Cardiac Control Entropy: Changes in complexity of

Cardiac Control Entropy: Changes in complexity of HR signal can be transient or sustained Stephen J. McGregor1, Erik M. Bollt2, Joe Skufca2, Melvyn Rubenfire3 1

School of Health and Human Performance, Eastern Michigan University, Ypsilanti, MI, 48197. Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY, 13699. 3 Division of Cardiovascular Medicine, 25 Frank Lloyd Wright Drive, Ann Arbor, MI 48106 2

Abstract: Understanding control of cardiac function is crucial for characterizing physiological responses, and for developing clinical tests. We present here the use of a recently developed statistic, control entropy (CE), which was designed to be free of the stationarity requirements of other entropy based methods used to identify changes in state of the cardiac control system. We show CE responses of ECG signal during two well characterized challenges to autonomic control of HR; postural change and the standardized Valsalva Maneuver. We also contrast data from different physiological conditions (hypotension, tachycardia, and normal) which may dictate differing control strategies identified by CE. This interesting link between function and output signals, and changes in healthy and subclinical disease populations suggests an exciting new direction for developing simple clinical tests.

Macklem posits that organized life and the processes that support it lie on a continuum of complexity between low entropy of stable, inaninimate objects on one end and high entropy, disorganized systems such as weather on the other. In the middle of this continuum lies ordered, adaptable, fluctuating life sustaining processes such as metabolism or other physiological processes/systems (12). Debate notwithstanding, direct evidence from physiological measures has supported this view to some extent (17, 21, 22). So, it would seem that healthy, vibrant, adaptable processes exhibit an entropy that is higher than some pathological conditions that may exhibit high order, but low adaptability, and yet, lower entropy than other conditions that exhibit more random behavior. Specifically with regard to cardiac rhythms, it has been argued that “healthy” hearts typically exhibit patterns with a higher entropy than unhealthy hearts (7, 16). The generalizability of this view is still not clear as in some cases, the entropy of patterns from unhealthy hearts is time scale dependent (e.g. atrial fibrillation) (7) and unhealthy hearts may also exhibit comparable or relatively high entropy (2, 7, 20). The technical innovation necessary to interpret the implication of the entropy of the process, is to associate an appropriate partition to the signal (4, 5). One problem that presents itself when using non-linear approaches, for the determination of healthy vs diseased hearts exhibiting various rhythms is the requirement of stationarity (16, 18). Since many physiological systems exhibit slowly moving parameters and changes on multiple time scales, the requirement of stationarity imposes serious limitations to the interpretation of many phenomena, and the assumption of stationarity may impart excess confidence in the conditions. Simply quantifying the entropy of a system over a given time frame may cloud interpretation of entropy measures by excluding, or attenuating changes in entropy that may occur during the supposed stationary period. Further, many physiological processes of

interest, particularly cardiac arrhythmias occur under clearly non-stationary, often fleeting, conditions, and therefore traditional approaches to entropy determination are not appropriate. This may help explain why previous entropy approaches have not found more wide acceptance in clinical disciplines. To address the problem of stationarity in entropy analysis, Bollt et al. (6) recently introduced a novel regularity statistic, Control Entropy (CE). There are several aspects of CE that make it unique among regularity measures, but most important here is the alleviation of dependence upon stationarity. CE is suited particularly to conditions of dynamically changing physiology, in particular, sub-clinical, but relevant phenomena of perturbed homeostasis. Furthermore, CE can be associated with the system’s control effort in the underlying process (6). Using ECG signal, we will attempt here to relate this interpretation to cardiac function, particularly differences in healthy and unhealthy individuals. To briefly restate the definition of control entropy (CE), recall the quantity at the center of information theory is the Shannon entropy (8), r

H S = −∑ pi ln pi i=1

which characterizes the amount of uncertainty indicated by a partition on symbols i=1, 2, ..., r, at least when each state has associated identically and independently distributed (i.i.d.) probabilities pi . The key problem is that i.i.d. must include stationary probabilities with time. Fundamentally, however, this basic and typically implicit assumption behind information theoretic analysis is contrary to our interest which is to study nonstationary systems. We have recently defined (6) a control entropy function,

CE[x(t);w](t) = SampEn[℘(

dx (t,t + w)] , dt

to indicate parametric changes in the scenario of varying parameters, under mild conditions on the dynamical variable x(t). Here, we denote a data set, x(t), together with its time domain as related to the sampling time duration, and w denotes a window length with respect to pre-history considered relevant at any instant t. Further, note the symbol ℘ in this equation is used to denote a partitioning step to associate probabilities to each state, which mathematically associates a metric topology. We (6) have shown that a SAX method (9, 10) is particularly useful, but other variations such as a continuous neighborhood approach may also apply. To demonstrate the utility, we first present the CE response of the ECG signal during two well characterized challenges to autonomic control of HR; postural change and the standard Valsalva Maneuver (VM) (Figure 1). It is evident that CE clearly identifies cardiovascular events such as posture changes and VM from the raw signal over the background noise. When either short window lengths (e.g. 801) corresponding to < 2 sec in duration or longer window lengths (e.g. 6401) corresponding to

approximately 13 sec in duration are used, CE “spikes” are observed during the actual postural changes and VM, as opposed to the more constant CE displayed during maintained posture. There is a change in CE from supine to standing (Figure 1 bottom) that might be identified by other measures of regularity (e.g. AE and SE) once conditions of stationarity are met, but it is unlikely that the clear changes in entropy identified during the postural changes and VM would be identified. In fact, it would not be appropriate to “look for them” as these are clearly non-stationary conditions.

Figure 1. ECG and CE of ECG during postural change and Valsalva Maneuver (VM). Top: Raw ECG collected at 500 Hz from healthy young subject. Middle: CE of unfiltered signal. Bottom: Exploded view of middle from 0-900 sec (Red- w = 801, White – w=6401). In all panels 0-300 and 600-900 sec = lying supine, 300-600 sec = standing, Yellow arrows indicate 20 sec standardized VM performed with 240 sec between.

Fourier Transformation is commonly used to analyze components of autonomic cardiovascular control in the frequency domain. In general, it is accepted that the high frequency domain (HF; 0.15 – 0.5 Hz) represents input from the parasympathetic and the low frequency domain (LF; 0.04-0.15 Hz) from the sympathetic, or a combination of sympathetic/parasympathetic arms of the autonomic nervous system respectively (1, 13). In Figure 2, we contrast CE of the ECG signal in a healthy subject (Green) with two individuals exhibiting clinical conditions, 1) orthostatic hypotension (OH; White) and 2) sinus tachycardia (ST; Red). In the bottom panel, we see that CE of HF signal is higher in the ST compared to healthy and OH, which are similar during standing. Likewise, the subject with sinus tachycardia exhibits the strongest CE spike in the HF signal during VM.

In contrast, in the top panel, CE of LF is similar in healthy (green) and ST (red), and lower in OH (white) during standing. The hypotensive subject exhibits the “weakest” CE response during VM in the HF frequency. Whether these differences are physiologically significant needs to be more clearly established, but as OH is associated with the lowest CE during standing in both HF and LF ranges, it seems apparent that this subject’s autonomic control system is in a low state of entropy and unable to appropriately adapt to rapid challenges to homeostasis. This is supported by the weak CE response in the standard VM maneuver. In the ST subject, CE is high in both the HF and LF ranges, and higher than the healthy individual in the HF range (Figure 2, bottom). As it would be assumed that changes in complexity, and healthy “high” entropy would be desirable in the HF range, this would be reflective of adaptability of the parasympathetic system during challenges to homeostasis under normal activities of daily living.

Figure 2 Control entropy (CE) of filtered ECG signal during postural change and standardized Valsalva maneuver (VM). Top: CE of low frequency (LF) ECG signal (0.04-0.15 Hz). Bottom: CE of high frequency (HF) ECG signal (0.15-0.5 Hz). Red – Sinus tachycardia (ST), White – orthostatic hypotension (OH), Green - Healthy normal

Is it possible that this elevated CE in ST is tending toward an elevated chronic complexity of “unhealthy”, unadaptable entropy as proposed by Macklem (12)? Further investigations of CE of ECG signal in clinical populations with established cardiovascular autonomic neuropathy would be necessary to say with certainty. To determine whether assessing cardiac autonomic control using control entropy could be clinically useful, the CE responses of healthy and diseased populations need to be catalogued under a variety of conditions and a signature developed for a “dynamical disease(s)” (11). In particular, the independence from the requirement of

stationarity could be exploited and the CE response of ECG during different dynamic challenges to homeostasis could be examined to collect a more robust catalogue of not only dynamical diseases, but “dynamical phenotypes” (3). An intriguing approach is the use of CE of ECG and other physiological variables during tests of exercise capacity, such as the VO2max. Exercise tests are used regularly in clinical settings and it would be exciting to determine the utility of a measure such as CE under such conditions. Preliminary data from our laboratory indicates CE can discriminate changes in regularity of cardiovascular and ventilatory control, which exhibit both coupling and diversion during intense exercise, which may provide both clinical and experimental insight to these responses. There has been interest in entropy analysis of physiological signals with clinical application since Pincus introduced AE (14, 15). It is likely the requirement of stationarity limits the pragmatic use of such measures in many clinical situations. With the relief of the requirement of stationarity with CE, a particularly exciting use may be to use CE as a “data mining” instrument in application to streaming high frequency signals such as ECG, ventilation, blood pressure etc. Sealy has argued for the value of nonlinear measures in the critical care setting (19), and there has been some interest in the use of regularity measures in anesthesiology . As can be seen in Figure 1, in high frequency, streaming, raw signal, events as subtle as VM and postural changes can clearly be identified over background noise. These changes can be distinguished in time frames as short as 5 sec (data not shown), yet, noise does not appear to contaminate the CE analysis. Further, for monitoring purposes and event identification, waveform interpretation is not necessary, thus making automated monitoring feasible. We have thus briefly reviewed here that CE is an exciting new tool capable of handling nonstationary physiological, particularly cardiac signals. The use of CE may offer insights into control issues of the heart, as well as lead to a clinical diagnostic tool.

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