Minimal Functional Clusters Predict the Probability of Reentry in Cardiac Fibrotic Tissue

Cardiac fibrosis is a well-known arrhythmogenic condition which can lead to sudden cardiac death. Physically, fibrosis can be viewed as a large number of small obstacles in an excitable medium, which may create nonlinear wave turbulence or reentry. The relation between the specific texture of fibrosis and the onset of reentry is of great theoretical and practical importance. Here, we present a conceptual framework which combines functional aspects of propagation manifested as conduction blocks, with reentry wavelength and geometrical clusters of fibrosis. We formulate them into the single concept of minimal functional cluster and through extensive simulations show that it characterizes the path of reexcitation accurately, and importantly its size distribution quantitatively predicts the reentry probability for different fibrosis densities and tissue excitabilities.

Figure 1

Action potential wave propagation and reentry formation in tissue with fibrosis. (a) A model with fibrosis (black) at $\varphi =0.5$. Snapshots of transmembrane voltage maps following wave initiation at the left boundary at $t=0$: (b) the occurrence of a unidirectional block (marked by the blue $T$), (c) reexcitation of the tissue behind the block (blue arrow), and (d)–(f) formation and persistence of the reentry. The blue box highlights the core area of the reentry.

Figure 2

Formation of functional clusters. (a) Map of randomly color-coded fibrotic clusters superimposed with the locations of the conduction blocks (red circles) following pacing from the left boundary ($\varphi =0.5$, $e=0.5$). (b) Corresponding map of functional clusters. (c) A representative functional cluster (right) formed from three smaller fibrotic clusters and two conduction blocks (left). (d) Minimal functional cluster (light colored) associated with the unidirectional block $A$. The dashed line shows the path of reexcitation.

Figure 3

Characterization of fibrotic and functional clusters. (a) Log-log plot of CCDF for a cluster size at $\varphi =0.5$. (b) Correlation length ($\xi$) vs $\varphi$. The dashed black line represents fibrotic and the blue lines the functional clusters at three values of $e$, respectively. Units of $\xi$ and $s$ are element edge length (0.25 mm) and area $(0.25\mathrm{mm}{)}^2$, respectively.

Figure 4

Wave delays and minimal clusters. Log-log CCDF plots of (a) delays [$\tau$] and (b) minimal functional cluster perimeters [${\mathrm{\Pi }}_{\min ,f}$] at different $\varphi$’s (blue shades) and for $e=0.5$. (c) Scatter plot of $\tau$ vs ${\mathrm{\Pi }}_{\min ,f}$ and (d) $\tau$ vs ${\mathrm{\Pi }}_{\min ,\mathrm{fib}}$, both at $\varphi =0.5$ and $e=0.5$. Unit of the perimeters is the element edge length (0.25 mm).

Figure 5

Estimation of reentry probability as a function of $\varphi$ and $e$. Dashed line, empirical; solid line, estimated probabilities based on (a) minimal functional clusters, (b) minimal fibrotic clusters, (c) total functional clusters, and (d) total fibrotic clusters. $\alpha =0.5$ in (a) and (b). Probabilities were calculated from $N=250$ realizations. Average RMSE between the two curves are 0.06, 0.24, 0.54, and 0.46 for (a)–(d), respectively.