Arrhythmogenicity of cardiac fibrosis: fractal measures and Betti numbers

TL;DR

This paper introduces mathematical markers like fractal dimension, lacunarity, and Betti numbers to characterize cardiac fibrosis patterns and predict their potential to cause arrhythmias based on extensive computational models.

Contribution

It develops new mathematical algorithms to quantify fibrotic tissue patterns and links these measures to arrhythmogenic risk in cardiac fibrosis.

Findings

High $eta_0$ correlates with increased arrhythmogenicity.

Small lacunarity parameter $b$ indicates higher risk.

Markers effectively differentiate fibrosis types in models.

Abstract

Infarction- or ischaemia-induced cardiac fibrosis can be arrythmogenic. We use mathematcal models for diffuse fibrosis ($\mathcal{DF}$), interstitial fibrosis ($\mathcal{IF}$), patchy fibrosis ($\mathcal{PF}$), and compact fibrosis ($\mathcal{CF}$) to study patterns of fibrotic cardiac tissue that have been generated by new mathematical algorithms. We show that the fractal dimension $\mathbb{D}$, the lacunarity $\mathcal{L}$, and the Betti numbers $\beta_0$ and $\beta_1$ of such patterns are \textit{fibrotic-tissue markers} that can be used to characterise the arrhythmogenicity of different types of cardiac fibrosis. We hypothesize, and then demonstrate by extensive \textit{in silico} studies of detailed mathematical models for cardiac tissue, that the arrhytmogenicity of fibrotic tissue is high when $\beta_0$ is large and the lacunarity parameter $b$ is small.