Vol. 279, Issue 4, H1579-H1590, October 2000
Virtual sources associated with linear and curved strands of
cardiac cells
Leslie
Tung1 and
André G.
Kléber2
1 Department of Biomedical Engineering, The Johns Hopkins
University, Baltimore, Maryland 21205; and 2 Department of
Physiology, University of Bern, 3012 Bern,
Switzerland
 |
ABSTRACT |
Transmembrane potential
(Vm) responses in cardiac strands with
different curvature were characterized during uniform electric-field stimulation with the use of modeling and experimental approaches. Linear and U-shaped strands (width 100-150 µm) were stained with voltage-sensitive dye. Vm was measured by
optical mapping across the width and at sites of beginning curvature.
Field pulses were applied transverse to the strands during the
action-potential plateau. For linear strands, Vm
contained 1) a rapid passive component (Vmar) nearly linear and symmetric
across the width, 2) a slower hyperpolarizing component
(Vmas) greater and faster on the
anodal side, and 3) at high field strengths a delayed
depolarizing component (Vmad) greater
on the anodal side. For U-shaped strands, Vm at
sites of beginning curvature also contained rapid and slow components (Vmbr and
Vmbs, respectively) that included
contributions from the linear strand response and from the fiber
curvature. Vmar,
Vmbr, and part of
Vmbs could be attributed to passive
behavior that was modeled, and Vmas,
Vmad, and part of
Vmbs could be attributed to active
membrane currents. Thus curved strands exhibit field responses
separable into components with characteristic amplitude, spatial, and
temporal signatures.
cultured cells; optical mapping; electric excitation; electric
shock; defibrillation
| |
INTRODUCTION |
ALTHOUGH MUCH
INVESTIGATIVE effort has been directed at the postshock events
that occur after an electric shock leading to defibrillation and
cardioversion, the mechanism(s) by which the applied electric field
changes the transmembrane potential (Vm) of the
cardiac cells is a critical, yet still poorly characterized, step
(21). Recent theoretical work in several laboratories, including our own on the "generalized activating function"
(23, 25), define the intimate role of tissue structure in
the Vm responses (26). Virtual
sources that drive the responses are created at fiber branches, bends,
and borders (21) and, indeed, anywhere where there are
gradients in intracellular conductivity or gradients in the
extracellular field (23). Despite the putative importance
of the tissue structure in shaping the shock response, the bulk of our
understanding has arisen from theoretical and computational studies,
whereas detailed experimental studies aimed at clarifying the
functional relationships have only recently emerged (7,
8).
The polarization of myocardial cells in the bulk of tissue distant from
the electrodes can result from gradients in electric field (direction
or intensity) in the presence of a homogeneous tissue structure, from
inhomogeneity in fiber structure in the presence of a uniform electric
field, or from some combination of the two (23). The
method of patterned cell culture (19) presents an
opportunity to study the second of the scenarios described above under
well-controlled experimental conditions. Indeed, the use of
voltage-sensitive dyes has enabled the detailed study of the pattern of
polarization around anatomical obstacles (6, 9) and fiber
bends and branches (8) in cultures of neonatal rat cardiomyocytes.
Computational studies on a tissue level, using the bidomain
model, have implicated two likely candidates for polarization of the
bulk tissue: surface polarization and fiber curvature
(26). The same concepts also apply to individual cardiac
fibers (strands of cardiac cells) across a length scale on the order of
millimeters. As is shown in this study, uniform electric fields produce
surface polarizations at the edges of the fibers as well as global
polarizations owing to fiber curvature. Accordingly, the goal of this
work is, first, to model the changes in Vm that
can be expected from passive strands of cardiac cells subjected to
uniform electric fields and, second, to obtain detailed, quantitative
measurements of such changes in corresponding structures of cultured
myocytes. Specifically, the study examines how the addition of fiber
curvature affects the temporal components of the responses. We find
that theoretical considerations based on passive tissue models predict primarily the early part of the observed responses. The active properties of the membrane play an additional important role in determining the response of the fiber that is attained at the end of
the field stimulus pulse. A preliminary version of this study was
presented in abstract form (28).
| |
METHODS |
Experimental setup.
The mapping system used in this study has been described previously
(5). Optical recordings of the cell strands were obtained by a 96-channel photodiode mapping system, at a spatial resolution of
15 µm, by use of a 40×, 1.3-numerical aperture oil-immersion objective. Signals were filtered in hardware by a 1.5-kHz low-pass filter and were sampled at a frequency of 25 kHz/channel with 12-bit resolution.
Patterned cell growth.
The method of patterned cell growth of neonatal rat myocytes has been
described previously (19). Cells were cultured as 100- to
150-µm-wide strands on 22-mm-diameter glass coverslips for 3-7
days after trypsin dissociation. Before the experiment was conducted,
individual strands were separated from the peripheral ring of cells by
scoring with a hypodermic needle. The coverslip containing the strands
was transferred to the experimental chamber, and the cells were stained
for 2-4 min by 2-4 µM of the voltage-sensitive dye RH237
(Molecular Probes, Eugene, OR). During the experiment, cells were
continuously superfused in Hanks' buffered solution (pH 7.4) at a
temperature between 31 and 35°C. The strand patterns used for these
experiments are shown in Fig. 1. They
consist of combinations of linear and semicircular segments with
different radii, ranging from 250 µm to 4.5 mm. U-shaped strand
geometries with a systematically varying radius of curvature
R were selected for study (Fig.
2). The strand shapes have
been superimposed, and the rectangular box shows the common region of
interest that was mapped for all of the strands. To minimize any
contamination of responses in the region of interest to virtual sources
lying off to the left of Fig. 2, the linear portions of each
strand extended for at least 2 mm outside the region of interest. For all of the U-shaped strands, the sites were located at the points of
beginning curvature. Vm responses were
quantified across the linear strand as a function of field intensity
and in the region of interest of U-shaped strands for a fixed field
intensity as a function of R.
View larger version (29K):
[in this window]
[in a new window]
|
Fig. 1.
Patterns used in the study. Three different patterns provided a
variety of 100- to 150-µm-wide strands of cells in various
combinations of straight line and semicircular segments.
|
|
View larger version (24K):
[in this window]
[in a new window]
|
Fig. 2.
Location of region of interest for U-shaped strands of
varying curvature. U-shaped strands with radii of curvature
(R) varying from 250 µm to 1.5 mm have been superimposed
here, along with a linear strand that contains an infinite
R. The region of interest is signified by the bold
rectangle. Optical recordings were obtained from 7-9 sites (shown
here for 7 sites, numbered 1-7), oriented transversely
across the strand and with a spatial resolution of 15 µm.
E, electric field intensity.
|
|
Field stimulus procedure.
Cell strands were stimulated by a 1-ms-duration S1 pulse applied
through a bipolar pipette electrode placed ~1-1.5 mm from the
recording site. S2 test pulses were applied through the bath by a pair
of parallel platinum plate electrodes ~2 × 18 mm in dimension
and spaced 24 mm apart on either side of a square-shaped experimental
chamber. The pulses were generated by discharging a custom-built pulse
generator that was designed to mimic a defibrillator. This resulted in
a low-tilt (16.0 ± 5.6%, measured in n = 200 trials) exponential waveform with a field intensity of 3-37 V/cm. The pulses had a duration of 8.4 ms and were applied during the action-potential plateau with a delay of ~10-15 ms after the
upstroke. Cell fluorescence was measured via epi-illumination, and for
each recording excitation, light was gated by a mechanical shutter for
a total duration of 50 ms, starting 10 ms before the onset of the S1
pulse. The field stimuli were applied in a direction transverse to the
linear strand and transverse to the linear portions of the U-shaped
strands. The field pulses were measured in the bath near the recording
site by a pair of silver wire electrodes spaced 3.3 mm apart and
connected to a differential amplifier. The amplitude of the S2 pulse
was quantified as the peak amplitude of the field waveform.
Waveform analysis.
The pattern of polarization across the width of the strands of
myocardial cells was recorded by selecting a series of sites along a
single column (or row, depending on the orientation of the strand) of
the photodiode array. If the S2 field response could not be read
cleanly at any one site owing to motion artifact or other corruption of
the optical signal, an adjacent site along a line parallel to the edge
of the strand and located no greater than 60 µm from the original
site was used as a substitute. For experiments in which measurements
were desired at the center of the strand, an average was taken of the
central two sites if the number of sites across the strand was even in number.
In plotting the S2 responses, linear corrections were made in each
recording for slopes in the optical signal arising either from baseline
change or from slope in the plateau. When determining the
action-potential amplitude to which each trace was normalized, a linear
ramp was subtracted from the recording to adjust the slope of the
signal to be zero during the 5- to 10-ms interval just before the
upstroke. The traces were then adjusted so that the slope of the signal
was zero over the 3- to 10-ms interval just before the S2 pulse, and
the S2 response was measured at various times during the S2 pulse
relative to the amplitude at 100 µs before the onset of the pulse.
Corrections were also made in some of the data analysis for variations
in strand width by dividing the center-to-center distance between the
recording sites (at both edges) by a nominal width of 120 µm and then
scaling the field intensities by this value.
After corrections were made, the optical responses to the S2 pulse were
measured at three different times during the S2 pulse: at 0.5-0.9
ms, at the peak minimum value of the response, and just before the end
of the pulse (8.1 ms). These data were then scaled as a percentage of
the measured amplitude of their respective action potentials. The fit
of the data by the nonlinear equation derived in the
APPENDIX was performed by use of the Solver function in
Excel (version 8.0 for Macintosh, Microsoft, Seattle, WA). The
statistical significance of differences in responses of curved fibers
and linear fibers was determined by use of a single-sided t-test in Excel. Values of P < 0.05 were
considered to be significant.
| |
RESULTS |
Linear strands.
Figure 3 shows the
Vm responses at nine sites across a
140-µm-wide linear strand for fields of +10.2 and 
9.9 V/cm applied transverse to the strand axis. As expected from previous work, depolarization occurs on the cathodal side and hyperpolarization on the
anodal side (9). When the direction of the field is reversed, the polarization pattern reverses. The patterns are nearly
identical, with small differences that may have resulted because the
field strengths were not identical, the sites were situated slightly
off center on the strand, or the strand itself might have been slightly
asymmetric in cellular morphology and intercellular connections. More
importantly, the center of the strand is not unaffected by the field
pulse but hyperpolarizes regardless of the direction of the field. This
observation shows that the center of the strand is not electrically
neutral. As will be seen later, the amount of hyperpolarization
increases with field intensity.
View larger version (55K):
[in this window]
[in a new window]
|
Fig. 3.
Spatial variation of the components of the total
transmembrane potential (Vm) response across a
purely linear strand. Shown here are the raw data obtained from 9 sites
across a 140-µm-wide strand during transverse, uniform field
stimulation with an 8.4-ms-duration pulse. The recordings were not
taken strictly along a single line, to avoid a location that had
excessive motion artifact. Uniform fields of opposite polarity were
applied. The white ring at the center of the image is an optical
artifact.
|
|
The detailed, temporal behavior of the responses just described is
shown in Fig. 4 for a 110-µm-wide
strand as the field intensity is varied. Previous theoretical and
computational analyses of field-stimulated cells (1, 11,
27) have shown that the response of the cell is a two-stage
process that consists of an initial polarization that occurs with an
ultrarapid time constant on the order of microseconds, followed by a
time-dependent change that reflects the polarization-induced changes in
ionic membrane currents. Although the strand is not a single cell, it
behaves like one in the transverse direction because the cells are
electrically coupled. However, the series resistances of the gap
junctions will slow down the speed of the initial polarization compared with that of the single cell, because they inhibit the intracellular redistribution of charge that leads to the polarization change. With
these theoretical considerations in mind, we measured the initial
polarization after a delay of 0.5-0.9 ms after the onset of the S2
pulse. The exact time that was chosen varied from strand to strand and
was selected to coincide with the initial peak values of responses in
the depolarizing direction and the inflection point of responses in the
hyperpolarizing direction. As shown for the traces at 21.0 V/cm, the
initial rapid responses during the first 0.6 ms [rapid component
(Vmar)] are delineated by the pair of
vertical lines at left. These responses are depolarizing at
one edge of the strand and hyperpolarizing on the other and appear to
be symmetric around the baseline. During the remainder of the S2
response, all of the traces move in the hyperpolarizing direction
[slow hyperpolarizing component (Vmas)]
and are delineated by the vertical lines at middle and
right. With fields of ~10 V/cm, the slow responses have
similar magnitudes and time courses across the strand, whereas at
higher field strengths, the responses diverge. Note, too, that at the
center of the strand, the initial response is nearly zero and
electrically neutral, whereas the slow response is not and moves in the
hyperpolarizing direction (Fig. 3). With higher fields of ~20 V/cm,
the slow response varies its characteristics from one side of the
strand to the other, becoming faster and larger on the anodal side. At
even higher field strengths, a third response [delayed depolarizing component (Vmad)] appears after a delay and
consists of a slow depolarization that is clearly seen for the traces
at 32.5 V/cm. After a local minimum is reached, which at the anodal
edge is 3.5 ms after the onset of the S2 pulse, the
Vm responses reverse from a hyperpolarizing slope to a depolarizing slope. This component,
Vmad, increases with further
increases in field intensity.
View larger version (35K):
[in this window]
[in a new window]
|
Fig. 4.
Recordings of the Vm responses
across the strand as a function of field strength. A 110-µm-wide
linear strand was stimulated by a uniform transverse electric field.
Optical recordings were obtained at 40× across the width of the strand
and measured from a row of 7 photodiodes, each with an image pixel size
of 15 µm. Action potentials were stimulated with a local S1 electrode
placed ~1.5 mm away, and an 8.4-ms S2 field pulse was applied.
Optical recordings were all obtained from the same strand for S2 shocks
of varying strength. Shown for the traces at 21.0 V/cm are the
definitions for the rapid (Vmar), slow
(Vmas), and delayed
(Vmad) components of the
Vm responses.
|
|
The data of Fig. 4 are plotted as a function of position in Fig.
5. Here, the total
Vm response is shown in the top left
panel and can be separated into the three components described
earlier. These components are plotted in the right panels as
a function of position across the strand for different field strengths.
As shown in the inset,
Vmar was measured as the change in
Vm during the first 0.6 ms of the S2 pulse and
was either positive or negative in polarity, depending on the position
within the strand. Vmas was measured
as the peak minimum amplitude of Vm,
relative to the amplitude at 0.6 ms, and was always negative.
Vmad was measured as the amplitude at
the end of the S2 pulse (at 8.1 ms), relative to the peak minimum, and
was always positive. Data were obtained at other field intensities but
are not shown for the sake of clarity. All Vm
components were normalized to the action-potential amplitude (APA).
Distance was taken to be the position of the center of the receptive
field of the photodiode relative to the upper edge of the strand. At
field strengths up to ~15 V/cm, Vmar is
approximately symmetric about the center of the strand. At higher field
strengths, a distinct inward rectification can be seen, such that the
hyperpolarizing responses are larger than the corresponding
depolarizing responses on the other side of the strand.
Vmas also shows two trends. At field
strengths up to ~20 V/cm, Vmas is
approximately constant across the cross section of the strand and
becomes more negative with increasing field intensity. At larger field
strengths, Vmas is still negative
across the strand but varies such that the hyperpolarization is greater
on the side of the strand facing the anode and less on the side facing
the cathode. Vmad begins to activate
above ~25 V/cm and increases in amplitude with field strength. It is
also nonuniform across the strand, with the depolarization being
greater on the side of the strand facing the anode.
View larger version (29K):
[in this window]
[in a new window]
|
Fig. 5.
The total Vm response and its
components at different field strengths, plotted as a function of
location across the strand. The total Vm
response was measured at the end of the S2 pulse at 8.1 ms, relative to
the 5-ms baseline of Vm just before the S2
pulse. Separation of the Vm responses into
different components was performed as illustrated in Fig. 4. All
components were normalized to the action-potential amplitude (APA),
measured from recordings that were adjusted in slope to be level during
the 5 ms just before the upstroke of the action potential. Distance was
taken to be the position of the center of the receptive field of the
photodiode relative to 1 edge of the strand.
|
|
The data of Fig. 5 are replotted in Fig.
6 as a function of field intensity along
with similar results from four other 125- to 150-µm-wide linear
strands. Because of the differences in strand width, field intensities
in a given experiment were normalized by a factor equal to the
center-to-center distance between recording sites at both edges of the
strand, divided by a nominal width of 120 µm. Shown in Fig. 6,
top left, are the total Vm responses at different S2 field strengths for sites on both edges and at the
center of the strand. The responses are asymmetric and biased in the
hyperpolarizing direction, such that the hyperpolarization of the edge
facing the anode is consistently greater at all field intensities than
the depolarization of the edge facing the cathode. Furthermore, the
center of the strand is not electrically neutral but exhibits a
hyperpolarizing response. These effects become magnified at higher
field intensities. Above 25 V/cm, there is an abrupt turn in the
central response back toward 0 mV.
View larger version (30K):
[in this window]
[in a new window]
|
Fig. 6.
The total Vm response and its
components at 3 sites on the strand, plotted as a function of field
strength. Data from 5 strands are summarized here, showing the total
Vm response measured at the end of the S2 pulse
at 8.1 ms, relative to the 3- to 10-ms baseline of
Vm just before the S2 pulse.
Vm is plotted as a function of field strength
for 3 sites: 1 at each edge of the strand and the 3rd at the center.
Separation of the Vm responses into different
components was performed as shown in Fig. 4.
Vmar was measured during the initial
0.5-0.9 ms of the response, Vmas was
measured during the remainder of the response until the time to peak
minimum, and Vmad was measured during the
remainder of the response after the peak minimum (if any). All
components were normalized to the APA, measured from recordings that
were adjusted in slope to be level during the 7-8 ms just before
the upstroke of the action potential. The dotted lines drawn in the
plot of Vmar are based on Eq. A3
(see APPENDIX), with y0 = ±60
µm and APA = 100 mV.
|
|
The total Vm response can be separated into the
three components described earlier and plotted in Fig. 6,
right. Of the three components, the rapid
Vmar most closely matches the
theoretical passive model. Two dotted lines based on Eq. A3
of the APPENDIX and having symmetric slopes are drawn for
comparison. Vmar at the two edges is
symmetric around the baseline at low field intensities but adopts a
bias in the negative direction at higher field intensities.
Vmar at the center of the strand
remains nearly zero throughout the entire range of field strengths
tested. Vmas is always negative and
increases in magnitude with field strength. It is nearly uniform across
the strand for fields up to ~15 V/cm, at which point the behavior of
the two edges and the center diverge, such that the hyperpolarization
increases monotonically from the edge facing the cathode to the
edge facing the anode. Vmad is always
positive and increases in magnitude with field strengths greater than
~20 V/cm. Vmad is nonuniformly
distributed across the strand and largest on the anodal side of the
strand. The significance of these three components will be discussed later.
U-shaped strands.
The effect of fiber curvature was studied in U-shaped strands with an
applied field of ~11 V/cm (Fig. 2). Figure
7 shows the responses measured across the
strand at the positions shown in the inset for different
R values of the semicircular bend. This situation is similar
to that analyzed earlier for the purely linear strand, except that the
recording sites are bound on one side by a semicircular bend. If the
radius becomes sufficiently large, one would expect the responses at
the recording sites to approach those of a purely linear strand.
However, because the responses with R = 1,500 µm are
not symmetric with reversal in the field polarity, even 1,500 µm is not "large enough." In all cases in which a positive field
was applied, a large net hyperpolarization appeared across all of the
sites by the end of the S2 pulse. This was sufficient to "reset"
(i.e., reactivate) the Na+ channels, so that a new action
potential was excited shortly after the termination of the S2 pulse in
all cases (not shown). With negative polarity fields, a net
depolarization appeared across the sites but only for the smallest
radius of 250 µm. No new action potential resulted after the end of
the pulse.
View larger version (26K):
[in this window]
[in a new window]
|
Fig. 7.
Spatial variation of the components of the total
Vm response from a semicircular bend. Shown here
are the raw data obtained from 7 sites across 3 120-µm-wide strands
having different radii. Uniform field stimulation with strength of
~11 V/cm was applied transversely with an 8.4-ms pulse with both
polarities of direction. The traces at 250 µm illustrate the
definitions for the rapid (Vmbr) and
slow (Vmbs) components of the
Vm responses.
|
|
As was the case with purely linear strands, there are rapid and slow
components to the membrane response, as illustrated for the traces for
a radius of 250 µm. We define Vmbr
to be the initial rapid response during the first 0.5-0.9 ms of
the S2 pulse and Vmbs to be the
residual slow response defined for the remainder of the S2 duration. We
see that component Vmbr resembles
component Vmar for the purely linear
strand and is roughly symmetric around the baseline, such that the
cathodal side depolarizes while the anodal side hyperpolarizes.
Vmbs differs from
Vmas and
Vmad for the purely linear strand,
because it changes its morphology with the direction of the field. For
positive (directed from bottom to top) fields at
R = 250 µm, component
Vmbs resembles
Vmas for the purely linear strand and
is in the hyperpolarizing direction all across the strand at all
curvatures tested. However, the magnitude of the polarization change is
substantially greater than for the purely linear strand (compare with
the traces of ~10 V/cm in Figs. 3 and 4). No component analogous to
Vmad in the purely linear
strand is present, presumably because the field is only ~11
V/cm. When the field is negative,
Vmbs is nearly absent at all
curvatures except for the smallest radius of 250 µm, where it is in
the depolarizing direction.
Similar data were obtained at field strengths of ~11 V/cm for a total
of 43 U-shaped 120-µm-wide strands of varying radii. In Fig.
8, the net Vm
responses to S2 at sites 1, 4, and 7 (the 2 edges and the center of the strand) are plotted on the
left for both polarities of field and as a function of
radius of the semicircular bend. The results from eight linear
120-µm-wide strands (corresponding to an infinite R) were
also measured and plotted as the leftmost points in all the
graphs for the purpose of comparison. Field strengths varied slightly
across all the strands: 11.5 ± 1.1 V/cm (n = 51)
for positive polarities and 11.1 ± 1.0 V/cm (n = 49) for negative polarities. The pattern of the responses is
strikingly different for the two field polarities, as described earlier
for the linear strand. With a positive polarity field (directed from
bottom to top), the responses (shown in Fig.
8A) are biased such that all of the sites across the strand
hyperpolarize relative to their values for the simple linear strand.
This bias becomes significant at radii of 
3,500 µm. When the field
is reversed (shown in Fig. 8B), the responses become biased
in the depolarizing direction and also become significant at radii
of <3,500 µm.
View larger version (28K):
[in this window]
[in a new window]
|
Fig. 8.
The total Vm response and its
components at 3 sites on the semicircular strand, plotted as a function
of the radius of the semicircular bend (R) of the adjacent
strand segment. Data from 51 strands are summarized here, showing the
net Vm responses to S2 measured at the end of
the S2 pulse at 8.1 ms, relative to the 5- to 10-ms baseline of
Vm just before the S2 pulse. The responses are
plotted for 3 of the sites, 1 at each edge of the strand (sites
1 and 7) and the third at the center (site
4), as a function of R. Means ± SD at each radius
were obtained from between 4 and 9 strands. Data obtained from 8 linear
strands are also included as the point with an infinite R.
The total Vm responses have also been separated
into rapid (Vmbr) and slow
(Vmbs) components. The dotted lines
drawn in the plots for Vmbr and
Vmbs are based on a least-squares fit
of the mean values of the data by the sum of a scaled version of
Eq. A16 (see APPENDIX), having a constant of
proportionality of  br and bs,
respectively, and the average values of
Vmar and
Vmas, respectively. The curve fits
have also been constrained to have the same space constant ( ) for a
given field polarity. For positive (upward) polarity fields,
E = 11.5 ± 1.1 V/cm (n = 51),
Vmar = 1.3 mV,
Vmbr = 21.2 mV, = 490 µm, br = 0.13, and bs = 0.54. For negative polarity fields, E = 11.1 ± 1.0 V/cm (n = 49), Vmar = 4.0 mV, Vmbr = 13.8 mV, = 364 µm, br = 0.17, and bs = 0.30. Outcome of single-tailed, equal-variance t-test
comparisons of data at different radii with that of the linear strand
(r =  ): * P < 0.05 and
** P < 0.01.
|
|
The data of Fig. 8 can be analyzed further in terms of the rapid
(Vmbr) and slow
(Vmbs) components plotted on the
graphs at right. Also plotted for comparison are the values
for Vmar and
Vmas for the linear strand. Both
components for both polarities of field could be fit by Eq. A16 of the APPENDIX, as indicated by the bold lines.
To constrain the permissible curve fits, the same value of space
constant () was assumed for each of the two field polarities, and
constants of proportionality ar, as,
br, and bs between each curve and
Eq. A16 were assumed for
Vmar,
Vmas,
Vmbr, and
Vmbs, respectively.
Vmbr varies widely across the strand,
is weakly sensitive to the radius of the semicircular bend, and shifts
in the hyperpolarizing direction with positive polarity fields and the
depolarizing direction with negative polarity fields. In contrast, the
slow component Vmbs varies little
across the strand but, like Vmbr,
shifts in the hyperpolarizing direction for positive polarity fields
(albeit much more strongly) and in the depolarizing direction for
negative polarity fields (with a magnitude comparable with Vmbr). The fits of the data by
Eq. A16 suggest that differs for the two polarities (490 vs. 364 µm, respectively) and that the passive response described by
Eq. A16 increases with time. Statistical tests indicate that
the means of the responses become significantly different from those of
the linear strand at small radii of curvature.
| |
DISCUSSION |
Extracellular field shocks used to defibrillate whole hearts
produce currents that flow across a complex structure involving extracellular and intracellular discontinuities at various structural levels. This study investigated the effect of transverse electric fields applied to synthetic cardiac strands. Strands and layers of
myocardial cells, showing typical curvature and branching patterns, form basic structural units of the myocardium in vivo (12,
24). They are likely to play an important role in the formation
of electric virtual sources (8). Such sources may exert a
defibrillatory effect via one of several mechanisms: 1)
extinction of the fibrillating wave front without induction of new ones
(2), 2) progressive excitation of the cell
membranes (4), or 3) local prolongation of the
refractory state (10).
The main findings of this study can be summarized as follows.
Transverse field stimulation of strands of myocardial cells with
rectangular pulses evokes Vm responses that
reproducibly arise from distinct structural features and are separable
on the basis of their temporal behavior. The most elementary response originates from the two edges of the strand. This situation is a
microscopic version of the surface polarization effect in tissue. For
all strands with widths in the range of 100-150 µm, there is a
rapid response (Vmar) with response
times of 0.9 ms or less that is hyperpolarizing on one side and
depolarizing on the other side of the strand. The rapid response is
accompanied by slower responses that develop during the remainder of
the field pulse. For field stimuli applied during the early plateau
phase of the action potential, and strands oriented perpendicular to
the field, the slower responses include a hyperpolarizing component
(Vmas) and also a depolarizing
component (Vmad) that appears only at
high field intensities. Both Vmas and
Vmad are larger on the anodal side of
the strand. The addition of higher-order structural features influences
both the rapid and the slow responses. Thus curved segments generate
additional rapid (Vmbr) and slow
components (Vmbs) that invade across
the width of the strand and are hyperpolarizing or depolarizing
depending on the field direction. This situation is a microscopic
version of the fiber curvature effect in tissue. The specific details
of these different components are discussed below.
Components of the field response of the linear strand.
The theoretical modeling culminating in Eq. A3 in the
APPENDIX describes the passive behavior of a narrow-width
strand (see Fig. 9A) after it has reached steady-state
conditions. It has been shown in a passive model of a short fiber that
the steady state is reached very rapidly, on the order of tens of
microseconds (1). In Figs. 4-6,
Vmar is the field response arising
within the first millisecond from virtual sources at the edges of the
linear strand. This response is determined by the onset of the
extracellular potential gradient in the bath that alters the
Vm values of the cells nearly instantaneously while the intracellular potential remains relatively constant during the millisecond interval. Hence,
Vmar would be expected to reflect the
passive properties of the strand and to be proportional to the
extracellular potential gradient, as given by Eq. A2 of the
APPENDIX. The dotted lines in the plot of
Vmar in Fig. 6 have been drawn with
slopes given by Eq. A3, if one assumes that APA is equal to
100 mV (19). The data fall close to but between the
theoretical lines for field strengths up to ~15 V/cm.
Vmar field responses of less than the
relation of Eq. A3 have also been observed in single
guinea pig ventricular cells (V. Sharma, S. N. Lu, and L. Tung,
unpublished observations) and may be the result of the development of
an internal electric field or of an attenuated response of the
membranes in the T-tubular network compared with that of the plasma
membrane of the cell. At higher field strengths, inward rectification
of Vmar is observed across the strand
(Fig. 5), a behavior not predicted by passive cable models. Thus active
membrane responses apparently can contribute to
Vmar during stronger field
stimulation.
View larger version (13K):
[in this window]
[in a new window]
|
Fig. 9.
Thin strand models. Linear and U-shaped geometries are
shown here. A: linear strand. The coordinate y is
defined transverse to the axis of the strand. B: U-shaped
strand. The coordinate s is defined along the length of the
strand, with the origin at the bottom of the semicircular loop and the
positive direction upward along the loop. The one-quarter-circular
bends on either corner of the U are separated by a linear base segment
of length L and have a radius R. The angle  is
defined such that s = R along the bend.
W, width between edges; Ey, field
intensity.
|
|
After the initial membrane responses, there are additional
and equally significant time-dependent responses,
Vmas and
Vmad, that arise during the field
stimulus pulse. These responses are presumably a "byproduct"
triggered by the rapid Vmar
polarization changes across the strand and are not readily explained by
passive models. The sequential interaction between (passive) rapid and
subsequent (active) slow components has been well characterized for the
field stimulation of single cells (3, 11, 27). Time-dependent currents are altered at both edges of the strand simultaneously but not with the same magnitude and type of ionic currents because of the polarity difference. Because the strands in our
experiments are relatively narrow (100-150 µm), the currents activated at the two edges interact electrotonically and produce a net
outward current that results in a hyperpolarizing response that
dominates across the entire strand. Theoretically, the hyperpolarizing Vmas response can correspond either
to a decrease of the inward current present before the pulse (e.g.,
Na+ window current or L-type Ca2+ current) or
to an increase of outward current (e.g., transient outward
current) or to an interaction of the two. The fact that Vmas is uniform across the strand for
field strengths up to ~15 V/cm makes it difficult to separate the
active components of Vmas initiated
by Vmar depolarization at the strand
border facing the cathode and by Vmar
hyperpolarization at the strand border facing the anode. Moreover, Vmas may arise from
Vmar-induced changes in current flow
through ion channels with unaltered biophysical behavior, or,
alternatively, it may reflect field-induced changes in kinetic and
steady-state properties or ion selectivity of the channels. Several
observations speak in favor of an important change of ion current at
the cathodal side of the strand. First, depolarization from the
action-potential plateau to values closer to the equilibrium potential
for Ca2+ or Na+ is expected to decrease the
inward current (and, hence, increase the net outward current) present
during the pulse. The fact that the membrane resistance
(rm) is relatively high during the plateau (29) implicates a strong electrotonic influence of this
change onto the other half of the strand hyperpolarized by
Vmar. Second, without a depolarizing
influence from the cathodal side of the strand, hyperpolarization of
the anodal side of the strand would be expected to activate
repolarizing K+ current, which has been shown to stabilize
Vm at its resting value (so-called "all-or-nothing"
repolarization) (14). However, in the present experiments,
Vm returned consistently to the plateau level. As mentioned
above, one has to envision the possibility that the application of a
strong extracellular field may modify the behavior of ionic channels in
a way that is not explained by current models of normal channel
behavior. In a recent simulation study, the hyperpolarizing
Vmas response of single cells to
extracellular field pulses was mimicked by a depolarization-induced
activation of an outward current outside the physiological range
(3). The present experiments also provide insight into the
"asymmetry" of the hyper- and depolarizing sources created by a
field pulse during the plateau (9, 30). The Vmas hyperpolarization adds to the
hyperpolarizing Vmar response at the
anodal half and subtracts from the depolarizing Vmar response at the cathodal half of
the strand.
At field strengths greater than ~20 V/cm (Figs. 4-6) in nominal
120-µm-wide strands, a further component,
Vmad, is observed. It is
characterized 1) by net depolarization and 2) by
the fact that the response is mainly confined to the anodal side of the
strand (Fig. 5). The inward current underlying this component results
in a nonuniform response across the strand, as was the case for
component Vmas, and appears to
activate only at high field strengths on the anodal side of the strand.
As of yet, the nature of this component is unexplained, and we do not
know to what extent it may be linked to
Vmas. The fact that the electrotonic
transmission of this component to the other edge of the strand is weak
suggests a reduction of . This could occur subsequent either to
partial cell-to-cell uncoupling or to a decrease in
rm. A delayed response similar to that of
Vmad has been modeled in single
cardiac cells under field stimulation with the use of a modified form
of the Luo-Rudy dynamic model (13) that incorporates an
electroporation current (3). However, in single guinea pig
cell experiments (V. Sharma, S. N. Lu, and L. Tung, unpublished
observations), the Vmad-like
component of the field response that appears at high field strengths is
not always followed by residual postshock depolarization, as would be
expected from electroporation. As an alternative, activation of a
pacemaker-like current that has been reported in ventricular cells
(17) could be postulated.
Finally, in this study, we have defined the various components of the
Vm response on the basis of temporal criteria.
Other decomposition schemes are possible, such as the "common-mode" and "differential-mode" approach used to describe electric-field responses of single guinea pig cardiac cells (22). Fast et al. (7) used the temporal and spatial patterns of the
Vm responses to distinguish low field (type I),
intermediate field (type II), and high field (type III) responses of
linear strands of cultured neonatal rat heart cells.
Effects of fiber curvature (U-shaped strand).
When the linear strand is joined to a semicircular U-shaped bend (Fig.
7), the responses across the strand shift. Depending on field polarity
and bend radius, either hyperpolarization or depolarization prevails
(Fig. 8), and the amount of shift depends on the bend radius. Unlike
the responses for the linear strand, where a strong hyperpolarizing
component is always present (Fig. 6), it is possible for all of the
sites to undergo a net depolarization by the end of the field pulse
(Fig. 8B, left; radius 250 µm).
The differential responses across the U-shaped strand are associated
mainly with a rapid component (Vmbr)
that resembles that in purely linear strands
(Vmar), whereas there is little
gradient in the slow component Vmbs
across the U-shaped strand (among sites 1,
4, and 7 in Fig. 8), similar to the case of the
slow Vmas response for the linear
strand. The Vmbr and
Vmbs responses both shift in the
hyperpolarizing direction for positive polarity fields and to a smaller
extent in the depolarizing direction for negative polarity fields. The
shifts increase and become significant as R decreases to
below a critical radius that depends on field polarity.
Thus the differences between the Vmar
and Vmbr responses and between the
Vmas and
Vmbs responses can be attributed to
the electrotonic effect of virtual sources generated in the curved
segment of the U-structure. The origin of the electrotonic current is
evident from the passive model (for analysis, see Fig. 9B).
The concept of the "activating function" relates the strength of
virtual sources lying along the strand to the first derivative of the
component of electric field tangential to the strand (20).
This concept can be used to derive the steady-state
Vm along a narrow passive strand with a
semicircular bend. We find that the global polarization predicted by
Eq. A16 of the APPENDIX is a good descriptor of
both the rapid and slow responses in Fig. 8. Hyperpolarizing currents
are produced at the anodal side of the semicircular bend and
depolarizing currents at the cathodal side.
For a given semicircular bend with radius R, the
Vm response is predicted to have the spatial
pattern shown (see Fig. 10) for the hyperpolarizing direction. First,
Vm increases in magnitude from zero at the
midpoint of the bend (where s = R
/2;
coordinate s is defined with the origin at the point of
beginning curvature, as shown in Fig. 9B. Second, at the end
of the bend (where s = 0), Vm
declines a small amount from its maximum value. Third, Vm falls exponentially to zero along the linear
segment outside the bend. Thus we can think of the semicircular bend as
having a pair of virtual sources, one hyperpolarizing and one
depolarizing, lying on either side of the midpoint of the bend and
driving the Vm responses. It has been shown that
the Vm responses to a pair of virtual sources
having opposite polarity will become slower as the separation between
sources increases (1, 25). Moreover, sources can be
located some distance away from a given recording site, so that spatial
diffusion of charge must occur before Vm will
change in response to the applied field. This results in further
slowing of the rise time of Vm. Together, these
effects imply that the steady-state response (plotted Fig. 10)
will be reached on a time scale much slower than that attained for the passive response of the purely linear strand. Therefore, we would expect both Vmbr and
Vmbs to be affected by the presence
of the semicircular bend, as suggested by our results (Fig. 8). Because
both components of the response may reflect the same set of virtual
sources, we chose to fit the data with scaled versions of Eq. A16, having the same for a given field polarity but with
different scaling factors (br and bs) to
account for changes in amplitude with time.
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 10.
Vm responses for the semicircular
strand. Plotted here are the model results for the hyperpolarizing
Vm responses at the bottom of the strand to a
uniform electric field directed in the positive y-direction.
The radius R of the strand has been varied from 0.5 to
10. The field strength has been assumed to be 10 V/cm and to be
0.4 mm. Each of the curves terminates on the right at the
abscissa for s equal to the midpoint of the semicircle.
|
|
It is important to note that, although the slow response of the
semicircular bend, Vmbs, can be
attributed to the structure of the bend and fitted by Eq. A16, the underlying mechanism may be more complicated. The fact that the electrotonic space constants are not identical for the hyperpolarizing and depolarizing responses (490 vs. 364 µm,
respectively) suggests that a passive model with constant membrane
resistance does not entirely account for the
Vmbs response. Additional active
currents may be involved, owing to the polarization represented by the
rapid component, Vmbr, just as was
the case of Vmas in response to
Vmar for the linear strand. The
relative amplitude of an active component of
Vmbs remains to be determined in
further studies.
To summarize, the rapid responses of narrow strands to field
stimulation are determined primarily by strand geometry and by orientation of the strand within the field. The slower responses largely reflect time-dependent changes owing to imbalances in hyperpolarizing and depolarizing ionic currents in the strand. In
linear strands, the membrane response is asymmetric at moderate field
strengths of ~11 V/cm, with the asymmetry arising primarily from a
slow active response. Addition of a semicircular bend to the strand
introduces additional passive and possibly active components to the
rapid and slow responses that are hyperpolarizing for one side of the
bend and depolarizing on the other. These responses become larger as
the radius of curvature R diminishes down to the point where
R 
. More generally, we would expect that
inclusion of any complex tissue geometry that changes the fiber
orientation with respect to the applied field would produce
higher-order complexities in the extent and polarity of the passive
components of the field responses that would lead in turn to further
complexities in the active components of the field responses.
In conclusion, electric-field responses of 100- to
150-µm-wide strands of myocardium consisting of straight lines or
U-shapes can be separated into components with characteristic temporal and amplitude signatures. With linear strands, the responses reflect differences at the two edges of the strand and consist of three components (rapid Vmar, slow
Vmas, and delayed
Vmad) that vary in amplitude and time
course with field strength. With U-shaped strands at field strengths of
~11 V/cm, the responses consist of a rapid component
(Vmbr) similar to
Vmar except at very small curvatures
as well a slow component
(Vmbs) more complex than
Vmas owing to the higher-order
curvature of the structure.
| |
APPENDIX |
Model of Passive Behavior of Vm
Vm across a linear strand.
Let us assume an infinitely long, linear strand aligned along the
x-direction with a uniform electric field oriented
perpendicular to the strand along the y-direction. A passive
cable model (with a constant membrane resistance
rm and geometry shown in Fig.
9A) predicts that
the polarizing responses of Vm to fields applied transverse to the strand should be symmetric at the two edges of the
strand and zero at the center. The theoretical changes in polarization
Vm(y) for a given field intensity
Ey are (16)
|
(A1)
|
where the width between the edges is W, h
represents hyperbolic, and the space constant is . If W
is considerably smaller than ,
Vm(y) across the strand (from
y = W/2 to W/2) is
approximately linear, and
|
(A2)
|
Thus the theoretical changes in polarization at the
strand edges at y = y0 and y = y0 (the recording sites near the edges) are
Vm(y0) = Vm(y0) = Eyy0, where
y0 will be slightly less than W/2. If
the polarization of the strand is normalized to APA, the slope
S of the linear change in Vmwith
E is equal to
![S=[V<SUB>m</SUB>(<IT>y<SUB>0</SUB></IT>)<IT>/</IT>APA]<IT>/E<SUB>y</SUB>=y<SUB>0</SUB>/</IT>APA](/content/vol279/issue4/fulltext/H1579/img003.gif)
|
(A3)
|
Vm across a strand consisting of a linear and curved
portion (U-shape).
The Vm of a narrow curved fiber lying in a
uniform electric field is derived here. We consider the passive
behavior for the general case of a U-shaped bend (Fig. 9B).
Assuming that the thickness of the fiber is small compared with the
characteristic dimensions of the bend (radius R and length
of base L), the variations in Vm
across the fiber will be neglected. The coordinate s is
defined with its origin on one side of the bend, corresponding to the site of the experimental measurements in this study. The electric field
E is assumed to be uniform in the y-direction
with intensity Ey.
For s < 0 and R/2 < s < (R+ L)/2,
Vm must satisfy the well-known one-dimensional
cable equation
|
(A4a)
|
where is the space constant. In the curved region
0 s < R/2
![V<SUB>m</SUB><IT>−&lgr;<SUP>2</SUP> </IT><FR><NU><IT>∂<SUP>2</SUP>V</IT><SUB>m</SUB></NU><DE><IT>∂s<SUP>2</SUP></IT></DE></FR><IT>=</IT>−<IT>&lgr;<SUP>2</SUP> </IT><FR><NU><IT>∂</IT>[<B>E</B><IT>·</IT><B>a</B><SUB><IT>s</IT></SUB>]</NU><DE><IT>∂s</IT></DE></FR>](/content/vol279/issue4/fulltext/H1579/img005.gif)
|
(A4b)
|
on the basis of the notion of the activating function (15,
18), defined to be the right side of Eq. A4b. The variable as is the unit
vector tangential to the fiber, and
E · as represents
the component of electric field directed along the fiber direction.
Note that the activating function is zero in Eq. A4a. At
s = 0, as is oriented
along the x-direction, and, therefore,
E · as is zero. At
s = R/2,
as is oriented along the
y-direction,
E · as is a
constant, and, therefore,
(E · as)/s
is zero. By use of the relation s = Rcos, Eq. A4b can be rewritten as
|
(A5)
|
The particular solution to Eq. A5 has the form
|
(A6)
|
Substituting Eq. A6 into Eq. A5 allows for
the determination of the coefficient A (20)
|
(A7)
|
The total solution for Vm requires the
addition of homogeneous solutions in the different portions of the
strand
|
(A8, A-C)
|
By symmetry, Vm must be symmetric about
the point s = (R+ L)/2 and is
equal to zero there. Thus Eq. A8c becomes
|
(A9)
|
At s = 0 and s = R/2, the transmembrane potential
Vm and intracellular current
Ii must be continuous. Because
|
(A10)
|
and because Ve/s is
continuous, continuity of Ii also
implies continuity of Vm/s,
where Ve is extracellular potential and
Vi is intracellular potential. The variable
ri in Eq. A10 is the specific
intracellular axial resistance. Thus the boundary conditions result in
the following equations at s = 0
|
(A11a)
|
|
(A11b)
|
and at s = R/2
|
(A12a)
|
|
(A12b)
|
The solving of Eqs. A11 and A12
for the coefficients B, C, D, and
E' and the substitution into Eqs. A8,
a and b, and A9 result in the total
Vm response
![V<SUB>m</SUB><IT>=</IT><FENCE><AR><R><C>−<FR><NU><IT>&lgr;<SUP>2</SUP>E<SUB>y</SUB>R</IT></NU><DE><IT>2</IT>(<IT>&lgr;<SUP>2</SUP>+R<SUP>2</SUP></IT>)</DE></FR> <FENCE><IT>1+</IT><FENCE><IT>e</IT><SUP>−[(<IT>R&pgr;</IT>+<IT>2L</IT>)<IT>/2&lgr;</IT>]</SUP><IT>+</IT><FR><NU><IT>&lgr;</IT></NU><DE><IT>R</IT></DE></FR> (<IT>1−e</IT><SUP>−(<IT>L/&lgr;</IT>)</SUP>)</FENCE><IT>e</IT><SUP>−(<IT>R&pgr;/2&lgr;</IT>)</SUP></FENCE><IT>e<SUP>s/&lgr;</SUP>,</IT></C><C><IT>s≤0</IT></C></R><R><C>−<FR><NU><IT>&lgr;<SUP>2</SUP>E<SUB>y</SUB>R</IT></NU><DE><IT>2</IT>(<IT>&lgr;<SUP>2</SUP>+R<SUP>2</SUP></IT>)</DE></FR> <FENCE><IT>2 </IT>cos <FENCE><FR><NU><IT>s</IT></NU><DE><IT>R</IT></DE></FR></FENCE><IT>−e</IT><SUP>−(<IT>s/&lgr;</IT>)</SUP><IT>+</IT><FENCE><IT>e</IT><SUP>−[(<IT>R&pgr;</IT>+<IT>2L</IT>)<IT>/2&lgr;</IT>]</SUP><IT>+</IT><FR><NU><IT>&lgr;</IT></NU><DE><IT>R</IT></DE></FR> (<IT>1−e</IT><SUP>−(<IT>L/&lgr;</IT>)</SUP>)</FENCE><IT>e</IT><SUP>[<IT>s</IT>−(<IT>R&pgr;/2</IT>)]<IT>/&lgr;</IT></SUP></FENCE><IT>,</IT></C><C><IT>0<s≤</IT><FR><NU><IT>R&pgr;</IT></NU><DE><IT>2</IT></DE></FR></C></R><R><C>−<FR><NU><IT>&lgr;<SUP>2</SUP>E<SUB>y</SUB>R</IT></NU><DE><IT>&lgr;<SUP>2</SUP>+R<SUP>2</SUP></IT></DE></FR> <FENCE><FENCE><IT>e</IT><SUP>−(<IT>R&pgr;/2&lgr;</IT>)</SUP><IT>−</IT><FR><NU><IT>&lgr;</IT></NU><DE><IT>R</IT></DE></FR></FENCE><IT>e</IT><SUP>−(<IT>L/2&lgr;</IT>)</SUP> sinh <FENCE><FR><NU><IT>s−</IT><FR><NU><IT>R&pgr;+L</IT></NU><DE><IT>2</IT></DE></FR></NU><DE><IT>&lgr;</IT></DE></FR></FENCE></FENCE><IT>,</IT></C><C><FR><NU><IT>R&pgr;</IT></NU><DE><IT>2</IT></DE></FR><IT><s≤</IT><FR><NU><IT>R&pgr;+L</IT></NU><DE><IT>2</IT></DE></FR></C></R></AR></FENCE>](/content/vol279/issue4/fulltext/H1579/img017.gif)
|
(A13, a-c)
|
For the case where L 
(i.e., a
one-quarter-circular bend only)
![V<SUB>m</SUB><IT>=</IT><FENCE><AR><R><C>−<FR><NU><IT>&lgr;<SUP>2</SUP>E<SUB>y</SUB>R</IT></NU><DE><IT>2</IT>(<IT>&lgr;<SUP>2</SUP>+R<SUP>2</SUP></IT>)</DE></FR> <FENCE><IT>1+</IT><FR><NU><IT>&lgr;</IT></NU><DE><IT>R</IT></DE></FR><IT> e</IT><SUP>−(<IT>R&pgr;/2&lgr;</IT>)</SUP></FENCE><IT>e<SUP>s/&lgr;</SUP>,</IT></C><C><IT>s≤0</IT></C></R><R><C>−<FR><NU><IT>&lgr;<SUP>2</SUP>E<SUB>y</SUB>R</IT></NU><DE><IT>2</IT>(<IT>&lgr;<SUP>2</SUP>+R<SUP>2</SUP></IT>)</DE></FR> <FENCE><IT>2 </IT>cos <FENCE><FR><NU><IT>s</IT></NU><DE><IT>R</IT></DE></FR></FENCE><IT>−e</IT><SUP>−(<IT>s/&lgr;</IT>)</SUP><IT>+</IT><FR><NU><IT>&lgr;</IT></NU><DE><IT>R</IT></DE></FR><IT> e</IT><SUP>[<IT>s</IT>−(<IT>R&pgr;/2</IT>)]<IT>/&lgr;</IT></SUP></FENCE><IT>,</IT></C><C><IT>0<s≤</IT><FR><NU><IT>R&pgr;</IT></NU><DE><IT>2</IT></DE></FR></C></R><R><C>−<FR><NU><IT>&lgr;<SUP>2</SUP>E<SUB>y</SUB>R</IT></NU><DE><IT>2</IT>(<IT>&lgr;<SUP>2</SUP>+R<SUP>2</SUP></IT>)</DE></FR> <FENCE><FENCE><FR><NU><IT>&lgr;</IT></NU><DE><IT>R</IT></DE></FR><IT>−e</IT><SUP>−(<IT>R&pgr;/2&lgr;</IT>)</SUP></FENCE><IT>e</IT><SUP>−[<IT>s</IT>−(<IT>R&pgr;/2</IT>)]<IT>/&lgr;</IT></SUP></FENCE><IT>,</IT></C><C><FR><NU><IT>R&pgr;</IT></NU><DE><IT>2</IT></DE></FR><IT><s</IT></C></R></AR></FENCE>](/content/vol279/issue4/fulltext/H1579/img018.gif)
|
(A14, a-c)
|
For the case of a semicircular bend in which L = 0, as in the experiments of this study, the total
Vm response reduces to
|
(A15, a and b)
|
Equation A15 is plotted in Fig.
10 for values of R/
ranging from 0.5 to 10. Because of symmetry, only the hyperpolarizing values of Vm have been plotted for
sR/2. Note that at s = 0, Vm is a nonmonotonic function of R
|
(A16)
|
and reaches the asymptotic relations
|
(A17)
|
From Fig. 10, we see that
Vm(0) is maximized when
R . Finally, we note that the maximum value for
Vm(s) at a given R does not occur at s = 0 but rather along the bend
where
|
(A18)
|
Equation A18 does not have a closed form solution but
can be solved numerically.
| |
ACKNOWLEDGEMENTS |
We are grateful to Stephan Rohr for helpful discussions, Regula
Flückiger Labrada for the cell cultures, and Vladimir Fast for
providing the data analysis software.
| |
FOOTNOTES |
This work was supported by National Heart, Lung, and Blood Institute
Grant HL-48266, the Tilghman Fund, the Swiss Heart Foundation, and the
Swiss National Science Foundation.
Address for reprint requests and other correspondence: L. Tung,
Dept. of Biomedical Engineering, The Johns Hopkins Univ., Rm. 703, Traylor Bldg., 720 Rutland Ave., Baltimore, MD 21205 (E-mail:
ltung{at}bme.jhu.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 22 December 1999; accepted in final form 20 April 2000.
| |
REFERENCES |
1.
Cartee, LA,
and
Plonsey R.
The transient subthreshold response of spherical and cylindrical cell models to extracellular stimulation.
IEEE Trans Biomed Eng
39:
76-85,
1992[Web of Science][Medline].
2.
Chen, PS,
Shibata N,
Dixon EG,
Martin RO,
and
Ideker RE.
Comparison of the defibrillation threshold and the upper limit of ventricular vulnerability.
Circulation
73:
1022-1028,
1986[Abstract/Free Full Text].
3.
Cheng, D,
Tung L,
and
Sobie EA.
Nonuniform responses of transmembrane potential during electric field stimulation of single cardiac cells.
Am J Physiol Heart Circ Physiol
277:
H351-H362,
1999[Abstract/Free Full Text].
4.
Dillon, S,
and
Kwaku K.
Progressive depolarization: a unified hypothesis for defibrillation and fibrillation induction by shocks.
J Cardiovasc Electrophysiol
9:
529-552,
1998[Web of Science][Medline].
5.
Fast, VG,
Darrow BJ,
Saffitz JE,
and
Kleber AG.
Anisotropic activation spread in heart cell monolayers assessed by high-resolution optical mapping. Role of tissue discontinuities.
Circ Res
79:
115-127,
1996[Abstract/Free Full Text].
6.
Fast, VG,
Rohr S,
Gillis AM,
and
Kleber AG.
Activation of cardiac tissue by extracellular electrical shocks.
Circ Res
82:
375-385,
1998[Abstract/Free Full Text].
7.
Fast, VG,
Rohr S,
and
Ideker RE.
Nonlinear changes of transmembrane potential caused by defibrillation shocks in strands of cultured myocytes.
Am J Physiol Heart Circ Physiol
278:
H688-H697,
2000[Abstract/Free Full Text].
8.
Gillis, AM,
Fast VG,
Rohr S,
and
Kleber AG.
Mechanism of ventricular defibrillation: the role of tissue geometry in the changes of transmembrane potential in patterned myocyte cultures.
Circulation
101:
2438-2445,
2000[Abstract/Free Full Text].
9.
Gillis, AM,
Fast VG,
Rohr S,
and
Kleber AG.
Spatial changes in transmembrane potential during extracellular electrical shocks in cultured monolayers of neonatal rat ventricular myocytes.
Circ Res
79:
676-690,
1996[Abstract/Free Full Text].
10.
Jones, JL,
and
Tovar OH.
The mechanism of defibrillation and cardioversion.
Proc IEEE
84:
392-403,
1996.
11.
Krassowska, W,
and
Neu JC.
Response of a single cell to an external electric field.
Biophys J
66:
1768-1776,
1994[Web of Science][Medline].
12.
Le Grice, IJ,
Smaill BH,
Chai LZ,
Edgar SG,
Gavin JB,
and
Hunter PJ.
Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog.
Am J Physiol Heart Circ Physiol
269:
H571-H582,
1995[Abstract/Free Full Text].
13.
Luo, CH,
and
Rudy Y.
A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes.
Circ Res
74:
1071-1096,
1994[Abstract/Free Full Text].
14.
McAllister, RE,
Noble D,
and
Tsien RW.
Reconstruction of the electrical activity of cardiac Purkinje fibres.
J Physiol (Lond)
251:
1-59,
1975.
15.
McNeal, DR.
Analysis of a model for excitation of myelinated nerve.
IEEE Trans Biomed Eng
23:
329-337,
1976[Web of Science][Medline].
16.
Plonsey, R,
and
Altman KW.
Electrical stimulation of excitable cells: a model approach.
Proc IEEE
76:
1122-1129,
1988.
17.
Ranjan, R,
Chiamvimonvat N,
Thakor NV,
Tomaselli GF,
and
Marban E.
Mechanism of anode break stimulation in the heart.
Biophys J
74:
1850-1863,
1998[Web of Science][Medline].
18.
Rattay, F.
Analysis of models for external stimulation of axons.
IEEE Trans Biomed Eng
33:
974-977,
1986[Web of Science][Medline].
19.
Rohr, S,
Schölly DM,
and
Kléber AG.
Patterned growth of neonatal rat heart cells in culture.
Circ Res
68:
114-130,
1991[Abstract/Free Full Text].
20.
Roth, BJ.
Mechanisms for electrical stimulation of excitable tissue.
Crit Rev Biomed Eng
22:
253-305,
1994[Web of Science][Medline].
21.
Roth, BJ,
and
Krassowska W.
The induction of reentry in cardiac tissue. The missing link: how electric fields alter transmembrane potential.
Chaos
8:
204-220,
1998[Web of Science][Medline].
22.
Sharma V and Tung L. Analysis of field-induced transmembrane
potential responses of single cardiac cells in terms of active and
passive components. CD-ROM Proceedings of the World Congress on Medical
Physics and Biomedical Engineering, July 23-28, Abstract no.
WE-E327-04, 2000.
23.
Sobie, EA,
Susil RC,
and
Tung L.
A generalized activating function for predicting virtual electrodes in cardiac tissue.
Biophys J
73:
1410-1423,
1997[Web of Science][Medline].
24.
Sommer, JR,
and
Scherer B.
Geometry of cell and bundle appositions in cardiac muscle: light microscopy.
Am J Physiol Heart Circ Physiol
248:
H792-H803,
1985.
25.
Susil, RC,
Sobie EA,
and
Tung L.
Separation between virtual sources modifies the response of cardiac tissue to field stimulation.
J Cardiovasc Electrophysiol
10:
715-727,
1999[Web of Science][Medline].
26.
Trayanova, N,
Skouibine K,
and
Aguel F.
The role of cardiac tissue structure in defibrillation.
Chaos
8:
221-233,
1998[Web of Science][Medline].
27.
Tung, L,
and
Borderies JR.
Analysis of electrical-field stimulation of single cardiac muscle cells.
Biophys J
63:
371-386,
1992[Web of Science][Medline].
28.
Tung, L,
Rohr S,
and
Kleber AG.
Transmembrane potential responses of linear and curved strands of cardiac cells to uniform field stimulation (Abstract).
Pacing Clin Electrophysiol
22:
746,
1999.
29.
Weidman, S.
Effect of current flow on the membrane potential of cardiac muscle.
J Physiol (Lond)
115:
227-236,
1951.
30.
Zhou, X,
Ideker RE,
Blitchington TF,
Smith WM,
and
Knisley SB.
Optical transmembrane potential measurements during defibrillation-strength shocks in perfused rabbit hearts.
Circ Res
77:
593-602,
1995[Abstract/Free Full Text].
Am J Physiol Heart Circ Physiol 279(4):H1579-H1590
0363-6135/00 $5.00
Copyright © 2000 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
