Evaluation of renormalised entropy for risk strati®cation using heart rate variability data N. Wessel 1 A. Voss 2 J. Kurths 1 A. Schirdewan 3 K. Hnatkova 4 M. Malik 4 1 Nonlinear Dynamics Group, Institute of Physics, University of Potsdam, Potsdam, Germany, 2 University of Applied Sciences, Jena, Germany, 3 Franz-Volhard-Hospital, Humboldt-University, Berlin, Germany 4 St. George's Hospital, Medical School, London, UK AbstractÐStandard time and frequency parameters of heart rate variability (HRV) describe only linear and periodic behaviour, whereas more complex relationships cannot be recognised. A method that may be capable of assessing more complex properties is the non-linear measure of `renormalised entropy.' A new concept of the method, RE AR , has been developed, based on a non-linear renormalisation of autoregressive spectral distributions. To test the hypothesis that renormalised entropy may improve the result of high-risk strati®cation after myocardial infarction, it is applied to a clinical pilot study (41 subjects) and to prospective data of the St George's Hospital post-infarction database (572 patients). The study shows that the new RE AR method is more reproducible and more stable in time than a previously introduced method (p50.001). Moreover, the results of the study con®rm the hypothesis that on average, the survivors have negative values of RE AR (0.11  0.18), whereas the non-survivors have positive values (0.03  0.22, p50.01). Further, the study shows that the combination of an HRV triangular index and RE AR leads to a better prediction of sudden arrhythmic death than standard measurements of HRV. In summary, the new RE AR method is an independent measure in HRV analysis that may be suitable for risk strati®cation in patients after myocardial infarction. KeywordsÐHeart rate variability, Renormalised entropy, Risk strati®cation Med. Biol. Eng. Comput., 2000, 38, 680±685 1 Introduction STANDARD TIME and frequency parameters of heart rate varia- bility (HRV) only describe linear and periodic behaviour, whereas more complex relationships and interactions are not addressed. At the same time, modulation of sinus rhythm involves many non-linear elements. Thus it is not realistic to restrict HRV analysis only to linear methods. The application of non-linear methods in addition to the traditional ones seems to be promising in this respect (GOLDBERGER et al., 1988; VOSS et al., 1993; 1996; 1998; KURTHS et al., 1995; SCHREIBER, 1997; MAKIKALLIO et al., 1997; SCHA È FER et al., 1998; WESSEL et al., 2000). Nevertheless, many non-linear methods require rather long, stationary time series and are not easily applicable to the data of cardiac periods. However, non-stationarities may play an important role in arrhythmogenesis. Thus, HRV analysis should not be restricted to stationary epochs. A method that may be capable of assessing more complex properties of cardiac periodograms is the non-linear measure of 'renormalised entropy'. The basic idea is to determine the complexity of cardiac periodograms based on a ®xed reference. Based on general considerations in thermodynamics, KLIMONTOVICH (1991) suggested comparing the relative degree of order of two different distributions by renormalising the reference distribution to a given energy. SAPARIN et al. (1994) proposed a procedure for calculating this quantity from time series and applied it to the logistic map. They showed that the renormalised entropy allows the degree of order to be compared, not only between chaotic and periodic series, but also between different periodic and chaotic regimes. KOPITZKI et al. (1998) applied this method to the data of invasive electro- encephalograph recordings. Their results suggested that renor- malised entropy may be a useful procedure for clinical applica- tions in this ®eld, such as seizure detection and localisation of epileptic foci. Applications of renormalised entropy to heart rate data based on the fast fourier transform (FFT) have been introduced previously (VOSS et al., 1993; 1996; WESSEL et al., 1994; KURTHS et al., 1995). However, this method suffers from a potential lack of reproducibility and time instability. To over- come these limitations, a new method was developed for the computation of renormalised entropy RE AR , based on an auto- regressive spectral estimation. Correspondence should be addressed to Dr N. Wessel; email: niels@agnld.uni-potsdam.de First received 25 April 2000 and in ®nal form 25 June 2000 MBEC online number: 20003523 ß IFMBE: 2000 680 Medical & Biological Engineering & Computing 2000, Vol. 38