Deep-learning-assisted detection and termination of spiral and broken-spiral waves in mathematical models for cardiac tissue

Unbroken- and broken-spiral waves, in partial-differential-equation (PDE) models for cardiac tissue, are the mathematical analogs of life-threatening cardiac arrhythmias, namely, ventricular tachycardia and ventricular-fibrillation. We develop (a) a deep-learning method for the detection of unbroken- and broken-spiral waves and (b) the elimination of such waves, e.g., by the application of low-amplitude control currents in the cardiac-tissue context. Our method is based on a convolutional neural network (CNN) that we train to distinguish between patterns with spiral-waves $\mathcal{S}$ and without spiral-waves $\mathcal{NS}$. We obtain these patterns by carrying out extensive direct numerical simulations of PDE models for cardiac tissue in which the transmembrane potential $V$, when portrayed via pseudocolor plots, displays patterns of electrical activation of types $\mathcal{S}$ and $\mathcal{NS}$. We then utilize our trained CNN to obtain, for a given pseudocolor image of $V$, a heatmap that has high intensity in the regions where this image shows the cores of spiral waves and the associated wavefronts. Given this heatmap, we show how to apply low-amplitude currents of a two-dimensional Gaussian profile to eliminate spiral-waves efficiently. Our in silico results are of direct relevance to the detection and elimination of these arrhythmias because our elimination of unbroken or broken-spiral waves is the mathematical analog of low-amplitude defibrillation.

Figure 1

Collages of illustrative pseudocolor plots of $V$ with (a) single-spiral ($\mathcal{S}$) images and (b) no spiral ($\mathcal{NS}$) images, which we use for training our CNN.

Figure 2

A schematic diagram of our CNN showing the Input, Middle, and Final layers; the details of each one of these layer are given in the supplemental material [36].

Figure 3

Pseudocolor plots of (left panel) $V$, showing broken-spiral waves, and (right panel) the heatmap $\mathcal{H}$ for the image in the left panel.

Figure 4

(a) Image from simulation data, (b) noisy image obtained from simulation data.

Figure 5

Schematic diagram showing the process of generating a heatmap for a noisy image. The SNR value of a noisy image in this particular case is $\mathrm{SNR}=0.369$.

Figure 6

Top row: Images of $V_m$ with added noise and different values of the SNR. Bottom row: The corresponding heatmaps generated from our trained CNN.

Figure 7

Pseudocolor plots that show (a) the defibrillation current as $I_{\text{ext}}$ (single 2D-Gaussian centered at the spiral core), whose $I_{\text{def}}=0.5$ pA/pF, $\sigma =0.375N\mathrm{\Delta }x$ (cm), and $N=256$. (b)–(d) $V$ of the Aliev-Panfilov model (Ref. [32]). (b) The spiral wave at the time of application, and (c),(d) after the defibrillation current is applied.

Figure 8

Pseudocolor plots of (a) $V$ of a broken-spiral wave, (b) the corresponding heatmap $\mathcal{H}$ [Eq. (4)], (c) the summed and normalized 2D Gaussians [$\mathcal{G}(i,j)$ in Eq. (6)], and (d) the Hadamard product $I_{\text{ext}}(i,j)$ [with $I_{\text{def}}=5$ pA/pF in Eq. (7)]. (e)–(h) Pseudocolor plots of $V$, at different representative times, showing the elimination of the broken spiral waves after the application of $I_{\text{ext}}(i,j)$ for a time of $t=t_{\text{def}}=120$ ms. (i)–(l) The analogs of (e)–(h) for the current-mesh control scheme of Ref. [20] (see the videos V1 and V2 in [36]).

Figure 9

Pseudocolor plots from (a) to (d) show phase diagrams in the $(t_{\text{def}},\sigma )$ plane showing parameter regions in which our Gaussian-control scheme succeeds (red) and does not succeed (blue) for different values of the current. Pseudocolor plots from (e) to (h) show another set of phase diagrams from in the $a,t_{\text{def}}$ plane showing parameter regions in which our Gaussian-control scheme succeeds (red) and does not succeed (blue) for different values $\sigma$.