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1 Todd Franklin Cardiac Research Laboratory, Children's Heart Center, Department of Pediatrics, Emory University, Atlanta, Georgia 30322; 2 Department of Medical Physiology and Sports Medicine, Utrecht University, 3584 CG Utrecht; and 3 Academic Medical Center, Department of Physiology, University of Amsterdam, 1105 AZ Amsterdam, The Netherlands
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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We have extended our "coupling clamp" technique, in which we couple a real cell to a real-time simulation of a model cell, to now incorporate a real cardiac cell as the central element of a two-dimensional sheet of model cells, in which the coupling conductances may be different in the x and y directions and a specific region of lack of coupling conductance may serve as a resistive barrier. We stimulated the real cell in the central location and determined the critical size of the real cell for successful activation of the entire sheet. We found that this critical size was decreased when anisotropy was present compared with the isotropic case and was further decreased when the central site of stimulation was close to the resistive barrier. The heart normally has some degree of anisotropy, and it has been shown that the remodeling that occurs in peri-infarction zones produces a particular loss of lateral connections compared with end-to-end connections among heart cells. We propose that the normal existence of anisotropy and enhancement of the degree of anisotropy both by loss of lateral gap junctions and the development of resistive barriers may play a facilitating role in the development of ectopic foci that may lead to cardiac arrhythmias.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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MANY EXPERIMENTAL AND THEORETICAL STUDIES have focused
on action potential propagation in cell pairs, linear strands, or
multidimensional structures of cardiac cells. Multidimensional cardiac
tissue differs in conduction behavior from the simple linear models in
several respects. Myocyte structure is asymmetric, with average adult myocytes being 100-150 µm in length and 15-25 µm in width
(11, 18), with the cellular arrangement of these cells giving the tissue a structural anisotropy. Anisotropy is further enhanced by the
pattern of gap junction distribution such that the propagation velocity
(
T) in the transverse direction is lower than
longitudinal velocity (L), with ratios of
L to T in the range of 2.7 in the
ventricle to as much as 12 in the atrial crista terminalis (3, 10, 14,
32). There are also further anisotropies in cardiac structure produced
by the presence of connective tissue strands and the occurrence of
scars from prior myocardial infarction (16, 26) that can be represented
as resistive barriers because of the effective absence of gap junctions
across such regions, which produces "discontinuous anisotropy"
(29).
Directional dependence of action potential propagation has been studied in both computer simulations (12, 17, 28) and cell culture systems with cultures grown on a directed collagen matrix to form an anisotropic network (5). Structural complexities affecting propagation and extracellular waveforms have been previously described for intact tissue in the atria and in the Purkinje fibers (27, 30-32) and at the Purkinje-ventricular muscle junction (34). Most of these studies have focused on the effects of the anisotropy and specific discontinuities on propagation of an already established wave of excitation, such as the formation of a "pivoting point" for propagation (4, 8, 23), propagation slowing through an "isthmus" of tissue (1, 2), or effects of sudden variations in strand geometry (6, 7, 22). We have studied many of these same phenomena with our "coupling clamp" technique, in which we either coupled two real cardiac myocytes together or coupled a real myocyte to a real-time simulation of a computer model of a myocyte and observed a common feature of alterations in propagation with changes in the ratio of the source current available to the current sink required to continue propagation (33, 35).
Whereas the multiple effects of anisotropy and resistive barriers on a propagating wave have received much attention, much less work has been done on the effects of anisotropy and resistive barriers on the initiation of a propagating wave of excitation. The initiation of a propagating wave from either a small region of direct stimulation or an automatically active focus involves many of the same general factors, such as the input resistance of the syncytial tissue, the ability of the focus region to serve as a current source, and the sensitivity of the focus region to electrotonic interactions with the surrounding quiescent region, which will tend to either abolish automaticity or prevent the initiation of propagation. One of the major problems in developing such an experimental system is that we have been restricted to either completely theoretical solutions of multidimensional arrangements of cell models or experimental models such as intact tissue or cell cultures in which neither the cell membrane properties nor the cellular connectivity can be directly measured. To study the way in which anisotropic connectivity and the presence of resistive barriers affect initiation of a propagating wave, we have extended our methodology of connecting a real cell to one or two model cells (35, 37) to enable the real-time simulation of a two-dimensional sheet of model ventricular cells into which a real ventricular cell is incorporated as the central element of the sheet.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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Cell isolation and electrodes.
Single ventricular myocytes were prepared from adult guinea pigs
weighing 300-500 g that were anesthetized (intraperitoneally) with
50 mg/kg pentobarbital sodium and 500 units of heparin. The heart was
rapidly removed via thoracotomy with artificial ventilation, and the
aorta was cannulated for Langendorff perfusion. Single cells were
isolated according to the method described previously (33). Briefly,
the isolated heart was mounted on a Langendorff apparatus and perfused
sequentially with normal Tyrode solution for 5 min, with nominally
Ca2+-free Tyrode solution for 6-7 min, with nominally
Ca2+-free Tyrode solution containing collagenase and
protease for 8-12 min, and with storage solution for 5 min at a
rate of 3-4 ml · g
1 · min1 at 35-36°C. For ventricular cell
isolation, the enzyme-perfused left ventricle was cut into small
pieces, stirred in storage solution, and filtered through nylon mesh.
The resulting suspension of cells was kept in storage solution at room
temperature. The isolated cells were transferred to an experimental
chamber containing normal Tyrode solution. The chamber was continuously
perfused with normal Tyrode solution at 2 ml/min, and the temperature
was maintained at 36 ± 0.5°C. Only cells that were quiescent and
had a rod-shaped appearance were used in this study. The pipettes were
pulled from borosilicate glass and, after fire polishing, had
resistances of 4-6 M
when filled with the internal solution.
High-resistance seals were formed with the cell membrane with the use
of light suction, and the membrane was disrupted by applying a
transient suction. The junctional potential was corrected by zeroing
the potential before the surface of the cell was touched by the pipette tip. Series resistance was carefully compensated for each cell by a
calibrated internal circuit of the amplifier used (Axoclamp; Axon
Instruments, Burlingame, CA). Repetitive stimulation of the guinea pig
ventricular cell was produced by depolarizing current pulses of 2 ms in
duration applied via the patch pipette.
Solutions. Normal Tyrode solution contained (in mM) 148.8 NaCl, 4.0 KCl, 1.8 CaCl2, 0.53 MgCl2, 0.33 NaH2PO4, 5 HEPES, and 5 glucose, with pH 7.4 adjusted with NaOH. The composition of Ca2+-free solution was the same as that of normal Tyrode solution except that CaCl2 was omitted. The enzyme solution contained 4-6 mg/100 ml collagenase (Yakult, Tokyo, Japan) and 0.5 mg/100 ml protease (type XIV, Sigma Chemical, St. Louis, MO) in Ca2+-free solution. The storage solution contained (in mM) 120 potassium glutamate, 5 MgCl2, 20 taurine, 0.5 EGTA, 10 glucose, and 10 HEPES, with pH 7.4 adjusted with KOH. The composition of the internal pipette solution (in mM) was 145 KCl, 5 Mg-ATP, 5.0 Na2 creatine phosphate, and 5.0 HEPES, with pH 7.2 adjusted with KOH. The external solution was normal Tyrode solution.
Real-time simulation of a sheet of cells with a real ventricular
cell.
We previously described the "coupling clamp" method used to
provide a coupling conductance either between two real cardiac cells
that are not directly in contact with each other or between a real
cardiac cell and a real-time simulation of a cardiac cell model (33,
37), and we recently extended this technique to include a real cardiac
cell within a linear strand of model cells (36). The common feature of
all of these methods is that at each time step, the computer samples
the membrane potential of the real cell (through an analog-to-digital
converter) and then uses this potential (Vcell) and
the membrane potential(s) of one or more model cells
(Vmodel) to which the real cell is connected to
compute the value of the coupling current leaving the real cell
(IC) from the sum of the coupling currents to each
of the model cells [each computed as (Vcell 
Vmodel) · GC,
where GC is the value of coupling conductance
previously selected]. During this same time step, a voltage
proportional to the coupling current is sent to the experimental setup
by a digital-to-analog converter and then converted to a current of
appropriate value by an amplifier and a voltage-to-current converter
for passage into the cell through the recording pipette. We also showed
that we could modulate the effective "size" of the real cell by
scaling the current passing into or out of the real cell via the
coupling circuit. Also, during the same time step, the computer solves
for the new value of membrane potential for each of the specified
models, using the computed coupling current from the real cell as well
as computed coupling currents among the model cells. As we discussed at
length in our earlier paper (37), this technique requires a fixed time
step (
t) which necessarily introduces some distortion into
the real-time simulation of the particular model system. With a
t of 80 µs we found that the cardiac membrane model of Luo
and Rudy (LR) (20, 21), which was specifically designed to represent
the membrane properties and intracellular ion concentrations of guinea pig ventricular cells, produces stable solutions that differ from their
solutions at a t of 1 µs by only a small percentage for such parameters as the maximum rate of change in potential
(dV/dt), conduction velocity, and stimulus threshold
(37).
t of 80 µs. Figure
1 shows a sheet composed of 7 × 7 = 49 elements. Each element (except those at the edges of the sheet) has
resistive coupling to four adjacent elements, and the coupling
conductances are assumed to be a constant coupling conductance
Gx along the x-axis and a possibly
different constant Gy along the
y-axis. Each element represents one cell, with the dimensions
and capacitance, as well as the numerical equations for membrane
conductances and ionic concentrations, as specified in the ventricular
membrane LR model (20). Elements are numbered along the x- and
y-axes as 3, 2, 1, 0, 1, 2, and 3 to
clearly identify the upper right quadrant of Fig. 1 as elements
(0, 0) through (3, 3), consisting of 16 elements for which the
real cell (in the center of the total sheet) is at the lower left
position of the upper right quadrant. We achieved horizontal and
vertical symmetry under the condition in which the only element that
directly received a stimulus or was spontaneously active was the
central element (0, 0). We also allowed the effective size of this
central element to be varied, as if this element represented a small
cluster of cells infinitely well coupled to each other. From the
condition of symmetry, we were then able to include all of the
interactions among elements in the entire sheet by solving only for the
potential and coupling currents of the 16 elements of the upper right
quadrant, even if Gx and
Gy were not equal to each other. If we let
Ix(i, j) = Gx · (Vi,j Vi+1,j) be
the current that flows to the right of element (i, j), then for the elements on the left edge of our upper right quadrant [elements (0, 0), (0, 1), (0, 2), and (0, 3)]
we simply doubled the computed Ix to
include the current that, from symmetry, would have flowed in the
opposite direction along the x-axis. Similarly, for the
elements along the bottom edge of our upper right quadrant
[elements (0, 0), (1, 0), (2, 0), and (3, 0)]
we let Iy(i, j) = Gy · (Vi,j Vi,j+1) be the current
that flows up along the y-axis, and we doubled this current for
these elements to include the current that, from symmetry, would have
flowed in the opposite direction down the y-axis. Please note
that the central element (which is the real cell) had both its
Ix and Iy
values doubled because it is connected to two elements [elements
(1, 0) and (0, 1)] that are in the quadrants
of the sheet not directly being simulated. For the other elements in
the upper right quadrant of Fig. 1, the definitions for
Ix and Iy are as given except for the edge elements, where the coupling current either to the right or upward is zero
[Ix = 0 for elements (3, 0),
(3, 1), (3, 2), and (3, 3) and Iy = 0 for elements (0, 3), (1, 3), (2, 3), and (3, 3)].
We thus reduced the problem of solving for a sheet of 49 elements to a
problem of solving for only 16 elements, one of which is a real cell.
We then further extended the method to include "resistive
barriers" within the sheet. This can be done as shown in Fig.
2 as long as the barriers used are in
pairs, with symmetry along either the vertical or horizontal axis. For the resistive barriers diagrammed in Fig. 2, we simply set
Gy equal to zero for elements
(1, 1), (0, 1), and (1, 1), and this also produced a
symmetric barrier as if Gy were equal to
zero for elements (1, 1), (0, 1), and
(1, 1).
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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We varied the size factor for the real guinea pig ventricular cell used
as element (0, 0) and paced this cell at 1 Hz while coupling the
cell to the sheet shown in Fig. 1 (with no barriers) for various values
of Gx and Gy.
Figure 3 shows successful propagation
through the sheet for Gx = 30 nS and
Gy = 30 nS, with a size factor of 7 for the
real cell. From bottom to top, each set of four plots of voltage versus time in Fig. 3 is for y values of 0, 1, 2, and 3, respectively, and the four action potentials shown on each plot are for x
values of 0, 1, 2, and 3, as labeled. For Fig. 3 as well as Figs.
4, 5, 8, and 9, the potential waveform of
the real cell is plotted as a thicker line. Propagation throughout the
sheet was symmetric, with the time of occurrence of activation for
element (1, 0) the same as that for element (0, 1), the time of
occurrence of activation for element (2, 0) the same as that for
element (0, 2), and so on. Along both the x- and
y-axes there was a delayed propagation from the real cell to
the adjacent model cells that is associated with a partial
repolarization in the real cell. When we then reduced the size factor
for the real cell to 5, we obtained the results shown in Fig. 4. At
this point the early repolarization in the real cell was more extreme
and led to failure of action potential propagation away from the real
cell into the model cells. For this particular real cell, we found that
the critical size for initiating a propagating action potential was
6.3. However, when we continued with a size factor of 5 for the real
cell and included the barriers as diagrammed in Fig. 2 into the sheet
representation, we obtained the results shown in Fig.
5. The values for
Gx and Gy in
Fig. 5 are the same as those in Figs. 3 and 4, but propagation is now
successful for a size factor of 5 (cf. Fig. 4), although with a
different spatial pattern. Note that the action potential in Fig. 5 for
y = 0 or y = 1 is propagating from left to right
(activating elements along the x-axis in the order 0, 1, 2, and
then 3), whereas the activation order for y = 2 or y = 3 in Fig. 5 is 3, 2, 1, and then 0 as the action potential propagates
counterclockwise around the barrier and then from right to left. The
peak amplitude of the action potentials for elements (0, 2) and
(0, 3) is larger because these action potentials are actually
"colliding" with action potentials propagating around the upper
barrier in a clockwise direction in the upper left quadrant of the
sheet (Fig. 2), which we are not directly simulating.
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We found a general phenomenon in that action potential propagation from
a central cell was more difficult (requiring a larger size factor for
the central real cell) when the conductances in the x and
y direction were equal than when they were unequal. The data
summarized in Fig. 6 were obtained from
eight guinea pig ventricular cells. For each cell, using a size factor
of 5, we systematically varied Gx and
Gy to test the success or failure of
propagation through the sheet when the real cell was repetitively
stimulated at 1 Hz. In Fig. 6, filled square symbols represent
combinations of Gx and
Gy for which propagation was successful and
open triangular symbols represent combinations for which propagation
failed. Fractions beside these symbols indicate the ratio of cells with
successful propagations to the total number of cells tested with the
indicated combination of Gx and
Gy. Note that the failures occur either
along the diagonal at which Gx equals
Gy or slightly off the diagonal
(Gy = 25 or 35 for
Gx = 30). For four of these cells we also
tested propagation along the diagonal values of
Gx = Gy with
the barriers present and found in each case that the presence of the
barriers converted failed propagation to successful propagation.
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To evaluate this phenomenon more quantitatively, we systematically
varied the size factor for the real cell to determine the critical size
for initiating successful propagation with a fixed value of
Gx (30 nS) and a variable value of
Gy from 20 to 40 nS. The results are shown
in Fig. 7, where the experimental results
for the critical size of the central element (mean ± SE) are shown as
filled square symbols. We included a horizontal dashed line at a size
of 5 to show that the mean values of critical size were <5 for
Gy values of 20 or 40 nS but >5 for
Gy values of 25, 30, and 35 nS with the
fixed Gx of 30 nS. We also did simulations
in which we replaced the real cell at (0, 0) with a model cell
identical to the other model cells of the sheet and then determined the
critical size of this central model cell. These results are shown as
open circular symbols in Fig. 7, showing lower values but the same
general phenomenon in that a greater critical size was required when
Gx was equal to
Gy.
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It may seem paradoxical that the critical size for initiating
propagation for a fixed value of Gx was
reduced when we either raised or lowered Gy
from the symmetric value. We show in Figs.
8 and 9 the
actual results obtained for the same cell as in Figs. 3-5 for
which propagation failed with Gx = Gy = 30 nS. For Fig. 8, we lowered
Gy by using settings of
Gx = 30 nS and
Gy = 20 nS. Note that activation
of the sheet became asymmetric, as expected. The propagation occurred
with less delay along the x-axis (y = 0) than along the
y-axis. A similar phenomenon occurred when we raised Gy with the fixed value of
Gx. We used settings of
Gx = 30 nS and
Gy = 40 nS and, again for the same cell,
got the results shown in Fig. 9. The activation sequence was again
asymmetric, with a more rapid initiation of propagation along the
y-axis, producing the same effect as in Fig. 8 of actually
diminishing the electrical loading on the central cell because of the
shorter delay in propagation (now along the y-axis), which
again produced the situation in which the cell that had been activated
was no longer a load on the central cell but actually helped to further
the spread of activation.
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The mechanisms for the lower critical size of the central element when
we either raised or lowered Gy from the
symmetric case can be more clearly seen in Fig.
10, in which we have included the voltage
waveforms of elements (0, 0), (0, 1), and (1, 0) as well as
the directional division of the currents associated with the central
element. We plotted the total coupling current for element (0, 0) in
the top left panel and the membrane potential of element (0, 0) in
the bottom left panel of Fig. 10. To represent the three sets of
coupling conductances, we used thick lines for the symmetric case
(Gx = Gy = 30 nS), thin lines for the condition in which
Gy increased to 40 nS, and dotted lines for
the condition in which Gy decreased to 20 nS. The total coupling current was plotted with a positive value
leaving the cell (0, 0), thus producing a repolarization. For
clarity, we omitted the stimulus current that was applied as a pulse of
2 ms in duration to initiate the action potential. For the symmetric
case, there was no activation of either element (0, 1) or element
(1, 0), and thus the coupling current leaving element (0, 0) is a
monotonically decreasing function. For each of the asymmetric cases,
because both elements (0, 1) and (1, 0) eventually activated,
there are significant changes in the time course of the coupling
current of element (0, 0). As either of the adjacent elements
(0, 1) or (1, 0) activate, there is a significant decrease in the
outward coupling current from element (0, 0) because these adjacent
elements are no longer an electrical load but actually supply current
back to element (0, 0). It is clear from the bottom left panel of
Fig. 10 that the activation of the adjacent elements is associated with
a termination of the fast repolarization process of element (0, 0)
and thus accounts for the successful activation of element (0, 0)
for either of the asymmetric conditions.
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The mechanisms for the activation of these adjacent elements are shown
in the middle and right panels of Fig. 10. In the middle panels, we
plotted the coupling current in the x direction for element
(0, 0) (top) and the membrane potential of element (1, 0), which
received one-half of this current [the other one-half went to
element (1, 0)] (bottom). Changing the value of
Gy from 30 nS to either 40 or 20 nS did not
significantly change the magnitude of the coupling current in the
x direction before activation of element (1, 0). However,
the response of the membrane potential of element (1, 0) to this
current was quite different. When Gy was
lowered to 20 nS, element (1, 0) had a larger voltage
response, which raised the membrane potential to threshold as
shown by the dotted line in the bottom middle panel of Fig. 10. The
larger voltage response of element (1, 0) when
Gy was lowered was produced by the
decreased load on this cell in the y direction through its
connections to elements (1, 1) and (1, 1) [the
elements above and below element (1, 0)], and thus
activation of element (1, 0) occurred with the decreased Gy value. This activation of element
(1, 0) then sent current back to element (0, 0) to assure
activation of element (0, 0) and thus produced a delayed activation
of element (0, 1), as shown by the dotted line in the bottom right
panel of Fig. 10.
The right panels of Fig. 10 show the mechanism of successful activation of the array when Gy was increased to 40 nS. The top right panel shows the coupling current for element (0, 0) flowing in the y direction. When Gy was raised or lowered, there was a significant increase or decrease, respectively, of this coupling current. Increases in this coupling current produced a greater response of the membrane potential of element (0, 1), as shown by the thin solid line of the bottom right panel, and thus led to activation of element (0, 1). This activation of element (0, 1) then sent current back to element (0, 0) and terminated the early repolarization of this element, as shown in the bottom left panel of Fig. 10, which then allowed more current to flow in the x direction and produced a delayed activation of element (1, 0), as shown by the thin solid line in the bottom middle panel. As shown in the top right panel of Fig. 10, decreases in Gy lowered the coupling current in the y direction and thus lowered the membrane potential response of element (0, 1), as shown in the bottom right panel. However, the activation of element (1, 0) led to a greater membrane potential in element (0, 0), which then activated, with delay, element (0, 1), as shown by the dotted line in the bottom right panel of Fig. 10.
We next compared the effects of a resistive barrier on the critical
size for the central element of the sheet to initiate propagation into
the sheet when we let Gx = Gy (isotropy) except for the specified conductances
of the barrier region. Figure 11 shows
the critical size as a function of Gx (or
Gy) for the condition with no barrier. Note
that a larger critical size is required when
Gx and Gy are
low than when they are higher. As shown in the outcome plot of Fig. 6, in which no points along the diagonal indicate successful propagation for a size of 5, all of the critical sizes for the sheets where Gx = Gy are
>5 (for which a horizontal line is drawn). Figure 11 also shows the
critical sizes determined for the same real cells when connected to
sheets where Gx = Gy but with a barrier added. Note that
there is a slight tendency for the critical size to decrease as
Gx and Gy
increase but that all of the values are much lower than for the
condition without the barrier and now indicate successful initiation of
propagation with a cell size of 5 for the central element.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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There are some limitations to our present implementation of a two-dimensional sheet of cardiac cells. The constraints of a fixed time step of 80 µs, which is required to solve the equations of Luo and Rudy (20, 21) for each of the 15 cell models as well as to provide the interaction with the real cell, introduce a small distortion into the rising phase of the model cell action potential, which we previously discussed (37). The same time considerations limit the size of the sheet that we can simulate (in real time) and thus indirectly limit the degree of anisotropy that we can represent. For very high degrees of anisotropy, the action potential propagates all the way to the edge of our sheet before propagating into the other direction, thus introducing "edge effects" into the results. Although we were not able to run a larger model system in "real time," we did implement the same two-dimensional sheet representation with a total of 13 × 13 = 169 elements in which we made the same assumptions of symmetry and solved for the upper right quadrant with all 49 of these elements represented by model cells. We repeated the determinations of critical cell size presented in the open symbols of Fig. 7 and found no difference to two decimal places with the larger sheet size, suggesting that for the effects on the central cell of the sheet, our real-time implement of a sheet of 49 cells is sufficient. We also emphasize that our simulation is entirely based on the assumption that a single cell is, for these values of coupling conductance, very nearly isopotential, as demonstrated by the theoretical work of Shaw and Rudy (24). Thus, for our studies, each "element" is an intact cell, and the actual spatial dimensions and orientation or the particular cells used does not affect the results. We deliberately chose lower values of coupling conductance than those occurring in "normal" ventricular tissue to emphasize the effects that might be observed in cells that are partly uncoupled, such as might have occurred by prior ischemia.
Our major results show that there is a critical size for the central element directly stimulated to produce propagation out into the two-dimensional sheet. For a continuous one- or two-dimensional system, this would be analogous to the "liminal length" that must be activated by a stimulus to produce propagation (9). To further the analogy for discontinuous tissue, the critical size of the central element might be considered the "liminal lump." Figure 7 shows that this phenomenon can be demonstrated by a simulation that contains only model cells (no real cell), but the numerical values for the critical sizes are significantly lower when the LR model cell is used as the central element. This is consistent with our earlier work (37), in which we showed that the LR model has activation characteristics (current threshold, maximum dV/dt, peak amplitude, etc.) quite similar to those of real guinea pig ventricular cells but does not exhibit as much early repolarization in response to an electrical load as real guinea pig ventricular cells (37). Thus the LR model cell is more effective as a leader cell for propagation than the real cell but serves very well as a follower cell. To some extent, this limitation of size would thus help to determine the critical size of an automatic focus region that might serve as an ectopic focus. However, for a spontaneously firing focus, the effects of the electrotonic interactions on the diastolic depolarization phase would also be of significance (13, 15, 35).
It is interesting that the size required for our central stimulated element to propagate into a symmetric sheet is higher than that required to propagate into an asymmetric sheet. It is clearly not the case that the critical size of the central element is determined only by the effective passive input impedance of the sheet, because the critical size is lowered (as shown in Figs. 8 and 9) by either lowering or raising one of the conductances from the symmetric value. For the effective electrical load imposed on an active element, the effect of repolarization of the central element (which ultimately determines the success or failure of propagation from the central element) depends not only on the conductance but also on the time interval for each current pathway. Our results show that some degree of asynchrony seems to favor activation from a central site. This then helps to explain the facilitating effects of the resistive barrier. As demonstrated in Fig. 5 and shown more quantitatively in Fig. 11, the presence of the resistive barriers as shown in Fig. 2 lowers the critical size of the central element of the sheet. In particular, for the case in which Gx = Gy, the presence of the barrier makes activation of the sheet asynchronous as well as restricting the current flow. The decrease in the critical size of the central element as the coupling conductances are raised without the barrier (Fig. 11) again demonstrates the difference between discontinuous and continuous conduction. If the sheet were a continuous structure and we simply measured the input current required for activation of the sheet, then the decreasing input impedance of the sheet as Gx and Gy were raised would raise the required input current. However, with discrete cellular elements the stimulus current activates the central element, which then serves as a complex source of current for the rest of the sheet, with the amount of current available to initiate propagation limited by the repolarization process of the central cell as the central cell is subjected to the electrotonic effects of the surrounding cells.
These experiments have demonstrated somewhat paradoxical phenomena in arrays of poorly coupled cells. The presence of anisotropy facilitates the initiation of a propagating action potential compared with the isotropic case. The presence of a resistive barrier near the site of stimulation does not inhibit but, rather, facilitates the initiation of a propagating action potential by reducing the critical size of the central element of the sheet. It has been shown in studies of the remodeling that occurs in peri-infarction zones that there is a particular loss of lateral connections compared with end-to-end connections among heart cells (19, 25). We propose that the normal existence of anisotropy and enhancement of the degree of anisotropy both by loss of lateral gap junctions and the development of resistive barriers may play a facilitating role in the development of ectopic foci that may lead to cardiac arrhythmias.
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ACKNOWLEDGEMENTS |
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This work was partially supported by National Heart, Lung, and Blood Institute Grant HL-22562 (R. W. Joyner), an American Heart Association Fellowship (M. B. Wagner), the Emory Egleston Children's Research Center, and Netherlands Organization for Scientific Research Grant 805-06-154 (R. Wilders).
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FOOTNOTES |
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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. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: R. W. Joyner, Dept. of Pediatrics, Emory Univ., 2040 Ridgewood Dr. NE, Atlanta, GA 30322 (E-mail: rjoyner{at}cellbio.emory.edu).
Received 4 March 1999; accepted in final form 3 August 1999.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
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, combinations of Gx
and Gy for which propagation was
successful; 
, combinations of Gx and
Gy for which propagation failed. Fractions
beside each symbol indicate ratio of number of cells with successful
propagations to total number of cells tested with that combination of
Gx and Gy.
, results with same protocol in which we replaced the real cell with
another model cell. Mean values of critical size were <5 (dashed
line) for Gy of 20 or 40 nS but 75 for
Gy of 25, 30, and 35 nS. LR model, Luo and Rudy
model.
