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Dr. Brown is Medical Director, New Mexico Center for Sleep Medicine, and Clinical Professor of Medicine, Division of Pulmonary, Allergy, and Critical Care, University of New Mexico School of Medicine.
Correspondence to: Lee K. Brown, MD, FCCP, Medical Director, New Mexico Center for Sleep Medicine, Lovelace Health Systems, 4700 Jefferson Blvd NE, Suite 800, Albuquerque, NM 87109; e-mail: lkbrown{at}alum.mit.edu
More years ago than I would care to admit, a class in thermodynamics was required as part of my undergraduate course of study in electrical engineering. All of us who were both engineering and premedicine students that year chose a particular class given by the chemistry department, since we could satisfy both an engineering requirement and a prerequisite for medical school at the same time, a form of "double-dipping" that was hard to resist. In short order, we realized that learning chemical thermodynamics was not a trivial undertaking, and many of us were fortunate to survive that class with our grade point averages more or less intact. Fast-forward 30-something years, and chaos theory has become the darling of applied mathematics, and a characteristic of chaos called entropy, which is dangerously close to a property taught in that dreaded thermodynamics class, is being used to describe all sorts of physical systems including biological systems. A case in point is the article appearing in this issue of CHEST (see page 80) by Burioka and colleagues reporting on measurements of the approximate entropy (ApEn) of respiration during wakefulness and sleep.
To the layman, chaos means undesired randomness or disorder. In mathematics, chaos theory (also known as dynamical instability) began as the study of the evolution in time of systems that are extremely sensitive to initial conditions. The usual example is how the flapping of a butterfly’s wings in South America can change the weather in Kansas. Chaos theory has evolved into the study of the behavior of physical systems that at first seem entirely random but in fact are not entirely so. Physical systems in general are said to inhabit "phase space," a multidimensional universe where each point corresponds to a fixed value for every variable describing the system, and the evolution in time of such a system can be described as a path (or trajectory) from one point to another. The physical systems described by chaos theory are deterministic, meaning that if it were possible to exactly quantify the variables describing one point, the trajectory leading to the next point in a time sequence could be entirely predicted. The basis of dynamical instability lies in the precept that these variables cannot be completely and exactly described; the initial conditions are subject to minute uncertainties, and thus the trajectory may change in a seemingly random manner. In general, any trajectory in phase space is said to occur in response to an "attractor," which determines the direction of movement. Trajectories that move toward an equilibrium position respond to a "point attractor"; trajectories that retrace a path in a strictly periodic movement are responding to "periodic attractors"; and trajectories that describe broad cycles of behavior within certain boundaries, but never exactly retrace the same path, are, by definition, responding to a "strange attractor." The latter systems are said to be chaotic but not random. Such a system is still deterministic but is nonlinear in that its future state cannot be precisely predicted over the long term. The beginning of chaos theory was prompted by the problem of predicting the movements of three interacting astronomical bodies. Examples of other such physical systems abound and include the stock market, traffic flow, fluid turbulence, population dynamics, and a myriad of biological processes.
Where does the study of thermodynamics, and the concept of entropy, fit into this field? Originally, entropy was a thermodynamic property that was used to describe a gas or other system of particles. In any fixed volume of gas, the particles comprising that gas could take on a variety of different arrangements within the volume, and the value of entropy describes the number of different arrangements that are possible for the system of particles within the volume (it is actually related to the natural logarithm of that value). Entropy is the subject of the second law of thermodynamics, which specifies that a system of particles will preferentially move toward a state in which entropy (or randomness) is maximal. For instance, a volume of gas will never, on its own and due to random movement of the gas particles, rearrange itself to occupy only a corner of the volume. The concept of entropy has been further extended by investigators of chaos theory to describe the predictability or randomness of physical systems as they change with time: the higher the value of entropy, the more random (or chaotic) the process.
The beat-to-beat variability of heart rate has probably been the most well-known medical application of chaos theory thus far. It has been evident for many years that this variability exists, but the mechanisms involved were only uncovered within the last decade or so as the necessary analytic techniques became available. Heart rate responds to the balance between sympathetic and parasympathetic input to the sinoatrial node, and this balance of tone normally varies with respiration, sleep state, circadian rhythm, posture, the renin-angiotensin system, behavioral state, and probably many other factors not yet identified.1 2 In fact, there are so many factors that normally determine heart rate that it has become evident that a complex cardiac rhythm is an indicator of cardiac health.1 2 Heart rate variability (HRV) increases with the gestational age of the fetus and during maturation in early infancy,1 and then begins a slow decline, starting in childhood, and continues to decline in the healthy elderly.3 Of clinical interest, fetal HRV decreases with antepartum distress,4 infant HRV in non-rapid eye movement sleep is lower in near-miss cases of sudden infant death syndrome than in healthy children,5 and HRV declines in patients with congestive heart failure6 and coronary artery disease, and after acute myocardial infarction. In the latter case, HRV has been correlated with outcome in that reduced indexes of complexity tend to predict poorer survival.7
The measurement of complexity also has been applied to the EEG, in which wakefulness is associated with a more chaotic EEG rhythm and the EEG during deep sleep and coma becomes more regular.8 Not surprisingly, general anesthesia results in a less chaotic EEG, and the deeper the anesthesia, the more regular cortical electrical activity becomes.9 10 Preliminary data11 also have suggested that certain disease states, such as dementia and Parkinson disease, can affect EEG regularity. Furthermore, chaos analysis applied to the dynamics of hormone release12 has been shown to distinguish healthy subjects from patients with acromegaly or acromegaly in remission,13 and healthy individuals from those with primary aldosteronism.14 In these examples, hormone concentrations were more chaotic in patients than in healthy subjects.
Not surprisingly, complexity also has been proposed as a tool for the evaluation of respiration. Several reports15 16 have suggested that baseline, awake respiration is chaotic in humans, but both used the calculation of Lyapunov exponents in their analyses, which describe the divergence of trajectories in phase space that begin at slightly different starting points. This type of analysis suggests the involvement of chaos in respiratory pattern but does not prove it. In addition, Sammon and Bruce17 analyzed respiratory pattern in rats both graphically and using numerical estimates of entropy, with results suggesting that the pattern is chaotic in vagally intact rats and becomes linear and strictly periodic after undergoing vagotomy. More recently, Engoren18 extended these observations in a practical direction, by calculating the ApEn of respiratory rate and tidal volume in patients being weaned from artificial ventilation. Breath-to-breath changes in tidal volume were more chaotic in all patients being weaned from mechanical ventilation compared with control subjects, and were more chaotic in patients who failed weaning than in those who were successfully liberated from mechanical ventilation.
The report of Burioka et al concerns the degree of chaos exhibited during sleep, which also has been examined previously. For instance, Pilgram and coworkers19 computed an index of complexity (ie, the correlation dimension) of the respiratory pattern in infants during rapid eye movement (REM) sleep and demonstrated a nonlinear deterministic process. Patzak et al20 analyzed the respiration of sleeping preterm infants (again using the correlation dimension) with similar results in REM sleep, while quiet sleep exhibited a lesser degree of complexity. Sako et al21 also previously reported on such an analysis using correlation dimension in respiratory recordings of young adult men and found the lowest degree of complexity during slow-wave sleep and the highest during REM sleep. The current report extends their findings in several important ways. The computation of ApEn is a measure of complexity that is said to more accurately reflect this property in noisy data ensembles of relatively short length.22 As previously noted, ApEn has been widely used in the study of cardiovascular and endocrine physiology and the EEG,3 4 5 9 10 12 13 14 but this is the first report of ApEn applied to respiratory patterns during sleep. In addition, Burioka et al computed the ApEn of the EEG and thus were able to compare trends in the complexity of both EEG and respiratory pattern simultaneously. Their finding that EEG complexity and respiratory pattern complexity change in the same direction makes intuitive sense, although it is not possible to conclude the direction of, or even the existence of, causality.
The knowledge that breathing is more regular, and less chaotic, during slow-wave sleep, more complex during lighter stages of non-REM sleep, and most chaotic during wakefulness and REM sleep is not by any means novel.23 The true significance of the work reported by Burioka et al lies in the promise of a new method with which to analyze the complexity of respiratory patterns during sleep. It is entirely possible that new insights into the neural control of respiration can be derived by the study of changes in the nonlinear dynamics of respiration with EEG, with various medications and respiratory loads, and with disease.
References
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