Quantification of long-range power law correlations among healthy and pathologic subjects using detrended fluctuation analysis and multifractal detrended fluctuation analysis

The healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium-like state. However recent studies reveal that under normal conditions, beat-to-beat fluctuation in heart rate display the kind of long-range correlations typically exhibited by the dynamical system far away from equilibrium. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of this long-range correlation behavior. Two different non-linear dynamic methods namely Detrended Fluctuation Analysis (DFA) and Multifractal (MF) DFA are used for the quantification of this correlation property in non-stationary physiological time series and it revealed the presence of long-range power law correlation for the group of healthy subjects while breakdown in the long-range power law correlation for the group of subjects with cardiac heart failure. Application of DFA analysis shows evidence for a crossover phenomenon associated with a change in short(αl) and long(α2) range scaling exponents. For healthy subjects, calculated value of αl and α2 (mean value ± S.D.) are 1.31 ± 0.17 and 1.00 ± 0.07 respectively. For subjects with cardiac heart failure calculated value of ctl and a2 is 0.71 ± 0.20 and 1.24 ± 0.07 respectively i.e. only one scaling exponent is not sufficient to characterize the entire heart-rate time series which resulted into MF-DFA approach. This suggested that there is more than one exponent values needed to characterize the heart rate time series. Multifractal DFA is based on generalization of DFA and a MATLAB code is developed to implement the MF-DFA algorithm and to identify whether the given time series under analysis exhibits multifractality or not by generating more than one exponent values for multifractal signal. The value of a for q\u3eO for healthy is 1.04 ± 0.02 and for CHF is found to be 1.32 ± 0.02 and the value of a for q\u3c0 for healthy subjects is 3.01 ±0.26 and for CHF subjects is found to be 3.53 ± 0.14 (mean value ± S.D.) The student\u27s ttest suggests that p-value is 0.00001 which is less than 0.05 thus the value of a for q \u3c0 and q\u3e0 among healthy subjects and CHF subjects are statistically different. Value of a for q\u3e0 is less than that for q\u3c0. And for q =2 MF-DFA retains monofractal DFA. Thus, MF-DFA is clearly able to discriminate among the healthy and CHF for q\u3c0 as well for q\u3e0. MF-DFA also determines which fluctuations i.e. (small or large) dominate for the given interbeat interval time series because for q\u3c0 the slow fluctuations dominate whereas for q\u3e0 large fluctuations dominate. DFA and MF-DFA were able to discriminate 23 Healthy subjects out of 26 Healthy subjects data sets i.e. true positive specificity is 0.89 and false negative specificity is 0.12 and 9 CHF subjects out of 11 CHF subjects data sets i.e. true positive specificity is 0.82 and false negative specificity is 0.19. These methods may be of use in distinguishing healthy from pathologic data sets based on the difference in the scaling properties