Spiral-wave dynamics depend sensitively on inhomogeneities in mathematical models of ventricular tissue

Every sixth death in industrialized countries occurs because of cardiac arrhythmias such as ventricular tachycardia (VT) and ventricular fibrillation (VF). There is growing consensus that VT is associated with an unbroken spiral wave of electrical activation on cardiac tissue but VF with broken waves, spiral turbulence, spatiotemporal chaos and rapid, irregular activation. Thus spiral-wave activity in cardiac tissue has been studied extensively. Nevertheless, many aspects of such spiral dynamics remain elusive because of the intrinsically high-dimensional nature of the cardiac-dynamical system. In particular, the role of tissue heterogeneities in the stability of cardiac spiral waves is still being investigated. Experiments with conduction inhomogeneities in cardiac tissue yield a variety of results: some suggest that conduction inhomogeneities can eliminate VF partially or completely, leading to VT or quiescence, but others show that VF is unaffected by obstacles. We propose theoretically that this variety of results is a natural manifestation of a complex, fractal-like boundary that must separate the basins of the attractors associated, respectively, with spiral breakup and single spiral wave. We substantiate this with extensive numerical studies of Panfilov and Luo-Rudy I models, where we show that the suppression of spiral breakup depends sensitively on the position, size, and nature of the inhomogeneity.

Figure 1

Panfilov-model spiral turbulence (ST). Transmembrane potentials for two dimension (pseudo-grayscale plots A–F) and three dimension (isosurface plots G and H). Two dimension: $200\mathrm{mm}\times 200\mathrm{mm}$ domain and a $40\mathrm{mm}\times 40\mathrm{mm}$ square obstacle with left-bottom corner at $(x,y)$. (A) no obstacle -ST; (B) $(x=160,y=100)$ ST persists; (c) $(x=150,y=100)$ ST replaced by RS (one rotating anchored spiral); (D) $(x=140,y=100)$ spiral moves away (medium quiescent). Three-dimensional analogs of (B) and (C): $(200\times 200\times 4)$ domain; an obstacle of height $4\mathrm{mm}$ and a square base of side $40\mathrm{mm}$ at (E) $(x=140,y=120,z=0)$ and (F) $(x=140,y=110,z=0)$.

Figure 2

Luo-Rudy-Model spiral turbulence. Pseudograyscale plots in a $90\times 90{\mathrm{mm}}^2$ illustrating how the initial condition (A) evolves, in the absence of obstacles, to (B) via the generation of spiral waves and their subsequent breakup. In the presence of a square obstacle of side $l$ placed with its bottom-left corner at $(x,y)$ we obtain the following: (C) $l=18\mathrm{mm}$ and ($x=58.5\mathrm{mm}$, $y=63\mathrm{mm}$) ST persists; (D) $l=22.5\mathrm{mm}$ and ($x=58.5\mathrm{mm}$, $y=63\mathrm{mm}$) RS (one spiral anchored at the obstacle); for $l=18$ and ($x=54\mathrm{mm}$, $y=63\mathrm{mm}$) spirals disappear leaving the medium quiescent (E) at $800\mathrm{ms}$ and (F) at $1000\mathrm{ms}$.

Figure 3

Panfilov-model stability diagram. The effect of an $40\times 40{\mathrm{mm}}^2$ obstacle in a $200\times 200{\mathrm{mm}}^2$ domain shown by small squares (side $l_p$) the colors of which indicate the final state of the system when the position of the bottom-left corner of the obstacle coincides with that of the small square (white, black, and gray denote ST, RS, and Q, respectively). (A) for $l_p=10\mathrm{mm}$. We get the fractal-like structure of the interfaces between ST, RS, and Q by zooming in on small subdomains encompassing parts of these interfaces [white boundaries in A, B, and C with (B) $l_p=5\mathrm{mm}$, (C) $l_p=2.5\mathrm{mm}$, and (D) $l_p=1\mathrm{mm}$].

Figure 4

The local time series, interbeat interval (IBI), and power spectrum of the transmembrane potential $V(x,y,t)$ at a representative point $(x,y)$ in the tissue. When the obstacle is at ($160\mathrm{mm}$, $100\mathrm{mm}$) a spiral turbulent state ST is obtained with the time series (A), and interbeat interval (B) showing nonperiodic chaotic behavior and a broad-band power spectrum (C). However, with the bottom-left corner of the obstacle at ($150\mathrm{mm}$, $100\mathrm{mm}$), the spiral wave gets attached to the obstacle after nine rotations $(\simeq 1800\mathrm{ms})$; this is reflected in the time series (D) and the plot of the interbeat interval (E); after transients the latter settles on to a constant value of $363\mathrm{ms}$; the power spectrum (F) shows discrete peaks with a fundamental frequency ${\omega }_f=2.74\mathrm{Hz}$ and its harmonics. Initial transients over the first $50000\delta t$ were removed before we collected data for calculating the power spectrum.

Figure 5

Inhomogeneities in ${ϵ}_1$. Inhomogeneities in the parameter ${ϵ}_1$ result in the coexistence of different types of spatiotemporal behavior in the same system. With ${ϵ}_1^{\mathrm{out}}=0.01$ and ${ϵ}_1^{\mathrm{in}}=0.02$ (see text), we obtain spatiotemporal chaos outside the inhomogeneity but quasiperiodic behavior inside it (A); the latter is illustrated by the power spectrum of $V(x,y,t)$ with discrete peaks (B) and the former by a broad-band power spectrum (C). With ${ϵ}_1^{\mathrm{out}}=0.03$ and ${ϵ}_1^{\mathrm{in}}=0.01$ and the left-bottom corner of the inhomogeneity placed at ($x=140\mathrm{mm}$, $y=140\mathrm{mm}$), single and broken spiral waves coexist in same medium (D), whereas, with the inhomogeneity at ($x=60\mathrm{mm}$, $y=50\mathrm{mm}$), a single rotating spiral gets anchored to the inhomogeneity (E, F) with quasiperiodic behavior illustrated by the interbeat interval (G) and the power spectrum (H). The power spectrum (H) shows six frequencies (${\omega }_1=4.06$, ${\omega }_2=5.56$, ${\omega }_3=6.57$, ${\omega }_4=7.05$, ${\omega }_5=8.58$, and ${\omega }_6=9.07\mathrm{Hz}$) not rationally related to each other; all other frequencies can be expressed as ${\sum }_{i=1}^6{n}_i{\omega }_i$, where the $n_i$ are integers. Initial transients over the first $50000\delta t$ were removed before we collected data for calculating power spectra.