Abstract
Heart failure is known to influence heart rhythm in patients. Complexity analysis techniques, including techniques associated with entropy, have great potential for providing a better understanding of cardiac rhythms, and for helping research in this area. We review the analysis principles of conventional time-domain analysis, frequency-domain analysis and of newer complexity analysis. We then illustrate the techniques using real clinical data, allowing a comparison of the techniques, and also of the differences between normal heart rate variability and that associated with heart failure.
Keywords
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Acharya, U.R., Joseph, K.P., Kannathal, N., Lim, C.M., Suri, J.S.: Heart rate variability: a review. Med. Biol. Eng. Comput. 44, 1031–1051 (2006)
Huikuri, H.V., Perkiömaki, J.S., Maestri, R., Pinna, G.D.: Clinical impact of evaluation of cardiovascular control by novel methods of heart rate dynamics. Philos. Trans. A Math. Phys. Eng. Sci. 367, 1223–1238 (2009)
Makikallio, T.H., Tapanainen, J.M., Tulppo, M.P., Huikuri, H.V.: Clinical applicability of heart rate variability analysis by methods based on nonlinear dynamics. Card. Electrophysiol. Rev. 6, 250–255 (2002)
Voss, A., Schulz, S., Schroeder, R., Baumert, M., Caminal, P.: Methods derived from nonlinear dynamics for analysing heart rate variability. Philos. Trans. A Math. Phys. Eng. Sci. 367, 277–296 (2009)
Weiss, J.N., Garfinkel, A., Spano, M.L., Ditto, W.L.: Chaos and chaos control in biology. J. Clin. Invest. 93, 1355–1360 (1994)
Brennan, M., Palaniswami, M., Kamen, P.: Poincare plot interpretation using a physiological model of HRV based on a network of oscillators. Am. J. Physiol. Heart Circ. Physiol. 283, H1873–H1886 (2002)
Kamen, P.W., Krum, H., Tonkin, A.M.: Poincare plot of heart rate variability allows quantitative display of parasympathetic nervous activity in humans. Clin. Sci. 91, 201–208 (1996)
Laitio, T.T., Mäkikallio, T.H., Huikuri, H.V., Kentala, E.S., Uotila, P., Jalonen, J.R., Helenius, H., Hartiala, J., Yli-Mäyry, S., Scheinin, H.: Relation of heart rate dynamics to the occurrence of myocardial ischemia after coronary artery bypass grafting. Am. J. Cardiol. 89, 1176–1181 (2002)
Kamen, P., Tonkin, A.: Application of the Poincare plot to heart rate variability: a new measure of functional status in heart failure. Aust. NZ J. Med. 25, 18–26 (1995)
Kobayashi, M., Musha, T.: 1/f fluctuation of heartbeat period. I.E.E.E. Trans. Biomed. Eng. 29, 456–457 (1982)
Lombardi, F.: Chaos theory, heart rate variability, and arrhythmic mortality. Circulation. 108, 8–10 (2000)
Peng, C.K., Havlin, S., Stanley, H.E., Goldberger, A.L.: Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos. 5, 82–87 (1995)
Huikuri, H.V., Mäkikallio, T.H., Peng, C.K., Goldberger, A.L., Hintze, U., Møller, M.: Fractal correlation properties of R-R interval dynamics and mortality in patients with depressed left ventricular function after an acute myocardial infarction. Circulation. 101, 47–53 (2000)
Mäkikallio, T.H., Høiber, S., Køber, L., Torp-Pedersen, C., Peng, C.K., Goldberger, A.L., Huikuri, H.V.: Fractal analysis of heart rate dynamics as a predictor of mortality in patients with depressed left ventricular function after acute myocardial infarction. Am. J. Cardiol. 83, 836–839 (1999)
Ivanov, P.C., Amaral, L.A.N., Goldberger, A.L., Havlin, S., Rosenblum, M.G., Struzik, Z.H.E.: Multifractality in human heartbeat dynamics. Nature. 399, 461–465 (1999)
Hadamard, J.: Les surfaces a courbures opposees et leurs lignes geodesiques. J. Math. Pures. Appl. 4, 27–73 (1898)
Kurths, J., Voss, A., Saparin, P., Witt, A., Kleiner, H.J., Wessel, N.: Quantitative analysis of heart rate variability. Chaos. 5, 88–94 (1995)
Voss, A., Kurths, J., Kleiner, H.J., Witt, A., Wessel, N., Saparin, P., Osterziel, K.J., Schurath, R., Dietz, R.: The application of methods of non-linear dynamics for the improved and predictive recognition of patients threatened by sudden cardiac death. Cardiovasc. Res. 31, 419–433 (1996)
Porta, A., Guzzetti, S., Montano, N., Furlan, R., Pagani, M., Malliani, A., Cerutti, S.: Entropy, entropy rate, and pattern classification as tools to typify complexity in short heart period variability series. I.E.E.E. Trans. Biomed. Eng. 48, 1282–1291 (2001)
Porta, A., Faes, L., Masé, M., D’Addio, G., Pinna, G.D., Maestri, R., Montano, N., Furlan, R., Guzzetti, S., Nollo, G., Malliani, A.: An integrated approach based on uniform quantization for the evaluation of complexity of short-term heart period variability: application to 24 h Holter recordings in healthy and heart failure humans. Chaos. 17, 015117 (2007)
Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Inf. Theory. 22, 75–81 (1976)
Abásolo, D., Hornero, R., Gómez, C., García, M., López, M.: Analysis of EEG background activity in Alzheimer’s disease patients with Lempel-Ziv complexity and central tendency measure. Med. Eng. Phys. 28, 315–322 (2006)
Sarlabous, L., Torres, A., Fiz, J.A., Morera, J., Jané, R.: Index for estimation of muscle force from mechanomyography based on the Lempel-Ziv algorithm. J. Electromyogr. Kinesiol. 23, 548–557 (2013)
Zhang, Y.T., Wei, S.S., Liu, H., Zhao, L.N., Liu, C.Y.: A novel encoding Lempel-Ziv complexity algorithm for quantifying the irregularity of physiological time series. Comput. Methods Prog. Biomed. 133, 7–15 (2016)
Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. U. S. A. 88, 2297–2301 (1991)
Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of biological signals. Phys. Rev. E. 71, 021906 (2005)
Liu, C.Y., Liu, C.C., Shao, P., Li, L.P., Sun, X., Wang, X.P., Liu, F.: Comparison of different threshold values r for approximate entropy: application to investigate the heart rate variability between heart failure and healthy control groups. Physiol. Meas. 32, 167–180 (2011b)
Pincus, S.M., Goldberger, A.L.: Physiological time-series analysis: what does regularity quantify? Am. J. Physiol. Heart Circ. Physiol. 266, H1643–H1656 (1994)
Richman, J.S., Moorman, J.R.: Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol. 278, H2039–H2049 (2000)
Lake, D.E., Richman, J.S., Griffin, M.P., Moorman, J.R.: Sample entropy analysis of neonatal heart rate variability. Am. J. Phys. Regul. Integr. Comp. Phys. 283, R789–R797 (2002)
Tuzcu, V., Nas, S., Börklü, T., Ugur, A.: Decrease in the heart rate complexity prior to the onset of atrial fibrillation. Europace. 8, 398–402 (2006)
Chen, W.T., Zhuang, J., Yu, W.X., Wang, Z.Z.: Measuring complexity using FuzzyEn, ApEn, and SampEn. Med. Eng. Phys. 31, 61–68 (2009)
Liu, C.Y., Li, K., Zhao, L.N., Liu, F., Zheng, D.C., Liu, C.C., Liu, S.T.: Analysis of heart rate variability using fuzzy measure entropy. Comput. Biol. Med. 43, 100–108 (2013)
Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy: a tool for complexity analysis of multichannel data. Phys. Rev. E. 84, 061918 (2011)
Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy analysis. IEEE Signal. Proc. Lett. 19, 91–94 (2012)
Azami, H., Escudero, J.: Refined composite multivariate generalized multiscale fuzzy entropy: a tool for complexity analysis of multichannel signals. Phys. A. 465, 261–276 (2017)
Arbolishvili, G.N., Mareev, V.I., Orlova, I.A., Belenkov, I.N.: Heart rate variability in chronic heart failure and its role in prognosis of the disease. Kardiologiia. 46, 4–11 (2005)
Nolan, J., Batin, P.D., Andrews, R., Lindsay, S.J., Brooksby, P., Mullen, M., Baig, W., Flapan, A.D., Cowley, A., Prescott, R.J.: Prospective study of heart rate variability and mortality in chronic heart failure results of the United Kingdom heart failure evaluation and assessment of risk trial (UK-Heart). Circulation. 98, 1510–1516 (1998)
Rector, T.S., Cohn, J.N.: Prognosis in congestive heart failure. Annu. Rev. Med. 45, 341–350 (1994)
Smilde, T.D.J., van Veldhuisen, D.J., van den Berg, M.P.: Prognostic value of heart rate variability and ventricular arrhythmias during 13-year follow-up in patients with mild to moderate heart failure. Clin. Res. Cardiol. 98, 233–239 (2009)
Malliani, A., Pagani, M., Lombardi, F., Cerutti, S.: Cardiovascular neural regulation explored in the frequency domain. Circulation. 84, 482–492 (1991)
Binkley, P.F., Nunziata, E., Haas, G.J., Nelson, S.D., Cody, R.J.: Parasympathetic withdrawal is an integral component of autonomic imbalance in congestive heart failure: demonstration in human subjects and verification in a paced canine model of ventricular failure. J. Am. Coll. Cardiol. 18, 464–472 (1991)
Poon, C.S., Merrill, C.K.: Decrease of cardiac chaos in congestive heart failure. Nature. 389, 492–495 (1997)
Woo, M.A., Stevenson, W.G., Moser, D.K., Middlekauff, H.R.: Complex heart rate variability and serum norepinephrine levels in patients with advanced heart failure. J. Am. Coll. Cardiol. 23, 565–569 (1994)
La Rovere, M.T., Pinna, G.D., Maestri, R., Mortara, A., Capomolla, S., Febo, O., Ferrari, R.,Franchini, M., Gnemmi, M., Opasich, C., Riccardi, P.G., Traversi, E., Cobelli, F.: Short-term heart rate variability strongly predicts sudden cardiac death in chronic heart failure patients. Circulation. 107, 565–570 (2003)
Hadase, M., Azuma, A., Zen, K., Asada, S., Kawasaki, T., Kamitani, T., Kawasaki, S., Sugihara, H., Matsubara, H.: Very low frequency power of heart rate variability is a powerful predictor of clinical prognosis in patients with congestive heart failure. Circ. J. 68, 343–347 (2004)
Guzzetti, S., Mezzetti, S., Magatelli, R., Porta, A., De Angelis, G., Rovelli, G., Malliani, A.: Linear and non-linear 24 h heart rate variability in chronic heart failure. Auton. Neurosci. 86, 114–119 (2000)
Mäkikallio, T.H., Huikuri, H.V., Hintze, U., Videbaek, J., Mitrani, R.D., Castellanos, A., Myerburg, R.J., Møller, M., DIAMOND Study Group: Fractal analysis and time- and frequency-domain measures of heart rate variability as predictors of mortality in patients with heart failure. Am. J. Cardiol. 87, 178–182 (2001)
Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett. 89, 068102 (2002)
Singh, J.P., Larson, M.G., Tsuji, H., Evans, J.C., O’Donnell, C.J., Levy, D.: Reduced heart rate variability and new-onset hypertension–insights into pathogenesis of hypertension: The Framingham Heart Study. Hypertension. 32, 293–297 (1998)
Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology: Heart rate variability: standards of measurement, physiological interpretation and clinical use. Circulation. 93, 1043–1065 (1996)
Goldberger, J.J., Ahmed, M.W., Parker, M.A., Kadish, A.H.: Dissociation of heart rate variability from parasympathetic tone. Am. J. Phys. 266, H2152–H2157 (1994)
Liu, C.Y., Li, P., Zhao, L.N., Yang, Y., Liu, C.C.: Evaluation method for heart failure using RR sequence normalized histogram. In: Murray, A. (ed.) Computing in Cardiology, pp. 305–308. IEEE, Hangzhou (2011a)
Ho, K.K., Moody, G.B., Peng, C.K., Mietus, J.E., Larson, M.G., Levy, D., Goldberger, A.L.: Predicting survival in heart failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indices of heart rate dynamics. Circulation. 96, 842–848 (1997)
Liu, C.Y., Zhang, C.Q., Zhang, L., Zhao, L.N., L C, C., Wang, H.J.: Measuring synchronization in coupled simulation and coupled cardiovascular time series: a comparison of different cross entropy measures. Biomed. Signal. Process. Control. 21, 49–57 (2015)
Liu, C.Y., Zhao, L.N.: Using Fuzzy Measure Entropy to improve the stability of traditional entropy measures. In: Murray, A. (ed.) Computing in cardiology, pp. 681–684. IEEE, Hangzhou (2011)
Zhao, L.N., Wei, S.S., Zhang, C.Q., Zhang, Y.T., Jiang, X.E., Liu, F., Liu, C.Y.: Determination of sample entropy and fuzzy measure entropy parameters for distinguishing congestive heart failure from normal sinus rhythm subjects. Entropy. 17, 6270–6288 (2015)
Zhang, X.S., Zhu, Y.S., Thakor, N.V., Wang, Z.Z.: Detecting ventricular tachycardia and fibrillation by complexity measure. I.E.E.E. Trans. Biomed. Eng. 46, 548–555 (1999)
Aboy, M., Hornero, R., Abasolo, D., Alvarez, D.: Interpretation of the Lempel-Ziv complexity measure in the context of biomedical signal analysis. I.E.E.E. Trans. Biomed. Eng. 53, 2282–2288 (2006)
Yentes, J.M., Hunt, N., Schmid, K.K., Kaipust, J.P., McGrath, D., Stergiou, N.: The appropriate use of approximate entropy and sample entropy with short data sets. Ann. Biomed. Eng. 41, 349–365 (2013)
Ruan, X.H., Liu, C.C., Liu, C.Y., Wang, X.P. and Li, P.: Automatic detection of atrial fibrillation using RR interval signal. In: 4th International Conference on Biomedical Engineering and Informatics: IEEE. pp. 644–647. (2011)
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We acknowledge and thank Dr. Charalampos Tsimenidis of Newcastle University for his helpful review of the manuscript.
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A. Appendix
A. Appendix
1.1 A.1 RR Sequence Normalized Histogram
The general construction procedure for the RR sequence normalized histogram is summarized as follows [53]:
Given an RR sequence {RR 1, RR 2,…, RR N }, where N denotes the sequence length. The maximum (RR max) and minimum (RR min) values were firstly determined to calculate the range in the sequence:
The threshold a = 0.1 × RR range is set and then the left-step parameter H l and right-step parameter H r can be calculated as
where RR mean denotes the mean value of the RR sequence. Based on these parameters, the RR i is divided into seven sections. Table 11.1 details the element division rules.
The element percentage p i in each of the sections is calculated as follows:
In a rectangular coordinate system, the p i corresponding to the seven sections (i.e. L1, L2, L3, C, R3, R2 and R1) is drawn to form the normalized histogram. Three quantitative indices can be obtained from the normalized histogram. They are respectively named as center-edge ratio (CER), cumulative energy (CE) and range information entropy (RIEn) and are defined as [53]:
1.2 A.2 Sample Entropy (SampEn)
The algorithm for SampEn is summarized as follows [29]: For the HRV series x(i), 1 ≤ i ≤ N, forms N − m + 1 vectors \( {X}_i^m=\left\{x(i),x\left(i+1\right),\cdots, x\left(i+m-1\right)\right\} \), 1 ≤ i ≤ N − m + 1. The distance between two vectors \( {X}_i^m \) and \( {Y}_i^m \) is defined as: \( {d}_{i,j}^m=\underset{k=0}{\overset{m-1}{ \max }}\left|x\left(i+k\right)-x\left(j+k\right)\right| \). Denote \( {B}_i^m(r) \) the average number of j that meets \( {d}_{i,j}^m\le r \) for all 1 ≤ j ≤ N − m, and similarly define \( {A}_i^m(r) \) by \( {d}_{i,j}^{m+1} \). SampEn is then defined by:
wherein the embedding dimension is usually set at m = 2 and the threshold at r = 0.2 × sd (sd indicates the standard deviation of the HRV series under-analyzed) [57, 60].
1.3 A.3 Fuzzy Measure Entropy (FuzzyMEn)
The calculation process of FuzzyMEn is summarized as follows [33, 56]:
For the RR or PTT segment x(i) (1 ≤ i ≤ N), firstly form the local vector sequences \( {XL}_i^m \) and global vector sequences \( {XG}_i^m \) respectively:
The vector \( {XL}_i^m \) represents m consecutive x(i) values but removing the local baseline \( \overline{x}(i) \), which is defined as:
The vector \( {XG}_i^m \) also represents m consecutive x(i) values but removing the global mean value \( \overline{x} \) of the segment x(i), which is defined as:
Subsequently, the distance between the local vector sequences \( {XL}_i^m \) and \( {XL}_j^m \), and the distance between the global vector sequences \( {XG}_i^m \) and \( {XG}_j^m \) are computed respectively as:
Given the parameters n L , r L , n G and r G , we calculate the similarity degree \( {DL}_{i,j}^m\left({n}_L,{r}_L\right) \) between the local vectors \( {XL}_i^m \) and \( {XL}_j^m \) by the fuzzy function \( \mu L\left({dL}_{i,j}^m,{n}_L,{r}_L\right) \), as well as the similarity degree \( {DG}_{i,j}^m\left({n}_G,{r}_G\right) \) between the global vectors \( {XG}_i^m \) and \( {XG}_j^m \) by the fuzzy function \( \mu G\left({dG}_{i,j}^m,{n}_G,{r}_G\right) \):
We define the functions ϕL m(n L , r L ) and ϕG m(n G , r G ) as:
Similarly, we define the function ϕL m + 1(n L , r L ) for m + 1 dimensional vectors \( {XL}_i^{m+1} \) and \( {XL}_j^{m+1} \) the function ϕG m + 1(n G , r G ) for m + 1 dimensional vectors \( {XG}_i^{m+1} \) and \( {YG}_j^{m+1} \):
Then the fuzzy local measure entropy (FuzzyLMEn) and fuzzy global measure entropy (FuzzyGMEn) are computed as:
Finally, the FuzzyMEn of RR segment x(i) is calculated as follows:
In this study, the local similarity weight was set to n L = 3 and the global vector similarity weight was set to n G = 2. The local tolerance threshold r L was set equal to the global threshold r G, i.e., r L = r G = r. Hence, the formula (11.21) becomes:
For both SampEn and FuzzyMEn, the entropy results were only based on the three parameters: the embedding dimension m, the tolerance threshold r and the RR segment length N.
1.4 A.4 Lempel-Ziv (LZ) Complexity
The calculation process of LZ complexity is summarized as follows [21, 59]:
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1.
For a binary symbolic sequence R = {s 1, s 2,…, s n }, let S and Q denote two strings, respectively, and SQ is the concatenation of S and Q, while the string SQπ is derived from SQ after its last character is deleted (π means the operation to delete the last character in the string). v(SQπ) denotes the vocabulary of all different substrings of SQπ. Initially, c(n) = 1, S = s 1, Q = s 2, and so SQπ = s 1;
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2.
In summary, S = s 1 s 2, …, s r , Q = s r+1, and so SQπ = s 1 s 2, …, s r ; if Q belongs to v(SQπ), then s r+1, that is, Q is a substring of SQπ, and so S does not change, and Q is updated to be s r+1 s r+2, and then judge if Q belongs to v(SQπ) or not. Repeat this process until Q does not belong to v(SQπ);
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3.
Now, Q = s r+1 s r+2, …, s r+i , which is not a substring of SQπ = s 1 s 2, …, s r s r+1,…, s r+i-1, so increase c(n) by one;
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4.
Thereafter, S is renewed to be S = s 1 s 2, …, s r+i , and Q = s r+i+1;
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5.
Then the procedures repeat until Q is the last character. At this time c(n) is the number of different substrings contained in R. For practical application, c(n) should be normalized. It has been proved that the upper bound of c(n) is
where ε n is a small quantity and ε n → 0 (n → ∞). In fact,
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6.
Finally, LZ complexity is defined as the normalized output of c(n):
where C(n) is the normalized LZ complexity, and denotes the arising rate of new patterns within the sequence.
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Liu, C., Murray, A. (2017). Applications of Complexity Analysis in Clinical Heart Failure. In: Barbieri, R., Scilingo, E., Valenza, G. (eds) Complexity and Nonlinearity in Cardiovascular Signals. Springer, Cham. https://doi.org/10.1007/978-3-319-58709-7_11
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