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(Circulation Research. 1998;83:1144-1164.)
© 1998 American Heart Association, Inc.

Original Contributions

Extracellular Discontinuities in Cardiac Muscle

Evidence for Capillary Effects on the Action Potential Foot

Madison S. Spach, J. Francis Heidlage, Paul C. Dolber, , Roger C. Barr

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

Top Abstract Introduction The Problem Part I. Experimental Procedures Part II. Biophysical Mechanisms Discussion References

 
Abstract—It has become of fundamental importance to understand variations in the shape of the upstroke of the action potential in order to identify structural loading effects. One component of this goal is a detailed experimental analysis of the time course of the foot of the cardiac action potential (V_m foot) during propagation in different directions in anisotropic cardiac muscle. To this end, we performed phase-plane analysis of transmembrane action potentials during anisotropic propagation in adult working myocardium. The results showed that during longitudinal propagation there was initial slowing of V_m foot that resulted in deviations from a simple exponential; corollary changes occurred at numerous sites during transverse propagation. We hypothesized that the effect on V_m foot observed in the experimental data was created by the microscopic structure, especially the capillaries. This hypothesis predicts that the phase-plane trajectory of V_m foot will deviate from linearity in the presence of a high density of capillaries, and that a linear trajectory will occur in the absence of capillaries. Comparison of the results of Fast and Kléber (Circ Res. 1993;73:914–925) in a monolayer of neonatal cardiac myocytes, which is devoid of capillaries, and our results in newborn ventricular muscle, which is rich in capillaries, showed drastic differences in V_m foot as predicted. Because this comparison provided experimental support for the capillary hypothesis, we explored the underlying biophysical mechanisms due to interstitial electrical field effects, using a "2-domain" model of myocytes and capillaries separated by interstitial space. The model results show that a propagating interstitial electrical field induces an inward capacitive current in the inactive capillaries that causes a feedback effect on the active membrane (source) that slows the initial rise of its action potential. The results show unexpected mechanisms related to extracellular structural loading that may play a role in selected conduction disturbances, such as in a reperfused ischemic region surrounded by normal myocardium.

Key Words: action potential foot • interstitial discontinuity • capillary • interstitial potential • electrical field effect

	Introduction

Top Abstract Introduction The Problem Part I. Experimental Procedures Part II. Biophysical Mechanisms Discussion References

 
It has become of fundamental importance to understand variations in shape of the upstroke of the action potential in order to identify structural loading effects on the action potential. For example, we used the finding of a greater value of the maximum rate of rise of the action potential (_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 (r_i) 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 r_i 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 (¯G_Na).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

Top Abstract Introduction The Problem Part I. Experimental Procedures Part II. Biophysical Mechanisms Discussion References

 
_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 (V_m foot) during anisotropic propagation. In the first application of dV/dt versus V_m 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 V_m 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 V_m foot in Purkinje strands. During LP in working myocardium, however, most sites demonstrated deviations from a linear trajectory because of variable slowing of V_m foot (eg, Figure 1B2). These deviations are the focus of this article.

View larger version (17K): [in this window] [in a new window]   Figure 1. Typical experimental interstitial (is) and intracellular (i) depolarization potential waveforms used to obtain the transmembrane potential Vm during LP in adult ventricular muscle. The is waveform in A1 was subtracted from the i waveform in A2 to obtain Vm (ie, Vm=i–is). A3 shows the best fit of the time course of Vm foot to a simple exponential (r=.98). The phase-plane graphs of dV/dt vs i and dV/dt vs Vm are shown in B1 and B2, respectively. The small vertical line in the dV/dt vs Vm display (*, B2) identifies the maximum difference between a linear trajectory (dashed line) and dV/dt of the actual Vm foot trajectory (–15.6 V/s).

Since these anisotropic variations in the trajectory of V_m foot were previously unknown (at least to us), we looked for past experimental evidence of nonlinear trajectories of V_m 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 V_m loops that they recorded in atrial cells, there was initial slowing of V_m foot similar to that which we encountered during LP in atrial and ventricular muscle (compare their Figure 8 in Reference 21²¹ and our Figure 1B2). Paes de Carvalho et al interpreted the initial slowing of the rise of V_m to be due to an abrupt decrease in internal resistance (r_i) secondary to cellular branching.²⁰

View larger version (24K): [in this window] [in a new window]   Figure 8. Influence of different densities of longitudinal capillaries on the phase-plane trajectory of Vm foot during LP. The different densities of the longitudinal capillaries were produced by varying the Ar(cpl)/Ar(myo) ratio from 0 to 0.81. In A, the numbers indicate the Ar(cpl)/Ar(myo) ratio for each dVm/dt vsVm display. In B, the same numbers denote the associated linear curves, which were obtained as the difference between dVm/dt of the linear trajectory (0 capillaries) and the resultant trajectory of each of the other dVm/dt vsVm displays.

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 Kléber^{9 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 V_m 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 V_m 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 V_m 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 phase–plane trajectory of V_m 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 r_i discontinuities do not account for the anisotropic variations in V_m 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 V_m 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 V_m 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 V_m 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

Top Abstract Introduction The Problem Part I. Experimental Procedures Part II. Biophysical Mechanisms Discussion References

 
Experimental Materials and Methods
Electrical Measurements
We studied in vitro atrial and ventricular preparations from the hearts of 15 adult dogs (weighing 10 to 22 kg) and ventricular epimyocardial preparations from 5 neonatal dogs 1 to 6 weeks of age. Adult ventricular preparations consisted of Purkinje strands (n=9), the endomyocardium (n=9), and the epimyocardium (n=9); atrial preparations included the interatrial band (n=4), crista terminalis (n=7), and pectinate bundles (n=7). All experiments conformed to the guiding principles of the Declaration of Helsinki. Each dog was anesthetized with pentobarbital sodium (30 mg/kg IV). The hearts were rapidly excised and a 5-mm-thick layer of ventricular tissue (or the atrial wall) was removed and pinned to the floor of a 15-cm tissue bath maintained at 36°C to 37°C. The composition of the bathing solution was (in mmol/L) 128 NaCl, 4.69 KCl, 1.18 MgSO₄, 0.41 NaH₂PO₄, 20.1 NaHCO₃, 2.23 CaCl₂, and 11.1 dextrose. Oxygen with 5% CO₂ was bubbled into the reservoir of perfusate, and a rapid rate of perfusion of the bathing solution was used to maintain the surface of each preparation as normal as possible. Four pairs of unipolar stimulus electrodes, each with a stimulus strength twice threshold, were located 3 to 5 cm apart to produce macroscopic planar wavefronts in either direction along the long axis of the fibers and across the fibers (4-way propagation).^{6 7} To ensure there was no effect of the local response to the stimulus³⁷ on the measured action potentials, the stimulating electrodes and the microelectrode impalement sites were separated by at least 10 mm during LP and 5 mm during TP (ie, greater than 6 resting space constants during LP and TP).³⁸

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 V_m 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 V_m=_i–_is, we were careful to subtract the interstitial potential _is at the membrane surface from the intracellular potential _i.^{1 31} Figures 1A1 and 1A2 illustrate the method for a typical set of _is and _i ventricular waveforms during LP, with the resultant V_m upstroke, and Figure 1A3 shows the time course of V_m foot and its best fit to a simple exponential (r=0.98). Figure 1B shows the phase-plane plots of _i and V_m. The dashed line superimposed on the ascending limb of the dV/dt versus V_m display represents a linear trajectory. As shown, the recorded deviation from linearity produced a concave-shaped trajectory of V_m 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 V_m foot trajectory (vertical mark at asterisk in Figure 1B2).

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 V_m 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 junction–interstitium 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).

View larger version (81K): [in this window] [in a new window]   Figure 9. Changes in collagenous septa along their lengths. Every 15th section from a set of 75 serial sections is shown. The short arrows and the arrowheads indicate marked variations in septal thickness along a fixed longitudinal axis; the long arrows in the last 2 panels indicate the sudden appearance of a collagenous septum. Bar=250 µm.

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 V_m takeoff potential varied between –80 and –88 mV, and all V_m foot deviations from a linear phase-plane trajectory occurred before V_m 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 V_m 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 V_m foot, we wish to define the descriptive terms that apply to the different shapes of the V_m 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 V_m 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 V_m depolarization (TP in Figure 2B); and (4) other V_m 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).

View larger version (24K): [in this window] [in a new window]   Figure 2. Typical dVm/dt vs Vm graphs of the depolarization phase of the action potential during TP and LP in adult atrial (A) and ventricular (B) muscle. In each preparation, the LP and TP action potentials were recorded during a single microelectrode impalement while the direction of conduction was changed. A1 shows superimposed dVm/dt vsVm graphs in which TP produced a linear trajectory of Vm foot (simple exponential), whereas LP produced a minor concave deviation from linearity during Vm foot. B1 shows superimposed TP and LP phase-plane graphs in which TP produced an initial slur in the trajectory of Vm foot (marked slowing of the rise of the initial 5 mV of depolarization), whereas LP produced a moderate concave deviation from a linear trajectory during Vm foot. Enlarged phase-plane graphs of the initial 17 to 23 mV of depolarization are presented in A2 and B2 to enhance comparison of the different shapes of the Vm foot trajectories during LP and TP. A and B also show that max was greater during TP than LP at each impalement site.

Table 1A lists the number of impalements in adult preparations that demonstrated each of the different shapes of the V_m foot trajectories. Purkinje strands demonstrated a linear phase-plane trajectory of V_m 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 V_m 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 V_m 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 V_m 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 V_m foot trajectory was identified (Figure 1B2). Considering all of the action potentials that demonstrated a concave deviation of the trajectory of V_m 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).

View this table: [in this window] [in a new window]   Table 1. Shapes of Vm Foot Phase-Plane Trajectories Observed Experimentally

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 1A3) to the initial 10- to 15-mV rise of V_m was used to obtain _foot. It is interesting that, although the results in working myocardium showed that there were clear deviations of V_m foot from a linear trajectory in the dV/dt versus V_m display, a comparison of the time course of V_m foot with a simple exponential during the initial 10- to 15-mV rise of V_m 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 V_m 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 V_m foot within working myocardium? When ¯G_Na is increased, the conduction velocity increases in association with a decrease in _foot and an increase in _max, and decreases in ¯G_Na 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 V_m 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.

View larger version (157K): [in this window] [in a new window]   Figure 3. Distribution of capillaries in canine ventricular muscle and in a Purkinje strand. The photomicrograph on the left is a cross section of left ventricular endomyocardium (V) that demonstrates the plethora of mostly longitudinally oriented capillaries (arrow) in ventricular muscle and the relative paucity of capillaries in an adjacent Purkinje strand (P). On the right, the electron micrograph of right ventricular papillary muscle shows that longitudinally oriented capillaries (arrow) produce multiple indentations along the perimeter of each myocyte. Variations of the width of interstitial space surrounding the capillaries and myocytes can be seen. Both sections were obtained from retrograde aortic perfused hearts using different fixatives: Bouin's fixative (left) and glutaraldehyde (right). The cross section on the left was stained with picrosirius red and viewed with fluorescence microscopy.41 Left bar=50 µm; right bar=10 µm.

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 V_m foot during LP and TP.

Experimental Support for Hypothesis
According to our hypothesis, the phase-plane trajectory of V_m 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 V_m display from the high-quality voltage V_m 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 V_m and dV/dt curves recorded during LP in a neonatal monolayer of cells (with the kind permission of the authors),⁹ and Figure 4B1 shows our reproduction of their curves. The associated dV/dt, V_m display in Figure 4B2 demonstrated that the trajectory of V_m foot was linear, as predicted. Contrariwise, in the neonatal ventricular epimyocardium LP produced very prominent concave deviations from a linear trajectory of V_m 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).

View larger version (23K): [in this window] [in a new window]   Figure 4. Experimental support for hypothesis of electrical field interactions between myocytes and capillaries that affect Vmfoot. A, Recordings of Fast and Kléber9 of high-quality Vm and dV/dt depolarization waveforms superimposed on optically recorded waveforms (F/F) during LP in neonatal cellular monolayers that have no capillaries. B, Our reproduction of their voltage waveforms (B1) and the associated dV/dt vs Vm phase-plane plot (B2), which had a linear Vm foot trajectory. C, Typical quite prominent concave deviations from linearity in the trajectory of Vm foot in phase-plane plots of the depolarization phase of action potentials recorded during LP in neonatal ventricular epicardium, which has the highest density of capillaries that occurs at any interval during development to maturity.52 53 The waveforms in panel A were reproduced from Reference 9 with the kind permission of the authors and the American Heart Association.

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 V_m foot. What if the phase-plane trajectory of V_m foot also varies in the neonatal cellular monolayers? Would multiple action potentials reveal concavities in the trajectory of V_m foot during LP and slurs during TP? To answer these questions, we identified numerous very high–quality optical recordings of V_m 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 V_m 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 V_m 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 V_m foot during LP in the 2 preparations (P<0.0001).

View larger version (59K): [in this window] [in a new window]   Figure 5. A 2-domain equivalent electrical circuit of anisotropic cardiac muscle. A, The intracellular domain (ri(L), ri(T)) was coupled to the interstitium (ris(L), ris(T), ris(V)) via the active myocyte membrane (Cm(myo), Rm(myo), H-H). The interior of the capillaries (ri(cpl)) was coupled to the interstitium via the capillary wall (Cm(cpl), Rm(cpl)). B, 2-domain equivalent electrical circuit in which intracellular space of the active layer was represented as a continuous anisotropic sheet (black layer). C, 2-domain equivalent electrical circuit in which a 2D cellular network7 represented the active layer in order to include microscopic ri cellular discontinuities produced by the gap junctions. The electrical value of each element of the equivalent electrical circuit is presented in Table 2.

View larger version (29K): [in this window] [in a new window]   Figure 7. Waveforms of the potential difference across the active myocyte membrane and across the capillary wall (A), with the associated current waveforms (B and C) and the spatial current field (D). Uniform LP and TP were produced in the 2-domain model with a continuous anisotropic active layer. In A, the temporal waveforms of the potential difference and the net current across the capillary wall, Vm(cpl) and Im(cpl), respectively, were obtained by Equations 4 through 7. Vm and the net transmembrane current of the sarcolemma (Im(myo)) were obtained by Equation 3. Each parameter of the 2-domain model was assigned the same baseline value as in Figure 6, except that the electrically active layer was represented by the 2D continuous anisotropic layer (Figure 5B). Maximum density of the longitudinal and transverse capillaries was present, and the capillary wall specific resistance was 8000 -cm2. In D, the current field linking the active membrane and the inactive capillary was reconstructed spatially from the potential and current waveforms shown for LP in panels A through C. For this, the temporal waveforms were multiplied by the longitudinal conduction velocity (0.5 m/s); ie, (dt)=dx. The lines with arrows represent the direction of current flow inside the myocyte layer and in the capillaries, as well as between these structures via interstitial space (interstitial longitudinal currents are not represented). The + and – signs denote the maximum and minimum density of current entering (–) and leaving (+) each structure. The misalignment of the maxima and minima of the active membrane and capillary wall was accounted for by the capillary capacitive current ICm(cpl) (ie, when ICm(cpl) was 0, the misalignment disappeared). The form of the result is similar to that obtained by Clark and Plonsey in a closely apposed active and inactive neuron in a bundle of nerves.50

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 r_i 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 V_m 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 V_m foot.

	Part II. Biophysical Mechanisms

Top Abstract Introduction The Problem Part I. Experimental Procedures Part II. Biophysical Mechanisms Discussion References

 
These new experimental results show that structural loading mechanisms can alter V_m foot and _max independently of one another. In contrast, in the classical relationship, V_m 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 r_i discontinuities,^{7 10} the major question now is the following: By what mechanism can the effective capacitance being discharged during V_m 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.

View this table: [in this window] [in a new window]   Table 2. Two-Domain Model Parameters With Symbols and Baseline Values

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, r_i(L) and r_i(T), respectively, produced a r_i(T)/r_i(L) ratio of 6. (Clerc measured a ratio of 9.4.⁵⁹ ) We used our 2D cellular model^{7 60} to represent the discontinuous active layer (Figure 5C). The interior of each myocyte was isotropic, and the conductances g_j 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, C_m(myo), R_m(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 R_is of 50 -cm.⁵⁷ The interstitial longitudinal resistance per unit length r_is(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 R_is value of 50 -cm and a cross-sectional area of 21 µm², the equivalent longitudinal resistance of interstitial space (r_is(L)) is 237 M/cm (Table 2B). We assigned a value of 787 M/cm to the equivalent transverse interstitial resistance (r_is(T)) to produce an r_isT/r_isL 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 (r_is(L1,L2), r_is(T1,T2), r_is(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 R_i(cpl) of 50 -cm. We characterized the inactive capillary wall as a distributed parallel resistance-capacitance network with a specific wall capacitance C_m(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 I_Cm=C_m(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 R_m(cpl) of the wall of intact myocardial capillaries, we used a value of 8000 -cm² for R_m(cpl) for all initial results. Then, we repeated each simulation by varying R_m(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 Kléber^{9 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 r_i 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 I_Na. Specifically,

(1)

where ¯G_Na is the maximal sodium conductance (28 mS/cm²), m and h are gating parameters, and V_Na is the sodium equilibrium potential (33.45 mV). With Ebihara-Johnson kinetics,⁶⁶ I_Na begins activation at a V_m value of approximately –44 mV, whereas in the Luo-Rudy⁴⁴ model I_Na begins to be activated at a V_m value of approximately –60 mV. To make certain that differences in the onset of I_Na activation did not affect the interpretation of the results, we repeated each type of simulation with the Luo-Rudy⁴⁴ model. The results showed no qualitative differences in the deviations from linearity in the trajectory of V_m foot for the 2 ionic models. Thus, we used the model as described above, with a single depolarization current and a single repolarization current, for most of the calculations. That provided the major advantage of considerably reducing the large amount of computer time required for the simulations. In this regard, it is worth noting that interaction with capillaries depends on dV_m/dt and thus depends on I_Na.

We approximated a repolarization current I_R by the following equation:

(2)

where ¯G_R is the repolarization conductance (0.05 mS/cm²) and V_R is the equilibrium potential of the repolarization current (–80 mV). Since the behavior of a cell can be affected by the size of the outward potassium current at V_m values near threshold, we computed the total K⁺ current at I_Na threshold in the Luo-Rudy⁴⁴ model and in our model during LP. The total K⁺ current and I_R had essentially the same small values (2.1 and 2.0 µA/cm², respectively) relative to the rapid and large increase in I_Na that occurred over the few millivolts of V_m change during which I_Na was activated.

The net transmembrane current per square cm of the sarcolemma, I_m(myo), was obtained as Hodgkin and Huxley¹⁷ described by adding the capacitive and ionic currents as

(3)

The potential difference across the capillary wall V_m(cpl) was obtained by

(4)

where _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, I_Cm(cpl) and I_R(cpl), respectively, were obtained by

(5)

where C_m(cpl) is the specific capacitance of the very thin capillary wall (1.0 µF/cm²),²³ and

(6)

where R_m(cpl) is the specific resistance of the capillary wall, with a baseline value of 8000 -cm². Thus, the net transmembrane current per square cm induced across the capillary wall, I_m(cpl), was

(7)

The mathematical formulation and computational procedures have been presented in detail in prior studies.^{7 36 60} The calculation steps were accomplished within a context that was a direct extension of well-established methods and models frequently reported for studying 1D propagation, which provided an advantage in checking for errors and in differentiating results. The computational procedure³⁶ involved the following 4 steps at each time increment of 0.25 µs. (1) The membrane and stimulus currents were used to calculate dV_m/dt, and then these values were used to update the membrane voltages. (2) The membrane voltages were used to update the gating variables for the active membrane model. (3) The active and passive (capillary) membrane voltages were used to solve (simultaneously) for the set of nodal potentials. (4) The nodal potentials were used to find the resulting active and passive membrane currents. The reference (ground) potential was obtained by setting the average of all interstitial space endpoints to 0 (Figure 5A). The matrix equations were solved by lower-upper triangular matrix decomposition, using routines written for the bordered-banded matrices characteristic of this type of problem.

The calculations were performed in 3-dimensional networks that contained 5500 to 85 000 segments (x–y 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 V_m, _m, I_Na, _is, and _i(cpl), from which subsequent analyses were done to derive V_m(cpl), I_Cm(cpl), and I_R(cpl). The following computed variables were saved at each segment to construct graphs: _max, time of _max (activation time), V_m peak potential, and the areas of the sodium conductance and the I_Na 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 R_m(cpl) of 8000 -cm². A concavity occurred in the trajectory of V_m foot during LP (Figure 6A1), and an initial slur occurred during TP (Figure 6A2). Although _max varied from site to site during both LP and TP, the trajectory of V_m foot remained the same along the axis of propagation (Figure 6A1 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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