© 2000 American Heart Association, Inc.
Research Commentary |
Influence of a Perfusing Bath on the Foot of the Cardiac Action Potential
From the Department of Physics, Oakland University, Rochester, Mich.
Correspondence to Brad Roth, Department of Physics, Oakland University, 190 SEB, Rochester, MI 48309. E-mail roth{at}oakland.edu
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Abstract |
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Abstract—Recently, Spach et al (Circ Res. 1998;83:1144–1164) measured the transmembrane action potential 150 to 200 µm below the tissue surface during longitudinal and transverse propagation. They found that "during longitudinal propagation there was initial slowing of Vm [action potential] foot that resulted in deviations from a simple exponential... " (p 1144). They attributed this behavior to the effects of capillaries on propagation. The purpose of this commentary is to show that the perfusing bath plays an important role in determining the time course of the action potential foot, even when the transmembrane potential is measured 150 µm below the tissue surface. Using numerical simulations based on the bidomain model, we find that the action potential foot for transverse propagation is nearly exponential (
foot=314 µs). For
longitudinal propagation, the action potential foot is not exponential
because of an initial slowing (best-fit foot=483 µs).
We conclude that the perfusing bath must be taken into account when
interpreting data showing differences in the shape of the action
potential foot with propagation direction, even if the transmembrane
potential is measured 150 µm below the tissue surface. The full
text of this article is available at http://www.circresaha.org.
Key Words: bidomain • action potential foot • perfusing bath • anisotropy
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Introduction |
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In 1981, Spach et al1 observed a smaller maximum rate of rise of the action potential,
max, and a larger time constant of the
action potential foot, foot, during
propagation parallel to the myocardiac fibers (longitudinal) than
during propagation perpendicular to the fibers (transverse). They
attributed these differences to the discrete cellular structure of the
myocardium. Their research has been cited widely and is
often taken as evidence for discontinuous propagation in cardiac
tissue.2 Several researchers3 4 5 6 7 8 9 10 11 have suggested that the observations of Spach et al1 may be caused by the bath perfusing the tissue rather than the discrete nature of the tissue itself. Recently, Spach et al12 presented additional evidence supporting their earlier data, but instead of measuring the transmembrane potential (Vm) at the tissue surface, as they did in 1981, they measured Vm 150 to 200 µm below the surface to eliminate bath perfusate effects. In their study, they emphasized the time course of the action potential foot. The purpose of this commentary is to model the experiment of Spach et al12 using a numerical simulation and to show that the perfusing bath plays an important role in determining the time course of the action potential foot, even when Vm is measured 150 µm below the tissue surface.
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Materials and Methods |
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The simulation is similar to that described by Pollard et al7 ; a slab of cardiac tissue is superfused by a conductive bath (Figure 1
). The bidomain
model13 represents the anisotropic
electrical properties of the cardiac tissue. This model is a continuum
description that does not take into account the discrete nature of
individual myocardial cells. The electrical potential in the isotropic
perfusing bath obeys Laplace’s equation. At the interface between the
tissue and the bath, the boundary conditions are continuity of the
extracellular (bath and interstitial) potential, continuity
of the normal component of the extracellular current density, and the
vanishing of the normal component of the intracellular current
density.14 All other boundaries are sealed.
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A planar wavefront propagates in the x direction, and the
z direction is perpendicular to the tissue-bath surface
(Figure 1
). Fibers are aligned in either the x
direction (longitudinal propagation) or the y direction
(transverse propagation). The tissue parameters are given
in the Table. The common scale factor of the 4 bidomain
conductivities15 is selected so that the resulting
propagation speed of the action potential is typical of that observed
in experiments.12
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The ionic current through the membrane is described as a passive leak term plus an active sodium channel.12 16 The sodium channel gates obey Ebihara-Johnson kinetics.17 We restrict our attention to the depolarization phase of the action potential.
We solve the bidomain equations for the tissue and Laplace’s equation for the bath by approximating the differential equations by finite differences.5 The time step is 2 µs. The space step in the z direction is 20 µm, and in the x direction is 50 µm for longitudinal propagation and 20 µm for transverse propagation. The boundary-value problem is solved iteratively using overrelaxation5 ; the iteration is terminated when the residual is <1 µV.
The membrane is at rest initially (Vm=-80 mV). At t=0, Vm along the left edge (x=0) is raised to 0 mV, initiating the action potential. Measurements of Vm and its derivative are made at the midpoint of the slab, where the action potential wavefront has reached a steady shape. The length of the slab is 15 mm for longitudinal propagation and 6 mm for transverse propagation (301 nodes in both cases). The slab is 0.5 mm thick, and its bottom surface is sealed. The transmembrane potential is measured at 3 depths: the tissue-bath surface, 150 µm below the tissue-bath surface, and at the bottom of the tissue. The bath is 1 mm thick.
The time constant of the action potential foot is calculated by fitting
a straight line to the phase-plane plot of dVm/dt
versus Vm over the range of
Vm from –79 to –65 mV (approximately the first
15 mV of depolarization). The reciprocal of the slope of this line is
foot.
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Results |
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Figure 2
shows the transmembrane
potential as a function of x and z, for
longitudinal and transverse propagation. In both cases, the wavefront
is curved, with the action potential at the surface leading the action
potential at the center. The spreading of the contours indicates that
the rate of rise of the action potential is lower at the surface than
in the bulk. The speed of longitudinal propagation is 0.552 m/s, and of
transverse propagation is 0.203 m/s. The peak-to-peak
interstitial potential, measured 150 µm below the
surface is 23.0 mV for longitudinal propagation and 11.6 mV for
transverse propagation.
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Figure 3A contains a phase-plane
plot of the action potential during longitudinal and transverse
propagation, for Vm measured at the tissue
surface. The rate of rise is 15% lower during longitudinal propagation
(
max=149 V/s) compared with transverse
propagation (max=175 V/s). The inset shows
a magnified view of the action potential foot. For propagation in
either direction, the action potential foot is not exponential (an
exponentially rising action potential foot would appear as a straight
line in a phase-plane plot). The best-fit value of
foot is 706 µs for propagation in the
longitudinal direction and 486 µs for propagation in the transverse
direction.
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The dotted curve in Figure 3A represents the action
potential calculated when the bath is not present. In this case,
the wavefront is not curved. The speed of longitudinal propagation is
0.505 m/s, and the speed of transverse propagation is 0.202 m/s. The
time course of the action potential is independent of the direction of
propagation. The action potential foot is exponential
(foot=294 µs), and
max (201 V/s) is greater than when the
bath is present.
Figure 3B contains similar data, but Vm is
measured 150 µm below the tissue surface. As in Figure 3A, max is less for longitudinal
propagation (196 V/s) than for transverse propagation (201 V/s),
although the difference between the two (2.5%) is smaller than when
Vm is measured at the surface. The action
potential foot for transverse propagation is nearly exponential
(foot=314 µs), although it contains a slight
"initial slur."12 For longitudinal propagation, the
action potential foot is clearly not exponential because of an initial
slowing (best-fit foot=483 µs).
At the bottom of the tissue (Figure 3C),
max is larger, and
foot is smaller, for longitudinal
propagation (max=214 V/s,
foot=272 µs) than for transverse propagation
(max=203 V/s,
foot=292 µs). The action potential foot is
nearly exponential, although there is a slight initial slur for
propagation in the longitudinal direction. Note that
max is greater than, and
foot is smaller than, if the bath were not
present.
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Discussion |
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The data of Spach et al1 are cited widely as evidence for discontinuous propagation in cardiac tissue.2 Their hypothesis of discontinuous propagation is supported by the following logic: (1) During 1-dimensional propagation in a tissue with continuous electrical properties, the time course of the action potential (including
max and
foot) does not depend on the intracellular and
interstitial conductivities18 ; (2) experiments
indicate that in cardiac tissue max and
foot differ with the direction of propagation
and therefore with conductivity1 ; and (3) therefore, the
conductivity of cardiac tissue is not continuous. A flaw exists in this
line of reasoning: when a conductive bath perfuses the tissue, the
propagation is not 1-dimensional. The extracellular conductivity is
higher for the tissue near the surface (adjacent to the bath) than it
is for the tissue far from the surface (deep within the bulk).
Therefore, gradients in Vm exist not only in the
direction of propagation, but also in the direction perpendicular to
the tissue surface. Reasoning based on the 1-dimensional cable model
(such as used in the first premise of the syllogism above) is not
applicable.
Several researchers3 4 5 6 7 8 9 10 11 have shown theoretically that the
presence of the perfusing bath may account for the difference in the
rate of rise with direction that was observed by Spach et
al.1 The high-conductivity bath causes the wavefront to be
curved (surface leading bulk) and the surface rate of rise to be
slowed. This effect is more dramatic for longitudinal propagation than
for transverse propagation because of the unequal anisotropy ratios of
the tissue. For longitudinal propagation, the intracellular and
interstitial conductivities are approximately the
same,15 so large interstitial potentials exist
in the bulk, although the potential in the high-conductivity bath is
small. For transverse propagation, the interstitial
conductivity is 
4 times greater than the intracellular
conductivity,15 so the extracellular potentials are small
both at the tissue surface and deep in the bulk. The smaller gradients
of the extracellular potential result in smaller gradients in the
transmembrane potential during transverse propagation compared with
longitudinal propagation. Our calculated changes in propagation speed,
max, and foot
measured at the tissue surface are qualitatively consistent
with previous numerical models3 4 5 6 7 8 9 10 11 and with experimental
data.1
Recently, Spach et al12 measured Vm
150 µm below the tissue surface, where they claim "there
should be minimal effects of the superfusate
solution." (p 1146). Although Spach et al12
recorded the action potential rate of rise, their main goal
was to present "a detailed experimental analysis of
the time course of the foot of the cardiac action potential
(Vm foot) during propagation in different
directions in anisotropic cardiac muscle." (p 1144). They observed
that "during longitudinal propagation there was initial slowing of
Vm foot that resulted in deviations from a simple
exponential; corollary changes occurred at numerous sites during
transverse propagation."12 (p 1144). They attributed
these results to an effect of capillaries on conduction.
The results in Figure 3B show that the influence of the
perfusing bath extends at least 150 µm below the tissue surface.
Furthermore, the bath causes the action potential foot to rise more
slowly than exponentially, and this slowing is greater for longitudinal
propagation than for transverse propagation. These results agree
qualitatively with the recent experimental data of Spach et
al.12 The action potential foot is particularly sensitive
to the perfusing bath, more so than other features of the action
potential.9 10 11 12 13 14 15 16 17 18 19
Quantitatively, the biggest discrepancy between our calculations and
the data of Spach et al12 lies not in the action potential
foot, but instead in max. Our calculations
indicate that max 150 µm below the
tissue surface is only 2.5% less for longitudinal propagation than for
transverse propagation, whereas the experimental data show an average
difference of 22%. The source of this discrepancy is unclear. It may
arise from the discrete nature of the tissue, from capillary effects,
from incorrect parameter values in the simulation, or from
of the presence of dead tissue 200 to 300 µm below the tissue
surface12 Our model does not incorporate a dead core of
tissue. According to Spach et al,12 the dead core has an
enlarged interstitial space, which might increase the
interstitial conductivity and cause the core to function
approximately in the same manner as the perfusing bath.
Spach et al12 supported their theory of capillary effects by comparing their data with that measured by Fast and Kléber20 in monolayers of neonatal cardiac myocytes. They suggested that because such monolayers are devoid of capillaries, the action potential foot should be exponential. The action potentials measured by Fast and Kléber20 do indeed have an exponential foot. However, the monolayers of Fast and Kléber20 are also devoid of "deep" tissue far from the perfusing bath, so there can be no gradients of Vm with depth. Therefore, the data of Fast and Kléber20 are also consistent with the hypothesis that the purfusing bath determines the shape of the action potential foot. Thus, data from monolayers does not distinguish between the capillary mechanism and the purfusing bath mechanism for slowing the action potential foot.
One way to distinguish between the 2 mechanisms (capillaries versus
perfusing bath) would be to repeat the experiments of Spach et
al1 12 with and without a perfusing bath present. The
tissue would have to be kept alive when the perfusing bath was absent,
perhaps by arterial perfusion. The results in Figure 3A indicate that when the bath is eliminated, the action
potential foot should become exponential, with no differences between
longitudinal and transverse propagation. Furthermore, the maximum rate
of rise of the action potential should increase and become independent
of propagation direction. Although this experiment is easy to conceive,
it would be susceptible to several sources of error. If
Vm were measured optically, the data would
represent an average over a depth of a few hundred
microns. Because the model predicts that
Vm changes dramatically over such distances, the
data would be difficult to interpret. Microelectrode measurements, on
the other hand, are sensitive to capacitative coupling to the perfusing
bath, and the degree of such coupling depends on the bath depth. The
rapid depolarization phase of the action potential is particularly
sensitive to electrode capacitance. Although it is possible to correct
the data for the influence of electrode capacitance, these corrections
would be crucial when comparing data measured at different bath
depths.
We cannot conclude from our study that capillaries are not important during action potential propagation. Nor can we conclude that discontinuous propagation does not occur (particularly in diseased tissue). These factors may well play a role in propagation. We can conclude, however, that the influence of a perfusing bath must be taken into account when interpreting data showing differences in the shape of the action potential foot with propagation direction, even if Vm is measured 150 µm below the tissue surface. Therefore, differences in action potential shape with direction1 12 cannot be taken as definitive evidence supporting discontinuous propagation or capillary effects if a perfusing bath is present. Finally, without additional experiments, we cannot exclude the possibility that in healthy tissue the difference in the shape of the action potential upstroke with propagation direction is simply an artifact of the way the tissue was perfused.
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Acknowledgments |
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This research was supported by NIH Grant RO1HL57207. We thank the School of Engineering and Computer Science at Oakland University for their computational support.
Received September 10, 1999; accepted October 27, 1999.
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