtual source interaction (Figures 2 and 3). The red curve shows the comparison
with the theoretical prediction (see online-only Data Supplement Equation I). Altogether, Figure 4 demonstrates that the
interaction of virtual sources at resistive boundaries has 2
major effects on Vm during an ECFS, (1) a decrease of the
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Figure 5. A, Original experimental traces recorded at the border of a 200 ␮m wide strand showing the action potential upstroke and
the changes in Vm induced by depolarizing extracellular field shocks (ECFSs; 6.36 V/cm) applied during the action potential plateau.
The upper trace depicts changes induced by the control pulse (3.5 ms), the lower trace changes induced by 4 pulses of 0.5 ms duration interrupted by I⫽0.5 ms. The test ECFSs are shown schematically in the inset of Panel A. B, Superimposition of computed
changes in Vm induced by 4 test pulses of 0.5 ms duration interrupted by I⫽0.1 ms (solid red curve) and a control pulse of equal duration (solid blue curve). Points connected by dotted lines depict experimentally recorded values (blue, control; red, test). Panels C and D
are the same as Panel B, but I⫽0.2 ms (Panel C) and I⫽0.5 ms (Panel D). Hash symbols (#) denote significant differences between test
and control (nonpaired t test, P⬍0.05). The values given at the 100% levels denote the absolute changes in percentage of action
potential amplitude. As expected, the changes in Vm decrease with increasing interpulse interval I. Of note is the observation that the
general time course of Vm (envelope) showing a rapid increase to a plateau does not change with increasing I.
maximal change in Vm, and (2) an increase in the relative rate
of change in Vm (see Discussion).
Comparison of the Effects of Single Versus
Multiple Pulses
Modifying the time course of ECFS pulses, and thus influencing Vm changes, may be useful to optimize the delivered
energy during electric pulses, and thus important for the
design of defibrillation devices. Figure 5A shows optical
signals depicting Vm changes caused by a continuous ECFS
(control) and an ECFS interrupted 3 times (I⫽0.5 ms). The
changes in Vm during a train of 4 pulses of 0.5 ms duration,
interrupted by 3 intervals I are shown in Panels B–D (Panel
B: I⫽0.1 ms, total duration 2.3 ms; Panel C: I⫽0.2 ms, total
duration 2.6 ms; Panel D: I⫽0.5 ms, total duration 3.5 ms).
These 3 protocols correspond to a decrease of delivered
energy by 13%, 23%, and 43%, with respect to control pulses
of the same total duration. Data are represented as mean⫾ SE
immediately before each interruption (n⫽6 for Panels B and
D, n⫽5 for Panel C). For each series, the corresponding
changes in Vm recorded during control experiments without
pulse interruption (blue) are superimposed on the changes
measured with pulse interruption (red). The computed changes
(see online-only Data Supplement Equation I; solid curves) are
superimposed on the experimentally determined values (circles
connected by dotted lines). Comparison of Figure 5B–5D shows
that increasing I from 0.1 ms to 0.5 ms, with a pulse duration of
0.5 ms, lowers the level of the maximal deflection of Vm but
does not affect the general time course of the voltage change.
The curves computed from Online Supplement equation I
confirm this general behavior, but show a faster rise in Vm
during control and test interventions at all intervals I. At I⫽0.1
ms, Vm at the end of the interruption and the plateau level are not
significantly different from control (Figure 5B). The plateau
level amounts to 77% of control (simulated value 91%), with
I⫽0.2 ms (Figure 5C), and to 70% of control with I⫽0.5 ms
(simulated value 85%; Figure 5D).
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sources with isopotential lines parallel to the tissue boundary.
This made it possible to apply a relatively simple
1-dimensional model for the computation of the theoretical
predictions and comparison with the experimental results.
Because in 1 dimension, the monodomain and bidomain
formulations are equivalent, this 1-dimensional simplification
permitted us to apply a monodomain approach. The quality of
the fit suggests that our experimental model is well represented by the electric circuit shown in Figure 1C.
The Effect of Strand Boundaries on the Size and
Time Course of Virtual Sources
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Figure 6. Determination of the space constant ␭ by a hyperpolarizing extracellular field shock (ECFS). A, Array of measuring
sites extending from the culture border (on the left). Two signals
showing hyperpolarization during am ECFS (6.36 V/cm) are
shown. B, Steady-state amplitudes of hyperpolarizing voltage
changes are plotted as a function of distance from the culture
border. The exponential fit (interrupted curve) using equation 2
yields a space constant of 152 ␮m (square).
Comparison of the Time Course of Vm During
Different Phases of the Action Potential and
Determination of the Length Constant ␭
To simulate the changes in Vm during the early phase of an
ECFS, it was necessary to determine the length constant ␭
based on experimental measurements. The length constant ␭
was determined from the decay of the steady-state Vm at the
end of a sufficiently long (8 –10 ms) hyperpolarizing pulse
applied during phase 4 of the action potential. Vm from up to
48 sites was recorded in each experiment and plotted as a
function of distance from the boundary (Figure 6). The
exponential fit (equation 2) in the experiment illustrated in
Figure 6B produced a value of ␭⫽152 ␮m. The mean value
of ␭ amounted to 159⫾6 ␮m (n⫽10). During the plateau
phase of the action potential, the well-known27 relative
increase in membrane resistance was responsible for the
increase in ␶. The associated increase in ␭ produced a very flat
profile of steady–state Vm along the optically mapped distance x,
which rendered an exponential fit analogous to Figure 6B
inaccurate. Therefore, the value for ␭plateau was calculated from
␭phase 4, ␶plateau, and ␶phase4; it amounted to 188 ␮m.
Discussion
A major purpose of this study was the determination of the
time course of the early change in Vm induced by an
extracellular field shock and its dependence on tissue boundaries. Application of a homogeneous extracellular field shock
perpendicular to a single tissue boundary produced virtual
One of the major advantages of recording Vm using voltagesensitive dyes is the absence of stimulation artifacts during
the application of ECFSs. In experiments in situ, this made it
possible to detect the virtual sources on the epicardial surface
and at intramural sites.2,6,7,9,11,28 It has been shown that virtual
sources and sinks coexist within distances of ⬍1 mm.7 A
similar close association between local hyper- and depolarizations more recently has been described in isolated perfused
rabbit hearts by Mowrey et al.28 Interestingly, these authors
found a strict inverse relationship between the maximal
amplitude of a virtual source caused by an ECFS and the rate
of rise, in correspondence with our results. This inverse
correlation was proposed as a general principle of tissue
behavior and independent of the application of drugs that
inhibited ion channels, suggesting indirectly that it might be
related to the intrinsic passive electric properties of the tissue.
The mechanistic explanation for this inverse relation is
provided in this work. Figure 7 depicts the theoretical
changes in the maximal amplitude and the time course of Vm
during an ECFS in absence and presence of interactions
between adjacent boundaries. It can be seen that the Vm
perpendicular to the border of the obstacle shows an exponential decay in the absence of interaction between the virtual
source and virtual sink at the boundaries (blue line), in
accordance with the experimental results (Figure 6).2–5 If the
length L of the excitable structure (corresponding to the
strand width in Figure 1B) decreases, the profile of Vm during
an ECFS assumes an increasingly linear shape (red curve)
and the absolute values of the amplitude maxima or minima
of Vm decrease, due to the electric interaction between the
strand borders. Concomitant with the decrease in amplitude,
source-sink interaction produces a more rapid change in the
transmembrane potential during the shock (Panel B).
Interruption of ECFS by Short Intervals
Changing the time course of ECFS of a given duration may be
useful to reduce the delivered energy, and it may be important
for the design of the charge-delivering device. The effect of
complex pulse forms on the efficiency of defibrillation was
investigated in detail by Malkin et al.19 These authors made
the empirical observation that the defibrillation efficiency of
an ECFS was highest in presence of a short initial peak in the
defibrillation pulse form. In our study, the maximal Vm
deflection in 200 ␮m wide strands was changed only to a
minor extent (⬍10%) if pulses of 2 to 3 ms in duration were
interrupted by 3 short (0.1– 0.2 ms) intervals, corresponding
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Figure 7. Computation of changes in Vm
(see online-only Data Supplement Equation I) illustrating the inverse relationship
between maximal amplitude and steepness of the change in Vm. A, Effect of
cable length on the amplitude at steady
state (t 3 ⬁) of the Vm response caused
by a rectangular extracellular field shock
(ECFS). With decreasing cable length,
the interaction between the flow of
transmembrane current at opposing
ends (virtual electrodes of opposite
polarity) produces a decrease in maximal
amplitude. B, Time course and amplitudes of Vm during ECFS in cables of
2 mm (blue) and 200 ␮m length (red).
Amplitude at L⫽2 mm is taken as 100%
(␭⫽188 ␮m, ␶⫽3.57 ms). C, Same computed signals as in Panel B, but with
normalized amplitudes.
to a decrease of the delivered energy by up to 23% (see the
online-only supplement for the calculation). Thus, in terms of
energy expenditure, the interrupted pulse represents an advantage. These experimental results and theoretical simulations taken together suggest that the interactions between
boundaries, producing a very rapid initial change in Vm,
should be taken into account when optimizing ECFS pulse
forms. Malkin et al19 attributed the observation that a short
initial peak in the defibrillation pulse form increases the
efficiency of defibrillation to the possible occurrence of
electroporation. We have previously shown that electroporation (which manifests itself by an internalization of the dye
Lucifer yellow8 or by a decrease in the amplitude of Vm late
during an ECFS17) occurs only at relatively high field
strengths (⬎20 V/cm) in our experimental model (see onlineonly supplement). A contribution of electroporation to the
observed changes in Vm in the present experimental setting is
therefore unlikely.
Limitations and Validity of the Cell Culture Model
The present study focuses specifically on the interaction
between virtual sources and sinks caused by the vicinity of
resistive boundaries during a single ECFS, and on the
theoretical basis underlying the observed Vm changes. Multiple important variables in defibrillation were not assessed,
such as the effect of (1) shape and the polarity of the shocks,
(2) the anisotropic architecture of cardiac tissue, (3) the
possibility that inhomogeneities in extracellular space resis-
tance create virtual sources, and (4) multiple pulses with
longer interpulse intervals. Indeed, such repetitive pulses
with intervals in the order of the defibrillation cycle length
have been efficiently used to sequentially interrupt reentrant circuits.9,21 A further aspect relates to the question of
the validity of the cell culture model for representing the
cellular network in vivo. It has been shown that the
interplay between the resistive properties of the extra- and
intracellular spaces is important for the formation of
virtual electrodes in the myocardium.5,13,29 Cultures from
neonatal rat myocytes have passive electric properties
different from adult tissue in vivo. The most important
difference is probably cell size.30 –33 Decreasing cell size
will decrease the spacing between cell borders, and consequently, increase intracellular resistance, ri (cytoplasmic
and intercellular resistance in series). This most likely
explains the observation that the length constant ␭ in rat
neonatal cultures is significantly smaller than in adult
hearts.34,35 Other factors, such as effects of the extracellular space resistance, ro, and ion channel expression (affecting membrane resistance rm) additionally will affect ␭.
Thus far, only 1 study dealt specifically with the determination of cable properties in cell cultures of rat cardiomyocytes.34 Irrespective of these considerations, an important
property of our experimental model is that it consists of a
cell monolayer and not of a multi-layered preparation, and
there is therefore no ambiguity in our optical measurements that might arise because of voltage-dependent fluorescence emitted from deeper tissue layers.
398
Circ Arrhythm Electrophysiol
April 2012
Potential Importance of Virtual Source-Sink
Interactions for Defibrillation
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Work in cell cultures and in whole hearts has demonstrated
the importance of intramyocardial virtual sources as site of
excitation waves, which may interrupt reentrant circuits
during defibrillation.9 The presence of resistive boundaries
caused by the myocardial architecture, blood vessels, and
bidomain tissue properties are important causes for the
formation of virtual sources. In larger mammals the space
constant, reflecting the degree of electric interaction between
adjacent myocardium, is larger than in murine hearts.1,35
Therefore, source-sink interaction of virtual sources as shown
in the present study is expected to occur between sources and
sinks located more than 1 mm apart in these species. The
interaction between physiological boundaries represented by
normal transmural architecture and blood vessels is therefore
likely to affect the maximal value of virtual sources and the
rate of rise of Vm. Moreover, the experimentally demonstrated dependence of the time course and amplitude of Vm on
source-sink interaction is expected to occur independently of
the mechanism causing virtual source formation.
Conclusion
The inverse relationship between the maximal Vm deflections
caused by sources and loads and the rate of rise of Vm during
the ECFS may have implications regarding the effect of
defibrillation shocks: First, single pulses of a duration shorter
than the membrane time constant are likely to produce
excitation at sites of source-sink interaction. Second, to
reach the threshold for excitation at such a site, increasing
the amplitude of a short ECFS will be more efficient than
increasing its duration for an equal amount of energy
delivered.
Sources of Funding
This study was supported by the Swiss National Science Foundation
(grant 310030 –120253 to A.G.K.) and by a grant from Schiller Inc.,
Wissembourg, France (to A.G.K.).
Disclosures
Dr André G. Kléber is the recipient of a research grant from Schiller
Incorporated, Wissembourg, France. Dr Jean-Philippe Didon is an
employee of Schiller Incorporated, Wissembourg, France.
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8. Gillis AM, Fast VG, Rohr S, Kleber AG. Spatial changes in transmembrane potential during extracellular electrical shocks in cultured
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Membrane Potential and Extracellular Shocks
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CLINICAL PERSPECTIVE
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The application of an electric pulse is routinely used in clinical practice to revert atrial and ventricular fibrillation to sinus
rhythm. Such pulses can be applied either externally or with implanted devices. Despite decades of research, the
complexities inherent to the mechanisms of defibrillation at the tissue and cell levels still remain elusive. The prevailing
hypothesis is that discontinuities in cardiac tissue structure lead to localized changes in membrane potential (Vm) during
extracellular field pulses, called virtual current sources and sinks. These changes may block existing or create new
excitation waves, which interfere with reentrant circuits. One of the aspects pertaining to improving defibrillation
techniques is to optimize electric charge delivery during the shock. In this study, we investigated the response of cultured
strands of neonatal rat ventricular myocytes to rectangular electric field shocks and to shocks separated by short
interruptions. In parallel, we simulated these responses using classical cable theory. In both the experimental and the
theoretical model, we observed that and explain why dynamic interactions between tissue boundaries markedly shorten the
rise time and decrease the amplitude of the Vm responses. This inverse relationship between the magnitude and rise time
of virtual sources indicates that very short shocks of adequate magnitude may produce changes in Vm efficient in
defibrillation. Moreover, brief interruptions in the shocks result in changes in Vm, which are minor in comparison to the
reduction of the delivered energy.
Virtual Sources and Sinks During Extracellular Field Shocks in Cardiac Cell Cultures:
Effects of Source-Sink Interactions Between Adjacent Tissue Boundaries
Aleksandar A. Kondratyev, Jean-Philippe Didon, Helene Hinnen-Oberer, Mathieu Lemay, Jan
P. Kucera and Andre G. Kleber
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Circ Arrhythm Electrophysiol. 2012;5:391-399; originally published online March 2, 2012;
doi: 10.1161/CIRCEP.111.968180
Circulation: Arrhythmia and Electrophysiology is published by the American Heart Association, 7272 Greenville
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Copyright © 2012 American Heart Association, Inc. All rights reserved.
Print ISSN: 1941-3149. Online ISSN: 1941-3084
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SUPPLEMENTAL MATERIAL
Virtual Sources and Sinks During Extracellular Field Shocks in Cardiac Cell
Cultures: Effects of Source-Sink Interactions Between Adjacent Tissue
Boundaries
Aleksandar A. Kondratyev, MD, PhD1; Jean-Philippe Didon, PhD2; Helene Hinnen-Oberer1;
Mathieu Lemay, PhD1; Jan P. Kucera, MD1; André G. Kléber, MD1
1
Dept. of Physiology, University of Bern, Switzerland;
2
Schiller Inc. Laboratory, Wissembourg, France
Stimulation and Optical Mapping of Transmembrane Potential
The culture dishes (22 mm in diameter) were mounted into a custom-built perfusion chamber
and perfused at 35oC with medium M199/7653 (Sigma) on the stage of an inverted
microscope (Nikon, TE-300, Nikon Switzerland). Transmembrane potentials were recorded
using the voltage-sensitive dye di-8-ANEPPS. The emitted light was collected through a 40x
objective and projected on an array of 380 optical fibers arranged in a hexagonal lattice and
individually connected to light-sensitive diodes (resolution 25
m). The induced
photocurrents were converted to voltage, amplified (frequency range 0 – 3.5kHz) and
digitally recorded at a frequency of 10 kHz from a selected subset of 128 fibers. Analysis
was perfomed using a custom-written software, based on LabView (National Instruments)
and embedded in MatLab (MathWorks).
Extracellular field shocks (ECFS’s) producing a homogeneous electrical field were applied
by a custom-built defibrillator to two platinum electrodes placed in the cell culture bath, as
described previously.1 Electrical field strength was measured in the bulk of the superfusate
using a bipolar electrode. The field strength in the bulk used for the electrophysiological
experiments, as measured at different locations in the bath amounted to 6,36±0.3 V/cm.
However, as suggested by the theoretical analyses of Krassowska and Neu,
2
the effective
field strength at the site of current injection into the tissue layer is likely to be smaller in our
experimental arrangement. The cultures were stimulated (S1) at 3 - 5 mm remote from the
mapping sites using a bipolar electrode (pulse duration 1 – 2ms, 1.5 threshold strength).
This bipolar electrode consisted of a patch electrode with a large opening (2 -5 m) serving
as a cathode and a platinum wire wound around the electrode shaft serving as an anode.
Computation of Time Dependent Changes in Transmembrane Potential (Vm),
Determination of the Space Constant
and the Time Constant
In our experiments we consistently applied ECFS exactly perpendicular to the tissue
boundaries. We have previously shown that such an arrangement produces changes in Vm
with isopotential lines parallel to the boundaries.
transmembrane potential across a patterned 200
1, 3
Figure 1S depicts the changes in
m wide strand. The isopotential map
showing depolarization (red) close to the cathode and repolarization (blue) close to the
anode caused by an ECSF of 6.36V/cm is superimposed on the outline of the strand (light
gray). The 2 signals in the inset show the transmembrane action potential with the change in
membrane potential during the plateau phase caused by the ECFS. The diode recorded the
potential from a circular area of 25 m.
Figure 1, on-line supplement: for explanation see text
This aspect of the isopotential lines parallel to the strand boundaries made it possible to use
a one-dimensional model to calculate the changes in Vm in the direction perpendicular to the
border(s) of the cell culture during the ECFS. For our computations, the general linear cable
equation was used.4 In the linear one dimensional model, as depicted in Figure 1C, the
change in transmembrane voltage, V, in response to an ECFS was represented as follows: 4
eq. I
where X is the position x normalized by the length constant
structure (distance between boundaries) normalized by
the ECFS normalized to the time constant
superposition and reflexion principles
4
(T = t/
X = x/ , L is the length of the
and T is the time after the onset of
The derivation of Eq. I applies the
and describes the response of a one-dimensional
structure of a finite length L to a stepwise injection of current at X = 0 (at one boundary) and
a simultaneous stepwise injection of the same current but with opposite polarity at X = L
(other boundary), starting at t = 0. This solution is obtained from the formalism describing an
imaginary infinite structure extending beyond the strand boundaries at x=0 and x=L. This can
be obtained formally by defining an infinite number of imaginary current sources obtained by
“reflecting” the structure of finite length an infinite number of times at the boundaries x=0 and
x=L (see reference 4 for a detailed description).
In their theoretical analysis of the response of cardiac tissue to an extracellular electrical field
using a passive bidomain model of cardiac tissue, Sobie et al.
5
have shown that if the
externally applied electric field is uniform and the tissue is isotropic, the activation function
(representing the spatial distribution of virtual sources and loads) is solely determined by the
gradient of intracellular conductivity. Assuming that our preparations were uniform, this
gradient is non-zero only at the boundaries of the strand and vanishes within the strand.
Therefore, the activation function corresponds to a source at x=0 and a load of opposite
polarity at x=L. Eq. I thus represents V(x,t) in a manner which is equivalent to that in the
bidomain model of Sobie et al. for the particular case of an infinitely long homogeneous and
isotropic strand of width L, subjected to an uniform extracellular field perpendicular to the
strand. For the case of a single boundary (half-plane monolayer), L → ∞.
In a similar way, the superposition principle was also used to compute the response of the
tissue to a sequence of short pulses.
Time course of Vm during the make and the break of ECFS
Application of rectangular ECFSs to whole hearts has recently been shown to produce a
hysteresis loop in the delivered current.6 Accordingly, it has been postulated that a change in
the resistive circuitry responsible for carrying electrical charge to the cell membranes might
be affected very early during short ECFSs. Electroporation of the cell membrane was
considered as the most likely mechanism involved. Supplemental Figure 2 shows a
transmembrane action potential with the change in Vm produced by an ECSF applied 40 ms
after the action potential onset (Panel A). Panel B depicts the enlarged segment of the signal
during the ECFS. The slow rise and decrease of Vm during and after the pulse (predicted by
eq.1 above) makes it highly unlikely that a shunt resistance, suggesting electroporation,
developed during ECFS application in our experiments. This is in line with the further
observations in our model that the repetitive application of pulses did not change their time
course, and that the internalization of the dye Lucifer Yellow and a decrease in Vm after
approximately 3 ms during an ECSF have been observed in our model only at substantially
higher field strengths (> 20 V/cm). 1, 7
Figure 2, on-line supplement:
for explanation see text.
References
1. Gillis AM, Fast VG, Rohr S, Kleber AG. Spatial changes in transmembrane potential
during extracellular electrical shocks in cultured monolayers of neonatal rat ventricular
myocytes. Circ Res. 1996;79:676-690.
2. Krassowska W, Neu JC. Response of a single cell to an external electric field. Biophys J.
1994;66:1768-1776.
3. Dosdall DJ, Fast VG, Ideker RE. Mechanisms of defibrillation. Annu Rev Biomed Eng.
2010;12:233-258.
4. Jack J, Noble D, Tsien R. Electric current flow in excitable tissues. Clarendon Press Oxford; 1975.
5. Sobie EA, Susil RC, Tung L. A generalized activating function for predicting virtual
electrodes in cardiac tissue. Biophys J. 1997;73:1410-1423.
6. Malkin RA, Guan D, Wikswo JP. Experimental evidence of improved transthoracic
defibrillation with electroporation-enhancing pulses. IEEE Trans Biomed Eng.
2006;53:1901-1910.
7. Tung L, Kleber AG. Virtual sources associated with linear and curved strands of cardiac
cells. Am J Physiol Heart Circ Physiol. 2000;279:H1579-1590.