#### What We Already Know about This Topic

#### What This Article Tells Us That Is New

*de novo*behavior; a phenomenon which has come to be known as “chaos.” Such stereotypes provide a unified description that closely parallels what is observed under incipient and established physiologic instability. However, in this model, seemingly random fluctuations may emerge as a direct consequence of the underlying dynamics, rather than as a pathologic breakdown of homeostasis leading to the unwanted transmission of exogenous influences. In a nonlinear model, physiologic equilibrium is in fact achieved through an ensemble of dynamical fluctuating processes, a situation known as homeokinesis. It follows that these fluctuations may contain important information regarding the dynamical state of the system that is otherwise not apparent from traditional observations. Unfortunately, such systems are inherently difficult to study, and recent interest has been fuelled in no small part by advances in mathematical tools and the availability of affordable computing power. Nevertheless, the application of sophisticated analysis techniques has unmasked previously “hidden information” in time-series data of a wide variety of natural processes.

^{1}

^{–}

^{7}The observed fluctuations exhibit characteristics of the nonlinear processes that generate them, thus the underlying physical state of the system can be inferred.

^{8}It has been demonstrated that changes in the underlying dynamical state may be detectable before they propagate as instability,

^{9}suggesting that bedside black-box nonlinear analysis may be a possibility for tracking changes and providing an early warning of impending physiologic collapse. Furthermore, bedside detection and categorization of different “unstable states” has the potential to guide patient care and gauge response to therapy.

^{3}weather patterns,

^{7}neuronal response in functional magnetic resonance imaging,

^{4}geophysics,

^{5}behavior of broadband internet traffic,

^{6}and highway congestion.

^{9}A fractal is a geometric structure constructed from an infinite set of increasingly small subunits. Each subunit is a scaled-down replica of the whole, producing a self-similar property across length scales,

*i.e.*, across many levels of magnification.

^{10}Fractal theory is concerned with structural self-similarity over scales of space or time and techniques have been developed to quantify the statistical self-similarity of a structure, recognizing subunits that share such statistical properties across orders of scale.

^{2},

^{10}The statistical notion of self-similarity is equally applicable in the time domain, permitting analysis of processes that are self-similar over many time scales.

^{2},

^{10}Fractals provide a natural nonlinear metric for the characterization of fluctuating processes.

*et al*.

^{1}demonstrated that the human heart rate, in health, has a fractal temporal structure, and that the fractal properties change in congestive heart failure. Subsequent work by the same and other authors have studied fractality for a number of physiologic processes, including heart rate in age and disease,

^{2}stride length in health and chronic neurologic disease,

^{2},

^{11}and middle cerebral artery blood flow in subarachnoid hemorrhage.

^{12}Importantly they have shown that fractal properties persist throughout episodes of rest and sleep and are not a result of superimposed complex physical or mental activity but instead reflect an emergent property of the underlying dynamical system.

^{10}Such systems are described as monofractal and several techniques for estimating the Hurst exponent have been developed, including detrended fluctuation analysis and rescaled-range analysis.

^{10},

^{13},

^{14}

^{1}it has been established that complex dynamical systems may instead result from a spectrum of processes with a range of different scaling parameters. Such systems with multiple scaling behaviors are called multifractal. These arise through several interacting processes, each with different self-similar behaviors acting in concert to produce the overall structure, or a single process whose self-similar statistical properties change within the timeframe under analysis. Parameterization of fractal properties, which can be thought of as a continuum from mono- to multifractal, provides an insight into the behavior and mechanisms of the underlying control systems.

^{2},

^{15}

*h*) can be interpreted in terms of the statistical properties of the time-series,

*viz*.

*h*< 0.5: Antipersistent behavior. An increase at one time interval is more likely to be followed by a decrease and

*vice versa*.

*h*= 0.5: Uncorrelated random walk. Increases and decreases are equally likely.

*h*> 0.5: Persistent behavior. An increase in one time period is more likely to be followed by a further increase in the next period.

*y*-axis charts the Hausdorff dimension,

*D(h).*The precise mathematical definition of Hausdorff dimension is rather involved; in essence it reveals the relative frequencies of the Hölder exponents, akin to a probability distribution. The Hölder exponent,

*hm*, with the greatest Hausdorff dimension is the most frequent and hence most dominant scaling behavior. In a manner common to other statistical distributions, the singularity spectrum is parameterized using measures of central-tendency and spread, conventionally

*hm*and width-at-half-height (

*WHH*), the width of the curve at half the maximal Hausdorff dimension (fig. 1A). Such parameterization permits the recording and comparison of multifractal properties within or across datasets. Multifractal techniques can also quantify monofractal scaling behavior, thus providing a very general approach. Relatively monofractal systems are characterized by narrow singularity spectra. In contrast, strongly multifractal systems display a wider distribution of Hölder exponents (fig. 1).

#### Materials and Methods

##### Patients and Data Collection

*inter alia*, for the intraoperative administration of cardiovascular drugs. Intraoperative analgesia was provided with remifentanil (0.1–0.2 μg

^{−1}· kg

^{−1}· min

^{−1}). Epidural anesthesia was not employed intraoperatively.

##### Multifractal Analysis

*hm*, and

*WHH*, as previously described (see Introduction and fig. 1). The spectral parameters for heart rate (

*hmHR*,

*WHHHR*

**)**and MAP (

*hmMAP*,

*WHHMAP*) were recorded in operation order and paired with the logbook of operative events for subsequent interpretation and analysis. Episodes of hypotension where α-1 agonist metaraminol (0.5 mg IV bolus) had been administered at the anesthetist's discretion were identified. Multifractal analysis was performed for the periods immediately before and after vasoconstrictor administration.

*et al.*for further detail.

^{16}

##### Statistical Analysis

*hmMAP*and

*hmHR*before and after metaraminol bolus are represented as mean ± SE. These are further subdivided upon whether the patient received treatment with atropine. A two-tailed, paired Student

*t*test is used to compare per-patient mean values of

*hmMAP*and

*hmHR*before and after metaraminol bolus (per-patient mean values are used to eliminate errors because of multiple comparisons). Robust hypothesis testing on

*hmHR*to compare patients based upon administration of atropine is difficult, as independence between measurements from the same patient cannot be assured. However, results of a two-tailed paired Wilcoxon signed-rank test (with 1,000-round Monte Carlo jitter modification for tied ranks) is presented under the assumption of independence for reference.

*WHHHR*and

*WHHMAP*. Further details on the quality of the

*WHH*results and the methodological issues surrounding

*WHH*estimation are included in the Discussion.

*P*< 0.05 was considered to be statistically significant.

#### Results

Table 2 Image Tools |
Table 3 Image Tools |

*e.g.*, fluid administration) or surgical perturbation was present. Summary statistics for the spectral parameters calculated from these events are detailed in tables 2 and 3.

*hmMAP*. Immediately after the bolus, a statistically significant increase in

*hmMAP*is seen (table 2), reflecting a change to a more correlated physiologic process with greater long-term memory behavior. During this time the system exhibits a short period of correlated behavior (Hölder exponents more than 1) or near- correlated behavior (Hölder exponents tending toward 1). Subsequently

*hmMAP*decreases to a value greater than its starting point, suggesting a more correlated underlying behavior persists. Metaraminol causes almost pure α-1 adrenergic mediated vasoconstriction, changing an under-filled vasculature into a much “tighter” system with fewer homeokinetic mechanisms in action. In other words, less homeokinetic complexity is present. Our observation of altered fractal behavior in response to pharmacological perturbation indicates a profound alteration in the nonlinear character of the ensemble of interacting physiologic homeokinetic mechanisms. Thus, cardiovascular pharmacology allows the clinician not only to manipulate traditional physiologic parameters, such as blood pressure or HR, but also to fundamentally control the stability of the system as a whole by modifying the total activity/number of homeokinetic mechanisms in operation.

*hmHR*is shown in figure 3B and behaves differently to that for blood pressure. Metaraminol has essentially no direct chronotropic effect and this may explain why the dynamical nature of heart rate homeostasis is unaffected. Statistical nonsignificance is demonstrated in table 2.

*WHHMAP*and

*WHHHR*) varied throughout the intraoperative period and were altered by metaraminol but without discernable pattern (table 3).

*hmHR*). This is clearly evident even before metaraminol is given. Atropine antagonizes the background parasympathetic tone and prevents further modulation of parasympathetic effects on the cardiac electrical systems. It could be considered as removing a degree of freedom from the nonlinear heart rate feedback, reducing the homeokinetic complexity and tending to make the system more correlated in its behavior. Following a metaraminol bolus, the dominant correlated behavior is sustained. In contrast, the effect of atropine is not clearly seen in MAP fractality (fig. 3A; identical marking), suggesting that heart rate variation is not a key homeokinetic mechanism in the determination of MAP.

*et al.*.

^{1}Beat-to-beat HR was converted to interheart beat interval and intervals of greater than three standard deviations were corrected through linear interpolation, including the point either side of the outlier. A similar approach was adopted for mean arterial blood pressure. The interbeat difference in mean arterial blood pressure was calculated and differences of greater than three standard deviations were corrected by linear interpolation. Overall approximately 3 or 4% of data points were interpolated by this method. Comparison of the fractal spectra for preprocessed and unprocessed data sets demonstrated that no significant changes occurred after preprocessing, except in blocks where many errant data points clustered. A number of the 256 data point blocks (12% of total blocks) contained a large proportion of interpolated points (up to 21% points were interpolated) and despite showing multifractal behavior in the unprocessed analysis they demonstrated monofractal behavior after linear interpolation. An unstable patient with frequent marked fluctuations in HR or blood pressure would be heavily interpolated by this technique, removing the fractal structures of interest. We therefore concluded that interpolation at the preprocessing stage was unnecessary as it did not provide a significant increase in the quality of results, neither were artifacts or ectopics found to be responsible for the vast majority of the detected fractal structure.

*hmHR*response similarly partitioning on prior administration of atropine. Limiting the blocks to a fixed time period includes more or fewer points than previously, hence small differences in the fractal estimation are to be expected. In practice, the length of each block could be fixed at any length. Our choice of 3.2 min was restricted by the timing and clustering of interventions during the operative period and was selected to ensure that the blocks were long enough to encompass the same behavior of interest as blocks in figure 3B.

#### Discussion

*et al.*demonstrated that the resting heart rate dynamics of young, healthy subjects show multifractal structure, suggesting that the classic understanding of “homeostasis” is too simple and underlying control systems are paradoxically “far from equilibrium.”

^{2}Periods of physical stress including vigorous exercise and critical illness demonstrate the great adaptability of homeostatic mechanisms; such dynamic scaling behavior may be conferred by the fractal properties of the physiologic feedback cascades. However, the self-similar scaling demonstrated in HR behavior degenerates with advancing age and chronic heart failure. Loss of fractal scaling ability is implicated in the reduced adaptability and consequent susceptibility of the elderly and comorbid at times of physiologic stress.

^{2}

*hmMAP*and

*hmHR*to vasoconstrictor therapy. Vasoconstriction led to an increase in

*hmMAP*(fig. 3A) but not in the case of

*hmHR*(fig. 3B). This suggests that vascular tone is an important homeokinetic mechanism in the control of MAP but not of heart rate, which seems reasonable. Conversely, administration of atropine is expected to eliminate HR variability from the ensemble of homeokinetic processes. Figure 3B demonstrates a consistently higher value of

*hmHR*for those patients pretreated with atropine. This implies a more correlated and less complex signal, consistent with a reduction in homeokinetic degrees of freedom. This partitioning is not evident for MAP data (fig. 3A), suggesting that pharmacological “paralysis” of HR variability does not much affect control and stability of MAP, which seems intuitively reasonable.

*et al.*

^{2}demonstrated that physiologic processes possess fractal structure and the fractal properties change in chronic disease, reflecting a reduction in homeokinetic repertoire in the failing organism. It seems plausible that the fractal properties of the same systems change or degenerate after acute insults. External perturbations such as changes in mechanical ventilation, surgical insults, and pharmacological interventions can have a marked effect on vascular physiology, and these alone might explain some of the detected changes in fractal structure. Analysis of how fractal properties change in response to a perturbation can also give useful insight into the underlying system. An extreme example occurs when nonlinear systems are on the verge of instability. A small external perturbation can have a destabilizing effect, leading to profound changes in behavior that perhaps in other situations would not occur.

*et al.*

^{17}demonstrated that multifractal analysis using short blocks of data produced errors in estimation of the multifractal spectrum tails, hence leading to errors in the calculation of

*WHH.*The short time intervals used in this study and the inherent inaccuracies in

*WHH*estimation are the most probable reason why trends are not seen in

*WHH*during periods of instability and pharmacological manipulation.

*hmHR*Hölder exponents are rather larger than those reported previously.

^{1},

^{2}We attribute this effect to the comparatively short blocks of length 2

^{8}we analyzed, much shorter than those used in other work.

^{1},

^{16},

^{17}This will tend to give larger, more correlated, estimates for

*hmHR*. As demonstrated by Oświęcimka

*et al.*

^{17}errors in the estimation of

*hm*are reasonably preserved with decreasing block length, and despite our absolute values being larger than previous works, we believe the overall trends demonstrated in this study are physiologically meaningful. Our measurements are therefore necessarily distinct from those of others

^{1},

^{2}in that they represent short-range scaling behavior and we cannot draw conclusions over longer time scales in our strongly perturbed systems where exogenous influences are likely to be causing significant changes of state. However, we suggest that it is the short-range scaling behavior that is of most clinical relevance in describing hemodynamic instability, which occurs over short time scales.

^{18}where external perturbations and clinical interventions are shown to move a patient from one state to another. Fractal properties are one measure of this state space and are shown in this study to change in association with external perturbations in a seemingly predictable way, with vasopressor agents and atropine producing more correlated and less complex fractal properties of blood pressure and HR control mechanisms, respectively. We suggest that studying the effect of perturbations to fractal systems is a valid and powerful technique for analyzing the underlying dynamical properties, giving information about current state and potential future behavior. Complex behavior is an emergent property of both the perturbing inputs and their interplay with nonlinearities: Many clinicians involved in the management of critically ill patients will recall patients who have responded unexpectedly to an intervention that should have improved their illness. An understanding of the state space and the ability in real-time to determine where in the state space a given patient lies may provide a powerful tool in guiding clinicians to the most appropriate package of interventions to restore the patient to a stable state.