Mathematical Medicine and Biology (2005) 22, 335−346 doi:10.1093/imammb/dqi013
Automating phase singularity localization in mathematical models of cardiac tissue dynamics S TEFFAN P UWAL AND B RADLEY J. ROTH† Department of Physics, Oakland University, Rochester, MI, USA AND
S ERGE K RUK Department of Mathematics and Statistics, Oakland University, Rochester, MI, USA [Received on 22 May 2005; revised on 8 August 2005; accepted on 23 August 2005] Electrical wave-fronts are responsible for contraction in heart tissue. Rotary wave-fronts break up into daughter waves and it is this break up that is believed to underlie ventricular fibrillation. Mathematical methods abound for simulation of fibrillation, and localizing the core of rotary wave-fronts (the phase singularities) is key to characterizing the state of fibrillation and effectiveness of defibrillation in these models. We present a formal method for automating this process in these various models. Automation will allow for side-by-side comparisons of suggested mechanisms of fibrillation, comparison of various models of these mechanisms and faster evaluation of defibrillation strategies making use of these models. Keywords: cardiac; fibrillation; phase singularity.
1. Introduction Numerous mathematical models exist that describe the electrophysiological behaviour of cardiac ventricular tissue (Fenton et al., 2002; Winfree, 1991). These models provide valuable insights into the dynamics of the heart. A reaction–diffusion equation describes the propagation of electrical wave-fronts through the tissue where an action potential (a potential of at least a threshold value that steeply rises to a plateau and then drops off—the reaction) diffuses through the tissue (Fenton et al., 2002; Fenton & Karma, 1998; Guyton & Hall, 2000). It is believed that rotary wave-fronts (spiral-shaped wave-fronts in 2D simulations) can break up into daughter wave-fronts through encountering regions of refractory tissue (tissue resistive to propagation or blocking it entirely) and that this continual break up is responsible for ventricular fibrillation (VF) (Fenton et al., 2002; Fenton & Karma, 1998). Localization of the core of these rotary waves (the phase singularity) is essential to characterize a state of fibrillation and the success of defibrillation strategies (Fenton & Karma, 1998; Bray & Wikswo, 2002; Iyer & Gray, 2001). The dynamics of fibrillation are not random but are, instead, chaotic (Garfinkel et al., 1997; Weiss et al., 1999; Bayly et al., 1998). Multivariable models are, therefore, able to describe this deterministic process. We shall specifically examine the Fenton–Karma model of spiral wave break up leading to fibrillation (Fenton et al., 2002; Fenton & Karma, 1998) and the FitzHugh–Nagumo model of membrane dynamics (Roth, 1998; Winfree, 1991). In the three-variable Fenton–Karma model (the potential and two gate variables describing the conductivity of the cell membrane) a phase space plot of a gate variable † Email:
[email protected] c The author 2005. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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against potential for a complete action potential produces ring-like figures (RLFs) whose relative size is dependent on the refractoriness of the tissue (Fenton et al., 2002; Bray & Wikswo, 2002; Iyer & Gray, 2001). In the two-variable FitzHugh–Nagumo model we shall evaluate the phase space of the single gate versus potential (ν(t) versus V (t)) and delayed potential versus potential (V (t + ∆t) versus V (t)). [Caution should be exercised in defining phase space as the exact effects on phase dynamics are not well understood (Bray & Wikswo, 2002).] Smaller RLFs are inscribed in larger RLFs and inscribed within all is a central region of phase space (see Fig. 1). Choosing a point (x ∗ , y ∗ ) inside the central region of phase space (any point will suffice, although a more central point is preferred) as the origin, the phase φ is defined to be the polar angle (four-quadrant inverse tangent) of the point on the trajectory the RLF traces out relative to this origin (Bray & Wikswo, 2002; Iyer & Gray, 2001); y − y∗ . (1.1) φ = Atan x − x∗ What is essential in choosing the origin is that the phase of the RLFs of all action potentials must progress through 2π radians. With a correctly calculated phase, one suggested method to localize the phase singularity relies on the fact that the closed line integral of gradient of phase will be a non-zero integer multiple of 2π for a path P that encompasses a phase singularity and is zero otherwise (Bray & Wikswo, 2002; Iyer & Gray, 2001; Gray et al., 1998). 2πn, n = ±1, ±2, . . . , if P encloses a phase singularity; ∇φ · d r= (1.2) 0, otherwise. P A time-consuming process when producing phase space plots is the visual inspection and selection of the origin (Bray & Wikswo, 2002; Iyer & Gray, 2001; Gray et al., 1998). Once an origin is chosen, localization of the phase singularity is a simple numerical process. Thus, automating the selection
FIG. 1. A phase space plot of the gate υ versus the scalable transmembrane potential over several action potentials for a point on a 2D tissue surface. The Fenton–Karma model, parameter set 6, is used. A plot of the action potentials associated with these phase space figures is given in Fig. 3.
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of the origin will allow the entire localization process to be automated. We foresee several immediate benefits of automating the process. First, the effect of the method used to produce a phase space plot can be evaluated by selecting the origin in a systematic manner, thus allowing for side-by-side comparisons of models of cardiac membrane dynamics. Additionally, defibrillation strategies can be evaluated on the basis of which mechanism of rotary wave break up and/or model is used in a much more time-efficient manner.
2. Methods Often in the literature the RLFs of phase space are described as circular (Gray et al., 1998). A cursory inspection of Fig. 1 shows that elliptic may be a more general description. The method we suggest is to best fit the smallest RLF to an ellipse (of which a circle is a special case) and use the center of this ellipse as the origin of phase space. Since these are the only two closed conic sections, this provides the most general best fit to a conic section (Arya, 1990). The smallest RLFs correspond to the smallest action potentials and the duration of an action potential (as well as its maximum potential) is dependent on the elapsed time between the wave-front of an action potential and the wave back of a previous action potential at a point on the tissue (a governance known as restitution) (Garfinkel et al., 2000). This restitution contributes to wave break up and VF (Fenton et al., 2002), although other factors also play a role (Cherry & Fenton, 2004; Tolkacheva et al., 2003; Cytrynbaum & Keener, 2002). The smallest action potentials are located near the phase singularity itself and it is the localization of these singularities that we are after. We select several points at random on the tissue and monitor their potential for a sufficiently long time. In this way, it is likely that a phase singularity will come very near one of these points at some time and a sufficiently small RLF can be obtained whose center will be in the central region of phase space. With time traces of potentials for several points over a long time, it is necessary to determine the minimax of the traces of potentials versus time. That is to say, the minimum peak value of all traces over all times. (Note: this is the smallest action potential of all traces observed, not just a single trace.) The action potential duration (APD) is the elapsed time between which the action potential was of a certain potential on its upstroke and of the same potential on the downstroke. This potential is generally indicated with a subscript such as APD80 , which would indicate the potential is 20% of its maximum possible value (80% recovered). In the scalable model (V = [0, 1]) APD80 would indicate 0.2 for this potential. We shall include action potentials for consideration as the smallest RLF that would be included in an APD80 restitution curve. To best fit to an ellipse, we shall use an APD90 action potential. Peaks of potentials are critical points and have potentials greater than resting potential. As we are using a numerical approximation, we suggest the critical point be defined as the point in time where ∂∂tV changes from increasing (>0) to decreasing (
