Original Contributions |
From the Departments of Pediatrics (M.S.S., J.F.H.), Surgery (P.C.D.), and Biomedical Engineering (R.C.B.), Duke University Medical Center, Durham, NC.
Correspondence to Madison S. Spach, MD, Box 3475, Duke University Medical Center, Durham, NC 27710. E-mail cspach{at}acpub.duke.edu
| Abstract |
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Key Words: action potential foot interstitial discontinuity capillary interstitial potential electrical field effect
| Introduction |
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max) during
transverse propagation (TP) compared with longitudinal propagation (LP)
to suggest that cardiac propagation is discontinuous at a microscopic
level because of recurrent discontinuities of internal resistance
(ri) produced by the gap
junctions.1 Other laboratories have repeatedly
reproduced these directional differences in
max in adult cardiac
muscle,2 3 4 and it is now generally accepted that
cardiac conduction is discontinuous at a microscopic
level.5 Recent results also show that in the
presence of uniform membrane properties, variations in cellular loading
generate variations in
max that are
dependent on the complex distribution of ri
discontinuities produced by the gap junctions in the path of an
advancing excitation wave.6 7 Thus, undulating
values of
max occur in any given direction
of propagation, and it is the average
max
value that is larger during transverse than
LP.6 7 8 9 10
We also applied the greater-TP-than-LP
max
relationship to predict that conduction would be depressed more along
the long axis of the fibers than across the fibers when the available
sodium conductance is decreased
(
¯GNa).1 As noted
by Vorperian et al,11 experiments with
Na+ channel blockers have shown depression of
conduction to be consistently greater in the longitudinal
direction.12 13 14 However, in our experiments with
premature beats, unidirectional longitudinal block that leads to
anisotropic reentry has occurred only in nonuniform anisotropic bundles
in which there has been loss of side-to-side connections between small
groups of cells over distances of multiple cell
lengths.15 Such results indicate that areas
without side-to-side electrical connections between fibers amplify the
stochastic microscopic loading effects of the normal distribution of
gap junctions and produce enhanced loading at a larger size
scale.7 In addition, Fast et
al16 have shown experimentally in neonatal
cellular monolayers that microscopic wavefront deviations and
collisions occur in the presence of obstacles. These general features
of anisotropic conduction have been produced by a texture model
developed by Pertsov in which longitudinally oriented impermeable
barriers produce a greater mean value of
max during TP than LP and accelerate
rotation of vortexlike reentry.10
| The Problem |
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max variations produced by
discontinuities of internal resistance are now well established.
However, there has been no detailed experimental analysis of
the time course of the foot of the transmembrane action potential
(Vm foot) during anisotropic propagation. In the
first application of dV/dt versus Vm phase-plane
analysis of the Hodgkin-Huxley model17 to
the heart, Paes de Carvalho et al18 noted that a
change in the time course from a simple exponential is easily
recognized in a dV/dt versus Vm display, which
converts a simple exponential into a linear
trajectory.19 Therefore, to extend our original
anisotropic analysis1 of
foot, we performed phase-plane
analysis of transmembrane action potentials measured at a depth
of
12 myocytes beneath the superfused surface of atrial and
ventricular preparations. The results demonstrated a linear
trajectory of Vm foot in Purkinje strands. During
LP in working myocardium, however, most sites demonstrated
deviations from a linear trajectory because of variable slowing of
Vm foot (eg, Figure 1B
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Since these anisotropic variations in the trajectory of
Vm foot were previously unknown (at least to us),
we looked for past experimental evidence of nonlinear trajectories of
Vm foot in phase-plane displays of cardiac action
potentials. Although we were unable to find such analyses of
anisotropic propagation, Paes de Carvalho et al noted that shifts of
the stimulus site often produced a change in the trajectory of the
ascending limb of the phase-plane display.20 21
Further, in some of the dV/dt versus Vm loops
that they recorded in atrial cells, there was initial slowing of
Vm foot similar to that which we encountered
during LP in atrial and ventricular muscle (compare their
Figure 8
in Reference 2121 and our Figure 1B
2). Paes de Carvalho et al
interpreted the initial slowing of the rise of Vm
to be due to an abrupt decrease in internal resistance
(ri) secondary to cellular
branching.20
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We hypothesized that the microscopic structure in the tissue that could produce an effect such as was being observed in the data was that of the capillaries. That is, the capacitance and resistance of the capillary wall separate the capillary lumen from interstitial space.22 23 Consequently, these electrical relationships are analogous to those of the first theoretical model by Medvinskii and Pertsov24 of ephaptic interactions between closely apposed normal (uninjured) cardiac fibers, one active and the other inactive. Suenson subsequently demonstrated that ephaptic transmission could be produced experimentally by pressing 2 papillary muscles together (to produce a high extracellular resistance between the 2 bundles) and initiating excitation in one of the bundles.25
Because extracellular discontinuities are unexplored, we initially present the experimental findings obtained in adult cardiac muscle (Part I). A recent comparison of results by Fast and Kléber9 26 and Fast et al16 in monolayers of neonatal cardiac myocytes and our results in newborn hearts provided an opportunity to determine whether experimental support for our hypothesis could be obtained that would validate an electrical model of working myocardium that includes extracellular discontinuities (in this case, the capillaries). We used the hypothesis to predict that there should be a marked difference in the phase-plane trajectory of Vm foot in the neonatal cellular monolayer, which is devoid of capillaries, and newborn ventricular muscle, which is rich in capillaries. Indeed, the 2 tissues showed dramatic differences in Vm foot. Since these findings provided experimental support for our hypothesis, we explore the biophysical implications of the experimental results with regard to mechanisms due to interstitial electrical field effects.27 For this, we examine anisotropic propagation in a "2-domain" model of myocytes and capillaries separated by interstitial space (Part II).
Microscopic resistive discontinuities (branching or discrete increases
in resistance due to the gap junctions) were considered but discarded
as an explanation for our Vm foot results.
Several resistive models were evaluated. These included the branching
model of Joyner et al,28 the 1-dimensional (1D)
model of Rudy and Quan,29 the 2-dimensional (2D)
models of Leon and Roberge8 and Fast and
Kléber,9 the texture model of
Pertsov,10 and our 2D cellular
model.7 Each model produced variations in
max, but each also produced a linear
phaseplane trajectory of Vm foot at all sites
for all directions of propagation. Further, although all of the 2D
anisotropic resistive models produced mean values of TP
max greater than LP
max, at all sites the value of
foot was
10% greater during TP than LP,
which is just the opposite of the original experimental
result.1 Thus, loading variations at a
microscopic level caused by discrete increases in internal resistance
due to the gap junctions,7 as well as the larger
macroscopic boundaries10 representing
an absence of side-to-side connections between narrow
areas,15 reproduce most known anisotropic
variations in
max. Nonetheless, models of
ri discontinuities do not account for the
anisotropic variations in Vm foot. The inability
to explain the foot with a resistive model was noted a decade ago when
we suggested that an as-yet-unidentified capacitive effect would have
to be included for a full accounting of the anisotropic behavior of
propagating depolarization.30
Also, the experimental variations in Vm foot that
were observed were not explained by computer simulations based on
bidomain models, even though some models have shown that the initial
portion of the action potential is influenced by the low specific
resistance of the fluid at the surface of superfused
preparations.31 32 33 34 For example, the results of
Roth's bidomain model of LP31 demonstrated a
slightly curvilinear phase-plane trajectory of Vm
foot at the surface, and the trajectory developed a marked initial slur
at increasing depths. In the bidomain models of a thin preparation by
Henriquez et al32 and by Pollard et
al,33 action potentials on the surface produced a
linear ascending limb of the phase-plane display during TP and a
moderately curvilinear trajectory during LP. Further, like these
bidomain models, the recent quasi 1D model of Wu et
al34 did not reproduce the variable concave
deviations during LP or the deviations from a linear
Vm foot trajectory during TP. Although the
results of these bidomain models demonstrated important effects of the
low specific resistance of the superfusate on the upstroke of
the action potential, they did not account for our experimental results
at a uniform depth of
12 cells beneath the surface, where there
should be minimal effects of the superfusate
solution.35 Barr and Plonsey have noted the
considerable strengths of the bidomain models at a macroscopic scale,
and they indicated that the macroscopic averaging that underlies the
bidomain formulation may not apply at the microscopic
level.36
| Part I. Experimental Procedures |
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To minimize possible curvature effects due to the low resistance
of the solution at the surface,31 32 33 39 we used
glass microelectrodes to measure Vm from cells at
a depth of 150 to 200 µm (
12 cells deep). To impale a cell,
the microelectrode tip initially was positioned at the surface of the
preparation. Using a Narishige micromanipulator, the tip was then
advanced 150 µm. If no injury potentials occurred, the tip was
advanced a few micrometers until a slight positive shift of
the ST segment on the device indicated that the tip was against the
membrane of a myocyte. This minimal shift disappeared in 1 to 2
minutes. Then, for each of 4 directions of conduction, the
interstitial potential waveforms
(
is) were recorded in reference to a
distant electrode in the bath. The microelectrode tip then was advanced
into the cell, and the intracellular potential
(
i) was recorded for each direction. The
interstimulus interval was maintained constant at 500 ms. The tip
impedance of the microelectrodes was 12 to 22 M
. The input impedance
of the amplifier was 1014
(rise time 30
µs). A computer recorded all waveforms at sampling rates between
62 500 and 100 000 Hz.
Microelectrode impalements of superfused tissues must be viewed with care, since the action potentials may be affected by hypoxia or local damage produced by the microelectrode, as noted by Tranum-Jensen and Janse.40 These effects are evidenced by a shift to a less negative resting potential or by a change in repolarization shape. We therefore continuously displayed the action potential to confirm that the resting potential, the repolarization shape, and the timing of depolarization did not change while waveforms were recorded for each direction of conduction. If a change occurred in these parameters, the prior recorded data were discarded. Each recording site thereby provided its own control for directional differences. Further, 2 unipolar tungsten wire electrodes were used to record extracellular waveforms at the surface to ensure that homogeneous conduction occurred at a macroscopic level during LP and TP (ie, there were no end effects).38
Because
Vm=
i
is,
we were careful to subtract the interstitial potential
is at the membrane surface from the
intracellular potential
i.1 31 Figures 1A
1 and 1A2 illustrate the method for a
typical set of
is and
i ventricular waveforms during LP,
with the resultant Vm upstroke, and Figure 1A
3
shows the time course of Vm foot and its best fit
to a simple exponential (r=0.98). Figure 1B
shows the phase-plane plots
of
i and Vm. The dashed
line superimposed on the ascending limb of the dV/dt versus
Vm display represents a linear
trajectory. As shown, the recorded deviation from linearity
produced a concave-shaped trajectory of Vm foot
during LP. To quantify the concave deviations from linearity, we
identified the maximum difference that occurred between dV/dt of the
linear trajectory and dV/dt of the actual Vm foot
trajectory (vertical mark at asterisk in Figure 1B
2).
To ensure that interpretation of the phase-plane plots was not
influenced by recording artifacts, we averaged the
is and
i waveforms
recorded at 10 sites for each of 4 directions of conduction; the
order of the directions was selected randomly at each site. Phase-plane
plots of individual Vm action potentials, as well
as plots of the average of 3 to 9 action potentials, showed a
reproducible trajectory of the foot for a given direction at each site.
Paired and unpaired t tests were used for statistical
analysis. Correlations were assessed by univariate
regression. Results were considered to be statistically significant
when P<0.05.
Morphological Studies
Fiber orientation was confirmed by histological
examination. The capillary distribution was examined by light
microscopy in an additional 8 canine hearts. Four hearts were perfused
with Bouin's fixative via the coronary arteries, and specimens
cut from the other 4 hearts were immersion fixed in Bouin's solution.
After at least 24 hours, specimens were embedded in paraffin and
sectioned at 7 µm. Sections were stained with picrosirius red
following phosphomolybdic acid treatment and examined for collagen
using bright-field or fluorescence
microscopy.41 An additional 4 adult hearts were
studied by transmission electron microscopy to evaluate the
relationship between the capillaries and myocytes. Two of these hearts
were perfused briefly with physiological saline and
then with 3% glutaraldehyde in 0.15 mol/L sodium
cacodylate buffer, pH 7.4. Left ventricular wall specimens
from 2 other hearts were immersion fixed with the same fixative. We
used the immersion-fixed specimens for a control study of the
capillaries as a corollary of the superfused experimental tissues.
After 24 hours, small samples were rinsed in 0.15 mol/L sodium
cacodylate, postfixed in 1% OsO4 in 0.15 mol/L
sodium cacodylate, rinsed in 10% sucrose, stained en bloc in uranyl
acetate, dehydrated, and embedded in Poly/Bed 812 (Polysciences).
Silver or silver-gray sections were cut and stained with uranyl acetate
and lead citrate and examined on a Zeiss EM10A electron microscope.
Additionally, 2 adult and 3 neonatal hearts were fixed by immersion in
1% paraformaldehyde in phosphate buffer and embedded
in paraffin. Double labeling with anti-connexin43 antibodies and wheat
germ agglutinin42 was applied to longitudinal
sections and serial cross sections to study gap junctioninterstitium
relationships.
Serial sections were viewed using video microscopy controlled through
the public-domain NIH Image program, version 1.61. The "Live Paste"
option and "Or" transfer mode of the NIH Image paste control were
used to obtain a best-fit alignment of each successive video image with
its predecessor. This process enabled study of the changing appearance
of the interstitium (or the gap junctions) along the course of the
myocytes (eg, Figure 9
).
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Experimental Results
Adult Preparations
In adult ventricular preparations, the average
conduction velocity was 0.49 m/s during LP and 0.20 m/s during TP, with
a LP/TP velocity ratio of 2.45. At the uniform depth of 150 to 200
µm, the peak-to-peak amplitude of the interstitial
potential
is varied between 16 and 30 mV
during LP; when the direction of conduction was changed to TP, the
peak-to-peak amplitude of
is decreased to a
range of 9 to 14 mV (P<0.01). The Vm
takeoff potential varied between 80 and 88 mV, and all
Vm foot deviations from a linear phase-plane
trajectory occurred before Vm reached 60 mV,
the threshold of the Na+ current in
ventricular muscle.43 44 Normal
action potentials occurred to a depth of 250 µm. Impalements at
greater depths evidenced "dying out" of the tissue during the first
hour of superfusion. After 1 hour of superfusion, there was no
electrical activity, and the potentials approximated 0 at depths
greater than 250 to 300 µm. Histological studies
showed that after ventricular preparations had been
superfused 4 to 6 hours, the myocytes were closely packed and had a
viable appearance to a depth of 250 to 300 µm (
20 cells
deep).45 In this viable layer we found no
histological evidence of abnormal myocytes. There was a
sharp demarcation, however, between these normal myocytes and the
deeper layer. In the deeper region the cells were separated from one
another by wide interstitial spaces (8 to 22 µm in
width),45 an appearance similar to that of
irreversibly injured myocytes with plasmalemmal disruption
described by Reimer and Jennnings46 in the region
of an infarction without blood supply for 1 to 2 hours or
greater.47
Representative phase-plane trajectories of
Vm foot during TP and LP in adult
ventricular and atrial muscle are shown in Figure 2
. Because we focus on the departure from
linearity in the phase-plane trajectory of Vm
foot, we wish to define the descriptive terms that apply to the
different shapes of the Vm foot trajectories that
occurred: (1) linear refers to the straight-line trajectory
of a simple exponential (TP in Figure 2A
); (2) concave
refers to a downward deviation from linearity (concavity up), which
occurred with different degrees of concavity because of minor (+),
moderate (++), considerable (+++), and marked (++++) slowing of the
rise of the initial 15 mV of Vm depolarization
(LP in Figure 2A
and 2B
); (3) initial slur is a very
prominent downward deviation from linearity due to marked slowing of
the rise of the initial 5 mV of Vm depolarization
(TP in Figure 2B
); and (4) other Vm
foot trajectories (not shown) were those few trajectories that had
unusual shapes when the direction of conduction was reversed along the
longitudinal axis of the fibers (eg, a concave shape in one direction
and a convex shape in the opposite direction).
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Table 1A
lists the number of impalements
in adult preparations that demonstrated each of the different shapes of
the Vm foot trajectories. Purkinje strands
demonstrated a linear phase-plane trajectory of
Vm foot at all impalement sites (n=9), a result
similar to the Purkinje phase-plane trajectories published by Pressler
et al.48 In ventricular and atrial
muscle, during TP the trajectory of Vm foot was
linear at most sites (n=35, 70%) (Figure 2A
), and at the remaining
sites (n=15, 30%) there was an initial slur in the trajectory of
Vm foot during TP (Figure 2B
). However, the most
prominent deviations from a linear trajectory occurred with propagation
along the longitudinal axis of the fibers. Most impalements
demonstrated a concave shape of the Vm foot
trajectory (n=40, 80%) during LP. The concave deviations varied from
site to site along the long axis of the fibers in each preparation, and
the degree of concavity appeared to be more pronounced in
ventricular than in atrial muscle (Table 1A
). To quantify
the concave deviations from linearity, the maximum difference in dV/dt
between a linear trajectory of the ascending limb of the phase-plane
display and the true Vm foot trajectory was
identified (Figure 1B
2). Considering all of the action potentials that
demonstrated a concave deviation of the trajectory of
Vm foot during LP in adult preparations (Table 1A
), the mean value of the maximum dV/dt difference from linearity was
9.4 V/s in atrial muscle and 15.1 V/s in ventricular
muscle (P<0.02).
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We also analyzed
foot and
max to determine whether the values of
these 2 standard measurements changed in the same way as found
previously with shifts from TP to LP.1 The best
fit of a single exponential (Figure 1A
3) to the initial 10- to 15-mV
rise of Vm was used to obtain
foot. It is interesting that, although the
results in working myocardium showed that there were clear
deviations of Vm foot from a linear trajectory in
the dV/dt versus Vm display, a comparison of the
time course of Vm foot with a simple exponential
during the initial 10- to 15-mV rise of Vm
demonstrated a high correlation coefficient (r=0.92 to 0.99,
n=20). Paired TP-LP observations (n=20) demonstrated that the mean
value of LP
foot (421 µs) was significantly
greater than the mean value of TP
foot (199
µs) (P<0.01), and the mean value of TP
max (202 V/s) was significantly greater
than LP
max (157 V/s)
(P<0.01). Although these results confirmed our previous
demonstration of anisotropic differences in
max and
foot in
adult cardiac bundles, there were site-to-site variations in
max along each axis of
conduction.7 For example,
max varied between 123 and 197 V/s during
LP, and during TP
max varied from 140 to
240 V/s. These
max variations occurred
independently of the trajectory of Vm foot (ie,
there was no relation between
max and
foot during LP [r=0.21], and
there was also a poor correlation between
max and
foot at
different sites during TP [r=0.50]).
A Hypothesis
What is the origin of the shape variations in the phase-plane
trajectory of Vm foot within working
myocardium? When ¯GNa is
increased, the conduction velocity increases in association with a
decrease in
foot and an increase in
max, and decreases in
¯GNa have the opposite
effect.1 Therefore, one possibility is that there
are spatial inhomogeneities in the available sodium conductance, such
as those that occur in the transition region between the
atrioventricular node and atrium, as well as in the transition area
between the atrioventricular node and His
bundle.18 This explanation, however, seems
unlikely in working myocardium, because each impalement
site served as its own control for different directions of conduction,
and in a given direction the site-to-site increases and decreases of
max occurred independently of changes in
Vm foot and without significant differences in
the takeoff potential. Considering the known morphology of cardiac
muscle (ie, restricted interstitial space that separates
the myocytes from a highly dense network of capillaries along the long
axis of cardiac fibers and a sparse network of capillaries across the
fibers), perhaps we are seeing the effects of extracellular
discontinuities caused by the capillaries that extend along the paths
of LP and TP. Figure 3
(left) shows a
cross section of adult ventricular muscle (V) that
illustrates the considerable number of longitudinally oriented
capillaries surrounding each myocyte, whereas in Purkinje strands (P)
capillaries are sparse.49 The electron micrograph
(Figure 3
, right) shows the restricted interstitial space
that separates the sarcolemma of each myocyte from several
capillaries.
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Crone and Christensen22 and Olesen and Crone23 demonstrated that the passive electrical properties of capillaries are accounted for by 1D cable theory.22 23 The capillary walls correspond to the insulating mantle that consists of a capacitance in parallel with a resistance,23 and the fluid within the lumen corresponds to the inner core. Consequently, when planar excitation waves propagate along the axis of the capillaries, interstitial electrical field variations create spatial variations in the current along the capillary wall. These conditions meet the requirements of ephaptic interactions (as distinct from synaptic or gap-junctional transmission) because of electrical field interactions that occur between closely apposed fibers, one active and the other unexcited. This phenomenon has undergone considerable theoretical study in adjacent nerve fibers with normal membrane (plasmalemma) passive properties,27 50 51 and the theory has been applied to cardiac muscle by Medvinskii and Pertsov.24 We therefore hypothesized that electrical field interactions between the active myocytes and the unexcited capillaries could account for the experimentally measured variations in Vm foot during LP and TP.
Experimental Support for Hypothesis
According to our hypothesis, the phase-plane trajectory of
Vm foot should deviate from linearity when LP
occurs in association with a high density of capillaries extending in
the longitudinal direction, whereas a linear trajectory should occur
during LP in the absence of capillaries. Therefore, to obtain
experimental support for the hypothesis, we constructed a dV/dt versus
Vm display from the high-quality voltage
Vm and dV/dt recordings made by Fast and
Kléber9 during LP in their neonatal
cellular monolayer, which has no capillaries. We then compared the
result with similar displays of action potentials that we recorded
during LP in neonatal ventricular myocardium,
which has the highest density of capillaries that occurs during any
time interval from early life to adulthood.52 53
Our immunolabeling with anti-connexin43 antibodies demonstrated
multiple gap junctions distributed evenly along the entire length of
the spindle-shaped myocytes in the neonatal canine ventricle (not
shown), which was similar to the distribution of the gap junctions in
the neonatal monolayer of Fast et al.16 Thus, at
a microscopic level, the structural difference between the 2
preparations was the presence or absence of capillaries.
Figure 4A
shows the original
Fast-Kléber Vm and dV/dt curves
recorded during LP in a neonatal monolayer of cells (with the kind
permission of the authors),9 and Figure 4B
1 shows
our reproduction of their curves. The associated dV/dt,
Vm display in Figure 4B
2 demonstrated that the
trajectory of Vm foot was linear, as predicted.
Contrariwise, in the neonatal ventricular
epimyocardium LP produced very prominent concave deviations
from a linear trajectory of Vm foot (Figure 4C
and Table 1B
). For all of the concave deviations from a linear
trajectory in neonatal ventricular muscle (n=18), the mean
value of the maximum dV/dt difference from linearity was 20.1 V/s,
which was a greater deviation from linearity than the mean value of
15.1 V/s of adult ventricular muscle
(P<0.02).
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One can question whether the comparison of data from 19 experiments in
neonatal myocardium (high capillary density) with data from
a single voltage action potential in the neonatal cardiac cell
monolayer (no capillaries) is sufficient to allow a conclusion that the
capillaries are responsible for the loading effect on
Vm foot. What if the phase-plane trajectory of
Vm foot also varies in the neonatal cellular
monolayers? Would multiple action potentials reveal concavities in the
trajectory of Vm foot during LP and slurs during
TP? To answer these questions, we identified numerous very
highquality optical recordings of Vm
upstrokes during LP and TP in the data that Fast and Kléber have
published from anisotropic monolayers.9 16 26 We
digitized 32 of their optical action potentials (18 during LP and 14
during TP) at rates between 11 000 and 24 000 Hz (see Figures 5
and 7
of Reference 1616 and Figure 3
of Reference 2626 ). Their optical action
potentials demonstrated results similar to those of their voltage
recording shown in Figure 4
; no concavities occurred in the
Vm foot trajectory during LP (n=18), nor were
there initial slurs during TP (n=14). During LP the maximum dV/dt
deviation from a linear trajectory of Vm foot was
minimal to 0 in the neonatal monolayer optical action potentials,
whereas the mean value of this deviation was considerable (20.1 V/s)
in action potentials of the neonatal epimyocardium, thus
indicating a quite different trajectory of Vm
foot during LP in the 2 preparations (P<0.0001).
|
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We also analyzed the standard indices of anisotropic conduction
in the neonatal ventricular epimyocardium. The
average LP velocity was 0.33 m/s, and the average TP velocity was 0.12
m/s with, a LP/TP velocity ratio of 2.75, similar to the LP/TP velocity
ratio in uniform anisotropic adult preparations.1
At each impalement site,
max increased or
decreased when the direction of propagation was changed from the
longitudinal to the transverse axis of the cells, a result similar to
that of Fast and Kléber.26 Further, the
mean value of TP
max (125 V/s) was not
significantly different from the mean value of LP
max (117 V/s) (P=0.34, n=19),
which was also similar to the anisotropic
max results of Fast and Kléber in
neonatal cellular monolayers.26 The lack of
significant difference in the mean LP and TP
max values in both types of neonatal
preparations was distinctly different from that of the adult uniform
anisotropic preparations, which demonstrated a significantly greater
mean value of TP
max than LP
max, as found
previously.1 7 Further, the similar distributions
of gap junctions in the 2 types of neonatal preparations were quite
different from the cellular distribution of the gap junctions in the
adult ventricle, in which the gap junctions are localized primarily to
the ends of the myocytes.54 55
Because variations in
max due to
ri discontinuities are dependent on the complex
spatial distribution of the gap junctions,6 7 we
concluded that the similar LP-TP
max
relationships in the neonatal cellular layers and the neonatal
ventricle were due to the similar distribution of the gap junctions in
the 2 types of preparations. (In unpublished work, we developed
a 2D neonatal cellular model based on disaggregated neonatal
ventricular myocytes and the aforementioned distribution of
gap junctions, an anatomic cellular substrate representing
the neonatal ventricle, as well as the neonatal cellular monolayers of
Fast et al.16 The 2D neonatal cellular model
reproduced the experimental LP and TP velocity differences of the
neonatal ventricular epicardium without a significant
difference in the mean values of LP and TP
max [n=175, P=0.18]. These
neonatal model results differ from those of our previous 2D cellular
model based on larger adult ventricular myocytes with the
gap junctions primarily at the ends of the cells. The adult model
produced significantly greater mean TP
max
than LP
max values
[P<0.01].7 ) However, the
considerable differences in the trajectory of Vm
foot in the neonatal monolayers versus the neonatal ventricle provide
experimental support for our hypothesis that electrical field
interactions between the myocytes and extracellular discontinuities
(ie, the capillaries) can produce variations in the rate of rise of
Vm foot.
| Part II. Biophysical Mechanisms |
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max independently of one another. In
contrast, in the classical relationship, Vm foot
and
max change in the same way when the
excitability of the membrane is altered.1 Since
max variations due to structural loading
appear to be well accounted for by ri
discontinuities,7 10 the major question now is
the following: By what mechanism can the effective capacitance being
discharged during Vm foot be different when the
direction of propagation is altered, and how can this effective
capacitance vary from site to site along a given conduction path?
Our hypothesis proposes that variable interstitial
electrical field interactions with the capillaries provide such a
mechanism. These interactions involve the following. (1) The density of
capillaries oriented along the longitudinal axis of the myocardial
fibers is considerably greater than that of capillaries oriented
transverse to the longitudinal axis.56 57 58 Also,
the frequency of transverse-oriented capillaries connecting the
longitudinal capillaries is quite
variable.56 57 (2) A greater magnitude and
spatial extent of the interstitial electrical field is
induced along the axis of the longitudinal capillaries during LP than
occurs along the axis of the transverse-oriented capillaries during TP.
The basis for invoking interstitial electrical field
interactions is that favorable structural conditions exist (namely a
restricted interstitial space and dense packing of the
myofibers and capillaries [ie, closely apposed myocytes and
capillaries]).24 36 50 51 Markin demonstrated
that electrical field interactions occur only when the axis of
propagation corresponds to the axis of the unexcited
cylinder.51 Accordingly, in anisotropic muscle
the spatial derivative of current (and potential) must differ from 0
along the axis of either the longitudinal or transverse-oriented
capillaries. However, with planar excitation waves in an idealized
anisotropic medium, spatial potential differences are absent along the
axis perpendicular to the direction of propagation. Therefore, during
LP the electrical field effects should occur along the longitudinally
oriented capillaries, but there should be no electrical field
interactions along the transverse-oriented capillaries, and vice versa
for TP (ie, current flow
0 along the axis perpendicular to the
direction of propagation).
A Two-Domain Model
Limitations at the experimental level thus far have prevented
measurements within the myocardium of currents induced in
capillaries by the interstitial electrical field during the
brief 1- to 2-ms interval of depolarization. This troublesome
problem has accompanied the experimental study in general of ephaptic
interactions that occur inside multifiber bundles. For such structures,
quantitative model studies have been used to great advantage based on
the rationale that electrical circuit analysis dictates that
some current must flow across the membranes of neighboring
structures.24 36 50 51 We therefore developed a
discrete anisotropic 2-domain model in which the electrical properties
of the microscopic structural components were represented
quantitatively on the basis of experimental values in the literature or
derived from our own morphologic measurements. We drew heavily from the
general model of 1D nerve fiber interactions by Clark and
Plonsey50 and of 1D cardiac fiber interactions by
Medvinskii and Pertsov.24
Cardiac muscle was represented as 2 anisotropic domains
that occupy separate but coupled spaces: (1) a layer of active cells,
and (2) extracellular space in which the active layer is coupled to the
inactive capillaries via the interstitium. Figure 5
shows an equivalent circuit for this
2-domain electrical representation of cardiac muscle, and Table 2
presents the symbols and the
baseline electrical values used to represent the microscopic
structural components of each domain.
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The electrically active layer was represented as
either continuous or discontinuous. In the continuous anisotropic layer
(Figure 5A
and 5B
), the different values of the longitudinal and
transverse internal resistances per unit length,
ri(L) and ri(T),
respectively, produced a
ri(T)/ri(L) ratio of 6.
(Clerc measured a ratio of 9.4.59 ) We used our 2D
cellular model7 60 to represent the
discontinuous active layer (Figure 5C
). The interior of each myocyte
was isotropic, and the conductances gj of the gap
junctions are indicated in Table 2A
. In both the continuous and
discontinuous representations, the active layer was divided
into segments with x-y plane dimensions of
100 µm2 (
x,
y=10 µm) and a depth of 11.3 µm to produce
a cross-sectional area of 113 µm2 and a
membrane surface area of 376 µm2 for each
segment. The sarcolemmal membrane was characterized as a parallel
resistance-capacitance network (Figure 5A
, Cm(myo), Rm(myo)) with fast
Na+ current kinetics.17
Each segment of active membrane was coupled to a 10x10-µm segment of
interstitial space that had an average specific resistivity
Ris of 50
-cm.57
The interstitial longitudinal resistance per unit length
ris(L) was assumed to be homogeneous
within a given region and to have values dependent on the
cross-sectional area of interstitial space, which varies
from region to region.57 61 In normal
ventricular adult muscle, Frank and Langer showed that an
interstitial width of 0.2 µm separates approximately
one third of the surface membrane and a portion of the circumference of
a capillary.62 Since there were multiple
longitudinal capillaries along each myocyte, we knew of no generally
accepted way to quantify the interactions between that portion of
active membrane closest to part of the wall of a capillary versus those
interactions that occur between the other parts of the capillaries and
the more distant sarcolemma of the myocyte. This is a problem common to
all models of ephaptic interactions and, as has been done by
others,24 36 50 51 we used the average width of
interstitial space to derive the average
interstitial resistance. To estimate this value for maximal
cellular packing, as well as to obtain values for the ratio of the area
of the wall of capillaries in relation to the myocyte membrane area
(Ar(cpl)/Ar(myo)), we
measured our own electron micrographs as well as those in the
literature.57 62 63 The average
interstitial width was 2.1 µm, which resulted in a
cross-sectional area of 21 µm2 for each
10 µm along the sarcolemma. When we assigned this
cross-sectional area to each segment of interstitial space,
the interstitial/intracellular volume ratio was 0.18,
similar to the 0.14 value measured by Frank and
Langer.62 For an Ris value
of 50
-cm and a cross-sectional area of 21
µm2, the equivalent longitudinal resistance of
interstitial space (ris(L)) is 237
M
/cm (Table 2B
). We assigned a value of 787 M
/cm to the
equivalent transverse interstitial resistance
(ris(T)) to produce an
risT/risL ratio of 3.3,
similar to the 2.6 ratio measured by Clerc.59
Figure 5A
shows how interstitial space was
represented by parallel connected
resistors36 (ris(L1,L2),
ris(T1,T2), ris(V)) to link
the active membrane (rectangles) to the passive capillary wall
(ovals).
The longitudinal and transverse-oriented capillaries were
represented as 1D cylinders.22 23 The
interior of the capillaries was assumed to be homogeneous
with a specific resistivity Ri(cpl) of 50
-cm.
We characterized the inactive capillary wall as a distributed parallel
resistance-capacitance network with a specific wall capacitance
Cm(cpl) of 1.0 µF/cm2
(Table 1C
). As Clark and Plonsey50 showed for
such a structure, during the propagation of depolarization capacitive
currents should be induced across the passive wall due to the local
circuit of the rapidly changing interstitial electrical
field of the advancing excitation wave. Further, Hodgkin and Huxley
used the fact that during the propagation of depolarization the
capacitive current
ICm=Cm(dV/dt).17
Although we know of no electrical measurements of the
specific resistance of intramyocardial capillaries, Fleischhauer et
al64 presented evidence that the
ventricular microvasculature "is electrically insulated
from interstitial space to a large degree." In the
absence of experimental data for the specific resistance
Rm(cpl) of the wall of intact myocardial
capillaries, we used a value of 8000
-cm2 for
Rm(cpl) for all initial results. Then, we
repeated each simulation by varying Rm(cpl)
between 1000
-cm2 and 68 000
-cm2, the results of which remained
qualitatively the same. We estimated the ratio of the area of the wall
of capillaries to the myocyte membrane to vary from 0 in the synthetic
neonatal cultures of Fast and Kléber9 16
and Fast et al26 to 0.1 in Purkinje strands and
to between 0.7 and 0.9 in adult ventricular muscle. To
evaluate the effects of the relative density of the capillaries, we
varied the
(Ar(cpl)/Ar(myo)) ratio
from 0 to 0.83 (Table 2C
).
Calculations and Data Output
We limited the study of propagation through the 2-domain network
(Figure 5
) to the depolarization phase of the action potential. Fast
and Kléber65 recently tested whether the
ionic models preferentially used in the literature for the description
of the ionic currents affected the action potential
parameters when conduction block occurred at a major
ri discontinuity (ie, the
Beeler-Reuter,43
Ebihara-Johnson,66 and
Luo-Rudy44 ionic models). These authors obtained
similar results independently of the different ionic models, and they
found especially close agreement between the results produced by the
Ebihara-Johnson66 and
Luo-Rudy44 ionic models. Fast and Kléber
attributed the close agreement to the similar description of the fast
Na+ current in the 2
models.65 We therefore used a model with
Hodgkin-Huxley form17 with Ebihara-Johnson
kinetics66 to represent the fast
Na+ current INa.
Specifically,
![]() | (1) |
We approximated a repolarization current IR
by the following equation:
![]() | (2) |
The net transmembrane current per square cm of the sarcolemma,
Im(myo), was obtained as Hodgkin and
Huxley17 described by adding the capacitive and
ionic currents as
![]() | (3) |
![]() | (4) |
is is the interstitial
potential and
i(cpl) is the potential inside
the capillary (both referenced to ground). Capacitive and resistive
currents across the capillary wall,
ICm(cpl) and
IR(cpl), respectively, were obtained
by
![]() | (5) |
![]() | (6) |
-cm2. Thus, the net transmembrane
current per square cm induced across the capillary wall,
Im(cpl), was
![]() | (7) |
The calculations were performed in 3-dimensional networks that
contained 5500 to 85 000 segments (xy plane)
with 4 vertical nodes coupling the active layer,
interstitial space, and capillaries of each segment (total
22 000 to 340 000 nodes). The shape of the array was arranged to
extend at least 8
(resting space constants of active layer) in the
direction of plane-wave propagation and 1
along an axis
perpendicular to the direction of propagation
(
L=1.1 mm and
T=0.4 mm). Plane-wave LP was initiated at
the right or left border of the model by an intracellular current
stimulus 2 times threshold along a line perpendicular to the
longitudinal axis of the cells. Plane-wave TP was initiated at the top
or bottom of the model by a twice-threshold stimulus along a line
parallel to the longitudinal axis of the cells. To ensure that the
stimulus current and the end boundaries of the model did not influence
the results, the area of observation was located more than 3 space
constants from the stimulus line.
The values of each variable were initially computed at 0.25-µs
intervals in all segments. We placed 10 to 1700 "observation sites"
at various segments of the array. The output stored for each of these
segments consisted of the values at each 20 µs of
Vm,
m,
INa,
is, and
i(cpl), from which subsequent analyses
were done to derive Vm(cpl),
ICm(cpl), and
IR(cpl). The following computed
variables were saved at each segment to construct graphs:
max, time of
max (activation time),
Vm peak potential, and the areas of the sodium
conductance and the INa curves.
Two-Domain Model Results
LP and TP
Figure 6
shows
representative results of the 2-domain model with a
cellular network (Figure 5C
) for a myocardial architecture that has
maximum cellular packing, maximum capillary density, and a capillary
wall specific resistance Rm(cpl) of 8000
-cm2. A concavity occurred in the trajectory
of Vm foot during LP (Figure 6A
1), and an initial
slur occurred during TP (Figure 6A
2). Although
max varied from site to site during both
LP and TP, the trajectory of Vm foot remained the
same along the axis of propagation (Figure 6A
1 and 6A2). Also, the
amplitude of the interstitial electrical field
(
is) was greater during LP than TP (Figure 6C
), as found experimentally.
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