Abstract—The fractal structure of heart rate is usually quantified by estimating a short-term (α1) and a long-term (α2) scaling exponent by Detrended Fluctuations Analysis (DFA). Evidence, however, has been provided that heart rate is a multifractal signal, better characterized by a large number of scaling exponents. Aim of this study is to verify whether two scaling exponents only from DFA provide a sufficiently accurate description of the possibly multifractal nature of cardiovascular signals. We measured ECG and finger arterial pressure in 33 volunteers for 10 minutes during each of 3 conditions: supine rest (SUP); sitting at rest (SIT); light physical exercise (EXE). DFA was applied on the beat-by-beat series of R-R interval (RRI) and mean arterial pressure (MAP). We then computed the local scaling exponent α(n), defined as the slope of the detrended fluctuation function F(n) around the beat scale n, in a log-log plot. If α1 and α2 correctly model the multiscale structure of blood pressure and heart rate, we should find that α(n) is constant over a short-term and a long-term range of beat scales.

Results show that only the long-term α2 exponent provides a relatively good approximation of the multiscale structure of RRI and MAP. Moreover, posture and physical activity have important effects on local scaling exponents, and on the range of beat scales n where α(n) can be approximated by a constant α2 coefficient.

I. INTRODUCTION ETHODS which quantify the fractal nature of heart rate may reveal aspects of cardiovascular regulation

not detectable by linear analysis. For this reason, the assessment of fractal (or self-similar) characteristics of heart rate is more and more employed to investigate physiology and pathology of cardiac control.

Self-similarity of fractal signals is often assessed by estimating the scaling coefficient (or Hurst exponent) by Detrended Fluctuation Analysis (DFA) [1]. Studies based on DFA of long-term heart rate recordings reported that the heart rate is characterized by two scaling coefficients, a short-term and a long-term exponent, respectively called α1 and α2. It has been shown that important information is contained in these exponents. In fact, they are influenced by

Manuscript received March 30, 2007. P. Castiglioni is with Polo Tecnologico (Biomedical Technology

Department), IRCCS “S.Maria Nascente”, Fondazione Don Gnocchi Onlus, Milan, Italy (fax: +39-024048919; e-mail: pcastiglioni@dongnocchi.it).

L. Quintin and A. Civijian are with University Lyon 1, Lyon, France. G. Parati is with Istituto Auxologico Italiano, Milan, Italy M. Di Rienzo is with Polo Tecnologico, IRCCS "S.Maria Nascente",

Fondazione Don Gnocchi ONLUS, Milan, Italy.

the night-day cycle [2],[3], physical exercise or head-up tilt [4]; moreover, they are altered by certain pathologies, like heart diseases [5]-[9] or atrial fibrillation [10].

The presence of two different scaling exponents suggests that the heart rate cannot be considered a fractionally homogeneous, or monofractal, process. This means that it cannot be described completely by a single fractal dimension, like the classic “1/f-noise”. On this regard, evidences have been recently provided that the heartbeat dynamics may belong to a special class of complex processes called multifractal [11],[12]. Multifractal signals are actually described by a large number of scaling coefficients, and methods different from DFA have been proposed to properly characterize the richness of heart-rate multifractal nature, as the large deviations spectrum [11] or the wavelet modulus maxima method [12]. These findings somehow challenge the use of DFA, or at least the identification of two scaling exponents only from DFA, when studying the self-similarity of heart rate.

Aim of this work is to investigate how accurately can DFA describe the possible multiscale nature of heart rate. In particular, the study addresses the following questions: are α1 and α2 a reasonable model of the scaling structure of heart rate? If it is so, what are the proper scale ranges for the identification of α1 and α2? Does posture, or physical activity, affect the identification ranges? To answer these methodological questions, we analysed heart rate data recorded in healthy subjects during different physiological conditions. The study is not limited to the analysis of heart-rate scaling structures, but it also includes the DFA of blood pressure, another cardiovascular signal whose dynamics has been described in terms of α1 and α2 scaling exponents [13]. This would allow a detailed comparison of the scaling structures of these two strongly interconnected signals.

II. METHODS

A. Subjects and Experimental Protocol Our study included 33 healthy, normotensive volunteers

(27males/6 females). Mean age was 32 (SD 8) yrs. Non-invasive arterial blood pressure at the finger artery and ECG were recorded in each subject for 10 minutes: 1) during supine rest (SUP); 2) while sitting at rest (SIT); and 3) during a mild upper limb exercise (EXE) at 5-10 watts performed in sitting position by means of an arm ergometer.

Local-Scale Analysis of Cardiovascular Signals by Detrended Fluctuations Analysis: Effects of Posture and Exercise

Paolo Castiglioni, Luc Quintin, Andrei Civijian, Gianfranco Parati, and Marco Di Rienzo, Member, IEEE

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Proceedings of the 29th Annual InternationalConference of the IEEE EMBSCité Internationale, Lyon, FranceAugust 23-26, 2007.

SaB07.1

1-4244-0788-5/07/$20.00 ©2007 IEEE 5035

These three conditions were characterized by a progressively higher degree of sympathetic activation (SUP=baseline tonic autonomic modulation; SIT=mild sympathetic activation due to blood redistribution associated with posture change; EXE=further sympathetic activation associated with physical activity). Signals were digitized at 200 Hz and 12 bits for off-line analysis.

B. Data Analysis The beat-by-beat series of mean arterial pressure (MAP)

and R-R interval (RRI) were derived from finger arterial pressure and ECG. Self-similarity was estimated for each series by the DFA algorithm available on the Physionet web site [14]. Briefly, given a beat-by-beat series x(k) of length N beats, first the integrated series y(k) was obtained after mean subtraction:

( ) ( )( )1

k

iy k x i μ

=

= −∑ (1)

with μ = ( )1

1

N

Nk

x k=

∑ . (2)

The integrated series was divided into boxes of equal length, n. In each box a least-square line yn(k), representing the trend in that box, was fit to the data. Then the integrated series was detrended by subtracting the local trend. The root-mean square of the detrended series, F(n), was calculated for each block size n between 4 and 128 beats:

( ) ( ) ( )( ) 2

1

1 N

nk

F n y k y kN =

= −∑ (3)

For a monofractal process with scaling exponent α, F(n) increases with the block size (or time scale) n as: F(n)≈ nα. This means that α can be estimated as the slope of the regression line between log(F(n)) and log(n). For real heart rate data, the slope is typically estimated over two ranges of the box size n, obtaining a short-range scaling exponent α1 and a long-range exponent α2. In this study, we extended this approach in order to estimate a local scaling exponent α(n) defined as a continuous function of the box size, or beat- scale, n. The local exponent is calculated as:

( ) [ ] [ ][ ] [ ]

log ( 1) log ( 1)log 1 log 1F n F n

nn n

α+ − −

=+ − −

(4).

Under the hypothesis that x(k) can be approximated by two scaling coefficients only, we should expect that

( )( )

1 1

2 2

for for

n n nn n n

α αα α

⎧ = ≤⎪⎨ = ≥⎪⎩

(5)

where n1 and n2 define two distinct ranges for the identification of short-term and long-term scaling coefficients.

C. Statistical Analysis For each condition (SUP, SIT and EXE) we computed the

mean over the group of the local scaling exponent function, α(n), with its 95% confidence intervals, separately for MAP and RRI. This allows to easily identify regions over which

α(n) can be approximated by constant values. We also compared the α(n) functions of MAP and RRI.

This was done separately in each of the three experimental conditions by computing the significance of the difference P (paired t-test) at each beat scale n. The threshold for statistical significance was P<5%.

Fig. 1. Root-mean square of detrended fluctuations F(n) plotted vs. the beat scale n on a log-log scale, during supine rest (SUP, blue line), sitting at rest (SIT, black line) and exercise (EXE, red line), for mean arterial pressure (MAP) and R-R interval (RRI). As reference, the ranges for the identification of short-term (α1) and long-term (α2) exponents as defined in [1] are also shown.

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III. RESULTS Figure 1 shows the average of the F(n) functions over the

group of 33 subjects, separately for each experimental condition. As to SIT and EXE conditions, F(n) of both MAP and RRI shows a clear change of slope from a beat-scale n =11 beats to about n=19 beats. This is the range where some studies separate the identification ranges of α1 and α2. However, figure 1 also suggests that the scaling structure may differ substantially during supine rest. In fact, as to RRI, the slope of F(n) in SUP condition seems to be the same over all the beat scales n. As to MAP, the slope changes more smoothly, and continuously, in SUP than in SIT and EXE.

This phenomenon appears more clearly when we plot the local scaling exponent α(n) vs. the beat scale n. Local exponents are shown in figure 2 as average and 95% confidence intervals. Clear differences exist among the three experimental conditions and between MAP and RRI.

At the lower level of sympathetic tone activation (SUP), the local scaling coefficient of RRI is relatively constant. In

fact, the 95% confidence limits lie between 0.8 and 1.0 over

the whole range of scales n, with the exception of the transition region 11<n<19, where the confidence limits fall below 0.8.

By contrast, blood pressure shows a completely different pattern during supine rest. The scaling exponent of MAP decreases progressively and at a constant rate from n=5 up to n=54, falling from about 1.3 to 0.8. When n > 54, α(n) tends to remain constant and equal to 0.8. Because of this trend, differences between local scaling exponents of MAP and RRI are particularly important at the shorter time scales; differences are significant up to the beat scale n=27.

The small sympathetic activation associated with sitting produces important changes in α(n) of both MAP and RRI (fig.2, central panel). The local exponent of RRI increases importantly between n=5 and n=11, reaching values greater than 1. The decrease of MAP α(n) from 1.3 to 0.8 is steeper in SIT than in SUP, and α(n) reaches a constant “plateau” at shorter time scales (n≥19).

The changes which follows the sympathetic activation from SUP to SIT are emphasized by the further sympathetic

activation associated with exercise (fig.2, right panel). The

Fig. 2. Upper panels: Local scaling exponents α(n) plotted vs. the time scale n for MAP (blue line, solid circle) and RRI (black line, open circle) separately in SUP, SIT and EXE conditions: average over the group of N=33 subjects and 95% confidence intervals of the mean. Lower panels: statistical significance P of the difference between MAP and RRI local scaling exponents, calculated at each time scale n; the dashed line represents the 5% threshold: when P is above the dashed line, the difference is statistical significant at the 5% level.

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local slope of MAP decreases even more rapidly during exercise, reaching a minimum value, lower than 0.7, at n=27. Exercise also further increases α(n) of RRI at the shorter time scales, so that the two local slopes are practically superimposable, with only small statistically significant differences in the 11<n<19 range.

IV. CONCLUSION By computing α(n) we confirm that the heart rate self-

similarity is characterized by a spectrum of short-term and long-term scaling exponents, nonunivocally influenced by changes of posture or physical activity. Results also indicate that care should be taken when the DFA scaling structure is modeled by two exponents only. In fact, α(n) of heart rate is approximated relatively well by a constant coefficient (α2) over long-term scales only, while at shorter scales the local exponent decreases too quickly, unless supine at rest conditions are considered. Similar considerations also regard blood pressure. In this case, however, also during supine rest a constant α1 exponent does not describe properly the short-term scaling structure of MAP. Moreover, the range n for α2 identification depends on posture or physical activity. In particular, during supine rest a constant plateau for α(n) does not exist at scales n≤54 beats.

From a methodological point of view, it should be considered that simulated data showed that estimation bias may affect α(n) when n is too small or too large [15]. Bias at small n is negligible when α >0.5, as in our data. By contrast, when n is larger than one-tenth of the signal length, α could be underestimated because of bias. This phenomenon may have slightly affected our α(n) estimates only at n≥91 (last three points). Considering that in this extreme range we even observed an increasing trend for α(n) of MAP, a possible underestimation does not affect our conclusions.

A final comment regards the comparison between blood pressure and heart rate. In the condition of lower sympathetic activation, MAP and RRI show quite different local exponents. Differences are particularly important up to scales of 30 beats. However, even modest sympathetic activations produce changes in both the scaling structures, which reduce the difference. At the sympathetic activation occurring during light physical exercise, the two local scaling exponents are practically superimposed. This suggests the existence of a strong common component in their complex dynamics, driven by the activation of sympathetic tone.

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