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RR slope analysis techniques
EDITORIAL COMMENTARY
Lissajous curves and QT hysteresis: A critical look at QT/RR slope analysis techniques Mari A. Watanabe, MD, PhD From the Cardiology Division, Department of Internal Medicine, St. Louis University, St. Louis, Missouri, and the Medizinische Klinik der Technischen Universität München, München, Germany. Cosine fitting (cosinor analysis) of circadian RR and QT values as conducted by E. Watanabe et al1 provides, in essence, the following 2 equations: RR ⫽ Cr * cos 共t兲, and QT ⫽ Cq * cos 共t ⫹ 兲, where t is a time variable, Cr and Cq are circadian amplitude coefficients, and (phi) is the phase difference between RR and QT, i.e., the difference in acrophase (time of peak value) between them. Figure 1A shows plots of the RR and QT over a 24-hour period for different and a Cq/Cr ratio arbitrarily chosen to be 0.5 for simplicity. If these 2 equations hold, then one should be able to plot QT versus RR automatically. In fact, a plot of 2 variables against each other, both of which are dependent on 1 independent time variable, belongs to a family of plots called Lissajous or Bowditch curves in mathematics, after the French and American mathematicians who investigated their properties in the 19th century. Most people will have seen Lissajous curves on oscilloscopes. They are symmetrical closed loops, varying from simple figure of 8s to complex intersecting ripples. If the periods of the 2 cosine functions are the same, as they are in our case (both are 24 hours), then the Lissajous curve is known to form an ellipse. Figures 1B through 1D show the Lissajous curves formed by changing the value of the phase difference . Figure 1B shows that if there is no phase difference between QT and RR, the ellipse collapses to a straight line. However, if there is a phase difference, there is hysteresis of QT on RR. As increases toward 6 hours, the ellipse widens to a maximum. Beyond 6 hours, the ellipse narrows and has a long axis with a negative slope. When QT and RR are completely out of phase (phase difference of 12 hours), the QT/RR plot is again a straight line. Figure 1E shows the slopes of QT against RR plotted against time for the different . For ⫽ 0, the QT/RR slope Address reprint requests and correspondence: Dr. Mari Watanabe, FDT 15, St. Louis University Hospital, 3635 Vista Avenue, St. Louis, MO 63110. E-mail address: [email protected].
is exactly Cq/Cr the entire day. For 0 ⬍ ⬍ 12, however, the QT/RR slope has discontinuities (values that go to infinity and negative infinity) and has the appearance of the trigonometric tangent function with a period of 12 hours. The over-the-day average of the QT/RR slope values (excluding the 2 hours in which the slope value is infinite) is shown plotted against in Figure 1F. The average slope is greatest when there is the least amount of hysteresis. What do Lissajous curves reveal about the study by Watanabe et al? It suggests that cosinor analysis should not be applied to the QT/RR slope, given that it should be a tangent function with a period of 12 hours, rather than a cosine function with a period of 24 hours. The last panel in their Figure 2 shows data points deviating widely from the cosine function of least squares fit, despite our assumption that the investigators are presenting their best plot, and the abstract assertion that all variables showed significant circadian rhythms. The selection of the wrong function for data fitting may be one reason why none of the circadian varying measures of QT/RR pattern, i.e., amplitude and acrophase, were found to have prognostic significance. The Lissajous analysis also provides a theoretical value for QT/RR slope average over the day: it should equal cos() * Cq/Cr. For example, regarding the values for cardiac death patients in their Table 2, 10 ms/53 ms ⫽ 0.19, which for a 2.3-hour acrophase difference should give a QT/RR slope average of 0.15. This is reasonably close to the value of 0.20 that is reported. However, for survivors, the predicted slope is 0.29, which is nearly twice the reported value of 0.17. The theoretical values predict a higher QT/RR slope for survivors than for nonsurvivors, counter to the results of most clinical studies, and reverse the reported relationship in this study. The discrepancy between theory and experimental data may be caused by the shoehorning of RR and QT intervals into a cosine wave, i.e., assuming smoothly changing data with a rotationally symmetric relationship about the mean value. Real data may require other methods of circadian analysis,2 when it has discontinuous slope and asymmetries, such as the shorter duration of sleep compared with wake. However, the more likely explanation is that the QT/RR slope defined by the Lissajous analysis roughly corresponds
1547-5271/$ -see front matter © 2007 Heart Rhythm Society. All rights reserved.
doi:10.1016/j.hrthm.2007.05.010
Watanabe
Editorial Commentary
1007
Figure 1 The relationship between cosine form circadian rhythm of QT and RR intervals and the QT/RR slope. A: Plots of cosine form RR and QT interval functions scaled to have amplitudes of 1.0 and 0.5, respectively, a 24-hour period, and RR acrophase of midnight. RR ⫽ cos (2 hour/24). The QT functions with 0-, 3-, 6-, 9-, and 12-hour phase difference from RR intervals are shown. Although the mean of both RR and QT values are set to 0 to keep the equations as simple as possible, the relative values of QT and RR including the slope will not be affected by shifting each mean to separate positive values. B-D: Lissajous curves of QT against RR for 0-, 3-, 6-, 9-, and 12-hour phase differences between QT and RR. The right-hand edge of the plots where the RR interval is greatest represents the RR and QT values at hour 0. The RR and QT values take the points around the ellipse over 24 hours. The Lissajous curves collapse to a straight line for phase differences of 0 and 12 hours, i.e., when QT and RR are completely in phase, or completely out of phase. E: The slope of QT vs RR as the Lissajous curves in panel B-D are tracked over 24 hours for 0-, 3-, 6-, 9-, and 12-hour phase difference between QT and RR. When QT and RR are completely in or out of phase, the slope is constant over the day. Values greater than the vertical axis limits are not shown in order to preserve the fine structure of the plot. F: The average of the discrete 22 QT/RR slope values computed for Figure 1E, as a function of the phase difference between QT and RR. The 2 hours at which the slope plot is discontinuous are excluded from the averaging. The continuous 24-hour average can be shown mathematically to equal 0.5 cos().
to the slope of the steady state action potential duration curve, and the slope measured by Watanabe et al, to the standard restitution curve. Figure 5 in a classic article by Elharrar and Surawicz3 shows this point. The two restitution curves have shallower slopes than the steady-state action potential duration curves. A more recent study illustrates this same point, but for a larger number of restitution curves.4 The restitution curve typically refers to a plot of action potential duration (surrogate for QT) against the preceding diastolic interval (surrogate for TQ), and the theory that steeper restitution curve slope is arrhythmogenic5,6 forms the basis of clinical studies seeking a relationship between QT (repolarization) dynamics and arrhythmic death. This discussion leads naturally to a final question. Which is the relevant QT/RR slope for sudden cardiac death prediction, the slope measured over 15-second (15- to 25-beat) time frames, or the slope over the entire day as predicted by the Lissajous figure? As mentioned in these pages before, the theoretical answer is still far from clear in that there are mathematical and animal experiment-based arguments in support of both theories.7 As for human electrocardiographic studies, analysis of QT intervals range from looking at a single QTc value, to calculation of the difference between QT intervals at the same heart rate,8 to sophisticated attempts to quantify and compensate for QT/RR hysteresis.9,10 The study by Watanabe et al adds yet another QT
risk predictor to this arsenal. However, as the current discussion shows, QT/RR slope depends on the time frame over which it is measured. It is my opinion that it is important for clinical investigators to step back and carefully consider what aspect of repolarization they want to measure before undertaking a study, and define this clearly when reporting results, whether in the introduction, or in the methods. This extra effort will be crucial in easing the way for future comparison between the prognostic value of various QT predictors, as well as in providing feedback to the theorists and bench scientists.
References 1. Watanabe E, Arakawa T, Uchiyama T, Tong M-Q, Yasui K, Takeuchi H, Terasawa T, Kodama I, Hishida H. Prognostic significance of circadian variability of RR and QT intervals and QT dynamicity in patients with chronic heart failure. Heart Rhythm 2007;4:999 –1005. 2. Watanabe MA, Alford M, Schneider R, Bauer A, Barthel P, Stein PK, Schmidt G. Demonstration of circadian rhythm in heart rate turbulence using novel application of correlator functions. Heart Rhythm 2007;4:292– 300. 3. Elharrar V, Surawicz B. Cycle length effect on restitution of action potential duration in dog cardiac fibers. Am J Physiol 1983;244(H13):H782–792. 4. Kalb SS, Tolkacheva EG, Schaeffer DG, Gauthier DJ, Krassowska W. Restitution in mapping models with an arbitrary amount of memory. Chaos 2005;15: 023701. 5. Gilmour RF Jr, Chialvo DR. Electrical restitution, critical mass, and the riddle of fibrillation. J Cardiovasc Electrophysiol 1999;10:1087–1089. 6. Gilmour RF Jr. Electrical restitution and ventricular fibrillation: negotiating a slippery slope. J Cardiovasc Electrophysiol 2002;13:1150 –1151.
1008 7. Watanabe MA. Standard restitution curves during action potential duration alternans. Heart Rhythm 2006;3:720 –721. 8. Browne KF, Prystowsky EN, Heger JJ, Cerimele BJ, Fineberg N, Zipes DP. Prolongation of the QT interval induced by probucol: demonstration of a method or determining QT interval change induced by a drug. Am Heart J 1984;107:680–684.
Heart Rhythm, Vol 4, No 8, August 2007 9. Pueyo E, Smetana P, Laguna P, Malik M. Estimation of the QT/RR hysteresis lag. J Electrocardiol 2003;36(suppl):187–190. 10. Lang CCE, Flapan AD, Neilson JMM. The impact of QT lag compensation on dynamic assessment of ventricular repolarization: reproducibility and the impact of lead selection. PACE 2001;24:366 –373.
Lissajous curves and QT hysteresis: A critical look at QT/RR slope analysis techniques Mari A. Watanabe, MD, PhD From the Cardiology Division, Department of Internal Medicine, St. Louis University, St. Louis, Missouri, and the Medizinische Klinik der Technischen Universität München, München, Germany. Cosine fitting (cosinor analysis) of circadian RR and QT values as conducted by E. Watanabe et al1 provides, in essence, the following 2 equations: RR ⫽ Cr * cos 共t兲, and QT ⫽ Cq * cos 共t ⫹ 兲, where t is a time variable, Cr and Cq are circadian amplitude coefficients, and (phi) is the phase difference between RR and QT, i.e., the difference in acrophase (time of peak value) between them. Figure 1A shows plots of the RR and QT over a 24-hour period for different and a Cq/Cr ratio arbitrarily chosen to be 0.5 for simplicity. If these 2 equations hold, then one should be able to plot QT versus RR automatically. In fact, a plot of 2 variables against each other, both of which are dependent on 1 independent time variable, belongs to a family of plots called Lissajous or Bowditch curves in mathematics, after the French and American mathematicians who investigated their properties in the 19th century. Most people will have seen Lissajous curves on oscilloscopes. They are symmetrical closed loops, varying from simple figure of 8s to complex intersecting ripples. If the periods of the 2 cosine functions are the same, as they are in our case (both are 24 hours), then the Lissajous curve is known to form an ellipse. Figures 1B through 1D show the Lissajous curves formed by changing the value of the phase difference . Figure 1B shows that if there is no phase difference between QT and RR, the ellipse collapses to a straight line. However, if there is a phase difference, there is hysteresis of QT on RR. As increases toward 6 hours, the ellipse widens to a maximum. Beyond 6 hours, the ellipse narrows and has a long axis with a negative slope. When QT and RR are completely out of phase (phase difference of 12 hours), the QT/RR plot is again a straight line. Figure 1E shows the slopes of QT against RR plotted against time for the different . For ⫽ 0, the QT/RR slope Address reprint requests and correspondence: Dr. Mari Watanabe, FDT 15, St. Louis University Hospital, 3635 Vista Avenue, St. Louis, MO 63110. E-mail address: [email protected].
is exactly Cq/Cr the entire day. For 0 ⬍ ⬍ 12, however, the QT/RR slope has discontinuities (values that go to infinity and negative infinity) and has the appearance of the trigonometric tangent function with a period of 12 hours. The over-the-day average of the QT/RR slope values (excluding the 2 hours in which the slope value is infinite) is shown plotted against in Figure 1F. The average slope is greatest when there is the least amount of hysteresis. What do Lissajous curves reveal about the study by Watanabe et al? It suggests that cosinor analysis should not be applied to the QT/RR slope, given that it should be a tangent function with a period of 12 hours, rather than a cosine function with a period of 24 hours. The last panel in their Figure 2 shows data points deviating widely from the cosine function of least squares fit, despite our assumption that the investigators are presenting their best plot, and the abstract assertion that all variables showed significant circadian rhythms. The selection of the wrong function for data fitting may be one reason why none of the circadian varying measures of QT/RR pattern, i.e., amplitude and acrophase, were found to have prognostic significance. The Lissajous analysis also provides a theoretical value for QT/RR slope average over the day: it should equal cos() * Cq/Cr. For example, regarding the values for cardiac death patients in their Table 2, 10 ms/53 ms ⫽ 0.19, which for a 2.3-hour acrophase difference should give a QT/RR slope average of 0.15. This is reasonably close to the value of 0.20 that is reported. However, for survivors, the predicted slope is 0.29, which is nearly twice the reported value of 0.17. The theoretical values predict a higher QT/RR slope for survivors than for nonsurvivors, counter to the results of most clinical studies, and reverse the reported relationship in this study. The discrepancy between theory and experimental data may be caused by the shoehorning of RR and QT intervals into a cosine wave, i.e., assuming smoothly changing data with a rotationally symmetric relationship about the mean value. Real data may require other methods of circadian analysis,2 when it has discontinuous slope and asymmetries, such as the shorter duration of sleep compared with wake. However, the more likely explanation is that the QT/RR slope defined by the Lissajous analysis roughly corresponds
1547-5271/$ -see front matter © 2007 Heart Rhythm Society. All rights reserved.
doi:10.1016/j.hrthm.2007.05.010
Watanabe
Editorial Commentary
1007
Figure 1 The relationship between cosine form circadian rhythm of QT and RR intervals and the QT/RR slope. A: Plots of cosine form RR and QT interval functions scaled to have amplitudes of 1.0 and 0.5, respectively, a 24-hour period, and RR acrophase of midnight. RR ⫽ cos (2 hour/24). The QT functions with 0-, 3-, 6-, 9-, and 12-hour phase difference from RR intervals are shown. Although the mean of both RR and QT values are set to 0 to keep the equations as simple as possible, the relative values of QT and RR including the slope will not be affected by shifting each mean to separate positive values. B-D: Lissajous curves of QT against RR for 0-, 3-, 6-, 9-, and 12-hour phase differences between QT and RR. The right-hand edge of the plots where the RR interval is greatest represents the RR and QT values at hour 0. The RR and QT values take the points around the ellipse over 24 hours. The Lissajous curves collapse to a straight line for phase differences of 0 and 12 hours, i.e., when QT and RR are completely in phase, or completely out of phase. E: The slope of QT vs RR as the Lissajous curves in panel B-D are tracked over 24 hours for 0-, 3-, 6-, 9-, and 12-hour phase difference between QT and RR. When QT and RR are completely in or out of phase, the slope is constant over the day. Values greater than the vertical axis limits are not shown in order to preserve the fine structure of the plot. F: The average of the discrete 22 QT/RR slope values computed for Figure 1E, as a function of the phase difference between QT and RR. The 2 hours at which the slope plot is discontinuous are excluded from the averaging. The continuous 24-hour average can be shown mathematically to equal 0.5 cos().
to the slope of the steady state action potential duration curve, and the slope measured by Watanabe et al, to the standard restitution curve. Figure 5 in a classic article by Elharrar and Surawicz3 shows this point. The two restitution curves have shallower slopes than the steady-state action potential duration curves. A more recent study illustrates this same point, but for a larger number of restitution curves.4 The restitution curve typically refers to a plot of action potential duration (surrogate for QT) against the preceding diastolic interval (surrogate for TQ), and the theory that steeper restitution curve slope is arrhythmogenic5,6 forms the basis of clinical studies seeking a relationship between QT (repolarization) dynamics and arrhythmic death. This discussion leads naturally to a final question. Which is the relevant QT/RR slope for sudden cardiac death prediction, the slope measured over 15-second (15- to 25-beat) time frames, or the slope over the entire day as predicted by the Lissajous figure? As mentioned in these pages before, the theoretical answer is still far from clear in that there are mathematical and animal experiment-based arguments in support of both theories.7 As for human electrocardiographic studies, analysis of QT intervals range from looking at a single QTc value, to calculation of the difference between QT intervals at the same heart rate,8 to sophisticated attempts to quantify and compensate for QT/RR hysteresis.9,10 The study by Watanabe et al adds yet another QT
risk predictor to this arsenal. However, as the current discussion shows, QT/RR slope depends on the time frame over which it is measured. It is my opinion that it is important for clinical investigators to step back and carefully consider what aspect of repolarization they want to measure before undertaking a study, and define this clearly when reporting results, whether in the introduction, or in the methods. This extra effort will be crucial in easing the way for future comparison between the prognostic value of various QT predictors, as well as in providing feedback to the theorists and bench scientists.
References 1. Watanabe E, Arakawa T, Uchiyama T, Tong M-Q, Yasui K, Takeuchi H, Terasawa T, Kodama I, Hishida H. Prognostic significance of circadian variability of RR and QT intervals and QT dynamicity in patients with chronic heart failure. Heart Rhythm 2007;4:999 –1005. 2. Watanabe MA, Alford M, Schneider R, Bauer A, Barthel P, Stein PK, Schmidt G. Demonstration of circadian rhythm in heart rate turbulence using novel application of correlator functions. Heart Rhythm 2007;4:292– 300. 3. Elharrar V, Surawicz B. Cycle length effect on restitution of action potential duration in dog cardiac fibers. Am J Physiol 1983;244(H13):H782–792. 4. Kalb SS, Tolkacheva EG, Schaeffer DG, Gauthier DJ, Krassowska W. Restitution in mapping models with an arbitrary amount of memory. Chaos 2005;15: 023701. 5. Gilmour RF Jr, Chialvo DR. Electrical restitution, critical mass, and the riddle of fibrillation. J Cardiovasc Electrophysiol 1999;10:1087–1089. 6. Gilmour RF Jr. Electrical restitution and ventricular fibrillation: negotiating a slippery slope. J Cardiovasc Electrophysiol 2002;13:1150 –1151.
1008 7. Watanabe MA. Standard restitution curves during action potential duration alternans. Heart Rhythm 2006;3:720 –721. 8. Browne KF, Prystowsky EN, Heger JJ, Cerimele BJ, Fineberg N, Zipes DP. Prolongation of the QT interval induced by probucol: demonstration of a method or determining QT interval change induced by a drug. Am Heart J 1984;107:680–684.
Heart Rhythm, Vol 4, No 8, August 2007 9. Pueyo E, Smetana P, Laguna P, Malik M. Estimation of the QT/RR hysteresis lag. J Electrocardiol 2003;36(suppl):187–190. 10. Lang CCE, Flapan AD, Neilson JMM. The impact of QT lag compensation on dynamic assessment of ventricular repolarization: reproducibility and the impact of lead selection. PACE 2001;24:366 –373.