Adaptive computation of approximate entropy and its application in integrative analysis of irregularity of heart rate variability and intracranial pressure signals

Abstract

The present study introduces an adaptive calculation of approximate entropy (ApEn) by exploiting sample-by-sample construction and update of nearest neighborhoods in an n-dimensional space. The algorithm is first validated with a standard numerical test set. It is then applied to electrocardiogram R wave interval (RR) and beat-to-beat intracranial pressure signals recorded from 12 patients undergoing normal pressure hydrocephalus diagnosis. The ApEn time series are further processed using the causal coherence analysis to study the interaction between ICP and RR interval. Numerical validation demonstrates that the proposed algorithm reproduces the known time-varying patterns in the test set and better tracks abrupt signal changes. It is also demonstrated that occurrences of large-amplitude ICP oscillation are associated with decreased ICP ApEn and RR ApEn for all 12 patients. The causal coherence analysis of ApEn time series shows that coherence between RR ApEn and ICP ApEn, after mathematically decoupling RR effect on ICP, is enhanced for the oscillatory ICP state and so is the amplitude of transfer function between ICP and RR interval. However, no enhanced coherence is observed after mathematically decoupling ICP effect on RR interval. In conclusion, the adaptive ApEn algorithm can be used to track nonstationary signal characteristics. Furthermore, interactions between dynamic systems could be studied by using ApEn time series of the direct observations of systems.

Fig. 1

Block diagram of the proposed adaptive ApEn computational algorithm. Data flow is represented using the dotted line and the procedural flow is represented using the solid line.

Fig. 2

Numerical test results. Each row corresponds to a test case with the raw signal shown in the first column, the ApEn time series using the windowing algorithm, and the adaptive algorithm in the second and the third columns.

Fig. 3

A detail illustration of the ApEn results for the third numerical test case as obtained using the sample-by-sample adaptive algorithm and the moving-window approach. The window width is 400 samples with a step size of 200.

Fig. 4

Four example illustrations of ApEn time series of RR interval and ICP signals. An oscillatory ICP episode is present in each example that is associated with a decrease of both ICP ApEn and RR ApEn. An increasing trend of ApEn time series is always present after oscillatory ICP state ends.

Fig. 5

A representative example of applying the adaptive algorithm to the data from seventh patient. The normalized signals were shown in the left panel with ICP shown in black line and RR interval in gray line. The corresponding ApEn time series are shown in the right panel. ApEn time series of both signals are similar even though their waveforms are different.

Fig. 6

Power spectrum density plots of the signals and time series shown in Fig. 5. Right panel shows results for the ApEn time series and the left panel shows results for the raw signals.