Centres of spiral waves can be detected as phase defect lines in optical voltage mapping data and numerical simulations
Authors: Kabus DA , Kabus DA , Arno L , Leenknegt L , Harlaar N , Dekker SO , Panfilov AV , De Vries AAF , Pijnappels DA , Dierckx H 
Published: May 2022

Abstract

Funding Acknowledgements
Type of funding sources: Public grant(s) – National budget only. Main funding source(s): KU Leuven and FWO-Flanders

Introduction
To study the excitation patterns of spiral waves in two spatial dimensions over time, it is useful to abstract them to much simpler mathematical structures. Previously, those waves have been abstracted to phase singularities: single points around which spiral moves [1,2]. Recently, it was proposed to upgrade these points to phase defect lines [3,4]. The dynamics of either of these structures can then be studied in place of the spiral waves.

Purpose
To work with phase defects, firstly, we need a reliable method of locating them.

Methods
We recorded the transmembrane voltage over space and time in a numerical simulation of the mono-domain model as well as by using optical voltage mapping data of 10 cm² monolayers of conditionally immortalised human atrial myocytes [5].
Optical voltage mapping was performed with a sampling time of 6 ms and 100×100 square pixels of 0.25 mm width. The recorded intensity were smoothed and rescaled to get an estimate of the transmembrane voltages. Detailed protocols can be found in [5].
For the numerical simulations, we have implemented the linear-core tissue model by Bueno-Orovio, Cherry, and Fenton (2008) [6] with finite differences on a 1200×1200 rectangular grid with a grid size of 0.25 mm and a time step of 9 μs. We normalised the transmembrane voltage u such that zero corresponds to the resting state and one to the plateau of the action potential.
A phase φ is an angle in radians which monotonously increases from zero to 2π during one action potential. Phases are defined in such a manner that, even for the sharp wave front of a cardiac action potential, this increase happens in a slow, gradual way. A phase can for instance be defined by using the local arrival time and the characteristic action potential duration [4].
At the centre of spiral waves, neighbouring parts of tissue have vastly different phase values φ. A phase singularity is a point where all phases meet. Phase defects are large jumps across space in phase. Phase defect lines can then be detected as lines in the phase defect field ρ with a value much larger than zero [7].

Results
For the simulation, we can clearly see that the linear core of the Bueno-Orovio, Cherry, and Fenton (2008) [6] model is identified as a phase defect line of up to 100 mm in length (Fig.1). The spiral wave moves along it and turns at the end points.
The same behaviour can be seen in vitro in the optical voltage mapping data (Fig.2). Multiple spirals can be observed, all of which revolve around phase defect lines of lengths of up to 4 mm.

Conclusion
In this study, we have shown that phase defect lines can be detected both in silico and in vitro. Their properties such as length, rotational frequency, and their dynamics can be used to gain additional insight into spiral waves in arrhythmias.
We have introduced multiple methods to localise phase defects for which the code is publicly available.