Studies on feedback control of cardiac alternans
Introduction
Sudden cardiac death due to ventricular fibrillation (VF) is the most common and often the first manifestation of coronary heart disease. It is responsible for about 50% of the mortality from cardiovascular diseases in the United States and other developed countries (Zipes & Wellens, 1998). In many cases VF is associated with, and may well be caused by, electrical alternans (Pastore, Girouard, Laurita, Akar, & Rosenbaum, 1999a). It has been shown experimentally that when cardiac tissue is stimulated by a rapid pacing protocol, the duration of the electrical excitation varies from beat-to-beat, and it is manifested as a variation in the action potential duration (APD), which may undergo a transition to the VF (Cao et al., 1999). These period doubling oscillations are referred to as “alternans” and on the scale of the whole heart these alternans are reflected in a beat-to-beat variation of the electrocardiogram (ECG) T-wave segments. In animal (Pastore, Girouard, Laurita, Akar, & Rosenbaum, 1999b) and clinical human research studies (Rosenbaum et al., 1994) it has been shown that even hearts with a small level of T-wave alternans in the ECG where at a higher risk to develop VF and sudden cardiac death.
A well-known way to produce temporal alternans in a single cell or spatiotemporal alternans in the cardiac tissue is to start pacing of the cardiac cell/tissue with sufficiently large pacing period and then to slowly decrease the length of the pacing interval until the critical pacing period at which alternation in the APD emerges. Typically, the end of action potential is followed by a period of rest, called diastolic period (DI), until the next excitable stimuli that activates new action potential propagation occurs, see Fig. 1. In the case of a short diastolic period, the cell/tissue does not have time to fully recover its resting electric properties before the next activation, which yields a shorter APD. Hence, these oscillations between patterns of a are of period doubling nature and their analysis was first demonstrated in the pioneering works of Nolasco and Dahlen (1968) where the basic mathematical analysis of alternans based on their analogy with electric systems was demonstrated. This instability of APD and DI patterns is crucially related to the onsets of VF, as it has been demonstrated by 2D numerical simulations in Karma, 1994, Qu et al., 2000. Specifically, large oscillations in the APD induce spiral wave breakup and wave turbulence. In particular, if there is a large spatial gradient among the regions of different APDs, the length of the diastolic interval falls below a critical value that is necessary for the next wave propagation. This brings about propagation failure, as the next wave front encounters the regions of the tissue that are less prone to undergo wave propagation. Propagation failure then results in the local wave break that initiates a spiral, which can initiate a similar wave break and by this mechanism invade the entire domain of the cardiac tissue. Since cardiac alternans is believed to be a precursor to VF and sudden cardiac death, an important question to address is whether in principle spatiotemporal alternans in cardiac tissue can be annihilated by means of feedback control, as this can represent an effective antiarrhythmic strategy.
In the past decade, most research efforts in the area of control of physiological systems were focused on the development of model-independent control techniques for chaotic systems, where the stabilization algorithm is based on the fact that there is an infinite number of unstable periodic orbits (UPOs) embedded in the chaotic attractor, so that small time-dependent perturbations to an accessible system parameter drive the system toward a desired UPO Hall and Gauthier, 1994, Hall and Gauthier, 2002, Hall et al., 1997, Ott et al., 1990, Socolar and Gauthier, 1998, Tolkaceva et al., 2004. In the same vein, Christini and Collins developed a novel real-time adaptive model-independent (RTAMI) control technique Christini and Collins, 1997, Jordan and Christini, 2004. All aforementioned control algorithms are model-independent and are based on the proportional perturbation feedback control paradigm which has been demonstrated to be suitable for some physiological systems. Specifically, experimental and theoretical works have demonstrated that simple closed-loop proportional perturbation feedback methods could be used to suppress the type of alternans that occur in a single cell, or in small pieces of tissue where significant spatiotemporal variations of repolarization and wave propagation dynamics do not take place Christini et al., 2006, Garfinkel et al., 1992, Hall and Gauthier, 2002, Hall et al., 1997. In the theoretical study of Echebarria and Karma (2002a) it has been demonstrated that alternans can be abolished in a small one-dimensional cable of cardiac cell tissue by applying pacing feedback produced by consecutive APD measurements at the pacing site. Analysis of this result is based on a small amplitude of alternans equation, which belongs to the class of parabolic partial differential equations (PDEs). However, this study did not provide an insight from a control point of view, which necessarily requires more detailed analysis of control-theoretic system properties (such as controllability and observability) and analysis of the closed-loop system under various feedback control laws. Specifically, the issues of implementing successful feedback control and handling inherent constraints present in the implementation of the feedback control algorithm need more careful and comprehensive examination. A possibility to suppress alternans by dynamic control methods in the human ventricles and possibly to prevent sudden cardiac death needs to be addressed from a control point of view accounting explicitly for the practical implementation of the controllers. Given that there is only a small number of electrodes that can be placed in the human ventricle for pacing purposes, and that there is only a small number of leads that can be placed in vivo in humans for the recording of cardiac activity, the main questions that need to be addressed are (1) Can in principle cardiac alternans be controlled? (2) What are the possible limitations on the applied control action? and (3) Are there any improvements in the pacing control algorithm that can be made on the basis of insight obtained by the experimental and theoretical works?
In this work, we provide quantitative answers to these questions by demonstrating first, real-time control of cardiac alternans in an extracted rabbit heart; second, by providing a numerical demonstration of the features of the pacing protocols applied; and third, by addressing analytical features of the feedback-based alternans annihilation problem. In the ensuing section, we provide features of the experimentally implemented real-time control system. The real-time control is realized by the perturbation of the basic pacing cycle length (PCL). The pacing protocol uses the feedback gain based on modulated surface optical mapping measurements of the APD duration in the vicinity of the pacing electrode. Experimental results demonstrate a marked reduction of the amplitude of alternans in the rabbit heart with a novel pacing protocol that utilizes large values of feedback gain; the controller prevents conduction block at the pacing site by using only positive basic pacing cycle length perturbations. In this way, the pacing protocol differs from previously proposed self-referencing gain feedback protocols Christini et al., 2006, Hall and Gauthier, 2002, Hall et al., 1997. In addition, in numerical studies associated with the experiment, successful alternans stabilization has been demonstrated with the use of the novel pacing protocol in the case of a 1D cable of cardiac cells (2.5 cm in length), which exceeds the length of experimentally considered rabbit heart tissue. The novel pacing protocol in numerical studies uses the measurements of APD alternans at the boundary where the pacing is applied and at a distant site away from the pacing boundary. In this way, the pacing protocol accounts for the spatial evolution of APD alternans in order to prevent occurrence of a conduction block. In Section 3, the experimental and numerical studies are complemented with an analysis of the associated amplitude of the alternans equation. Specifically, the amplitude of alternans linear parabolic PDE which includes a Sturm–Liouville spatial differential operator is considered (Ray, 1981). First, a modal representation of the linear parabolic PDE is computed. The analysis demonstrates that the spatial operator of the amplitude of alternans contains a few unstable modes that can be stabilized by means of boundary control. Namely, only a few unstable modes are stabilized by the standard pole placement technique while the remaining infinite-dimensional modal complement remains stable under feedback. In the same section, a basic condition for the controllability and observability required for an output feedback controller to achieve closed-loop stability is given. Simulation studies initially demonstrate agreement among results obtained by numerical simulations of 1D ionic cardiac cell cable model when the pacing protocol using the measurement of alternans along the cable is applied, and further results demonstrate exponential stabilization of the amplitude of alternans parabolic PDE by full state and output boundary feedback control. Finally, we point out critical features of the experimental, numerical and analytical study in order to use control to suppress alternans in the human myocardium.
Section snippets
Cable of cardiac cells model and pacing control
The one-dimensional (1D) homogeneous cable of cardiac cells of length L paced at one end is considered, and it is described by the following parabolic PDE:where the membrane current is given by the Noble model (Noble, 1962), represents the voltage that is supplied by the pacer which generates voltage pulses in the form of square pulses (amplitude: 3.5 mV; duration: 1 ms and period ms), cm2/ms and
Analysis of amplitude of alternans PDE
In this section, we provide an analysis of the amplitude of alternans equation by analyzing the structure of the alternans equation and studying the ability of model-based control to suppress alternans. Namely, associated with Eqs. (1), (2) is the small amplitude of APD alternans PDE which was developed by Echebarria and Karma (2002a). The small amplitude of alternans parabolic PDE is linearized around the spatially uniform steady state and takes the following form:
Concluding remarks
Our experimental work has demonstrated real-time stabilization of the cardiac alternans in an intact rabbit heart. Although our experimental work addresses only alternans stabilization that is recorded at the surface of the optically mapped rabbit heart, we believe that the interior of the heart close to the recording sites also undergoes successful alternans stabilization. In our experimental findings the alternans annihilation is demonstrated with model-independent self-referencing
Acknowledgment
Financial support from AHA (American Heart Association) Post-Doctoral Grant Award 0725121Y for Stevan Dubljevic is gratefully acknowledged.
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