Detrended Fluctuation Analysis (DFA) is a popular method for assessing the fractal characteristics of biosignals, recently adapted for evaluating the heartrate multifractal and/or multiscale characteristics. However, the existing methods do not consider the beatbybeat sampling of heart rate and have relatively low scale resolutions and were not applied to cardiovascular signals other than heart rate. Therefore, aim of this work is to present a DFAbased method for joint multifractal/multiscale analysis designed to address the above critical points and to provide the first description of the multifractal/multiscale structure of interbeat intervals (IBI), systolic blood pressure (SBP), and diastolic blood pressure (DBP) in male and female volunteers separately. The method optimizes data splitting in blocks to reduce the DFA estimation variance and to evaluate scale coefficients with Taylor’s expansion formulas and maps the scales from beat domains to temporal domains. Applied to cardiovascular signals recorded in 42 female and 42 male volunteers, it showed that scale coefficients and degree of multifractality depend on the temporal scale, with marked differences between IBI, SBP, and DBP and with significant sex differences. Results may be interpreted considering the distinct physiological mechanisms regulating heartrate and bloodpressure dynamics and the different autonomic profile of males and females.
Beatbybeat measures of cardiovascular variables show an intrinsic variability, even when the cardiovascular system is observed in steadystate conditions. These spontaneous changes may reflect the processes underlying the cardiovascular homeostasis. Components of this variability show a fractal nature and in the last two decades different authors suggested that, at least for the heart rate, such components may be the output of a complex system that generates selfsimilar signals [
The first descriptions of cardiovascular selfsimilarity were based on modeling the heart rate as a time series belonging to the families of fractional Gaussian noises or of fractional Brownian motions and on estimating the corresponding Hurst exponent [
For these reasons, the more recent research in the field of heartrate variability is aimed at proposing methods that take into account both the scale dependency of selfsimilarity and its multifractal nature [
Our estimator of the multifractalmultiscale characteristics of beatbybeat cardiovascular signals was based on detrended fluctuation analysis (DFA), a method originally proposed for calculating a scale exponent,
Given a time series of the cardiovascular variable
Data splitting was repeated for block sizes
According to the multifractal approach for DFA [
The multifractal exponent
Since
The
Finally, we defined a concise index of multifractality, function of the scale
The power spectrum of each cardiovascular series was also calculated. Beattobeat series were interpolated linearly at 10 Hz and resampled at 5 Hz. The Welch periodogram was estimated by splitting the resampled series in 50% overlapping Hann windows of 1638.4 s duration, by computing the FFT spectrum in each window and by averaging the spectra over all the windows. The final periodogram was smoothed with a broadband procedure [
We considered recordings previously collected in two studies aimed at evaluating the influence of sodium sensitivity on the cardiovascular control in normotensive [
General characteristics of participants by sex.

Age (years)  Body mass index (kg/m^{2})  Prevalence of hypertension  

Females  42  34.3 (9.7)  22.6 (2.8)  38.1% 
Males  42  34.4 (10.1)  23.6 (2.4)  38.1% 

0.96  0.09 
Each participant was studied in a quiet environment in the morning, after 5 days of lowsalt diet (30 mmol NaCl per day) to minimize the confounding effects of dietary sodium on cardiovascular variability. Continuous finger arterial blood pressure was recorded for about two hours, in sitting position at rest, by Portapres model2 (Finapres Medical Systems B. V., Amsterdam, Netherlands). The finger cuff was placed on the mid finger of the left hand. SBP, DBP, and IBI (calculated as time interval between consecutive SBP values) were derived beatbybeat for the whole duration of the recording. Brachial blood pressure was measured simultaneously with a cuff on the right arm every 15 minutes, and the SBP and DBP readings of the brachial device were used to calibrate beatbybeat SBP and DBP values from the finger cuff.
We described statistical patterns in
Figure
Figure
Similarly, comparing IBI and DBP,
Also SBP and DBP scale coefficients differ importantly. For
These results make it clear that the degree of multifractality is a function of
Figure
We presented a novel algorithm for quantifying cardiovascular complexity based on previous researches that in various ways adapted DFA for assessing multifractal or/and multiscale aspects of heartrate variability. By applying our method to data collected in healthy volunteers, we provided the first detailed description of differences in multifractal and multiscale features among those cardiovascular time series more often recorded in clinical settings or in physiological studies: IBI, SBP, and DBP.
Three were the main results of our study. First, not only do selfsimilarity coefficients depend on the observational scale, but also the way
We showed that
Interestingly, we found gender differences in
We found marked differences between
A final consideration regards the different physiological information derivable from traditional spectral analyses and from a multifractalmultiscale analysis of cardiovascular signals. We found gender differences in
To evaluate the range of scales where our algorithm provides reliable estimates of multifractal coefficients, we applied it on synthesized series with known selfsimilarity structure. For this aim, we generated 100 series each of
The multifractal index of monofractal signals should, in theory, be equal to zero. In practice, however, we expect
As reference, we also performed a traditional multifractal analysis for each cardiovascular series. For this purpose, we calculated the generalized Hurst exponents,
The authors declare that there are no conflicts of interest regarding the publication of this paper.