Ionic Mechanisms of Propagation in Cardiac Tissue
Roles of the Sodium and L-type Calcium Currents During Reduced Excitability and Decreased Gap Junction Coupling
Abstract In cardiac tissue, reduced membrane excitability and reduced gap junction coupling both slow conduction velocity of the action potential. However, the ionic mechanisms of slow conduction for the two conditions are very different. We explored, using a multicellular theoretical fiber, the ionic mechanisms and functional role of the fast sodium current, INa, and the L-type calcium current, ICa(L), during conduction slowing for the two fiber conditions. A safety factor for conduction (SF) was formulated and computed for each condition. Reduced excitability caused a lower SF as conduction velocity decreased. In contrast, reduced gap junction coupling caused a paradoxical increase in SF as conduction velocity decreased. The opposite effect of the two conditions on SF was reflected in the minimum attainable conduction velocity before failure: decreased excitability could reduce velocity to only one third of control (from 54 to 17 cm/s) before failure occurred, whereas decreased coupling could reduce velocity to as low as 0.26 cm/s before block. Under normal conditions and conditions of reduced excitability, ICa(L) had a minimal effect on SF and on conduction. However, ICa(L) played a major role in sustaining conduction when intercellular coupling was reduced. This phenomenon demonstrates that structural, nonmembrane factors can cause a switch of intrinsic membrane processes that support conduction. High intracellular calcium concentration, [Ca]i, lowered propagation safety and caused earlier block when intercellular coupling was reduced. [Ca]i affected conduction via calcium-dependent inactivation of ICa(L). The increase of safety factor during reduced coupling suggests a major involvement of uncoupling in stable slow conduction in infarcted myocardium, making microreentry possible. Reliance on ICa(L) for this type of conduction suggests ICa(L) as a possible target for antiarrhythmic drug therapy.
Although the time and voltage-dependent kinetics of the major ionic membrane currents of ventricular cells are well described, their functional role in propagation of the action potential is not fully understood. The major cardiac inward currents are the fast sodium current, INa, and the L-type calcium current, ICa(L). Typically, INa is responsible for excitability and conduction, and ICa(L) is responsible for maintaining the action potential plateau. We sought to elucidate the relative role of INa and ICa(L) in maintaining multicellular ventricular propagation under both normal physiological conditions and two pathophysiological conditions: reduced membrane excitability and reduced intercellular coupling.
Reduced membrane excitability, caused by reduced availability of INa, is present during conditions of acute myocardial ischemia,1 2 3 tachycardia,3 and treatment with class I antiarrhythmic agents.4 Reduced excitability lowers the safety factor for conduction (safety factor is a dimensionless parameter that indicates the margin of safety with which the action potential propagates relative to the minimum requirements for sustained conduction). However, the precise relationship between safety factor and excitability has not been studied: it is not known whether the safety factor drops monotonically with decrease in INa availability, nor has the steepness of this relationship been characterized. For instance, does a 50% reduction in excitability bring the membrane very close or only marginally close to conduction block? Safety factor considerations usually address INa only. At highly reduced membrane excitability, it is possible that ICa(L) contributes significantly to the safety factor. In this study we attempt to determine the contribution of ICa(L) to the safety factor and its role in conduction.
Similar to the condition of reduced membrane excitability, reduced intercellular coupling at gap junctions decreases velocity of the propagating action potential. However, the magnitude of velocity reduction and the effect on safety factor for conduction may be vastly different for the two conditions. Several theoretical5 6 7 and experimental8 9 10 studies have suggested that reduced coupling can cause major reductions in velocity before conduction block and that velocity in the context of reduced coupling can potentially be much slower than velocity in the context of reduced excitability. Despite slowed conduction, evidence tends to suggest that propagation across cells with reduced coupling occurs with a high safety factor.11 12 These reports of high propagation safety with low intercellular coupling are not universal, and there exists evidence to the contrary.9
Long intercellular propagation delays associated with highly reduced gap junction coupling also suggest a potential role for ICa(L) in supporting conduction.13 14 With long intercellular delays, it is feasible that the downstream neighboring cell remains unexcited and constitutes a current sink when the upstream depolarized cell (current source) is already in its plateau phase. During the plateau phase, ICa(L) is the inward source current of the upstream cell and could be important to sustain cell-to-cell conduction. The degree of coupling reduction necessary before ICa(L) becomes a major contributor to both conduction velocity and safety factor has not been determined. Also, the phenomenon of calcium-dependent ([Ca]i) inactivation of ICa(L) suggests that there can be direct coupling between [Ca]i and propagation safety. It was recently shown in neuronal preparations (rat ganglion cells) that calcium overload reduces safety factor for conduction.15
We characterized propagation in the setting of reduced membrane excitability and reduced gap junction coupling with simulations using detailed cellular ionic models and the appropriate intercellular coupling conditions of a multicellular fiber. The individual cells were represented by the dynamic Luo-Rudy (LRd) ventricular cell formulation,16 17 18 interconnected by resistive pathways representing gap junctions. In the context of this study, it should be emphasized that the LRd model incorporates important kinetic properties of INa and ICa(L). These include fast and slow inactivation processes of INa, an order-of-magnitude faster ICa(L) activation than in previous models (eg, Beeler and Reuter19 ), and calcium-dependent inactivation of ICa(L). Safety factor of conduction was formulated using a modified version of formulations previously introduced.9 12
The results demonstrate that a decrease in membrane excitability (eg, due to ischemia, tachycardia, or treatment with antiarrhythmics) causes a monotonic decrease of the safety factor for conduction and a modest decrease of conduction velocity before the occurrence of block. In contrast, a decrease in intercellular coupling (eg, due to anisotropy of the myocardial structure, aging, elevated intracellular calcium, or infarction) can have a profound depressant effect on conduction velocity that is accompanied by a paradoxical augmentation of the safety factor for conduction. The interplay between membrane factors and gap junction factors has significant relevance to conduction under pathological conditions in which one pathophysiological change (reduced coupling) may compensate for another (reduced excitability) in terms of their opposite effects on propagation safety. Detailed studies of the ionic mechanisms of normal and pathological conduction identify the relative importance of INa and ICa(L) in supporting conduction under a given set of conditions. Such insight could be very helpful for informed treatment of cardiac arrhythmias.
Materials and Methods
The LRd cell model and its use as the cellular element of a multicellular fiber have been described previously.16 18 20 The following is a brief description of the cell and fiber models and formulation of the safety factor of conduction (SF). In view of the strong emphasis in this study on the role of gap junctions in propagation and the detailed cable analysis required for SF computation, related methodology is presented in “Appendices 1 and 2.” A description of simulation protocols is also included below.
Multicellular Fiber Model
For each cell in the fiber, the LRd model is used to compute ionic currents and concentration changes. In this model, the guinea pig–type ventricular action potential is numerically constructed on the basis of experimental data. Included in the model are the membrane ionic channel currents, represented mathematically by a Hodgkin-Huxley type formalism, as well as ionic pumps and exchangers. In addition, processes that regulate ionic concentration changes, especially dynamic changes of intracellular calcium concentration, are introduced. A diagram of the cell model is provided in Fig 1A⇓, with the two major excitatory currents (INa and ICa(L)) highlighted. Detailed tables of equations governing the model are provided in References 1616 to 18.
The theoretical fiber (Fig 1B⇑) is composed of serially arranged ventricular cells, each of LRd formulation. The temporal transmembrane current fluxes of the LRd model are related to spatial (axial) current flow by a finite difference approximation6 of the cable equation21 22 23 where Iion represents the individual membrane ionic current densities (μA/μF) of the LRd model, Is is the stimulus current density (μA/μF), a is the radius of the fiber (11 μm), Cm is the membrane capacity (1 μA/μF), Vit is membrane potential at segment i and time t, Δx is the discretization element (μm), RCG is the ratio between capacitive and geometrical areas (RCG=2), and Ri is the axial resistance per unit length (Ωcm). Ri is composed of myoplasmic resistance (Rmyo=150 Ωcm) and gap junction resistance (normal Rg=1.5 Ωcm2, which corresponds to a conductance of gj=2.5 μs). Direct explicit solutions to Equation 1 would necessitate a very small time increment (Δt) for numeric convergence and could not guarantee stability. For faster computing with a larger Δt, the relation in Equation 1 can be expressed in implicit Crank Nicolson24 form by replacing the second central difference with the average of the second central difference at time t and t+Δt.6 The Crank and Nicolson method is second-order accurate and is unconditionally stable. Its solution entails, at any given time step, simultaneous solution of membrane potential over the entire fiber once individual ionic currents are computed.25 We always use the Crank and Nicolson implicit method to solve for the relation in Equation 1. Extracellular resistance, Ro, is neglected (the fiber is assumed to be in an extensive medium). No flux (sealed ends) boundary conditions are used by setting ∂Vm/∂x=0 at the beginning and end of the fiber.
Spatial discretization and distribution of fiber resistivity require special consideration for a discontinuous fiber, as discussed in “Appendix 1.” For computations in this report, Δx=100 μm (entire cell length) is used as the discretization element with Ri reflecting the lumped contribution of axial and gap junction resistance (Ri=Rmyo+Rg/Δx). When intracellular detail is sought, a smaller Δx of 21 computation elements per cell (Δx=4.76 μm) is used with Rg concentrated at the edge elements of the cells. All simulations were tested for numeric convergence and accuracy.
Regardless of fiber discretization, the intracellular calcium transient [Ca]i, intracellular calcium buffering, and sarcoplasmic reticulum (SR) calcium fluxes are always computed for the entire cell as one unit. When Δx is less than one cell length, sarcolemma calcium currents from each element are summed for net calcium entry into the cell. Unless otherwise specified, the middle element of a multi-element cell is always used for computation of upstroke velocity, dVm/dt.
Safety Factor for Conduction
On the basis of the approach of others,9 12 we define the safety factor for conduction (SF) as the ratio of charge generated for the fiber by cell excitation to the minimal amount of charge required to cause the excitation. An SF >1 indicates that more charge was produced during cellular excitation than charge required to cause excitation. The fraction of SF >1 indicates the margin of safety. When SF falls <1, the charge requirements are not met and conduction fails. The equation used to compute SF is where Ic is the capacitive current of the cell in question, Iout is the axial current between the particular cell and its downstream neighbor, Iin is the axial current that enters the particular cell from its upstream neighbor, and Qm is the integral of transmembrane current over time. A, the integration limit for SF, is the period that the cell in question remains a sink, the period during which the membrane has consumed more charge than it has produced. The beneficial aspect of our approach is that SF, computed at a particular cell, takes into consideration charge lost and gained with respect to the entire fiber. This definition accounts for excitability changes due to charging time (ie, strength-duration) and for effects of changes in intercellular coupling. A detailed explanation of SF is provided in “Appendix 2.”
The theoretical fibers used in this study were between 70 and 100 cells (0.70 to 1.0 cm) in length. Rmyo (myoplasmic resistivity) was 150 Ωcm and gj (gap junction conductance) varied between 0.005 and 2.5 μs (corresponding to resistance changes between Rg=760 and 1.5 Ωcm2, respectively). A variable time step was introduced that tracks the propagating action potential, using small time steps (Δt=0.005 ms) when membrane activity is high and large time steps (Δt=1.0 ms) when membrane activity is low. The variable time step algorithm sums the change in membrane voltage for nine cells in each direction from the cell under consideration. Each millisecond, the sum is evaluated for all cells of the fiber. If a cell’s sum is greater than a preset threshold (20 mV) or the cell is within 15 ms of calcium release, then the cell’s time step for the next millisecond is small. Otherwise the time step is large. The variable time step is used for membrane current computations only. Axial currents are always computed every 0.005 ms. The variable time step was always in effect, except where indicated.
The fiber was excited by applying a 0.5-ms superthreshold stimulus to cell 1 and, when multiple (n) elements per cell were used, to the middle element of cell 1. Stimulus strength (μA/μF) varied with n and gj. For n=1 and gj=2.5 μs, stimulus was −600 μA/μF. Results are independent of the stimulus used for distances greater than one space constant (about 9 cells) from the stimulus site because propagation depends on fiber axial current, not on the applied stimulus, for excitation. End effects were also restricted to within 9 cells from the fiber end.
Propagation in a Multicellular Fiber
The theoretical fiber we used is composed of interconnected cell models rather than a membrane model with finite spatial dimensions. The difference is that individual cell models contain both membrane and intracellular processes. Excitation in the multicellular fiber by a propagating action potential results in both membrane and intracellular events (eg, the transmembrane action potential and the intracellular calcium transient). Fig 2A⇓ shows the profile of a computed action potential (AP) in space at time=t and time=t+3 ms. The distance covered in 3 ms is 1.6 mm (16 cells), which is observed in Fig 3⇓ and can be computed from the product of the time period (3 ms) and conduction velocity (54 cm/s). At the head of the action potential there is a slow electrotonic depolarizing phase (AP “foot”). Slightly further upstream (to the left) cells are undergoing a sharp upstroke caused by the fast sodium current (INa) depolarizing local membrane. Further upstream (left) of the upstroke, late upstroke, and early plateau depolarization is occurring, which is due to the L-type calcium current (ICa(L)).
Spread of electrical excitation in Fig 2A⇑ initiates a chain of events that lead to calcium-induced calcium release (CICR) and a rise in intracellular calcium. Fig 2B⇑ shows the calcium transient in space for the same time instants as the AP profiles in Fig 2A⇑. At each time instant (t and t+3) the AP upstroke (wave front) is about 25 cells ahead of the calcium transient. This spatial difference results from the period of calcium accumulation that triggers CICR16 and the dynamics of the SR calcium release process.
The coupling between membrane excitation and calcium release is bidirectional. Calcium accumulation from calcium entry through ICa(L) during the late upstroke initiates CICR. The resulting myoplasmic calcium transients decrease local ICa(L) via calcium-dependent inactivation. Under normal conditions, CICR is sufficiently delayed from the upstroke to not interfere with conduction. However, when conduction is slowed, CICR leading to calcium-dependent inactivation of ICa(L) may become an important factor.
Fig 3⇑ shows action potential profiles in two neighboring cells under conditions of normal gap junction coupling (panel A) and a marginal (10-fold) decrease in coupling (panel B). For normal coupling (gj=2.5 μs), the gap junction conductance between cells is equal to the myoplasmic conductance of the entire cell. The result is equal conduction time of 0.09 ms between cells as in crossing the cell length. On a macroscopic (many cells) scale, the result is an apparent uniform spread of excitation as seen in Fig 3A⇑. With only 10-fold reduction in coupling (Fig 3B⇑) a large (0.5 ms) conduction delay is introduced at each intercellular junction, while the entire cell depolarizes almost simultaneously. The almost simultaneous depolarization of the cell is due to increased confinement of depolarizing current to the cell when intercellular coupling is reduced, decreasing loading effects and current leakage out of the cell. Note (Fig 3B⇑) that propagation under these conditions is nonuniform (“discontinuous”), with long delays at gap junctions. In fact, the macroscopic conduction velocity over many cells is determined by the gap junction delays rather than by the (negligible) time spent in traveling across individual cells. With even further reduction in coupling, the intercellular delay can be on the order of milliseconds, providing time for calcium released from the SR to contribute to ICa(L) inactivation, altering the balance of currents at the head of a traveling action potential. The issues of propagation down a fiber with reduced coupling and interaction between the calcium transient and ICa(L) are explored in later sections of this report. Principles observed in Figs 2⇑ and 3⇑ are that propagation is a multicellular phenomenon that involves interaction between membrane currents, gap junction properties, and intracellular ionic processes. The interactions between these multiple factors must be considered when propagation is analyzed and when mechanisms of normal and abnormal conduction are investigated.
Reduced Membrane Excitability Lowers Conduction Velocity and Propagation Safety, With Little Effect From ICa(L)
We investigated the effects of reduced membrane excitability (eg, due to acute ischemia or class I drugs) on conduction of the action potential impulse. The parameters used to characterize conduction with changes in excitability are conduction velocity (θ), safety factor for conduction (SF), maximum upstroke velocity ([dVm/dt]max), and peak sodium current (INa,max). These computed parameters are shown with different degrees of membrane excitability in Fig 4⇓. Because excitability is determined by availability of fast sodium channels, membrane excitability is reduced by lowering ¯gNa, the maximum conductance of INa. All four indices of conduction decrease monotonically with excitability reduction. This result is to be expected; lower sodium channel availability causes slower conduction of an action potential that has a slower upstroke and is less safe. Fig 4A⇓ shows that the minimum conduction velocity obtained before conduction failure is 17 cm/s, indicating that the minimum possible conduction velocity with reduction in excitability is only one third of control velocity (54 cm/s).
Safety factor of conduction, also in Fig 4A⇑, decreases slowly with excitability reduction, indicating a relative insensitivity to moderate changes in membrane excitability. It can be recalled from the discussion in “Methods” that SF is the ratio between charge gained by the fiber from cell excitation to charge required for cell depolarization. Our simulations reveal that reducing sodium channel availability only marginally reduces charge gained by the fiber from excitation (numerator of SF) because (1) capacitive charge gained (first numerator term) is proportional to peak membrane potential, which is not reduced significantly by reducing INa, and (2) subthreshold depolarization time is not greatly reduced, resulting in only small changes to the charge delivered downstream (second numerator term). Our simulations also indicate that charge required to depolarize the fiber (the denominator of SF) is not significantly increased. Therefore less than extreme reductions in INa availability do not constitute a great detriment to SF. At INa availability <11.25%, the cell membrane has difficulty reaching threshold, depolarizing charge requirements from upstream fiber start to increase dramatically (denominator), and SF drops precipitously toward 1. The rapid decrease of SF toward 1 in Fig 4C⇑ is indicative of the nonlinear “all or none” response of the cell membrane.
The data in Fig 4B⇑ suggest that INa dominates conduction for almost the entire range of membrane excitability. Maximum upstroke velocity, an indicator of the depolarizing current during the upstroke, follows INa,max at all levels of reduced excitability. However, as membrane excitability is reduced below 30% ¯gNa, INa,max decreases faster than (dVm/dt)max, suggesting additional support from another inward current (ICa(L)). We also found that when ICa(L) is removed, conduction fails at slightly higher membrane excitability (¯gNa=15% as compared with 11% when ICa(L) is present). Therefore we examined, in Fig 5⇓, the role of ICa(L) in action potential conduction under conditions of depressed membrane excitability. In Fig 5A⇓, SF is computed for a control fiber with all membrane currents intact (solid line) and for a fiber with ICa(L) removed (dashed line). For most values of membrane excitability, the presence of ICa(L) does not influence SF. At extreme reductions of excitability (<30% availability), ICa(L) augments SF slightly, resulting in conduction failure at the slightly lower value of % ¯gNa.
At extremely low membrane excitability, in the vicinity of ¯gNa=20%, a slight positive influence of ICa(L) on SF is evident in Fig 5A⇑. We investigated the mechanism by which ICa(L) increases the safety of highly depressed conduction. Fig 5B⇑ shows action potential upstrokes from the middle cell of the fiber computed with ICa(L) (solid line) and without ICa(L) (dashed line), both at 20% ¯gNa. Time 0 is the time of (dVm/dt)max for the middle cell. The action potential upstrokes, as reflected in (dVm/dt)max, are slightly faster for the fiber computed with ICa(L) (46 versus 35 V/s). The difference in (dVm/dt)max is not due to different INa,max, which is actually higher for the fiber without ICa(L) (−48 versus −54 μA/μF, not shown), nor is the difference due to direct local action of ICa(L), which does not significantly activate at the negative voltage range (around −33 mV) at which (dVm/dt)max occurs. (dVm/dt)max is somewhat higher in the fiber with ICa(L) because ICa(L) increases slightly the action potential amplitude in the adjoining upstream cell (compare action potentials with and without ICa(L) in Fig 5⇑). Higher action potential amplitude acts to increase the potential gradient and the electrotonic driving force between an excited cell and its downstream neighbor that is still undergoing depolarization. The result is a slight increase in axial current and somewhat greater axial charge delivery that contributes directly to the depolarization of the downstream cell.
The quantitative role of ICa(L) in augmenting electrotonic driving force and downstream depolarization can be established by computing the charge (current integrated over time) generated by a cell membrane between its own (dVm/dt)max and (dVm/dt)max of its downstream neighbor (identified by a thin vertical line in the figure). This computation is a measure of charge generated by a source cell to depolarize its downstream neighbor. Note that INa and ICa(L) are integrated over a very short interval (to the thin vertical line) because the downstream neighbor is activated early during the plateau of the upstream source cell (a long intercellular delay at the gap junction is not present). During the short interval, INa is near its maximum value and ICa(L) is only beginning to activate. In the inset of Fig 5B⇑, we show relative charge generated by INa and ICa(L) at 20% ¯gNa. The computed charge ratio QNa:QCa is 75:1. Thus it can be concluded that the predominant current responsible for maintaining conduction, even at 20% ¯gNa, remains INa.
The results of this section demonstrate that despite a slight influence from ICa(L) in supporting axial charge delivery at extreme levels of membrane depression, in an otherwise well-coupled fiber excitability and conduction are determined by INa. Decreased excitability decreases all parameters of conduction, ultimately leading to conduction block. In the following section we evaluate the effect of decreased intercellular coupling while the membrane currents are not directly modified.
Reduced Intercellular Coupling Reduces Conduction Velocity and Has a Biphasic Effect on Propagation Safety, With Major Effect from ICa(L)
Propagation transverse to fiber orientation and propagation across areas of a healed infarct are examples of normal and pathophysiological propagation in which intercellular coupling is reduced. Fig 6⇓ contains the parameters of propagation plotted against changes in intercellular coupling conductance, gj. Conduction velocity decreases monotonically with reduction in intercellular coupling. This behavior is more dramatic but qualitatively similar to that of reductions in membrane excitability. However, as seen in both Fig 6A⇓ and 6B⇓, safety factor for conduction and maximum upstroke velocity of the action potential do not decrease with decreasing levels of intercellular coupling. Both parameters display a biphasic behavior, increasing then decreasing, in response to reduction of gj. Across the entire range of gj, SF is greatest at gj=0.023 μs and (dVm/dt)max is greatest at gj=0.040 μs. Therefore, maximum values for both parameters occur when intercellular coupling is reduced by a factor of about 15 relative to its normal value of 2.5 μs.
The slight monotonic decrease in INa,max with decrease in gj, seen in Fig 6B⇑, establishes that membrane currents are not the cause of the biphasic changes in SF and (dVm/dt)max. Instead, the changes of intercellular coupling affect source-load relationships in the fiber, which feed back on the action of membrane currents. Inward membrane current either acts to depolarize local membrane or generates axial current flow downstream to act on distant membrane. When coupling is reduced, less inward current is shunted downstream (less load), effectively increasing availability of inward current for local depolarization. As a result of increased availability of local depolarizing current, safety factor and (dVm/dt)max increase. The descent of safety factor and (dVm/dt)max at very low levels of coupling is due to reduced availability of INa source-current under these conditions. INa,max is small at very low levels of intercellular coupling due to a long subthreshold depolarization phase (slow charging process due to the small axial current when resistance between cells is large) that provides for dynamic inactivation of sodium channels before reaching their activation threshold. This results in reduced sodium channel availability at threshold and a small INa,max. Ultimately, for very low degrees of intercellular coupling, subthreshold dynamic inactivation of INa is not compensated for by conservation of current for local membrane due to the reduced cellular coupling. The result is decreased safety factor, decreased (dVm/dt)max, and with sufficiently reduced coupling, conduction block. The data in Fig 6B⇑ demonstrate that when coupling is extremely reduced, INa,max is almost the sole determinant of (dVm/dt)max (the two curves practically overlap at this range), as occurs in an isolated space-clamped cell (a situation that is approached by the highly uncoupled fiber).
The results of Fig 6⇑ illustrate that altered intercellular coupling can cause major changes of the action of membrane currents even when the intrinsic properties (ie, density and gating) of these membrane currents remain unchanged. We introduced earlier the possibility that ICa(L) can affect conduction when coupling is reduced. This hypothesis is tested by computing the safety factor for conduction over a full range of intercellular coupling with and without the presence of ICa(L). Results are shown in Fig 7A⇓ that show that the two safety factor curves have the same biphasic shape. However, a striking difference between the curves is that conduction failed at gj=0.0056 μs for the fiber with all ionic currents intact and at a much higher gj=0.0197 μs for the fiber computed without ICa(L). The finding that the presence of ICa(L) can support conduction that otherwise fails over more than a threefold decrease in gap junction coupling strongly suggests that ICa(L) is a major determinant of propagation under conditions of highly reduced intercellular coupling and long propagation delays across gap junctions.
The interplay between INa and ICa(L) during conduction with decreased intercellular coupling is further explored in Fig 7B⇑, which contains an action potential upstroke at gj=0.020 μs, computed with (solid line) and without (dashed line) ICa(L). Gap junction conductance of gj=0.020 μs is used because it is slightly greater than the conductance at which ICa(L) becomes essential for conduction. As was explained in the context of reduced excitability, ICa(L) does not play a direct role in local excitation but, by maintaining a higher plateau, ICa(L) enhances the axial driving force and the resulting electrotonic source current. When decreased coupling causes long propagation delays, ICa(L), which is the major depolarizing current during late upstroke and early plateau, becomes extremely important. The role of ICa(L) in augmenting the driving force and the electrotonic current is confirmed by the postupstroke membrane potentials of Fig 7B⇑. The action potential that included ICa(L) maintains an early plateau at significantly higher potential than the action potential without ICa(L). It is the higher plateau that increases the driving force, forcing more source current to the adjacent downstream cell. The bar graph inset in Fig 7B⇑ indicates the increasing role of ICa(L) in supporting conduction under conditions of decreased coupling. The bars, like those of Fig 5⇑, show the relative charge contribution to sustain conduction from INa and from ICa(L) and are computed by integrating these currents over time, until the downstream neighboring cell is activated. Note that due to the long delay at the gap junction, the downstream neighbor is excited when the upstream source cell is well into its plateau (thin vertical line in Fig 7B⇑). Therefore, ICa(L), which is active during the plateau, is integrated over a much longer time than in the case of depressed membrane (compare with Fig 5B⇑), and INa, which is inactivated at the plateau, does not contribute additional charge beyond its early contribution during this interval. As a result, unlike the INa dominance conditions of reduced excitability (Fig 5⇑), charge contributions from both currents are almost equal (QNa:QCa(L)=1.47:1) for reduced coupling of gj=0.020 μs. Therefore, ICa(L) under such conditions is almost as important as INa in sustaining conduction. With further reductions in coupling, charge contribution from ICa(L) needed to support conduction exceeds that from INa (ie, at gj of 0.010, 0.006, and 0.0057 μs, QNa:QCa(L) charge ratios are 0.81, 0.26, and 0.16, respectively; the case of gj=0.006 μs is shown in Fig 7C⇑). Therefore, in the situation of highly reduced coupling, INa is necessary to bring the membrane into the activation range of ICa(L), but the major source of depolarizing charge and the most significant current to sustain propagation is ICa(L).
Intracellular Calcium Can Influence Conduction
There are numerous pathological conditions during which ventricular cells experience an increase in intracellular calcium, a condition known as calcium overload. Because [Ca]i has a direct effect on ICa(L) via calcium-dependent inactivation, it follows that changes in intracellular calcium may affect conduction. We explored the effect of elevated [Ca]i on action potential propagation by comparing propagation in a fiber with control values of resting [Ca]i=0.12 μM to a fiber with elevated resting [Ca]i=0.25 μM. Considering that ICa(L) affects propagation when coupling is reduced, we computed the safety factor of each fiber over a full range of intercellular coupling with a protocol similar to that of Fig 7A⇑. We found (not shown) that calcium overload begins to lower SF after a 100-fold reduction in gj and causes conduction failure at slightly higher intercellular conductance (failure occurs at gj=0.007 μs and gj=0.0057 μs at elevated and normal [Ca]i, respectively). Therefore, under extreme conditions, when dependence on ICa(L) to sustain conduction is a major factor, decreasing ICa(L) via elevated [Ca]i will decrease propagation safety.
To confirm that [Ca]i action on ICa(L) is the mechanism of decreased propagation safety under conditions of calcium loading, we examined, in Fig 8⇓, the action potential upstroke and related calcium current that occur at low intercellular coupling (gj=0.008 μs). Two fibers were used, one with normal [Ca]i (solid line) and one with elevated [Ca]i (dashed line). The action potential upstrokes are followed by a dip, which (as recognized from previous figures) is caused by the charging of the adjoining downstream cell. During the dip, ICa(L) reaches a maximum value and then decreases. The maximum values of ICa(L) are coincident with calcium release from the SR and the following decline is due mostly to calcium-dependent inactivation. ICa(L) decreases more for the cell with elevated [Ca]i, reflecting greater calcium-dependent inactivation. Greater inactivation of ICa(L) causes (Fig 8A⇓) faster action potential repolarization, decreased axial driving force, and longer charging period of the adjacent downstream cell. The inset of Fig 8B⇓ shows that the charge from ICa(L) needed to excite the adjoining cell is approximately the same for both fibers. Since ICa(L) is smaller in the overloaded fiber, longer intercellular charging time is needed, leading to decreased sodium channel availability (greater dynamic inactivation) and reduced SF.
This study establishes two types of seemingly paradoxical behavior for propagation of excitation in multicellular cardiac fibers. The first is that as conduction velocity is slowed by decreased gap junction coupling, propagation safety is increased. Conduction in the milieu of decreased coupling (such as in a zone of infarction) is slow but safe. The second is that the functional role of excitation currents (ie, INa and ICa(L)) is determined more by passive structural factors external to the membrane than by intrinsic membrane factors. Decreased intercellular coupling, not decreased sodium channel availability (reduced membrane excitability), results in a major functional role for ICa(L) in propagation. The significance of these behaviors is discussed below.
Slow Conduction and Propagation Safety
Both reduction in membrane excitability and reduction in intercellular coupling reduce conduction velocity. However, the mechanisms by which each pathology causes conduction slowing are fundamentally different. Reduced excitability decreases availability of the depolarizing membrane current, INa, slowing membrane depolarization. Reduced coupling increases intercellular conduction time by limiting (reducing) electrotonic axial current flow. By also decreasing electrical load, decreased coupling confines current to the depolarizing cell, causing an enhancement of its depolarization.
Safety factor data for the two different pathologies are compiled with respect to conduction velocity in Fig 9⇓. The data establish that during conduction slowing decreased coupling increases safety factor for conduction, whereas decreased excitability decreases safety factor. The opposite effects on safety factor are evident from the minimum conduction velocity obtained before block. Reduced excitability can reduce conduction velocity only threefold (to 17 cm/s) before block occurs, whereas uncoupling can reduce conduction velocity to an extremely low value of 0.26 cm/s before conduction fails.
There are differences in the literature as to whether conduction slowing with reduced coupling occurs with higher or lower safety. In a classic study, Spach et al8 found that cardiac conduction in the longitudinal (well-coupled) direction had higher conduction velocity but was less safe than conduction in the transverse (less-coupled) direction. Safety was evaluated by susceptibility to block with premature stimulation. Delgado et al9 observed the opposite, that conduction failed preferentially in the transverse direction. In their study, block was obtained by elevation of extracellular potassium concentration. A possible reason for the disparity between the two studies is the technique used to obtain block. Elevation of extracellular potassium concentration, unlike premature stimulation, alters the potassium reversal potential and depolarizes the resting membrane potential.26 27 As a result, not only do sodium channels inactivate, but distance between resting potential and threshold potential is reduced.27 28 In addition, conduction near block due to elevated extracellular potassium depends on the L-type calcium current to aid in excitation.20 Therefore an evaluation of safety factor based on elevating [K]o may include important contribution from ICa(L) and might differ from the safety factor evaluated by premature stimulation.
It is worthwhile to emphasize the magnitude of the safety factor changes caused by reduced excitability versus reduced coupling. Reduced excitability causes a monotonic reduction in SF, lowering it by 60% (from 1.6 to 1), before causing conduction failure. In contrast, reduced coupling increases SF by 200% before causing a drop in SF and block. This is evidence that, in terms of safety factor for conduction, membrane excitability properties are less influential than passive fiber properties. A slight decrease in coupling could compensate for a major decrease in membrane excitability. For example, the ischemic combination of acidosis (pH=6.5) and hyperkalemia ([K]o=10 mmol/L) decreases resting INa availability by 60%,20 lowering SF from 1.6 to 1.4. Merely decreasing gap junction coupling by a factor of three will compensate for such ischemic decrease in safety. Uncoupling is typically considered to be an attempt of the myocardium to isolate damaged cells.29 However, because reduced coupling improves the safety factor for conduction, a certain degree of uncoupling may be an important tissue-level response to ensure conduction despite depressed membrane excitability.
Roles of INa and ICa(L) in Propagation
Under all conditions other than reduced intercellular coupling, INa is the major ionic current that determines excitability. This includes slow conduction due to depressed membrane. When intercellular coupling is even marginally reduced, we found that conduction relies on ICa(L) as well as INa. Conduction block occurred, with all membrane currents intact, at gap junction conductance gj=0.0057 μs. When ICa(L) availability was reduced to zero, conduction block occurred at gj=0.0197 μs, three times the intercellular conductance that caused block in the presence of ICa(L). The important role of ICa(L) under conditions of decreased coupling is also evidenced by the total charge generated to excite an adjoining cell. At control conditions (gj=2.5 μs) the charge ratio QNa:QCa is 166:1, indicating INa dominance. At gj=0.020 μs, the ratio decreases to 1.47:1, reflecting equal roles of INa and ICa(L) in conduction. The charge ratio decreases further with further decreases in coupling, and at gj=0.0057 μs, QNa:QCa is 0.16:1 indicating ICa(L) dominance and an order of magnitude greater charge contribution by ICa(L) than by INa.
The effect of ICa(L) at low intercellular coupling has been studied in isolated cell pairs (between two real cells, a real cell and a model cell, and two model cells) by Joyner and colleagues.13 30 They determined that if the leader cell is unable to excite a follower cell due to limited intercellular conductance or size related source-load mismatch, ICa(L) enhancement in the leader cell could successfully restore excitation in the follower cell. The converse was also true, ICa(L) inhibition in the leader cell could block otherwise successful follower cell excitation. For all conditions, ICa(L) was effective when intercellular delay was on the order of 5 ms (equivalent to conduction velocity of about 2 cm/s).
In the multicellular fiber, we found that ICa(L) influenced safety factor for conduction when gj was as high as 0.5 μs (one fifth of control), which causes an intercellular conduction delay of only 0.3 ms. As gj decreases and the conduction delay increases, the downstream cell depolarizes when the upstream source cell is “deeper” into the plateau phase of its action potential. At this phase, ICa(L) is the depolarizing current and the major contributor of depolarizing charge to the downstream cell (note that charge involves integrating the current over time). Therefore, with longer intercellular conduction delays, more of the depolarizing charge delivered to the downstream cell is generated by ICa(L). Two- and three-dimensional structural anisotropy as well as nonmyocyte inhomogeneities (eg, connective tissue septae) may cause local conduction delays even when global conduction velocity is not drastically reduced. For instance Spach and Heidlage,31 32 using a detailed two-dimensional cellular model with realistic cell geometry and gap junction distribution, observed microscopic transcellular conduction delays on the order of 1 ms. Rohr and colleagues14 33 34 found, with cultured rat heart cells patterned into a narrow strand opening into a large rectangular area (large load), that ICa(L) block caused conduction failure when activation delay at this opening was in the millisecond range. Conversely, when block of conduction from the small strand to the large area preexisted, ICa(L) enhancement with Bay K 8644 reestablished successful conduction.33 34 Fast and colleagues35 36 37 have observed in cultured rat cell monolayers that nonuniformities on different-size scales (eg, uneven gap junction distribution, connective tissue sheets, nonexcitable vascular cells) may represent predilective sites for conduction block.35 On the basis of these studies and our results, ICa(L) may then be a major component of microscopic propagation and of the safety factor for conduction at these sites even when overall macroscopic conduction velocity is not significantly reduced. ICa(L) is likely to play an important role in any situation when propagation is associated with long local conduction delays (eg, tissue expansion at Purkinje muscle junctions, increase in load on a wave front while circulating around a pivot point during reentry).
There is strong experimental evidence that the electrophysiological characteristics of conduction within a healed infarct zone are the result of reduced intercellular coupling. Fractionated electrograms, observed in infarcted regions,10 38 are associated with separation and altered orientation of myocardial fibers.10 39 In these cases slowed activation and very slow conduction are likely a result of decreased intercellular coupling.38 On the basis of our simulations (Fig 9⇑), the very slow velocities (<1 cm/s) that are measured in healed infarcts and that make reentry in a small region (“microreentry”) possible must involve long local conduction delays and cannot be supported by depressed excitability (which cannot support velocity slower than 17 cm/s). Slow and discontinuous conduction within surviving muscle of epicardial infarcts has been identified as a mechanism of reentrant arrhythmias.40 41 42 43 The newly recognized importance of ICa(L) in highly discontinuous propagation that involves long conduction delays identifies the L-type calcium channel as a possible target for antiarrhythmic therapy under such conditions.
With our discussion focusing on the role of ICa(L), we hasten to emphasize that INa, even with reduced availability, is always required for conduction. ICa(L) activates between Vm=−40 and −30 mV.16 INa is still needed to bring the membrane potential to within the range of ICa(L) activation. [K]o elevation, and consequent depolarization of resting membrane potential, is the only scenario we know in which ICa(L) may be the sole excitatory current.20
Intracellular Calcium Transients and Propagation
Through the calcium-induced calcium release process, excitation-contraction coupling involves coupling between sarcolemmal calcium flux via ICa(L) and intracellular calcium transients. That is, intracellular calcium responds to sarcolemmal currents. In the reverse direction, ICa(L) and other currents (eg, INaCa, IKs) are modulated by [Ca]i. The simulations of this study indicate a possible functional role for ICa(L) modulation by [Ca]i in propagation of the cardiac action potential.
The L-type calcium current can be inactivated by free intracellular calcium ions.44 Sipido et al45 have shown in guinea pig cells that ICa(L) inactivation is proportional to [Ca]i. Grantham and Cannell46 found in guinea pig cells (using the ICa(L) formulation from the LRd model) that ICa(L) behavior during an action potential cannot be reconstructed correctly without calcium-dependent inactivation. Lüscher et al47 showed that replacement of calcium ions by strontium ions (also a divalent positive charge carrier) prevented action potential conduction failure in dorsal root ganglion cells, suggesting that [Ca]i-dependent inactivation of ICa(L) can affect conduction (strontium causes minimal inactivation of ICa(L)). More recently, the same team15 found that flash photolytic liberation of a calcium buffer, during trains of action potentials, which partly failed to invade the cell body, immediately lowered intracellular [Ca]i and restored safe action potential propagation. Also, the velocity of the propagated action potential was reduced when intracellular calcium was increased by flash photolysis.15
Our results demonstrate that elevated intracellular calcium can both decrease conduction velocity and lower the safety factor for conduction during highly discontinuous ventricular propagation, which depends on ICa(L). Doubling the free intracellular calcium concentration (to 0.25 μmol/L) resulted in conduction failure at gj≤0.007 μs, 23% higher than the conductance at which it otherwise occurred. Although, for the uniform fiber, reduction of gap junction coupling by two orders of magnitude is needed before the effects of [Ca]i on propagation are apparent, the principle of [Ca]i modulated conduction may have much greater significance in conduction across tissue inhomogeneities where large source-load mismatches exist and cause very long conduction delays.34 By studying the direct response of ICa(L) to changes in [Ca]i, we established that the ionic mechanism of [Ca]i modulated propagation is, as suggested by Lüscher et al,15 47 calcium-dependent inactivation of ICa(L).
The study emphasizes that reduced membrane excitability (eg, during acute ischemia) cannot support very slow conduction and is mostly associated with conduction failure. In contrast, slow conduction due to gap junction decoupling is safe, and very slow velocities can be supported by such structural changes (eg, due to infarction).
Our results suggest that the passive myocardial structure has a major effect on the role of membrane currents in propagation of the cardiac action potential. We find it striking that a nonmembrane change (ie, decreased intercellular coupling) can cause the membrane to switch to a different process (calcium current instead of sodium current) as the major mechanism for supporting conduction. A control fiber relies on INa to sustain conduction and, except for extreme circumstances, a fiber with depressed membrane excitability still relies on INa for action potential conduction. However, reduced gap junction coupling (a nonmembrane structural change) changes the dynamics of membrane ionic currents during the action potential and switches the membrane to an ICa(L)-dependent mode of operation. This finding illustrates the strong bidirectional interaction between the membrane (source) and passive myocardial structure (sink) during action potential propagation.
In addition to its direct feedback influence on membrane dynamics, reduced gap junction coupling modulates the effects of interaction between intracellular ionic processes (eg, the calcium transient) and membrane ionic currents (eg, ICa(L)). In this study, [Ca]i-dependent inactivation of ICa(L) was shown to influence propagation only when long conduction delays at gap junctions were involved (the condition for ICa(L) involvement in sustaining conduction). This example emphasizes even more the highly interactive nature of action potential propagation in cardiac tissue.
The fact that ICa(L) plays an important role in conduction only where propagation is very slow and highly discontinuous is encouraging from the perspective of antiarrhythmic drug therapy. Such conduction requires a pathological substrate (eg, infarcted region), implying that conduction depends on ICa(L) only where the pathology exists. In healthy myocardium with normal intercellular coupling, ICa(L) is not needed to maintain propagation of excitation. With appropriate levels of universally applied ICa(L) block, conduction will be modified in the vulnerable infarcted region only, without affecting the normal myocardium, thereby ensuring specific drug action. In the same context of antiarrhythmic drug therapy, the study demonstrates that the intervention should consider the underlying pathophysiological state of the myocardium. For example, slow conduction during most phases of acute ischemia involves only INa, whereas slow conduction in a healed infarct involves long local delays and therefore ICa(L). Consequently, the targets for intervention (drug treatment, genetic modification) might be very different in treating propagation-related arrhythmias (eg, reentry) in the context of different pathophysiological conditions.
Spatial computational discretization and distribution of fiber resistivity require two separate considerations, one of numeric accuracy (convergence) and one of anatomic correctness. For a continuous fiber, numeric accuracy requires that the spatial discretization element (Δx) be no longer than one tenth of the fiber space constant so that the entire element is essentially equipotential.7 However, a ventricular fiber is discontinuous, consisting of intracellular myoplasm of low resistivity separated by intercellular gap junctions of (comparatively) high resistivity. Under normal coupling conditions, resistance over the entire 100 μm of myoplasm is equal to the resistance across the gap junction,48 49 which occupies no more than 80 Å. To select a Δx that is both numerically accurate and reflects the discontinuous structure of a cardiac fiber, we computed key physiological parameters for different values of gap junction resistance (Rg) at different levels of spatial discretization (Δx). Results are shown in the Table⇓, which is a compilation of peak membrane potential (Vm,peak), maximum upstroke velocity ([dVm/dt]max), action potential duration at 90% repolarization ([APD]90), and conduction velocity (θ), all computed for an action potential traveling down a 7-mm fiber. Simulations were repeated for three types of fiber discretization: (1) a discontinuous fiber of 21 computational elements per cell length (Δx=4.76 μm) with Rg confined to the edge elements of the cell (representing the local gap junction resistance); this high level of discretization ensures convergence and provides a gold standard for propagation in a discontinuous fiber; (2) a continuous fiber also discretized to 21 elements/cell length (Δx=4.76 μm) but with Rg evenly distributed across the entire cell; this provides an accurate model of propagation in a continuous fiber; and (3) a continuous fiber discretized to computational elements equal to the entire cell length (Δx=100 μm) with Rmyo and Rg lumped together at each discretization point. All simulations were performed with implicit Crank Nicolson24 solution of the cable equation. When the fiber is discontinuous, the partial differential equations are modified for the prejunctional and postjunctional nodes to match boundary conditions and account for Rg. The Crank Nicolson solution and discretization of a discontinuous fiber are detailed clearly in Reference 6. The computed parameter values are shown together with percent error relative to the truly discontinuous, highly discretized (21 elements/cell) fiber. Cellular data were computed at the middle element of cell 35 in the discontinuous fiber and at the same location in the continuous fibers. Conduction velocity was computed from the time difference between the occurrence of maximum upstroke velocity in cells 20 and 50.
The Table⇑ reveals that for the highly discretized discontinuous fiber, (dVm/dt)max (the most sensitive parameter of computational accuracy) increases with increasing intercellular resistance (Rg), whereas for the highly discretized continuous fiber, (dVm/dt)max is constant. Conduction velocity for both fibers decreases with increasing Rg. These results are expected [in the continuous fiber, eg, an axon, (dVm/dt)max is determined only by the membrane excitability; in the discontinuous fiber, cellular current confinement due to gap junctions acts to modify (dVm/dt)max5 ] and establish the necessity of using a discontinuous fiber model for simulating propagation in the multicellular myocardial tissue. A very important observation, based on the data in the Table⇑, is that the fiber that is formulated as continuous but with a large Δx=100 μm (equal to the entire cell length of the discontinuous fiber) approximates very closely the accurate discontinuous fiber with 21 elements per cell (Δx=4.76 μm). The greatest error of computed parameters is for (dVm/dt)max at Rg=1.5 Ωcm2, which differs by <1% (actual error is 0.61%) from the accurate value obtained with the highly discretized discontinuous fiber.
The reason that Δx=100 μm (the entire cell length) is an adequate discretization is different for simulating discontinuous propagation at different values of gap junction coupling resistance. When Rg is very small (Rg≅0), the fiber is effectively continuous with large axial conductance (large space constant). Under such conditions, Δx=100 μm is approximately one tenth of the space constant and serves as an adequate discretization that insures convergence and accuracy. When Rg is relatively high (ie, Rg ≥5 Ωcm2), the gap junctions are the primary sites of potential changes and the entire cell becomes effectively isopotential. Under these conditions, Δx=100 μm (the cell) is accurate because the requirement that the discretization element be isopotential is fulfilled. The greatest error for the Δx=100 μm discretization is expected in the middle range of Rg, when Rg is large enough to reduce the space constant (λ) so that Δx>λ/10, yet it is not so high that the cell becomes isopotential. The greatest error within this region (0.61% at Rg=1.5 Ωcm2) is small enough for accurate computation. For this reason we use Δx=100 μm in our simulations (it translates into tremendous savings in computing time, which is 5% of that for the highly discretized fiber with 21 elements per cell). When knowledge of variation of dVm/dt and other parameters within the cell is required, Δx is reduced to subdivide the cell with desired resolution. Otherwise Δx=100 μm is an adequate discretization for the multicellular discontinuous cardiac fiber. All our simulation protocols were checked for accuracy and convergence with 21 elements per cell but in general we use Δx=100 μm, with Ri reflecting the lumped contribution of axial and gap junction resistance (Ri=Rmyo+Rg/Δx).
Our definition of safety factor is based on the net membrane charge (Qm) that is generated over the course of an action potential for any cell in the fiber. Qm quantifies a balance between charge produced and charge consumed by the cell membrane. When Qm is positive, the cell membrane has consumed more charge than it has produced, constituting a charge sink for the fiber. When Qm is negative, the membrane has functioned as a net charge source for the fiber.
Over the course of an entire action potential, a cell membrane alternates between being a sink and a source. The currents involved and the changes in Qm are illustrated in Fig 10⇓. As a propagating action potential approaches a cell (from the left in Fig 10A⇓), a positive voltage gradient is established between the cell and its upstream neighbor. The voltage gradient causes axial current flow, Iin, into the cell. A smaller gradient is formed between the cell and its downstream neighbor, causing axial current, Iout, to leave the cell. Since the system is conservative, the difference between Iin and Iout must cross the membrane as the transmembrane current, Im, which charges the membrane capacitance to cause depolarization. The charge, Qm, consumed by the membrane is given by the time integral of Im. As the membrane continues to depolarize, contribution to Im from axial current decreases. This decrease is due to increased voltage gradient between the cell and its downstream neighbor and decreased voltage gradient between the cell and its upstream neighbor, acting to increase Iout and decrease Iin and, consequently, to decrease Im=Iin−Iout. At the same time, inward membrane ionic currents (ie, Iion, mainly from the fast sodium current) activate and become the major source of depolarizing current. As a result, total transmembrane current, Im, becomes negative causing Qm (the integral of Im) to decrease. In Fig 10B⇓, this is indicated by a decreasing Qm beyond the midpoint of the upstroke. The second half of the upstroke can be viewed as the period during which the membrane replaces the charge it initially consumed to reach its activation threshold. When Qm returns to zero, which occurs near peak membrane potential (Fig 10B⇓), the membrane has restored the charge it consumed and represents neither a net sink nor a net source to the fiber.
The return of Qm to zero is an appropriate time to analyze the margin of safety because it marks the instant that membrane excitation is complete and the membrane is no longer in debt. The total charge the cell consumed from the fiber is the time integral of Iin, or Qin. The total charge that the cell contributed to the fiber is the time integral of Iout, or Qout. A third important charge is the capacitive (potential) energy stored in the membrane as a result of excitation, energy that will be contributed to the fiber as the membrane repolarizes back to its rest potential. The capacitive charge of the membrane, Qc=Cm · (Vm−Vrest), is equal to the time integral of the membrane capacitive current, Ic. Therefore, safety factor, which is a ratio of total charge produced (Qout+Qc) to charge consumed (Qin), is formulated as (evaluated from the start of membrane depolarization to the point when Qm returns to zero, ie, over the domain Qm>0).
In formulating the safety factor of propagation, we borrowed heavily from two definitions that exist in the literature.9 12 Delgado et al9 defined safety factor as the total axial charge provided to a cell divided by the axial charge necessary to sustain propagation. Their safety factor is therefore based on the margin of extra charge provided to a cell, after a threshold amount has been provided: where Iin is the axial current into the cell, as defined earlier. SFDel is a quantitative formulation of the intuitive understanding of safety factor (the fraction of charge delivered to the cell beyond the membrane threshold is the propagation safety). However, there are two difficulties when computing safety factor with this definition. The first difficulty is with the integration limit of the denominator. Inward charge is computed from the beginning of axial current flow to “threshold.” Threshold is determined by severing the fiber at the junction between the cell in question and its upstream neighbor at different time points until the earliest time is found at which the severed fiber can sustain conduction. A difficulty with this concept of “threshold” is that in a multicellular fiber the cell never functions independently. Propagation after severing the fiber and forming a sealed end will have different velocity and conduction parameters than propagation in the intact fiber.
A second more significant difficulty with SFDel is that it does not account for axial charge that leaves the cell to downstream fiber (Iout). By not accounting for Iout, SFDel implicitly assumes that all axial current that enters the cell remains in the cell and charges the local membrane. This assumption is not valid under most conditions. If, for instance, membrane excitability is reduced, the slope of the action potential upstroke will be slower below and above threshold. Greater charging time will cause both numerator and denominator of SFDel to increase, the numerator more so because it integrates Iin over a longer time period. The result will be a nonphysiological increase in SFDel with decreased membrane excitability. This can be seen in Fig 10C⇑ (dotted line). Increased safety factor with decreased excitability is indicative that the propagation safety is not computed correctly.
The other formulation of safety factor for multicellular cardiac propagation was developed by Leon and Roberge.12 These authors took a local, cellular approach to safety factor that consisted of the ratio of charge generated by inward Iion (Qion) to total membrane charge generated by inward Im (Qm). The mathematical formulation of the Leon and Roberge safety factor is: The middle and right hand sides of SFLeon indicate the equivalence between total transmembrane current (Im) and the sum of ionic and capacitive currents (Iion+Ic). Therefore, SFLeon is based on the efficient use of local ionic current (Iion) for local depolarization (Ic). Safety factor for a segment of membrane will increase, due to a smaller denominator, when a large portion of its ionic charge is used for membrane depolarization. For an isolated cell, load effects are zero, (Iion=−Ic), and SFLeon is infinite.
A difficulty with SFLeon is the integration limit of the denominator. Im is integrated only when it is negative, or inward. Im becomes inward after excitation threshold is reached, when Iion is greater than the requirements of depolarization. By computing Im only above threshold, SFLeon does not account for the charging requirements of the membrane before reaching threshold. Like SFDel, this limitation can cause error in the computed safety factor. For instance, when membrane excitability is reduced, Iion as a source of fiber charge is reduced. When Iion is reduced, the integral of Iion+Ic (the denominator of SFLeon) is reduced by a greater fraction than the integral of Iion alone (the numerator of SFLeon) because, in general, the same absolute reduction (reduced Qion) of a smaller value (Qm=Qion+Qc; Qion and Qc are of opposite sign) causes a greater fractional reduction of this value. As a result, SFLeon will increase when membrane excitability is reduced. This behavior can be observed for an actual simulation (Fig 10C⇑, solid line), where, similar to SFDel, SFLeon increases with decreasing membrane excitability.
In summary, SFDel does not account for downstream loading effects while SFLeon does not account for charge supplied by upstream fiber. The definition of SF developed here takes these factors into account. The response of our formulated SF to changes in excitability is shown in Fig 4⇑ and discussed in the corresponding text. That SF decreases with reduction in membrane excitability and drops to unity just before propagation fails support the validity of its formulation. More detailed discussion of SF behavior during changes in membrane excitability and gap junction coupling are presented in the “Results” and “Discussion” sections of this report.
This work was supported by the National Institutes of Health grants HL-49054 and HL-33343 (National Heart, Lung, and Blood Institute).
Previously published in abstract form (Biophys J. 1997;72:A110).
- Received February 13, 1997.
- Accepted August 21, 1997.
- © 1997 American Heart Association, Inc.
Veenstra RD, Joyner RW, Wiedmann RT, Young ML, Tan RC. Effects of hypoxia, hyperkalemia, and metabolic acidosis on canine subendocardial action potential conduction. Circ Res.. 1987;60:93-101.
Bigger JT Jr, Hoffman BF. Antiarrhythmic drugs. In: Gilmam AG, Rall TW, Nies AS, Taylor P, eds. The Pharmacological Basis of Therapeutics. New York, NY: Pergamon Press; 1990:840-873.
Rudy Y, Quan WL. A model study of the effects of the discrete cellular structure on electrical propagation in cardiac tissue. Circ Res.. 1987;61:815-823.
Joyner RW. Effects of the discrete pattern of electrical coupling on propagation through and electrical syncytium. Circ Res.. 1982;50:192-200.
Spach MS, Miller WT III, Geselowitz DB, Barr RC, Kootsey JM, Johnson EA. The discontinuous nature of propagation in normal canine cardiac muscle: evidence for recurrent discontinuities of intracellular resistance that affect membrane currents. Circ Res.. 1981;48:39-54.
Delgado C, Steinhaus B, Delmar M, Chialvo DR, Jalife J. Directional differences in excitability and margin of safety for propagation in sheep ventricular epicardial muscle. Circ Res.. 1990;67:97-110.
de Bakker JM, van Capelle FJ, Janse MJ, Wilde AA, Coronel R, Becker AE, Dingemans KP, van Hemel NM, Hauer RN. Reentry as a cause of ventricular tachycardia in patients with chronic ischemic heart disease: electrophysiologic and anatomic correlation. Circulation. 1988;77:589-606.
Spach MS, Dolber PC, Heidlage JF, Kootsey JM, Johnson EA. Propagating depolarization in anisotropic human and canine cardiac muscle: apparent directional differences in membrane capacitance: a simplified model for selective directional effects of modifying the sodium conductance on Vmax, tau foot, and the propagation safety factor. Circ Res.. 1987;60:206-219.
Leon LJ, Roberge FA. Directional characteristics of action potential propagation in cardiac muscle: a model study. Circ Res.. 1991;69:378-395.
Lüscher C, Lipp P, Lüscher HR, Niggli E. Control of action potential propagation by intracellular Ca2+ in cultured rat dorsal root ganglion cells. J Physiol. 1996;490.2:319-324.
Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential, I: simulations of ionic currents and concentration changes. Circ Res.. 1994;74:1071-1096.
Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential, II: afterdepolarizations, triggered activity, and potentiation. Circ Res.. 1994;74:1097-1113.
Zeng J, Laurita KR, Rosenbaum DS, Rudy Y. Two components of the delayed rectifier K+ current in ventricular myocytes of the guinea pig type: theoretical formulation and their role in repolarization. Circ Res.. 1995;77:140-152.
Shaw RM, Rudy Y. Electrophysiologic effects of acute myocardial ischemia: a mechanistic investigation of action potential conduction and conduction failure. Circ Res.. 1997;80:124-138.
Weidmann S. The electrical constants of Purkinje fibers. J Physiol Lond.. 1952;118:348-360.
Hodgkin AL, Rushton WAH. The electrical constants of a crustacean nerve fibre. Proc R Soc B.. 1946;133:444-479.
Jack JJB, Noble D, Tsien RW. Electric Current Flow in Excitable Cells. Oxford, UK: Clarendon Press; 1975:502.
Sewall G. The numerical solution of ordinary and partial differential equations. Boston, Mass: Academic Press/Harcourt Brace Jovanovich; 1988:271.
Harris AS, Bisteni A, Russel RA, Brigham JC, Firestone JE. Excitatory factors in ventricular tachycardia resulting from myocardial ischemia: potassium a major excitant. Science. 1954;119:200-203.
Dominguez G, Fozzard HA. Influence of extracellular K+ concentration on cable properties and excitability of sheep cardiac Purkinje fibers. Circ Res.. 1970;26:565-574.
Gettes LS, Buchanan JW Jr, Saito T, Kagiyama Y, Oshita S, Fujino T. Studies concerned with slow conduction. In: Zipes DP, Jalife J, eds. Cardiac Electrophysiology and Arrhythmias. Orlando, Fla: Grune & Stratton; 1985:81-87.
Page E. Cardiac gap junctions. In: Fozzard HA, Jennings RB, Haber E, Katz AM, Morgan HE, eds. The Heart and Cardiovascular System. New York, NY: Raven Press, Publishers; 1992:1003-1047.
Spach MS, Heidlage JF. The stochastic nature of cardiac propagation at a microscopic level: electrical description of myocardial architecture and its application to conduction. Circ Res.. 1995;76:366-380.
Rohr S, Kucera JP. The calcium inward current can play a critical role for the success of impulse propagation across abrupt expansions of cardiac tissue in the presence of the sodium inward current. Circulation 1995;92(suppl I):I-432. Abstract.
Rohr S, Kucera JP, Fast VG, Kleber AG. Paradoxical improvement of impulse conduction in cardiac tissue by partial cellular uncoupling. Science.. 1997;275:841-844.
Fast VG, Darrow BJ, Saffitz JE, Kléber AG. Anisotropic activation spread in heart cell monolayers assessed by high resolution optical mapping: role of tissue discontinuities. Circ Res.. 1996;79:115-127.
Fast VG, Kleber AG. Microscopic conduction in cultured strands of neonatal rat heart cells measured with voltage-sensitive dyes. Circ Res.. 1993;73:914-925.
Fast VG, Kleber AG. Anisotropic conduction in monolayers of neonatal rat heart cells cultured on collagen substrate. Circ Res.. 1994;75:591-595.
Ursell PC, Gardner PI, Albala A, Fenoglio JJ Jr, Wit AL. Structural and electrophysiological changes in the epicardial border zone of canine myocardial infarcts during infarct healing. Circ Res.. 1985;56:436-451.
Gardner PI, Ursell PC, Fenoglio JJ Jr, Wit AL. Electrophysiologic and anatomic basis for fractionated electrograms recorded from healed myocardial infarcts. Circulation.. 1985;72:596-611.
El-Sherif N, Scherlag BJ, Lazzara R, Hope RR. Reentrant ventricular arrhythmias in the late myocardial infarction period, I: conduction characteristics in the infarction zone. Circulation.. 1977;55:686-702.
Wit AL, Allessie MA, Bonke FI, Lammers W, Smeets J, Fenoglio JJ Jr. Electrophysiologic mapping to determine the mechanism of experimental ventricular tachycardia initiated by premature impulses: experimental approach and initial results demonstrating reentrant excitation. Am J Cardiol.. 1982;49:166-185.
Mehra R, Zeiler RH, Gough WB, El-Sherif N. Reentrant ventricular arrhythmias in the late myocardial infarction period, IX: electrophysiologic-anatomic correlation of reentrant circuits. Circulation.. 1983;67:11-24.
Dillon SM, Allessie MA, Ursell PC, Wit AL. Influences of anisotropic tissue structure on reentrant circuits in the epicardial border zone of subacute canine infarcts. Circ Res.. 1988;63:182-206.
Sipido KR, CalleWaert G, Carmeliet E. Inhibition and rapid recovery of Ca2+ current during Ca2+ release from sarcoplasmic reticulum in guinea pig ventricular myocytes. Circ Res.. 1995;76:102-109.
Grantham CJ, Cannell MB. Ca2+ influx during the cardiac action potential in guinea pig ventricular myocytes. Circ Res.. 1996;79:194-200.
Lüscher C, Streit J, Quadroni R, Lüscher HR. Action potential propagation through embryonic dorsal root ganglion cells in culture, I: influence of cell morphology. J Neurophys.. 1994;72:622-633.
White RL, Spray DC, Campos de Carvalho AC, Wittenberg BA, Bennett MVL. Some electrical and pharmacological properties of gap junctions between adult ventricular myocytes. Am J Physiol.. 1985;249:C447-C455.