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Electrocardiogram-Based Restitution Curve A Illanes Manriquez1 , Q Zhang1 , C Medigue2 , Y Papelier3 , M Sorine2 1 INRIA-IRISA, Rennes, France 2 INRIA Rocquencourt, Le Chesnay, France 3 o e e Hˆ pital Antoine-B´ cl` re, Clamart, France Abstract the relationship between the QT interval and the preced- ing TQ interval of the ECG recorded under the handgrip This work investigates the relationship between the QT isometric exercise. In an electrical point of view, the TQ interval and its preceding TQ interval of the electrocar- interval is a rest period, while in a mechanical point of diogram (ECG) recorded under the handgrip isometric ex- view it represents the ventricular diastole (ventricular ﬁll- ercise. The experimental results at the ECG scale in this ing period). Such a relationship is similar to the well estab- work show that the relationship between these intervals is lished restitution curve observed in isolated cells, between similar to the well established restitution curve observed the APD and the preceding diastolic interval (DI). Two fac- in isolated cells, between the action potential duration and tors are essential for the success of these experimentations: its preceding diastolic interval. This result opens new per- the isometric handgrip test exercised by the tested subject spectives for the analysis of ECG signals. which ensures a sufﬁcient variation of the QT interval [6], and a reliable algorithm for the detection of T-wave end in non stationary ECG signals [7]. 1. Introduction The results show that the curve obtained by plotting QT The electrocardiogram (ECG) has a great clinical value interval against the preceding TQ interval has a shape sim- for diagnostic of disorders in heart rate and anomalies in ilar to that of the cellular restitution curve, despite the electric conduction. Each cardiac cycle is characterised by difference between the macroscopic measurements based successive waveforms, known as P wave, QRS complex on ECG and the cellular measurements. The curve ob- and T wave. tained is referred as the ECG-based restitution curve in Time intervals deﬁned between onsets and offsets of dif- the sequel. Following this similarity, a parametric resti- ferent waves are signiﬁcant because they reﬂect physiolog- tution curve model, derived from a two-current model of ical processes of the heart and autonomous nervous system cell membrane action potential, is ﬁtted to the ECG-based (ANS). One of the most important intervals is the QT in- restitution curve with a satisfactory accuracy. terval, which reﬂects, in an electrical point of view, the ventricular depolarization and repolarisation duration and, The paper is organized as follows. In section 2 the in a mechanical point of view, the ventricular systole (con- method to obtain the ECG-based restitution curve, includ- traction). This interval is calculated on the ECG signal as ing the isometric handgrip test and the parametrical resti- the time distance from the onset of the QRS complex to the tution curve derived from an action potential model, is de- end of the T wave. scribed. In section 3 the experimental results are presented. Although the QT interval reﬂects the duration of global The concluding remarks and discussion are shown in sec- ventricular electrical activity, its relationship with ventric- tion 4. ular cellular action potential duration (APD) in the heart is in general complex [1]. The APD represents the time required for a cardiac cell to achieve the repolarisation fol- 2. Methods lowing a depolarizing stimulus in the cell, and the QT in- terval recorded at the body surface is related to the APDs of a large number of cells which vary from site to site in the In this section the method to obtain the ECG-based resti- ventricle [2, 3, 4]. Moreover, many of ECG expressions of tution curve is presented. First, the handgrip test is ex- electrical systole may be cancelled by the multidirectional plained followed by the procedure to obtain the curves and nature of the process of activation and recovery [1, 5]. the conﬁrmation of the relationship between ECG scale In this paper, we report experimental results revealing and cellular scale. ISSN 0276−6547 493 Computers in Cardiology 2006;33:493−496. In the sequel the obtained experimental restitution curve will be called ECG-based restitution curve 2.3. Analytic restitution curve Some analytical models for cellular restitution curve are well known in the literature. By similarity we look for an analytical model for the ECG-based restitution curve. One possibility is to derive such a model from an action potential (AP) model. A model for electrical activity of cardiac membrane [8], which incorporate an inward and an outward current, is used in this work. The model contains two functions of time, the action potential (AP) v(t) and a gating variable h(t) as shown in equations (1) and (2). Figure 1. RR interval, heart rate (HR), and systolic blood pressure (SBP) time series during ﬁve handgrip bouts dv = Jin (v, h) + Jout (v) + Jstim (t) (1) which are separated by recovery bouts, in one subject. dt 1−h dh τopen si v < vgate = −h (2) 2.1. The ECG signal during the Handgrip dt τclose si v > vgate test 2 (1−v) where Jin (v) = hv τin represents the enter current and v Handgrip test is an isometric exercise, which triggers Jout = − τout represents the outward current, τin , τout , in a short term strictly autonomic responses. It produces τopen and τclose are time constants. The stimulus current heart rate and arterial pressure increases [6]. This cardio- Jstim is an external current applied by the experimenter. vascular response (alteration of the baroreﬂex functioning) Typically, it consists of a periodic train of brief pulses with is thought to be mediated by the voluntary central com- duration of 1 ms [9], mand, which arises at the onset of any kind of exercise. One of the advantages of this model, that will be ex- This command leads to a synchronous activation of the ploited later, is that it naturally gives rise to an explicit motor and cardiovascular system. Such an activation re- formula for the restitution curve that can be derived from sults in an indirect but very fast stimulation of the car- the model. This restitution curve is qualitatively similar to diovascular system as though the system anticipated the the commonly used exponential restitution curve [10, 11]. hypothetical oxygen muscle needs. The ﬁrst minute of The action potential duration at time instant n + 1 de- a handgrip test is an exceptional physiological condition rived from the model of equations (1) and (2) is deﬁned at of pure parasympathetic response, resulting in cardiac ac- follows: celeration, blood pressure augmentation, and alteration of − τDIn 1 − (1 − hmin )e open cardiovascular parameters variability (see Figure 1). AP Dn+1 = τclose ln (3) hmin 2.2. Experimental restitution curves where DIn is the preceding diastolic interval at time in- stant n, hmin = 4 ττout corresponds to the minimum of the in In order to obtain the experimental restitution curve dv nullcline dt = 0 in equation (1). from ECG the following steps are required: The analytical restitution curve of equation (3) is used to • Recording of the ECG during handgrip test. ﬁt the ECG-based restitution curve obtained in paragraph • Computation of the QT interval and its preceding TQ 2.2. In order to ﬁt this curve, the non linear least squared interval at each cardiac cycle. method is used to minimize the error between f (T Qn ) of • Plot of QTn+1 and the preceding T Qn . the equation (3) and the QT interval at cardiac cycle n + 1 For the computation of QT, instead of directly detecting of the ECG signal. This is deﬁned as the onset of QRS, we detect R peaks and shift them to ob- N tain the QRS onset, by considering that the QR interval is min (f (T Qn , θ) − QTn+1 )2 (4) constant in an ECG record. This is a usually used method θ n=1 for QT interval computation, because it is easier to detect R peak than the Q wave onset. The T wave end is detected where f (QTn , θ) correspond to the equation (3), θ = at each cardiac cycle with the algorithm presented in [7]. [τclose hmin τopen ] correspond to the parameter vector of 494 RR T1 Hon T2 Hoff 343 1200 1100 339 × × 1000 × × ×× ×× × 335 × ××× × × ××× ××××× × ×× × 900 ××× × × × ×× ×× × × ×× × × × 800 331 × ×× 700 × × 600 327 ×× 0.0 42.9 85.7 128.6 171.4 214.3 257.1 300.0 ×× 323 ××× ×× QT × × ×× ×× ×× 340 T1 Hon T2 Hoff 319 ×××× × ×× ×× ××× 330 315 × × × 320 ××× 310 311 × ××× 300 ×× 307 × × 290 280 303 0.0 42.9 85.7 128.6 171.4 214.3 257.1 300.0 200 300 400 500 600 700 800 900 RR 1300 350 1200 T1 Hon T1 Hoff ×× × × ×× ××× × × × × × × ×× × × 1100 × × ×× × × Figure 2. AP model stimulated by a train of pulses of 1000 900 340 × × × × 800 × ×× frequency similar to the heart rate. 700 600 × × × ×× × × ×× ××× × 330 ×× ×× ×× × 500 270 290 310 330 350 370 390 410 430 450 470 ×× ×× ×× × ×× ××× QT ×× × 360 × × T1 Hon T1 Hoff 320 ×× 350 equation (3) that we desire to estimate, and N is the num- × 340 ×× × ×× ×× ××× ×× ×× 330 ×× × 310 ber of cardiac cycles. 320 310 × ×× × In order to conﬁrm this analytic model, the parame- 300 290 270 290 310 330 350 370 390 410 430 450 470 300 200 300 400 500 600 700 800 900 ters estimated by solving the minimization problem (4) are used to simulate action potentials with equations (1) and Figure 3. At the left are shown the RR and QT variation, (2). To do this, the AP model is stimulated with a se- the time instant of onset and offset of the handgrip (Hon quence of brief pulses. After QT computing in the same and Hof f ), and the time intervals taken to plot the ECG- ECG record used to ﬁnd the ECG-based restitution curve, based restitution curve (T1 and T2 ) during acceleration of we use the Q wave onset instants as the instants of im- heart rate for two ECG records. At the right are shown the pulse occurrences as shown in Figure 2. To simulate the respectively ECG based restitution curves (crosses) and the AP model we need the four time constants of equations (1) parametric restitution curve of equation (3) (solid line). and (2), however, from the minimization problem (4) only two time constants, τclose and τopen can be estimated. For τin and τout it is sufﬁcient to estimate only one of these tained on an isolated cell [12, 13]. In the same ﬁgure time constants because they are related with the third esti- is shown as well (in solid line) the parametric restitution mated parameter, hmin , by the equation hmin = 4 ττout . To in curve of equation (3) using the estimated parameters of do this, the minimization of the mean square error (MSE) equation (4) that ﬁts the ECG-based restitution curve. The between the QT interval of the ECG and the APD issue results show that the parametric restitution curve ﬁts the from the stimulated AP model has been carried as follows. ECG based restitution curve satisfactorily. N 1 2 min QTn − AP Dn (τout ) (5) τout N n=1 3.2. AP model simulation 3. Results Some results concerning the APD variability obtained 3.1. ECG-based restitution curve from the model of equations (1) and (2) as explained in section 2 are shown. In this section the results concerning the obtained ECG- based restitution curve during handgrip test and the para- In ﬁgure 4 at left are shown, for two ECG records, the metric restitution curve that ﬁts this curve are shown. variability of the QT interval (dot line) of the ECG-based In ﬁgure 3 at the left, two RR and QT time series for two restitution curve and the APD variability (solid line) cal- ECG record during handgrip are shown. At the beginning culated from the model of equations (1) and (2) after stim- of the handgrip test the heart rate increases while the QT ulation of the AP model and optimization of τout (τin ). At interval decreases. At the time that handgrip test stops, the the right of the same ﬁgure, the respective restitution curve heart rate recovers very quickly while the QT interval takes plotting DIn versus AP Dn+1 (crosses), both calculated more time to recover. from the model, and the ECG-based restitution curve plot- ting T Qn versus QTn+1 (triangles) are shown. At the right of the ﬁgure 3 (crosses), two ECG-based restitution curves obtained during acceleration of the heart The results show that the time course of the model’s rate (between time instants T1 and T2 on the ﬁgure) are APD follows in a very good way the time course of the shown. These obtained curves have similar shape to the the QT interval of the ECG during Handgrip, moreover, rstitution curve between APD and its preceding DI ob- both restitution curves have similar shapes. 495 338 343 334 339 ∇ ∇ ity of the recover phase resulting in small quantity of data ×∇ × ∇××× ×× ∇ ∇ × ××× ∇ × ××××× ∇ to analyze. Despite that, in certain cases at the recovery × × × × × × 335 ∇ ∇××× × ∇ ∇××× × ∇∇∇∇ ××∇ × ∇ ×× ∇ ∇×× 330 ∇ × ×∇∇ ∇ ∇ ∇∇ ∇ ∇ ∇∇ ∇ ∇ 331 ∇ ∇∇× phase restitution curves are observed. ×× ×× ∇ ∇ 326 × 327 ∇∇ × ∇∇ ×× 322 323 ×∇∇ ∇ ×∇∇ × × ∇∇×∇∇ ∇∇ ∇ ×× ∇ × 319 ∇∇ ∇× ∇ ∇×∇∇ ∇ References 318 × ∇× ∇ ∇× ∇ ×∇ 315 ∇ ∇× ∇×× 314 ∇ ∇∇ × 311 ∇∇ ∇∇ × ∇×∇ 310 307 ∇ ∇ 306 303 [1] Kautzner J. QT interval measurements. Cardiac Electro- 0 20 40 60 80 100 120 200 300 400 500 600 700 800 900 350 350 × × × ∇ ∇∇× × × × ×∇ × ××× × ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ × × ×× ∇ ∇ ∇ physiology Review 2002;6:273–277. × ∇ [2] Franz M, Swerdlow C, Liem B, Schaefer J. Cycle length de- ∇ ∇ ∇∇ ∇ ∇ × ∇ ∇ ∇ ×× ∇ 340 340 pendence of human action potential duration in vivo. The × ∇ ∇ ×∇∇ ∇ ∇ ×× ∇ ∇ × ∇ ×∇ ××∇ ∇∇∇ american society for clinical investigation 1988;82:972– ∇× ∇ ∇∇ 330 330 × ×∇ ∇ ∇× ∇ ∇ ∇ ∇× ∇ ∇ ∇ ∇ ∇∇ ∇ ×∇∇∇ ∇ ×∇∇ ×∇ ∇ 979. × × × × × ∇ ∇× ∇ × ∇ ∇× 320 320 ∇∇ ∇ × × ∇∇× [3] Watanabe T, Rautaharju P, McDonald T. Ventricular action ∇×× ∇∇ ∇ × × × × × × × × × × ∇∇∇ ∇∇ ∇ ∇∇× ∇∇ ∇∇ 310 310 ∇ ∇∇ ∇ potentials, ventricular extracellular potentials, and ECG of 300 0 20 40 60 80 100 120 300 200 300 400 500 600 700 800 guinea-pig. Circulation research 1985;57:362–373. [4] Abildskov J. The sequence of normal recovery of excitabil- Figure 4. Results for two ECG registers. At the left, the ity in the dog heart. Circulation research 1975;52:442–446. QT time course (dot line) and the AP D time course (solid [5] Burgess M, Millar K, Abildskov J. Cancellation of electro- line) issue from the AP model. At the right, the respective cardiographic effects during ventricular recovery. Journal ECG restitution curve (triangles) and AP model restitution of electrocardiology 1969;2:101–107. curve (crosses). [6] Medigue C, Papelier Y, Bise S, M.Sorine. Short term con- trol of the cardiovascular system: Assessment with the iso- metric handgrip exercise. Technical report, INRIA, 2004. 4. Discussion and conclusions [7] Zhang Q, Illanes Manriquez A, Medigue C, Papelier Y, Sorine M. Robust and efﬁcient location of T-wave ends in electrocardiogram. Computers in Cardiology IEEE Com- In this paper, experimental results revealing the relation- puter Society 2005;32:711–714. ship between the QT interval and its preceding TQ inter- [8] Mitchell C, Schaeffer D. A two-current model for de dy- val have been presented. This relationship is similar to namics of cardiacs membrane. Bulletin of Mathematical the restitution curve observed in isolated cells. Despite of Biology 2003;65:767–793. the difference between macroscopic measurements of the [9] Tolkacheva E, Schaeffer D, Gauthier D, Mitchell C. Anal- ECG during the handgrip test and cellular measurements, ysis of the Fenton-Karma model through an approximation we obtain satisfactory results reﬂecting the relationship be- by a one-dimensional map. Chaos 2002;12(4):1034–1042. tween ECG scale and cell scale. Following this similarity [10] Nolasco J, Dahlen R. A graphic method for the study of a parametric model of a restitution curve has been devel- alternation in cardiac action potentials. Journal of applied oped. This model is derived from a two current based car- physiology 1968;25:191–196. diac cellular action potential model. In addition, the action [11] Guevara M, Ward G, Shrier A, Glass L. Electrical alternans and period doubling bifurcations. Computers in cardiology potential model has been stimulated using a train of im- 1984;167–170. pulse with frequency equal to the heart rate frequency. This [12] Koller M, Riccio M, Gilmour R. Dynamic restitution of ac- simulation indicates that APD model variation time course tion potential duration during electrical alternans ans ven- track the QT interval variation time course conﬁrming the tricular ﬁbrillation. American journal of physiology 1998; relationship between ECG scale and cell scale. 275(2):H1635–H1642. ECG-based restitution curves have been observed under [13] Elharrar V, Surawicz B. Cycle length effect on restitution the isometric handgrip test in most of situations. However, of action potential duration in dog cardiac ﬁbers. American the ECG-based restitution curve is not always observed. journal of physiologie 1983;244:H782–H792. In acceleration phase the results show that the restitution curves, similar to that seen on isolated cells, are obtained if a large QT variability is observed during handgrip test. Address for correspondence: Otherwise, no restitution curve is observed if an increase in Alfredo Illanes Manriquez heart rate is not followed by persistently increasing QT in- IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France. tervals. In the recover phase, when the handgrip test stops aillanes@irisa.fr and the heart rate decreases, the results show a correlation between T Qn and QTn+1 . However, in contrast to the heart rate acceleration stage, these curves have not a shape similar to that seen on isolated cells in most of the cases. This difference of shapes can be explained by the rapid- 496