513
Virtual Cathode Effects During Stimulation of
Cardiac Muscle
Two-dimensional In Vivo Experiments
John P. Wikswo Jr., Todd A. Wisialowski, William A. Altemeier, Jeffrey R. Balser,
Harry A. Kopelman, and Dan M. Roden
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We have found that when suprathreshold cathodal stimuli were applied to the epicardium of
canine ventricle, impulse propagation originated at a "virtual cathode" with dimensions
greater than those of the physical cathode. We report the two-dimensional geometry of the
virtual cathode as a function of stimulus strength; the results are compared with the predictions
of an anisotropic, bidomain model of cardiac conduction recently developed in our laboratories.
Data were collected in six pentobarbital-anesthetized dogs by using a small plaque electrode
sewn to the left ventricular epicardium. Arrival times at closely spaced bipolar electrodes
oriented radially around a central cathode were obtained as a function of stimulus strength and
fiber orientation. The dimensions of the virtual cathode were determined by linear backextrapolation of arrival times to the time of stimulation. The directional dependence of the
conduction velocity was consistent with previous reports: at 1 mA, longitudinal (00) and
transverse (90°) velocities were 0.60±0.03 and 0.29+±0.02 m/sec, respectively. At 7 mA, the
longitudinal velocity was 0.75+±0.05 m/sec, whereas there was no significant change in the
transverse velocity. In contrast to conduction velocity, the virtual cathode was smallest in the
longitudinal orientation and largest between 45° and 600. Virtual cathode size was dependent on
both orientation and stimulus strength: at 00, the virtual cathode was small (-1 mm) and
relatively constant over the range of 1-7 mA; at oblique orientations (45°-90°), it displayed a
roughly logarithmic dependence on stimulus strength, -1 mm at 1 mA and -3 mm at 7 mA.
The bidomain, anisotropic model reproduced both the stimulus strength and the fiberorientation dependence of the virtual cathode geometry when the intracellular and extracellular
anisotropies were 10: 1 and 4: 1, respectively, but not when the two anisotropies were equal. We
suggest that the virtual cathode provides a direct measure of the determinants of cardiac
activation; its complex geometry appears to reflect the bidomain, anisotropic nature of cardiac
muscle. (Circulation Research 1991;68:513-530)
R eliable determination of conduction velocity
in electrically excitable biological tissue requires more than simply measuring the time
difference between a stimulus artifact and a single
electrogram recorded at a known distance from the
cathodal stimulus site. For example, in measurements on peripheral nerves suspended in air, the
elapsed time between the application of a cathodal
From the Living State Physics Group, Department of Physics
and Astronomy, Vanderbilt University, and Departments of Pharmacology and Medicine, Vanderbilt University School of Medicine, Nashville, Tenn.
Supported in part by a grant from the US Public Health Service
(HL-36724). J.R.B. is supported by an institutional award from the
Medical Scientist Training Program (GM 07347).
Address for reprints: John P. Wikswo Jr., Department of Physics
and Astronomy, Vanderbilt University, Box 1807 Station B, Nashville, TN 37235.
Received November 29, 1989; accepted October 8, 1990.
stimulus at one electrode and the appearance of a
propagating action potential at another electrode
several centimeters away has an easily observed
dependence on the strength of the stimulus.1 For low
stimulus strengths, the onset of propagating activation is delayed by the latency of an excitable membrane near threshold; hence, the elapsed time between the stimulus and the arrival of activation at a
distant recording site is lengthened. In contrast, for
high stimulus strengths, 1-2 cm of nerve can be
directly depolarized by the stimulus pulse'; in this
case, propagation begins during or soon after the
stimulus, not at the stimulus electrode but instead at
some distance from the electrode. The region of
nerve that is depolarized rapidly by the stimulus
current and from which propagation proceeds has
been termed the virtual cathode.2
Virtual cathode effects in cardiac muscle are more
difficult to understand than those in nerves because
514
Circulation Research Vol 68, No 2, February 1991
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of the three-dimensional, syncytial, anisotropic characteristics of cardiac tissue.3-7 When ventricular myocardium is stimulated, impulses propagate both
along and across myocardial fibers, with conduction
velocity longitudinal to fiber orientation being over
twice that transverse to it.3,4,6-8 Thus, with epicardial
stimulation, isochrones have been found to assume
the ellipsoidal configuration with their major axis
aligned along local fiber orientation. Multiple bipolar
electrodes on the epicardium around a cathodal
stimulus site can be used to determine longitudinal
and transverse conduction velocities.9-12 In this
study, we used the measured velocities and backextrapolation of the trajectory of the wavefront to
determine the dimensions of the virtual cathode. We
could thus study the size and shape of the virtual
cathode as a function of both the stimulus strength
and the angle between the direction of propagation
and the myocardial fiber axis. Intuitively, one might
assume that the virtual cathode in cardiac tissue
would assume the shape of a prolate ellipsoid, with
the long axis aligned along the orientation of rapid
propagation. However, our findings indicated that
the virtual cathode dimensions assumed an unexpected direction dependence.
Numerical simulations with a linear, bisyncytial,
anisotropic, two-dimensional, finite-element model
of cardiac tissue13 have reproduced many of the
geometric complexities of the biological data we now
present. These calculations suggest that the effects
reported here are at least in part the direct result of
the differing anisotropies of the intracellular and
extracellular domains, with the intracellular conductivity more anisotropic than the extracellular one.13
Because of these differing anisotropies, existing analytical cable models for cardiac tissue14-16 may not
adequately describe all aspects of the response of
cardiac tissue to the injection of current, and nonlinear models may be required to understand fully the
virtual cathode effects (J.P. Wikswo Jr., J.P. Barach,
unpublished results, 1988).
Methods
The experiments described here were conducted in
six mongrel dogs (weight 17-39 kg). Animals were
anesthetized with sodium pentobarbital (30 mg/kg),
intubated, and ventilated with 02 (95%) and CO2
(5%) using a respirator (Harvard Apparatus, South
Natick, Mass.). Supplemental doses of pentobarbital
were given as needed during the experiment to
maintain deep anesthesia. The right femoral artery
was cannulated to monitor continuously mean blood
pressure and to obtain blood samples. Arterial blood
gases were monitored frequently, and ventilator settings were adjusted to maintain Po2>70 mm Hg and
pH in the 7.35-7.45 range. A left lateral thoracotomy
was performed in the fourth intercostal space, and a
pericardial cradle was created.
To measure virtual cathode effects, we constructed
a small planar electrode array with four pairs of
bipolar electrodes equally spaced along each of four
15mm
FIGURE 1. The electrode array used to record the arival
time of the activation wavefront at different distances from the
stimulus electrode. The central electrode (S) was fabricated
from 250-,um titanium wire and was used for stimulation. The
bipoles were 1.0-mm long and were made of two 250-p.m
diameter silver wires separated by 0.6 mm and threaded
through a thin Plexiglas disk, soldered to flexible insulated
leads, sealed with epoxy, and chlorided. The bipoles are
located 1.5, 3.0, 4.5, and 6.0 mm from the stimulating
electrode.
perpendicular axes, as shown in Figure 1. Bipolar
electrodes were used to allow accurate arrival time
determination with a minimum of data processing,
and to reduce the stimulus artifact. The 0.6-mm
spacing of the two electrodes was chosen to maximize
the size of the signal without significant loss in spatial
resolution.9 The bipoles were closely spaced so that
all of the measurements could be made close to the
stimulating electrode, thereby minimizing distortion
of the expanding wavefront as it propagated into
deep layers of myocardium whose fiber orientation
differed from that at the surface. The end of a
250-,um diameter titanium wire at the center of the
array served as the central, stimulating cathode. A
Lucite ring, which was fabricated to support the
planar electrode array yet allow it to be rotated, was
first sewn to the left ventricular epicardium lateral to
the left anterior descending artery. Next, the orientation of the array in the ring was adjusted until
bipolar signals recorded at the most distant sites 900
from each other (e.g., sites A4 and B4 in Figure 1)
had the maximal separation in time, thereby defining
the orientation of the fastest, longitudinal propagation (00) and slowest, transverse propagation (900).
The electrode array was then fixed in place on the
Lucite ring by using set screws. The outermost bipoles were 7 mm from the sutures that held the
Lucite ring in place. The stimulating anode, a stain-
Wikswo et al Cardiac Virtual Cathode Effects
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less steel pacemaker wire, was sewn to the right
ventricular epicardium. Finally, gauze moistened
with saline was placed over the heart, and the edges
of the thoracotomy were apposed. After data were
acquired with the electrode in the 00 and 900 positions, signals were recorded from the 1800 and 2700
electrodes. The set screws were then loosened, the
array was rotated 300 or 450, the set screws were
retightened, and data were acquired from along each
of the four new directions. This electrode array is a
significant improvement over the one used in our
earlier studies,12 which had fewer electrodes, had
120° (rather than 90°) angular separation, and did not
have the flexibility of adjustable orientation.
To acquire data, the central electrode of the array
was stimulated at a cycle length of 300 msec with
current pulses of 0.5-0.7-msec duration and 0.115-mA strength by using a programmable stimulator
and a constant-current stimulus isolation unit (Bloom
Associates, Ltd., Narberth, Pa.). No electrode polarization effects were detected, and the stimulus threshold was stable over the course of an experiment, which
involved an average of 16,000 stimuli over 80 minutes
(i.e., 200 beats/min). The signals from four bipolar
electrodes in each of two orthogonal directions were
filtered (0.01 Hz to 1 kHz) and amplified (x 10) by
eight differential amplifiers. The eight signals were
simultaneously sampled at a minimum rate of 2.0 kHz,
held, and then sequentially digitized with 12-bit resolution by an MDAS 7000 data acquisition system
(Kaye Instruments, Bedford, Mass.), which was controlled by an IBM AT microcomputer. With a trigger
signal provided by the stimulator, the data acquisition
system collected 10 consecutive waveforms and computed their average. The average waveforms were
transferred in analog form to an oscilloscope (Hitachi,
Woodbury, N.Y.) for immediate viewing and then
transferred in digital form to the microcomputer via a
Hewlett-Packard Interface Bus (HPIB) for storage
and off-line analysis.
To determine virtual cathode size, we assumed
that impulse propagation velocity was constant after
activation at the edge of the virtual cathode. The
distance between the recording electrode and the
stimulating cathodal electrode was plotted (ordinate)
as a function of the time difference between the end
of the stimulus and the arrival of the peak of bipolar
signal (abscissa). A linear, least-squares fit to these
data yielded a slope that was the conduction velocity
in that direction and an intercept on the distance axis
that was taken to be the edge of the virtual cathode.'
At low stimulus strengths, virtual cathode size determined by this technique was negative and, as discussed further below, must instead be interpreted as
latency of activation after the end of the stimulus
pulse.
In an attempt to quantify the relation between the
virtual cathode size, rvy and the strength of the current stimulus, Is, we examined two different analytical
expressions derived from cable theory. First, when a
current, Is, is applied intracellularly to a one-dimen-
515
sional, passive cable, the transmembrane potential,
Vm (measured relative to the resting potential), exhibits a spatial variation given by17"18
Vm(x)=KIse /A
(1)
where x is the distance from the stimulus site, K is a
constant related to the resistances of the cable, and A
is the length constant of the cable. The edge of the
virtual cathode, x,c, is determined by the point at
which the transmembrane voltage, Vm(xvc), equals
the membrane threshold potential, Vth, in which case
Equation 1 becomes
Vth=KIse 'vJI
(2)
=e(x-/A+
(3)
or
I
where
P3 is a hybrid constant given by
I3=ln(Vth/K)
(4)
Equation 3 can be solved for the virtual cathode size
as a function of stimulus current
(5)
x,,= A[lnIS-ln(Vth/K)] = A[ln(Il)-,f]
Thus the virtual cathode size in a one-dimensional,
passive cable exhibits a logarithmic dependence on Is.
The situation in three-dimensional, anisotropic
cardiac tissue is more complex in that the virtual
cathode size, as measured radially from the stimulating electrode, will vary with angle relative to the fiber
axis. We examined two approaches to the problem.
First, we considered the possibility that, in any direction, Equation 5 for a linear cable could be applied,
with the virtual cathode radius rv, the length constant
A, and the constant f3 all depending on the angle 4
between rvc and the fiber direction. Alternatively, it
has been proposed that the relation among Vm, I, and
rv, in three dimensions may be better characterized
as19,20
Vth=KIs
e- r1/A
(6)
where Vth, K, A, and r,, may all depend on 4. In this
case, the relation between Is and r, can be rewritten
as
ls=e [rJA+In(r,,,)+,81
(7)
A simplex nonlinear fitting routine was used to fit the
parameters A and 63 in Equations 3 and 7 to the r,
versus Is data.
All data are expressed as mean±+1 SD.
Results
A typical set of electrograms recorded simultaneously in the longitudinal (00) and transverse (90°)
orientations at low (0.6 mA, threshold) and high (3
mA, five times threshold) stimulus strengths are
presented in Figure 2. For a given stimulus and
Circulation Research Vol 68, No 2, February 1991
516
AVA H
mV
l
100
LONGITUDINAL
Is= 3 mA
a)
LONGITUDINAL
IS = 0.6 mA
100
\1 2
tsp -I
l
23 A
FIGURE 2. Unprocessed, digitally recorded data from a typical experiment. Panels a and b: Eight electrograms recorded
C)
simultaneously as voltage differences AVA
|| \1 A Ab bipolar electrodes Al-A4 (as labeled in
3
the rapidly propagating
direction parallel to the fiber axis and as
AVB by electrodes BJ-B4 along the slowly
propagating transverse direction for threshold stimuli (100 psec at 0.6 mA). In panel
a, the time, tsp, between the end of the
stimulus and the peak of the signal is shown
Figure 1) along
~b)
TRANSVERSE
Is = 0.6 mA
AV8
TRANSVERSE
I|gs 3 mA
2
t1
3
d)
=
2
4
mV
4
for the electrogram recorded from bipole
0
A4. Panels
0
20
10
Time, ms
30
0
10
20
Time, ms
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direction, the peaks of the electrograms were equally
spaced, indicating a conduction velocity that was
independent of distance from the stimulus electrode.
For the threshold electrograms in Figures 2a and 2b,
the time interval between the end of the stimulus and
the peak of the signal (ranging from 4.0 to 12.2 msec
for the four longitudinal recordings and 13.6 to 28.6
msec for the transverse) represented the combined
effects of the latency of activation within the virtual
cathode and the delay associated with propagation
between the edge of the virtual cathode and the
recording electrode. When the stimulus was increased to five times threshold (Figures 2c and 2d),
there were minimal changes in the spacing between
electrograms; that is, conduction velocity was essentially constant. However, the time intervals were
reduced (ranging from 1.9 to 7.9 msec longitudinal
and 3.2 to 20.1 msec transverse), indicating the
presence of the virtual cathode.
Figure 3 shows data obtained from each of four
longitudinal and four transverse recording sites for
stimulus strengths ranging from threshold to 15 mA.
The dependence of the time interval (tsp of Figure 2)
on the stimulus current is readily apparent: in the
longitudinal direction, the time interval decreased
rapidly as the stimulus current was increased from
threshold to 1 mA and less rapidly for stimulus current
greater than 1 mA; in the transverse direction, the
time interval decreased more uniformly over the entire range of stimulus current values. As the stimulus
was increased, the signal from the propagating wavefront merged with the stimulus artifact for recording
sites close to the stimulus electrode (panels a, e, and
f), indicating that the edge of the virtual cathode had
extended out to that bipole. The merger of the waveforms with the stimulus artifact in panel f but not in
panel b demonstrates that the virtual cathode was
actually larger in the transverse direction.
Figure 4 uses pairs of simultaneously recorded
electrograms to demonstrate that for large stimulus
currents, the virtual cathode was larger in the transverse direction than in the longitudinal one. For a
0.8-mA stimulus (panel a), the longitudinal activa-
30
c
and d:
Comparable electro-
grams recorded for a 100-psec, 3.0-mA
stimulus. Is, stimulus current.
tion wavefront reached the third bipole (4.5 mm from
the stimulus, 1800) almost 6 msec before the transverse activation (2700) reached the corresponding
bipole, consistent with the rapid conduction along
the fiber axis. However, for a 15-mA stimulus (panel
b), the slower, transverse activation reached the third
bipole 1.5 msec before the longitudinal one. This
reversal of arrival times suggests that the transverse
propagation began closer to the third bipole than did
longitudinal propagation.
Figure 5 shows that electrograms recorded from
sites in the longitudinal orientation occasionally became biphasic at higher stimulus strengths. As the
current was increased from 2 to 10 mA, the electrograms recorded 4.5 mm from the stimulus (panel a)
arrived earlier, had a decreasing amplitude, and
eventually exhibited two separated peaks. The electrogram recorded 1.5 mm further away (panel b)
showed similar behavior for 2 and 5 mA but was
clearly biphasic for a current of 8 or 10 mA. Such
changes of waveform shape as a function of stimulus
strength occurred only with longitudinal propagation.
As will be discussed below, this may be the direct
result of the activation wavefronts having to propagate through or around a hyperpolarized region lying
along the fiber axis that is beyond the virtual cathode.
A different phenomenon was often seen in the
transverse direction: arrival times were not always a
continuous function of stimulus strengths. For the
specific example shown in Figure 6, the arrival time
in the transverse direction (90°) dropped abruptly by
5-6 msec in all four bipoles as the stimulus increased
from 0.9 to 1.0 mA. Because this change was the
same in all four transverse electrodes, the causal
event must have occurred within the 1.5 mm between
the cathodal electrode and the first bipole.
Several of the effects described in Figures 2-6 can
be explained by the virtual cathode having a complex
shape that depends on stimulus current. To demonstrate this, we performed a quantitative analysis of
virtual cathode size and shape. Figure 7 plots the
bipole signals to demonstrate how the least-squares
fit of the time of the peak and the propagation
Wikswo et al Cardiac Virtual Cathode Effects
LONGITUDINAL
517
TRANSVERSE
e)
Electrode Dl
50
mV
'I,
~/°v
I
0
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FIGURE 3. Typical bipolar
electrograms recorded by electrodes A1-A4 along the fiber axis
and electrodes D1-D4 transverse
to it for stimulus currents (I)
ranging from threshold to 15 mA.
15
o
Time,
20
ms
25
30
5
15
10
Time,
distance was used to determine the propagation
velocity and the combined effect of the virtual cathode size and the latency of activation. The linear
dependence of the peak location on time in Figures
7a-7c illustrates that, within the accuracy achieved
with these bipole recordings, the conduction velocity
was constant in the region between the edge of the
virtual cathode and the outermost electrode. In analyzing the data from all of the experiments, we could
20
25
30
ms
not identify any significant, systematic deviation from
linearity.
For longitudinal propagation after a threshold
stimulus (Figure 7a), propagation appeared to have
begun 1.1 msec after the end of the 0.5-msec stimulus, as indicated by the straight line originating at the
open square on the time axis. As stimulus current was
increased to 1.0 mA (Figure 7b), longitudinal propagation appeared to have been initiated at the end of
Circulation Research Vol 68, No 2, February 1991
518
a)
100-
Is = 0.8 mA
longitudinal
transverse
AV
mV 50
-
jr O
/0
0-L
,' 0
0
IsS
100 --
mV
50
0
b
15 mA
-
0-
0
4
12
8
16
20
/
24
~rvO
Time, ms
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
FIGURE 4. A typical set of simultaneously recorded electrogram pairs that illustrate how the longitudinal signals arive
before the transverse one at low stimuli (Is, 0.8 mA), while the
reverse is the case at high stimuli (15 mA).
0
6CO
_.
the stimulus, originating at the edge of the virtual
cathode as indicated by the solid square at 0.8 mm on
the distance axis. The virtual cathode was even larger
for larger stimulation current values. No latency was
seen in the transverse direction for threshold stimuLONGITUDINAL ELECTRODE 3
100
r
A~~~~~~~~~~~~
AV
fi \ l /
mV
-
\
a)
2 mA
5 mA
8 mA
-O8 mA
/ r )CV
_.
1a
LOGIUA
E/
0
100
0
LONGITUDINAL ELECTRODE 4
r
&
\~~~~~b
,'
5
10
15
20
25
'T
,'
30 0
Time, ms
FIGURE 6. An example of sudden decreases in transverse
arrival times. The sets of electrograms are plotted as functions
of stimulus current (I), from the same experiment as in Figure
3, but for the four electrodes at 90°.
mv
'
lation current (Figure 7d), but instead propagation
appeared to have been initiated 0.6 mm from the
electrode. For 10 mA (Figure 7c), prop~J
-........stimulating
0
agation was first seen at the 3.0 mm electrode in the
longitudinal direction as indicated by the distinct
0
peak for that electrode and at 4.5 mm in the trans12
8
10
2
4
6
verse direction (Figure 7f). The virtual cathode at
Time, ms
10.0 mA was twice as large in the transverse direction
FIGURE 5. An example from a single experiment of how a
as compared with the longitudinal one.
small chanpge in stimulus current (I,) can affect the shape of
These data also illustrate how the waveforms
the longitutdinal electrograms distant from the stimulus site.
change when the virtual cathode approaches a rePanel a: E lectrograms recorded 4.5 mm from the stimulus
cording bipole. In each panel of Figure 7, the trace
electrode fo)r four different stimulus currents. Panel b: Correnearest the time axis corresponds to the signal from
sponding el ectrograms recorded simultaneously at 6.0 mm.
A / \
; / \ /\
-
-
s- |
Wikswo et al Cardiac Virtual Cathode Effects
LONGITUDINAL
TRANSVERSE
a)
d)
50
mV
e)
r
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10
20
Time,
30
ms
O
o
X~
C) 4 e
0
C
c)
0
519
f)
O
-
A
C)
0e
0
0
10
20
Time,
300
ms
FIGURE 7. Graphic presentation of typical data from a single experiment illustrating the spatial variation in electrograms with
distance from the stimulus electrode for three different stimulus currents (I,) in longitudinal (panels a-c) and transverse (panels
d-f) directions. Data are from a typical experiment. The peaks of the signals are used to determine the best-fit line describing
propagation of the wavefront, with the slope in the time-distance plane corresponding to the conduction velocity. The intercept on
the time axis (panel a, open square) corresponds to a latency of activation, whereas an intercept on the distance axis (panels b-f
solid square) is the virtual cathode size.
the recording closest to the stimulus. At 0.6 mA, a
distinct action potential is evident. At 1.0 mA, the
action potential appears on the shoulder of the
stimulus artifact, indicating that the virtual cathode
has extended almost to the first electrode pair. When
the stimulation current is 10 mA, no action potential
is evident, and the virtual cathode encompasses the
first electrode pair. The electrogram morphology in
Figure 7c arises from the proximity of the virtual
cathode to the innermost bipole in the longitudinal
direction and thus provides an estimate of virtual
cathode size that is independent of the assumptions
regarding uniform propagation or latency.
The data analysis technique illustrated in Figure 7
was then used to derive the size of the virtual cathode
(intercept) and the conduction velocity (slope) as a
function of stimulus strength for each individual set
of electrograms, with the summary data from a single
experiment shown in Figure 8 for three electrode
orientations. At low stimuli, this procedure will result
in a negative virtual cathode size. It is more realistic
to interpret negative virtual cathode sizes instead as
an activation latency, or activation time
(8)
where both rvc and can depend on Is. For the
example in Figure 8, at 900 and a threshold stimulus
of 0.6 mA, rv, was -2.5 mm, which corresponds to a
latency of 8.6 msec. At 00, r, was only -0.8 mm,
corresponding to a latency of 1.4 msec. While latency
may exist at high stimulus strengths, the time delay is
small compared with the reduction in time interval
associated with the large virtual cathodes. The latency effect is most significant at low stimulus
ta= -rvc/o
520
Circulation Research Vol 68, No 2, February 1991
4
VIRTUAL CATHODE SIZE
a)
(P
3
2
rvc
mm
=
0'
CONDUCTION VELOCITY
d)
-It
0.8
2
l
O
=
0.6
0
m/s
-1
0.4
0.2
-2
-3
4
0
4)
3
=
4 5-
*
t t
2
U
rvc
mm
b)
. .
0.8
a
0.6
0
m/s
-1
(
=
45*
[
0.4
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0.2
-2
-3
4
0
c)
(D
3
2
rvc
mm
=
90
f)
0.8
'
U2
1
t t
0.6
.
.
¢
0
-1
0
5
1
0.2
-2
5--
-3
.
0
0
5
10
15
0
Is, mA
strengths. The sudden change in the slope of the r,,
Is curves that occurs between 1.5 and two
times threshold in Figures 8a-8c is a manifestation of
the shift from a latency-dominated regime to a virtual
cathode-dominated one. In our subsequent analysis
of the current dependence of rv, we have included
only those values of I, for which rvc was positive,
typically two to three times threshold.
The solid squares in Figures 8a-8c identify the
value of the current for which the propagating electrogram seemed to disappear into the stimulus artifact for a particular recording electrode. The distance between this electrode and the stimulus
electrode then gives a measure of virtual cathode size
in that direction. While this determination of virtual
cathode size is more subjective than the least-squares
fit, it was consistent with that estimate of virtual
cathode size for the corresponding current.
Figure 9 shows the average dependence of virtual
cathode size and conduction velocity on stimulus
strength for all measurements made on six dogs, with
data from measurement angles in the second through
fourth quadrants mapped into the first quadrant; it
was assumed for this summary plot that the data were
symmetrical about the longitudinal and transverse
versus
9
m/s 0.4
FIGURE 8. The virtual cathode
size (rvc panels a-c) and conduction velocity (0, panels d-f)
as a function of stimulus current
(I) for three directions in a typical experiment. The error bars
show the uncertainties as determined during the least-squares
fit, when the peak times were
assumed to be measured without
error, and allpoints are weighted
equally. The dependence of virtual cathode size on current is
also estimated by determining
for a particular recording electrode the lowest current for
which there is no minimum in
the electrogram immediately after the stimulus artifact, shown
by a solid square with the bipole
number above or below it. 0 is
the angle between the direction
of the virtual cathode measurement and the fiber direction.
5
10
15
Is, mA
axes
for normal cardiac tissue. Note that for stimulus
current greater than 2 mA, the virtual cathode is
larger in the transverse direction than in the longitu-
dinal one. At high stimulus strengths the virtual
cathode was twice as large for 30°, 450, and 90° as it
was for 0°. Figures 9f-9j demonstrate that the measured conduction velocity has only a slight variability
near threshold and a clear dependence on stimulus
current that is strongest for small angles.
The one-dimensional function in Equation 3 was
then fit to each individual set of data by using a
simplex algorithm, with the results shown in Table 1
and as the solid lines in Figure 9; as can be seen, A
varies as a function of orientation with the smallest
value in the longitudinal direction and its greatest
value at 450. The fits to the bidomain model are
discussed below. The relative uncertainty in the value
for is substantially larger than that for A. Because /3
depends on both Vth and K, no interference on any
orientation dependence of these two parameters can
be made from this analysis.
When the three-dimensional function in Equation
7 was fit to these data, as shown in Table 1, the
residual sums of squares were similar to the onedimensional case, so we cannot at this point definitely
Wikswo et al Cardiac Virtual Cathode Effects
VIRTUAL CATHODE SIZE
4
(P
CONDUCTION VELOCITY
a)
O0
=
521
1.01
It=Q;
f)
30O
9)
0.8
3
/ 0.6
m/s
mm 2
r~---4 0..
0
1
0
0.410.21-
4
1.0
3
0.8
=
4
I 0.6 m/s
mm 2
M----1 r-l
r---j L---i
0.4
0.2
1
0
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
0
1.0
4
m 0.6 h
in/s
rvc2
mm 2
h)
4, = 45-
0.8
3
1
0.4 -
1
FIGURE 9. The average virtual cathode
size (rc) and conduction velocity (0) vs.
stimulus current (I,) for the combined data
from all experiments. The angle (4) is
shown in each panel. The data points represent the unweighted averages, and the
error bars are the standard deviations of the
data from this average. The solid lines in
panels a-e are the nonlinear, least-squares
fits to the equation rvc(4)=A(4)[ln(Is)+
K(4i)J, while the lines in panels f-j are the
linear least-squares fit.
1
0.2 0
4
1.0
3
0.8
0
[
4'
=
60'
4,
=
90*
I)
m 0.6 h
in/s
rvc2
mm 2
0.4
0.2
o
0
e)
4
=
90O'
*=
1
CD
rvc 3-
1.0r
0.8 0.6 -
mm 2-
m/s
0.4 I
+
0.2
o
1
2
3
4
i)
5
6
7
o
1
T
T
2
Is
distinguish which function is a better characterization of cardiac tissue. The length constant A was
again smallest in the longitudinal orientation and
greatest at 450; interestingly, the fitted values of A for
4.300 were threefold to fivefold greater than in the
one-dimensional fit. The relative uncertainties in P3
are even larger than those for the one-dimensional
fit. Again, since P3 depends on both Vth and K, we
draw no inference on any derived value for Vth or K
from this fit.
3
4
t
s
6
7
Is
While the average virtual cathode size varied by
than a factor of 4 over the range of stimulus
currents, the propagation velocity varied at most by
25% in the longitudinal direction, and by less than
10% for angles between 450 and 90°. A linear regression analysis was used to determine the statistical
significance of the dependence of conduction velocity
on stimulus strength with the results shown in Table 2
and as the straight lines in Figures 9f-9j. The increase
in propagation velocity with increased stimulus
more
522
Circulation Research Vol 68, No 2, February 1991
TABLE 1. Average Parameters From the Fit of the Data From All Six Dogs and the Fit to the Bidomain Model
From measured data
Bidomain model
A (mm)
4)
n
A (mm)
/3
/3
One-dimensional cable Is = e[rvc(6)/A(k)+0(6)]
12
0.85+0.55
0o
30`
15
1.10+0.19
45`
8
1.42+0.34
1.28+0.26
60`
15
12
1.13±0.13
90°
Three-dimensional cable Is = e{rve(O)A(d)+ln[rvc()]+P(O)}
-0.24±0.48
-0.42+0.41
0.10+0.03
-0.57±0.89
-0.37±0.49
0.03
1.11
1.30
1.23
1.15
-25.9
0.11
0.09
0.04
0.02
0.04
0.58±0.27
-622.5
-0.20+2.05
2.88±1.13
2.54
0.21
-0.06+0.61
0.66±0.24
2.98
8.91+3.78
0.12
8.05±6.06
2.70
0.11
0.11+0.81
3.83±1.75
2.49
0.14
0.08+0.86
at
(4).
n is the total number of separate sets of measurements made each angle
The uncertainties are the standard
deviation. (), Angle between virtual cathode measurement and fiber direction; A, length constant of cable; /3, constant
defined in Equation 4.
00
30`
45°
60°
900
12
15
8
15
12
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
strength was significant (p<0.005) for 0° and 300. The
propagation velocity for 450 and 600 had no distinct
dependence on stimulus current. At 90°, there was a
slight (10%) decrease in conduction velocity that was
statistically significant (p<0.005) as the stimulus current was increased from 1 to 7 mA.
The angular dependence of the virtual cathode
size is shown in Figure 10 for each of the six dogs
studied. The virtual cathode assumed the shape of an
indented oval, or "dog bone," at high stimulus currents, in that the minimum virtual cathode size
occurred in the orientation exhibiting the highest
conduction velocity, while the maximum occurred at
approximately 450 to 600 from this axis.
Discussion
We have used a small, radial array of bipolar,
epicardial electrodes to determine the time when a
cardiac activation wave front arrives at up to 48
locations within a radius of 6 mm from a small,
cathodal epicardial stimulating electrode. Other researchers have used measurements of time differences between multiple electrodes and a regression
analysis to determine conduction velocities, but such
analyses consider only the slope of the regression line
and ignore the distance- or time-axis intercepts. In
our studies, we interpreted a time-axis intercept as a
latency between stimulation and initiation of propagation, while we treat a distance-axis intercept as the
virtual cathode size. For stimuli greater than two to
three times threshold, latency effects were negligible.
Thus our least-squares fit of the data provided both
the conduction velocity and the effective propagation
distance along eight to 12 radial lines that were at
different angles with respect to the fiber direction, as
defined by the direction of fastest propagation. As
expected from published measurements,3-8 the conduction velocity exhibited a clear dependence on the
angle of propagation relative to the fiber direction.
We found, as in measurements on nerves,' that when
TABLE 2. Conduction Velocities Versus Angle for Three Different Stimulus Currents for the Averaged Data From
All Dogs
4)
00
30`
450
600
900
n
12
15
8
15
12
Is=1 mA
0.59+0.09
0.46+0.07
0.41±+0.12
0.29±0.05
0.25+0.05
0 (m/sec)
I=4 mA
0.70+0.15
0.51+0.11
0.40+0.13
0.31±+0.09
0.24±0.05
Slope
I=7 mA
0.73+0.06
0.57±0.03
0.43±0.04
0.32+0.02
0.23+0.02
([m/sec]/mA)
0.029+0.004
0.022+0.003
-0.002±0.003
0.003±0.001
-0.004+0.001
p
<0.005
<0.005
<0.4
<0.05
<0.005
0.60+0.03
0.71±+0.03
0.75+0.05
...
...
0.29±0.02
0.29±0.02
0.31±+0.02
...
...
2.1+0.2
2.5+0.3
2.4±0.3
...
...
0JOL/
n is the total number of separate sets of measurements made at each angle (4)). The uncertainties are the standard
deviation. 4, Angle between virtual cathode measurement and fiber direction; 0, conduction velocity; OL and 0r,
longitudinal and transverse conduction velocities, respectively; I, stimulus current.
OL
Br
Wikswo et al Cardiac Virtual Cathode Effects
(a)
4
:
I
523
(d)
Is= 2 mA
Is= 03 mA
: s= 3 mA
l Is= 05 mA
Is=4 mA
LElIs=07 mA
is ~5 mA
I 10mA
Els=
I
-4
4
-4
4
-4 ±
-41-
(e)
S=03 mA
4T
-4-.
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
Is= 05 MA
Li
.:.:.:. ...g
Is= 09 mA
....
lI=l1mA
-.-.....,.. ::::
16=3mA
EI¶Its=SmA
.Is= 13 mA
lEl
-4
4
4
-4'
41
4
K.
::x:::::>
(c)
(f)
n Is=3mA
Li Is= 1.5 mA
M Is=4mA
aI1s= 3.0 mA
E,
m Is= 5mA
vl Is= 5.0 mA
Is=46 mA
E Is= 7.0 mA
4
-4
l7A
4
-4
FIGURE 10. Plots showing the shape of the virtual cathode
along the axes are in millimeters.
the epicardium was stimulated by a current pulse two
to three times the threshold value, propagation was
initiated not at the cathodal electrode but at the edge
of the larger virtual cathode. The size of the virtual
cathode increased approximately logarithmically as
the strength of the stimulus current was increased.
This was consistent with observations on nerves.' An
unexpected result was that the virtual cathode exhibited a complex shape that depended on both the
stimulus current and the direction of propagation.
-4
over a range
of stimulus
currents
(I) for all six dogs. The distances
The virtual cathode was smaller in the longitudinal
direction than in the slower, transverse direction.
This is counterintuitive given that the elliptical shape
of the activation wave fronts is well known. However,
as discussed below, our conduction velocity data are
consistent with published results by others, and the
observed shape and current dependence of the virtual cathode are consistent with predictions based on
our mathematical models of current flow and propagation in cardiac tissue.13
524
Circulation Research Vol 68, No 2, February 1991
Figure 10 shows that for stimuli greater than 3 mA,
the virtual cathode had the same dog-bone shape for
each of the six dogs studied, with the exception of the
first-quadrant data recorded from the first dog (Figure 10a). That may reflect regional ischemia associated with the sutures holding the mounting ring. The
dog-bone shape in Figure 10b is slightly skewed,
possibly the result of underlying fibers not having the
same orientation as those on the surface or of some
other variation in fiber architecture or tissue anisotropy. At the lowest stimulus strengths, the virtual
cathode often exhibited a complex shape, undoubtedly the result of activation latency discussed above.
With these exceptions, the virtual cathode shape is
symmetrical across the fast and slow axes.
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
where 0L and 0T are the longitudinal and transverse
conduction velocities, respectively. The observed angular dependence of the conduction velocity, as
determined from the averaged data in Figure 9, is
illustrated in Figure 11 for three different stimulus
currents. A nonlinear least-squares fit to Equation 9
was used to determine OL and 0T, and these values,
given in Table 2, were used to obtain the solid lines in
the figure. The data are in excellent agreement with
the theoretical prediction of Equation 9. The value of
0T is independent of Is; there is a 25% increase in OL
when the stimulus was increased from 1 to 7 mA,
with the trend significant at the level of p<0.05. The
longitudinal velocity was between 2.0 and 2.5 times
greater than the transverse velocity. Overall, the
values of OL and 6T are consistent with values reported in the literature.3-8
Comparison With a Numerical, Bidomain Model
While the equations describing a two-dimensional
bidomain can be solved analytically for equal and
reciprocal anisotropy ratios,22-24 there are no known
analytical solutions to the bidomain equations for
generalized anisotropies with physiologically realistic
values for the conductivities. The angular dependence of the virtual cathode size has been examined
theoretically by using a linear, time-dependent (passive), finite-element model of a two-dimensional,
anisotropic cardiac bidomain.13 In these calculations,
a point electrode at the origin was used to inject
constant current into the extracellular space, and the
resulting distribution of the transmembrane potential
was then computed for different values of the electrical anisotropies of the intracellular and extracellular spaces. With the assumption that propagation
would be initiated at the locus of points where the
=
1 mA
Is
=
4 mA
0.6
rn/s
b)
0.9
\
0.6
9
m/s
(9)
Is
9
Angular Dependence of Conduction Velocity
One test of the accuracy of our measurements is to
assess how well the angular dependence of the
propagation velocity agrees with theoretical predictions. The conduction velocity should exhibit an
angular dependence given by2'
O()=OLOT [(OL sin4f)2+(OT Cos4f)2]-1/2
a)
0.9
0.3 oL-
c)
0.9
0.6
m
rn/s
0.3
0
30
60
90
(D, degrees
FIGURE 11. The angular dependence (4) of the conduction
velocity (6), as deternined by the least-squares fits in Figure 9,
for three different stimulus currents (IJ. The solid lines are the
least-squares fit of these data to Equation 3. The resulting
values of the longitudinal and transverse conduction velocities,
0L and OT, are listed in Table 2.
transmembrane potential exceeded a threshold value
(assumed to be 20 mV above the resting transmembrane potential), the threshold (20 mV) isopotential
contour predicted by this linear model should correspond to the edge of the virtual cathode. Figure 12a
shows the shape of the virtual cathode for five
different stimulus currents predicted by their model,
computed for a 5.7: 1 anisotropy ratio (uxlry) for both
the intracellular and extracellular spaces. In this
equal-anisotropy case, the virtual cathode is always
elliptical, with the major axis of the ellipse oriented
along the direction of fastest propagation. The transmembrane potential is positive everywhere outside of
the virtual cathode. Figure 12b shows the extreme
case of reciprocal anisotropies, where the anisotropy
ratio is 10: 1 for the intracellular space and 1: 10 for
the extracellular. The virtual cathode, shown as a
complete curve only for the 1-mA stimulus, is a dog
bone with the long axis along the direction of slowest
propagation. In addition, the model predicts that
there will be a region of hyperpolarization beyond
the virtual cathode centered on the longitudinal axis.
Wikswo et al Cardiac Virtual Cathode Effects
525
FINITE ELEMENT BIDOMAIN MODEL
90
90*
(a)
(b)
3
3-
Y, 2
Y, 2
mm
'_-4---
mm
1
1
1 6 mA
\~~~~~
0
1
2
O~.
_
0
3
3
X, m m
X, mm
Fiber direction
2
1
(C)
(d)
90.
3
,-, 6o0
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
,.,>.:-
.-
-45~
rvc 2
Y,
mm
mm
1
u
1
1.
0 12 4
16
8
X, mm
I
rrnA
EXPERIMENTAL DATA
Fiber direction
(f)
(e)
go
90
,--", 60
3
45
,.-_ *~ 30
3-
0/
Y, 2
r vc
2
[
FIGURE 12. Panel a: The shape of the
virtual cathode for five different stimulus
currents, as predicted by a linear, twodimensional, anisotropic bidomain model
of cardiac tissue with equal anisotropy
ratios of 5.7:1 in the intracellular and
extracellular spaces. The plots are symmetrical about the x and y axes. Panel b: Solid
lines, the virtual cathode shape for an
intracellular anisotropy ratio of 10:1 and
an extracellular anisotropy ratio of 1:10;
dashed lines, the boundaty of the region
where the transmembrane potential is hyperpolarized to 20 mV below the resting
potential. For five different stimulus currents. Panel c: The virtual cathode (solid
lines) and hyperpolarized contours
(dashed lines) for the nominal anisotropy
ratios of 10:1 for the intracellular space
and 4:1 for the extracellular space (adapted from Reference 13 with copyright permission of the Biophysical Society). Panel
d: The dependence of virtual cathode size
(rvJ) on stimulus current (I,) for the model
for five different angles, as determined
from panel c. Panel e: The shape of the
virtual cathode from the average data for
five different stimulus currents, as reconstructed by applying Equation S to the data
in Table 1. Panel f: The dependence of rvc
on I, from the average data for five different angles, as determined from panel e.
mm
mm
1
0
1
2
X,
mm
3
0 12 4
8
IsIm1A
This region extends further out as the stimulus
current is increased. The physiologically realistic, or
nominal, case is shown in Figure 12c, in which the
anisotropy ratio is 10:1 for the intracellular space
and 4: 1 for the extracellular space.25 The dog bone is
more concave than for the reciprocal case, and the
hyperpolarized region is smaller. When the stimulus
current is increased, the hyperpolarized region extends both further inward and further outward. Figure 12d shows the predicted dependence of virtual
cathode size on stimulus current for nominal anisotropies. Figures 12c and 12d can be compared with
16
Figures 10 and 9a-9e. The qualitative agreement is
excellent.
One of the most important results of the theoretical analysis of the bidomain model is that the
dog-bone shape does not appear if the intracellular
and extracellular anisotropies are equal.13 From this,
we can conclude that the dog-bone shape observed in
Figure 10 arises in part from the differing anisotropies between the intracellular and extracellular
spaces in the myocardium.
While the results of the bidomain model for nominal anisotropies are obviously not consistent with a
526
Circulation Research Vol 68, No 2, February 1991
Downloaded from http://circres.ahajournals.org/ by guest on June 17, 2017
simple cable response, it is useful to examine the fit
of the model data to both the one- and threedimensional equations. Table 1 gives the values of A
and /8 obtained by fitting Equation 3 to the model
data in Figure 12b. Both the model and the experimental data have the largest length constant at 450
and the smallest at 0°. Adjusting the value assumed
for the threshold potential in the model by +5 mV
affects the model values slightly, but the trends are
consistent. Table 1 shows the result of the fits of the
data and the model to the three-dimensional cable
equation given by Equation 7. The same trends occur
as with the one-dimensional fit.
The finite-element model can also be compared
with the experimental data by using the results of
Table 1 and Equation 5 to reconstruct average experimental plots. The reconstructed data from the
one-dimensional fit are shown in Figures 12e and 12f.
Above 2 mA, the virtual cathode is smallest at 00 for
both the nominal-anisotropy bidomain model and the
experimental data. For stimuli between 2 and 8 mA,
the virtual cathode is larger for 60° than for 30° or
90°. Because data were recorded at 450 for only the
first two dogs (Figures 10a and 10b), the 45° rvc data
in Figures 12c and 12d have a larger uncertainty.
From Figures 10 and 12d, we can conclude that the
largest virtual cathode size occurs in the vicinity of
450-600, depending on the strength of the stimulus.
Ideally, a larger number of closely spaced bipoles
would allow simultaneous measurement of the virtual
cathode radius at 150 angular increments.
Comparison of Length Constants
In our virtual cathode experiments, currents well
above threshold were applied extracellularly, and the
propagation of the action potential rather than direct
measurement of the electrotonic potentials was used
to determine a length constant. The agreement of
our data with both the simple cable equations (Figure 9) and the nominal-anisotropy bidomain model
(Figure 12) suggests that the virtual cathode effect
arises from the cablelike properties of cardiac tissue,
as we expected. However, the interpretation of the
cable parameters A and ,3 and comparison of these
with those measured by conventional techniques26-32
is difficult because the two types of measurements
are made under entirely different conditions. The
majority of conventional cable measurements are
made on quiescent fibers, often with hyperpolarizing
pulses. The electrical parameters so derived may not
be appropriate for tissue that is actively propagating
an action potential. As shown by Pressler,33 the
unidirectional space constant, input resistance, and
membrane time constant for Purkinje fibers were
significantly larger for a quiescent fiber than for a
repetitively stimulated fiber that is examined in its
diastolic phase. He also indicated that measurement
of passive electrical properties during cellular activity
have several theoretical limitations, including longitudinal and time-dependent voltage gradients. Our
measurements of the virtual cathode effect clearly
involve the initiation of a propagating impulse and
hence may reflect a length constant that is affected by
factors not present during the application of a hyperpolarizing pulse to a quiescent fiber. This may explain
the fact that the length constant we measure in the
direction of fastest conduction is shorter than in the
transverse direction, whereas the opposite is the case
for nonstimulating measurement techniques.4'26'34,35
The effect of the active membrane response on the
observed length constants was also examined by
Roberge et al.36 They used a nonlinear, numerical
model to demonstrate that the subthreshold response
of an active cable to a rectangular pulse applied at a
point differs significantly from the response of a
passive cable, with the attenuation of the electrotonic
response with distance being much more rapid in the
passive cable. They concluded that for active cables,
the magnitude and spatial extent of the electrotonic
events near threshold cannot be explained solely in
terms of linear cable theory. They also pointed out
that it may be valid to use a one-dimensional cable to
interpret a three-dimensional system, since the more
rapid falloff of transmembrane potential associated
with a linear, three-dimensional cable (Equation 6)
as compared with the one-dimensional cable (Equation 2) will be more than compensated for by the
available regenerative Na+ current. If the stimulus
maintains a region of tissue depolarized near threshold for any length of time, active sodium influx will
dramatically alter the voltage profile from that obtained in a passive cable calculation (J.P. Wikswo Jr.,
J.P. Barach, unpublished results, 1988).
Implications of the Virtual Cathode Shape on the
Propagation of Activation
Given that the dog-bone shape of the virtual
cathode is real, we can then inquire how this affects
the propagation of cardiac activation. Activation of
longitudinal sites by wave fronts converging from the
adjacent 450 edges of the virtual cathode is a likely
explanation of the unusually shaped electrograms in
Figure 5. The fact that the 5 mA electrogram was
monophasic at 4.5 mm (Figure 5a, dotted line), while
it was biphasic at 6 mm (Figure 5b, dotted line)
suggests that this convergence occurred between 4.5
and 6 mm. When the stimulus was increased to 10
mA, the electrogram at 4.5 mm was distorted (Figure
Sa, solid line), while that at 6 mm was nearly normal
(Figure 5b, solid line), suggesting that with stronger
stimuli, the region where anomalous conduction occurred had moved inward.
The discontinuous propagation suggested by Figure
6 may be related to the discontinuous propagation
documented by Spach et al,3738 but the 1.5-4.5-mm
distances over which we made our measurements and
the size of our present electrodes precludes a more
definitive identification of the source of the discontinuities that we observed.
In Figure 9, we showed that the conduction velocity in the longitudinal direction increased slightly
with stimulus current. The bidomain calculations
Wikswo et al Cardiac Virtual Cathode Effects
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predicted that, as a result of the unequal anisotropy
ratios, there would be a small region along the
longitudinal axis for which a cathodal stimulus would
produce hyperpolarization,13 shown by the dashed
contours in Figures 12b and 12c. While we have yet to
obtain independent confirmation of the existence or
extent of this hyperpolarization, it is possible that the
waveforms in Figure 5 and the dependence of longitudinal conduction velocity on stimulus current could
be the result of propagation of activation through a
slightly hyperpolarized region. As the stimulus
strength was increased, this region would be expected
to become larger, extending both further inward and
further outward. Additional studies will be required
to determine whether such hyperpolarization would
effectively create a "barrier" to propagation in the
longitudinal direction, perhaps even establishing the
substrate for reentry with the application of high
current strengths.
An alternative explanation for the dependence on
stimulus strength of the conduction velocity along the
axis is that the concavity of the virtual cathode about
00 led to a transient concavity of the propagating
depolarization wavefront. This concavity could be
expected to increase the propagation velocity over
that expected for a planar wavefront moving at the
same angle.39-42 As stimulus current was increased
from 1 to 7 mA, the concavity increased and may
have resulted in a corresponding increase in the 00
conduction velocity.
We have assumed in our least-squares fit that the
depolarization wavefront propagated with a constant
velocity after activation at the edge of the virtual
cathode. For low stimulus strengths, some latency was
certainly present in these experiments (e.g., Figures
8a-8c) and, in fact, led to the clearly untenable result of
a negative virtual cathode size. Hence, particularly with
stimuli delivered around threshold, the assumption of a
constant conduction velocity may contribute an experimental error to these results. Based on the high
correlation coefficients obtained in the least-squares fit
and the agreement of the disappearance of the propagating electrograms at the edge of the virtual cathode,
we do not think this was a significant problem at the
scale of the present measurements. As we discussed
above, we might expect that the local conduction velocity was affected by the radius of curvature of the
activation wavefront. In the transverse direction, this
radius was large, and transverse propagation might be a
close approximation to that of a straight wavefront for
propagation over the entire range covered by the
electrode. In contrast, for angles within 450 of the fiber
axis, the wavefront was sharply curved, with a local
curvature that decreased as the wavefront expanded.
An even more closely spaced electrode array would be
required to determine whether the conduction velocity
changes significantly within the first millimeter of wavefront propagation,7,36,43 but this presents a number of
practical problems. These effects might also be examined using a nonlinear mathematical model1436 but
expanded to two or three dimensions.21'43-46
527
Long-range Effects As Seen in Propagating
Action Potentials
The phenomena we have described were observed
within 4 mm of the stimulus electrode. However, the
effects of the irregular virtual cathode shape can apparently be seen at greater distances. Witkowski and
Penkoske47 have reported measurements of the
shape of epicardial isochrones resulting from a point
stimulus by using a rectilinear electrode array with
more recording sites but with larger interelectrode
spacing. Their data, as well as that of Roberts et al,7
demonstrated that the eccentricity of the activation
wavefront increases for the first several milliseconds
after stimulation. Figure 13 demonstrates similar
effects in our data. For all four stimulus currents, the
late isochrones were elliptical, as would be expected
given the anisotropy in conduction velocity. However,
at the highest currents, the early isochrones exhibit
the dog-bone shape that would be expected for
propagation away from a similarly shaped virtual
cathode. With increasing time, these early isochrones
evolve smoothly into an elliptical shape, with the long
axes of the ellipsoid oriented along the axes of rapid
propagation. The differences between the angular
dependence of the virtual cathode and the subsequent free propagation indicate that the mechanisms
governing how tissue is brought to threshold by a
point stimulus are different from those responsible
for subsequent propagation.
Figure 13 also provides corroborating evidence
regarding the validity of our least-squares fit to
determine the size of the virtual cathode. The plots
shown in Figure 10 indicate the shape of the virtual
cathode at the time of stimulation and are determined only after significant data analysis that includes back-extrapolation to the end of the stimulus
and an assumption that there is no activation latency
after the end of the stimulus. The isochrones in
Figure 13 were obtained simply by linear interpolation of the peak times as shown in Figure 7, in that
the least-squares conduction velocity along a particular direction for the current shown was used to
determine the location of the depolarization wavefront each millisecond. Both analysis techniques
demonstrate the existence of the dog-bone shape.
The dog-bone shape is readily apparent only when
the stimulus strength is two or three times the
threshold value, and the isochrones of the propagating wavefront are nonelliptical only for the first
millisecond or two, when the activation has yet to
propagate more than 1 or 2 mm from the stimulus
site. Because stimulus artifacts are less troublesome
for bipolar stimulating electrodes (for which the
virtual cathode will be significantly smaller) and for
recordings made several millimeters or more from
the stimulus electrode (and hence several milliseconds after the end of the stimulus), and because
epicardial electrode arrays with millimeter electrode
separation are difficult to construct, the paucity of
reported departures from the commonly observed
528
Circulation Research Vol 68, No 2, February 1991
-
X
FIGURE 13. A plot of the isochrones describing the
propagation of the depolarization wavefront past the
electrode array for a single dog and a 10-mA stimulus
current. The darkened region is the virtual cathode.
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-6
mm
elliptical isochrones is not surprising. There is, however, the possibility that the anisotropic bisyncytial
effects that are the cause of the dog-bone shape may
also manifest themselves in other ways. Spach et a16
used experimental data and a numerical model to
describe the intracellular and transmembrane currents in anisotropic muscle. Maps of the excitation
sequence at 110 positions within an area 12 x 10 mm
after a stimulus of 1.5 times threshold showed that
the 2-, 4-, and 6-msec isochrones were approximately
elliptical. Classification of the shape of the extracellular potential waveforms showed that there were
four different waveform morphologies that allowed
clear delineation of seven regions within the map.
The central region surrounding the stimulus site had
a dog-bone shape, with a minimum extent of 0.5 mm
in the fast direction, a 2-mm extent in the slow
direction, and a slightly greater extent along the
diagonal. Thus the dog-bone shape was not present
in the observed isochrones (.2 msec) but was readily
apparent in the spatial variation of the electrogram
morphology. No explanation was offered as to the
origin of the shape of the central region; it remains to
be seen whether there is any connection between the
shape of this region and the virtual cathode effects
we have observed.
A transversely oriented dog-bone shape is also
apparent in the repolarization map data presented by
Osaka et al.48 It was stated that the transverse
orientation of the repolarization isochrones resulted
from the repolarization process proceeding more
rapidly in the transverse direction than in the longitudinal one. No explanation was offered for the
dog-bone shape evident in their 370-msec isochrone.
Implications for Measurements of
Conduction Velocities
Our study also provides some guidance on the
measurement of conduction velocities. The measurement of propagation times between two separate
bipolar recording electrodes will minimize both virtual cathode and latency effects. In our earlier experiments,1 we found that a concentric bipolar stimulating electrode can reduce the size of the virtual
cathode to less than 0.2 mm. If it is not possible to use
a bipolar stimulating electrode, latency effects will be
small for currents greater than 2-3 mA, for which the
virtual cathode will have a radius of 1-3 mm. In this
case, the proximal bipole of a pair of bipoles used to
determine conduction velocities should be no less
than 3 mm from the stimulating electrode.
Liminal Length
The relation between the virtual cathode and the
liminal length49-51 is complex and is discussed in
more detail by Wikswo and Barach (unpublished
results, 1988). In the simplest approximation, the
liminal length represents the smallest possible virtual cathode. Because of the various uncertainties
in interpreting our arrival-time data at threshold
stimuli, we cannot yet determine whether the liminal length differs in the longitudinal versus transverse directions, that is, whether the threshold
virtual cathode is circular or of another shape. The
major findings of this paper are for stimuli more
than twice threshold, for which the concept of
liminal length is not appropriate.
Conclusions
In these experiments, we have demonstrated that
the virtual cathode effect develops in cardiac muscle
Wikswo et al Cardiac Virtual Cathode Effects
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with an unanticipated directional dependence. Numerical simulations of current flow in cardiac muscle
suggest that the angular dependence of the virtual
cathode effect may reflect the bisyncytial, anisotropic
nature of cardiac muscle and indicate clearly that the
determinants of virtual cathode size are considerably
more complex than in one-dimensional cablelike
preparations. The logarithmic dependence of the size
of the virtual cathode on the strength of the stimulus
current can be characterized by a length constant
that is largest at 450 from the direction of fastest
propagation and smallest at 00.
The relation between this length constant and
that determined by subthreshold pulses applied to
quiescent fibers is unclear. Whether the virtual
cathode phenomenon is likely to contribute to
perturbations of cardiac electrophysiology that are
important in humans remains to be studied. It is, for
example, possible that under appropriate conditions, the dog bone and the adjacent hyperpolarized
regions could form the substrate for reentry, so that
impulses propagate through a localized area of
unidirectional block. Thus reentrant arrhythmias
associated with the application of higher stimulus
strengths may reflect the angular dependence of the
virtual cathode. Moreover, virtual cathode effects
may play a role in determining the effect of a large
stimulus on the success of defibrillation. Indeed, we
have recently observed that a drug that has been
associated with impaired defibrillation decreased
virtual cathode size, while an agent that facilitates
defibrillation increased virtual cathode size.52,53
Thus, further evaluation of the determinants of the
virtual cathode are desirable from both the theoretical and practical points of view.
Acknowledgments
We are indebted to John Barach and Nestor
Sepulveda for lengthy discussions regarding the interpretation of these data, and to Leonora Wikswo
for her comments on this manuscript. Computer time
was provided by the College of Arts and Science,
Vanderbilt University, Nashville, Tenn.
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KEY WORDS * virtual cathode * cardiac excitability * latency
* cardiac propagation
Virtual cathode effects during stimulation of cardiac muscle. Two-dimensional in vivo
experiments.
J P Wikswo, Jr, T A Wisialowski, W A Altemeier, J R Balser, H A Kopelman and D M Roden
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Circ Res. 1991;68:513-530
doi: 10.1161/01.RES.68.2.513
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