The Pinwheel Experiment Revisited

J. theor. Biol. (1998) 190, 389–393

The Pinwheel Experiment Revisited B J. R* Department of Physics & Astronomy, Vanderbilt University, Box 1807, Station B, Nashville, TN 37235, U.S.A. (Received on 9 June 1997, Accepted in revised form on 9 October 1997)

The critical point hypothesis explains the origin of some cardiac arrhythmias, and the bidomain model describes electrical stimulation of the heart. In this paper, the critical point hypothesis is combined with the bidomain model. The result is four new predictions about the pinwheel experiment, a fundamental experiment in cardiac electrophysiology. These are: (1) The duration of the vulnerable period during cathodal S2 stimulation is longer for an S1 wavefront propagating perpendicular to the fibers than for an S1 wavefront propagating parallel to the fibers. (2) For anodal S2 stimulation with the S1 wavefront propagating parallel to the fibers, the vulnerable period splits into two periods with an ‘‘invulnerable period’’ between them. (3) For anodal S2 stimulation with the S1 wavefront propagating perpendicular to the fibers, the vulnerable period consists of only one period. (4) A previously suggested mechanism for the upper limit of vulnerability (S2 is so strong that the entire tissue is depolarized by an amount greater than S*) is no longer applicable. 7 1998 Academic Press Limited

1. Introduction How does an applied electrical current initiate a cardiac arrhythmia? Answering this question is essential for understanding important medical devices, such as the cardiac pacemaker and defibrillator (Zipes & Jalife, 1995). Many arrhythmias arise when wavefronts of electrical activity propagate through the heart along abnormal paths. In *E-mail: roth.compsci.cas.vanderbilt.edu †The term ‘‘pinwheel experiment’’ originates in Winfree’s publications (1980, 1983, 1987, 1989). He gave the experiment this name ‘‘because it was designed to nucleate a pinwheel-like circulating wave’’ (Winfree, 1989). Winfree defines the term in a general way (see the glossary of his 1987 book, p. 296), but his primary example of a pinwheel experiment is one in which a planar S1 wavefront interacts with a point S2 stimulus (see the illustrations of such an experiment in his 1987 book, pp. 128–131). In my paper, I use the term ‘‘pinwheel experiment’’ to refer to this specific experiment, in which S2 is delivered through a point electrode. ‡Winfree (1983) used the term ‘‘singular point hypothesis,’’ but Frazier et al. (1989) later renamed it the ‘‘critical point hypothesis’’ and it is their name that is now widely used. §I restrict my discussion to two dimensions, ignoring complications such as the rotation of myocardial fibers across the heart wall. 0022–5193/98/040389 + 05 $25.00/0/jt970565

particular, propagation can occur around one or more closed loops, forming a re-entrant circuit. My goal is to understand how an electrical stimulus initiates a re-entrant arrhythmia. Specifically, I combine the critical point hypothesis for re-entry induction (Winfree, 1989) with the bidomain model of electrical stimulation (Sepulveda et al., 1989). The result is four new predictions about a fundamental experiment in cardiac electrophysiology: the ‘‘pinwheel experiment.’’† 2. The Critical Point Hypothesis Winfree (1983, 1987, 1989) proposed a mechanism for the induction of cardiac arrhythmias by electrical stimulation: the critical point hypothesis.‡ The pinwheel experiment in Fig. 1 illustrates this hypothesis. A planar action-potential wavefront (S1) propagates to the left through a two-dimensional sheet of cardiac tissue.§ The thick line corresponds to the depolarization phase of the action potential (the leading edge of the wavefront), and the shaded region corresponds to the plateau and repolarization phases. 7 1998 Academic Press Limited

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390

T

*

S*

F. 1. The pinwheel experiment. An S1 wavefront propagates to the left: dashed line = edge of tissue; thick line = depolarization front; shaded region = refractory tissue; T* = critical phase; dot = stimulus electrode, S* = critical depolarization induced by the S2 stimulus. The S2 stimulus initiates two rotors, located where S* and T* intersect. One rotor rotates counterclockwise about the upper intersection, and the other rotates clockwise about the lower intersection.

The tissue is refractory within the shaded region. Winfree (1989) defines the ‘‘critical phase’’, T*, as the time during the cardiac action potential when a stimulus of appropriate strength results in an arrhythmia. As a rough approximation, T* occurs near the end of the cardiac refractory period. The dot in Fig. 1 indicates the position of a stimulus electrode. As T* is passing near the electrode, apply a stimulus current pulse (S2). In Winfree’s discussion of the pinwheel experiment, he assumed that a cathodal stimulus current depolarizes the tissue; the strength of the depolarization falls off with distance; and the effect of the stimulus is isotropic, so that transmembrane potential contours are circles concentric with the electrode (Winfree, 1989). There exists some critical depolarization S* (circle in Fig. 1) that corresponds to threshold for cardiac tissue that is in the state T*. The critical point hypothesis is that each point where S* intersects T* becomes a rotor—the core of a rotating wave (Winfree, 1987). Rotors can exist in cardiac tissue (Davidenko et al., 1992) and they may underlie many arrhythmias (Zipes & Jalife, 1995). Two rotors exist in Fig. 1, located where S* and T* intersect. One rotor rotates counterclockwise around the upper intersection, and another rotates clockwise around the lower intersection. As long as the two rotors are far enough apart that they do not mutually annihilate each other, they coexist and form a stable figure-of-eight re-entrant circuit (Zipes & Jalife, 1995). Shibata et al. (1988) verified the critical point hypothesis by performing the pinwheel experiment in a dog heart.

Winfree (1989) used the critical point hypothesis to estimate the ‘‘vulnerable period duration.’’ As the S1 action potential propagates past the electrode, there exists only a brief window of time, called the ‘‘vulnerable period,’’ for which an S2 stimulus can initiate a rotor. If the S2 stimulus occurs too early or too late, T* is too far to the right or left of the electrode to intersect S*. The time when S* intersects T* is the vulnerable period. The duration of the vulnerable period is approximately equal to the diameter of S* divided by the conduction velocity of the S1 wavefront (see Winfree, 1989, fig. 11a). An S2 stimulus strength exists below which an arrhythmia or fibrillation occurs, but above which they do not. In the cardiac literature, this stimulus strength is called the ‘‘upper limit of vulnerability’’ (Shibata et al., 1988). Winfree (1987, 1989) suggested a mechanism for this phenomenon: ‘‘if [the S2 stimulus] were so strong that the entire medium is stimulated beyond S*, there would be no intersections [of S* and T*]. Thus, there is an upper limit to vulnerability’’ (Winfree, 1989). 3. The Bidomain Model My plan is to modify Winfree’s pinwheel experiment by using a more accurate shape for S* determined from the bidomain model (Henriquez, 1993). The bidomain model is a two- or three-dimensional cable model that takes into account the anisotropy of the intracellular and extracellular spaces (Roth, 1992). It represents cardiac tissue as a macroscopic continuum rather than as a microscopic collection of discrete cells (Neu & Krassowska, 1993). The electrical properties of the tissue depend upon direction because of the underlying geometry of the myocardial fibers. The ratios of conductivities parallel and perpendicular to the fibers differ in the intracellular and extracellular spaces (Roth, 1997a), a condition known as ‘‘unequal anisotropy ratios.’’ Many of the interesting predictions of the bidomain model arise because of unequal anisotropy ratios (Roth & Wikswo, 1996). Sepulveda et al. (1989) used the bidomain model to calculate the transmembrane potential distribution induced when a two-dimensional sheet of cardiac tissue having unequal anisotropy ratios is stimulated at a point. (This corresponds to unipolar stimulation with a cathode, where the distant anode is far enough away that it plays no role in our discussion.) They found that under the cathode the contours of depolarization are neither circles nor ellipses, but have a ‘‘dog-bone’’ shape (see contour +1 in Fig. 2). The depolarization extends farther in the direction

    1 0

0

2 3 4

–1

–1 –2

–3

–3

–2

F. 2. The steady-state transmembrane potential induced by an extracellular unipolar electrode in a two-dimensional, passive bidomain with unequal anisotropy ratios. The dot indicates the position of the electrode. Fibers are oriented horizontally. The units of transmembrane potential are arbitrary. Tissue having a depolarization greater than four is shaded. o = 0.75, a = 1, and lL/lT = 2.5 [using parameters defined by Roth (1997a)]

perpendicular to the myocardial fibers than in the direction parallel to them. This orientation is opposite that which might be expected from the anisotropy of the conduction velocity, which is faster parallel to the fibers than perpendicular to them. Furthermore, regions of hyperpolarization exist near the cathode along the fiber direction (Fig. 2). If the unipolar stimulus electrode is an anode rather than a cathode, then the stimulus hyperpolarizes the tissue under the anode and depolarizes the tissue along the fiber direction. These predictions have been verified theoretically (Roth, 1992, 1997b) and experimentally (Wikswo et al., 1991; Neunlist & Tung, 1995; Knisley, 1995; Wikswo et al., 1995). 4. Critical Point Hypothesis and the Bidomain Model Combined The combination of the critical point hypothesis and the bidomain model provides important new insights into the electrical behavior of cardiac tissue (Roth & Krassowska, 1998). In particular, it results in new predictions about the pinwheel experiment. Consider an experiment that is identical to the one described in Fig. 1, except that the critical depolarization contour, S*, is one of the transmembrane potential contours shown in Fig. 2. Because S* is not radially symmetric about the electrode, the response of the tissue to electrical stimulation depends on the direction of propagation of the S1 wavefront. Because the tissue is depolarized at different locations during cathodal and anodal stimulation, the response of the tissue depends on the polarity of the S2 stimulus. And because not only the size but also the shape of the contour lines change for different values

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of the transmembrane potential (Fig. 2), the response of the tissue depends on the strength of the S2 stimulus. Consider a strong cathodal S2 stimulus and an S1 wavefront propagating parallel to the myocardial fibers [Fig. 3(a) and (b)]. In this case, S* has a dog bone shape, like the contour +1 in Fig. 2. S* and T* intersect at either two or four locations. Early and late in the vulnerable period the stimulus creates four rotors [Fig. 3(b)], and during the middle of the vulnerable period the stimulus creates two rotors [Fig. 3(a)]. S* is relatively narrow in the direction parallel to the fibers. This shape for S*, together with the relatively fast conduction velocity parallel to the fibers, implies a short duration of the vulnerable period. If the S1 wavefront propagates perpendicular to the myocardial fibers, a strong cathodal S2 stimulus initiates at most two rotors [Fig. 3(c)]. S* is relatively wide in the direction perpendicular to the fibers, and the conduction velocity in that direction is relatively slow, implying a long duration of the vulnerable period. Therefore, my first prediction is that the duration of the vulnerable period during cathodal S2 stimulation is longer for an S1 wavefront propagating perpendicular to the fibers than for an S1 wavefront propagating parallel to the fibers. How much longer? The height of S* in the direction transverse to the fibers is about 1.4 times greater than the maximum width of S* in the direction parallel to the fibers (Fig. 2). The conduction velocity is about 2.5 times faster parallel to the fibers than perpendicular to them (Roth, 1997a). Thus, the duration of the vulnerable period should be about 3.5 times greater for S1 propagating perpendicular to the fibers than for S1 propagating parallel to them. This is a rough estimate only, because the shape of S* differs in the two directions, and the location and number of rotors may differ in the two cases. A quantitative determination of the vulnerable period is impossible without performing more detailed simulations or experiments. During anodal stimulation, S* consists of two distinct curves, displaced from one another in the direction parallel to the fibers (Fig. 4). For an S1 wavefront propagating parallel to the fibers, S* intersects T* in two locations if the S2 stimulus occurs early or late in the vulnerable period [Fig. 4(a)]. But S* does not intersect T* if the S2 stimulus occurs when T* is under the anode [Fig. 4(b)]. Therefore, my second prediction is that for anodal S2 stimulation with an S1 wavefront propagating parallel to the fibers, the vulnerable period splits into two periods, with an ‘‘invulnerable period’’ between them when T*

. . 

392 (a)

T*

S*

is under the anode.† If the S1 wavefront propagates perpendicular to the fibers, S* and T* intersect at four locations throughout the vulnerable period [Fig. 4(c)]. Therefore, my third prediction is that for anodal S2 stimulation with an S1 wavefront propagating perpendicular to the fibers, the vulnerable period consists of only one period, centered on the time when T* passes under the anode. The shape of S* has implications for the mechanism of the upper limit of vulnerability. The

(a) (b) T* S

*

S*

S*

T*

(b) T*

(c)

S*

S*

S*

(c) T

*

S*

F. 3. Initiation of rotors after a cathodal S2 stimulus. The dot indicates the position of the electrode. Fibers are oriented horizontally. When the S1 wavefront propagates parallel to the fiber direction (to the left), a dog-bone shaped S* results in either (a) two, or (b) four rotors. (c) When the S1 wavefront propagates perpendicular to the fiber direction (downward), only two rotors occur.

†The vulnerable period may also be split if the critical stimulus contour intersects the boundary of the tissue, as shown by Winfree in fig. 6.5 of his 1987 book. However, my prediction holds even for a very large tissue, in which boundary effects are unimportant.

S*

T*

F. 4. Initiation of rotors after an anodal S2 stimulus. The dot indicates the position of the electrode. Fibers are oriented horizontally. The S* contour consists of two distinct curves, one to the right and one to the left of the stimulus electrode. When the S1 wavefront propagates parallel to the fiber direction (to the left), the S2 stimulus initiates either (a) two or (b) no rotors. (c) When the S1 wavefront propagates perpendicular to the fiber direction (downward), four rotors occur.

    mechanism suggested by Winfree (1987, 1989) required that S2 be so strong that S* moved outside the tissue boundary. However, the bidomain model suggests that S* always passes near the electrode regardless of the stimulus strength or polarity (Fig. 2). Therefore my fourth prediction is that when the bidomain model is combined with the critical point hypothesis, the mechanism suggested previously by Winfree (1987, 1989) for the upper limit of vulnerability (S2 is so large the entire tissue is depolarized by an amount greater than S*) is no longer applicable. However, interesting behavior occurs as the outermost sections of S* approach the edge of the tissue. For both cathodal and anodal stimulation, a very strong S2, for which the outer edge of S* leaves the tissue, decreases the maximum number of rotors from four to two. The remaining two rotors may be too close together to survive. If so, then an upper limit of vulnerability exists, although it arises by a somewhat different mechanism than Winfree postulated. In addition, for a very strong cathodal S2 stimulus and an S1 wavefront propagating parallel to the fibers [Fig. 3(a)], the vulnerable period splits into two periods with an intermediate invulnerable period, just as for anodal stimulation. I thank Art Winfree for his comments and suggestions about this manuscript. His original analysis of the pinwheel experiment motivated this study. This research was supported by the American Heart Association–Tennessee Affiliate and the Whitaker Foundation.

REFERENCES D, J. M., P, A. V., S, R., B, W. & J, J. (1992). Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355, 349–351. F, D. W., W, P. D., W, J. M., T, A. S, L., S, W. M. & I, R. E. (1989). Stimulus-induced critical

393

points: mechanism for electrical induction of reentry in normal canine myocardium. J. Clin. Invest. 83, 1039–1052. H, C. S. (1993). Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21, 1–77. K, S. B. (1995). Transmembrane voltage changes during unipolar stimulation of rabbit ventricle. Circ. Res. 77, 1229–1239. N, J. C. & K, W. (1993). Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21, 137–199. N, M. & T, L. (1995). Spatial distribution of cardiac transmembrane potential around an extracellular electrode: dependence on fiber orientation. Biophys. J. 68, 2310–2322. R, B. J. (1992). How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle. J. Math. Biol. 30, 633–646. R, B. J. (1997a). Electrical conductivity values used with the bidomain model of cardiac tissue. IEEE Trans. Biomed. Eng. 44, 326–328. R, B. J. (1997b). Approximate analytical solutions to the bidomain equations with unequal anisotropy ratios. Phys. Rev. E 55, 1819–1826. R, B. J. & K, W. (1998). The induction of reentry in cardiac tissue. The missing link: How electric fields alter transmembrane potential. Chaos (in press). R, B. J. & W, J. P. J. (1996). The effect of externally applied electrical fields on myocardial tissue. Proc. IEEE 84, 379–391. S, N. G., R, B. J. & W, J. P. J. (1989). Current injection into a two-dimensional anisotropic bidomain. Biophys. J. 55, 987–999. S, N., C, P.-S., D, E. G., W, P. D., D, N. D., S, W. M. & I, R. E. (1988). Influence of shock strength and timing on induction of ventricular arrhythmias in dogs. Am. J. Physiol. 255, H891–H901. W, J. P. J., W, T. A., A, W. A., B, J. R., K, H. A. & R, D. M. (1991). Virtual cathode effects during stimulation of cardiac muscle. Circ. Res. 68, 513–530. W, J. P. J., A, R. A. & L, S.-F. (1995). Virtual electrodes in cardiac tissue: A common mechanism for anodal and cathodal stimulation. Biophys. J. 69, 2195–2210. W, A. T. (1980). The Geometry of Biological Time. New York: Springer-Varlag. W, A. T. (1983). Sudden cardiac death: A problem in topology. Sci. Am. 248, 144–161. W, A. T. (1987). When Time Breaks Down. Princeton, NJ: Princeton University Press. W, A. T. (1989). Electrical instability in cardiac muscle: phase singularities and rotors. J. theor. Biol. 138, 353–405. Z, D. P. & J, J. (1995). Cardiac Electrophysiology, From Cell to Bedside, 2nd Edn. Philadelphia: W. B. Saunders Company