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1 Istituto Superiore di
Sanità, The possibility of computing a cardiac age on
the basis of spectral analysis of healthy individual tachograms was
confirmed and facilitated by the use of a nonlinear technique:
recurrence quantification analysis. The age of 112 subjects was
predicted by this technique in terms of a progressive increase in the
deterministic character of the heartbeat. This result confirms the
"random-walk" character of the heartbeat as predicted by the
terminal dynamics paradigm, thus allowing for a simple and
comprehensive model of the effect of aging on cardiac dynamics: as age
progresses, heart rate dynamics become increasingly predictable
(constrained) on a beat-to-beat basis. This implies a basically
stochastic nature of heart rate dynamics, probably reflecting the
continuous adjustments to an unpredictable internal environment.
recurrence quantification analysis; nonlinear dynamics; heart rate
variability; terminal dynamics
IN A RECENT WORK (5) we derived a "cardiac age"
estimator based on the spectral analysis of the tachograms of healthy
individuals in resting and tilt phases. This estimator was derived by
means of a multiple linear regression model using the chronological age
of individuals as dependent variable and 4 principal components extracted from 24 variables stemming from the Fourier analysis of the
tachograms (5). The cardiac age estimator showed a remarkable correlation with the actual age of individuals
(r = 0.74).
In another work (7) we demonstrated the possibility of a
straightforward representation of cardiac dynamics in terms of a
first-order Markov model. According to this model, heartbeat dynamics
could be considered a random walk, where the system at each beat was
presented with three alternatives:
1) remain in the same state (i.e.,
having a beat of a length very similar to the previous one),
2) shift to the higher class of beat
duration, or 3) shift to the lower
class of beat duration.
Previously (7), we demonstrated that this model was effective for
describing the nonoscillatory component of heart rate variability
(HRV); in this work we wanted to prove the usefulness of the model to
provide a simple description of the effect of senescence on HRV.
According to this model, we hypothesized that senescence, reducing the
system plasticity, makes the dynamics select
alternative 1 over
alternative 2 or
3, with a consequent increase in
predictability (i.e., deterministic, rule-obeying character) of the
tachograms.
To assess this model, we analyzed the tachograms with a nonlinear
analysis tool [recurrence quantification analysis
(RQA)1]
that, in our opinion, given its independence of stationarity and its
intrinsic discrete character, is particularly suited for the analysis
of HRV (10, 15) and allows for a direct quantification of the
deterministic character of the studied series. RQA allowed us to build
a cardiac age indicator consistent with that derived from Fourier
analysis, requiring only one component, which allowed for a
straightforward interpretation of the effect of senescence on HRV in
terms of an increase in the deterministic, rule-obeying character
of beating. The terminal dynamics paradigm (7, 14, 17) offers a
physical basis for the observed results.
Data collection protocol.
To rely on a normal healthy population, the following exclusion
criteria were adopted (5): history or actual evidence of cardiovascular, respiratory, renal, liver, or gastrointestinal disease;
diastolic blood pressure >90 mmHg or systolic blood pressure >150
mmHg; body mass index >26 kg/m2;
smoking; diabetes; plasma cholesterol >5.7 mmol/l; arrhythmias or
conduction abnormalities; echocardiographic evidence of wall motion
abnormalities or valvular disease; and ultrasound evidence of
significant carotid stenosis. Of the 455 subjects initially enrolled
for this study, only 141 completed the study protocol; data from 112 of
141 subjects who completed the initial study (5) were analyzed in the
present study (29 subjects were lost for the present study because of
accidental loss of relative data resulting from a computer crash). Both
populations had very similar statistical characteristics.
ABSTRACT
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Abstract
Introduction
Methods
Results
Discussion
Appendix
References
INTRODUCTION
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Abstract
Introduction
Methods
Results
Discussion
Appendix
References
METHODS
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Abstract
Introduction
Methods
Results
Discussion
Appendix
References
RQA. RQA was first introduced in physics by Eckmann et al. in 1987 (6) as a purely graphic technique. Five years later, Zbilut and Webber (16) enhanced the technique by defining five nonlinear quantitative descriptors of the recurrence plot that were found to be diagnostically useful in the quantitative assessment of time series structure in fields ranging from molecular dynamics to physiology (8, 10, 12, 13).
This technique has been demonstrated to be particularly useful in quantifying transient behavior far from equilibrium (12); this feature is especially important in dealing with complex systems that can hardly be considered at equilibrium such as heart rate. The RQA is based on the computation of a distance matrix between the rows (epoch) of the embedding matrix of the tachogram at unitary lag (given the discrete character of the heartbeat). This distance matrix is computed making use of Euclidean metrics; after this first step, the distance matrix, having as rows and columns the subsequent epochs of the series of length equal to the chosen embedding dimension (in this case set to 15), is colored, with darkening of the pixels located at specific (i,j) coordinates, which correspond to a distance between i and j rows (epochs) lower than a predetermined radius (for details see Refs. 8, 13, and 16). The features of the distance function make the plot symmetrical (Di,j) = D( j,i) and with a darkened main diagonal corresponding to the identity line (Di,j = 0, j = i). The darkened points individuate the recurrences (recurrent points) of the dynamic process, and the plot can be considered a global picture of the autocorrelation structure of the system at hand (6, 16). In other words, a recurrence plot visualizes the distance matrix between the epochs (rows) of the embedding matrix, which, in turn, represent the autocorrelation in the signal at all the possible time scales. In fact, it is important to note that the distance is computed for all the possible pairs of epochs: the elements near the principal diagonal of the plot corresponding to short-range correlations (the diagonal marks the identity in time) and the long-range correlations corresponding to points distant from the main diagonal. Besides the global impression given by the graphic appearance of the plot (see Fig. 1 for the RQA plot of 2 typical tachograms), the indexes developed by Trulla et al. (12) and Webber and Zbilut (13, 16) allow for a quantitative description of the recurrence structure of the plot. The RQA descriptors (4 of 5) used in this work are as follows:
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Data analysis. The RQA variables were supplemented by the computation of mean and standard deviation of the tachograms and by the differences between rest and tilt phases for all the computed indexes (the differences are shown by the prefix "dif" followed by the name of the corresponding variable).
As a first step in the analysis, the Pearson correlation coefficients (r) between the tachogram variables and the chronological age of patients (AGE) were computed. All the computed variables, except the dif variables, given their low correlation with age, were then submitted to a principal component analysis (PCA; see Refs. 3 and 5 for details of this widely used statistical technique). Here it is sufficient to stress that this technique allows us to build general synthetic indexes collecting all the relevant information present in the original variables. These indexes are by construction independent between them, allowing separation of the different aspects of the studied data set. The extracted components were then correlated with age. Only the first, most informative component (PC1) was used to build a simple bivariate regression model to predict the actual age of individuals, with AGE as dependent variable and PC1 as regressor (5). The estimate of chronological age given by this model was called cardiac age (ESTAGE). To compare the results of this work with the previous study (5), the data set was classified by means of a cluster analysis procedure (see Ref. 5 for explanation of the k-means clustering technique) into homogeneous classes of age. AGE was compared with ESTAGE at the cluster level to appreciate the general trend of cardiac regulation aging over the years.| |
RESULTS |
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Top
Abstract
Introduction
Methods
Results
Discussion
Appendix
References
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Simple correlations of the measured parameters with age. The entire population consisted of 112 healthy individuals 20-85 yr of age [52.42 ± 17.95 (SD) yr]. Table 1 shows the Pearson correlation coefficients of the studied parameters with the AGE variable. All the RQA parameters and SD values exhibit high and significant correlations with age, whereas the heart rate (MEAN) is uncorrelated with age. The difference parameters are generally uncorrelated with age, except for weak correlations for difMEAN and difSD. Given this result, the dif variables were excluded from subsequent analyses.
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PCA. The results of PCA are reported in Table 2 in terms of factor loadings (correlation coefficients between experimental variables and components), percentage of variability explained by each single component, and correlation coefficients of each component with AGE as an external variable.
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Setup of a cardiac age estimator. The high correlation between AGE and PC1 allowed us to build a cardiac age index in terms of the best linear model (in a least-squares sense) able to predict the actual chronological age of the individuals on the basis of the relative PC1 score.
The application of a least-squares regression model to our data set generated the subsequent linear expression for cardiac age (ESTAGE)
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Clustering of ages. The application of the k-means algorithm to the age distribution of the population under study produced a five-class solution (Table 3) almost identical to that described previously, explaining ~94% of the total variability of AGE.
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DISCUSSION |
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Methods
Results
Discussion
Appendix
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Besides the confirmation of the possibility to compute a cardiac age estimator for clinical and screening purposes and the existence of a saturation effect in cardiac age probably due to selection biases (i.e., the implicit selection linked to the fulfillment of the inclusion criteria at older ages), the main added value of this work is the elucidation of the dynamic character of the effect of aging on heart rate regulation and, more in general, in the proof of a "random-walk" hypothesis for heartbeat dynamics. The emerging picture of heart rate regulation at the beat-to-beat level is that of a stochastic process. In this frame the age-dependent increase of determinism has to be interpreted as an increase in the correlation time of the heartbeat (i.e., residence in the same quasi-state) with a consequent increase in the general predictability of the signal. This picture comes from the consistency of the observed results with the Markov formalization introduced previously (7). It is important to stress that a Markov model is only a phenomenological description: what appears stochastic to us could be the result of complex adaptations to the internal state; moreover, any natural process has inertia, so two nearby states are, on average, more similar than two that are far apart. In these terms, the effect of senescence on cardiac dynamics can simply be expressed in terms of decreased "sensitivity" of the system to the changes of internal state.
In this work, age was by far the main-order parameter of the total variability, and, in contrast to the previous work, its effect on HRV was not dispersed over different components but was concentrated on a unique factor. This factor (PC1) measures the influence of senescence on HRV and appears as a global determinism factor (Table 2).
With increasing age, heartbeat dynamics become increasingly deterministic and predictable. Recently, Giuliani et al. (7) proposed a "quantumlike" regulation of heartbeat with the heart oscillating between a few discrete states corresponding to different modal frequencies. The oscillation followed quantumlike dynamics with only two possible choices at each time step (beat): 1) remaining in the last visited state and 2) shifting to an adjacent state, i.e., a simple first-order Markov model. This kind of behavior, based on the theoretical framework of terminal dynamics [an innovative physical paradigm developed by Zak et al. (14), which models the dynamics of complex systems as short sequences of deterministic paths interspersed with purely stochastic singularities], was demonstrated for the first eigenvector of heartbeat sequences of rats and humans (7, 14, 17). The first eigenvector of the tachogram can be considered a filtered version of the original signal retaining the major dynamic features of the raw series (7). Given this general pattern of functioning, an increase in determinism corresponds to an increase in the relative probability for each single beat to remain in the same state (length class) as the previous one (increased probability for choice 1).
From an RQA point of view, this corresponds to an increase in the DET parameter (and, consequently, ENT and MAXLINE variables).
Terminal dynamics give us the physical rationale of how it is possible to exert such control. Randomness in such a paradigm is generated by the dynamics themselves and not from some error function. Furthermore, complex nonlinear interactions can be obtained at the singular points, whereas between these singular points the dynamics are constrained in a deterministic way. The advantage of this paradigm is that the organism is allowed to adjust the dynamics according to environmental influences at the singular points, whereas the deterministic character of a classical chaos paradigm would limit the adaptability of the system, given that, by definition, the deterministic character of the model makes the system strictly dependent on initial conditions. It may be argued that these objections can be countered by the existence of "control parameters," which can retune the system but, given the tremendous amounts of noise in biological systems and the extreme sensitivity to initial conditions of chaotic dynamics, the energy expended to run adaptive controllers would be considerable (7, 14, 17).
In this work we demonstrated that aging acts on cardiac dynamics constraining the heart random walk on the last visited state, thus allowing us to generalize the Markov-like function to the heartbeat as a whole and not only to its first eigenvector.
This fact, in turn, implies that the optimal functioning of HRV regulation (younger ages) corresponds to a more stochastic character of HRV, paralleling a maximum adaptation power to an unpredictably changing environment. In older ages this adaptability is drastically reduced with a consequent increase in the deterministic character of the dynamics.
Our results parallel the findings of Mestivier and colleagues (10), who demonstrated an increase in heartbeat determinism in terms of MAXLINE in diabetic neuropathy.
More generally, our findings suggest that heart rate regulation is intrinsically stochastic and that the view of a basically deterministic (attractor-like) system with superimposed noise is basically incorrect. Besides the analysis of data and theoretical considerations (14, 17), our view is supported by the growing body of evidence (2, 4, 9) on the role of stochastic resonance effects in physiological systems.
From a practical point of view, our results may allow clinicians to compute in a very straightforward way (see APPENDIX) the cardiac age of patients, which could be useful for general screening purposes. Screening situations in which the computed cardiac age could be useful include the epidemiologic and biomonitoring studies carried out to discover potentially neurotoxic effects of environmental hazards (1). In these cases, knowing the expected effect of age could greatly improve the power of the study. Obviously, the need to extend the method to the study of pathological situations remains crucial.
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APPENDIX |
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Methods
Results
Discussion
Appendix
References
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Computation of ESTAGE (cardiac age by means of RQA). The subsequent steps indicate how to compute the cardiac age score. Obviously, all the registration parameters must be set as indicated in METHODS. Otherwise it is best to generate another score following the procedure we outlined in METHODS. The RQA software (RQD program, version 4.0) necessitates definition of the measuring parameters: in this case, lag = 1, emb = 15, norm = Euclid, dist = absolute, radius = 5, line = 5.
Given these premises, the first principal component is computed as
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ACKNOWLEDGEMENTS |
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We thank Charles L. Webber and Joseph P. Zbilut for helpful discussions.
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
1 RQA software, developed by C. L. Webber and J. P. Zbilut, can be downloaded from the following URL: http://homepages.luc.edu/~cwebber/.
Address for reprint requests: A. Giuliani, Istituto Superiore di Sanità, TCE Lab, Viale Regina Elena 299, 00161 Rome, Italy.
Received 17 February 1998; accepted in final form 17 June 1998.
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REFERENCES |
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Discussion
Appendix
References
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