In this study, we present a new methodology for time-varying characterization of cardiovascular (CV) control, which includes RR interval (RRI), systolic arterial pressure (SAP), respiration (RSP) and pulse transit time (PTT). Within a multivariate model, CV dynamics are represented as stochastic point processes whose means has a tetravariate autoregressive structure. Such framework provides the simultaneous time-frequency assessment of: (i) both arms of the SAP-RRI loop, along baroreflex and mechanical feedforward pathways; (ii) Respiratory sinus arrhythmia (RSA), through the direct evaluation of the interactions between RSP and the RRI; (iii) the coupling between cardio-respiratory activity and vascular tone through quantification of the interactions between PTT and the other CV variables. We validated the model by characterizing CV control in 16 healthy subjects during a tilt table test, and we were able to confirm a satisfactory model's goodness-of-ft. We further estimated transfer function gains, instantaneous powers and directed coherences, and observed that RSP strongly drove respiratory-related oscillations in all the other CV variables, and that PTT depended on RRI dynamics rather than blood pressure variations. During head-up tilt, baroreflex sensitivity and RSA decreased, while the gain from RRI to SAP increased, thus confirming previous physiological characterizations. © 2012 CCAL

Barbieri, R

Citi, L

Orini, M

Valenza, G

Orini, Michele

Citi, Luca

Barbieri, Riccardo

VALENZA, GAETANO

Archivio della Ricerca - Università di Pisa

Tetravariate point-process model for the continuous characterization of cardiovascular-respiratory dynamics during passive postural changes

University of Essex Research Repository

Tetravariate Point-Process Model for the Continuous Characterization ofCardiovascular-Respiratory Dynamics during Passive Postural ChangesMichele Orini1, Gaetano Valenza2,3, Luca Citi2, Riccardo Barbieri21GTC, I3A, University of Zaragoza, and CIBER-BBN, Spain2Neuroscience Statistics Research Laboratory, Harvard Medical School, Massachusetts GeneralHospital, Boston, MA, USA, and Massachusetts Institute of Technology, Cambridge, MA, USA3Interdepartmental Research Center "E Piaggio", University of Pisa, ItalyAbstractIn this study, we present a new methodology for time-varying characterization of cardiovascular (CV) control,which includes RR interval (RRI), systolic arterial pres-sure (SAP), respiration (RSP) and pulse transit time (PTT).Within a multivariate model, CV dynamics are representedas stochastic point processes whose means has a tetravari-ate autoregressive structure. Such framework provides thesimultaneous time-frequency assessment of: (i) both armsof the SAP-RRI loop, along baroreflex and mechanicalfeedforward pathways; (ii) Respiratory sinus arrhythmia(RSA), through the direct evaluation of the interactions be-tween RSP and the RRI; (iii) the coupling between cardio-respiratory activity and vascular tone through quantifica-tion of the interactions between PTT and the other CV vari-ables. We validated the model by characterizing CV con-trol in 16 healthy subjects during a tilt table test, and wewere able to confirm a satisfactory model’s goodness-of-fit. We further estimated transfer function gains, instan-taneous powers and directed coherences, and observedthat RSP strongly drove respiratory-related oscillations inall the other CV variables, and that PTT depended onRRI dynamics rather than blood pressure variations. Dur-ing head-up tilt, baroreflex sensitivity and RSA decreased,while the gain from RRI to SAP increased, thus confirmingprevious physiological characterizations.1. IntroductionThe assessment of cardiovascular (CV) control, both inhealth and disease, is of primarily importance to improveour understanding and early detection of CV dysfunctions.For a comprehensive characterization of CV functioning,a multivariate non-stationary framework is needed. In thisstudy, we propose a multivariate point-process model tostudy the CV system, which takes into account the auto-nomic control of the circulation. The model includes fourCV variables, namely, the RR interval (RRI), systolic ar-terial pressure (SAP), respiration (RSP) and pulse transittime (PTT). With respect to previous multivariate models[1–3], in this framework we directly considered the influ-ence of respiration, through the input RSP, and of vasculardynamics, through the inclusion of PTT, which is strictlyrelated to pulse wave velocity and arterial stiffness [4].Additionally, the multivariate point process approach al-lows to give model’s goodness-of-fit measures, and offersthe possibility of estimating time-varying indices of causalcoupling [5] for the respiratory and low frequency (LF)spectral bands separately. We used this model to charac-terize changes in the dynamic interactions between all fourCV variables during a tilt table test.2. Methodology2.1. The tetravariate point-process modelLet’s define as tRn and tPn the time occurrence of the n-th R wave in the ECG and the arrival time of the systolicpressure peak to the finger, respectively (see Fig. 1); bewRRIn = tRn+1− tRn and wPTTn = tPn− tRn the n-th RRI and PTT,respectively. Be xRSPn = xRSP(tRn) and xSAPn = xSAP(tPn), therespiratory signal sampled at heart beat occurrence and thesystolic pressure value estimated at the finger, respectively(see Fig. 1).The probability density function of RRI and PTT can bedescribed for any t > tRn by an inverse Gaussian (IG) dis-tribution [6]:f IGJ (t) =√λJ(t)2π[t− tRn]3exp(−λJ(t)[t− tRn − μJ(t)]22μ2J(t)[t− tRn ])(1)where J ∈ {RRI, PTT}, and μJ(t) and λJ(t) are the meanand the shape parameters. Note that when (1) is used to de-scribe the statistical structure of PTT, f IGJ (t) is well definedfor tRn ≤ t ≤ tPn. Whithin this framework, the variance of273cinc.org Computing in Cardiology 2012; 39:273-276.PTT and RRI can be estimated as σ2J = μ3J(t)/λJ(t) [6].The probability density function of SAP and RSP is de-scribed by a Normal distribution:fNJ (t) =√12πσ2J(t)exp(− [xJn − μJ(t)]22σ2J(t))(2)where J ∈ {SAP,RSP} (see Fig. 1). In this case, thechoice of a normal distribution, rather than an IG, is mo-tivated by the fact that these variables do not representthe inter-event-interval of a point process but rather mea-surements of signals for which an additive Gaussian noisemodel can be assumed.The considered CV variables are continuously interacting,so that at any time the value of one variable depends onits recent history as well as on the recent history of anyother variable. This dependence is embedded in the point-process framework by assuming a multivariate autoregres-sive structure for any μJ(t):M(t) = A0(t) +Q∑k=1Ak(t)W n−k (3)where M(t) = [μRRI(t), μPTT(t), μSAP(t), μRSP(t)]T is thevector of the means of distributions (1)-(2), Ak,ij(t) =[a(ij)k (t)] is the 4 × 4 matrix of the time-varying modelcoefficients, and W n = [wRRIn , wPTTn , xSAPn , xRSPn ]T is the vec-tor of the observed regressors. Importantly, coefficientsa(ij)k (t) provide a time-varying quantification of the inter-action along j → i. For example, a(13)2 (t) quantifies thelinear contribution that xSAPn−2, i.e. SAP value at tPn−2, hason μRRI(t), with tRn < t < tRn+1. Thus, a(31)k (t), a(13)k (t)and a(14)k (t) correspond to pathways that can be associatedto the mechanical feedforward influence of RRI on SAP,baroreflex feedback from SAP to RRI, and respiratory si-nus arrhythmia (RSA), respectively.The order of the model was empirically set at Q = 5,while the temporal resolution was 0.001 s. Time-varyingmodel coefficients are identified by Newton-Raphson max-imization of the local likelihood, which also includes rightcensoring, and using a weighting function W (t − tRn) =0.98(t−tRn), with t− tRn ≤ 120 s [6].To assess the capability of the model to describe thestatistical properties of the multivariate point processes,i.e. RRI and PTT, the model’s goodness-of-fit is quan-tified. Goodness-of-fit is evaluated by representingthe Kolmogorov-Smirnov (KS) plot which measures thelargest distance between the cumulative distribution func-tion of RRI and PTT series transformed to the interval(0,1] by using the time rescaling theorem [6], and the cu-mulative distribution function of a uniform distribution on(0,1]. The smaller the KS distances, the closer is the agree-ment between original RRI and PTT series and the pro-posed model. If the model completely captures the sta-tistical properties of RRI and PTT, the transformed seriesRegistro-01M ECGtRn−2 tRn−1 tRn tRn+1 tRn+2tPn−2 tPn−1 tPn tPn+1 tPn+2xSAPn−2 xSAPn−1 xSAPn xSAPn+1 xSAPn+2xRSPn−2xRSPn−1 xRSPnxRSPn+1xRSPn+2xECG(t)[au]xSAP(t)[au]xRSP(t)[au]Time [s]Figure 1. An example of RSP, ECG and SAP signals.Circles and crosses mark the occurrence time of a R wave,tRn, and arrival time of pulse wave at the finger, tPn, used asreference to estimate the PTT.should be uncorrelated [6]. To quantify the degree of cor-relation in the rescaled series, two indices were estimated:the number of the lag points for which the correlation wasoutside the confidence interval, c#, and the ratio betweenthe highest correlation and the confidence level, cR.2.2. Time-frequency CV assessmentThe non-stationary transfer function of the system,H(t, f) = [Hij(t, f)], is estimated as:H(t, f) =[I −Q∑k=1Ak(t)e−i2πfk]-1(4)where I is the identity matrix. Time-frequency spectra,Sij(t, f), and directed coherence, γDCij (t, f), are definedfor {i, j} ∈ {1, . . . , 4}, as [2, 5]:Sij(t, f) =4∑m=1Him(t, f)σ2m(t)H∗jm(t, f) (5)γDCij (t, f) =σj(t)Hij(t, f)√∑4m=1 σ2m(t)|Him(t, f)|2(6)Directed coherence γDCij (t, f) represents the ratio betweenthe part of Sii(t, f) due to process j and Sii(t, f) [5]. Bydefinition,∑4m=1 |γDCim (t, f)|2 = 1.2.3. Estimation of the indices of interactionTo estimate the time course of the indices of interaction,two TV spectral bands are localized in the time-frequencydomain. The first one, related to respiratory activity, is de-fined as: ΩRSP = {(t, f)| f ∈ [fRSP(t) ± ΔF]}, where274fRSP(t) is the respiratory rate, estimated from S44(t, f),and ΔF = 0.05 Hz. The second spectral band, relatedto LF spectral range, is defined as ΩLF = {(t, f)| f ∈[0.04,min(0.15, fRSP(t)−ΔF)]}. Note that ΩRSP and ΩLFcannot overlap, and when fRSP(t) is low, ΩRSP can includeportions of the traditional LF band. If for a given t0, thewidth of ΩLF is lower than ΔF, then ΩLF|t=t0 = ∅.γDCij,B(t) is estimated as the global maximum inside eachspectral band ΩB, while transfer function gains, |Hij,B(t)|,and instantaneous powers, Pii,B(t), are estimated by av-eraging |Hij(t, f)| and Sii(t, f) inside ΩB, with B ∈{LF, RSP}.3. Experimental settingSixteen volunteers (28.6±2.9 years, 10 males) withoutany previous cardiovascular history underwent a head uptilt table test according to the following protocol: 4 min inearly supine position (TES), 5 min tilted head up to an angleof 70 degrees (THT) and 4 min back to later supine position(TLS) [7, 8]. The pressure signal was recorded at the fingerby the Finometer system with a sampling frequency (fs) of250 Hz and without correction for the hydrostatic gradientchange during tilt, whereas standard lead V4 ECG signalwas recorded with fs=1 KHz. Time occurrence of each Rwave in the ECG, tRn, and systolic peaks in the pressure sig-nal, {tPn, xSAPn }, were automatically determined and manu-ally revised. The respiratory signal was recorded througha strain gauge transducer, with fs=125 Hz.4. ResultsTable 1 reports the mean and standard deviation of theKS distance, c#, cR estimated for all subjects, and showsthat the model was able to capture the statistical propertiesof the CV variables.The median time courses, estimated across subjects, of theindices characterizing the most relevant CV interactionsare shown in Fig. 2. Upper, middle and lower graphics re-fer to γDCB,ij(t), |HB,ij(t)| and PB,ii(t), respectively, whilethe first and the last three columns report indices estimatedin ΩLF and ΩRSP, respectively. Note that the indices ofthose subjects for which a given index was not estimatedfor more than 75% of the duration of the experiment (whenΩLF is too narrow or when there is not a maximum insidethe spectral band) were not included in the calculations.To assess whether the changes in the indices were statis-tically significant, temporal median values were estimatedin TES, THT and TLS and compared by signed rank test.To exclude fast changes during transients, the first and last30 s of each interval were not considered in the statisticalanalysis, and significance was assumed for P<0.05.In ΩLF, PTT depended on RRI dynamics rather than bloodpressure variations (Fig. 2(a),(c)); The mutual influenceTable 1. Goodness of fit as mean ± standard deviation ofKS distances, c# and cR.Model’s goodness-of-fitSignal KS-dist c# cRPTT 0.087±0.028 1.375±1.258 1.189±0.254RRI 0.051±0.022 1.313±1.352 1.215±0.355of SAP and RRI was high, and during tilt directed co-herence from RRI to SAP decreased (Fig. 2(b)). Dur-ing tilt, |H21(t)| and |H31(t)|, representing RRI→PTT andRRI→SAP interactions, increased, while |H13(t)|, repre-senting baroreflex pathway, decreased. Interestingly, thedecrease in |H13(t)| is faster and happens earlier than theincrease in |H21(t)| and |H31(t)| (Fig. 2(g)-(h)). Thismay be explained by a slower response of the mechanicalfeedforward pathway with respect to the faster neurally-mediated feedback baroreflex. During tilt, LF power ofPTT and SAP increased (Fig. 2(o)-(q)). In ΩRSP, RSPhad great influence on RRI, PTT and SAP, while, as ex-pected, these three variables had virtual no influence onRSP (Fig. 2(d)-(f)). A decrease in RSA, |H14(t)|, suggeststhat parasympathetic activity is reduced during tilt. Notethat by construction the increase in RSA is independentfrom the increase in respiratory power observed during tilt.5. Discussion and conclusionsIn this study, we presented an innovative point-processmodel for the time-varying characterization of CV regula-tion. The model provides the simultaneous assessment of:(i) both arms of the SAP-RRI loop, along baroreflex andmechanical feedforward pathways; (ii) RSA, through thedirect evaluation of the interactions between the respira-tory input and the RRI; (iii) the coupling between cardio-respiratory activity and the vasculature, which is possi-ble thanks to the quantification of the interactions betweenPTT and the other variables. In addition, since the CV vari-ables are embedded in a point process framework, modelcoefficients and parameters are estimated in continuoustime, respecting the physiological order of discrete-timeevents such as systole and pulse arrival time. Importantly,this framework offers the possibility of quantifying themodel’s goodness-of-fit [6]. The assumption of a tetravari-ate autoregressive structure for the model allows to ana-lyze the causal interactions between the CV variables inthe time-frequency domain [2,5]. Results confirm that res-piration can be considered as a critical external input whichdrives respiratory-related oscillations in other CV variables[9]. Head-up tilt provoked a decrease in the baroreflex sen-sitivity and in RSA, and a simultaneous increase in the gainof the feedforward mechanical effect. Also of relevance,275Indices estimated in ΩRSPIndices estimated in ΩLFFigure 2. Median time course of the indices estimated across subjects. Upper graphics: directed coherence, γDCij (t);pathways j → i and i → j are plotted in blue and red, respectively. Graphics on the middle: gains of the transfer functions,|Hij(t)|. Lower graphics: instantaneous powers, Pii(t). Gains and powers are in arbitrary units [au], for easier comparison(we are mainly interested in temporal changes). Vertical lines mark TES, THT and TLS.autonomic-mediated baroreflex changes were faster thanvasculature-mediated changes along the RRI→SAP direc-tion.Future works should include the exploration of additionalvariables, such as diastolic arterial pressure or central ve-nous pressure, different model orders, the assessment ofthe latencies between coupled oscillations, and other in-dices of interaction, including statistical analyses to assesssignificant strength of the directional couplings.AcknowledgementsThis work was supported in part by NIH grantR01-HL084502, and by Ministerio de Ciencia y Tec-nología, FEDER under Project TEC2010-21703-C03-02,and TRA2009-0127, and by ARAID and Ibercaja underproject Programa de apoyo a la I+D+i.First and second authors contributed equally to this work.Address for correspondence:Luca Citi, PhD, 55 Fruit Street, Jackson 4, Boston, MA02114. email: lciti@neurostat.mit.eduReferences[1] Saul JP, Berger RD, Albrecht P, Stein SP, Chen MH, CohenRJ. 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Tetravariate point-process model for the continuous characterization of cardiovascular-respiratory dynamics during passive postural changes

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