Electrical stimulation of cardiac myocytes

Annals of Biomedical Engineering, Vol. 23, pp. 812-821, 1995 0090--6964/95 $10.50 + .00 Printed in the USA. All rights reserved. Copyright �9 1995 Biomedical Engineering Society

Electrical Stimulation of Cardiac Myocytes

RAVI RAN JAN a n d NITISH V. THAKOR

Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, MD

Abstract - -The influence of nonuniform cell shape and field orientation on the field stimulation thresholds of cardiac myocytes was studied both experimentally and comput~ttionally. The percent change in excitation threshold, which was studied with patch clamp technique, was found to be 182 +- 83.1% (mean + SD) higher when the electric field (EF) was parallel to the transverse cell axis versus the longitudinal axis (t7 < 0.0006). On reversing the polarity of the applied EF, the percentage change in threshold was observed to increase by 98.9 +-- 71.0% (p < 0.0002), implying asymmetry of the stimulation threshold of isolated myocytes. Finite element models were made to investi- gate the distribution of the transmembrane potential of these experimentally studied myocytes. A typical cell model showed that the maximum transmembrane potential induced on opposite ends of the cell was 39.1 mV and -46 .5 mV for longitudinal field (aligned with the long axis of the cell), but was 40.5 mV and -44 .8 mV for transverse field (aligned with the short axis of the cell). More significantly, it was found that the maximum transmembrane potential occurred at discrete points or "hot spots" on the cell membrane. It is hypothesized that the depolarization of the cell initiates at the hot spot and then spreads over the entire cell. The different excitation thresholds for different polarities of the applied EF can be explained by the different maximum induced at the opposite ends of the cell.

Keywords~Cardiac myocyte, Cell stimulation, Stimulation threshold.

INTRODUCTION

During electrical defibrillation, an electric field is applied across the heart and according to the most commonly accepted hypothesis, ventricular defibrillation is produced by exciting cells in a critical mass of the ventricle (30). In the past few years, improvement in defibrillation shock efficacy has been reported with different lead orientations (2,12,13), different kinds of wave forms (6,14,15,24), and sequential shock delivery (3,11,12). However, the mechanisms underlying these improvements are still under investigation.

Acknowledgment The experiments reported in this paper were conducted during the sabbatical of one of the authors (N.V.T.) at the Eli Lilly Research Laboratories. The authors thank Drs. P. Reid and R. Sweeney for their collaboration and support for the experimental study.

Address correspondence to N.V. Thakor, Biomedical Engineering, Johns Hopkins University School of Medicine, 720 Rutland Avenue, Baltimore, MD 21205, U.S.A.

Received 13Mar95, Revised 30Jun95, Accepted 21Ju195

A defibrillation shock exerts its electrophysiological effect through its influence on the transmembrane potential. The shock current generated by defibrillation passes between the electrodes through the intervening heart tissue. This current is not confined to the extracellular con- ducting fluid spaces but passes into its intracellular conductive pathways as well. This current from the external electrodes eventually leaves the intracellular conductive pathways by passing through the myocardial cell mem- branes (20,29). It is this transmembrane current flow that causes the spatially varying membrane potential and the eventual excitation of the voltage gated channels, which results in the electrophysiological responses from the myocardium. It has been shown through optical recordings (10,18) and models (17,21,22,26) that the application of an electric field results in the depolarization of one end of a cell and the hyperpolarization of the other. Still, the overall spatial distribution of the transmembrane potential around a cardiac cell remains to be studied.

Due to the transmural variation of fiber orientation along the myocardium (9,23), the applied shock results in cardiac cells being exposed to an electric field of varying magnitude and direction. This anisotropic orientation of the fiber with respect to the applied field direction has been found to have a significant influence on the tissue response. The excitation threshold has been found to be lower for field oriented along the fiber axis (1,8,19,27). An early model of spheroidal cells in a uniform electric field showed that a higher maximum transmembrane potential is developed when the field is parallel to the longer side of the cell (17). With the applied electric field in three dimensions and nonuniform throughout the heart volume, it is difficult to know the field across each cell. As a result, it is easier to study single cell preparations for the effect of applied field on the cell, and even though defibrillation applies to tissue and not to single cells, effects of applied field on single cells can be used as indicators of the field effects on the whole heart.

In the present study, both experimental and finite element modeling techniques were used to explore the influence of cell shape, cell orientation, and the polarity of the applied field on the excitation thresholds of cardiac cells in uniform external fields. Experiments were done on isolated cardiac myocytes to examine the effects of electrical

812

Electrical Stimulation of Cardiac Myocytes 813

shock. Then finite element models (FEM) of the cells were constructed to explain the experimental findings and explore the effect of (i) irregularities on the cell circumference, (ii) the length-to-width radio of the cell, and (iii) the presence of a neighboring cell on the spatial distribution of the transmembrane potential along the cell circumference and hence on the excitation threshold of the cell.

METHODS

Experimental

Experimental studies were conducted on myocytes isolated from the right ventricles of mongrel dogs. Tissue pieces were isolated from the endocardial surface of the right ventricle. Cells were enzymatically dissociated and after a period of equilibration were transferred to a per- fused chamber on an inverted microscope. The cell chamber was designed by machining a shallow well with ta- pered edges from a Plexiglas sheet. The chamber was instrumented with two pairs of orthogonally placed min- iature plate electrodes located 1 cm apart. These plate electrodes, also called the field electrodes, were connected to a custom-built instrument designed to deliver electrical field pulses of either polarity in either of the two orthogonal directions (Fig. 1). A reference electrode was posi- tioned using a micromanipulator in close vicinity to the cell so as to sense the potential at the cell, which was then connected to an active circuit to drive the plate potential symmetrically around this reference electrode. This ar- rangement ensured that the stimulus artifact in the patch clamp measurement was minimal. The field electrodes were driven by a stimulator interfaced to a Macintosh personal computer (PC). The PC was equipped with a 12-bit data-acquisition system programmed with LabView software (National Instruments, Austin, TX) to deliver the S1 stimulus pulse to the patch pipette and $2 pulse to

$1

s2 Vplate

(Vp) from

computer

VSense

~- .~. \ / EPC 7

" - . \ I I /

o Electric shock plates

Sense electrode

FIGURE 1. Schematic of the experimental set-up used for single cell patch clamp experiments.

either of the field electrode pairs. The patch clamp experiments were carried out in the conventional fashion with borosilicate glass pipettes, which were pulled by a two- stage puller, polished and back-filled. The pipette resistance typically ranged from 2 to 8 Mohms. After a giga- ohm seal was achieved, the membrane was perforated by a combination of electrical pulse and pressure to achieve a whole cell current clamp configuration using an EPC-7 patch clamp amplifier (Fig. 1, inset). A rapid series of measurements were made on the patched cell using com- binations of S 1 pulses (to initiate the action potential) and $2 pulses (to simulate defibrillation-like external field). Most measurements were carried out at room temperature, and for various pulse strengths, polarities, and directions; at least five recordings were obtained in response to $2 pulses and were later averaged.

The S 1 pulse strength determined the threshold of stimulation, while variation of the $2 pulse strength and duration permitted measurement of action potential duration and reexcitation of the cell by the external field pulses. The field electrodes were placed in the bath to apply the field longitudinally (parallel to long cell axis) as well as transversely (parallel to short cell axis). The polarity was switched under computer control. Thus, experimental measurements were obtained on anodal and cathodal pulses applied both longitudinally and transversely. The cells were imaged by a video camera attached to the microscope and a video frame grabber interfaced to the PC. The digitized images were later transferred to the finite element software for theoretical analysis of effects of the applied external electrical fields.

All comparisons were made with the use of paired sam- ple t tests, and values of p < 0.005 were considered significant.

Computational

Two-dimensional finite element models (FEMs) of cardiac myocytes were constructed from digitized video images of cardiac cells used for experimental study. FEM was used because it is one of the suitable numerical techniques for solving partial differential equations and also because of its flexibility to deal with curved boundaries and complicated boundary conditions. For our problem, we used the commercially developed software package ANSYS (Swanson Analysis Inc., Houston, PA). The model was used to determine the passive steady-state response of the cell to externally applied EF.

Theory. The electric field in a two-dimensional conductor can be represented by

f . (~v3 = o (1)

V : ~ x i + - - ' 3y J '

814 R. RANJAN and N. V. THAKOR

TABLE 1. The excitation threshold for four cells with the applied field along the longitudinal and transverse axis of

the cell.

Excitation Threshold (V/cm)

Field along Field along Longitudinal Transverse Axis Axis of Cell of Cell

Cell 1 Polarity 1 3 6 Polarity 2 - 2 - 6



Cell 4 Polarity 1 2 7

Polarity 1 and 2 refer to the excitation threshold for opposite polarities of the applied field.

where V is a gradient vector, or is the electrical conductivity, and V is the scalar potential. Equation 1 assumes that there are no internal current sources present in the cell and the potential field is generated only by the externally applied source. Assuming the cell membrane to be iso- tropic (of uniform conductivity), Eq. 1 becomes

( o2g O2V~ or" \ ~ x2 + 0 - 7 } = 0, (2)

which is the Laplace equation. This equation was solved with appropriate boundary conditions for the entire bath with the cell placed in it. The Laplace equation could easily be solved for regularly shaped domains, but because of the complex geometry of the cells, an analytical solu- tion was not possible. Hence, an FEM was used to solve Eq. 2.

TABLE 2, The time for 90% repolarization (APD 90) of the action potentials for $2 field direction along the longitudinal

and transverse axes of the cell.

Cell #

APD 90 (ms)

Field along the Field along the Longitudinal Transverse Axis

Axis of the Cell of the Cell

Cell 1 1138 1061 Cell 2 821 684 Cell 3 1214 940 Cell 4* 744 837 Cell 5 800 672 Cell 6 1259 1107 Cell 7 822 653 Cell 8 1002 809 Cell 9 521 461

*Unusual cell showing a favorable response for field along the transverse cell axis.

Real Cell Model. To construct the computer model, video images of cells used in the experimental study were digitized and the cell outline was derived from these images. From this outline, a uniformly thick cell membrane was created and the cell was placed in a square bath of dimensions at least five times larger than the largest dimension of the cell. The conductivity of the cell membrane region in the model was fixed at 0.0005 S/cm 2, while the conductivity of the extracellular and intracellular media was fixed at 0.05 S/cm (17). The application of an electric field (5 V/cm) was simulated by assigning the boundary condition of a fixed applied voltage to the nodes at both of the edges of the bath in which the cell was placed.

Ideal Cell Model. Most cardiac cells are typically cylin- drical in shape: 10 to 20 ~m in diameter and 50 to 100 p~m in length (28) with irregularities on the surface. To com- pare and contrast the effect of these irregularities on the spatial distribution of the transmembrane potential, a model of an "ideal" cell of rectangular shape with smooth parallel sides was made. As in the case of ventricular myocytes, the length-to-width ratio of the rectangular cell was fixed at 5:1.

Paired Cell Model. Heart cells are coupled, forming a network known as the syncytium. Conduction in this syncytium occurs from individual cells to their neighbors. To enable extrapolating the results of the single cell to the entire syncytium, models of paired cells were made. The paired cell model allowed a comparison of the changes in spatial distribution of transmembrane potential across each cell due to the presence of a neighboring cell. Video im-

>

150 -

100

50

0

-50

-100 -

-150 -50

- - Longitudinal . . . . . . . . . Transverse

, [ ~ i , l , , ~ l , , t i , , , l , , , l ~ , , l ~ t , l ~ , l , , , l

350 750 1150 1550 1950 time(ms)

FIGURE 2. Action potential recorded during field stimuli (S2) oriented along the longi tudinal (solid) and transverse (dashed) axes of the cell. $2 was of equal magnitude in both cases. Note the greater extension of repolarization for field along the longitudinal axis of the cell.


ages of paired cells were used to construct the actual models.

RESULTS

Experimental

Effect of Field Orientation. The effect of cell orientation on the excitation threshold was studied by applying 10 msec pulses on 4 myocytes (Table 1). The change in excitation threshold was found to be 182 -+ 83.1% (mean -+ SD) higher when the applied EF was along the transverse rather than along the longitudinal axis of the cell (p < 0.0006). To determine the percent change, the threshold for field along the longitudinal axis was considered to be 100% and then the relative increase in the threshold for transverse field was computed. This finding agrees with earlier studies (1,27). Also, the effect of the shock $2 applied during repolarization was studied on 9 myocytes (Table 2). The shock applied during the repolarization phase of the action potential resulted in rescheduling or prolongation of the repolarization, as reported earlier in studies on whole heart (5,7,31). The extent of rescheduling, measured in terms of 90% repolarization (APD 90) time, was 15.6 --+ 11.4% more for longitudinal fields than transverse fields for the same field strength (p < 0.007). An example of this effect is shown in Fig. 2. With one exception, the cells always responded more favorably to longitudinal than to transverse fields (Table 2). The cell showing the unusual behavior of preferred response to transverse stimulation is analyzed later in the article.

Effect of Polarity of the Field. From the study of excitation threshold carried out in 12 cells, the change in threshold was found to increase by 98.9 --- 70.1%, just by reversing the polarity. Table 3 shows the excitation threshold for the 12 cells for opposite polarities, with polarity 1 defined postexperimentally as the one with higher threshold. The rescheduling of repolarization due to reversing the polarity was studied on 10 myocytes (Table 4). A reversal of the polarity of the applied field was found to change the 90% repolarization time (APD 90) by 30.0 --- 17.4% in the cells. Figure 3 shows the difference in rescheduling of the action potential in a cell due to the polarity reversal. The paired t test shows that the difference in both cases is significant (p < 0.0002).

Computational

Transmembrane Potential Distribution along the Cell Membrane. Model simulations show that a higher transmembrane potential is induced at cell ends when the applied EF is along the longitudinal cell axis rather than along the transverse cell axis. The results of the models are tabulated in Table 5. Figure 4 shows the distribution of isopotential lines for a uniform 5-V/cm field applied along

TABLE 3. Excitation threshold for different polarities of the applied field.

Excitation Threshold Absolute Value

(V/cm)

Polarity 1 Polarity 2

Cell 1 10 7 Cell 2 9 5 Cell 3 4 2 Cell 4 4 2 Cell 5 5 4 Cell 6 7 3 Cell 7 4 3 Cell 8 5 3 Cell 9 6 2 Cell 10 3 2 Cell 11 4 2 Cell 12 3 2

The absolute value of the applied electric field is tabulated with polarity 1 defined (postexperimentally) as the one with higher threshold for each cell.

the longitudinal (Fig. 4, Panel A) and the transverse (Fig. 4, Panel B) axes of a typical cell (Table 5, Cell 1). The sides or edges of the cell being depolarized or hyperpolarized depended on the polarity of the applied field. More significantly, from a plot showing the variation of the transmembrane potential with polar angle (Fig. 5) it is found that the maximum occurs at discrete spots or regions on the membrane, called "hot spots." The points of maximum transmembrane potential are marked (a-d) on Figs. 4 and 5. The distribution of transmembrane potential along the cell outline shown in Fig. 5 was generated by taking the difference in the potential computationally determined by the model for corresponding nodes outside and inside the cell. The inside of the cell was found to be isopotential. The polar angle in Fig. 5 refers to the angle made by the line joining the point on the cell circumfer-

TABLE 4. The 90% repolarization time (APD 90) of the action potential observed in isolated cardiomyocytes for

opposite polarities of the applied field.

APD 90 (ms)

Polarity 1 Polarity 2

Cell 1 992 903 Cell 2 817 592 Cell 3 994 824 Cell 4 993 609 Cell 5 1107 808 Cell 6 940 868 Cell 7 1167 982 Cell 8 837 509 Cell 9 745 509 Cell 10 924 829

The 90% repolarization t ime (APD 90) of the action potential observed in isolated cardiomyocytes for opposite polarities of the applied field.


g

;>

150 -

100

50

0

-50

-100

-150 -50

- - control (no $2) I ................. $2 =-5 V/cm i . . . . . $2 = 5 V/cm

. . J . . . . . . I t . _ _

350 750 1150 1550 1950 time(ms)

FIGURE 3. Action potential recorded with S2's of equal strength but opposite polarities. Note the difference in the rescheduling of repolarization in the two cases.

ence to the center of the cell, with the line joining the middle of the right side of the cell to the center.

The cell that showed the unusual behavior (Table 5, Unusual Cell) of responding favorably to transverse fields was also modeled and the results are discussed below under the heading "Model of the 'Unusual' Cell."

Model of an "Ideal" Cell. Figures 6 and 7 show the distribution of the isopotential lines and the variation of the transmembrane potential with the polar angle, respec- tively, for the " ideal" rectangular cell. The maximum transmembrane potential for the rectangular cell is 44.4 mV(a) for longitudinal field and 44.1 mV(b) for transverse field (Table 5). Unlike the real cell, the opposite sides of the ideal cell have the same transmembrane potential induced across them due to symmetry in the cell.

Model of the "Unusual" Cell. During the experimental

Transmembrane Potential Induced on Opposite Ends of Cell in

mV for a 5-V/cm Applied Field 1 1 4

Field along Field along Longitudinal Transverse

Cell Axis Cell Axis

Cell 1 39.1 46.5 40.5 44.8 Cell 2 45.1 49.3 47.8 46.0 Cell 3 51.3 39.4 46.0 44.1 Cell 4 (ideal cell) 44.4 44,4 44.1 44.1 Unusual cell 40.5 39.8 43.8 36.2

The opposite ends of the cell were depolarized/hyperpolarized depending on the polarity of the applied field. The table shows the absolute values of the induced transmernbrane potential.

study one myocyte showed the unusual behavior of responding favorably to field across the transverse cell axis rather than across the longitudinal axis of the cell (Table 5). This cell was further investigated through a model made from the image of this cell. For this cell the maximum transmembrane potential induced for transverse field was found to be 8.2% higher than the maximum induced for longitudinal field (Table 5). This induction of a higher transmembrane potential for transverse field may contrib- ute to the "unusual" behavior of the cell. The effect of cell outline irregularities and the length-to-width ratio of this cell on the magnitude and the spatial distribution of the transmembrane potential were also studied. The effect of irregularities was studied by smoothening the edges of the cell through software interpolation at certain faces. The results are tabulated in Table 6. Figure 8 shows the various sections of the cell edges (a-d) that were smoothened. With most of the cell boundary irregularities smoothened, a higher maximum transmembrane potential was induced for longitudinal field. Changing the length- to-width ratio of the cell by stretching the cell also had a significant impact on the transmembrane potential distribution (Table 7).

Paired Cell Response to the Applied Field. To understand better the effect of the applied field in a syncytium, we made models of paired cells. The distribution of isopotential lines for a paired cell for a uniform 5-V/cm applied field is shown in Fig. 9. For the cells marked A and B in the figure, the maximum transmembrane potentials induced on the opposing ends, with and without the adjoining cell, are tabulated in Table 8. For Cell A, the maximum induced transmembrane potential decreased by 17% for field along the longitudinal cell axis without the ad-

TABLE 5. The transmembrane potentials induced on opposing ends of the modeled cell for the applied field

along the longitudinal and transverse cell axis,

50 ~m

FIGURE 4. The distribution of isopotentiat lines given by the FEM model for a 5-V/cm (=0.5 mV/mm) field applied along the (A) longitudinal and (B) transverse axes of the cell. The separation between the isopotential lines equals 5 mV. Points a-d show the "hot spots" or the points of maximum transmem- brahe potential induced: a, 39.1 mV, b, 46.5 mV; c, 40.5 mV; and d, 44.8 mV. The depolarizing or hyperpolarizing end is determined by the polarity of the applied field. The distribution of the transmembrane potential along the cell outline is shown in Fig. 5,


6 0 . . . . . . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, . ~ 40 ~

20

~

-2(1

4 o i i i iT . . . . . . . . . . . . . . . . . '

I I I

600 100 200 300 400 Polar Angle (degrees)

0 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

:2

= t

40 ~ . . : . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

20 .................. i ................. .................. i

, 0 1 ............... ...... il ................. 6 0 r . . . . . . . . . , . . . . . .


FIGURE 5. (A) The variation of transmembrane potential with polar angle for field along the longitudinal axis of the cell. Points a and b are the "hot spots" or the points where maximum transmembrane potential is induced: a, 39.1 mV; b, 46.5m. (B) The variation of transmembrane potential with polar angle for field along the transverse axis of the ceil. Points c and d are the "hot spots" or the points where maximum transmembrane potential is induced: c, 40.5 mV; d, 44.8 mV.

joining cell for the same EF. This induction of a lower maximum in an isolated cell agrees well with the findings of Gaylor et al. (8).

DISCUSSION

Lower Excitation Threshold for Field along the Longitudinal Axis of the Cell

The experimental results indicate that the excitation threshold for externally applied EF depends on both the polarity and direction of the applied field with respect to the cell axis, as well as on the cell shape. FEM modeling, which gives the spatial distribution of the transmembrane

potential across the cell circumference shows that the cells that have a favorable response to field along their longitudinal axis also have a higher transmembrane potential induced at their ends for this field orientation. This induction of a higher maximum transmembrane potential for field along the longitudinal axis of the cell might explain the experimentally observed lower excitation threshold for this field orientation: the critical threshold for depolarization (16) will be reached for a lower external applied field when the applied field is along the longitudinal rather than transverse axis of the cell.

From the spatial distribution of the transmembrane potential, it was found that the points of maximum transmembrane potential are isolated to discrete points on the cell (Figs. 4 and 5). It is hypothesized that these discrete "hot spots" are the sites of initial depolarization within the cell because it is here that the critical threshold for depolarization will be first reached. Then the excitation will spread to cover the entire cell.

Response to Opposite Polarities of the Applied Field

Reversing the polarity of the applied field had a significant effect on the response of the cell in terms of both the rescheduling the repolarization of the action potential and the excitation threshold (Fig. 3, Tables 3 and 4). The FEM model showed that the transmembrane potentials induced at the two opposite ends of the experimental cells were not the same. Recently, Knisley et al. (18) observed that, when electrically stimulated, one end of an isolated cardiac myocyte is hyperpolarized and the other depolarized in response to an applied field. They also observed that unequal magnitudes of transmembrane potential are induced in the opposing ends of the cell, though the difference was statistically insignificant. Based on the results of

50 p_m

FIGURE 6. The distribution of isopotential lines for a modeled ideal cell due to a 5 V/cm field applied (A) along and (B) across the cell axis. Points a and b show the points of maximum transmembrane potential. The distribution of the transmembrane potential along the cell outline is shown in Fig. 7.


~" 60 ......

s o

- 2 0 . . . . . . i ........................ i

~ "600 . . . . . 200' 300' 400 '100 . . . .

Polar Angle (degrees)

"-" 60

g 40 : b

~ -40

L I


FIGURE 7. (al The variation of transmembrane potential with polar angle for a 5-V/cm field applied along the longitudinal axis of the ideal cell, The "hot spots" for this cell are marked by a. Note that the opposing sides have the same maximum induced potential. (B) The variation of transmembrane potential with polar angle for a 5-V/cm field applied along the transverse axis of the ideal cell. The "hot spots" for this cell are marked by b. Note that the opposing sides have the same maximum induced potential.

Unusual Cell: Cell Responding Favorably to Field along the Transverse Cell Axis

The model of the unusual cell showed that the maximum transmembrane potential induced in this cell was higher for field along the transverse axis of the cell axis rather than for field along the longitudinal cell axis (Table 5, Unusual Cell). This reversal could explain the unusual behavior of the cell. Increasing the length and width of the cell resulted in a higher transmembrane potential being induced across the edges of the cell (Table 7). Changing the length-to-width ratio of the cell and smoothening the cell outline irregularities resulted in marked differences in the transmembrane potential induced across the cell edges. All of which suggests that the cell shape has an important bearing on the spatial distribution of the transmembrane potential and the magnitude of the induced hot spots, and hence on the excitation thresholds.

It has been suggested that the transmembrane potential induced at the ends of the cell depends in part on the product of the electric field strength and the length of the cell along the direction of the applied field (18,27). The higher transmembrane potential induced at the cell ends for longitudinal fields for most of the cells studied, as well as the increase in the transmembrane potential on increasing the length-to-width ratio of the cell in the model, seems to validate this suggestion. However, the unusual results obtained from one of the cells suggest that the cell axis is not the only factor determining the maximum induced potential. Investigating this cell shape further, we found that by artificial smoothening of the irregularities on

TABLE 6. The maximum transmembrane potential induced at the opposite ends of the modeled "unusual cell" after

smoothening portions of the edges of the cell.

Transmembrane Potential Induced on Opposite Ends of Cell in

mV for a 5-V/cm Applied Field

FEM cell models in this study, the differences in the transmembrane potential induced at the opposing ends were not statistically significant, but the experimental results suggest a significant influence on the cell response. Depend- ing on the polarity of the applied field, if the end of the cell with the larger transmembrane potential across it becomes the depolarizing edge, the cell would be stimulated at a lower external applied field. The model for the "ideal" rectangular cell with uniform edges has the same transmembrane potential drop across the opposing ends of the cell. This suggests that the shape of a cell is a significant factor in determining the magnitudes of the hyperpolarization and depolarization induced in the opposing ends of the cell, and hence the different cell response for opposite polarities of the applied field.

Portion of Field along Field along Edges of Cell Longitudinal Transverse Straightened a Cell Axis Cell Axis (*)

d 39.2 39,3 43.1 35 -9.78 b, d 39.2 39.8 41.5 37.3 -4 .14 b 40.6 40,3 42.1 38.5 -4.57 a, b 40.8 39.9 41.0 39.4 -0.59 a, b, d 39.4 39.5 40.4 38.2 -2.25 a, b, c, d 37.7 39.8 39.2 37.8 + 1.4

The opposite ends of the ceil were depolarized/hyperpolarized depending on the polarity of the applied field. The table shows the absolute values of the induced transmembrane potential. (*), the percent difference between the maximum transmembrane potential induced for field along the longitudinal and transverse cell axes. Negative percentage difference indicates that a higher maximum transmembrane potential was induced for transverse field. 8See Fig. 8.

819

3- I a

/

b

Electrical Stimulation of Cardiac Myocytes

T d

2

c ______#

the cell boundary in the model, the maximum induced potential occurred when the field was along the longitudinal axis. This observation suggests that the unusual behavior of the cell is forced by the irregularities on the cell boundary.

FIGURE 8. The outline of the "unusual" cell that responded favorably to field along the transverse axis of the cell. Sec- tions a-d indicate the section of the edges smoothened by software interpolation to study the effect of irregularities on the distribution of transmembrane potential (Table 6). For smoothening a region of the cell, the end points of the region were interpolated by a straight line.

TABLE 7. The maximum transmembrane potential induced at the opposite ends of the modeled "unusual cell" for

different length to width ratios of the cell.

Transmembrane Potential Induced on Opposite Ends of Cell in mV for

a 5-V/cm Applied Field

Length and Field along Field along Width of Longitudinal Transverse

Cell Model Cell Axis Cell Axis (*)

L W 40.5 39.7 43.8 36.2 - 8.2 L 2W 51.6 51.0 55.2 47.0 - 7.07 2L W 71.3 68.3 71.5 65.2 -0.25

The opposite ends of the cell were depolarized/hyperpolarized depending on the polarity of the applied field. The table shows the absolute values of the induced transmembrane potential. L, length in control cell; W, width in control cell; (*), the percent difference between the maximum transmembrane potential induced for longitudinal and transverse field. Negative percent difference indicates that a higher maximum transmembrane potential was induced for transverse field.

FIGURE 9. The distribution of isopotential lines given by the FEM model for the paired cell for a 5-V/cm field applied along the (A) longitudinal and (B) transverse cell axis. The different cells are marked A and B. The separation between the isopotential lines is 5 mV. Points a-f mark the "hot spots" on the cell boundary: a, 50.4; b, 59.6; c, 46.6; d, 61.9; e, 34.3; f, 30.1 (all in mV). The separation between isopotential lines is 5 mV,


TABLE 8. The maximum transmembrane potential induced at the opposite ends of each cell in the paired cell model, both with and without the adjoining cell.

Transmembrane Potential Induced on Opposite Ends of Cell in mV for a 5-V/cm Applied Field

Paired Cell Isolated Cell

Field along Longitudinal Cell Axis

Field along Transverse Cell Axis

Field along Longitudinal Cell Axis

Field along Transverse Cell Axis

Cell A 59.6 50.4 34.3 44.3 45.1 49.4 47.8 46.0 Cell B 61.9 46.6 44.4 30.1 51.3 39.3 45.9 44.1

The opposite ends of the cell were depolarized/hyperpolarized depending on the polarity of the applied field. The table shows the absolute values of the induced transmembrane potential.

Paired Cell Model

To extrapolate the results of field stimulation of single cells to that of cardiac tissue, we took an intermediate step of modeling pairs of cells. Comparing the maximum transmembrane potential induced in the cell, with and without the adjoining cell, suggests that a lower field should be required for cells in the syncytium when the field is along the cell axis. Also, as the fibers in the heart have a trans- murally varying helix angle, a shock results in a markedly uneven distribution of potential gradients for epicardial electrodes (4). This fiber geometry very likely causes a varied response from the cells, which depends on their position relative to the defibrillation electrodes.

Defibrillation is thought to be successful through a modification of the prior electrical behavior of the myocardial cells as a result of the applied electric shock (30). In this study, we have shown the effect of the applied shock on cardiac cells at a cellular level. One end of the cell is depolarized and the other end hyperpolarized depending on the polarity of the applied field. This depolarization/hyperpolarization of the ends of the cell leads to a periodic component in transmembrane potential induced, over a distance. Based on modeling studies this "saw- tooth" behavior is observed more prominently around the middle of the fiber, away from the externally applied sources (25). The dependence of the threshold on the direction and polarity of the applied electric field observed in cells both experimentally and computationally stresses the importance of orientation of the defibrillating electrodes.

The cell models were two-dimensional, and as a result the model shows the distribution of the transmembrane potential induced along a plane formed by the projection of the three-dimensional cell outline in two dimensions. While this limits the accuracy with which we can simulate the three-dimensional experiments, the two-dimensional models highlight the importance of the effects of irregularities on cell boundary better than three-dimensional models with spherical geometry and could also possibly explain the different response to fields of opposite polar-

ity. It would be instructive to make three-dimensional models with real cell shape and look at the time-dependent behavior.

It is necessary to exercise caution when extrapolating results from the single cell model to the entire tissue. The extracellular space in a myocardium is not an infinite iso- tropic conductive medium as in the model, but is severely restricted by the presence of adjacent cells. Moreover, the presence of gap junctions between the ceils will lead to the coupling of adjacent cells, which in turn affects the potential inside the adjacent cells. The overall distribution of the transmembrane potential is still expected to be similar because it is primarily dependent on the spatial organiza- tion of the cell membrane which has a much lower conductivity than does the inside of the cell.

CONCLUSIONS

The myocardium is composed of fiber bundles within which the cells show preferred axis of orientation and somewhat distinctive shapes. As a result, any applied external field affects each cell differently. It is observed that any applied field leads to high transmembrane potential being induced at a point, or hot spot, on the cell boundary. These hot spots could be the sites of initiation of depolarization of the cell, which would then propagate to the entire cell. The magnitude of the hot spot is observed to be higher for field along the longitudinal axis of the cell than field along the transverse axis of the cell, a condition contributing to the preferred response of most cells to longitudinal fields. The cells respond differently to fields of the same strength but opposite polarity. This difference could be because of the unequal transmembrane potential induced at the opposite ends of the cell as a result of asymmetric cell shape.

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Electrical stimulation of cardiac myocytes

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