Spiral-wave dynamics in a mathematical model of human ventricular tissue with myocytes and Purkinje fibers

We present systematic numerical studies of the possible effects of the coupling of human endocardial and Purkinje cells at cellular and two-dimensional tissue levels. We find that the autorhythmic-activity frequency of the Purkinje cell in a composite decreases with an increase in the coupling strength; this can even eliminate the autorhythmicity. We observe a delay between the beginning of the action potentials of endocardial and Purkinje cells in a composite; such a delay increases as we decrease the diffusive coupling, and eventually a failure of transmission occurs. An increase in the diffusive coupling decreases the slope of the action-potential-duration-restitution curve of an endocardial cell in a composite. By using a minimal model for the Purkinje network, in which we have a two-dimensional, bilayer tissue, with a layer of Purkinje cells on top of a layer of endocardial cells, we can stabilize spiral-wave turbulence; however, for a sparse distribution of Purkinje-ventricular junctions, at which these two layers are coupled, we can also obtain additional focal activity and many complex transient regimes. We also present additional effects resulting from the coupling of Purkinje and endocardial layers and discuss the relation of our results to the studies performed in anatomically accurate models of the Purkinje network.

Figure 1

Schematic diagram of a small part of our square simulation domain consisting of endocaridal and Purkinje cells with $\mathcal{R}=2$; here ${\mathcal{R}}^2$ is the ratio of the total number of the endocardial (or Purkinje) cells to the total number of sites where the Purkinje cells are coupled to the endocardial cells. If $\mathcal{R}=1$, then every Purkinje cell is connected directly to the endocardial cell below it. The abbreviations $D_{ee},D_{pp}$, and $D_j$ stand for diffusion constant in the endocardial layer, diffusion constant in the Purkinje layer, and coupling strength between an endocardial cell and a Purkinje cell in our bilayer domain.

Figure 2

Autorhythmic activity of a Purkinje cell in an endocardial-Purkinje composite; here squares $\left(\blacksquare \right)$ and triangles $\left(▴\right)$ represent coupled Purkinje cells with $D_j=0.001D_{ee}$ and $D_j=0.003D_{ee}$, respectively, and circles $\left(•\right)$ denote data points from an uncoupled Purkinje cell. (a) Plots of the Purkinje action potential $V_p$ in a composite. (b) Plots of the current $\kappa (V_e-V_p)$ between endocardial and Purkinje cells; such plots show that the endocardial cell in this composite acts like a current sink. (c) Plots of the frequency $f$ of the autorhythmic activity of the Purkinje cell in a composite versus the coupling $D_j^{\mathrm{norm}}=D_j/D_{ee}$. Note that $f$ decreases as $D_j$ increases, and, finally, autorhythmicity is eliminated after $D_j\simeq 0.0021D_{ee}$.

Figure 3

Action potentials for endocardial and Purkinje cells in a composite and their conduction-delay times $\mathrm{\Delta }t$. (a) Plots of the action potentials $V_p$ $(•$ or $\blacksquare )$ and $V_e$ $(◯$ or $\square )$, when a stimulus is applied to both uncoupled endocaridal and Purkinje cells, i.e., $D_j=0$ $(•,◯)$ and the endocardial cell in our composite with $D_j=0.01D_{ee}$ $\left(\blacksquare ,\square \right)$. (b) Plots as in (a), but with a stimulus applied to the Purkinje cell in our composite. Insets in (a) and (b) show the depolarization phases of action potentials on an expanded scale; these inset plots show the conduction delay from an endocardial to a Purkinje cell, or vice versa, depending on whether the stimulus is applied to an endocardial cell or to a Purkinje cell in a composite. Such inset plots show that, for a given value of $D_j$, the conduction delay time is more when the stimulus is applied to the Purkinje cell than when it is applied to the endocardial cell in a composite. (c) Plots of the conduction-delay time $\mathrm{\Delta }t$ from the endocardial to the Purkinje cells versus $D_j^{\mathrm{norm}}=D_j/D_{ee}$, when the stimulus is applied to the endocaridal cell of a composite. (d) Plots as in (c) but of the conduction delay from the Purkinje to the endocardial cell, when the stimulus is applied to the Purkinje cell of a composite. Insets in (c) and (d) show the plots of such delay on an expanded time scale; these insets also show that, as we decrease $D_j$, the failure of the initiation of the action potential in a Purkinje cell, when the stimulus is applied to the endocardial cell in a composite, occurs earlier than it does when the stimulus is applied to the Purkinje cell in a composite.

Figure 4

Action potential duration restitution (APDR) and its slope for an endocardial cell in a composite with different sets of endocardial parameters; here circles $\left(•\right)$ represent an uncoupled endocardial cell, and squares $\left(\blacksquare \right)$ and triangles $\left(▴\right)$ are used, respectively, for an endocardial cell in a composite with $D_j=0.01D_{ee}$. The plots in (a), (b), and (c) show the APDR for parameter sets $\mathcal{P}1,\mathcal{P}2$, and $\mathcal{P}3$ (see Table 1), respectively; inset plots show the slopes of the APDR curve. The steepness of the APDR curve decreases, for all the parameter sets, as we increase $D_j$ in a composite. The maximum slopes of the APDR are $\simeq 2.0,\simeq 1.1$, and $\simeq 1.1$, for an uncoupled endocardial cell $\left(•\right)$ with parameter sets $\mathcal{P}1,\mathcal{P}2$, and $\mathcal{P}3$, respectively; however, the maximum slope of the APDR for an endocardial cell in a composite decreases as $D_j$ increases $\left(\blacksquare ,▴\right)$ because of the coupling to the Purkinje cell.

Figure 5

Spiral-waves dynamics in isolated endocardial and Purkinje 2D simulation domains. Pseudocolor plots of the transmembrane potential $V_m$ for the endocardial layer with parameter sets (a) $\mathcal{P}1$, (b) $\mathcal{P}2$, and (c) $\mathcal{P}3$, and (d) the Purkinje layer. Panels (e), (f), and (g) show, respectively, the time series for the parameter sets $\mathcal{P}1,\mathcal{P}2$, and $\mathcal{P}3$, recorded from a representative point $(x=125\mathrm{mm},y=125$ mm) that is marked by an asterisk in (a), (b), and (c); (h) the time series recorded from Purkinje layer, marked by an asterisk in (d). (i), (j), (k), and (l) show power spectra calculated by using these time series (see Methods for the lengths of time series). The irregular time series in (e) and the broad-band nature of the power spectrum in (i) are characteristic of a spiral-turbulence (ST) state in the endocardial layer with the parameter set $\mathcal{P}1$; the pseudocolor plot in (a) shows that this ST state is maintained by multiple waves, i.e., it is a multiple-spiral-turbulence (MST) state. For the parameter set $\mathcal{P}2$, the irregular time series in (f) and the development of subsidiary peaks in $E\left(\omega \right)$ in (j) confirm that the temporal evolution of the state is chaotic; the pseudocolor plot in (b) shows the existence of single, meandering, spiral turbulence (SMST), in which the spiral arms and core evolve chaotically in space and time. The pseudocolor plot in (c) shows a rotating spiral (RS) for the parameter set $\mathcal{P}3$; the periodic time series in (g) and the discrete, strong peaks in $E\left(\omega \right)$ at 4.75 Hz and its harmonics in (k) provide evidence of a single-rotating-spiral-periodically (SRSP) for the parameter set $\mathcal{P}1$. The peudocolor plot in (d), the periodic time series in (h), and discrete peaks in the power spectrum, with fundamental frequency ${\omega }_f\simeq 2.75$ Hz in (l), show the existence of an SRSP in the Purkinje layer.

Figure 6

Spiral-wave dynamics in our composite bilayer for the $\mathcal{P}1$ parameter set for the endocardial cells with $D_j=0.1D_{ee}$ and $\mathcal{R}=1$ (left column), $\mathcal{R}=2$ (middle column), and $\mathcal{R}=4$ (right column). The first row shows pseudocolor plots of the endocardial layer transmembrane potential $V_e$; the plots in the second row are the exact analogs of these for the Purkinje-layer transmembrane potential $V_p$. The third row shows plots of the time series for $V_e$ $\left(•\right)$ and $V_p$ $\left(◯\right)$, which are recorded from the representative points $(x=125$ mm, $y=125$ mm), in both endocardial and Purkinje layers; these are marked by asterisks in (a) to (f). The fourth row shows plots of the power spectra $E\left(\omega \right)$ versus the frequency $\omega$ for the endocardial layer $\left(•\right)$ and the Purkinje layer $\left(◯\right)$. A multiple-spiral-turbulence (MST) state in an isolated enocardial layer [Fig. 5] is converted to a single-rotating-spiral-periodic (SRSP) state, as shown in (a) and (b), or a spiral-absorption (SA) state, as shown in (c), by the inclusion of Purkinje cells. The periodic time series in (g) and (h) and the regular peaks in the power spectra in (j) and (k) give additional evidence of the existence of such SRSP states in the endocardial layer with $\mathcal{R}=1$ and 2. The time series of $V_e$ and $V_p$ in (i) provides evidence for the disappearance of spiral waves in both the endocardial and Purkinje layers simultaneously at time $t\simeq 5.7$ s for $\mathcal{R}=4$.

Figure 7

Spiral-wave dynamics in our composite bilayer for the $\mathcal{P}1$ parameter set for the endocardial cells with $D_j=0.1D_{ee}$ and $\mathcal{R}=8$ (left column), $\mathcal{R}=16$ (middle column), and $\mathcal{R}=32$ (right column); this figure is the analog of Fig. 6. An MST state in an isolated endocardial layer [Fig. 5] still remains in the MST state on the inclusion of Purkinje cells, as shown by the pseudocolor plots of $V_e$ in (a)–(c). The irregular time series obtained from a representative point in the endocardial layer [marked by asterisks in (a) to (f)] in (g)–(i) $\left(•\right)$ and the broad-band power spectra of such time series in (j)–(l) $\left(•\right)$ confirm the existence of such MST states in the endocardial layer of our composite bilayer. However, the Purkinje layer in the composite bilayer shows focal-wave activations as shown in the pseudocolor plots of $V_p$ in (d)–(f); such focal-wave activity occurs periodically as shown by the time-series plots in (g)–(i) $\left(◯\right)$; and the focal-wave frequency decreases as $\mathcal{R}$ increases, as shown by the power spectra in (j)–(l) $\left(◯\right)$.

Figure 8

The dependence of spiral-wave dynamics on (i) myocyte parameters, (ii) normalized endocardial and Purkinje coupling parameters $D_j^{\mathrm{norm}}$ (i.e., $D_j^{\mathrm{norm}}=D_j/D_{ee})$, and (iii) the number of Purkinje-ventricular junctions, which we measure by $\mathcal{R}$. (a) This stability diagram displays various spiral states in the $D_j^{\mathrm{norm}}$ and $\mathcal{R}$ parameter space for the $\mathcal{P}1$ myocyte parameters; circles $\left(•\right)$, triangles $\left(▴\right)$, diamonds $\left(♦\right)$, pentagrams $\left(★\right)$, and squares $\left(\blacksquare \right)$ represent, respectively, a single-rotating-spiral-periodic (SRSP) state, a single-rotating-spiral-quasiperiodic (SRSQ) state, a single-meandering-spiral-turbulence (SMST) state, a multiple-spiral-turbulence (MST) state, and a state with spiral-absorption (SA); in the absence of the Purkinje-fiber layer, the endocardial layer displays an MST state. Panels (b) and (c) are the exact analogs of (a), but for the $\mathcal{P}2$ and $\mathcal{P}3$ parameter sets, respectively, for endocardial cells. An isolated endocardial layer displays SMST and SRSP states, respectively, for the $\mathcal{P}1$ and $\mathcal{P}2$ parameter sets.