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Anesthesiology:
doi: 10.1097/ALN.0b013e31826a4d49
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Fractals in Clinical Hemodynamics

Buchman, Timothy G. Ph.D., M.D., F.A.C.S., F.C.C.P., M.C.C.M.

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IN this issue of ANESTHESIOLOGY, Bishop et al. report on a novel approach to the interpretation of hemodynamic analysis, namely fractal and multifractal.1 Although the readers of this journal are intimately familiar with the cardiovascular system and its gyrations in response to stress, surgery, and anesthetics, the application of fractals and specifically of multifractals to data synthesis and interpretation initially might seem beyond the reach of, or relevance to, practitioners. My purpose in writing this commentary is to offer context and perspective to what the authors have done and what it might mean to today's clinician, and to tomorrow's.
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“… multifractal variable physiology … is not only a signature of health but also a potential therapeutic tool …”
First, let us observe that those of us who care for sick patients in highly technical environments are data-rich and information-poor. We who care for sick patients—whether in the operating room on in the intensive care unit—cannot possibly accumulate, absorb, and synthesize the streaming waveforms or their 6-s scalar snapshots (the glowing digits that announce a heart rate to be 82 beats/min, a blood pressure to be 116/54 mmHg, and so on) that dance across the monitor. Interpretive strategies abound, the simplest of which is to create a time series of values and then attempt to give some statistics on the time series. Technological considerations aside, modern hardware and software make it easy to determine the time intervals between heartbeats (or its reciprocal, the instantaneous heart rate), the series of systolic and diastolic blood pressures, and so forth.
Anesthesiologists are rightly most concerned about gross trends: the heart rate going up and the blood pressures going down signals a problem with intravascular volume, the blood pressure going up and the heart rate going down suggests an intracranial catastrophe, and so forth. But what if there is no apparent trend? What if the values are seemingly scattered around some value we call “normal”? Isn't this just “regular rate and rhythm”?
It is not. Normal hemodynamics (and several other aspects of physiology) are inherently—if subtly—variable.2 Moreover, changes in the character of that variability often herald substantial changes in patient condition.3 Therefore, it is a matter of some interest to characterize that variability and more importantly to detect and describe those changes before gross deterioration of the patient.4
The general structure of hemodynamic time-series variability is fractal. Like branching from a tree-trunk (each branching level produces more, but smaller, branches), the variability of physiology is characterized by coarser and finer excursions. Conventional measures of dispersion do not do us much good in talking about instantaneous heart rates, any more than they help us pick out a good-looking tree at the horticultural center. The key point is that the variability of health physiology can be analyzed on different temporal scales. One way to go about assessing variation across scales of time is to “coarse-grain” the data, that is to average adjacent data points (pairs, threes, fours, described as scale 2, scale 3, scale 4, and so forth) and reassess the variability at scale. This is a basis for multifractal assessment.5 The analysis of data from chronic illnesses (such as congestive heart failure) produces very different patterns across scales, typically reflecting loss of variability. Moreover, such analyses seem to predict important clinical outcomes.6
All of this is prelude to the work presented in this issue of ANESTHESIOLOGY. Bishop et al. reasoned that the stress of operation and the interventions of the anesthesiologist might acutely alter the mechanisms that give rise to multifractality in human physiology. They acquired the available signals—electrocardiograph and continuous blood pressure—and processed those signals into beat-to-beat time series. Next, they used wavelets to identify how much fractality was present in the time series at different points in care.
Why wavelets? It's easiest to ask why not a Fourier transformation. Recall that any time series (or for that matter, any polynomial) can be represented as a sum of sine waves of different frequencies. That is the Fourier transformation. We “hear” the Fourier transform every time we play music on an MP3 player—the digital bits that we download are no more than the coefficients that set the strengths of different tones (sine waves) that are combined to hear our favorite tune.
Wavelets accomplish much the same thing, with an important exception. Fourier coefficients localize frequency across an entire time series, but wavelets localize their information in both frequency and time and thus can encode localized variability within the time series. Of course, just encoding the variability is not enough, one has to have a standard way of finding and describing that localized variability. That way is the tongue-twisting “wavelet modulus maxima technique” used by the authors. It turns out that just knowing the “wavelet modulus maxima” is enough to almost completely reconstruct a time series. It also enables one to characterize how much “multifractality” there is in a time series through display of a “singularity spectrum.” The “singularities” that are being characterized are the steps (slow excursions) and cusps (rapid excursions) in the time series.
A singularity spectrum typically looks like a parabola. It has a maximum (the peak) and a spread (a convenient way of describing the spread of a parabola is to describe its width at half height). Where that maximum occurs characterizes the dominant fractal dimension. The width at half height gives a sense of multifractality (fig. 1 in article by Bishop et al.).
Evidence continues to accumulate that variable physiology, especially multifractal variable physiology, is not only a signature of health but also a potential therapeutic tool when applied via a physiologic support device, such as a mechanical ventilator or cardiopulmonary bypass device.79 Novel ventilator modes, such as adaptive servo ventilation, demonstrate that application of variable support in one physiologic system (respiratory system) promotes variable physiology in a linked system (cardiac).10
What can we conclude from the article by Bishop et al., which shows us that a vasoconstrictor alters the fractal dimension of the blood pressure (but not the heart rate) while atropine alters the fractal dimension of the heart rate (but not the blood pressure)? What can we conclude from the failure to detect a change in multifractality with either agent? Simply that acute management of anesthetized and critically ill patients alters the physiologic regulatory mechanisms in ways that cannot otherwise be detected with conventional displays. As the authors freely acknowledge, this is a pilot study, and much remains to be done. As readers, and more importantly as clinicians, we should understand that hidden within the rivers of data marching across the monitors is additional information about the compensatory mechanisms that drive patient physiology and our care. Fractal analysis of physiologic data is not yet ready for use at the bedside. But it is coming in our clinical lifetimes.
Timothy G. Buchman, Ph.D., M.D., F.A.C.S., F.C.C.P., M.C.C.M., Emory Center for Critical Care, Woodruff Health Science Center, Atlanta, Georgia. tbuchma@emory.edu
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References

1. Bishop SM, Yarham SI, Navapurkar VU, Menon DK, Ercole A: Multifractal analysis of hemodynamic behavior: Intraoperative instability and its pharmacologic manipulation. ANESTHESIOLOGY 2012; 117:810–21

2. Goldberger AL, Amaral LA, Hausdorff JM, Ivanov PCh, Peng CK, Stanley HE: Fractal dynamics in physiology: Alterations with disease and aging. Proc Natl Acad Sci USA 2002; 99(Suppl 1):2466–72

3. Buchman TG: The community of the self. Nature 2002; 420:246–51

4. Buchman TG: Nonlinear dynamics, complex systems, and the pathobiology of critical illness. Curr Opin Crit Care 2004; 10:378–82

5. Stanley HE, Amaral LA, Goldberger AL, Havlin S, Ivanov PCh, Peng CK: Statistical physics and physiology: Monofractal and multifractal approaches. Physica A 1999; 270:309–24

6. Ho YL, Lin C, Lin YH, Lo MT: The prognostic value of non-linear analysis of heart rate variability in patients with congestive heart failure: A pilot study of multiscale entropy. PLoS One 2011; 6:e18699

7. Graham MR, Gulati H, Kha L, Girling LG, Goertzen A, Mutch WA: Resolution of pulmonary edema with variable mechanical ventilation in a porcine model of acute lung injury. Can J Anaesth 2011; 58:740–50

8. Spieth PM, Carvalho AR, Güldner A, Pelosi P, Kirichuk O, Koch T, de Abreu MG: Effects of different levels of pressure support variability in experimental lung injury. ANESTHESIOLOGY 2009; 110:342–50

9. Singal RK, Docking LM, Girling LG, Graham MR, Nickerson PW, McManus BM, Magil AB, Walker EK, Warrian RK, Cheang MS, Mutch WA: Biologically variable bypass reduces enzymuria after deep hypothermic circulatory arrest. Ann Thorac Surg 2006; 82:1480–8

10. D'Elia E, Vanoli E, La Rovere MT, Fanfulla F, Maggioni A, Casali V, Damiano S, Specchia G, Mortara A: Adaptive servo ventilation reduces central sleep apnea in chronic heart failure patients: Beneficial effects on autonomic modulation of heart rate. J Cardiovasc Med (Hagerstown) 2012 [Epub ahead of print]

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