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Implicit and Explicit Model-Based Signal Processing for the Analysisof Short-Term Cardiovascular Interactions ALBERTO PORTA, GIUSEPPE BASELLI, AND SERGIO CERUTTI Invited Paper

The paper stresses the importance of model-based signal processing in the analysis of the cardiovascular regulation mechanisms. It is remarked that even traditional signal processing implicitly assumes a model and interprets data according to it. Therefore, traditional signal processing is here referred to as implicit model-based signal processing in contrast with explicit model-based signal processing directly stemming from modeling considerations. The paper points out the advantages that can be achieved by rendering explicit the underlying model structure when it is implicit and by jointly applying implicit and explicit model-based signal processing approaches when dealing with complex, interacting, closed-loop biomedical systems. These advantages are rendered practical over several examples derived from the study of the short-term cardiovascular regulation performed by the autonomic nervous system. Keywords—Autonomic nervous system, biomedical signal processing, cardiovascular control, cardiovascular variability, physiological modeling.

I. INTRODUCTION Model-based signal processing (MBSP) is an emerging approach in describing physiological systems. The MBSP approach has been found useful in the study of the short-term cardiovascular control. The observation that cardiovascular variables such as heart period (here indicated as RR interval) and systolic arterial pressure (SAP) exhibit, when considered on a beat-to-beat basis, small, nonrandom, changes around their mean values with a frequency well below the heart rate (below 0.5 Hz in humans) has intrigued physicians,

Manuscript received September 10, 2005; revised November 20, 2005. A. Porta is with the Dipartimento di Scienze Precliniche, LITA di Vialba, Laboratorio di Modellistica di Sistemi Complessi, Universitá degli Studi di Milano, Milano 20157, Italy (e-mail: [email protected]). G. Baselli and S. Cerutti are with the Dipartimento di Bioingegneria, Politecnico di Milano, Milano 20133, Italy (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/JPROC.2006.871774

physiologists and biophysicists (see [1] for a review of these pioneering studies) and prompted them to interpret these variations as an evidence of the cardiovascular regulation. More recently these oscillations have been termed as low frequency ( 0.1 Hz in humans, LF) and high-frequency (HF, around the respiratory rate) rhythms. The complexity of the cardiovascular control has been clearly outlined by Koepchen [2] and regulation of cardiovascular variables is now assumed to be the result of the action of multiple feedback systems (e.g., chemoreceptive and baroreceptive) and self-sustained autonomous oscillators situated not only at brain stem level (e.g., respiratory and vasomotor centers) but also at spinal and at peripheral level (e.g., vasomotor districts). We refer to [2, Fig. 4] and [3, Fig. 1] for schemes summarizing short-term cardiovascular regulatory mechanisms. Therefore, MBSP appears to be a natural way to approach this complexity to disentangle mechanisms, simulate effects and describe physiological interactions. For example, several very simple feedback models, both linear with a delay [4] and nonlinear [5], have indicated the negative feedback and the latency of the baroreflex circuit as one of the main mechanisms enhancing LF oscillations [6], [7]. The possibility offered by the MBSP approach to create more complex structures describing several concurring regulatory mechanisms has been exploited to take into account both RR-SAP closed-loop interactions and the effect of respiration [8]–[10] and even to describe regulatory mechanisms in LF band independent of the baroreflex circuit [11], [12]. In parallel with MBSP, traditional signal processing (SP) has been applied with a similar intention (disentangling mechanisms and describing physiological interactions) but also with the more practical aim to extract simple descriptive indexes. For example, an SP approach based on spectral analysis and applied to RR interval and SAP variabilities permitted to address the noninvasive measure of the status

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of the autonomic nervous system [13]–[16] and of the baroreflex sensitivity [17], [18]. Usually traditional SP and MBSP are not carried out together missing several advantages related to the combination of a deeper comprehension of the mechanisms that may be reached through a MBSP approach with the simplicity in interpreting results and in providing quantitative indexes typical of the SP techniques. The aim of this paper is to overcome the conceptual separation between SP and MBSP and to stimulate a joint application of both methodologies. Indeed, the separation between them is artificial and a unique common framework can be easily set if it is considered that both share data processing and modeling skills. Indeed, in traditional SP the underlying model is present, although implicit in the hypotheses of the technique, while in MBSP the model is usually rendered fully explicit. The framework set in this paper adheres to the current perspective in biomedical engineering that proposes to face spatiotemporal complexity in integrative science by exploiting contemporaneously several nonredundant approaches [19]–[23]. Consequently, in this paper, SP and MBSP will be indicated as implicit MBSP (IMBSP) and explicit MBSP (EMBSP), respectively, to stress the common background. Next, advantages and shortcomings of both approaches will be reviewed and the advantages of a joint application will be elucidated. Several examples relevant to the application of IMBSP and EMBSP approaches to the analysis of cardiovascular control will be given in Section III and Section IV, respectively, with a special focus on the model structure and implications that the assumed model structure produced on the interpretation of data. Section V will be devoted to examples of joint application of IMBSP and EMBSP approaches with a special focus on the advantages that this integrated approach might provide. The applications are mainly biased on the experience of our group: some of them have been previously published and some are only sketched to stimulate further research in the area. The aim of the paper is not to furnish a complete review of IMBSP and EMBSP methods in the field of short-term cardiovascular regulation but to stress the need to associate both IMBSP and EMBSP to obtain a more powerful methodology and avoid to focus only an aspect of the problem (i.e., the IMBSP approach is more focused on index extraction, while the EMBSP one on the description of complexity of the physiological system under investigation). It is worth pointing out that the proposed MBSP approach is more helpful in describing the interactions among systemic cardiovascular variables (e.g., RR interval and SAP) and regulatory mechanisms at a level that may hide specific physiological knowledge (e.g., different neurotransmitter dynamics and saturation of the transduction mechanisms). However, it may be rendered closer to specific physiological mechanisms by simply introducing in the same framework less systemic cardiovascular variables (e.g., efferent vagal and sympathetic neural activity directed to heart, afferent neural baroreceptive activity, flows in specific peripheral districts).

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II. IMPLICIT AND EXPLICIT MODEL-BASED SIGNAL PROCESSING A. IMBSP Traditional SP can be defined as an approach that manipulates data in a specific domain and that does not apparently assume any particular description of the data generation system. Therefore, SP techniques are very smooth with respect to data and they require solely the fulfillment of general requirements to be applied. For example, parametric statistical tests assume Gaussian distribution [24], spectral analysis presumes linearity [25], local nonlinear prediction hypothesizes the presence of a continuous nonlinear function linking past samples to future values [26], and transfer function analysis assumes a linear time-invariant relationship between input and output signals [27]. However, the general requirement about data structure represents the implicit model underlying the SP technique, thus rending any SP method a special MBSP approach using this implicit model to interpret reality. B. EMBSP MBSP is based on an explicit particular description of the system producing the signals (i.e., the model) and of the interactions among them. The model, usually belonging to a general class (e.g., linear time-invariant or nonlinear feedback models), is rendered closer to the data by introducing a priori information about the system under study. The larger the amount of a priori knowledge included, the more detailed and specific the description of the system. Usually the parameters of the model can be a priori set based on physiological considerations or can be directly identified from the real data via identification techniques mainly based on minimization procedures. The adequacy of the model structure is evaluated by comparing, e.g., by root mean squared error (rmse), original data, and signals generated by the model. C. Theoretical Advantages and Possible Shortcomings of the IMBSP Approach The main advantage of the application of IMBSP approaches is that they provide directly indexes describing dynamics. These indexes are quite general because the implicit model underlying any IMBSP method usually covers a large class of physical systems and robust in case of changes of the model structure inside the general adopted class. In addition, the IMBSP approaches are based on classical, reliable, well-defined tools that do not need much dedicated competence to be adapted to the specific problem and, generally, give reliable and robust results. There are two main disadvantages in the application of the IMBSP approaches. The first shortcoming is that, as most of the IMBSP methods are based on simple single-input single-output models, they cannot describe complex, concomitant interactions among several different mechanisms. The second shortcoming is a byproduct of their easiness in being applied. Indeed, when the specific IMBSP method is decided without considering its implicit assumptions or

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Fig. 1. (a) Implicit model underlying AR power spectral estimation: the zero mean rr series is seen as the output of an all-pole linear time-invariant filter fed by the white noise w . (b) RR series recorded from a healthy human at rest with (c) its AR power spectral density. The spectral peaks are labeled as LF and HF solely according to their position on the frequency axis. Nonlinearities, if present in RR series, remain confined in the residual w .

when the assumptions are not explicitly taken into account in interpreting the results, the IMBSP approach may lead to erroneous conclusions and misleading interpretations. For example, when the results of a cross-spectral analysis between two signals, and , are interpreted as the estimate of the transfer function, the causal relationship establishing which is the input and which is the output and the absence of feedbacks have to be a priori assumed and are hardly inferred by the analysis results. Indeed, as the method is based on the decomposition into sinusoids and considers each frequency separately, the translation of phase relationships into causal delays is ambiguous. Let us, e.g., consider a phase shift of signal in respect to signal at frequency , this would fit with (and is often interpreted as) a causal delay with positive correlation from to . However, it of is immediate to show that a delay of in the opposite causal direction (i.e., from to ) is possible as well. D. Theoretical Advantages and Possible Shortcomings of the EMBSP Approach The main advantages of the EMBSP approach are related to the possibility to render the underlying model as complex as necessary to take into account several different interacting regulatory mechanisms. The elucidation of the structure of the data generation system allows one to easily link the observed dynamics to mechanisms that can generate it and the relevant changes to modifications of specific model parameters. As a consequence, the EMBSP methods are often utilized to verify if an activation of certain mechanisms is compatible with the observed dynamics. On the contrary, they are less utilized for their ability to provide parameters closely linked to physiological mechanisms according to the imposed model structure. This limited utilization is the result not only of the difficulty to single out, derive, and validate useful parameters from complex model structures but also of the typical attitude of the model developer that usually stresses more the phenomenological aspect of the model and less its possibility to extract synthetic indexes. This lack reduces the chance of utilization of the EMBSP approach in routine clinical applications and limits comparisons among different models of the same phenomenon.

E. Theoretical Advantages of Jointly Applying IMBSP and EMBSP Approaches The theoretical advantages of a joint application of IMBSP and EMBSP methods come out from the different center of attention of both techniques. The focus on modeling provided by EMBSP should prevent from the rough and indiscriminate application of any IMBSP method and provide the contemporaneous description of several interacting mechanisms. On the contrary, the focus of IMBSP on the extraction of general descriptive indexes may stimulate a condensed description of the interacting mechanisms more suitable for clinical and epidemiological applications over large trials, for comparing different models of the same phenomenon, for testing refined versions of the same model and for sensitivity analysis. III. APPLICATIONS OF THE IMBSP APPROACH TO SHORT-TERM CARDIOVASCULAR REGULATION A. Quantification of Sympatho-Vagal Balance via Power Spectral Analysis The relative engagement of two branches of the autonomic nervous system can be evaluated in the frequency domain via power spectral analysis applied to consecutive RR intervals sequences of about [i.e., RR RR where represents the progressive cardiac beat number, Fig. 1(b)] [16]. The ratio between the LF and HF powers (i.e., LF/HF ratio) [15] and the HF power expressed in absolute units both under free [13] or controlled breathing [28] are the most utilized indexes. The former indicates the prevalence of sympathetic modulation and the latter the importance of parasympathetic activation (both under free and controlled respiration). Spectral analysis is an IMBSP method that implicitly assumes a linear time-invariant model generating the series and fluctuations are interpreted as a linear superposition of a finite number of basic functions. In specific power spectral estimation methods the linear time-invariant model has a characteristic, well-defined form. For example, in autoregressive (AR) power spectral estimation the RR interval rr , fluctuations around its mean, rr are modeled in the time domain as a linear combination of

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Fig. 2. (a) Implicit model underlying local nonlinear prediction method: the continuous function f ( ) maps p past rr values (i.e., rr (n))) onto the one step-ahead value rr(n + 1). (b) RR series from a healthy human during controlled respiration at 10 breaths/min, (c) the predicted series, and (d) the prediction RR )=q , where error. The predicted series is derived after a uniform partition of the p-dimensional phase space in q hypercubes of size " = (RR RR and RR are the maximum and the minimum RR intervals and q is the number of quantization levels (here q = 6). Two patterns rr (n) are defined to be similar (i.e., close in the phase space) if they lay in the same hypercube. A figure of merit is used to detect the best embedding dimension p (here p = 3). Further details can be found in [34].

0

past samples weighed by the coefficients mean white noise

rr

rr

’s plus a zero

(1)

Using the representation of the AR process in the domain rr

(2)

is the one delay operator (i.e., where ), the rr process can be seen [Fig. 1(a)] as the output of the linear time-invariant model described by the transfer function (3) where

(4) The power spectral density rr [Fig. 1(c)] is derived directly from this model structure as rr

(5)

is the variance of and T is the mean heart pewhere can be factorized riod. It can be easily shown [12] that in a sum of terms that, anti-transformed in the time domain, correspond to basic AR processes with one real pole or a pair of complex and conjugated poles. Therefore, any AR process can be seen as a linear superposition of a certain number of basic AR processes whose power can be easily calculated using the residue theorem [29]. The assumption of linearity 808

does not mean that a nonlinear process cannot be analyzed by means of spectral analysis but it signifies only that results furnished by this approach miss nonlinear mechanisms. For and , the detection of example, given spectral peaks at and cannot be interpreted spectral peaks at as the effect of nonlinear interactions, but as oscillations inand . Overlooked informadependent from those at tion relevant to nonlinearities remains confined in the phase spectrum in the case of nonparametric spectral methods (i.e., the phases of the oscillations are not random) [27] and in in the case of parametric spectral the residual process method [25]. B. Detection of Nonlinearities in Heart Rate Variability via Local Nonlinear Prediction To detect the presence of nonlinearities is necessary to use an IMBSP method that removes the assumption of linearity. One of the most utilized IMBSP approaches to detect nonlinearities in the analysis of short term cardiovascular oscillations is a technique based on local nonlinear prediction. Local nonlinear prediction uses only a few past samples to foresee future values. Given past rr samples organized as rr rr rr an ordered sequence rr representing a point in a p-dimensional reconstructed the one-step state space [30] and indicated with rr , the local nonlinear predicahead sample following rr , will produce tion method affirms that close states, rr similar future values, rr [26], [31]. This statement holds under the implicit assumption that there is a determinmapping past samples onto istic continuous function rr ), Fig. 2(a)]. No future values [i.e., rr assumption about the form of is required. The funcis locally approximated, thus deriving the prediction tion rr , with piecewise constant models (e.g., the weighed mean [32], [33] or the median [34]) or piecewise first order linear model [31] applied to a set of next selected among all because they have simvalues rr ilar previous patterns rr . Defined the prediction error as rr rr , the validation of the prediction is PROCEEDINGS OF THE IEEE, VOL. 94, NO. 4, APRIL 2006

Fig. 3. (a) Implicit model underlying transfer function estimation: a linear time-invariant single-input single-output model with a noise w additive to rr and j independent of sap changes. (b) and (c) RR and SAP series recorded from a healthy human at rest. (d) The magnitude of RR-SAP transfer function jH is plotted (solid line) with the squared coherence function, K , (dashed line) and its threshold of significance derived from a surrogate data approach (dotted line). jH j is sampled at LF and HF (solid circles) detected on SAP power spectrum. These values, provided that K exceeds the threshold of significance as in this example, are taken as indexes of the baroreflex sensitivity (jH (LF)j = 7:5 and jH (HF)j = 8 ms/mmHg). RR-SAP transfer function is estimated via bivariate AR approach as in [42].

usually performed by calculating the mean squared prediction error (MSPE). MSPE is calculated “out-of-sample” (i.e., the first half of the series is utilized to evaluate the prediction, , but this value is compared with samples belonging to the second half of the series) [31], [33] or “in-sample” using ad hoc figures of merit compensating for the MSPE negative bias due to overfitting [34]. This procedure provides also the optimal number of previous samples p (i.e., the optimal embedding dimension) that allows the best prediction. MSPE measures predictability of rr series but nothing can be said about nonlinearity yet. Indeed, nonlinearities can be detected only if MSPE is significantly smaller than that can be calculated using a linear model like e.g., the AR one setting that

rr

(6)

Usually the direct comparison with the best AR fitting (e.g., according to Akaike figure of merit [35]) is infrequently utilized and it is preferred a surrogate data approach [36], [37]. Surrogate data are a set of data artificially created from the original signal to check for a specific problem (i.e., the null hypothesis to be rejected). They share some properties with the original dynamic (e.g., mean, variance, static distribution, power spectrum) but others are completely destroyed. To detect nonlinearities in RR interval variability series phase-randomized or amplitude-adjusted phase-randomized surrogates were utilized [34], [38]. These surrogates have been designed to destroy nonlinear properties. Unfortunately, it was found that they have some drawbacks [39]: phase-randomized surrogates preserve power spectrum (i.e., linear properties) but not static distribution, while amplitude-adjusted ones do exactly the inverse. The drawback of the amplitude-adjusted surrogates is more limiting as linear dynamical properties are not preserved. Therefore, iteratively refined amplitude-adjusted surrogates [39], [40] that try to preserve both power spectrum and static distribution are more suitable than amplitude-adjusted ones in the context of checking the null hypothesis of a

linear Gaussian process, eventually measured through a nonlinear static invertible transformation capable of distorting Gaussian distribution. Therefore, if MSPE is significantly smaller in the original data than in iteratively-refined amplitude-adjusted surrogate data set, nonlinearity is detected and cannot be ascribed to the presence of a nonlinear static invertible transformation but it is a specific property of the dynamics. The analysis of nonlinearities in heart rate variability data revealed that nonlinear dynamics are not ubiquitous in short-term heart rate variability recordings but depend on the type of cardiovascular variability and on the experimental condition [34]. Indeed, they are more present in RR interval variability than in AP variability and during controlled respiration, mostly at slow breathing rates [Fig. 2(b)], thus indicating that a forcing periodical input like respiration may produce nonlinear phenomena. After checking iteratively refined amplitude-adjusted surrogate data, it is found that the series depicted in Fig. 2(b) exhibits significant nonlinear dynamics. C. Evaluation of Baroreflex Gain via Spectral and Cross-Spectral Methods The variability of RR interval and AP (usually systolic, rr SAP) around their mean values [rr and sap sap , Fig. 3(b) and (c)] are utilized to derive a noninvasive, more physiological estimation of the baroreflex sensitivity (i.e., without any administration of vasodilatative or vasoconstrictive drugs). Frequency-domain noninvasive estimates of baroreflex sensitivity have been calculated using power spectra [17], [18] and cross-spectra [41]. These approaches evaluate as the RR-SAP transfer function magnitude the square root of the ratio between RR and SAP power and , or as the ratio between the RR-SAP spectra, cross-spectrum modulus and (Fig. 3(d), solid is usually sampled at LF and HF (i.e., line). and HF ) and these values are taken as indexes of baroreflex sensitivity. Both parametric (based on bivariate AR model) [42] or nonparametric (i.e., based on FFT traditional approach) [18], [41] methods are

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utilized to estimate power spectrum and cross-spectrum. A estimate usual prerequisite for the reliability of the is that the squared coherence function (Fig. 3(d), dashed line)

[9], [11], [45] or exploit only the fraction of RR interval variations really buffering SAP changes [42]. The use of these EMBSP methods provides baroreflex estimates significantly different (usually smaller) from those based on spectral and cross-spectral IMBSP methods and only partially correlated with them [50].

(7)

measures the is significantly different from zero. degree of linear association between rr and sap series independently of the causal direction of the interactions and it ranges from zero (perfect uncorrelation) to one (perfect corindicates that the two relation). Therefore, series exchange a certain amount of information, but nothing can be said about the direction of the information exchange: can be the result of the activation of indeed, the feed-forward, mainly mechanical, path (from rr to sap) or of the feedback baroreflex path (from sap to rr) or of both is (closed-loop interactions). The significance of decided by checking whether it exceeds a threshold at the frequencies of interest (i.e., LF and HF in cardiovascular variabilities). This threshold can be frequency-independent (basically a large and predetermined value, i.e., 0.5 [43] or a value derived by taking into account the degrees of freedom of power spectral analysis [44]) or frequency-dependent and derived from a surrogate data approach as in Fig. 3(d) (dotted line) [45], [46]. The implicit model underlying this IMBSP method is single-input single-output, linear time-invariant with a noise additive to rr and independent of sap changes (Fig. 3(a), in the case in which transfer function estimation is derived as the ratio of the power spectra). This approach assumes an open loop model (i.e., RR series cannot contribute to SAP changes). Under this assumption, measures the degree of the linear association along the causal direction from sap to rr because the reverse causal path is excluded. This assumption is manifestly erroneous as a prolongation of the RR interval determines a decrease of diastolic arterial pressure and probably an SAP decrease (if not completely compensated by an increase of contractility). In other words, this method ascribes to the feedback the part of correlation due to the disregarded feed-forward path. As a consequence, a reliable estimation of the baroreflex sensitivity can be obtained from the transfer function approach only when the contribution of the feed-forward arm is actually negligible (i.e., if RR interval actually buffers SAP changes). In addition, the method neglects completely the influences of respiration on RR interval not mediated by baroreflex (e.g., due to respiratory-related modulations of efferent vagal activity independent of baroreflex [28], to cardiopulmonary reflexes activated by respiratory-related changes of the intrathoracic pressure [47], [48] and to periodical mechanical stretch of the sinus node tissue [49]). According to these observations more sophisticated EMBSP approaches (see Sections IV-A and IV-C) have been utilized. They explicitly take into account the closed-loop nature of the RR-SAP relationship and the influences of respiration 810

IV. APPLICATIONS OF THE EMBSP APPROACH TO SHORT-TERM CARDIOVASCULAR REGULATION A. Multivariate Linear Models for the Evaluation of Short-Term Cardiovascular Regulation One of the main advantages of the EMBSP methods lays in the description of the interactions among several, contemporaneously recorded, signals. Linear time-invariant multivariate models have been largely utilized to describe the complexity of the interactions in short term analysis of cardiovascular series [9], [11], [42], [51]–[56]. In the vast majority of the applications they belong to the class of multivariate dynamic adjustment (MDA) models [57] defined in the time domain as

(8)

where y is the vector of the signals, u is the vector of the AR noises described as in (2) as

(9)

with and A(z) is the polynomials of order p

matrix of

(10)

outside the main diagonal (

, with

) and

(11)

along the main diagonal. Therefore, the elements of are causal finite impulse response filters linking the current to past samples of (if , sample describes the autoregression of over its own past). If all are modeled as white noises, the MDA model is residuals reduced to a multivariate AR model (MAR) (see an application in Section V-B). The MDA class can be customized [12] by deciding: 1) which interactions are present (i.e., which with ); 2) which auto-loops are present (i.e., which ); 3) which signals are exogenous , but for any (i.e., if is exogenous, PROCEEDINGS OF THE IEEE, VOL. 94, NO. 4, APRIL 2006

Fig. 4. (a) MDA model describing the closed-loop rr-sap interactions and the effect on both of respiration as proposed by Baselli et al. [9]. (b)–(d) RR, SAP, and respiration series from a healthy human at rest. (e) Response of the block A to a unitary ramp mimicking the rise of SAP (triangles) and its best linear fit (solid line). The slope of the best fit over the first 15 points is taken as a measure of the baroreflex sensitivity (here 11.7 ms/mmHg).

); 4) which immediate effects are present (i.e., which with ); 5) which delay in the closed-loops and where (e.g., if acts on with a delay then with ); and 6) which residual is white. This tuning of the model structure allows one to easily introduce a priori information, thus rendering the model more specific to the particular application. Using the representation in the domain, the MDA model becomes (12) Therefore, the joint process y can be seen as the output of the linear time-invariant model described by the transfer fed by a vector of white noises with function matrix (13) is a where I is the identity matrix and diagonal matrix containing along the main diagonal in position (m,m) and 0 outside the main are single-input single-output diagonal. The elements transfer functions with both zeroes and poles typical of autoregressive moving average (ARMA) processes. For example, the model proposed by Baselli et al. [9] [Fig. 4(a)] is described by

(14)

is the zero mean respirawhere tory series derived from sampling the respiratory signal at the first peak defining RR . This structure allows to describe the closed-loop interactions between rr and sap by means of and , the self-regulation of sap via the blocks , the exogenous influences of r on rr and sap series and but and ( ), the fast vagal action via the baroreflex , the delay in rr-sap closed loop is inserted in the block by setting with ), the regulatory action of rhythmical inputs external to rr-sap closed-loop inand . teractions via B. Classification of Different Sources of Cardiovascular Variability One major issue in the study of cardiovascular variabilities is to link oscillations to mechanisms (i.e., oscillation sources) that are capable to produce fluctuations. According to Koepchen [58] these mechanisms can be subdivided mainly in two classes: 1) feedback circuits (e.g., baroreflex, chemoreflex and brain perfusion control systems) capable to produce oscillations due to resonance properties or to the delay inherent to circuit and 2) native self-sustained oscillators (e.g., vasomotor and respiratory centers, peripheral vasomotion). Identifying and distinguishing these mechanisms passes through the possibility to mark oscillations not only based on their frequency, but also on the specific mechanism producing that rhythm. None of the IMBSP approaches allows this type of classification as mechanisms are not precisely modeled. Labeling rhythms is very easy in the EMBSP approach based e.g., on the MDA model by marking poles (i.e., the entities responsible for describing

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oscillations in the MDA model as it happens in the AR model, see Section III-A, although in the MDA model also the zeroes play a role in shaping fluctuations). As clearly shown from (13) in a MDA model the poles are the zeroes , of the determinant of the transfer function matrix termed closed-loop poles (cl-poles) as they are generated from interactions of all the series, and the zeroes of each , termed -poles as they describe oscillations and generated outside the model present in the input different classes of poles can be structure. Therefore, classified [12]: one class of cl-poles that can be associated with feedback circuits described by the MDA model and classes of -poles that can be associated with the activity of autochthonous oscillators acting on the MDA structure. If signals are exogenous (i.e., they act on the first signals but they cannot be influenced by) and uncorrelated can be factoreach other, the determinant (det) of and ized as the product of the determinant of as

(15) represents the transfer funcIn this case tion matrix describing the interactions among the first codescribes the transfer operating signals and . According to function generating the exogenous signal new classes of poles, this structure it can be detected termed as -poles and describing the oscillations of the exogenous signals, one class ) and classes of cl-poles (i.e., the zeroes of -poles [12]. For example, in the MDA of model proposed by Baselli et al. [9] [see (14) and Fig. 4(a)] r series is exogenous for rr and sap variabilities and poles are -poles, -poles, and r-poles (i.e., labeled as cl-poles, they are the zeroes of , , and , respectively). The pole classification evidenced that cl-poles are always present in the LF band both in humans [59] and -poles were found at rest in dogs [11], [60]. In humans in half of the subjects and less during sympathetic activation induced by standing and mild bicycle exercise [59]. In dogs -poles were repeatedly found during the sympathetic activation induced by coronary artery occlusion [11], [60] and in humans they were often detected and their importance, in terms of amplitude of the correspondent oscillations, increased during mild bicycle exercise [59].

lengthening [Fig. 4(e)] is taken as an estimate of the baroreflex sensitivity [9]. In dogs this closed-loop index termed , was about 14 ms/mmHg at control and decreased after nitroglycerine infusion (this drug produces tachycardia as a reflex response of SAP decrease) and coronary artery occlusion (this intervention induces tachycardia leaving SAP unchanged) and it was practically abolished after total aortic baroreceptor denervation (this intervention opens the RR-SAP loop at the level of the baroreceptor feedback) was around 5 ms/mmHg at rest and [11]. In humans only slightly decreased after standing and significantly and progressively decreased during incremental bicycle exercise [59]. V. JOINT APPLICATIONS OF IMBSP AND EMBSP APPROACHES TO SHORT-TERM CARDIOVASCULAR REGULATION A. Multivariate Partial Spectral Decomposition of Interacting Cardiovascular Signals An IMBSP method like spectral analysis in connection with an EMBSP approach based on MDA modeling permits to derive spectral indexes that are more insightful than monovariate ones derived in Section III-A. Indeed, the power spectrum of a signal can be decomposed in a sum of partial power spectra due to each input of the MDA model. In the MDA model the spectral density matrix (16) is the transpose of and is the variwhere , ance matrix of w, exhibits the auto-spectrum of , along the main diagonal and the cross-spectrum between and , , outside the main diagonal. Under the assumption of uncorrelation between each white noise pair, the has the form of (i.e., it shows variance matrix , , along the main diagonal and zero the variance of outside the main diagonal). Therefore, each power spectrum can be decomposed [12] as

(17) and each cross-spectrum

as

C. Closed-Loop Evaluation of Baroreflex Gain The EMBSP approach can be utilized to improve the baroreflex gain estimation by explicitly taking into account the RR-SAP closed-loop interactions and the influences of respiration on both SAP and RR interval variabilities [Fig. 4(a)] [9]. After identifying all the blocks of the model is via generalized least squares [57], [61], the block fed by an artificial unity ramp simulating the SAP rise and the slope of its response describing the correspondent RR 812

(18) Instead of decomposing each in the frequency domain, decomposition can be carried out in the time domain as

(19)

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Fig. 5. (a) Model decomposing beat-to-beat RT variability series in RR-related and RR-unrelated parts as proposed by Porta et al. [62]. (b) and (c) RR and ) RT series from a healthy human at rest. (d) Power spectrum of RT series (S ) is the sum of the partial power spectra of RT series due to RR changes (S . (e) and (f) S and S , respectively. and independent of RR fluctuations (S

where represents the partial process due to white noise and it is described as (20) with (21)

For example, power spectral decomposition has been applied in connection with the MDA model [Fig. 5(a)] that describes the dynamical interactions between RR interval [Fig. 5(b)] and ventricular repolarization duration [approximated by the time distance between R and T wave apexes, Fig. 5(c)] variabilities [62]. In this MDA model

B. Coherence and Causal Coherence Analyses and Their Implication in the Evaluation of the Baroreflex Gain indicates the amount As reported in Section III-C, of information exchanged along the baroreflex path (from sap to rr) only if the assumption of the open loop relationship depicted in Fig. 3(a) is verified. However, this hypothesis cannot be checked through the IMBSP methods proposed in Section III-C and it can be imposed only by opening pharmacologically or surgically the feedforward path from rr to sap [63]. On the contrary, the use of an EMBSP technique based, for example, on a MAR model like the bivariate AR model, can permit to check whether the exchange of information occurs along the baroreflex path (i.e., the assumption of the IMBSP methods of Section III-C) [45] and might allow a more faithful estimation of the baroreflex sensitivity [64]. The bivariate AR model [Fig. 6(a)] is defined by

(23) (22) In healthy humans at rest, it was found that [62] RR-unrelated RT variability is negligible in LF and HF bands [Fig. 5(f)] and conversely RR-related RT variability is dominant [Fig. 5(e)]. On the contrary, the contribution of RR-unrelated RT variability is large at very low frequency [Fig. 5(f)]. A maneuver capable of inducing reflex sympathetic activation like the 90 head-up tilt did not reduce the tight link between RR and RT variabilities and increased the percentage of RR-related RT variability in the LF band [62].

Based on this model (7) becomes (24) (at bottom of the can be accordingly derived as next page) and the (25) (at bottom of the next page) Equations (24) and (25) clearly evidence that both the and contribute to the calculation of blocks coherence and transfer function from sap to rr, thus mixing feed-forward and feedback paths and preventing to privilege neither directions of interactions [45]. However, thank to the model structure, this problem can be overcome simply by switching off the block describing the feed-forward path ), thus artificially opening the closed loop. (i.e.,

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Fig. 6. (a) Explicit model underlying the calculation of K and H : both the mechanical feed-forward (A ) and the baroreflex feedback (A ) are modeled with the two blocks describing the autoregression of rr and sap on its own past values (A and A , respectively) as proposed in Porta et al. [45]. (b) After identifying the parameters of the bivariate AR model, the block A is constrained to 0 and jH j (solid line), K (dashed line) and its threshold of significance based on a surrogate data approach (dotted line) are derived. jH j is sampled at LF and HF (solid circles) as detected on SAP power spectrum. Only at HF K (HF) is larger than the threshold of significance. jH (HF)j is 7 ms/mmHg.

Therefore, a measure of the information exchange along the baroreflex path, termed casual coherence function from sap to rr [45], is defined as (26) and, consequently, the RR-SAP transfer function that excludes the feed-forward mechanisms [64] is (27) Equation (26) can be used even in connection with traditional estimates of the baroreflex sensitivity based on the IMBSP methods of Section III-C [17], [18] to check the assumption of open loop relationship from sap to rr. Fig. 6(b) (solid line), (dashed shows an example of line) and its threshold of significance as a function of the frequency (dotted line) calculated over the same variabiliis larger ties series depicted in Fig. 3(b) and (c). than the threshold only at HF, thus indicating that RR and SAP variability series are linked along the baroreflex path at HF but not at LF (as detected on SAP power spectrum) and stressing that baroreflex circuit might not essential for conHF is veying LF rhythms [65]. While in Fig. 3(d)

HF is slightly smaller 8 ms/mmHg, in Fig. 6(b) (7 ms/mmHg). Causal coherence analysis indicated that at rest in humans, in the LF band, RR-SAP variability interactions occurred mainly along the feed-forward path, while the feedback path gained more importance during head-up tilt [66]. A more balanced situation between the two paths was observed in the HF band [66]. The correspondent baroreflex gains were different from those derived from simpler IMBSP approaches described in Section III-C, thus suggesting that part of the variability observed in spontaneous baroreflex estimates [67] might be explained in terms of different capabilities of the method to account for causality [42]. C. Analysis of the Residuals After Multivariate Model Identification An EMBSP approach always permits the calculation of the residuals (i.e., the difference between the original signals and those predicted or simulated by the model). In MDA models (Section IV-A) residual w can be easily derived by whitening the data y as (28) where is the transfer function matrix defined in (13). The calculation of the residuals is carried out after the iden-

(24)

(25)

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tification of . The hypothesis that the residuals are white and uncorrelated each other has to be checked to ensure that model structure has captured the dynamics of the series and their interactions (in other words it has to be checked that is diagonal). Therefore, an IMBSP the variance matrix approach in the time domain is usually applied. This method and checks that the normalized autocorrelation cross correlation between residuals is zero for (even at in the case of ) [12], [68]. However, other IMBSP techniques might be fruitful to apply to residuals. Indeed, if nonlinear interactions among cardiovascular variabilities are present, a linear MDA model cannot capture them and evidence of them may remain in the residuals. Therefore, a comparison of the residuals with ad hoc surrogates might allow the rejection of the null hypothesis of linear interactions among variabilities. D. Evaluation of Synchronization Between Several Peripheral Vascular Districts Interactions among various active components (neural circuits, systemic reflexes, and peripheral regulation mechanisms), being the source of the complexity of autonomic cardiovascular regulation [69], cannot be explored generally using linear tools (e.g., in the case of synchronization). In this perspective, IMBSP methods based on nonlinear local prediction [70]–[72] or conditional entropy [73], [74] can be powerful tools to explore causal interdependencies, synchronization and coordination. The common feature to all these applications is the use of past samples of a signal (e.g., rr rr rr rr ) to foresee ) and the the next value of another signal (e.g., sap exploitation of the degree of unpredictability, quantified via MSPE [70]–[72] or conditional entropy [73], [74] to measure uncoupling. These applications make the same assumption reported in Section III-B when considering that, now, maps past samples of a signal onto one the function sample of another one and require the same care as in Section III-B concerning the validation of prediction (or the calculation of sample entropy). In the following lines we present an example of application of a IMBSP method based on conditional entropy [73] to evaluate synchronization between data produced by a simulation model which tries to mimic (though in a very simplified way) the complexity of the regulation of peripheral flow distribution and its effects on systemic parameters without and with a neural modulation [75]. This example stresses again the advantage of applying IMBSP methods to EMBSP structures to quantify the modeled interactions without giving up to the interpretation the underlying complex mechanisms. Briefly, in this example eight peripheral vascular districts (PVDs) are considered at the leaves of a resistant/compliant arterial tree. Each PVD contains a nonlinear feedback for flow compensation against pressure changes. As to details, reference is made to [75]. PVDs were shown to have the capability of developing spontaneous LF oscillation (mimicking periodic vasomotion). Therefore, PVDs are represented as LF oscillators in Fig. 7(a). PVDs were tuned

Fig. 7. (a) Simulation model of arterial dephasing/rephasing mechanism contributing to LF AP wave suppression/enhancement. See [75] for details. Dephased data are generated with neural modulation off, while partially rephasing is induced by switching on a neural modulation. (b) and (c) Superposed flow signals at the third and eighth leaves (solid and dotted lines) with neural modulation off and on, respectively. (d) Uncoupling function UF with neural modulation off: UF is flat, thus indicating that the two flows are uncoupled. (e) Uncoupling function UF with neural modulation on: UF exhibits a deep minimum, thus indicating that the two flows are synchronized. (f) The time course of the uncoupling index UI (i.e., the UF minimum) over the entire simulation. The instant of application of the neural input is marked with an arrow (at 400 s). Uncoupling index UI is close to one before applying the neural input and significantly lower after.

to natural frequencies of 0.1 Hz with a 20% random dispersion. The behavior of PVDs, attached to common arterial compliances and competing for local flow (weak coupling mechanism), was shown to privilege phase opposition in the case of two PVDs and phase dispersion in the case of many. Consequently, vasomotion is masked by negative interference at systemic level. Conversely, systemic AP waves appear when dephasing is disrupted by a common neural periodical outflow. This partial synchronization was proposed as a mechanism contributing to the enhancement of AP waves at LF [75]. In the simulated data set, the system is started without any neural input to the vessels. After a transient lasting about 100 s, peripheral activity sets on a dephased behavior [see, e.g., the flows at the third and eighth leaves in Fig. 7(b)] with negligible AP waves (not shown in Fig. 7) though the powerful oscillations in peripheral flows. After 400 sec a sinusoidal

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input (0.1 Hz, amplitude 30% of baseline) is applied. Interestingly, this input is not sufficient to create synchronization. However, the partial disruption of dephasing and the partial synchronization of small PVDs groups [e.g., the flows at the third and eighth leaves become more synchronized, Fig. 7(c)] are capable to provide positive interference sufficient to create noticeable AP waves (not shown in Fig. 7). Quantification of coupling is performed via an uncoupling function [73] ranging from zero (full synchronization) to one (full uncoupling). Before switching on the neural modulation, the uncoupling function is flat, thus indicating that the increase of the number of past samples of one signal (i.e., the pattern length ) is not helpful in reducing the uncertainty about future values of the other one and, thus, the two flows are uncoupled [Fig. 7(d)]. On the contrary, after switching on the periodical neural input, the uncoupling function exhibits an important minimum [Fig. 7(e)], thus stressing that the use of past samples of one signal is helpful in reducing the entropy of the other one and in improving prediction. Fig. 7(f) shows the iterate application of this synchronization analysis over the entire simulation (the analysis was carried out on signal windows of 300 s moved ahead in steps of 1 s). This application allows the continuous monitoring of the coupling between the two flow signals. Before switching on the neural input at 400 s [this event is marked with an arrow in Fig. 7(f)], the uncoupling index (i.e., the minimum of the uncoupling function) is close to one. After the application of the neural modulation the uncoupling index starts decreasing (actually it starts decreasing 150 s before the arrow, since signal windows at that time start including samples resulting from the influence of neural input). Maximum synchronization is abruptly reached at 560 s. VI. CONCLUSION IMBSP and EMBSP methods should not be applied separately but jointly in order to maximize the information that can be derived from data. Indeed, this combined use might allow the exploitation of the possibility offered by the IMBSP approach to derive descriptive indexes over a specific structure describing the signal interactions set by the EMBSP approach. This joint application might be particularly fruitful when analyzing multivariate recordings and when the underlying system is composed by several interacting and/or competing subsystems as it occurs in the cardiovascular control system. REFERENCES [1] M. A. Cohen and J. A. Taylor, “Short-term cardiovascular oscillations in man: Measuring and modeling the physiologies,” J. Physiol., vol. 542, pp. 669–683, 2002. [2] H. P. Koepchen, “History of studies and concepts of blood pressure waves,” in Mechanisms of Blood Pressure Waves, K. Miyakawa, C. Polosa, and H. P. Koepchen, Eds. Berlin, Germany: SpringerVerlag, 1984, pp. 3–23. [3] G. Baselli, E. Caiani, A. Porta, N. Montano, M. G. Signorini, and S. Cerutti, “Biomedical signal processing and modeling in cardiovascular systems,” Crit. Rev. Biomed. Eng., vol. 30, pp. 55–84, 2002.

816

[4] D. E. Burgess, J. D. Hundley, S.-G. Li, D. C. Randall, and D. R. Brown, “First-order differential-delay equation for the baroreflex predicts the 0.4 Hz blood pressure rhythm in rats,” Amer. J. Physiol., vol. 273, pp. R1878–R1884, 1997. [5] J. V. Ringwood and S. C. Malpas, “Slow oscillations in blood pressure via a nonlinear feedback model,” Amer. J. Physiol., vol. 280, pp. R1105–R1115, 2001. [6] A. C. Guyton and J. H. Harris, “Pressoreceptor-autonomic oscillation: A probable cause of vasomotor waves,” Amer. J. Physiol., vol. 165, pp. 158–166, 1951. [7] K. Sagawa, O. Carrier, and A. C. Guyton, “Elicitation of theoretically predicted feedback oscillation in arterial pressure,” Amer. J. Physiol., vol. 203, pp. 141–146, 1962. [8] R. W. De Boer, J. M. Karemaker, and J. Strackee, “Hemodynamic fluctuations and baroreflex sensitivity in humans: A beat-to-beat model,” Amer. J. Physiol., vol. 253, pp. H680–H689, 1987. [9] G. Baselli, S. Cerutti, S. Civardi, A. Malliani, and M. Pagani, “Cardiovascular variability signals: Toward the identification of a closed-loop model of the neural control mechanisms,” IEEE Trans. Biomed. Eng., vol. 35, no. 12, pp. 1033–1046, Dec. 1988. [10] R. I. Kitney, T. Fulton, A. H. McDonald, and D. A. Linkens, “Transient interactions between blood pressure, respiration and heart rate in man,” J. Biomed. Eng., vol. 7, pp. 217–224, 1985. [11] G. Baselli, S. Cerutti, F. Badilini, L. Biancardi, A. Porta, M. Pagani, F. Lombardi, O. Rimoldi, R. Furlan, and A. Malliani, “Model for the assessment of heart period and arterial pressure variability interactions and respiratory influences,” Med. Biol. Eng. Comput., vol. 32, pp. 143–152, 1994. [12] G. Baselli, A. Porta, O. Rimoldi, M. Pagani, and S. Cerutti, “Spectral decomposition in multichannel recordings based on multivariate parametric identification,” IEEE Trans. Biomed. Eng., vol. 44, no. 11, pp. 1092–1101, Nov. 1997. [13] S. Akselrod, D. Gordon, F. A. Ubel, D. C. Shannon, R. D. Berger, and R. J. Cohen, “Power spectrum analysis of heart rate fluctuations: A quantitative probe of beat-to-beat cardiovascular control,” Science, vol. 213, pp. 220–223, 1981. [14] P. G. Katona and F. Jih, “Respiratory sinus arrhythmia: Noninvasive measure of parasympathetic cardiac control,” J. Appl. Physiol., vol. 39, pp. 801–805, 1975. [15] M. Pagani, F. Lombardi, S. Guzzetti, O. Rimoldi, R. Furlan, P. Pizzinelli, G. Sandrone, G. Malfatto, S. Dell’Orto, E. Piccaluga, M. Turiel, G. Baselli, S. Cerutti, and A. Malliani, “Power spectral analysis of heart rate and arterial pressure variabilities as a marker of sympatho-vagal interaction in man and conscious dog,” Circulat. Res., vol. 59, pp. 178–193, 1986. [16] Task force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology, “Standard of measurement, physiological interpretation and clinical use,” Circulation, vol. 93, pp. 1043–1065, 1996. [17] M. Pagani, V. K. Somers, R. Furlan, S. Dell’Orto, J. Conway, G. Baselli, S. Cerutti, P. Sleight, and A. Malliani, “Changes in autonomic regulation induced by physical training in mild hypertension,” Hypertension, vol. 12, pp. 600–610, 1988. [18] H. W. J. Robbe, L. J. M. Mulder, H. Ruddel, W. A. Langewitz, J. B. P. Eldman, and G. lder, “Assessment of baroreceptor reflex sensitivity by means of spectral analysis,” Hypertension, vol. 10, pp. 538–543, 1987. [19] J. L. Coatrieux, “Issues on signal processing and physiological modeling. Part I: Surface analysis,” Crit. Rev. Biomed. Eng., vol. 30, pp. 9–35, 2002. [20] ——, “Issues on signal processing and physiological modeling. Part II: Depth model-driven analysis,” Crit. Rev. Biomed. Eng., vol. 30, pp. 37–54, 2002. [21] ——, “Integrative science: A modeling challenge,” IEEE Eng. Med. Biol. Mag., vol. 23, no. 1, pp. 12–14, Jan.–Feb. 2004. [22] ——, “Integrative science: Biosignal processing and modeling,” IEEE Eng. Med. Biol. Mag., vol. 23, no. 3, pp. 9–12, May–Jun. 2004. [23] ——, “Integrative science: Place and future of the model-based information processing,” IEEE Eng. Med. Biol. Mag., vol. 23, no. 4, pp. 19–21, Jul.–Aug. 2004. [24] M. Bland, An Introduction to Medical Statistics. New York: Oxford Univ. Press, 2000. [25] S. M. Kay and S. L. Marple, “Spectrum analysis: A modern perspective,” Proc. IEEE, vol. 69, no. 11, pp. 1380–1418, Nov. 1981. [26] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge: Cambridge Univ. Press, 1997.

PROCEEDINGS OF THE IEEE, VOL. 94, NO. 4, APRIL 2006

[27] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. [28] J. A. Taylor and D. L. Eckberg, “Fundamental relations between short-term RR interval and arterial pressure oscillations in humans,” Circulation, vol. 93, pp. 1527–1532, 1996. [29] L. H. Zetterberg, “Estimation of parameters for a linear difference equation with application to ECG analysis,” Math. Biosci., vol. 5, pp. 227–275, 1979. [30] F. Takens, “Detecting strange attractors in fluid turbolence,” in Lecture Notes in Mathematics, D. Rand and L. S. Young, Eds. Berlin, Germany: Springer-Verlag, 1981, p. 366. [31] J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett., vol. 59, pp. 845–848, 1987. [32] G. Sugihara and R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series,” Nature, vol. 344, pp. 734–741, 1990. [33] K. Kanters, M. V. Hojgaard, E. Agner, and N.-H. Holstein-Rathlou, “Short- and long-term variations in nonlinear dynamics of heart rate variability,” Cardiovasc. Res., vol. 31, pp. 400–409, 1996. [34] A. Porta, G. Baselli, S. Guzzetti, M. Pagani, A. Malliani, and S. Cerutti, “Prediction of short cardiovascular variability signals based on conditional distribution,” IEEE Trans. Biomed. Eng., vol. 47, no. 12, pp. 1555–1564, Dec. 2000. [35] H. Akaike, “A new look at the statistical novel identification,” IEEE Trans. Autom. Control, vol. AC–19, no. 6, pp. 716–723, Dec. 1974. [36] J. Theiler, S. Eubank, A. Longtin, and J. Galdrikian, “Testing for nonlinearity in time series: The method of surrogate data,” Physica D, vol. 58, pp. 77–94, 1992. [37] D. T. Kaplan and L. Glass, Understanding Nonlinear Dynamics. New York: Springer-Verlag, 1995. [38] K. Kanters, N.-H. Holstein-Rathlou, and E. Agner, “Lack of evidence for low-dimensional chaos in heart rate variability,” J. Cardiovasc. Electrophysiol., vol. 5, pp. 591–601, 1994. [39] T. Schreiber and A. Schmitz, “Improved surrogate data for nonlinearity tests,” Phys. Rev. Lett., vol. 4, pp. 635–638, 1996. [40] D. Kugiumtzis, “Testing your surrogate data before you test for nonlinearity,” Phys. Rev. E, vol. 60, pp. 2808–2816, 1999. [41] J. P. Saul, R. D. Berger, P. Albrecht, S. P. Stein, M. H. Chen, and R. J. Cohen, “Transfer function analysis of the circulation: Unique insights into cardiovascular regulation,” Amer. J. Physiol., vol. 261, pp. H1231–H1245, 1991. [42] A. Porta, G. Baselli, O. Rimoldi, A. Malliani, and M. Pagani, “Assessing baroreflex gain from spontaneous variability in conscious dogs: Role of causality and respiration,” Amer. J. Physiol., vol. 279, pp. H2558–H2567, 2000. [43] R. W. De Boer, J. M. Karemaker, and J. Strackee, “Relationships between short-term blood pressure fluctuations and heart rate variability in resting subjects I: A spectral analysis approach,” Med. Biol. Eng. Comput., vol. 23, pp. 352–358, 1985. [44] J. A. Taylor, D. L. Carr, C. W. Myers, and D. L. Eckberg, “Mechanisms underlying very-low frequency RR-interval oscillations in humans,” Circulation, vol. 98, pp. 547–555, 1998. [45] A. Porta, R. Furlan, O. Rimoldi, M. Pagani, A. Malliani, and P. van de Borne, “Quantifying the strength of the linear causal coupling in closed loop interacting cardiovascular variability signals,” Biol. Cybern., vol. 86, pp. 241–251, 2002. [46] L. Faes, G. D. Pinna, A. Porta, R. Maestri, and G. Nollo, “Surrogate data analysis for assessing the significance of the coherence function,” IEEE Trans. Biomed. Eng., vol. 51, no. 7, pp. 1156–1166, Jul. 2004. [47] T. H. Desai, J. C. Collins, M. Snell, and R. Moscheda-Garcia, “Modeling of arterial and cardiopulmonary baroreflex control of heart rate,” Amer. J. Physiol., vol. 272, pp. H2343–H2352, 1997. [48] D. Lucini, A. Porta, O. Milani, G. Baselli, and M. Pagani, “Assessment of arterial and cardiopulmonary baroreflex gains from simultaneous recordings of spontaneous cardiovascular and respiratory variability,” J. Hypertens., vol. 18, pp. 281–286, 2000. [49] L. Bernardi, F. Keller, M. Sanders, P. S. Reddy, B. Griffhth, F. Meno, and R. Pinsky, “Respiratory sinus arrhythmia in the denervated human heart,” J. Appl. Physiol., vol. 67, pp. 1447–1455, 1989.

[50] D. Laude, J. L. Elghozi, A. Girard, F. Bellard, M. Bouhaddi, P. Castiglioni, C. Cerutti, A. Cividjian, M. di Rienzo, J. O. Fortrat, B. Janssen, J. M. Karemaker, G. Leftheriotis, G. Parati, P. B. Persson, A. Porta, L. Quintin, J. Regnard, H. Rudiger, and H. M. Stauss, “Comparison of various techniques used to estimate spontaneous baroreflex sensitivity (the EuroBaVar study),” Amer. J. Physiol., vol. 286, pp. R226–R231, 2004. [51] M. H. Perrott and R. J. Cohen, “An efficient approach to ARMA modeling of biological systems with multiple inputs and delays,” IEEE Trans. Biomed. Eng., vol. 43, no. 1, pp. 1–14, Jan. 1996. [52] D. J. Patton, J. K. Triedman, M. H. Perrott, A. A. Vidian, and J. P. Saul, “Baroreflex gain: Characterization using autoregressive moving average analysis,” Amer. J. Physiol., vol. 270, pp. H1240–H1249, 1996. [53] J. K. Triedman, M. H. Perrott, R. J. Cohen, and J. P. Saul, “Respiratory sinus arrhythmia: Time domain characterization using autoregressive moving average analysis,” Amer. J. Physiol., vol. 268, pp. H2232–H2238, 1995. [54] T. J. Mullen, M. L. Appel, R. Mukkamala, J. M. Mathias, and R. J. Cohen, “System identification of closed loop cardiovascular control: Effects of posture and autonomic blockade,” Amer. J. Physiol., vol. 272, pp. H448–H461, 1997. [55] R. Mukkamala, K. Toska, and R. J. Cohen, “Noninvasive identification of the total peripheral resistance baroreflex,” Amer. J. Physiol., vol. 284, pp. H947–H959, 2003. [56] A. Porta, N. Montano, M. Pagani, A. Malliani, G. Baselli, V. K. Somers, and P. van de Borne, “Non-invasive model-based estimation of the sinus node dynamic properties from spontaneous cardiovascular variability series,” Med. Biol. Eng. Comput., vol. 41, pp. 52–61, 2003. [57] T. Soderstrom and P. Stoica, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1988. [58] H. P. Koepchen, “Physiology of rhythms and control systems: An integrative approach,” in Rhythms in Physiological Systems, H. Haken and H. P. Koepchen, Eds. Berlin, Germany: SpringerVerlag, 1991, pp. 3–20. [59] G. Baselli, A. Porta, S. Cerutti, E. Caiani, D. Lucini, and M. Pagani, “RR-arterial pressure variability relationships,” Autonom. Neurosci., vol. 90, pp. 57–65, 2001. [60] G. Baselli, A. Porta, G. Ferrari, S. Cerutti, M. Pagani, and O. Rimoldi, “Multi-variate identification and spectral decomposition for the assessment of cardiovascular control,” in Computer Analysis of Blood Pressure and Heart Rate Signals, M. di Rienzo, G. Mancia, G. Parati, A. Pedotti, and A. Zanchetti, Eds. Amsterdam, The Netherlands: IOS, 1995, pp. 95–104. [61] L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1999. [62] A. Porta, G. Baselli, E. Caiani, A. Malliani, F. Lombardi, and S. Cerutti, “Quantifying electrocardiogram RT-RR variability interactions,” Med. Biol. Eng. Comput., vol. 36, pp. 27–34, 1998. [63] S. Akselrod, D. Gordon, J. B. Madwed, N. C. Snidman, D. C. Shannon, and R. J. Cohen, “Hemodynamic regulation: Investigation by spectral analysis,” Amer. J. Physiol., vol. 249, pp. H867–H875, 1985. [64] L. Faes, A. Porta, R. Cucino, S. Cerutti, R. Antolini, and G. Nollo, “Causal transfer function analysis to describe closed loop interactions between cardiovascular an cardiorespiratory variability signals,” Biol. Cybern., vol. 90, pp. 390–399, 2004. [65] A. Malliani, M. Pagani, F. Lombardi, and S. Cerutti, “Cardiovascular neural regulation explored in the frequency domain,” Circulation, vol. 84, pp. 482–492, 1991. [66] G. Nollo, L. Faes, A. Porta, R. Antolini, and F. Ravelli, “Exploring directionality in spontaneous heart period and systolic arterial pressure variability interactions in humans: Implications in the evaluation of baroreflex gain,” Amer. J. Physiol., vol. 288, pp. H1777–H1785, 2005. [67] R. D. Lipman, J. K. Salisbury, and J. A. Taylor, “Spontaneous indexes are inconsistent with arterial baroreflex gain,” Hypertension, vol. 42, pp. 481–487, 2003. [68] S. M. Kay, Modern Spectral Analysis: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1989. [69] A. Malliani, Principles of Cardiovascular Neural Regulation in Health and Disease. Boston, MA: Kluwer, 2000. [70] H. D. I. Abarbanel, T. L. Carroll, L. M. Pecora, J. J. Sidorowich, and L. S. Tsimring, “Predicting physical variables in time-delay embedding,” Phys. Rev. E, vol. 49, pp. 1840–1853, 1994.

PORTA et al.: MODEL-BASED SIGNAL PROCESSING FOR THE ANALYSIS OF SHORT-TERM CARDIOVASCULAR INTERACTIONS

817

[71] R. Q. Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger, “Performance of different synchronization measures in real data: A case study on electroencephalographic signals,” Phys. Rev. E, vol. 65, p. 041 903, 2002. [72] E. Pereda, D. De la Cruz, L. De Vera, and J. J. Gonzalez, “Comparing generalized and phase synchronization in cardiovascular and cardiorespiratory signals,” IEEE Trans. Biomed. Eng., vol. 52, no. 4, pp. 578–583, Apr. 2005. [73] A. Porta, G. Baselli, F. Lombardi, N. Montano, A. Malliani, and S. Cerutti, “Conditional entropy approach for the evaluation of the coupling strength,” Biol. Cybern., vol. 81, pp. 119–129, 1999. [74] A. Porta, S. Guzzetti, N. Montano, M. Pagani, V. K. Somers, A. Malliani, G. Baselli, and S. Cerutti, “Information domain analysis of cardiovascular variability signals: Evaluation of regularity, synchronization and co-ordination,” Med. Biol. Eng. Comput., vol. 38, pp. 180–188, 2000. [75] G. Baselli, A. Porta, and M. Pagani, “Coupling arterial windkessel with peripheral vasomotion: Modeling the effects on low frequency oscillations,” IEEE Trans. Biomed. Eng., vol. 53, no. 1, pp. 53–64, Jan. 2006.

Alberto Porta was born in 1964. He graduated in electronic engineering and received the Ph.D. degree in biomedical engineering from the Politecnico di Milano, Milano, Italy, in 1989 and 1998, respectively. He was a Research Fellow on automatic control and system theory in the Dipartimento di Elettronica per l’Automazione, Universitá di Brescia, Brescia, Italy, from 1989 to 1994. Since 1999, he has been with the Dipartimento di Scienze Precliniche, Universitá degli Studi di Milano. From 1999 to 2004, he taught a course on biomedical instrumentation at the Dipartimento di Elettronica per l’Automazione, Universitá di Brescia. His primary interests include time series analysis, system identification, and modeling applied to cardiovascular control mechanisms.

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Giuseppe Baselli was born in 1958. He received the Italian degree in electronic engineering at the Politecnico di Milano, Milano, Italy, in 1983. He subsequently worked at the Politecnico di Milano as Research Fellow in biomedical signal processing. From 1986 to 1999, he was a Researcher at the Università di Brescia, Brescia, Italy. In 1998, he became Associate Professor in the Bioengineering Department of the Politecnico di Milano and Full Professor in 2001. He is a member of the board of the Ph.D. track in bioengineering, and he is President of the course in biomedical engineering at the same university. His research interests are in the field of biomedical signal and image processing and in its relationships with the linear and nonlinear modeling of physiological systems, mainly applied to the study of autonomic cardiovascular regulation and to neurosensory systems.

Sergio Cerutti (Fellow, IEEE) is Professor of Biomedical Signal and Data Processing in the Dipartimento di Bioingegneria, Politecnico di Milano, Milano, Italy. Since 2000, he has also been Chairman of the same department. Since March 1983, he has also taught a graduate-level course on biomedical signal processing at engineering faculties (Milano and Roma) and at specialization schools of medical faculties (Milano and Roma). His research interests are mainly in the following topics: biomedical signal processing (ECG, blood pressure signal and respiration, cardiovascular variability signals, EEG and evoked potentials), cardiovascular modeling, neurosciences, medical informatics, and regulation and standards in medical equipment and devices. Dr. Cerutti is Associate Editor of IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. He was an Elected Member of the IEEE Engineering in Medicine and Biology Society (IEEE-EMBS) AdCom (Region 8) from 1993 to 1996. He is a member of the Steering Committee of the IEEE-EMBS Summer School on Biomedical Signal Processing; he was the local organizer of three IEEE-EMBS Summer Schools held in Siena, Italy, in 1995, 1999, and 2003.

PROCEEDINGS OF THE IEEE, VOL. 94, NO. 4, APRIL 2006