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IEEE TRANSACTIONS OU BIOMEDICAL ENGINEERING. VOL. 41, NO, 5, MAY 1994

Electrical Coupling and Impulse Propagation in Anatomically Modeled Ventricular Tissue

Barbara J. Muller-Borer, Donald J. Erdman and Jack W. Buchanan

Abstract-Computer simulations were used to study the role of resistive couplings on flat-wave action potential propagation through a thin sheet of ventricular tissue. Unlike simulations using continuous or periodic structures, this unique electrical model includes random size cells with random spaced longitudi- nal and lateral connections to simulate the physiologic structure of the tissue. The resolution of the electrical model is ten microns, thus providing a simulated view at the subcellular level. Flat- wave longitudinal propagation was evaluated with an electrical circuit of over 140,000 circuit elements, modeling a 0.25 mm by 5.0 mm sheet of tissue. An electrical circuit of over 84,000 circuit elements, modeling a 0.5 mm by 1.5 mm sheet was used to study flat-wave transverse propagation. Under normal cellular coupling conditions, at the macrostructure level, electrical conduction through the simulated sheets appeared continuous and directional differences in conduction velocity, action potential amplitude and T ;,,,, were observed. However, at the subcellular level (10 pm) unequal action potential delays were measured at the longitudinal and lateral gap junctions and irregular wave- shapes were observed in the propagating signal. Furthermore, when the modeled tissue was homogeneously uncoupled at the gap junctions conduction velocities decreased as the action potential delay between modeled cells increased. The variability in the measured action potential was most significant in areas with fewer lateral gap junctions, i.e., lateral gap junctions between fibers were separated by a distance of 100 pm or more.

I. INTRODUCTION

INCE THE late 1970’s experimental research has provided S evidence that the distribution of electrical couplings be- tween cells may contribute to changes in conduction velocities, conduction pathways and waveshape [ 11-15]. Furthermore, recent experimental measurements by Israel et al. [6] report that conduction disturbances in cardiac tissue may take place on a microscopic scale at the level of individual intercel- lular connections. The present simulation study examines the consequence of cellular uncoupling on impulse prop- agation in ventricular tissue using an anatomically based, 2-Dimensiona1, electrical network model.* Similar periodic networks have been used to study action potential propagation in a modeled tissue sheet. Yet, discrete resistive changes in cell couplings at the microstructure level have not been addressed

Manuscript received October 19. 1992; revised January 2.5. 1994. B. J . Muller-Borer i s with the Division of Cardiology, University of North

D. J. Erdman is with the SAS Institute Inc., Cary, NC 275 13 USA. J. W. Buchanan was with the Departments of Medicine and Biomedical

Engineering, University of North Carolina at Chapel Hill. He is now with the Departments of Biomedical Engineering, Medicine and Physiology, University of Tennessee, Memphis, TN 38 163 USA.

Carolina at Chapel Hill, Chapel Hill. NC 27599-7075 USA.

IEEE Log Number 9401317. *Portions of this work have been presented in preliminary thnn [ 101

in a 2-Dimensional structure. Furthermore, other 1- and 2- Dimensional models do not include anatomic geometries such as random cell length and random placement of transverse cellular interconnections. It is possible that the random struc- ture plays an important role in the generation of conduction disturbances not observed in periodic or continuous structures.

The goals of this project were twofold. The first was to construct a network of a physiologically significant size, i.e., the experimental papillary tissue preparation [7 ] , [8], with discrete units based on the anatomic cellular microstructure 191. Using the circuit analyzer with macromodeling (CAzM) on a Convex I1 mini-supercomputer, an electrical network model with over 140,000 circuit elements was studied. For this analysis, nonlinear components modeling the tissue’s mem- brane response were designed. Random cell length and the random placement of intercellular connections were preserved. The second goal was to evaluate the role of the magnitude of the resistive couplings in the modeled ventricular tissue within physiologic bounds and determine their influence on longitudinal and transverse electrical conduction at both the macro and micro structure level.

11. METHODS

A . Two-Dimensional Network

An electrical network models a uniformly anisotropic sheet of ventricular myocardium. A compartmental approach de- fines discrete membrane elements of the tissue in I O pm units. The top portion of Fig. 1 illustrates the basic network structure used to model the tissue, with variable length cells and random spaced interconnections between modeled fibers. The electrical network is assembled from identical, excitable membrane elements (m) , 10 pm in length. The membrane elements connect longitudinally through passive resistive el- ements ( T , , , ~ ~ ) to form random length modeled cells. The length of the modeled cells range from 30 /Lm to 130 pm, normally distributed, with a constant diameter of 10 pm. This size is consistent with experimental measurements reported by Sommers and Scherer [SI. The random length modeled cells connect longitudinally, through a resistive element (rgap) oriented parallel to the longitudinal axis of the modeled cardiac myocyte forming a modeled fiber of cells. Each modeled cell is connected laterally to an adjacent modeled cell through a resistive element (,rgap) forming a modeled tissue sheet as shown in Fig. I . The lateral gap junctions are spaced at random intervals ranging from 30 pm-I 30 pm, normally distributed, along the fibers. In defining the tissue structure

00 I X-9?94/94$04.00 0 1994 IEEE

446 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 41, NO. 5, MAY 1994

Basic Nefwork Slructure

Modeled Fiber Sheet \

Fig. 1. This figure illustrates the modeled ventricular tissue network, with the network components for the membrane element ( n z ) , myoplasmic resistance ( T ~ ~ , ~ , , ) , and gap junction interconnections (rgap). Note the variable cell length and spacing of lateral connections (top). The bottom figure is an example of a modeled tissue sheet of 25 parallel fibers with 500 cellular elements per fiber.

and placement of the lateral gap junctions, the modeled cells in fiber 1 are connected to the modeled cells in fiber 2, the modeled cells in fiber 2 are connected to the modeled cells in fiber 3 , etc., to form the modeled tissue sheet. Each modeled cell is connected to a minimum of three adjacent cells (i.e., two longitudinal connections and one lateral connection). With such connectivity, the lateral and longitudinal gap junction resistances of the network are quantitatively and qualitatively equal. By connecting the elements as described, two sheets of modeled tissue are created. Longitudinal flat-wave propagation is evaluated using a sheet of 25 fibers with 63 f 3 cells per fiber (i.e., a 0.25 mm by 5.0 mm sheet of 12,500 membrane elements). A sheet of 50 fibers with 20 2z 2 cells per fiber is used to evaluate transverse flat-wave propagation (i.e., a 0.5 mm by 1.5 mm sheet of 7,500 membrane elements). C code utilizing a pseudorandom number generator routine defines the network and electrical connections within the specified bounds (i.e., the cell length and placement of lateral connections fall within a normal distribution). The boundary elements of the network (i.e., all cells at the transverse and longitudinal edges) are connected to ground through a resistive element (rmyo). The nominal model parameters are given in Table I with the resistive and capacitive values representing physiologic measurements reported in the literature [7], [SI, [ I l l , [12], 1131.

B. Membrane Element Puramefers A mathematical model describing the ventricular electrical

activity defines the electrical configuration of each membrane element in the network. For each membrane element an electrical device for the time and voltage dependent ionic currents, sodium (Na+), calcium (Ca2+) and potassium (K+), and the time dependent ionic current, potassium (Kx), are created. Fig. 2 illustrates the general circuit for each membrane element with an example of a device circuit. The current into each membrane element (Il,>) is equal to the total ionic current

Iin

I I I I I

Iout a. G d

.?4JG b. c

Fig. 2. (a) Schematic of the membrane element with four ionic devices, modeling the ionic response. (b) Example of a multitenninal ionic device (Z) with three gating parameters (y1, y2, y3) which define the ionic response following a stimulus. (c) A four terminal device representing the gating parameter.

TABLE I

Intracellular Resistivity (rhomy,) 209 I? cm Intracellular Resistance (rlnyo) **266 kQ Gap Junction Resistance (rsap) 1.331 Mf2 Membrane Capacitance (Cnl) 1 .O pF/cm2

Sodium Equilibrium Potential ( V N ~ ) Sodium Conductance ( g ~ ~ ) 30.0 mS/cm2 Calcium Equilibrium Potential (V,,)* Calcium Conductance (gc.,) 0.09 mS/cm’

40.0 mV

Membrane Element Length Membrane Element Diameter Modeled Cell Length

10 p m 10 fim 30 pm-130 [tm

Network 1 NPtwnrk 2

0.25 mm x 5.0 mm 0.50 mm x 1.5 mm

*determined by a series of equations as defined by Beeler-Reuter [15] **scaled to the 10 pm membrane element

(I;) plus the capacitive current (Icm) as represented in (1)-(3).

(1)

Icm = C,dVm/dt. ( 3 )

I in = 1; + I c m IL = (INa+) + (ICa2+) f ( I K + ) f (Ik’x) (2 )

Where cm is the membrane capacitance, V, is the membrane voltage and t is time. The ionic responses of the cell membrane modeled in the electrical circuit are defined by modified Beeler-Reuter [ 141 mathematical equations with the equations for the sodium response, established by Drouhard and Roberge [15]. The ionic changes of the membrane define the transient electrical response described by (4)-(7).

.

MULLER-BORER et al.: ELECTRICAL COUPLING AND IMPULSE PROPAGATION 441

V N ~ , the sodium equilibrium voltage and gNa and gca, the conductances for the respective ions are constants. Vi,, the calcium equilibrium voltage, is dependent on the membrane voltage which changes during the action potential response. The nonlinear response of the sodium, calcium and potassium ions are determined by the dimensionless gating parameters for activation (m, d , kl) and inactivation ( h , j , f ) which range between zero and one. These parameters are described by first order differential equations based on the formalism originally proposed by Hodgkin and Huxley [ 161. The general equation used to determine the value of these parameters is

dYldt = ay + Y(Py - a y ) . (8)

Where alpha ( a ) and beta (p) are rate coefficients derived from experimental measurements. These coefficients are dependent on the membrane potential and correspond to the opening and closing of the gates.

The circuit representation for each membrane element is a collection of devices connected at nodes as shown in Fig. 2(a). The general device for the membrane element is defined mathematically as

dq("J1, u 2 , . . ."Jn)/dt + f(V1.2 '2 . . . . % I n ) = i('UI,712,. . ."Jn)

(9) with y the charge component, f the steady state or resistive component, v the node voltages and 2 the current associated with the device. An example of a circuit used to model the nonlinear ionic response is shown in Fig. 2(b). Each gating parameter is translated into a four terminal device as shown in Fig. 2(c). The equation for each four terminal device takes the form

(10) csdyi/dt + f ( ~ i > K. V z ) 1 0

f ( ~ l , V , , , V z ) =Cs[~ly l (Vm~Vz)(1-~1) -Oy,(vm,Vz)~l]

where

(11) and C, is a scaling factor (1 nF). This circuit has current flowing through two of its edges. The current at terminal y1 is f(y1, b k , Vz) and the current at ground is - f (y l , Vm. Vz). No charge is associated with this device, i.e., q is zero. The equation for the capacitor is

CsdVm/dt = tc,. (12)

A four terminal device for each gating parameter is connected as shown in Fig. 2(b) to construct a multiterminal nonlinear de- vice. This method is used to construct multiterminal nonlinear devices for the sodium, calcium and potassium ions.

C. Circuit Simulator

The determination of this mathematical model requires the simultaneous solution of a coupled system of differential equa- tions. CAzM a general purpose circuit simulator, provides the

speed and accuracy necessary to accommodate the goals of this study. The simulator requires that the circuit be represented in the form of a system of differential algebraic equations, (9), described by Erdman [ 171. As previously described, each ionic device is written as a subcircuit that becomes a part of the total membrane circuit. The capability of the circuit simulator to establish each ionic response as a special device within the membrane circuit provides the flexibility to change constants within the ionic equations and network structure (i.e., equilibrium voltages and conductance values) without recompiling CAzM for each simulation. CAzM uses a variable time step during integration to provide a small At s) during depolarization and larger At (10-1 s) during repolar- ization of the action potential. However, at any given time during the simulation the minimum time step is used, e.g., if one area of the network is depolarizing while another area is repolarizing the smaller time step of integration is used to determine the voltage. During depolarization this process is CPU and memory intensive, however, during repolarization larger time steps are used, reducing the memory and CPU requirements.

A 520 ms simulation of 140,000 elements requires 4.7 CPU hours on a Convex I1 for a network with nominal resistive couplings and increases to 5.4 CPU hours when the coupling resistances are increased.

D. Simulations

Simulations of the propagating action potential response resulting from a current stimulus are evaluated for conditions of nominal cell coupling and cellular uncoupling when a stimulus is delivered a) at the transverse edge (Network 1) and b) along the longitudinal edge (Network 2) as shown in Fig. 3. A 190.0 nA current stimulus is delivered for 1.0 ms to initiate flat-wave longitudinal propagation in Network 1. The circular stimulus area is 120 pm in diameter (i.e., 12 membrane elements), centered equidistant from the lateral boundaries and 1.0 mm from the proximal boundary. The dimensions of the stimulus were selected to match the size of microelectrodes used experimentally [18]. The shaded areas in Fig. 3 denote the areas in each network where steady state propagation was observed and well within bounds to avoid stimulus artifact and edge effects. The measurement area in Network 1 was approximately one space constant beyond the stimulus site (1.2 mm) and a space constant away from the distal boundary (1.0 mm). An increase in conduction velocity and action potential amplitude (APA) observed in regions 0.5 mm from the distal and proximal ends of Network 1 were defined as edge effects. To initiate flat-wave transverse propagation the same current stimulus is delivered along the sixth fiber at cellular elements 15 to 135 (120 membrane elements) of Network 2. The shaded area of Network 2 is approximately 0.12 mm from the stimulus sight and 0.1 mm from the distal boundary. Increased conduction velocity and action potential amplitude observed in fibers 1-3, and fibers 47-50 of Network 2, were defined as edge effects. For each simulation:

1) Vmax, measured in V/s, records the maximum change in the voltage during the upstroke of the action potential.

448 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 41, NO. 5, MAY 1994

a. Network 1 Direction of Action Potential Propagation

2 3 4 5 sheet length (mm) stimulus

b. Network2

+ stimulus

Fig. 3. Simulation protocols. (a) Network I evaluates flat-wave longitudinal propagation through a 5.0 mm by 0.25 mm sheet. (b) Network 2 evaluates flat-wave transverse propagation through a 1 .S mm by 0.5 mm sheet.

tLxI,ax, measured in ms, is the time when Vi,,, is recorded at each cellular element. This value determines activation time along the network. APA, records the maximum value of the action potential upstroke amplitude. Longitudinal conduction velocity ( O l ) , is measured by taking the difference in the t-l&, between elements 2 2 e 4 0 0 in fibers 5-20, along the longitudinal axis. Transverse conduction velocity ( O t ) . is measured by taking the difference in t-Vnla,, between fibers 2 0 4 0 in elements 20-130 along the transverse axis. Action potential delay, measured in I L S , records a) the difference in activation time between adjacent modeled cells at the longitudinal gap junctions when a longi- tudinal stimulus is applied and b) the difference in activation time between adjacent cell fibers at the lateral gap junctions during transverse stimulation.

Two simulations are evaluated for each stimulus condition. The first is the nominal case in which all gap junction resistance values are 1.331 MS1. All the gap junction resistances are increased by a factor of ten to simulate cellular uncoupling for the second series of simulations.

111. RESULTS

A . Conduction Velocity und Deluy

Longitudinal signal propagation through Network 1 is shown in Fig. 4 for the tightly coupled (top) and uncoupled conditions (bottom). The t - ~ l l a , of each 10 pm element in the network is plotted to illustrate the propagating wavefront. In the top figure the isochrones represent a time difference of one ms, while the isochrones in the bottom figure represent a time difference of two ms. For the tightly coupled condition, the average longitudinal conduction velocity of the propagating wavefront measured 41.6 f 0.5 cm/s. Few discontinuities in propagation are observed at this resolution. As the coupling resistances increased by a factor of ten, the discontinuities in

2 Elapsed Time

0 0 1.0 2.0 3.0 4.0 5.0

Network Length (mm) _____)

Direction OF Action Potential Propagation

Fig. 4. Isochronal bands illustrating flat-wave longitudinal propagation through Network I . Signal conduction through the tightly coupled network (top) was faster with fewer discontinuities than in the uncoupled network (bottom).

Control Modeled Simulation Uncoupling

Time Time I- 0.6 ms -1 I- 2.0 ms -1

1- zmpn -1 1- 200pm -1 - Direction of Action Potential Propagation

Fig. S. Enlarged view of electrical propagation through a 0.2 mm by 0.25 mm section of Network I . Discontinuous propagation is observed in the uneven edges of the isochrones. Each isochronal band represents approximately one cell. The random cell lengths are clearly observed in the tightly coupled network (left), while the inhomogeneities of the network are amplified during simulated uncoupling (right).

the spread of the action potential at the modeled gap junctions are better observed. The conduction velocity of the action potential during simulated cellular uncoupling decreased to 17.4 k 0.5 cm/s. Conduction velocity was constant in both simulations, as shown by the approximately equal width of the isochrones. Fig. 5 is an enlarged view of Network 1 showing longitudinal electrical propagation through a 0.2 mm by 0.25 mm area. This section of 20 membrane elements in 25 fibers illustrates propagation through three modeled cells, as represented by the shaded isochrones. At this resolution, discontinuous propagation is observed during both coupling conditions as noted by the uneven edges of the isochrones. The uncoupled network on the right depicts a greater degree of inhomogeneity in action potential propagation, with areas of fast and slow propagation and reduced effect of the transverse connections.

The discrete organization of the network allows observations of changes in signal propagation and waveshape at the sub- cellular level. The upstroke of the action potential propagating through a 0.5 mm section of fiber 16 in Network 1 is shown in Fig. 6. Conduction through this section of the network is shown for the nominally coupled network (left) and uncoupled network (right). The electrical signal reaches this section of

MULLER-BORER el a1 : ELECTRICAL COUPLlhG AND IMPULSE PROPAGATION 449

Longiludinal Activation Delay

I 1

- > - 2 c

-810

4 0 4 U 5.6 6 4 7 2 U 0 U U 9.6 104 11.2 120 1211 136 14.4 I S 2 16.0 16.8 I76 I U 4 192

Simulated Uncoupling Tim.sn.reurr~nlslimulus (ms)

Cmllrol simulslian

Fig. 6. The upstroke of the action potential as the electrical signal propagates through SO cellular elements (elements 31G360) in fiber 16 of Network I . Propagation through the tightly coupled network is shown on the left and simulated cellular uncoupling is shown on the right.

Nelwork Length (mm)

Fig. 7. Isochronal bands illustrating flat-wave transverse propagation through Network 2 during normal coupling (top) and simulated cellular uncoupling (bottom). Signal conduction through the tightly coupled network was faster as shown by the wider bands.

the sheet at approximately 4.8 ms in the tightly coupled network and at 11.0 ms in the uncoupled network. The dark bands in Fig. 6 represent signal conduction within a modeled cell, while the gap junctions are represented by the white space. In the tightly coupled network, activation times across the gap junctions measured 74 f 12 ps. Under simulated cellular uncoupling, the average delay across the modeled gap junctions increased to 382 f 102 p s , while conduction time within the cell was observed to decrease.

Transverse signal propagation through Network 2 is shown in Fig. 7 for the tightly coupled network (top) and uncoupled network (bottom). The t-lkax of each 10 pm element in the network is plotted illustrating flat-wave propagation through the network. The isochrones represent a one ms (top) or two ms (bottom) time difference. Conduction velocity of the propagating wavefront in the transverse direction measured 6.2 f 0.2 cm/s for the tightly coupled network. When the gap junction resistance was homogeneously increased, conduction velocity of the action potential in the transverse direction decreased to 1.9 f 0.1 cm/s. The decreased velocity is represented in the increased number of isochrones as the wavefront passed through the network (bottom figure). At the macrostructure level action potential propagation through the network appears to be continuous although a few dis-

continuities are observed in the uneven edges of the of the isochrones. Fig. 8 illustrates propagation at the subcellular level during nominal coupling (left) and simulated cellular uncoupling (right). Membrane elements 54, 81 and 100 in fibers 20 through 29 illustrate conduction variability and changes in waveshape in the transverse direction. Activation delay between elements measured at transverse gap junctions increased from 187.5 f 82.7 p s to 553.3 & 214.7 ps during simulated uncoupling. In Fig. 8 membrane elements at or near gap junctions are identified by faster activation times between membrane elements in adjacent fibers, while those membrane elements with large separations between transverse connections are noted by the longer delays. For example, element 54 is directly connected through transverse couplings at fibers 2 1-22 (+), while the distance between transverse connections at fibers 25-26 (A) are separated by 120 pm.

B . Action Potential Amplitude, Rnax and Wuivshape

The APA in Network 1 and in Network 2 decreased as a result of simulated cellular uncoupling. The decreases in APA measured during simulated uncoupling were statistically significant ( p < 0.001) when compared to the APA of the tightly coupled conditions. Average APA in Network 1 decreased from 86.5 f 0.6 mV to 80.6 f 0.9 mV. Average APA in Network 2 decreased from 85.6 f 0.1 mV to 85.0 f 0.5 mV. V,,,,, increased in Network 1 and Network 2 during simu-

lated cellular uncoupling. The increases in Vma, when com- pared to the tightly coupled conditions were statistically sig- nificant ( p < 0.001) for each network. In Network I , ~,,,, measured 83.1 f 6.5 V/s with normal coupling and increased to 85.0 & 12.6 V/s during simulated cellular uncoupling. This represented a. 3% increase. Fig. 9 illustrates the variability in longitudinal V,,,, across 100 membrane elements in fiber 18 of Network 1. Under normal coupling conditions V,,,,, tended to reach a maximum value at the axial gap junctions (*) and reach a minimum at the transverse gap junctions (A). In contrast, during simulated cellular uncoupling abrupt changes in G;,,, are observed at the axial gap junctions while no change or small decreases in V,,,,, are observed at the transverse gap junctions.

When Network 2 was stimulated transversely, a 17%' in- crease was observed as average v,,, measured 126.8 f 17.0 V/s with normal coupling and increased to 152.2 f 25.6 V/s during cell-to-cell uncoupling. An illustration of the changes in V,,,,, during transverse stimulation are shown in Fig. 10. Fig. IO(a) shows the variability in i/,,,,, as the electrical signal propagates through membrane elements 25-125 in fiber 30 (i.e., perpendicular to the direction of propagation) of Network 2. The transverse (A) and axial (*) couplings have a small, but similar effect on V,,, during nominal coupling conditions as observed during longitudinal stimulation, i.e., increased V,,,, at the axial gap junctions and decreased V,,, at the lateral gap junctions. During homogeneous cellular uncoupling abrupt changes in V,,,,, are associated with the axia1,gap junctions (*). The effec! of the lateral gap junctions on V,,,,, is shown in Fig. 10(b). V,,, of membrane element 120 in the

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Transverse Activation Delay

Fig. 8. Transverse propagation through elements 54, 81 and 100 (fibers 20-29) with normal coupling (left) and simulated cellular uncoupling (right). The membrane elements directlv connected to adiacent fibers throueh transverse couplings ( * ) and with large separations (> 120 mm) between transverse couplings (A) are highlighted.

-

TABLE I1

Network 1 Network 2 707 membrane elements 2553 membrane elements Coupled Uncoupled Coupled Uncoupled

0 (cm/s) 41.6f0.5 17.4f0.5 6.2f0.2 1.960.1

74.06 12.0 382.0f102.0 187.5f82.7 553.3f214.7

APA (mV) 86.5f0.6 80.660.9 85.6f0.1 85.0f0.5 i/niax (V/s) 83.1f6.5 85.0f12.6 124.3ik16.3 154.1525.8

contiguous fibers 25 through 40 (i.e., parallel to the direction of propagation and perpendicular to the fiber orientation) under tightly coupled and uncoupled conditions are shown. V,,, tends to be greater under uncoupled conditions, but this increase is moderated by the spatial distribution of the transverse gap junctions as previously shown. The modeled tissue surrounding element 120 in fibers 27 and 30 contain transverse junctions with separations greater than 100 pm. In contrast, the modeled tissue surrounding element 120 in fibers 33 through 39 include lateral junctions with separations less than 100 pm. Similarly, fibers 32-33 and fibers 34-35 are connected by lateral gap junctions.

The discrete nature of the network model provided the abil- ity to observe characteristics in the action potential waveshape resulting only from increased cellular uncoupling and network geometry. With increased gap junction resistance notching in the upstroke of the action potential was observed during both longitudinal and transverse stimulation (see Figs. 6 and 8). Notching in the action potential upstroke was similarly associated with the spatial distribution of the gap junction ele- ments. During longitudinal and transverse stimulation notched upstrokes were most often observed in modeled cells where the spatial separation between lateral junctions was greater than 100 pm. Measured results are summarized in Table 11.

IV. DISCUSSION This research presents a method for studying action potential

propagation resulting from homogeneous cellular uncoupling in a 2-Dimensional, discrete network of electrically coupled excitable cells. Previous modeling approaches have evaluated action potential propagation through modeled tissue, but have

m 105 T 100

95 - 90 s 85

80

75

70

Fig. 9. Longitudinal f;,,,, in membrane elements 300400 along fiber 18 during normal cellular coupling and simulated cellular uncoupling in Network 1. The axial gap junctions ( * ) are clearly distinguished during cellular uncoupling by the sudden decrease or increase in I;,,,,. The transverse gap junctions (A) are identified in the tightly coupled network as a decrease in I;,,,, within a modeled cell.

failed to incorporate the tissue variability simulated with this approach. By representing the physiologic organization of ventricular tissue at the subcellular level and the random interconnections of the tissue's microstructure the variability in propagating action potentials of this electrical model are similar to those observed experimentally. The variability in the measured parameters, although minimal in tightly cou- pled tissue, increased as result of cell-to-cell uncoupling. Furthermore, the simulation results suggest that the spatial resolution of the gap junction resistors F e an important factor contributing to the variability of V,,, and action potential amplitude. Increased directional differences of the action potential waveshape and variability in signal conduction demonstrated during simulated cellular uncoupling may be contributing factors in the development of arrhythmic events during ischemia.

A. Conduction Velocity Longitudinal conduction velocities are consistent with sim-

ilar models of 2-Dimensional, discrete periodic structures [ 191 and measured longitudinal conduction in ventricular sheets [20]. Yet, longitudinal conduction velocity is slower than

MULLER-BORER et al.: ELECTRICAL COUPLING AND IMPULSE PROPAGATION 45 I

160

240

80.W 1

: s E 3 2 4 z s z X 1 z C 4 : S

- nomsi - “MDyphd

Fiber Piumber

(b)

Fig. 10. Discrete I;,,,, during transverse propagation in Network 2. (a) The top figure shows the variability in I;,,,, as the electrical signal travels through membrane elements 25-125 in fiber 30 (perpendicular to the direction of propagation) during the normal and uncoupled conditions. (b) The bottom figure illustrates how the spacing of the transverse couplings effect the magnitude of the change in r;,,,, during the normal and uncoupled conditions. Membrane element 120. fibers 2 5 4 0 are shown.

simulations in continuous 2-Dimensional periodic sheets. Leon et al. [2 1 I recorded longitudinal conduction velocities ap- proximately 37% greater, in a continuous fiber model which used the same mathematical algorithm to model the action potential response. This difference in conduction velocity may be attributed to the combined intercellular junction re- sistance and myoplasmic resistance of the continuous fiber. This lumped resistive parameter does not account for the variable activation delay between modeled cells resulting from recurrent discontinuity in current flow as shown in Fig. 5. The discontinuities in longitudinal propagation may appear negligible for a tightly coupled network but become significant as the activation delay between cells increases.

The generation of flat-wave longitudinal propagation im- plies that each fiber along the transverse edge of the network is simultaneously activated. In a periodic structure longitudinal propagation along each fiber would be synchronized with a transverse potential gradient of zero, i.e., the propagating wavefront would be independent of transverse couplings. In this study the circular stimulus approximates an edge stimulus. Yet with the modeled anatomical structure of random spaced interconnections the modeled fibers of the electrical network are not isolated during flat-wave longitudinal propagation, i.e., the transverse potential gradient is not zero. This is shown in the variability in longitudinal conduction velocity and discrete differences in activation delays between modeled cells. Levine et al. [22] suggest that transverse current flow during longitu- dinal conduction provides a drag effect leading to a decrease in longitudinal conduction velocity. Prior simulations in our lab-

oratory have shown longitudinal propagation to be faster in a single fiber network than in a multifiber network under tightly coupled conditions, supporting this observation. Furthermore, in the multifiber network, under tightly coupled conditions, longitudinal conduction across the gap junctions was often more rapid than through the modeled cells suggesting faster parallel resistive pathways, i.e., transverse gap junctions. How- ever, during cellular uncoupling simulation results showed conduction within the modeled cells to increase while signal conduction between cells decreased. These results suggest a preferred longitudinal path and reduced influence of the transverse potential gradient. Correspondingly, our research shows that during simulated cellular uncoupling differences in longitudinal conduction velocities between single fiber and multifiber networks are negligible.

At the whole network level, under tightly coupled con- ditions, smooth contour bands of equal widths suggest a functional syncytia, i.e., the electrical impulse generated in one area propagated in an orderly sequence through the network. What appears as continuous propagation is an averaging of the conduction velocities, with the electrotonic response of each membrane element influencing its neighbor. As the resistivity of the network increased activation delay between modeled cells was observed. The longitudinal activation delay between modeled cells closely approximate experimental measurements in tissue preparations. Israel et al. [6] measured activation in prepared ventricular tissue with electrodes spaced at 20 pm centers, reporting delays of 60 z t 7 p s across implied gap junctions under normal cellular coupling. This variability in action potential conduction has also been reported by Spach et al. [23]. Microscopic mapping of longitudinal conduction velocities with electrode spacing less than 200 pm produced local longitudinal velocities between 23 cm/s to 52 cm/s with tightly coupled tissue. Secondly, the increase in activation delay measured in the network model when cell-to-cell cou- pling resistance increased ten fold compares to the increased activation delay of 320 7 ~ r s measured by Israel et al. [6] during induced ischemia.

Transverse conduction velocities in the tightly coupled net- work compare to measured velocities in similar 2-Dimensional discrete periodic models [ 191, continuous models [24] and experimental preparations [ 2 ] . Flat-wave transverse propaga- tion implies that each node along the longitudinal edge of the network is simultaneously activated and is directly determined by the value and spacing of the lateral gap junctions, i.e., trans- verse conduction perpendicular to the longitudinal axis of the cardiac myocyte, is governed by the lateral discontinuities. The geometric differences in the longitudinal and lateral connec- tions accounts for the slower transverse conduction velocities. At the whole tissue level, the wavefront appears continuous as the electrical signal advances through the network with little discontinuity in the isochrones. However, at the cellular level variability in activation times for equal elements in adjacent fibers are shown to exist. This variability at the discrete level increases during simulated cellular uncoupling.

Transverse signal propagation slowed significantly during homogeneous cellular uncoupling. When the gap junction resistances were homogeneously increased conduction velocity

452 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 41, NO. 5. MAY 1994

of the flat longitudinal wavefront decreased by 57% while conduction velocity of the flat transverse wavefront decreased by 82%. Experimental measurements of transverse conduction during induced cellular uncoupling report velocities around 7 cm/s [2]. With the discrete electrical model the signal continued to propagate without decremental conduction or block at 2 cm/s. This may be attributed to the homogeneous “normal” ionic properties throughout the network. The results indicate that cellular uncoupling within physiologic bounds contributes to conduction slowing, a prerequisite for reentrant arrhythmias. However, the resistive changes alone do not result in longitudinal or transverse conduction block. These results are in accord with a modeling study by Rudy and Quan [25] using a one dimensional discrete fiber. This study reported that within a normal physiologic range of coupling resistance the velocity of propagation followed the inverse square root relation of continuous cable theory. Furthermore, decremental conduction and conduction block were observed only at high levels of coupling resistance.

While these simulations address only flat-wave propagation the results suggest that indeed altemate pathways to propa- gation exist and are available due to the geometry of the tissue. Unlike periodic structures in which the role of the transverse or longitudinal couplings are minimized during flat-wave propagation the importance of these junctions are shown. The size of the stimuli modeled in this study was chosen to approximate the size of microelectrodes, therefore electrical propagation from a point stimulus is not shown. A point source stimulus would result in an elliptical wave- front radiating away from the stimulus. It is expected that propagation velocities would increase as the potential between fibers increased, offering faster alternative pathways.

B. APA, V,, and Waveshape

The structural organization and resistive interconnections of the network were responsible for the measured changes in APA, V,,, and waveshape as the active membrane properties remained uniform throughout the network. As cell-to-cell coupling was homogeneously decreased the spatial distribution of the gap junctions played a significant role in the variability of the measured parameters. Areas of the network with a greater concentration of lateral cell-to-cell couplings exhibited less variability in the propagating action potential wavefront than those areas with fewer lateral connections.

Experimental studies [2], [26] report that in anisotropic ventricular tissue V,,,, is directionally dependent, i.e., V,,, is greater in the transverse direction than in the longitudinal direction. The simulation results are in accord with these ex- perimental findings. Additionally, increased variability in V,,,, measured during cellular uncoupling has also been observed by Delmar et al. [2]. Unlike continuous models, the variability in V,,, is shown to exist in action potentials measured during longitudinal and transverse propagation. In the tightly coupled networks the variability in V,,, was coincident with the placement of the axial and lateral junctional elements. The location of the junctional elements could be noted by a rise or fall in measured V,,,,,. When the networks were uncoupled the

greatest discrete changes in V,,,,, were measured at the axial gap junctions. However, the spatial frequency of the lateral gap junctions tended to moderate the discrete changes in V,,,,,, i.e., the magnitude of change in V,, was less in areas of the network with close lateral couplings (A less than 100 pm).

Notches observed in the action potential upstroke are often recorded in experimental measurements during increased axial resistivity [4]. In this study, notching was observed in both the transverse and longitudinal directions as. the networks became uncoupled. As with the changes in V,,,,, areas of each network with closely spaced junctional couplings showed less notching during simulated cellular uncoupling than those areas with a greater spacing (> 100p m) between junctional couplings. The spatial resolution of the lateral gap junctions is an important factor contributing to the appearance of contin- uous propagation. The current entering a given node arrives from different sources at different times, however the time delays are minimized in areas with more lateral couplings. Further study of the effects of the time delays resulting from uncoupling on the action potential waveshape are warranted.

Gap junction resistance increased by a factor of ten resulted in a small change in APA. These results suggest a minimal effect of cellular uncoupling on APA. As predicted, when the gap junction resistances were increased the action potential amplitude tended to decrease, most notably in the longitudinal direction. As longitudinal conduction velocities decreased less current was available to depolarize each cell, contributing to the decrease in APA. Yet, during forced transverse propagation little change in APA was observed when the network was uncoupled. This small change in transverse APA may be explained by the network geometry and higher side to side resistivity created by uncoupling at the gap junctions [5].

C. Network Assumptions and Limitations

In applying the simulation results toward the understanding of observed physiologic phenomena it is important to note the assumptions and limitations of this approach. First, this study describes flat-wave propagation in a multicellular sheet of tissue with no connections to the extracellular environment. The extracellular environment is assumed to be isopotential, making no significant contribution to the propagating action potential. This is a simplified assumption as animal studies suggest that extracellular resistance contributes to a slowing of conduction velocities in early ischemia [27]. Modeling results using a bidomain structure (i.e., internal and external continuous structure) also express the importance of the effect of the extracellular medium on the propagating wavefront [28]. Our current computational resources could not accommodate the addition of an extracellular resistive network, however, it is theorized that adding a homogeneous extracellular resistive network to the network model would contribute to an overall slowing in conduction velocities. Presently, this 2-Dimensional model cannot adequately predict propagation changes resulting from an inhomogeneous extracellular network.

The dimensions of the network and stimulus were selected to approximate physiologic experimental preparations. The spatial properties of the modeled cells (i.e., assumed cell

MULLER-BORER et al.: ELECTRICAL COUPLING AND IMPULSE PROPAGATION 453

lengths and distances) are included in the mathematics which govem the resistance and conductance of each membrane unit. This elementary characterization assumes that the space be- tween modeled cells at the axial and lateral junctions are equal, i.e., the gap junctions represent pathways for signal conduction and do not represent a defined length. Homogeneously altering the dimensions of the network to simulate additional spacing between cells and fibers would result in a homogenous change in the measured flat conduction velocities. However, the spatial properties and implied lengths are important factors to address when evaluating conduction velocities originating from a point stimulus. Hoyt et al. 1291 reports that a sheet of cardiac tissue is composed of irregular shaped cells oriented along the longitudinal axis and connected end to end with no true lateral gap junctions. This structure implies that longitudinal and transverse current follows pathways through the end-to- end contacts. In the network model, due to the method which generated each modeled tissue sheet, the random placement of the lateral gap junctions did not place all the lateral gap junctions at the modeled cell ends. Rather, the structural configuration of the network model, i.e., random size cells and random spaced interconnections, was an initial attempt to model an inhomogeneous structure not observed in periodic models. Furthermore, Hoyt er al. 1291 reported that individual myocytes are connected to an average of 9.1 f 2.2 other myocytes. This level of cellular interconnection exceeds the level of the modeled tissue, suggesting a more homogeneous structure. These structural differences are important factors to consider as the model is advanced.

Like other modeling approaches, the transmembrane action potentials are governed by mathematical equations derived from voltage clamp experiments of healthy cells. The ionic properties are those of a single cell response and do not describe complex ionic changes in a multidimensional sheet of tissue during propagation or ischemia. While the action potentials generated by this network model do not duplicate the physiologic parameters V,,, and APA of membrane action potentials derived from voltage clamp experiments, the V,,, and APA are comparable to propagating signals measured in experimental tissue preparations 121. Subsequent simulations in our laboratory showed that when the sodium conductance was increased by 50(% conduction velocity, V,,, and APA also increased to values generated by other models. There is no question that the ionic response of the cell membrane plays a significant role in action potential shape and conduction, however, i t is assumed that homogeneous changes in the ionic parameters within normal bounds would not affect the qualitative results presented here.

This approach shows that the microstructure of a physiolog- ically significant size tissue sheet can be studied with the use of a circuit simulator. The 10 pm cellular element resolution pro- vides observations of inhomogeneities occurring at the cellular level which may give rise to arrhythmic activity. Presently this model allows measurement of activation between modeled cells as well as overall conduction velocities. As with experi- mental preparations conduction velocity between two points in two dimensions must be approximated. The electrical signal may not travel linearly between two points, however, it is

assumed that the signal will travel a path of least resistance. In tightly coupled tissue the transverse connections provide lower resistance pathways through parallel resistive interconnections. Signal conduction through this network is faster even though the distance traveled is longer. As the network is uncoupled the transverse junctions do not provide faster alternative pathways as a result of increased time delays at the cellular junctions. Conduction velocity through the network then decreases.

The electrical network can be used to study potential path- ways around high resistive barriers, variations to the propa- gating action potential due to combined ionic and resistive modifications and the interaction of these membrane param- eters at modeled ischemic border zones. The dimensions of the network do not allow for reentry or collision of propa- gating wavefronts around a resistive barrier modeled within physiologic bounds, i.e., action potential activation is rapid through the network (- 20 ms) and action potential duration is too long (- 300 ms) for a second depolarization to occur. However, signal conduction through high resistive barriers can be simulated and the path of the propagating action potential studied in two dimensions. Simulations have shown the action potential response sensitive to changes in ionic conductances. Increased conduction velocity and V,,, result from increased sodium conductance and decreased action potential duration are observed with a decrease in calcium conductance. The variability introduced by an increase in resistive properties in the presence of ionic and metabolic changes may lead to lethal arrhythmic activity. It is expected that this electrical model will be useful in evaluating inhomogeneities in active and passive membrane properties.

ACKNOWLEDGMENT

I would like to thank Timothy A. Johnson, Ph.D. and Wayne E. Cascio, MD for their critical review and scientific contributions in the preparation of this paper.

REFERENCES

J. W. Buchanan and T. Fujino, “Ventricular Muscle as a Function- ally Continuous Medium for Electrical Propagation: Experiments and Simulations,” Imaging, Analysis and Simulation of the Cardiac System, S. Sideman, R. Beyar, Eds. London: Freund Publishing House, pp. 699-717, 1990. M. Delmar, D. C. Michaels, T. Johnson and J. Jalife, “Effects of Increas- ing Intercellular Resistance on Transverse and Longitudinal Propagation in Sheep Epicardial Muscle,’’ Circ. Res . , vol. 60, pp. 780-785, 1987. P. J. Diaz, Y. Rudy and R. Plonsey, “Intercalated Discs as a Cause for Discontinuous Propagation in Cardiac Muscle: A Theoretical Simula- tion,” Ann. Bio. Eng., vol. I I . pp. 177-189, 1983. J. Jaliffe, S. Sicouri, M. Delmar and D. C. Michaels, “Electrical Uncoupling and Impulse Propagation in Isolated Sheep Purkinje Fibres,” Heurt Circ,. Physiol., vol. 26. pp. h179-hl88, 1989. M. S. Spach, W. T. Miller 111. P. C. Dolber, M. Kootsey, J. R. Sommer and C. E. Mosher Jr.. “The Functional Role of Structural Complexities in the Propagation of Depolarization in the Atrium of the Dog.” Cirr,. Res., pp. 175-191, 1990. D. A. Israel, D. J. Edell and R. G. Mark, “Time Delays in Propagation of Cardiac Action Potentia1,”Am. J . Ph~~siol. , vol. 258, pp. HI906-HI917. 1990. W. E. Cascio, G. Yan and A. G. Kleber, “Passive Electrical Properties, Mechanical Activity. and Extracellular Potassium in Arterially Perfused and Ischemic Rabbit Ventricular Muscle,” Circ. Res., vol. 66, pp. I46 I -I 473, 1990.

454 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 41, NO. 5 . MAY 1994

[8] A. G. Kleber and C. B. Riegger, “Electrical Constants of Arterially Perfused Rabbit Papillary Muscle,” J . Physiol., vol. 385, pp. 307-324, 1987.

[9] J. R. Sommer and B. Scherer, “Geometry of Cell and Bundle Apposi- tions in Cardiac Muscle: Light Microscopy,” Am. J . Physiol., vol. 248, pp. H792-H803, 1985.

[IO] B. J. Muller, D. J. Erdman and J. W. Buchanan, “Discrete Bioelectric Model of Ventricular Tissue,” Proc. of the 13th annual Int. Conf., IEEE Engineering in Medicine and Biology Society, pp. 5 10-5 I I , I99 1.

[ I 11 L. Clerc, “Directional Differences of Impulse Spread in Trabecular Muscle from Mammalian Heart,” J . Physiol., vol. 255. pp. 335-346, 1976.

[I21 J. Daut, “The Passive Electrical Properties of Guinea Pig Ventricu- lar Muscle as Examined by Voltage Clamp Techniques,” J . Physiol. (Londonj, vol. 330, pp. 221-242, 1982.

[I31 D. L. Roberts and A. M. Scher, “Effect of Tissue Anisotropy on Extracellular Potential Fields in Canine Myocardium in Situ,” Circ. Res., vol. 50, pp. 342-351, 1982.

[I41 G. W. Beeler and H. Reuter, “Reconstruction of the Action Potential of Ventricular Myocardial Fibres,’’ J . Physiol., vol. 268, pp. 177-210, 1977.

[I51 J. P. Drouhard and F. A. Roberge, “Revised Formulation of the Hodgkin-Huxley Representation of the Sodium Current in Cardiac Cells,” Computers and Biomedical Research, vol. 20, pp. 333-350, 1987.

[I61 A. L. Hodgkin and A. F. Huxley, “A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve,’’ J . Physiol., vol. 117, pp. 500-544, 1952.

[I71 D. J . Erdman, N. Hooke and B. J. Muller-Borer, “Simulation of the Cardiac Action Potential Using a Table-Based Circuit Simulator,” Technical Report tr90-10. Microelectronics Center of North Carolina. 1990.

[I81 T. A. Johnson, C. L. Engle, R. P. Kusy, S. B. Knisley, C. A. Graebner and L. S. Gettes, “Fabrication, Evaluation and Use of Extracellular K+ and H+ Ion-Selective Eectrodes,” Am. J . Physiol., vol. 258, pp. HI 224-H 123 I , 1990.

[I91 F. A. Roberge, A. Vinet and B. Victorri, “Reconstruction of Propagated Electrical Activity with a Two Dimensional Model of Anisostropic Heart Muscle,” Circ. Res., vol. 58. no. 4, pp. 46147.5, 1986.

(201 C. Delgado, B. Steinhous, M. Delmar, D. R. Chialvo and J . Jalife, “Directional Differences in Excitability and Margin of Safety for Prop- agation in Sheep Ventricular Epicardial Muscle,” Circ. Res., vol. 67, pp. 97-1 10, 1990.

[21] L. J. Leon and F. A. Roberge, “Directional Characteristics of Action Potential Propagation in Cardiac Muscle, A Model Study,” Circ. Res.. vol. 69, pp. 378-39.5, 1991.

[22] J. H. Levine, N. E. Moore. A. H. Kadish. M. Stern, W. Balke and J. F. Spear, “Elliptic Drag During Longitudinal Conduction in Anistropic Canine Myocardium” (abstract). Circ., vol. 76, Suppl. IV. p. IV-431,

~~

1987. M. S. Spach, P. C. Dolber and J. F. Heilage, “Resolution of Dis- continuous Versus Continuous Propagation: Microscopic Mapping of the Derivatives of Extracellular Potential Waveforms,” Ch. 16 Cardiac Electrophysiology: From Cell to Bedside. Zipes. Jalife Eds. W. B. Saunders Co., 1990. L. J. Leon and F. A. Roberge, “Structural Complexity Effects on Transverse Propagation in a Two-Dimensional Model of Myocardium,” IEEE Trans. Bio. Eng., vol. 38. pp. 997-1009, 1991. Y. Rudy and W. Quan, “A Model Study of the Effects of the Discrete Cellular Structure on Electrical Propagation in Cardiac Tissue,” Circ. Res., vol. 61, pp. 815-823, 1987. M. S. Spach, W. T. Miller, 111, D. Geselowitz, R. C. Barr, J. M. Kootsey and E. A. Johnson, “The Discontinuous Nature of Propagation in Normal Canine Cardiac Muscle,” Circ. Res., vol. 48, pp. 39-54, 1981.

[27] A. G. Kleber, C. Riegger and M. J. Janse. “Electrical Uncoupling and Increase of Extracellular Resistance after Induction of Ischemia in Isolated, Arterially Perfused Rabbit Papillary Muscle,” Circ. Res., vol. 61, pp. 271-279, 1987.

[28] C. S. Henriquez and R. Plonsey, “Simulation of Propagation Along a Cylindrical Bundle of Cardiac Tissue-11: Results of Simulation,” IEEE Trans. Bio. Eng., vol. 37, no. 9, pp. 861-875, 1990.

[29] R. H. Hoyt, M. L. Cohen and J. E. Saffitz. “The Distribution and Three Dimensional Structure of Intercellular Junctions in Canine My- ocardium,” Circ. Res., vol. 64, pp. 563-574, 1988.

Barbara J. Muller-Borer received the B.S. degree in engineering from Tufts University, Medford, MA in 1979, and M.S. and Ph.D. degrees in biomedical engineering from the University of North Carolina at Chapel Hill in 1986 and 1991, respectively.

Since 1991 she has held an appointment as a Postdoctoral Fellow with the Experimental Cardi- ology Group in the Department of Medicine at the University of North Carolina at Chapel Hill. Her main areas of research include computational and experimental electrophysiology.

Donald J. Erdman was born in Jackson, MI on March 6, 1962. He received the B.S. degree with high honors in 1984 from Eckerd College in St. Petersburg. FL where he majored in mathematics and chemistry, He received the M.S. and Ph.D. in computer science from Duke University in 1986 and 1989. respectively.

From 1989 to 1991 he was employed at MCNC in Research Triangle Park, NC. In 1991 he became a Senior Research Statistican at SAS Institute in Cary, NC. His research interests include computational

methods for large nonlinear differential algebraic systems and estimation and simulation of nonlinear systems of equations.

Jack W. Buchanan was born on August 21, 1945 in Crossville, TN. He received B.S.E.E., M.S.E.E. and M.D. degrees from the University of Ken- tucky in 1967, 1969. and 1975, respectively. He received internal medicine and cardiac electrophysi- ology training at the University of Texas at Houston, the University of Kentucky and the University of North Carolina at Chapel Hill.

From 1982 to 1990. he was Research Assistant Professor of Medicine and Biomedical Engineering at UNC-CH. In 1990. he became Associate Profes-

sor of Biomedical Engineering, Medicine, Health Informatics and Physiology at the University of Tennessee, Memphis, and Staff Physician (Cardiology) at the Memphis VA Medical Center.