THE HUMAN HEART A TRANSFER FUNCTION MODEL FOR A V CONDUCTION IN

Transcription

1 / Automedica, 1983, Vol. 4, pp /83/ \ $18.50/0 \ 1983 Gdon and Breach Science Pllblisher" lnc. Printed in Great Britain 225 A TRANSFER FUNCTION MODEL FOR A V CONDUCTION IN THE HUMAN HEART I. VAN DER TWEEL, J. N. HERBSCHLEB and F. L. MEIJLER Department o(cardiolog, Universit Hospital, P.O. Bo /6250,3500 CC Utrecht, The Netherlands (Received November 18, 1982) INTRODUCTION 2 METHODS Contraction of cardiac muscie is triggered b depolarization of the cell membrane. The depolarization front is spread in a rapid codinated fashion through the mocardium [IJ and through a specialized conducting sstem of the heart. From the sinoatrial (SA) node, the nmal pacemaker of the heart, the ecitation wave spreads throughout the atria. At the base of the atria, the ecitation wave enters a small group of specialized cells which fm the atrioventricular (A V) node. This node and the bundie of fibres called the His bundie stemming from it, constitute the onl conducting link between the atria and the ventricles. This ensures that ecitation can onl travel from atria to ventricles through the A V node. After leaving the A V node, the impulse travels along the bundie of His to the base of the papillar muscles and then spreads throughout the left and right ventricles. Little is known about the wa the A V node conducts the impulses, about the possible influence of previous impulses and especiall about the time intervals between these impulses. Earlier studies were done with isolated rat hearts (see Heethaar [2]) resulting in an empirical model that showed dependence of the time interval between a stimulus (S) and the resulting ventricular ecitation (S-R interval) of, at most, five preceding stimulus intervals (S-S intervals). We made an efft to eplain the conduction through the A V node during random stimulation b fitting a transfer function model to the S-S and S-R intervals. 251 During cardiac catheterization and after infmed consent, apatient's heart was stimulated via a bipolar stimulation catheter on the right atrium, thus eliminating the influence of the SA node. Time intervals between stimuli were random and followed a nmal distribution. The mean and standard deviation of the chosen distribution were patient-dependent to avoid blocking in the A V node escape beats from the SA node. Random intervals were chosen because of underling clinical considerations that atrial fibrillation can be viewed as a series of random stimuli. The analog signals (stimulus and QRS complees) were digitized with a sampling frequenc of 400 Hz, sted on magnetic tape and analsed b computation of - auto- and cross-crelation functions; - transfer function modeis, accding to Bo and Jenkins [3]. Up till now three patients have been analsed, resulting in four series of observations. 3 ST A TISTICAL ANAL YSIS F the mathematical analsis the measured time intervals (in msec) were considered as equidistant events with amplitudes cresponding to the length of the intervals.

2 VAN DER TWEEL, J. N. HERBSCHLEB AND F. L. MEIJLER impulse response function input dnamic sstem output FIGURE 1 Input to, and output from a dnamic sstem. s s s atria AV node ventricles R R R FIGURE 2 Schematic representation of AV conduction. Xt, S-S interval ; t, S-R interval ; Yt_l, preceding S-R interval ; Zt, R-R interval.

3 MODEL FOR AV CONDUCTION Auto- and Cross-con'elation FlIl1ction The auto-ere1ation funetion is a measure of the relation between sueeeeding observations of a variabie. Likewise, the eross-ere1ation funetion measures the relation between sueeeeding observations of two variables. 3.2 Transfer Flinction Model Suppose pairs of observations (Xt, Yt) are available of an input and an output Y f some dnamie sstem. Xt and Yt must be stationar proeesses, which can be derived b differencing of the series. If Xt and Yt are deviations fr om their respeetive means and ji, the output deviation Yt ean be represented as a linear aggregate of input deviations Xt, Xt- I, Xt- 2,... in the fm (Figure 1) : Yt = VOXt + V1Xt- 1 + V2Xt Yt = v(b)t (B is the baekward shift operat, defined b BXt=Xt- I). The weights Vo, VI, V2,... are ealled the impulse response weights. When there is no immediate response of the output to the input, one me of the initial v's will be equal to zero. Theeticall, the impulse response funetion v(b) is an infinite funetion and thus diffieult to estimate. Therefe, and f reasons of parsimon of the parameters, the above mentioned relation between output and input ean also be written as 8(B)Yt = w(b)xt- b + I1t. This noise I1t is assumed to be generated b an ARIMA-proeess (autegressive integrated moving average proeess), statisticall independent of the input Xt. 4 RESULTS Initiall we looked at the auto-ere1ation funetion of R-R intervals. The problem, however, was th at R-R intervals contain me infmation than ean be eplained as resulting from the A V node, because an R-R interval also depends on the preeeding S-R interval, as it satisfies the equation (Figure 2) : Zt = Xt + )'t - Yt - 1. So we deeided to leave R-R intervals and look at S-Rintervals and their relation to preeeding SoS intervals. The auto-ere1ation funetion of the S-Rintervals shows a clear relation between sueeeeding intervals (see, e.g., Figure 3). When the eross-ere1ation funetion between SoS and S-R 8(B)Yt = w(b)xt-b in whieh lag b = lag parameter Now v(b) is equal to the quotient of two fini te funetions web) and 8(B) of the der respeetivel s and r, v(b) = w(b)/8(b). These funetions ean be estimated (see Bo and Jenkins [3]). In praetiee, the sstem will be disturbed b noise, whieh erupts the output Yt b an amount nt. Thus, the model looks like : FIGURE 3 Autocrelation function of S-R intervals with cresponding confidence bounds.

4 254 I. VAN DER TWEEL, J. N. HERBSCHLEB AND F. L. MEIJLER intervals is considered, a negative crelation coefficient can be seen at lag 0 and one two less negative but significant coefficients at lags 1 and 2 (see, e.g., Figure 4). These cross-crelation coefficients indicate a transfer function model with two three impulse response weights and a lag parameter b equal to zero, i.e., )'t = VOXt + Vl Xt -l + V2Xt- 2 + nt Transfer function model identification and estimation of the four series of observations in all cases led to this tpe of model. Generall, we found I1t to confm to an autegressive process of the fm rewritten as 8(B)Yt = w(b)xt + nt rp(b)8(b)yt = rp(b)w(b )Xt + at and rafter estimation of 8(B ) and web)] as rp(b)yt = rp(b)v(b)xt + at In the case of two impujse response weights and two autegressive noise parameters, this resujts in )'t = rplyt - l + rpzyt- 2 + VOXt - (VOrpl - VI) Xt- l - in which at is uncrelated white noise and rp(b) is of the der two three. F better understanding the above mentioned model can be Parameter values belonging to the four series can be found in Table 1. Computer simulation of the model with the known stimulus series as input lag -1 FIGU R E 4 Crosscrelation function of S-S and S-R interval s.

5 MODEL FOR AV CONDUCTION 25 5 TABLE I Parameters f the fitted transfer fllnction models (3 patients; 4 series of observations) Patient empiri cal empirical nr. S- S S- R = = (n=106) s = s = impulse respons e noise stand. de. weights parameters of at - v = = O 1 s =3.64 v = a 4l = v = = = (n=202) s = s = v =-O = s =5.90 O 1 a v = l2= v = Ä - - = = (n=201 ) s = s = v = l =0. s =2.49 O 1 a v = l = l = = = (n=224) s = s = v = O 4J 1 =0. s =2. 66 a v = J = J 3=0. ielded theetical S-R and R-R intervals with the same mean and standard deviation as the empirical found intervals. 5 DISCUSSION B fitting a transfer function model to eplain the relation between S-S and S-Rintervals and thereb the conduction through the A V node in case of random stimulation, a model is fitted with known statistical properties. This is a great advantage over empirical modeis. The four series of obser vations belonging to three patients showed a relation between an S-R interval, one two preceding S-Rintervals and one two preceding S-S intervals. Patient no. 3 (series 3a and 3b) shows a smaller range of S-Rintervals than patients 1 and 2 and two impulse response weights instead of three. Nevertheless, the two series satisf the same tpe of model. When we look at the residuals (i.e., the difference between the empirical and theetical S-Rinterval), there still eists a slightl negative crelation between residuals and theetical S-R intervals. This could possibl be eplained b the knowledge that a long S-S interval in general will be followed b a sht S-R interval and vice versa. Further studies will concern me patients, as weil as dogs and isolated rat hearts and also step response weights in transfer function modeis. ACKNOWLEDGEMENT We wollld like to thank Dr. J. Oosting and other members of the Department of Medical Statistics of the Universit of Amsterdam f their helpful comments. REFERENCES J. D. Durrer, R. Th. Van Dam, G. E. Freud, M. J. Janse, F. L. Meij ler and R. C. Arzbaecher, Tata! ecitation of the isolated human heart, Circa/ation, 41, (1970). 2. R. M. Heethaar, A mathematical model of A-V conduction in the rat heart, Ph.D. Thesis (1972). 3. G. E. P. Bo and G. M. Jenkins, Time Series Analsis, Fecasting and Con/rol, Holden-Da (1970).