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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 95, Issue 3, 1138-1150, 1 August 2008

doi:10.1529/biophysj.107.128207

Biophysical Theory and Modeling

Representation of Collective Electrical Behavior of Cardiac Cell Sheets

Seth Weinberg^*, Shahriar Iravanian^† and Leslie Tung^*, ,  

^* Department of Biomedical Engineering, The Johns Hopkins University, Baltimore, Maryland
^† Division of Cardiology, Emory University, Atlanta, Georgia

Address reprint requests to Dr. Leslie Tung.

The electrical activity of the heart produces electric potentials that can be measured at the body surface. Potential differences between points on the body surface provide the electrocardiogram (ECG), a time-dependent signal that encompasses the collective electrical behavior of the heart. The ECG is used clinically to distinguish among different conduction patterns such as sinus rhythm, tachycardia, and fibrillation 1. For in silico, in vitro, or ex vivo cardiac model systems, the pseudo-ECG (pECG) provides a measure of the collective electrical behavior of the system. It has been applied to one- and two-dimensional computational models and two-dimensional slices or sheets of cardiac tissue. The value of the pECG, as evidenced by its usage in the literature, is its representation of tissue-level effects as a clinical-like waveform even though the waveform contains less information than complete maps of V_m. In doing so, results obtained from simplified experimental systems can be interpreted and understood from a clinical perspective. Furthermore, like the ECG, the presence of beat-to-beat variations in activation and/or repolarization patterns at a global level become apparent.

Experimentally, pECG has been measured in ex vivo preparations by bipolar electrodes placed on or around the tissue 2,3,4. However, when such recordings are unavailable or unfeasible to make, pECG can be computed from spatial maps of transmembrane potential (V_m). One approach that has been used is an ad hoc, difference method, which subtracts the average V_m from one-half of a myocardial tissue layer from the average of the other half, as was done for thin ventricular epicardial slices and atrial tissue 5,6,7,8.

An alternative method used a theoretical approach to compute the extracellular potential at a particular point in space (i.e., for a unipolar lead), based on the model's spatial distribution of V_m, 9. A unipolar lead voltage was computed at a certain distance from the end of a one-dimensional multicellular fiber of coupled cardiac cells meant to represent the transmural heterogeneities in the ventricles. A unipolar lead has also been utilized in two-dimensional computation models 10,11.

In this study, we theoretically derive the bipolar and unipolar pECGs for any desired lead location from optical recordings of V_m in a two-dimensional isotropic monolayer of cardiac cells. Further, we demonstrate that our expressions simplify to a weighted sum of V_m, scaled by two geometric functions. We then discuss the relative strengths and weaknesses of bipolar and unipolar pECGs, through examples of different patterns of electrical propagation in cultured cardiac cell monolayers. We show that computation of the pseudo-vectorcardiogram (pVCG), based on an orthogonal pair of bipolar leads, clearly illustrates the collective electrical behavior of the cells and can be used to distinguish among different types of electrical activity.

A computationally simple pECG can be obtained by subtracting the average of V_m from one-half of a cell sheet from that of the other half 6,12, which we refer to as the difference method. The pseudo-lead voltages along the x and y axes, , are defined as

(1)

(2)

We previously used the concept of a lead field to compute a bipolar pECG for cell monolayer experiments 13,14. Here, we generalize our approach and determine the pECG for any desired placement of lead electrodes. Let the cardiac cells occupy U, a circular two-dimensional area of radius R lying on the bottom of a semiinfinite bath. A bipolar lead is placed in the x direction, with electrodes at (a,0,h) and (−a,0,h), respectively (Figure 1AB). We define the pseudo-lead voltage, pV_x, as

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

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Figure 1

(A) Top view of monolayer and (B) side view of monolayer and bipolar lead electrode position. (C) Bipolar lead field, (D) lead field magnitude along primary axis (x axis in A), and (E) secondary axis (y axis in A) of bipolar lead field for increasing h (at a=R). (F) Lead field magnitude along primary axis for increasing a at height . (G) Unipolar lead field. (H) Lead field magnitude along radial axis of unipolar lead field for increasing h (at a=R). For panels D–H, h is plotted from in steps of with shown as the bold trace. The case of is also plotted as the dashed trace. For panel F, a is plotted from R to 2R, in steps of 0.1R.

The method outlined above can be generalized to any placement of electrode leads. For example, a bipolar lead oriented in the y direction and situated a distance h above the monolayer, with electrodes a radial distance of a from the center, would have a pseudo-lead voltage of

(11)

(12)

(13)

(14)

To facilitate their calculation, pV_x and pV_y can be expressed as integrals of V_m rather than integrals of derivatives of V_m. Green's first identity for the divergence theorem in two dimensions (for any two continuous functions, ϕ and ψ) is

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

The ad hoc bipolar lead voltages () are also integrals (sums) of V_m (see Eqs. (1)) but have different weighting functions for V_m ( and ) compared with γ_x and γ_y, and do not include a boundary term at the perimeter of the monolayer (first term of Eq. (17)). The value is a two-dimensional sign function, positive on one-half of the cell sheet and negative on the other half (Figure 2A). It is a coarse approximation of γ_x (see Eq. (19)) at (Figure 2C). The value γ_x is also shown at other values of h (normalized to the maximum of γ_x at ). It increases sharply near the electrodes at (Figure 2B) and becomes much flatter and linear in shape at (Figure 2D). At large , γ_x approaches a flat plane tilted around the y axis (Figure 2E). The value α_x has a maximum amplitude directly below the electrodes, at θ=0 and π, and is zero in the y direction at θ=π/2 and −π/2 (Figure 2F). At (Figure 2F, dashed black trace), α_x has sharper peaks and a larger amplitude below the electrodes, compared with (Figure 2D, solid black trace). At (Figure 2F, dashed shaded trace), α_x has a flatter angular dependence and a lower amplitude under the electrodes. At (Figure 2F, solid shaded trace), α_x is near zero at all locations. The amplitude of α_x has been scaled by the ratio of the monolayer circumference to area (and normalized to the maximum of γ_x at ) to allow comparison with γ_x. At , the maximum value of α_x is <10% of γ_x. For increasing h, the amplitude of γ_x decreases at a faster rate than that of α_x and therefore the relative weight of the boundary term increases.

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Figure 2

Weighting functions for bipolar pECGs. The value γ_x, computed by (A) difference method and (B–E) theoretical method. For panels B–E, , , , and , respectively. (F) The value α_x, computed by theoretical method is shown on a polar plot, with radial axis ranging from −0.2 to 0.2. For panel F, (dashed black trace), (solid black trace), (dashed shaded trace), and (shaded trace).

The pseudo-vectorcardiogram (pVCG) is obtained by plotting pV_y against pV_x. For comparison, the pseudo-vectorcardiogram (pVCG^D) is also obtained by plotting against .

A unipolar (single electrode) lead, placed at (a,b,h) with reference at infinity, can be used for pECG measurements instead of the bipolar leads. The derivation of the unipolar lead voltage pV₀ closely follows that of the bipolar lead voltage. The unipolar lead was placed over the center of the monolayer (a=0, b=0) unless otherwise stated. The unipolar lead field, L₀, is assumed to be for the case where the reference electrode is at infinity, and is given by

(23)

(24)

The unipolar lead voltage, pV₀, is given by

(25)

(26)

(27)

(28)

(29)

Our cell culture procedure to create cell monolayers has been previously described 11,12. Briefly, neonatal rat ventricular myocytes were dissociated from two-day-old Sprague-Dawley rat hearts with the use of the enzymes, trypsin, and collagenase. The resulting cell suspension was plated at high density onto plastic coverslips to form monolayers that became confluent after 3–4 days of culture. Experiments were performed on days 4–9 after plating. The data for each example presented is from a different cell monolayer.

Our method of optical mapping of cell monolayers has been previously described 17,18. Briefly, maps of transmembrane potential were recorded by placing the cell monolayer directly on top of a bundle of 253 optical fibers 1-mm in diameter, arranged in a tightly packed, 17-mm-diameter hexagonal array. During experiments, the cell monolayers were stained with 10μM di-4-ANEPPS, a fluorescent voltage-sensitive dye, and continually superfused with warmed (36±0.5°C) oxygenated Tyrode's solution (in mmol/L: 135 NaCl, 5.4 KCl, 1.8 CaCl₂, 1 MgCl₂, 0.33 NaH₂PO₄, 5 HEPES, 5 glucose). The fluorescent dye signal was relayed by the optical fiber bundle to an array of photodetectors and amplifiers, processed by custom-written software and converted into pseudo-colored maps of V_m.

To compare pVCGs calculated using the difference and theoretically derived methods, we first centered each pVCG at the origin by subtracting from V_x and V_y their respective mean values. We then determined the maximum excursion of the pVCG from the origin and normalized the radial distance of each pVCG point to that value. In this way, the pVCG lay within a unit circle. Using Eq. (30), we computed the root mean-square difference (RMSD) by taking the square root of the mean of the square of the difference (in percent) of the two traces at each time point,

(30)

We show an example in Fig. 3 of a planar wave propagating across the monolayer that was initiated from a line electrode on the left side. The voltage map and isochrone map are shown in Figure 3AB, respectively. The wave is propagating across the monolayer primarily in the positive x direction, with a small component in the positive y direction. Both pV_x and pV_y are initially zero (Figure 3C a) and become positive (or negative) when the wavefront (or waveback) propagates across the monolayer. pV_x and pV_y are periodic at the pacing rate, larger during wavefront than waveback propagation (because of the larger transmembrane potential gradient), and approximately in phase with the same sign. It is possible for pV_x and pV_y to have opposite sign if the direction of propagation is toward the upper left or lower right (second or fourth quadrants) so that pV_x is negative while pV_y is positive, or vice versa. pVCG also reflects the direction of propagation and is oriented primarily in the x direction with a small component in the y direction (Figure 3D). The unipolar lead voltage pV₀ (Figure 3C b) captures the general activity of planar wave propagation because there exists a component of the unipolar lead field (Figure 3H) that is oriented along the direction of propagation. However, pV₀ does not indicate what the direction of propagation is, and it has a similar appearance for a wave propagating along any other direction across the monolayer.

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Figure 3

(A) Normalized voltage map for planar propagation. The color bar on the right indicates the relative amplitude of V_m (scaled from 0 to 1). (B) Isochrone map. (C) (a) Bipolar pECGs (pV_x and pV_y) and (b) unipolar pECG (pV₀). The colored vertical lines in panel C correspond to isochrones in the isochrone map in panel B. (D) pVCG. Arrows indicate the direction of pVCG with time.

We show a radial wave propagating outwardly from a point stimulus in Figure 4A. The position of the stimulus is slightly off-center by ∼1mm, toward the upper-left quadrant of the monolayer. Propagation terminates slightly sooner in the upper-left region, as seen in the isochrone map (Figure 4B). Because of the earlier termination of propagation in the upper-left region, residual propagation persists for a short time with an average direction toward the lower-right region. The x component (or y component) of the bipolar lead field is oriented in the same (or opposite) direction as the x (or y) component of the residual wave. Therefore, pV_x (or pV_y) shows positive (or negative) deflections during depolarization and negative (or positive) deflections during repolarization. pV_x and pV_y are approximately in phase with one another but have opposite signs (Figure 4C a, solid traces). The residual activity vector leads to large deflections in the pVCG in the lower right direction during depolarization and small deflections in the upper left direction during repolarization (Figure 4D, solid trace). To compute more balanced pECGs and pVCG, we can shift the lead placement so that the stimulus site is centered and then equalize the amount of tissue on all sides by taking an appropriate subset of optical recording sites. With this adjustment, the morphologies of pV_x and pV_y are altered (Figure 4C a, shaded traces), and pVCG remains close to the origin during wavefront propagation (Figure 4D, shaded trace). However, because the monolayer repolarizes nonuniformly in this example, the waveback propagates more slowly in the lower-right direction. This results in a pVCG that retains a small deflection in the upper-left direction (Figure 4D, shaded trace). pV₀ measured with the lead placed over the middle of the monolayer (Figure 4C b, solid trace) or centered over the stimulus site (Figure 4C b, shaded trace) measures the radial component of propagation, and the two are nearly identical. In the latter case, the unipolar lead field (Figure 1G) is completely aligned with the direction of propagation.

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Figure 4

(A) Normalized voltage map for radial propagation. (B) Isochrone map. (C) (a) Bipolar pECGs (pV_x and pV_y) for leads placed asymmetrically around (solid traces) or centered on (shaded traces) the stimulus site, and (b) unipolar pECG (pV₀) for lead offset from (solid trace) or centered on (shaded trace) the stimulus site. The shaded trace is covered by the solid trace at nearly all points and is difficult to see. (D) pVCG for bipolar leads placed asymmetrically (solid trace) or symmetrically (shaded trace) around the stimulus site. Different colors for the isochrones in panel B correspond to the instants of time shown in panel C.

In Fig. 5, a spiral wave is pinned to and propagating around a 3.5-mm-diameter hole near the center of the monolayer. pV_x and pV_y are periodic (at the rotation frequency of the spiral wave), sinusoidal, roughly equal in magnitude, and have an ∼90° phase difference (Figure 5C a, solid traces). pVCG is roughly circular (Figure 5D, solid trace), although with distinct bends because the hole is off-center and the amount of tissue differs on opposite sides of the hole. With a shift in the lead placement so that the hole is centered with equal amounts of tissue on all sides (like in the case of point stimulation), pV_x and pV_y are shifted slightly in time (Figure 5C a, shaded traces), and pVCG is more circular in shape (Figure 5D, shaded trace). Plotting pVCG in time (not shown) also reveals the direction of rotation (clockwise for this example). Slight variations in pV_x and pV_y amplitude and pVCG shape over successive cycles are due to small variations of the relative timing of the wavefront and waveback from cycle to cycle. With a unipolar lead placed directly above the center of the monolayer, a periodic waveform similar to albeit less sinusoidal than that obtained with the bipolar leads is observed (Figure 5C b, solid trace). Although the spiral wave moves primarily in a tangential direction perpendicular to the radially oriented unipolar lead field (Figure 1G), pV₀ oscillates at the spiral wave period because of the offset position of the unipolar lead from the center of the hole, which renders components of the lead field to be more sensitive to the wave movement. However, if the lead position is shifted so that it lies over the center of the hole, the lead field is relatively insensitive to wavefront propagation. pV₀ acquires a low-amplitude, harmonic component (Figure 5C b, shaded trace) at twice the reentry rate, which is a typical observation at the core of a spiral wave 6. Regardless of lead placement, pV₀ does not contain information regarding the direction of rotation.

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Figure 5

(A) Normalized voltage maps of spiral wave anchored to 3.5-mm-diameter hole. (B) Isochrone map. (C) (a) Bipolar pECGs (pV_x and pV_y) for leads placed asymmetrically (solid traces) or symmetrically (shaded traces) around hole, and (b) unipolar pECG (pV₀) for lead offset from (solid trace) or centered on (shaded trace) the hole. (D) pVCG for bipolar leads placed asymmetrically (solid trace) or symmetrically (shaded trace) around the hole. Different colors for the isochrones in panel B correspond to the instants of time shown in panel C.

A figure-eight reentry was initiated by rapid pacing followed by a premature stimulus. The reentry wave consists of a pair of entrained spiral waves that rotate in opposite directions and merge during part of the cycle along a common pathway. In the example shown in Figure 6A, the axis of the common pathway (the open line delineating the collision site of the two spiral wave bands) happens to slowly rotate in the clockwise direction with successive cycles. The lead voltages are periodic at the rotation frequency of the reentry, and the phase difference between pV_x and pV_y varies during each cycle and from cycle to cycle (cycles shown with different colors, Figure 6B a). pVCG is elliptical in shape, with the primary axis parallel to the common pathway (Figure 6C). The primary axis of the ellipse rotates in the clockwise direction with successive cycles. pV₀ is a noisy, irregular trace with varying frequency and amplitude, and information concerning the rotation frequency or common pathway of the reentry is absent (Figure 6B b).

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Figure 6

(A) Normalized voltage maps of rotating figure-eight reentry. (B) (a) Bipolar pECGs (pV_x and pV_y) and (b) unipolar pECG (pV₀). (C) pVCG. Different colors for the times in panel A correspond to different cycles and are the same as in panels B and C.

In Figure 7A, point stimulation near the edge produces a wave propagating mostly in the positive y direction, with a small component in the negative x direction (0–60ms). The wave moves through a heterogeneous region in the center of the monolayer, where conduction velocity is slower. The next stimulated wave breaks (80–100ms) and forms a spiral wave anchored to the heterogeneous region (120–210ms). pV_x and pV_y have broad deflections during paced propagation and appear sinusoidal during spiral wave propagation (Figure 7B a). pVCG mirrors the direction of paced propagation−pV_x is increasingly negative and pV_y is increasingly positive during wavefront propagation, leading to a deflection in the upper-left direction. pV_x, pV_y, and pVCG all demonstrate the transition from paced propagation to a spiral wave and is clearest for pVCG (Figure 7B aC). The transition is also apparent in the phase difference between the two lead voltages, changing from ∼0° (with opposite sign) to ∼90°, as indicated by the decreased tilts of the vertical timing marks (compare blue versus corresponding red marks). Like pV_x and pV_y, the value pV₀ shows a change in morphology upon the transition, with sharp deflections during paced propagation and a sinusoidal shape during the spiral wave (Figure 7B b).

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Figure 7

(A) Normalized voltage maps during transition from paced wave to spiral wave. (B) (a) Bipolar pECGs (pV_x and pV_y) and (b) unipolar pECG (pV₀). Vertical bars have been added to mark the relative timing of the peaks and valleys. (C) pVCG. Different colors in the times in panel A correspond to different propagation patterns and are the same as in panels B and C.

A more complex transition between electrical behaviors is illustrated in Fig. 8. As shown in Figure 8A, the monolayer was initially paced from a point stimulus electrode near the edge (0ms). A heterogeneous region in the center of the monolayer caused wavebreak (20ms), leading to a figure-eight reentry after pacing ended (60–235ms). The figure-eight reentrant wave eventually transitioned into a single spiral wave rotating around the center region (705–955ms). For clarification, the times are colored differently during the three propagation patterns. The bipolar lead voltages transition from single deflections to sinusoids, as behavior changes from paced propagation to reentry (Figure 8B). The lead voltages are in phase with opposite sign during paced propagation and are on average ∼90° out of phase during spiral wave propagation, as shown by the decreased tilts of the vertical timing marks. The phase difference varies during figure-eight reentry. Both transitions are difficult to identify from a single lead voltage trace. However, pVCG differentiates the three behaviors clearly (Figure 8C). The initial pacing is illustrated by a fairly linear pVCG (red trace), which becomes elliptical for the figure-eight reentry (blue trace) and circular for the single spiral wave (green trace). Regarding pV₀, it is not clear from this signal alone what type of propagation is initially present, although some kind of transition is evident by the large jump in amplitude after ∼600ms that is followed by a more sinusoidal shape. The transition from figure-eight reentry to single spiral wave also cannot be specifically identified, although some kind of transition is apparent by the appearance of a high frequency component during the last 700ms of the trace (Figure 8B b).

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Figure 8

(A) Normalized voltage maps during transition from paced wave to figure-eight to spiral wave. (B) (a) Bipolar pECGs (pV_x and pV_y) and (b) unipolar pECG (pV₀). Vertical bars have been added to mark the relative timing of the peaks and valleys. (C) pVCG. Different colors in the times in panel A correspond to different propagation patterns and are the same as in panels B and C.

For comparison, pVCG was computed at other heights (, and ) for the six examples presented earlier. pVCGs at (Figure 9A) are distorted from the pVCGs at (Figure 9B). For the planar wave, pVCG loses its linear shape. The spiral wave pVCG is less circular and has sharper corners. The figure-eight pVCG does not have a distinctly elliptical shape. The transition from paced propagation to spiral wave reentry (Transition 1) is identifiable, since the pVCG shape remains very different for the two behaviors. However, both the linear and circular shaped regions are distorted from their shapes at . Finally, the transition from paced propagation to figure-eight reentry (Transition 2) is not discernible. pVCGs at (Figure 9CD) are similar in all cases to pVCGs at , although with lower amplitudes. Hence, we conclude that is an effective operating height for the bipolar lead that yields lead signals that represent the global electrical behavior of the cell sheet in a manner akin to those of remote bipolar leads, while at the same time maintaining the amplitude of the signals (Fig. 1).

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Figure 9

pVCGs calculated at (A) , (B) , (C) , (D) , and pVCG^D (E) for six cases of wavefront propagation. The example for the radial wave has been centered. Each plot has been normalized to its own peak amplitude. pVCGs computed at , , , and , have unnormalized relative amplitudes of ∼2.5, 1, 0.3, and 0.0042, respectively.

pVCG^D values calculated using the difference method are shown in Figure 9E for the six examples presented earlier using the theoretical approach. For the most part, pVCG^D is similar to pVCG at , with some notable differences. For the example of radial propagation, pVCG^D does not show the small deflection found in pVCG owing to nonuniform repolarization, and for the anchored spiral wave, pVCG^D is rotated relative to pVCG. For Transition 2, pVCG^D is not as linear as pVCG during paced propagation, is more circular during the figure-eight reentry, and does not have as distinct a transition between figure-eight and single spiral wave reentry. We computed the root mean-square difference between normalized pVCG at and normalized pVCG^D, and the average percent error was 10.5% for all examples, with the largest differences for the radial (19.6%) and spiral (12.4%) waves.

Although not in prevalent use today 19, the VCG has been used clinically to provide morphological interpretation of the electrical phenomena of the heart and continues to have diagnostic advantages over the ECG in certain situations 20. The VCG was first constructed in the frontal plane, based on the Einthoven limb leads 21. Since then the VCG was expanded to three dimensions 22, and compound lead systems were proposed to generate a set of orthogonal, corrected leads that compensated for internal inhomogeneities in torso conductance and geometry 23,24,25. Lead field analysis of the commonly used VCG systems quantified the uniformity and sensitivity of the leads in each dimension 26. Although the two-dimensional sheet lying in a bath is much simpler compared with the whole heart lying in the torso, lead field analysis has not been applied in this context and can be used to determine the sensitivity of the recording leads for the pseudo-ECG to the underlying cellular transmembrane potentials.

In this article, we present a formulation of bipolar pECG and pVCG measurements of a two-dimensional cardiac sheet (cell monolayer) and analyze their dependence on lead placement. Further, we determine a mathematical relation for an effective operating height for the bipolar lead placed over opposite edges of the cell monolayer. Lead placement at half the effective height provides a pVCG that overweights the contribution of cells near the lead electrodes (Fig. 1), whereas pVCGs for bipolar leads at twice or 10-times is similar to that at (Fig. 9) but with lower amplitude. Therefore, can serve as the operating height that balances global sensitivity and amplitude for pECG and pVCG computation.

Numerically calculated bipolar and unipolar leads have been used previously to approximate the clinical ECG for one- and two-dimensional studies 9,10,11,13. The numerical bipolar leads are analogous to the Einthoven limb leads, which measure the global activity of the heart. The numerical unipolar lead is analogous to the precordial leads, V₁–V₆ of the 12-lead system, that are placed on the chest to detect localized activity in the region of the heart closest to the lead 27,28. We find that for the case of a unipolar lead situated above a cardiac cell sheet, the unipolar pECG is most sensitive to activity near the electrode, but importantly, it is insensitive to activity at the point directly underneath the electrode (Figure 1H). This is because at that location, electrical propagation is perpendicular to the three-dimensional lead field. However, in the three-dimensional heart, electrical activity can propagate parallel to the lead field, and therefore, the precordial leads are most sensitive to activity directly underneath that is propagating toward (or away from) the electrode.

There is growing use of monolayers as an in vitro system for investigating cardiac electrical behavior 29,30, and optical maps of transmembrane potential have enabled the study of different patterns of electrical propagation, as well as transitions between them in the context of arrhythmia 31. Electrical recordings of the VCG would be a valuable adjunct to optical maps because of their succinct representation of the electrical behavior, but are technically very difficult to obtain, owing to the extremely small extracellular potentials generated by a monolayer of cells in a bath. If we consider a 17-mm-diameter cardiac monolayer with intracellular conductivity σ_i=2.5mS/cm 32, a monolayer thickness of 10μm 33, placed in a bath with conductivity σ_b=20mS/cm 34, and assume an action potential amplitude of 100mV 35, a maximum transmembrane voltage upstroke velocity, (dV_m/dt)_max, of 125V/s 36, and a conduction velocity of 25cm/s 37, an ECG measurement would have a peak value of ∼4μV. Instead, pECG (and pVCG) can be determined from the optical maps of V_m using the methodology described in this study. Further, Iravanian and Christini have recently shown that it is possible to process optical signals for real-time control capability 38. Since pECG is a weighted sum of V_m values, it can also be computed in real-time.

For two-dimensional propagation in cardiac cell monolayers, we have shown that the bipolar pVCG capably represents electrical propagation associated with plane waves, radial waves, reentrant spiral waves, figure-eight spiral waves, and transitions between them. Transitions from normal pacing behavior to a reentrant wave, or from a reentrant wave to multiple waves, are crucial cardiac events related to arrhythmia, and their detection is important. Plane waves appear as linear trajectories, radial waves as trajectories localized to the origin, single spiral waves as circles, and figure-eight spiral waves as ellipses. Phase differences between the bipolar lead voltages can also help to identify propagation patterns. The phase difference is ∼0° during paced propagation, 90° during spiral wave propagation, and variable during figure-eight reentries. In contrast, the unipolar pECG does not represent some of these electrical activities well nor the transition between them. Key information such as direction of propagation or frequency of reentry is missing or may be distorted.

Finally, with the advent of optical mapping of V_m from tissue surfaces, bipolar pECGs (lead voltages ) were calculated in previous studies by using an ad hoc method that subtracts the sum of V_m on one-half of the mapped area from the sum of the other half 5,6,7,8,12. However, we have shown that two sets of geometric functions are necessary to correctly sum V_m for circular cardiac cell monolayers (sheets)—one set (γ_x and γ_y) that is applied over the area of the monolayer and the other set (α_x and α_y) that is applied over the perimeter of the monolayer. Their usage produces bipolar lead voltages pV_x and pV_y that are biophysically based, and also fast to compute, compared with alternative calculations based on the gradients of V_m.