Author + information
- Received December 20, 1994
- Revision received May 21, 1996
- Accepted June 3, 1996
- Published online October 1, 1996.
- ETTORE PETRUCCI* ()
- ↵*Address for correspondence: Dr. Ettore Petrucci, Servizio di Cardiologia, Ospedale A. Bellini, 21019 Somma Lombardo (Varese), Italy.
- LUCA T. MAINARDI*
- ANNA M. BIANCHIb
- SERGIO CERUTTIa
Objectives. We sought to evaluate changes in RR interval variability during dipyridamole infusion and dipyridamole-induced myocardial ischemia.
Background. Myocardial ischemia and the autonomic nervous system can be mutually interdependent. Spectral analysis of RR interval variability is a useful tool in assessing autonomic tone.
Methods. We used a time variant autoregressive spectral estimation algorithm that could extract spectral variables even in the presence of nonstationary signals. Two groups were considered: group A (patients with ischemia, n = 15) with effort or mixed angina, angiographically assessed coronary artery disease and positive exercise and dipyridamole echocardiographic test results, and group B (control subjects, n = 10) with normal exercise and dipyridamole echocardiographic test results. We investigated the following variables: RR interval mean and variance, low frequency (LF) and high frequency (HF) power in normalized units, LF ratio (LF/LFbasal power), HF ratio (HF/HFbasal power) and LF/HF ratio. For each test epoch, we calculated for group A and group B the mean value ± SE of all indexes considered. Differences due to an effect either of group (ischemic vs. control) or of time (including both drug and ischemia effects) were analyzed by using analysis of variance for repeated measurements.
Results. Dipyridamole injection was characterized by a reduction of all spectral components in negative test. The LF ratio was the only variable able to discriminate patients with ischemia from control subjects (p < 0.05), whereas a time effect was evident for both mean RR interval and high frequency power in normalized units (p < 0.05). The LF ratio decreased in group B from 1 ± 0.00 (basal) to 0.31 ± 0.22 (peak), and increased in group A from 1 ± 0.00 to 15.41 ± 6.59, respectively. Results of an unpaired t test comparing the peak values of the two groups were also statistically significant (p < 0.01).
Conclusions. Our data show that time variant analysis of heart rate variability evidences an increase in the low frequency ratio that allows differentiation of positive from negative test results, suggesting that the electrocardiogram may contain ischemia information unrelated to ST-T variations, even if their enhancement requires a more complex data processing procedure.
The autonomic nervous system activity modulates the intrinsic sinus rate, inducing cyclic variations of the RR intervals. Frequency domain analysis of the RR series allows detection and quantification of two main spectral components (Fig. 1): a low frequency (LF) and a high frequency (HF) component, centered, respectively, around 0.1 Hz (10-s period) and 0.3 Hz (3-s period). The LF component increases in the presence of increased sympathetic tone, whereas the HF component is synchronous with respiration and can be assumed to be a marker of parasympathetic activity [1–5]. A third component, the very low frequency (VLF) component (<0.03 Hz) can be detected in the spectrum, but its physiologic significance is still uncertain .
Analysis of heart rate variability provides a useful, noninvasive tool for studying autonomic activity. Although changes in spectral components have been documented in many experimental and clinical conditions involving variations of autonomic function [2, 3, 7–12], only a few studies have examined the spectral modifications during transient myocardial ischemia in humans. Imbalance between sympathetic and vagal activity before and during attacks of vasospastic angina has been documented [13, 14], whereas the study of heart rate variability during transient regional ischemia obtained by angioplasty gave ambiguous results . The available data reflect both the complexity of the relations between the autonomic nervous system and myocardial ischemia [16–21], and the limitations of classic (batch) algorithms for spectral analysis, which cannot provide reliable results when the RR series is nonstationary .
In this study, we applied a recently developed, more appropriate time variant approach [23, 24] to describe dynamic changes of spectral components in a reproducible clinical model of transient myocardial ischemia, the dipyridamole echocardiography test . We studied a group of patients with coronary artery disease and positive test results, and a control group with negative test results, and we correlated spectral patterns with echocardiographic, electrocardiographic (ECG) and clinical events occurring during the test. We attempted to evaluate whether time variant spectral variables are affected by dipyridamole infusion or by dipyridamole-induced myocardial ischemia, or both, and to assess whether the time variant variables make it possible to differentiate ischemic from nonischemic responses.
Patients. The study patients underwent diagnostic ECG exercise testing and dipyridamole echocardiography testing to evaluate a chest pain syndrome . All patients gave written informed consent to the examinations, according to the routine hospital protocol.
Two groups of subjects were selected: Group A (ischemia) included 15 patients in clinically stable condition with a history of mixed or effort angina, cycle ergometer exercise test results positive for both ST segment changes and angina, dipyridamole echocardiography test results positive for kinetic and ECG changes and angina and angiographically assessed significant coronary artery disease (coronary angiography was performed within 1 month of the dipyridamole test). All tests were executed after an appropriate therapy-free interval for all cardioactive drugs. Three patients had a history of previous myocardial infarction.
Table 1 shows demographic, angiographic and dipyridamole echocardiography test data for group A patients. Group B (control group) included 10 subjects with atypical chest pain, normal stress test, normal echocardiogram and dipyridamole echocardiography test negative for kinetic and ECG changes and symptoms. Groups A and B did not differ in age (mean ± SD [range]: 57.9 ± 10.5 years [42 to 72] vs. 56.3 ± 10.1 years [42 to 68]; 95% confidence interval of the difference −8.5 to 11.7) or gender (11 men and 4 women; vs. 8 men and 2 women).
Inclusion criteria for both groups were no contraindications for dipyridamole echocardiography test; good echocardiographic window; sinus rhythm throughout the test with ≤1% ectopic beats and no repetitive arrhythmias; and absence of diseases or situations potentially affecting RR interval variability, such as left ventricular dysfunction (ejection fraction <50%) or a history of hypertension, diabetes, alcoholism. All subsequent subjects meeting the inclusion criteria were entered into the study.
Dipyridamole echocardiographic test. Two-dimensional echocardiographic monitoring was performed in combination with dipyridamole infusion as follows: 0.56 mg/kg body weight during 4 min, no dose for 4 min, then, if the test result remained negative, 0.28 mg/kg during 2 min [25, 27]. The cumulative dose was therefore 0.84 mg/kg during 10 min. During each minute of the procedure, blood pressure was recorded by means of an arm cuff sphygmomanometer.
Two-dimensional echocardiogram and 12-lead ECG were continuously monitored and recorded during and up to 20 min after dipyridamole administration. Commercially available imaging systems (model SSD-870 ALOKA and Cardioline Stress Test System Remco Italia) were used. The left ventricle was divided into 13 segments . Segmental wall motion was graded according to current American Society of Echocardiography recommendations . The wall motion score index was derived by summation of individual segment scores divided by the number of interpreted segments. The dipyridamole echocardiography test was terminated at the end of the protocol or with the occurrence of one of the following events: 1) change of two or more kinetic levels for at least one left ventricular segment; 2) ST segment displacement of 2 mm in one lead, and 3) unequivocal anginal chest pain. Pain or discomfort that can arise aspecifically during dipyridamole infusion was not considered in itself a reason for terminating the test. Patients with negative test results received aminophylline (40 to 70 mg intravenously during 1 min, 7 min after high dose administration) to reverse or prevent side effects . In patients with positive test results, dipyridamole was no longer infused when any sign of evolving ischemia appeared, and the ECG, echocardiogram and patient symptoms were accurately monitored. When one of the termination criteria occurred, the test result was declared positive and aminophylline was promptly injected; otherwise, the patient was monitored so long as monitoring was considered clinically useful, and aminophylline was administered before test termination. In the latter case, the final diagnosis was decided off-line.
Data from tests fulfilling the study inclusion criteria were reviewed off-line by two observers independently (differences were resolved by consensus), and the onset of ischemia-related echocardiographic, ECG and clinical events was assessed, respectively, as 1) the first appearance of any evidence of kinetic abnormalities, 2) 1-mm ST segment displacement in one or more leads, and 3) the first appearance of any kind of chest discomfort. These points are indicated as D, E and P respectively, in Fig. 3Fig. 6 and will be assumed as time references throughout the text.
RR interval analysis. During the dipyridamole echocardiography test, standard ECG leads were sampled at 500 Hz, 12-bits precision, stored on hard disk and processed off-line on a microcomputer. QRS detection and RR interval measurement were automatically performed by a derivative/threshold algorithm, looking for the R wave peak as a reference point. Each QRS complex was then interpolated by a parabolic curve. The R point was chosen to correspond with the maximum of the interpolating parabola to improve the accuracy of detection of the peak R wave . Premature beats, missed beats and artifacts were visually identified with use of an interactive graphic interface and corrected by the operator. In this way, an RR tachogram was obtained, that is, a discrete series of successive RR intervals as a function of the number of recognized QRS complexes (Figs. 2 and 3A).
Spectral analysis: time variant algorithm. We used a time variant spectral approach based on an autoregressive model whose parameters are recursively updated according to the current signal characteristics, whereas the importance of past signal characteristics is progressively reduced by the action of a forgetting factor ω . This method allows one to update the spectral characteristics up to one spectrum for each new RR interval, leading to an evaluation of the classic spectral variables (LF power, HF power and LF/HF ratio) on a beat to beat basis. We used the recursive least square algorithm [23, 24] with a constant forgetting factor ω = 0.98, and a model order of 12. For a more detailed description of the algorithm see the Appendix.
For each spectrum, a spectral decomposition algorithm  automatically detects the oscillatory components by evaluating their power (as the area covered by each spectral peak) and their center frequency. Two main frequency bands were selected: 0.03 to 0.15 Hz and 0.18 to 0.45 Hz for the LF and HF components, respectively .
Spectral variables. The series of the estimated spectra can be looked at under the so-called compressed spectral array form , as shown in Fig. 2B and Fig. 3B, where the series of successive spectra are plotted as a function of the beat number, starting from up downward. (The horizontal and the vertical axes show the same scales used in Fig. 1). This three-dimensional view allows a semiquantitative evaluation of spectral changes during evolving events.
Quantitative information is obtained by the spectral variables extracted from each spectrum (i.e., for each RR interval). The following variables were available on a beat to beat basis: mean and variance of RR series, LF and HF power and the LF/HF ratio.
To reduce the large amount of data, the beat to beat spectral variables were averaged on successive temporal windows of 30 s. This arbitrarily selected time window is suitable for comparing spectral changes with echocardiographic, ECG and clinical events during the dipyridamole echocardiography test.
Moreover, a normalization of LF and HF power was required to decrease the great interpatient spread of these variables. Different normalizations have previously been proposed ; here, we describe changes in LF and HF power by using two normalization procedures: The first is the percent of LF and HF power with respect to the total power minus the VLF component (i.e., the “classic” normalized units), the second is the ratio between actual and basal absolute power values, the latter being the mean value during the 4 min preceding the beginning of dipyridamole infusion. The latter will be indicated as LF ratio (LF/LFbasal power) and HF ratio (HF/HFbasal power) throughout the text.
In summary, the following variables were investigated, as obtained every 30 s: RR mean and variance, LF and HF power in normalized units; LF and HF ratios; LF/HF ratio.
Statistical analysis. Mean and standard deviation were computed for basal and peak values of systolic pressure, diastolic pressure and rate-pressure product. Differences between groups were assessed by using an unpaired t test. For each 30-s period, we calculated, for group A (patients with ischemia) and group B (normal control group) the mean value ± SE for all considered RR interval variability indexes. Further statistical analysis was planned taking into account that ischemia occurred in group A patients at different times during the test. 1) The effects of dipyridamole administration in negative tests were assessed in group B subjects by comparing the peak results with basal values, by means of a paired t test. Presumptive peak pharmacologic effect was defined as 2 min after the end of the high dose injection in group B and 2 min after the end of dipyridamole injection in group A (high dose in 2 cases and low dose in 13).
2) The differences due either to group effect (ischemic vs. nonischemic responses) or to time effect (including both drug and ischemic effects), as well as the interaction between these two sources of variation, were analyzed by using analysis of variance (ANOVA) for repeated measurements on group A and B data. To minimize the risk of spurious significant results due to multiple comparisons, we sampled the data every 2 min, and we used the most conservative computation for the numbers of degrees of freedom for critical values of the Fisher F test related to groups, times and the interaction of groups per times. Statistical significance was defined as a p value < 0.05.
3) Only variables showing statistically significant differences between groups were further investigated with a Bonferroni-corrected unpaired t test, comparing the peak values obtained in the two groups.
For the same variables, all 30-s data were analyzed to evaluate more carefully temporal changes due to both dipyridamole infusion and development of ischemia. In the control group we estimated, for each 30-s period, the mean value and the 95% confidence interval of the mean value, obtaining a plot showing the “physiologic” response to dipyridamole administration as a function of time. In group A, a similar pooling of data at fixed time intervals would merge the drug and ischemic effects, because ischemia develops with varying delay from the beginning of the test (Table 1), inducing different interactions with dipyridamole-related changes. To overcome this problem we superimposed each patient's trend on the control group trend to graphically show, patient by patient, the different behavior induced by ischemia.
All group B (control) subjects completed the entire test. In group A, ischemia developed after high dose infusion in one patient, whereas another (Patient 13) received the high dose infusion because of real-time underestimation (for technical reasons) of apical dyssynergy. Side effects due to dipyridamole were always minor and well tolerated. Chronologic links between dipyridamole infusion and echocardiographic, ECG and clinical changes in group A patients are reported in Table 1. The wall motion score index in group A patients was 1.39 ± 0.104 (mean ± SD, range 1.23 to 1.53).
During the test, heart rate significantly increased from the basal level to peak effect (see “Mean RR Interval” in Table 2) without differences between groups. Similarly, no difference between groups A and B was found in basal and peak values of systolic arterial pressure (basal 158 ± 26.5 vs. 153.5 ± 21.8 mm Hg, p = NS; peak 145 ± 31 vs. 147 ± 17.6 mm Hg, p = NS); diastolic arterial pressure (basal 91 ± 8.4 vs. 90 ± 5 mm Hg, p = NS; peak 83 ± 11 vs. 83.5 ± 10.5 mm Hg, p = NS); and rate-pressure product (basal 12,150 ± 2,950 vs. 11,800 ± 2,180, p = NS; peak 15,250 ± 3,600 vs. 14,700 ± 2,900, p = NS).
Qualitative evaluation. Spectral changes during dipyridamole echocardiography testing can be qualitatively evaluated by visual inspection of compressed spectral array plots. The pattern of changes is similar in all group B patients. Dipyridamole infusion is always followed by a progressive reduction in both LF and HF components, with a shift of the residual power toward the VLF band. This typical pattern is shown in Fig. 2. The difference between the basal conditions (upper spectra) and the dipyridamole infusion (middle spectra) is clearly evident; the two spectral peaks (LF and HF peaks) that were initially present disappear after dipyridamole injection and are restored only by aminophylline administration (lower spectra). In some patients, just before high dose infusion, there is a slight reappearance of variability, which is suddenly depressed by the second dipyridamole injection. Aminophylline administration promptly restores the basal variability pattern.
In group A, too, low dose dipyridamole infusion induces a reduction in both LF and HF power. However, when ischemia occurs, an increase in LF power is invariably found in all patients. The effect of ischemia is evidenced by the comparison of Fig. 2 (negative test response) and Fig. 3 (positive test response); in synchrony with echocardiographic dyssynergy, the compressed spectral array plot of Fig. 3 shows the reappearance of a clearly evident LF peak, maintaining itself throughout the ischemic period. Aminophylline administration further increases the power in the LF band and restores the HF component.
Quantitative analysis.Table 2 and Fig. 4 show, for groups A and B, the mean value ± SE for all the spectral variables, sampled at relevant test periods: before testing, end of low dose infusion, end of high dose infusion (or the corresponding dipyridamole time in 13 patients of group A who did not receive the high dose infusion because of ischemia) and peak pharmacologic effect (as previously defined). Because of excessive differences in time of administration and dosages, aminophylline data were not included.
In patients with a negative test response, dipyridamole administration induced (basal vs. peak values, paired t test, see Table 3) a reduction in the mean RR interval (p < 0.05), RR variance (p < 0.01) and LF ratio (p < 0.01). The HF ratio and the HF power in normalized units slightly decreased, whereas the LF power in normalized units and the LF/HF ratio slightly increased (p = NS).
The results of ANOVA (Fisher F values) are shown in Table 4. The only variable able to statistically discriminate patients with ischemia from control subjects was the LF ratio, which decreased in group B from 1 ± 0.00 (basal) to 0.31 ± 0.22 (peak) and increased in group A from 1 ± 0.00 to 15.41 ± 6.59, respectively (Table 2, Fig. 4). Results of the unpaired t test comparing the peak values of the two groups also showed statistically significant difference (p < 0.01).
A time (or pharmacologic) effect was evident for mean RR interval and HF power in normalized units, which decreased during dipyridamole infusion in both groups.
The trend of the 30-s LF ratio changes in group B is shown in Fig. 5. The decrease in LF power induced by low dose and high dose dipyridamole administration is clear, as is the aminophylline action, which restores a nearly basal condition.
The patient by patient behavior of LF ratio in group A is shown in Fig. 6. The increase of this variable is present in all patients, irrespective of echocardiographic location of ischemia, and it appears to be an early phenomenon; in 13 of our patients it apparently developed before or in synchrony with dyssynergy, maintaining itself throughout the ischemic episode. An ST shift >1 mm, pain and aminophylline administration were almost always associated with a further increase in LF power.
Spectral analysis of RR interval variability is a well established method of studying the autonomic modulation of sinus rate [1–5]. A limitation of traditional batch analysis algorithms is that they require a stationary input signal; that is, the RR series has to maintain the same statistical characteristics (at least mean value and variance) throughout the entire analyzed data segment . Thus, they cannot provide a reliable estimation of spectral variables during nonstationary phenomena such as transient myocardial ischemia.
Recently, time variant algorithms, able to quantify spectral changes even in the presence of nonstationary events, have been proposed [23, 34–36]. Assuming that spectral changes are correlated with neural control of the cardiocirculatory system, these algorithms could be used to noninvasively monitor autonomic variations associated with myocardial ischemia in humans. This approach could lead to a better understanding of an autonomic role in triggering ischemic episodes and eventually provide a further clinical marker of ischemia.
In this report we focused on a well standardized, reproducible, noninvasive clinical model of transient myocardial ischemia, the dipyridamole echocardiography test. In this model, ischemia is induced through a metabolically mediated redistribution of coronary flow, with only minor changes in myocardial oxygen demand , and is detected with high temporal and spatial resolution by the echocardiographic appearance of regional dyssynergy.
Interplay between the autonomic nervous system and transient myocardial ischemia. The relations between the autonomic nervous system and myocardial ischemia are complex and difficult to study. Although autonomic tone can trigger an ischemic episode, influencing both coronary vasomotion and myocardial oxygen demand , ischemic myocardium evokes autonomic reflexes, which in turn can influence the ischemic event [16–18]. Coronary occlusion in animals [16, 20] has shown that the neurovegetative response to transient myocardial ischemia is modulated by both activation of polymodal receptors in the ischemic myocardium and baroreflex response to ischemia-related hemodynamic compromise [17, 20, 21]. Such neurovegetative response is related to ischemia location, extent and duration; blood flow reduction; and impairment of autonomic nerve endings . In the clinical setting, other variables are probably involved, including the various pathophysiologic features of each ischemic episode, the role of anginal pain  and interpatient differences in autonomic status.
Neurovegetative changes during dipyridamole testing. Dipyridamole inhibits cellular uptake of adenosine , leading to its accumulation. The main cardiac effect of adenosine is arteriolar dilation, with a secondary inappropriate increase and redistribution of coronary flow . Adenosine also has several direct and indirect effects on the cardiovascular system and on nervous system regulatory mechanisms [39–43]. It inhibits norepinephrine release from efferent sympathetic nerves [40, 41], has a negative chronotropic, inotropic and dromotropic action  and produces sustained hypotension without compensatory tachycardia in patients devoid of autonomic reflexes and in anesthetized patients [44, 45]. Conversely, adenosine infusion in unanesthetized normal subjects increases systolic arterial pressure and heart rate [46–48], suggesting that the net result is a stimulation of the sympathetic output . This activation can be attributed to both baroreceptor unloading  and direct chemoreceptor stimulation .
In the presence of an epicardial coronary stenosis, adenosine-mediated redistribution of coronary flow can result in underperfusion of the subendocardium and, eventually, in true ischemia. As previously described, ischemia induces a series of complex autonomic responses that are modulated by multiple afferences from myocardial receptors, baroreceptors and probably the central nervous system, especially if pain is present .
Spectral patterns during dipyridamole testing. The main effect of dipyridamole infusion on heart rate variability variables is a decrease in LF and HF ratio components. Conversely, when ischemia develops, the LF and HF ratios reverse this decreasing trend and begin to increase. In patients with positive and negative test results, LF power in normalized units slightly increases and HF power in normalized units significantly decreases; therefore, in both groups the LF/HF ratio increases with respect to the basal level but does not discriminate positive from negative tests. ANOVA does not demonstrate differences related to the occurrence of ischemia in mean RR interval and in classic spectral variables (variance, LF and HF power in normalized units and LF/HF ratio), reflecting only a time (or drug) effect that could be interpreted as a sympathetic activation. In our study, the only variable able to statistically differentiate ischemic from nonischemic responses was the LF ratio, introduced to reduce interpatient differences and to normalize temporal changes with respect to each patient's pretest pattern. The increase in LF ratio seemed to start before obvious echocardiographic dyssynergy and ST segment displacement, at least in some of our patients.
Spectral patterns and autonomic behavior. Although pathophysiologic considerations were not among our study objectives, we can speculate about the relations between autonomic behavior and spectral patterns. Assuming that both dipyridamole and ischemia stimulate a sympathetic response [16, 20, 49], we may conclude that the different spectral patterns found in this study reflect different expressions of sympathetic prevalence. Baroreflex activation, reflexes from ischemic myocardium or anginal pain may influence sympathetic output. Because no differences in arterial pressure or heart rate were found between our two groups, a different activation of baroreflexes seems unlikely. Moreover, because the increase in the LF ratio occurs earlier than the development of pain, the implication of a cardiac reflex seems the most favorable hypothesis. Stimulation of cardiac receptors could explain why the increase in LF ratio appeared before clear echocardiographic dyssynergy in some of our patients, suggesting that changes in heart rate variability could reflect an early, perhaps biochemical stage of the ischemic cascade.
Study limitations. One limitation of our study is the use of the echocardiographic technique, which, although highly sensitive and specific, is indeed qualitative and subjective in detection of ischemia. To overcome this problem, we included only patients with coronary artery disease, low threshold ischemia and clear changes in kinetics during dipyridamole testing, and we reduced subjectivity by comparing the analyses of two independent observers.
A second limitation is that for ethical reasons the absence of significant coronary artery disease was not assessed angiographically in the control group. In fact, our control subjects had no clinical indication for coronary angiography because the simultaneous presence of a negative response on both exercise testing and dipyridamole testing has been associated  with a very low incidence of significant coronary artery disease. Although these patients cannot be regarded as “true normal” subjects (autonomic response may be abnormal in the presence of atypical chest pain), the absence of induced ischemia (our end point in this study) make them a useful control group for this study.
Future developments. In our selected patients, time variant spectral analysis of RR interval variability showed an increase in LF ratio that allowed discrimination between positive and negative test responses, suggesting that the ECG can provide information on ischemia unrelated to ST-T wave variations, even if its disclosure requires a more complex data processing procedure. These results can be used as a starting point for further investigations in a larger group of unselected patients and, possibly, of other provocative tests.
The new method itself requires additional refinements, especially concerning the management, synthesis and quality control of the great flow of data as well as the definition of the most significant spectral indexes in this new time variant context. Finally, because of its recursive nature, our algorithm could be implemented to run in real time, provided that the problems of artifacts and arrhythmias are correctly addressed on-line. This possibility, and its noninvasive nature, make it potentially useful for clinical applications, especially in conditions (bad echocardiographic window, conduction defects, repolarization abnormalities in basal ECG) in which traditional markers of ischemia are unreliable.
We thank the Cardioline-Remco Italia s.p.a., San Pedrino di Vignate, Milan, Italy for technical assistance in ECG data storage.
A discrete time series y(n) can be viewed as the output of a linear system where w(n) is the white noise, with zero mean and variance σ2, driving the system, whose transfer function H(z) contains all the input/output relations defining the y(n)-generating mechanisms. A simple linear system structure is given by the autoregressive (AR) model, defined by the following expression:where the ak coefficients are the parameters identifying the model, k is the discrete time index, and p is the model order. For an AR(p) form, the transfer function becomes
It is worth remembering that knowledge of the model coefficients and the estimated input noise variance univocally define the properties of the generated process, in particular its spectrum S(ω), which can be obtained aswhere H* is the complex conjugate of H. In the classical batch least-square method , the parameters of the model can be estimated from the data by minimizing the following figure of merit J:where N is the number of samples, and e(n) is the prediction error series, obtained as the difference between the real and the estimated signal values. The uniform windowing in figure of merit J makes the batch least-square approach able to identify the single model that gives the best whole data fitting. In this way, to obtain a consistent model estimation, an unchanged signal generation mechanism must be hypothesized for the entire data set; thus, the process has to remain in a stationary condition.
If this assumption is not realistic, the signal will not remain in a single steady state, and a unique model cannot be representative for the evolving phenomenon. Thus, the model is made time variant, and its parameters are updated on a beat to beat basis in order to follow the modified process characteristics.
A time-variant identification can be carried out by minimizing a slightly different figure of merit:where ω (0 < ω < 1) is a real scalar called a forgetting factor. The exponential weight gives decreasing importance to the past terms, highlighting the current ones and making the model able to fit the more recent process properties.
The choice of ω is usually a crucial point and must be made by considering the best compromise between fast convergence and noise sensitivity. In fact, for too high ω values the algorithm is not able to follow the faster signal dynamics, whereas for too low values the identification becomes too sensitive to the superimposed noise. It has been found  that a good compromise between these conflicting aspects is obtained for RR variability by setting ω = 0.98.
The model parameter vectors are estimated in a recursive way according to where k(t) is the gain of the algorithm and differs passing from the recursive least-square  method to directional or Fortescue methods [51, 52].
B.1 Abbreviations and Acronyms
ANOVA = analysis of variance
ECG = electrocardiogram
HF = high frequency
LF = low frequency
VLF = very low frequency
↵1 This report was partially supported by a grant from the Italian Ministry of University (MURST 40%—Special Project on Cardiovascular System).
- Received December 20, 1994.
- Revision received May 21, 1996.
- Accepted June 3, 1996.
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