INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Clinical observations (Pratt et al., 1983; Nikolic et al., 1982) indicate that ventricular fibrillation (VF) is almost always preceded by ventricular tachycardia (VT) of variable duration, ranging from a few to many beats. High-resolution cardiac mapping studies show that electrically induced VF starts as a rotating reentrant wave which is unstable and rapidly begins to meander and/or break up into multiple wavefronts, producing the electrocardiographic patterns of polymorphic VT (Gray et al., 1995) and VF (Chen et al., 1988). However, the mechanisms of the transition from VT to VF remain poorly understood. It is essential to gain a better understanding of this process to develop effective pharmacological antifibrillatory therapy.

During VT, ventricular myocytes are driven at rapid rates at which the fraction of the cardiac cycle spent in diastole decreases, leaving less time for intracellular Ca²⁺ removal and relaxation. Rapid pacing leads to elevated diastolic intracellular Ca²⁺ levels, and in many species greater filling of sarcoplasmic reticulum (SR) Ca²⁺ stores (Bassani et al., 1995; McCall et al., 1998). It is well established (Wit and Janse, 1993) that intracellular Ca²⁺ accumulation predisposes the myocardium to abnormal electrical activity such as delayed and early afterdepolarizations (DADs and EADs), and may initiate VF by itself (Koretsune and Marban, 1989; Lakatta and Guarnieri, 1993). Intracellular Ca²⁺ modulates the shape and duration of action potential (AP) through its effects on different membrane currents such as the Na⁺-Ca²⁺ exchanger current (I_NaCa), the L-type channel current (I_Ca(L)), and the Ca²⁺-activated nonselective current (I_ns(Ca)), thereby altering electrophysiological properties such as refractoriness and membrane depolarization rate. Thus, it is reasonable to postulate that if intracellular Ca²⁺ overload develops during VT, it could facilitate the transition into VF. To address this issue we used a combined experimental and simulation approach. First we measured intracellular Ca²⁺ transients in rabbit myocytes during pacing at rapid rates comparable to VT. Using these results, we modified the LR ventricular action potential model (Luo and Rudy, 1994; Zeng et al., 1995) to match its behavior to the experimentally obtained myocyte data. To accomplish this, we initially reformulated relaxation of intracellular Ca²⁺ transient and the mechanisms of Ca²⁺ release from the SR to incorporate recent physiological findings (Bassani et al., 1995; López-López et al., 1995; Yao et al., 1997) into the model's intracellular Ca²⁺ dynamics.

We then simulated wave propagation in a two-dimensional (2D) uniform isotropic cardiac tissue using the modified ventricular action potential model, and examined the influence of intracellular Ca²⁺ dynamics on the stability of spiral wave reentry as a model of the VT to VF transition. The results suggest that intracellular Ca²⁺ dynamics independently affect the stability of spiral wave reentry, which may contribute to the breakup of reentrant wavefronts that converts VT into VF.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Cell isolation and patch clamp methods

Ventricular myocytes were isolated enzymatically from the hearts of 2-3-kg adult New Zealand white rabbits using standard methods as described previously (Goldhaber and Liu, 1994). The cells were stored until use in standard Tyrode's solution containing (in mM): 136 NaCl, 5.4 KCl, 0.33 Na₂PO₄, 1.8 CaCl₂, 1 MgCl₂, 10 dextrose, and 10 HEPES-NaOH, pH 7.4. This storage solution was also used as the experimental bath solution. All experiments were carried out at 33-35°C.

We used the amphotericin perforated patch technique (Rae et al., 1991) to obtain whole-cell recordings of membrane voltage under current clamp conditions or to apply an action potential voltage clamp. Briefly, patch electrodes with a tip diameter of 2-3 µm and a resistance of 2-3 M were dipped for ~10 s into a pipette solution containing (in mM): 140 potassium aspartate, 5 NaCl, 10 HEPES, 1 EGTA, 5 MgATP, 5 creatine phosphate, 0.05 cAMP pH 7.2 with HCl. The electrode was then back-filled using the same pipette solution containing 240 µg/ml amphotericin-B (A4888; Sigma, St. Louis, MO) added. A gigaseal was rapidly formed using standard techniques (Hammil et al., 1981). Typically, 10 min later, amphotericin pores lowered the resistance sufficiently to current or voltage clamp the cells. Membrane current and voltage were measured with an Axopatch 200 patch clamp amplifier controlled by a personal computer using a Digidata 1200 acquisition board driven by pCLAMP 6.0 software (Axon Instruments, Foster City, CA).

Fluorescence measurements

Cells were loaded with the calcium indicator fura-2 by incubating them for 20 min in bath solution containing 5 µM fura-2 AM (Molecular Probes, Eugene, OR) and 0.016% (wt/wt) pluronic (Molecular Probes). Cells were then washed twice with indicator-free bath solution and placed in a chamber mounted on the heating stage of an inverted microscope modified for simultaneous patch clamping and high-speed intracellular Ca²⁺ measurements using fura-2 (Goldhaber et al., 1991). The intensity of fura-2 fluorescence emission at 510 nm was measured by a photomultiplier during alternate excitation (1200 Hz) at 335 nm and at 405 nm. The ratio (R) of the two fluorescence wavelengths (335/405) is a reflection of intracellular Ca²⁺ (Grynkiewicz et al., 1985). Fluorescence ratios were pseudo-calibrated to intracellular [Ca²⁺] using the following method. We assumed systolic and diastolic Ca²⁺ values of 820 and 200 nM, respectively, during pacing at 1 Hz. These values were chosen to correspond to the values used in the computer simulation. Fura-2 fluorescence ratios (R) are approximately linearly related to [Ca²⁺] up to 10 times the K_d for Ca²⁺ binding. Thus we were able to relate the R values at systole and diastole to systolic and diastolic Ca²⁺ using the following equations:

(1)

(2)

(3)

where Ca_systole is the diastolic [Ca²⁺] during pacing at 1 Hz and Ca_diastole is the diastolic [Ca²⁺]. R_systole is the value of the fluorescence ratio during systole, and R_diastole is the value of the ratio during diastole. The accuracy of this method was checked against the standard Ca²⁺ calibration equation described by Grynkiewicz et al. (1985), and gave essentially identical results.

Pacing protocols

Myocytes were either paced in the current clamp mode using a twice diastolic threshold 2 ms current pulse or in the voltage clamp mode using as the command waveform an action potential waveform recorded previously from a healthy rabbit ventricular myocyte. The action potential waveforms were recorded at several different pacing cycle lengths so that waveforms with a range of action potential durations could be used for pacing under voltage clamp conditions.

The pacing protocol was as follows: cells were paced sequentially for ~16 s at the following cycle lengths (CL) (in ms): 1000, 500, 400, 300, 180. Fura-2 fluorescence was recorded intermittently for 4-s periods to limit exposure of the cells to the ultraviolet excitation beam.

Simulation methods

Ventricular action potential generation and propagation model

(4)

Reformulation of intracellular Ca²⁺ handling

View larger version (9K): [in this window] [in a new window]   FIGURE 1   Probability P(V) of a localized Ca2+ spark as a function of membrane potential. The black points show the data obtained experimentally by López-López et al. (1995) and the solid line shows our fit to it (see expression for P(V) in Appendix).

Computer implementation

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Intracellular Ca²⁺ dynamics in paced isolated ventricular myocytes: comparison with the model

Fig. 2 A shows APs and intracellular Ca²⁺ transients recorded at different pacing rates in an isolated rabbit ventricular myocyte, loaded with fura-2 AM, using the perforated patch clamp technique at 35°C. Note that as pacing rate increased, diastolic and systolic Ca²⁺ increased progressively. Rapid pacing eventually led to alternation of the AP and intracellular Ca²⁺ transient. To determine whether alternation was caused by the action potential or by intracellular Ca²⁺ handling, myocytes were paced in the voltage clamp mode using an action potential waveform. Under these conditions, membrane voltage was identical from beat to beat. Therefore, any beat-to-beat alterations in the intracellular Ca²⁺ transient must be an inherent property of intracellular Ca²⁺ dynamics, and not secondary to the alterations in the AP. Fig. 2 B shows that under these conditions, alternation of the intracellular Ca²⁺ transient still occurred at rapid pacing rates, indicating that cellular Ca²⁺ handling processes are inherently capable of nonlinear behavior. Note that at CL = 1000 ms the AP digitized for the AP clamp in Fig. 2 B was longer than the paced free-running AP in Fig. 2 A (due to the substantial cell-to-cell variation APD at long CLs in rabbit myocytes), at shorter CLs the AP clamps and paced free-running APs were comparable.

View larger version (32K): [in this window] [in a new window]   FIGURE 2   Changes in the AP and [Ca2+]i transient during rapid pacing in isolated rabbit ventricular myocytes. (A) Membrane potential (V), and [Ca2+]i transients during rapid pacing at cycle lengths (CL) indicated. (B) Same as A, except the myocyte is paced with an AP clamp. Note that alternation in the transient still occurs at CL = 180 ms, despite a fixed APD.

Fig. 3 A shows simulated AP for the modified LR model incorporating our changes in intracellular Ca²⁺ dynamics and paced at 1000 ms CL. The AP waveform and duration, as well as the maximum membrane depolarization rate, are very close to that of the original LR model. The values of diastolic and systolic [Ca²⁺]_i at this CL lie in a typical experimentally observed range (see Fig. 3 B). In contrast to the original LR model, however, the new model produces fractional rather than complete Ca²⁺ release from the JSR (~52% is depleted; see Fig. 4 C), consistent with physiological data (Bassani et al., 1995).

View larger version (34K): [in this window] [in a new window]   FIGURE 3   Validation of the computer model. A-AP at CL = 400 ms. B-minimum and maximum [Ca2+]i during pacing with various CLs ranging from 280 ms to 1000 ms. Circles indicate experimental results (solid and unfilled symbols are from different myocytes); lines indicate computer simulation. C-[Ca2+]i transients at CL = 400 ms (left panel) and CL = 250 ms (right panel). Transients display alternans at CL = 250. D-comparison of experimentally measured relaxation of [Ca2+]i transients in isolated myocytes (left panel from Yao et al.) with our computer model (right panel); thin curve, control; thick lines with Na+-Ca+ exchange blocked. E-effect of -[Ca2+]i transients on the shape of AP. The short AP (thin line) is asociated with the small -[Ca2+]i transient (thin line) in both model (left) and an isolated ventricular myocyte (right) in which two successive beats during alternans were superimposed.

View larger version (38K): [in this window] [in a new window]   FIGURE 4   Computer model during pacing at CL = 150 ms. (A) AP. (B) Free [Ca2+] in myoplasm. (C) Free [Ca2+] in JSR. (D) Spontaneous Ca2+ release current. (E) Superimposed INaCa and Ins,(Ca) as functions of membrane potential during 13th an 14th APs from A. Currents during 14th (longer) AP are shown by bold lines.

We next compared the values of diastolic and systolic [Ca²⁺]_i at various pacing CLs with the experimental data from isolated rabbit ventricular myocytes. The results, summarized in Fig. 3, B and C, show comparatively good agreement with experimental data. Ca²⁺ dynamics show the phenomenon of Ca²⁺ accumulation and alternans at shorter CLs. Alternans of [Ca²⁺] transient is caused by activation of the spontaneous Ca²⁺ release current, and leads to a slight (2 ms) alternans in APD. The onset of alternation in the AP and Ca²⁺ transient occurs at a bit shorter CL for the myocytes, however (see Fig. 2), and has larger amplitude.

Fig. 3 D demonstrates how relaxation of Ca²⁺ transient in the model (right panel) compares with experimental data (left panel) from Yao et al. (1997). We obtained close agreement for both normal relaxation and relaxation with Na⁺-Ca²⁺ exchanger blocked. For normal relaxation we used CL = 4000 ms, [Ca²⁺]_o = 1.08 mmol/l, [Na]_i = 10 mmol/l, [Na]_o = 125 mmol/l, [K]_i = 134 mmol/l, [K]_o = 3.7 mmol/l. Na⁺-Ca²⁺ exchanger was blocked by setting I_NaCa to zero. The time constants obtained for the slow (major) phase of Ca²⁺ transient relaxation with and without Na⁺-Ca²⁺ exchanger block were ₁ = 300 ms and ₂ = 225 ms, respectively, while fits to experimental data gave ₁ = 290 ms, ₂ = 225 ms.

Fig. 3 E compares the effect of intracellular Ca²⁺ transient on the waveform of the AP. There is general agreement in details between the superimposed APs and Ca²⁺ transients in the model (left panel) and the myocyte (right panel). Both simulation and experiment show significant prolongation of APD corresponding to the larger Ca²⁺ transient. It is important to recognize the dual effect of cytoplasmic free [Ca²⁺] on the APD. On one hand, a large Ca²⁺ transient tends to shorten APD by increasing [Ca²⁺]-dependent inactivation of the L-type channel. On the other hand, it tends to prolong APD by enhancing inward Na⁺-Ca²⁺ exchange and nonspecific Ca²⁺-activated currents. The net result, however, in both model and experiment favored APD prolongation with the large [Ca²⁺] transients during alternans (see Fig. 3 E). Although I_ks has some dependence on [Ca²⁺]_i in the LR model, its effects on AP configuration were found to be insignificant.

Intracellular Ca²⁺ dynamics at very rapid pacing rates

During electrical induction of VF, the cycle length of the initial VT is typically in the range of 150 ms before it degenerates to VF. Neither intact ventricular tissue nor isolated ventricular myocytes can be reliably paced at such short CLs with one-to-one capture. (During the VT phase of VF, this is a reason that wavebreak occurs, causing the transition to VF.) In the model, as the CL was progressively shortened toward this range, we observed a progressive increase in the amplitude of alternans of the AP and Ca²⁺ transient (beginning at CL = 300 ms), leading to a gradual distortion of Ca²⁺ transients and AP configuration. Fig. 4 shows the time course of [Ca²⁺]_i and [Ca²⁺]_JSR as well as AP and I_spon for a CL of 150 ms. After an initial transient period lasting ~4000 ms, stable quasiperiodic oscillations of Ca²⁺ in myoplasm and JSR were established. The quasiperiodic oscillations were caused by oscillations in spontaneous SR release, I_spon (see Fig. 4 D). The latter had a period of 600 ms (four times the period of stimulation) and began only after a significant increase in diastolic value of [Ca²⁺]_i had occurred. Diastolic values of [Ca²⁺]_i rose above 1 µmol/l. The AP was modulated by [Ca²⁺]_i oscillations through I_NaCa, I_ns(Ca) (see Fig. 4 E), and I_Ca(L). In Fig. 4 E, I_NaCa and I_ns(Ca) were plotted as functions of membrane potential during both depolarization and repolarization phases of the 13th and 14th APs (from Fig. 4 A). The current traces diverge during the repolarization phase. Clearly, for a range of values of membrane potential, inward components of I_NaCa, I_ns(Ca) are larger for larger [Ca²⁺]_i transient, thus prolonging APD. Rapid pacing also caused a significant decrease in the maximum absolute value of I_Na (from 360 µA/µF for CL = 1000 ms to 10 µA/µF for CL = 150 ms). Qualitatively similar oscillations in [Ca²⁺]_i with a quasiperiodic appearance were occasionally observed in rabbit ventricular myocytes during pacing. We found that the onset of quasiperiodic oscillations during rapid pacing was not very sensitive to major parameters of intracellular Ca²⁺ dynamics, although the amplitude of the oscillations was. For example, increasing K_NaCa and maximum rate of SR Ca²⁺ pump by 50% and G_rel by 30-50% did not qualitatively affect the model behavior.

Effects of intracellular Ca²⁺ dynamics on wave propagation in a 2D cardiac tissue model

We next examined the effects of intracellular Ca²⁺ dynamics on spiral wave reentry in simulated 2D cardiac tissue. We chose the diffusion coefficient D to give a conduction velocity of ~55 cm/s for a solitary plane wave. To obtain a reentrant spiral wave, the rectangular region behind the tail of the rectilinear wave was excited. Because the excitation could not spread into an unrecovered region, a point (q) appeared where the front was adjacent to the tail. This led to the curving of the wavefront around this point (see Fig. 5 A), which formed the tip of a spiral wave (Fig. 5, B and C). The spiral wave made four rotations with a period of 170 ms and diastolic interval of 20 ms, and then became nonstationary, with the wavefront progressively deteriorating for ~3000 ms (Fig. 5 D). The breakup of the spiral wave was sensitive to the "gain" in the [Ca²⁺]_i sensitivity of various Ca²⁺-sensitive ionic currents. For example, increasing the [Ca²⁺]_i sensitivity of I_ns(Ca) by decreasing K_m,ns(Ca) from 1.2 µmol/l to 1.0 µmol/l facilitated breakup of the wavefront into the fibrillation-like state (Fig. 5 E). During spiral wave rotation, each cell in a tissue model is being rapidly excited (CL = 170 ms), which we showed in single cell simulations was sufficient to cause the conditions of intracellular Ca²⁺ overload and spontaneous Ca²⁺ release.

View larger version (109K): [in this window] [in a new window]   FIGURE 5   Ca2+ dynamics cause spiral wave breakup in simulated 2D cardiac tissue model. (A-E) Initiation of a spiral wave following a premature stimulus (A and B). After the spiral wave becomes established (C, t = 1275 ms), deformations develop (D, t = 4010) leading to spiral wave breakup (E). After spontaneous Ca2+ release was blocked, the multiple reentrant waves coalesced back into a single spiral. Blue indicates V < 60 mV; green, V  60 mV; red, absolute value of Imemb > 10 µA/µF.

Analysis of APs and Ca²⁺ transients recorded from a local site in the tissue showed that the transition from the stationary to nonstationary regime started abruptly with an unusually large Ca²⁺ transient (see Fig. 6 A) due to spontaneous Ca²⁺ release. This caused substantial prolongation of the AP (Fig. 6 A, top graph), due to the increase in inward components of I_NaCa and I_ns,Ca, which led to marked shortening of the subsequent diastolic interval. The short diastolic interval dramatically decreased the depolarization rate of the subsequent AP (by ~5-fold) due to the incomplete recovery of I_Na from inactivation, which slowed the conduction velocity of the wavefront. The short diastolic interval also shortened the duration of the subsequent AP due to its restitution properties, markedly altering the wavelength (the product of APD and conduction velocity) in this region. Conversely, the short diastolic interval and altered AP affected the intracellular Ca²⁺ transient of the next beat, which further modified the AP via its feedback on Ca²⁺-sensitive currents.

View larger version (51K): [in this window] [in a new window]   FIGURE 6   Traces of membrane potential V (top), and [Ca2+]i (bottom) measured at a site (Nx = 100, Ny = 100; origin is at top left corner) in a simulated 2D tissue. Quasiperiodic oscillations disappeared when Ispon was blocked, as in Fig. 5 F. (A) V and [Ca2+]i during development of spiral wave breakup. (B) V and [Ca2+]i after the block of spontaneous Ca2+ release.

If the interaction between [Ca²⁺]_i and the Ca²⁺-sensitive currents affecting the AP and conduction velocity is sufficiently strong, variations of restitution properties along the arm of the spiral wave grow to the point where the excitation wave can no longer propagate. Note particularly in Fig. 5 E that the wavebreaks occur at wavefront/waveback interactions (see arrows). This causes the spiral wave to break up, leading to the fibrillation-like state. In this scenario, spontaneous Ca²⁺ release acts as a gain-enhancing mechanism between [Ca²⁺]_i and Ca²⁺-sensitive currents. As we mentioned above, if this gain was decreased by reducing the Ca²⁺ sensitivity of Ca²⁺-sensitive currents (reducing the amplitudes of APD oscillations), then spiral wave breakup was prevented. Likewise, if the gain was decreased by eliminating spontaneous Ca²⁺ release, spiral wave breakup also did not occur. In fact, if spiral wave breakup was allowed to develop with spontaneous Ca²⁺ release mechanism intact, its subsequent elimination caused the multiple reentrant wavefronts to coalesce back into a single stationary spiral wave (Figs. 5 F and 6 B). In this case, the tip of the reformed spiral wave was shifted with respect to its original position (before start of nonstationary reentry), due to the redistribution of recovery processes during the nonstationary regime.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Intracellular Ca²⁺ overload and VF are interrelated phenomena. Indeed, increased [Ca²⁺]_i appears to play an important role in initiating VF (Wit and Janse, 1993; Lakatta and Guarnieri, 1993), and VF by itself may lead to Ca²⁺ overload conditions (Koretsune and Marban, 1989), which further maintains VF. Therefore, it is important to understand the role of Ca²⁺ dynamics in the transition from VT to VF. We investigated this issue using a combined experimental and mathematical modeling approach. Our results show that 1) rapid pacing of rabbit ventricular myocytes leads to increased intracellular Ca²⁺ levels and complex patterns of variability in the AP and the intracellular Ca²⁺ transient (Fig. 2 A); 2) the complex patterns of the intracellular Ca²⁺ transient arise directly from the dynamics of intracellular Ca²⁺ regulation, and are not merely passive responses to beat-to-beat alterations in AP (Fig. 2 B); 3) these complex Ca²⁺ dynamics have been simulated in a version of the LR ventricular action potential with modified intracellular Ca²⁺ dynamics, and show agreement with the experimental findings in isolated myocytes; and 4) when incorporated into simulated 2D cardiac tissue, this action potential model produces a form of spiral wave breakup in which intracellular Ca²⁺ dynamics play a key role through their influence on Ca²⁺-sensitive membrane currents such as I_Ca(L), I_NaCa, and I_ns(Ca). To the extent that spiral wave breakup is useful as a model for the transition from VT to VF, these findings suggest that intracellular Ca²⁺ dynamics can play an important role in the destabilization of VT and its degeneration into VF.

Effects of rapid pacing on intracellular Ca²⁺

The effects of pacing on intracellular Ca²⁺ in cardiac muscle have been studied before, both by direct measurements of intracellular Ca²⁺ with fluorescent dyes and by using tension as a surrogate. Alternation of the AP and Ca²⁺ transient during pacing have been analyzed in numerous studies. In most, pharmacological interventions were applied to determine whether the alternation arose primarily from electrical or mechanical processes. For example, Saitoh et al. (1989) provided pharmacologically based evidence that Ca²⁺ dynamics are the main cause of electrical and mechanical alternans in ventricular muscle, whereas recovery properties of membrane currents are the main determinants in His-Purkinje tissue. However, pharmacologic tools are never completely specific, and the range of heart rates studied in these preparations were generally slower than the typical rate of VT before it degenerates to VF. In this study, the ability to rapidly pace using action potential clamps provides a major advantage. By preventing changes in the AP from secondarily influencing intracellular Ca²⁺ dynamics, it allows direct assessment of the stability of intracellular Ca²⁺ dynamics without having to resort to pharmacological interventions. In addition, the use of the perforated patch technique ensures minimal disturbance of the cytoplasm and preservation of intact intracellular signaling pathways. Finally, we investigated a range of pacing rates directly relevant to VT and VF.

The observation that stable alternation of the Ca²⁺ transient occurred during rapid pacing with the AP clamp (in which waveform of AP is fixed) indicates that the restitution of the Ca²⁺ transient is independently influenced by other factors besides the diastolic interval. This explains the quasiperiodic behavior of the AP and Ca²⁺ transients during regular pacing (Fig. 4, left column): AP restitution properties contribute the oscillations of APD, which are modulated by a oscillations arising independently from the restitution properties of the intracellular Ca²⁺ transient. Perhaps it is not surprising that the interaction between these processes, when distributed spatially over a 2D tissue, can amplify the electrophysiological heterogeneities that lead to spiral wave breakup (Fig. 5).

Validity of the action potential model

The relevance of our analysis of the role of intracellular Ca²⁺ dynamics in the transition from VT to VF depends on the validity of the action potential model and its ability to accurately simulate the behavior of cardiac muscle at very rapid pacing rates. The LR model is becoming the most widely used model of ventricular AP, and therefore we chose to use it after reformulating the intracellular Ca²⁺ dynamics. Mainly, we replaced the discontinuous formulation of Ca²⁺ release from the SR in the original model by a continuous one. The advantages of this modification are that it reflects more closely the physiological mechanisms of Ca²⁺ release, and also eliminates mathematical artifacts associated with threshold and time delay properties of the model (Luo and Rudy, 1994). Our formulation provides graded depletion of the SR Ca²⁺ content (see Fig. 4 C), while most other models deplete the SR almost completely with each excitation, contrary to experimental data (Bassani et al., 1995).

The model parameters were adjusted to reproduce as closely as possible the experimental behavior of the isolated rabbit ventricular myocytes during rapid pacing, particularly the accumulation of intracellular Ca²⁺ and the development of alternans. In the model, the rise in the diastolic values of [Ca²⁺]_i with rapid pacing is attributed to the fact that Ca²⁺ is entering the cell through the L-type channel more rapidly than it is extruded. The increased [Ca²⁺]_i accelerates I_Ca,(L) inactivation and enhances the I_NaCa and I_ns(Ca), consequently affecting the shape of the AP. At a rate comparable to the rate of VT (150 ms), [Ca²⁺]_i reached a level sufficient to induce spontaneous SR Ca²⁺ release. The latter occurred periodically but with a period different from the CL. The interaction of two periodic processes resulted in a quasiperiodic regime. Thus, spontaneous SR Ca²⁺ release is a candidate for the postulated additional factor suggested by the experimental results which, independent of the diastolic interval, controls Ca²⁺ transient restitution. Further studies are necessary to test this possibility. It is worthwhile to note that the original LR model produces chaotic, rather than quasiperiodic, oscillations with much larger amplitudes under the same pacing conditions (Chudin et al., 1998). This difference is caused by the discontinuous threshold nature of CICR formulation in the original LR model, which is avoided in our reformulation.

Role of intracellular Ca²⁺ dynamics in spiral wave breakup

When the modified LR model was used to simulate 2D cardiac tissue, we found that Ca²⁺ dynamics were directly responsible for causing the transition from stationary to violently meandering spiral wave reentry promoting wavebreak and a fibrillation-like state. This occurred because the complex temporal intracellular Ca²⁺ dynamics resulted in spatial inhomogeneities in intracellular [Ca²⁺]_i, which amplified the inward component of the Na⁺-Ca²⁺ exchanger current and the nonspecific Ca²⁺-activated current to produce in turn electrophysiological inhomogeneities by prolonging APD. These spatial regions of prolonged repolarization interact with the wavefront during the next rotation of the spiral wave, sharply decreasing conduction velocity and causing wavebreak. This is illustrated in Fig. 5, in which the red color represents the points on the wave front where absolute value of I_memb current (see Eq. 4) is greater than 10 µA/µF (the approximate threshold corresponding to significant participation of I_Na in wavefront propagation). Breaks in the red line indicate slow propagation supported by the L-type Ca²⁺ current (where I_Na is highly inactivated) or conduction failure. We verified that the qualitative nature of our results remained robust with respect to various aspects of CICR current such as expressions for P(v), , and value of G_rel. Moreover, despite the different formulation of intracellular Ca²⁺ dynamics and different morphology of Ca²⁺ transient, the original LR model gave qualitatively similar results; that is, when its Ca²⁺ dynamics were operational, spiral wave breakup occurred due to the Ca²⁺ instability (Chudin et al., 1998).

These findings are generally consistent with studies implicating cardiac restitution properties as key determinants of spiral wave instability and breakup (Koller et al., 1998; Karma, 1994; Weiss et al., 1999). We suggest that the effects of intracellular Ca²⁺ may operate dynamically by promoting functional electrophysiological heterogeneities. By modulating various Ca²⁺-sensitive currents, intracellular Ca²⁺ levels locally alter cardiac restitution properties. If the "gain" between intracellular Ca²⁺ and Ca²⁺-sensitive currents affecting restitution is sufficiently high, intracellular Ca²⁺ dynamics may promote instability.

It is interesting to compare the results of our study with findings of Merillat et al., (1990), who demonstrated that Na-K pump inhibition (which leads to Ca²⁺ overload) produced VF even when spontaneous Ca²⁺ oscillations were blocked. In our study we demonstrate that such spontaneous Ca²⁺ oscillations may cause transition from VT to VF, but this does not imply that they are the only mechanism (e.g., see Weiss et al., 1999).

Limitations and implications of the study

A number of limitations should be noted in considering the implications of this study for clinical VF. We have only simulated homogeneous isotropic 2D cardiac tissue, whereas real hearts are 3D, anisotropic, and inhomogeneous. Nevertheless, if intracellular Ca²⁺ dynamics promote spiral wave instability in such a simple model, they are likely to contribute to instability in more complex models as well.

Our reformulation of intracellular Ca²⁺ dynamics improves but still grossly simplifies the real physiological situation, about which many controversies are still unsettled. The AP model contains the major, but not all of the known, Ca²⁺-sensitive currents in heart such as Ca²⁺-sensitive Cl current (Zygmunt and Gibbons, 1991) and Ca²⁺ sensitivity of I_K1 (Zaza et al., 1998). Finally, it cannot be directly ascertained whether spontaneous Ca²⁺ release from the SR actually causes Ca²⁺ alternans in cardiac myocytes during rapid pacing, but our model reproduces Ca²⁺ alternans phenomenologically. It may be possible to gain further insight into these mechanisms by altering elements of intracellular Ca²⁺ regulation with various drugs and hormones, and studying the interplay among intracellular Ca²⁺ dynamics, the AP, and wave propagation using a combined experimental and simulation approach.

Also, our model, like most others, assumes constant [Na⁺] and [K⁺] in myoplasm. Unfortunately, our experience shows that allowing changes of these ionic concentrations causes slowly developing instabilities due to the loss of ionic balance. Undoubtedly, incorporation of [Na⁺] and [K⁺] dynamics into the cell model can illuminate other interesting phenomena. Finally, the myocyte experiments were performed at slightly lower temperature (33-35°C) than physiological, because myocytes did not tolerate sustained rapid pacing at 37°C.

Despite these limitations, our results suggest that intracellular Ca²⁺ dynamics must be considered as a potentially important factor promoting the transition of VT to VF. Altered intracellular Ca²⁺ dynamics by factors promoting intracellular Ca²⁺ overload (e.g., digitalis toxicity) or by autonomic factors (e.g., enhanced sympathetic tone) could play a role clinically in the predisposition to VT/VF and sudden cardiac death.