Phase-2 reentry in cardiac tissue: Role of the slow calcium pulse

Phase-2 re-entry is thought to underlie many causes of idiopathic ventricular arrhythmias as, for instance, those occurring in Brugada syndrome. In this paper, we study under which circumstances a region of depolarized tissue can re-excite adjacent regions that exhibit shorter action potential duration (APD), eventually inducing reentry. For this purpose, we use a simplified ionic model that reproduces well the ventricular action potential. With the help of this model, we analyze the conditions that lead to very short action potentials (APs), as well as possible mechanisms for re-excitation in a cable. We then study the induction of re-entrant waves (spiral waves) in simulations of AP propagation in the heart ventricles. We show that re-excitation takes place via a slow pulse produced by calcium current that propagates into the region of short APs until it encounters excitable tissue. We calculate analytically the speed of the slow pulse, and also give an estimate of the minimal tissue size necessary for allowing reexcitation to take place.

Figure 1

(a) Comparison between the five-variable model and an experimental action potential in ventricular tissue, in a cell paced with a stimulation period of 1 s. (b) Action potentials for the standard, or wild-type (WT), parameters (dotted line) and modified parameters showing a shortening of the APD obtained by increasing the transient outward current conductance $g_{to}$: $g_{to}=2.45\times g_{to}^{WT}$ (dot-dashed line); $g_{to}=2.46\times g_{to}^{WT}$ (dashed line); $g_{to}=2.56\times g_{to}^{WT}$ (solid line).

Figure 2

(Color online) Values of the APD as a function of several parameters of our model. Black regions means no AP, blue intermediate regions depict domeless AP, and red regions correspond to normal AP. We show in the upper row (a, b, and c) the results for normal epicardium, while in the lower row (d, e, and f) we have introduced the following changes: (d) and (f) ${\tau }_{h_{-}}=2.31\text{ms}={\tau }_{h_{-}}^{WT}/1.5$, (e) $g_{fi}=2.5{\text{ms}}^{-1}=g_{fi}^{WT}/1.39$.

Figure 3

Value of the APD at all spatial locations in a fiber of length $L=3\text{cm}$. All points in the cable are stimulated at once. The spatial distribution of the parameter $g_{to}$ follows a ramp given by $g_{to}=4.5{\text{ms}}^{-1}+G\left(L/2-x\right)$. Here we show the influence of the gradient (slope) $G$. The ramp connects to the constant values: $g_{to}=5.5{\text{ms}}^{-1}$ at the left side of the cable and to $g_{to}=3.5{\text{ms}}^{-1}$ at the right side of the cable. Note that for the smallest gradient in (b) $\left(\text{G}=0.07{\text{ms}}^{-1}{\text{cm}}^{-1}\right)$, we have taken $g_{to}=5.16{\text{ms}}^{-1}+G\left(L/2-x\right)$, so that it could fit in the same figure as with the other gradient parameter. The difference between (a) and (b) lies in the fact that in (a) we take $g_{si}=g_{si}^{WT}$, while in (b) $g_{si}=1.08\times g_{si}^{WT}$.

Figure 4

(Color online) Space-time plots showing the evolution of a pulse in a 1D fiber with a gradient of $I_{to}$ conductance, or $I_{fi}$ inactivation times. The length of the cable is $L=7.5\text{cm}$, and the total simulation time is 1 s. Depending on the conductance of the slow inward current $g_{si}$ the system may present no reexcitation (a) and (b), or multiple reexcitations (FR or BR), with or without recovering the dome. The values of the parameters are for (a) and (b) $g_{si}=0.9\times g_{si}^{WT}$; in (c) and (d), $g_{si}=g_{si}^{WT}$; in (e) and (f), $g_{si}=1.08\times g_{si}^{WT}$. For (a) to (f), one sets $g_{to}^{min}=3.5$ and $g_{to}^{max}=5.5{\text{ms}}^{-1}$ in the two sides of the cable and ${\tau }_{h_{-}}={\tau }_{h_{-}}^{WT}/1.5$. In panels (g) and (h), one sets $g_{si}=1.1\times g_{si}^{WT}$ and $g_{to}=4.5{\text{ms}}^{-1}$, with ${\tau }_{h_{-}}^{max}=3.47$ and ${\tau }_{h_{-}}^{min}=1.735{\text{ms}}^{-1}$ in each side of the spatial domain.

Figure 5

Critical value of $g_{si}$ necessary to obtain reexcitation as a function of the strength of the gradient in $g_{to}$. We take $g_{to}=3.5+2/\left(1+\text{exp}\left[\left(x-L/2\right)/s\right]\right)$. In the simulations with a shallow gradient the length of the system is increased as to reach values of $g_{to}$ within 1% of the maximum and minimum.

Figure 6

(Color online) (a) Pulse with multiple BR, obtained with the parameters $g_{si}=1.07\times g_{si}^{WT}$, ${\tau }_{h_{-}}={\tau }_{h_{-}}^{WT}/1.5$, and the rest of parameters are those given in Table of Appendix . The gradient in $g_{to}$ is centered at $x_d=27\text{cm}$ with $g_{to}^{max}=6.2{\text{ms}}^{-1}$ and $g_{to}^{min}=3.5{\text{ms}}^{-1}$. In (b), simulation with $I_{fi}$ and $I_{to}$ initially turned off. The system size is $L=15\text{cm}$ and $g_{to}=6.2{\text{ms}}^{-1}$, $g_{si}=1.07\times g_{si}^{WT}$, ${\tau }_{h_{-}}={\tau }_{h_{-}}^{WT}/1.5$ in the whole domain. After an initial excitation (at time $t=0$), one sees the propagation of a slow pulse due solely to the slow inward current $I_{si}$. At time $t=600\text{ms}$, indicated by the vertical line, the currents $I_{fi}$ and $I_{to}$ are turned on, after which periodic reexcitations appear. Note that the slow pulse remains practically unperturbed for $t>600\text{ms}$.

Figure 7

(a) Velocity of the slow pulse as a function of the slow inward current conductance $g_{si}$. Note that here $g_{si}$ stands for $g_{si}\times g_{si}^{WT}$ and the rest of parameters are those given in Table of Appendix . These curves are obtained from the analytical theory (see Appendix ) by solving Eqs. (B17, B18). For each value of $g_{si}$ there are two possible values of $c_{si}$ corresponding to a stable (high $c_{si}$) and to an unstable (low $c_{si}$) branch. The slow pulse solution disappears through a saddle-node bifurcation as $g_{si}$ is decreased below the critical value $\left(g_{si}\approx 0.91\right)$. The speed $c_{si}$ depends also on the state of recovery of the gate $f$ (inactivation gate of $I_{si}$) on the tissue ahead the slow pulse. For comparison sake, the triangles correspond to the velocity obtained from numerical simulations where none of the simplifying assumptions of the analytical theory are taken (see Appendix ). In (b), we compare the analytical (lower and upper branch) and the numerical slow pulse profile for parameter $g_{si}=1.15\times g_{si}^{WT}$. The agreement for the shape is not as good as the agreement with the velocity due to some of the simplifying assumptions of the theory.

Figure 8

Spatial distribution of voltage (solid line) and gate h (dashed line), at different times, corresponding to the simulation shown in Fig. 6a. The time difference between successive frames is 50 ms.

Figure 9

(Color online) (a) Definition of the different times and distances used in the text. (b) Time evolution of the voltage and the sodium inactivation gate $h$ during several reexcitations, for the simulation displayed in Fig. 6 (cut at $\text{x}=25.5\text{cm}$).

Figure 10

(Color online) (a) Simulation of AP propagation in a cable with heterogeneous distribution of $g_{to}$. The cable is stimulated at $x=0$ at time $t=0$. In the central region, delimited by the two Boltzmann gradients of Eq. (2), $g_{to}$ is set to $g_{to}^{min}=3.5{\text{ms}}^{-1}$, and in the rest of the cable $g_{to}^{max}=5.5{\text{ms}}^{-1}$. The other parameters are those given in Table , except that ${\tau }_{h_{-}}={\tau }_{h_{-}}^{WT}/1.5$, and $g_{si}=1.07\times g_{si}^{WT}$. The slow pulse propagates from the spike-and-dome central region to the periphery “domeless” area, resulting in backward reexcitation (BR, indicated by the lower arrow) and forward reexcitation (FR, indicated by the upper arrow). In (b), the time evolution of the membrane potential is shown at several spatial locations.

Figure 11

Minimal size of the central region of Fig. 10 with $g_{to}^{min}=3.5{\text{ms}}^{-1}$ necessary to obtain a (a) forward or (b) backward reexcitation, for a pacing frequency of 1 Hz. For this minimal size, the minimal distance at which a FR or a BR is produced is plotted in (c) for $g_{to}^{max}=5.5{\text{ms}}^{-1}$, and in (d) for $g_{to}^{max}=6.2{\text{ms}}^{-1}$.

Figure 12

(Color online) Structure of the two ventricles where the three-dimensional (3D) simulations were performed. The color scale (online) shows the normalized linear gradient of $g_{to}$ from the minimum value (0) to its maximal value (1) that is reached at the reference point of coordinates (124,64,120).

Figure 13

(Color online) Two different viewpoints of the membrane potential $V$ distribution taken at time $t=120\text{ms}$ after the simulation starts in the case of parameter $g_{to}∊\left[3.5–6.5\right]{\text{ms}}^{-1}$. Re-entrant activity is clearly visible in the upper RV due to the contrast of the $I_{to}$ between the RVup region and the rest of the heart.

Figure 14

Computed ECGs from the 3D simulations for the parameters $g_{si}=1.08\times g_{si}^{WT}$, ${\tau }_h={\tau }_h^{WT}/1.5$, and $g_{to}$ varying linearly in the following ranges: (a) $g_{to}∊\left[3.5–4.5\right]{\text{ms}}^{-1}$ (solid line); (b) $g_{to}∊\left[3.5–5.5\right]{\text{ms}}^{-1}$ (dashed line); (c) $g_{to}∊\left[3.5–6.5\right]{\text{ms}}^{-1}$ (dot-dashed line). The lead shown here is the standard ECG first lead I [42]. Note that at time $t=120\text{ms}$, one clearly distinguishes the reexcitation shown in Fig. 12 in the corresponding ECG (dot-dashed line).