Linear multivariate models for physiological signal analysis: applications

Computer Methods and Programs in Biomedicine 51 (1996) 121-130

Linear multivariate models for physiological signal analysis: applications

Ilkka Korhonen” a , Luca Mainardib, Giuseppe Basellib, Anna Bianchib, Pekka Loula”, Guy Carrault ’

a VIT Information Technology, Multimedia Systems, Tampere, Finland bPolytechnic University of Milan, Department of Biomedical Engineering Milan, Italy

clJnive~ite de Rennes, Laboratoire Traitement du Signal et de L’lmage, Rennes. France

Abstract

Some applications of linear multivariate modelling methods in the analysis of physiological signals are presented. These applications illustrate the methods in the analysis of cardiovascular dynamics, which has been one of the main application fields of the multivariate modelling during the last ten years. It is demonstrated that physiologically meaningful information about the causal interactions in the cardiovascular system can be drawn from the routinely available clinical signals. Both static and dynamic conditions are considered.

Keywords: Multivariate; Linear; Modelling; Spectral analysis; Spectral decomposition; Dynamic

1. Introduction

During the last decade, multivariate modelling of physiological systems has gained increasing at- tention. Different applications have been proposed, ranging from the description of real physiological systems and phenomena to the utilisation of intersignal correlation to enhance or empha- sise some characteristics of signals such as rupture detection and segmentation.

Among the applications, multivariate modelling of the cardiovascular system function has been

*Corresponding author.

the one of the main application fields [l-5]. In the cardiovascular system, the multiple variables observed are coupled together by the neural and mechanical effects. The traditional spectral analysis does not provide information on the causal relationships between the signals involved. To achieve this information, closed-loop modelling is needed [5,6]. Therefore, multivariate models have been used to extract important parameters relevant to cardiovascular control by modelling the interactions between blood pressure (BP) variability, heart rate (HR) variability and respiration CR) [5,7-121. Other cardiovascular variability signals can be considered, too, such as cardiac output or stroke volume, direct or indirect contractility in-

0169-2607/96/$15.00 0 1996 Elsevier Science Ireland Ltd. All rights reserved. PlISOl69-2607(96)01767-l

122 1. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (1996) 121-130

dexes, venous return, peripheral flow or resis- tances, oxygen saturation, vagal and s~pa~etic discharge, conduction and repolarization dura- tions, etc.

In some other applications, the aim of the modelling is to utilise the intersignal properties of the multivariate data in a way which is not directly related to the physiology. For example, the interlead correlation in the m~tichannel EEG may be utilised to enhance the rupture detection sensitivity [13]. In this case, the model serves the means of data compression, producing some phys- iologica~ arbitrary parameters which depict the properties of the signals. The monitoring of these parameters in time indicates the changes in the system function in a non-specific manner, and the relations to the ph~iolo~ are derived by other methods.

In the present paper, the utilisation of the multivariate linear modelling in the field of physiological signal analysis is presented. The application examples selected use different kinds of multivariate models in the analysis of cardiovascular dynamics. Indeed, one of the main features of multivariate models is the gexibility of the structure and the way in which it can be adapted to the signals available and to the a priori assump- tions to be included. In this paper, a few examples of simple model structures, which have been suc- cessfully employed in the simultaneous description of cardiovascular variability series, will be considered. These model structures belong to the more general classes of the multivariate autoregressive (MAR) models and the multivariate dynamic adjustment (MDA) models which were presented in the companion paper 1141.

2. Applications

2.1. ~~~v~~te autoregressive modelling of cardiovascular interactions

MAR modelling enables the analysis of interactions between all the variables included f14,15]. The utilisation of the MAR model in the analysis of cardiovascular variability signals was proposed by Kalli et al. [%I. Kalli et al. utilised the conventional, strictly causal MAR model structure [14]:

y(k) = - fJ a(i)y(k - i) +e(k) * i= 1

y(k) =A-‘tqletk) (1)

where y represents the set of signals measured, A represents the MAR model coefficients, e is the modelling error, 4 is the unit delay operator, A4 is the model order and k is the sample delay. Because of the strong inter-signal correlation of HR and BP, the noise sources e in the model became dependent, which introduced some error in the modelling results, e.g. noise source contributions [4,17]. This has been reducing the us- ability of the method.

By introducing a modified MAR model the noise sources of the model may be made independent [14,15]. The method utilises the Cholesky decomposition [18] in the transformation of the conventional MAR model into a new one with implicit independent noise sources 1151:

A,,,,(q) = PA(q)

where A,,, is the modified model coefficient matrix and S results from the Cholesky decomposition of the original noise source covariance matrix

C-SDS” (31

S is hence a lower triangular matrix and D (the noise covariance matrix of the new model) is a diagonal matrix implying noise source indepen- dence [15]. This procedure, which transforms the noise source dependence to the new model coefficients, enables the use of spectral de~m~sition and noise source contribution ratio (NSCR) [15] in the analysis of intercorrelated signals.

This kind of MAR modelling may be applied, for example in the analysis of ~te~ctio~ between HR, diastolic BP (DBP), systolic BP @BP) and R during stationary conditions. As an example, stationary (run test, P < 0.05; [19]) signals of five minutes length were recorded from a single subject during quiet supine, sitting (Fig. 1) and stand-

I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (1994) 121-130 123

Fig. 1. Tracings of R, HR, SBP and DBP from a healthy subject during sitting position. R was measured by transthoracic impedance method. DBP and SBP were extracted from a non-invasive continuous BP signal and HR from the R-waves of electrocardiogram (ECG). The sampling rate was 200 Hz for all the signals. After linear interpolation of the consecutive heart beats [20], the time series were low-pass filtered and resampled at 1 Hz.

ing positions. Any low-frequency trends and DC- components were removed from the signals before the analysis. When fitted to the data, the new MAR model with independent noise sources (Eq. 2) contained zero delay transfer paths between the variables to the predefined directions. The directions of the transfer paths on the zero delay were chosen according to the a priori knowledge: R was able to affect all the other variables, HR was able to affect DBP and SBP, and DBP was able to affect SBP. The model was validated a posteriori by computing the prediction ratios (range: 84-96%) [21] and the whiteness of the residuals (P < 0.05) [21].

The NSCR analysis was carried out in two frequency bands: 0.05-0.12 Hz (low frequency, LF) and 0.12-0.35 Hz (high-frequency, HF), corresponding to the vasomotor and respiratory bands, respectively [22]. The NSCR analysis showed that the variability of R was mainly con- tributed by R’s own noise source in both LF (NSCR: 59-70%) and HF (NSCR: 50-67%, data not shown). The NSCR (Fig. 2) analysis of HR, DBP .and SBP indicated changes in the cardiovascular regulation as the subject moved from one

NSCR to HR

OJ’ ” ” ’ I I, I, L SUP SIT STA SUP SIT !3A

NSCR to DBP

SUP SR STA SUP S4T STA

NSCR to SBP l.F HF

SUP SIT !XA SUP SIT STA

0 R •i HR n DBP n SBP

Fig. 2. Noise source contribution ratios to HR, DBP and SBP in supine (SUP), sitting (SIT) and standing (STA) positions describe the relative amount of variability present in each signal that is arising from the other signals. Note the dominance of HR in LF range in supine position, replaced by the dominance of BP (SBP + DBP) in standing position. In HF range, R and HR are the main sources of variability.

position to another. When moving from supine to sitting or standing position the role of BP (DBP + SBP) increased whereas the role of HR de- creased, indicating the shift of weight from HR to BP control mechanisms in the cardiovascular regulation. This can be explained by the increased circulatory demands and therefore increased sympathetic tone [23]. The DBP was more dominant than SBP in the LF range, especially in the standing position. This cannot be clearly explained, since, according to our knowledge, this is the first study in which both SBP and DBP are included in the same closed loop model. Thus, further studies are needed.

124 I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (1996) 121-130

In the HF range, HR was the dominant factor in the supine position in both HR and BP variability. In the sitting and standing positions, the R took the dominant role on the HF range. This means that in the supine position the role of respiratory sinus arrhytmia (NSCR from HR to BP) was more dominant in the respiratory BP variability than the direct mechanical effects (NSCR from R to BP): in the sitting and standing positions, the situation was the opposite. This finding is well in line with the decreasing role of HR in the cardiovascular control when moving from supine to upright position [23]. The effect of R was found to disperse, to some extent, to the LF range, too. This finding is in agreement with some previous studies in which spontaneous breathing is found to affect HR and BP variability in large frequency range, including LF [lo]. This suggests that it is important to control R when analysing HR and BP interactions.

These findings were confirmed by the partial spectral analysis (Fig. 3).

The results drawn can be considered physiologically reasonable. In addition, the objective numerical validation criteria (prediction ratio, whiteness of residuals, stationarity) justify the presumptions made. Hence, MAR model may be considered a reasonable approximation of the cardiovascular dynamics during stationary conditions. The characteristics of the model may be illustrated by the NSCR’s and partial spectra estimates, which provide interpretable information. The MAR model identification may be done by effective standard algorithms to any given set of signals. These facts suggest the conclusion that the MAR modelling method may provide a prac- tical tool for assessing the dynamics of a multivariate physiological system, e.g. during intensive care or anesthesia.

2.2. Multivariate dynamic modelling of cardiovascular control

The main advantage of MDA modelling is the ability to detect and classify different rhythmo- genie sites present in the cardiovascular system. In the following examples, some different kind of MDA models are presented to illustrate this idea.

HR partial spectra

DBP partial spectra

SBP partial spectra

H?!

0 R fiii HR n DBP n SBP

Fig. 3. Partial spectra of HR, DBP and SBP in sitting position. The areas of shadings denote the effects of the variables on each spectrum according to the model Note the dominant effect of R at the respiratory frequency 0.19 Hz. In vasomotor band, around 0.05-0.12 Hz, HR plays the dominant role. The partial spectral analysis may be used to complete the NSCR analysis.

In the applications, some transfer paths are sup- pressed or set to identity according to the knowledge of the cardiovascular system, in order to highlight some peculiar relationships between signals and to quantify the different mechanisms of control of the cardiovascular magnitudes.

The beat-to-beat variability series were extracted from continuous non-invasive recording of ECG, BP and R signals. In particular, the tachogram (t> was obtained as the beat-to-beat

I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 @J96) 121-1.30 125

series of the RR intervals, the s~togram (8) was the series of SBP values, and the respirogram (r) was the series obtained by sampling the respiration signal in correspondence with each R-wave of ECG 151. The identification procedure was carried out according to Baselli et al. IS]. The model order was chosen according to the multivariate extension of Akaike’s information crite- rion f24J and the correctness of the prediction was assessed by verifying the identification hypothesis (i.e. whiteness of each noise, uncorrelation between each pair of inputs, and ~agonali~ of prediction error covariance matrix] via Anderson test f251.

Considering s and t series contemporaneously, the principal characteristic is in their closed-loop interaction, as depicted in Fig. 4a. In fact, t af- fects s through me~hani~l effects (block &,I while baroreflex mechanisms perform a feedback (block tr,,). A muitivariate identification permits the disentangling of the two branches of the loop simply by observing the interactions between the two spontaneous variabilities. After closed-loop identi~ation, the block H,, can be extracted and used as a quantitative model of a cardiac baroreceptive response (Fig. 4b). The simulated ramp of t plotted against the ramp of s (Fig. 4~) permits

Fig. 4. ia) A closed loop mode! for the description of the inte~~tions between t and s. The ~denti~catjon procedure allows the identification of the transfer functions H,, and H,,, thus separating the simultaneous interactions between the two signals. u, and U, account for the external sources of variability. (b) After the identification of the H,, and H,, blocks it is possible to extract the former and use it to simulate the rise in t series (msec) in response to a pressure ramp in the input s (mmH@. fcl The value of LY-cI gain can be obtained by regression of the simulated ramp of I <y-axis) plotted versus the ramp of s (x-axis) (from {I’Z],.

the extra&an of a sIope parameter (cr-cl) which quantiges the block gain.

The simple model in Fig. 5a is particularly suited when the effect of R on HR variability has to be considered in order to provide a quantitative evaluation of respiratory a~h~ia and its spectral content. The model separates the effects af r, taken as an exogenous input, from different sources of sinus a~h~hmia described by the random input. So the spectrum of t is divided into the sum of two partial spectra as depicted by Fig. 5b. The partial spectrum due to I is also referred to as the part of the spectrum of t that is coherent with t (CRP]. This procedure is particularly useful when clear HF and LF rhythms are dif- ficult to recognize; e.g. with a highly reduced variability consequent to diabetic neuropathy [‘7]= In these cases a separation into a non-coherent

a~-I;

0 d 0 .12 0.24 033 0.48 om

% CRP a NCRP l-lZ

Fig. 5, (a) Model of t variability which considers the effect of r as an exogenous input. wt is the white noise input to the system. The block R, estimates the open&q transfer function from r ta t while the block H,, describes the variability of t not coherent to r. (b) The model allows the division of the power spectrum density @‘SE>) of the t variability into two partial spectra which are, resp&iveIy coherent @XP) and not-coherent (NCRP) with respiration.

126 I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (I 996) 12I -130

and a coherent part can substitute the usual decomposition into an LF and a I-IF component, respectively, in order to assess the sympatho-vagal balance.

The model shown in Fig. 6 contains all the elements of the more simple models described above and is specifically designed to study the interactions among t, s and r [S,lO,ll]. The double-loop structure accounts both for the t-s interactions and for the s to s closed loop regulation. Hence, it is possible to describe better the baroreceptive mechanisms (block H,,), to simplify the block H,, to a single mechanical parameter, and to evaluate the resonance of s to s regulation (block H,,). In addition, the model is able to detect the external oscillations which enter to the considered loops and which modulate either t (respiration block R,, and other sinus node mod-

r&?-l Hrl ?A.-!& !-y--+J U*

r

H 0 .. “1% . . . . . . . . . **g wt t

Fig. 6. Model of the interactions between t and s with r as an exogenous input. The basic elements of the models presented in Figs. 4 and 5 may be recognised in this complex structure. The circular arrows indicate the possible sources which may be responsible for the origin of the various rhythms in the cardiovascular variability series. IV,, u, and u, correspond to those of the previous figures, while ws and wr are the white noises acting directly on s and r, respectively. H,, represents the monovariate autoregressive (AR) model of the signal r (ARXAR model structure, see the companion paper [14]), while If,,,, and H,,,, define the monovariate AR models which account for external contributions to t and s, respectively. For the transfer functions H,,, H,,, H,,, R, and R,?, see text (from [l I]).

Fig. 7. Spectral decomposition of the spectrum of t into partial spectra. S,, is the global spectrum. S,,Ut (white area) is the contribution of u, to the global spectrum and may be considered as the effect of the sinus-node modulation (see Fig. 6 for the definition of the different sources of variability). s ,,Us (white area) is the contribution of u,$ to the global spectrum, reflecting both the effect of vasomotor modulation and the mediation of the closed-loop structure, Finally, S,,r (white area> is the contribution of r to the global spectrum (respiratory arrhythmia, denoted by r-pole). In all the figures, the shadowed area evidences the contribution of the closed- loop poles (dl-poles) on the overall variability. The closed-loop is the core of the model and it may generate or amplify certain oscillations (from [I 11).

ulations, block H,,,,) or s (direct respiratory BP modulation, block R,, and other vascular modulation, block H,,,,). In fact the model classifies four types of poles and spectral components: (i) double-loop poles (dl-poles), relevant to the resonances of the modelled interactions; (ii) us-poles, relevant to vascular modulation; (iii) ut-poles, relevant to sinus-node modulation; and (iv) r-poles, relevant to respiratory rhythms. The consequent spectral decomposition (Fig. 7) is useful to study the different mechanisms that can contribute to the spontaneous LF oscillations: resonances of BP regulation (dl-poles), vasomotion (u,-poles) and central oscillations (mainly z+poles, if a modulation of the autonomic outflow to the heart can be hypothesized).

2.3. Time-variant multivariate autoregressive modelling of heart rate and blood pressure

Multivariate time-variant identification is needed when the system under study exhibits

I. Korhonen et al. /ComputerMethods and Programs in Biomedicine 51 (1996) 121-130 121

dynamic changes in time. In the present example, the beat-to-beat model parameters (mainly the gain of H,,, a-cl) are used to monitor changes in the interactions between t and s series during episodes of vasovagal syncope. Syncope was in- duced by rest-to-tilt manoeuvre in a group of young healthy subjects according to a co-oper- ative research project with the Department of Internal Medicine, University of Milan 1261.

The time-variant identification of the MAR model was carried out by a recursive least square algorithm [26]. The model order was set to 8: this order value has been found appropriate to describe cardiovascular variability signals. The for- getting factor was set to 0.98 according to Bianchi et al. [9], which roughly corresponds to a temporal window of 50 samples exponentially weighed.

Figures 8a and 8b show the t and s variability signals during syncope. Two arrows mark the in- stant of tilt (T-arrow) and the beginning of syncope (S-arrow). The beat-to-beat spectral parameters, estimated from the model, provide relevant information on the cardiovascular mechanisms of control at sinus node (Fig. 8c), on the vessel (Fig. 8d), and on the ar-ci gain (Fig. 8e). Sympathetic tone increases after the tilt, as evidenced by the increase of LF component on the tachogram (Fig. 8~). A further increase brings this parameter up to over 0.99 in normalised units, reaching its maximum a few minutes before the syncopal event. The abnormal sympathetic activity driven both at cardiac (Fig. 8c) and vascular (Fig. 8d) level can be related with the mec~nisms which trigger the concerning syncopal event. It is re- markable that after the beginning of the with- drawal in vasomotion (decrease in LF power on the systogram, Fig. 8d) the low frequency sinus node modulation (Fig, 8c) still increases, suggest- ing an uncoupling between sympathetic central drive and vascular control in the first phases of syncope. The reduction in tr-cl values seems to confirm such an hypothesis.

3. Discussion

The applications presented were dealing with cardiovascular variability signals. This is due to the importance of the between-signal relation-

ships in cardiovascular dynamics. Beat-to-beat fluctuations in HR and BP, analysed by mono- [6,27-301 or multivariate [3-5,7,8,10-12,17,22,31] methods, provide quantitative and quali~t~e information of the functioning of the cardiovascular control and especially on the state of the autonomous nervous system. When causal interactions of the system are to be studied, closed-loop methods are needed [5,6]. The methods presented in this and the companion 1141 paper provide tools for this purpose. By MAR or MDA modelIing of the BP, HR and R interactions, it was shown how the changes in the autonomous nervous system state were reflected in the NSCR estimates, sympatho-vagal balance, or cu-cl gain in different conditions (position changes, vasovagal syncope).

The application examples presented in this paper were shown in the studies of vasovagal syncope or position changes, but several other fields of applications are possible: when monitoring patient status in ICU, during clinical surgery (angioplasty) or during provocative tests (stress test, drug infusion). Model parameters are obtained by post-processing the commonly recorded cardiovascular signals (ECG, arterial BP and respiratory signal) and they are able to provide non- invasive indices on both the autonomic nervous system control status and the different relationships among cardiovascular signals. Such indices may help the monitoring of patient status by providing further parameters of clinical interest.

The cardiovascular signals used in this paper, i.e., HR and BP, as well as R, represent signals which are commons recorded in the intensive care unit (ICU) or anesthesia, together with several other related signals (e.g. cardiac output, stroke volume, peripheral flow or resistance, etc.). Until now, the main emphasis of the multivariate methods has been on the applications utilising only a limited range of signals. The use of extra signals, available for example in KU, may provide more robust info~ation on the autonomous nervous system state, and hence the patient status. As the methods, especially MAR modelling, are easily adapted to any set of signals, this provides a potential subject for further studies.

In addition to cardiovascular and related signals, many other ph~iological signals, e.g. EEG,

128 I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (I 996) 121-130

0.8

0.6

?ooo - 2ooo 2500

beat number

S

beat number

Fig. 8. Beat-to-beat spectral parameters during an episode of vasovagal syncope as obtained by the presented multivariate time-variant identification procedure (see the companion paper [141X The continuous tracking of the relevant spectral parameters is obtained by the simultaneous analysis of (a) t and (b) s variability series. The estimated spectral parameters are: Cc) LF component power calculated on the t series in normalized units (n.u., i.e., the percentage value of the LF power in respect to the total power without the very-low frequency component); (d) LF component calculated on the s series (in absolute units), whose increase indicates an increased sympathetic drive on the vessel; and finally (e) a-cl gain which can be considered as a measure of the baroreceptive response. The spectral trends give information on the changes in the autonomic nervous system ~ntroliing mechanisms after tilt (73 and before syncope C-3.

I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (I 996) 121-130 129

electrogastrogram (EGG), electromyogram (EMG), etc., exhibit rhythmic-like fluctuations, in the analysis of which second-order properties are of great importance. The multivariate modelling of EEG provides information on the synchronisa- tion properties of the different channels in it, and thus aids rupture detection and segmentation [13]. Likewise, the multivariate analysis of other physiological signals, too, may provide a new insight into a problem in hand. For example, in recent studies it has been shown that synchronous analysis of EEG, tracheal pressure, and HR, reveal clinically significant findings which may not other- wise be made [32,33]. This example shows that the utilisation of the cross-variable interactions and dependency may lead to significant improvement in the employment of the information content of the data, and thus to new or enhanced clinical findings.

As stated above, multivariate modelling is a promising method to apply in the analysis and monitoring of the physiological state of the subject. When the multivariate modelling methods are to be applied in patient monitoring in ICU or during anesthesia, certain matters are to be considered. Firstly, the key phenomenon is the time dependency of the signals. Thus, the methods must be applied as their time-variant versions. Usually, the state of the patient is stable, and what is to be detected is the rupture from the stationary conditions, indicating the change in physiological conditions. In this procedure, multivariate rupture detection methods may be utilised 1131. Secondly, in ICU and in anesthesia, the signals obtained typically contain noise and artefacts, which seriously disturb the modelling. To handle these, special methods must be developed to classify the detected ruptures as of physiological origin or artefacts. Together with large within- and between-subject variability, this sets a great challenge to the development of the intelligent adaptive clustering algorithms utilising multivariate modelling methods.

References

[l] RI. Kitney and C.M. Gerveshi, System identification of the blood pressure control system by low frequency neural stimulation, Trans. Inst. M C 4 (1982) 203-212.

01

[31

[41

[51

b1

(71

[81

[91

II01

1111

I121

[131

1141

MS. Leaning, H.E. Pullen, E.R. Carson and L. Finkel- stein, Modelling a complex biological system: the human cardiovascular system - 1. Methodology and model description, Trans. Inst. M C 5 (1983) 71-86. R.W. deBoer, J.M. Karemaker and J. Strackee, Hemo- dynamic fluctuations and baroreflex sensitivity in humans: a beat-to-beat model, Am. J. Physiol. 253 (1987) 680-689. S. Kalli, J. Griinlund, H. Ihalainen, A. Siimes, I. Valimlki and K. Antila, Multivariate autoregressive modeling of cardiovascular control in neonatal lamb. Comp. Biomed. Res. 21 (1988) 512-530. G. Baselli, S. Ccrutti, S. Civardi. A. Malliani and M. Pagani, Cardiovascular variability signals: towards the identification of a closed-loop model of the neural control mechanisms. IEEE Trans. BME 35 (1988) 1033-1046. S. Akselrod, D. Gordon, J.B. Madwed, N.C. Snidman, D.C. Shannon and R.J. Cohen, Hemodynamic regulation: investigation of spectral analysis, Am. J. Physiol. 249 (1985) M867-M875. A. Bianchi, B. Bontempi, S. Cerutti, P. Gianoglio, G. Comi and M.G Natali Sora, Spectral analysis of heart rate variability signal and respiration in diabetic subjects, Med. Biol. Eng. Comp. (1990) 28-205. G. Baselli, L. Biancardi, A. Porta and S. Cerutti, Multi- channel parametric analysis of the coupling between respiration and cardiovascular variabilities, J. Ambulat. Monit. 5 (1992) 153-165. A. Bianchi. L. Mainardi, E. Petrucci, M. Signorini, M. Mainardi and S. Cerutti, Time-variant power spectrum analysis for the detection of transient episodes in HRV signal, IEEE Trans. BME 40 (1993) 136-144. G. Baselli, S. Cerutti, F. Badilini, L. Biancardi. A. Porta, M. Pagani, F. Lombardi, 0. Rimoldi, R. Furlan and A. Malliani, Model for the assessment of heart period and arterial pressure variability interactions and of respiration influences, Med. Biol. Eng. Comp. 32 (1994) 143-152. G. Baselli. A. Porta, G. Ferrari, S. Cerutti, M. Pagani and 0. Rimoldi. Multivariate identification and spectral decomposition for the assessment of cardiovascular control, in Computer analysis of blood pressure and heart rate signals, eds. M. Di Rienzo, G. Mancia, G. Parati, A. Pedotti. and A. Zanchetti, pp. 95-103 (10s Press. Amsterdam, 1995). G. Baselli, A. Porta and G. Ferrari, Models for the analysis of cardiovascular variability signals. in Heart rate variability, eds. M. Malik, and A.J. Camm, pp. 135-145 (Futura Publishing Company, Armonk, NY, 1995). I. Gath, C. Feuerstein, D.T. Pham and G. Rondouin, On the tracking of rapid dynamic changes in seizure EEG, IEEE Trans. BME 30 (1992) 952-958. I. Korhonen, L. Mainardi, P. Loula, G. Carrault. G. Baselli and A. Bianchi. Linear multivariate models for

130

[151

[161

1171

[I81

1191

UOI

Ml

WI

El

WI

DSI

I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (1996) 121-130

physiological signal analysis: theory, Comp. Meth. Prog. Biomed. (this issue). I. Korhonen, R. Takalo and V. Turjanmaa, Multivariate autoregressive model with immediate transfer paths for assessment of interactions between cardio-pulmonary variability signals, Med. Biol. Eng. Comp. (in press). S. Kalli, R. Suoranta and M. Jokipii, Applying a multivariate autoregressive model to describe interactions between blood pressure and heart rate, in Measurement in clinical medicine, pp. 77-82 (The Institute of Mea- surement and Control, London, 1986). S. Kalli, V. Turjanmaa, R. Suoranta and A. Uusitalo, Assessment of blood pressure and heart rate variability using multivariate autoregressive modelling, Biomed. Meas. Inf. Contr. 2 (1988) 46-56. SM. Kay, Modem spectral estimation: theory and application, 543 p. (Prentice-Hall, Englewood Cliffs, New Jersey, 1988). J.S. Bendat and A.G. Piersol, Random data. Analysis and measurement procedures, 2nd edition, 566 p. (John Wiley and Sons, New York, 1986). H. Luczag and W. Laurig, An analysis of heart rate variability, Ergonomics 16 (1973) 85-97. L. Ljung, System identification. Theory for the user, 519 p. (Prentice-Hall, Englewood Cliffs, New Jersey, 1987). B. Pomeranz, R.J.B. Macaulay, M.A. Caudill, I. Kutz, D. Adam. D. Gordon, KM. Kilbom, A.G. Barger, D.C. Shannon, R.J. Cohen and H. Benson, Assessment of autonomic function in humans by heart rate spectral analysis, Am. J. Physiol. 248 (1985) H151-H153. V. Turjanmaa, S. Kalli, M. Sydanmaa and A. Uusitalo, Short-term variability of systolic blood pressure and heart rate in normotensive subjects, Clin. Phys. 10 (1990) 389-401. H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Contr. AC-19 (1974) 716-723. G.E.P. Box and G.M. Jenkins, Time series analysis.

&I

t271

[%I

[291

[301

[311

WI

[333

forecasting and control (Holden-Day, San Fransisco, 1970). LT. Mainardi, A.M. Bianchi, R. Barbieri, V. Di Virgilio, S. Piazza, R. Furlan, G. Baselli and S. Cerutti, Time- variant closed-loop interaction between HR and ABP variability signals during syncope, in Computers in cardi- ology, pp. 699-702 (IEEE Computer Society Press, Los Alamitos, CA, 1993). B.W. Hyndman, R.I. Kitney and B.McA. Sayers, Sponta- neous rhythms in physiological control systems, Nature 233 (1971) 339-341. B.McA. Sayers, Analysis of heart rate variability, Er- gonomics 16 (1973) 17-32. S. Akselrod, D. Gordon, F.A. Ubel, DC. Shannon, A.C. Berger and R.J. Cohen, Power spectrum analysis of heart rate fluctuation: a quantitative probe of beat-to- beat cardiovascular control, Science 213 (1981) 220-222. M. Pagani, F. Lombardi, S. Guzzetti, 0. Rimoldi, R. Furlan, P. Pizzinelli, G. Sandrone, G. Malfatto, S. Dell’Orto, E. Piccaluga, M. Turiel, G. Baselli, S. Gxutti and A. Malliani, Power spectral analysis of a beat-to-beat heart rate and blood pressure variability as a possible marker of sympatho-vagal interaction in man and con- scious dog, Circ. Res. 59 (1986) 178-193. G. Baselli, S. Cerutti, S. Civardi, D. Liberati, F. Lom- bardi, A. Malliani and M. Pagani, Spectral and cross- spectral analysis of heart rate and arterial blood pressure variability signals, Comput. Biomed. Res. 19 (1986) 520-534. V. Jantti and A. Yli-Hankala, Correlation of instanta- neous heart rate and EEG suppression during enflurane anaesthesia: synchronous inhibition of heart rate and cortical electrical activity, Electroencephalograph. Clin. Neurophysiol. 76 (1990) 476-479. A. Yli-Hankala, T. Randell, T. Seppala and L. Lindgren, Increases in hemodynamic variables and catecholamine levels after rapid increase in isoflurane concentration, Anesthesiology 78 (1993) 266-271.

Documents

Linear multivariate models for physiological signal analysis: applications