Precision of a Parabolic Optimum Calculated from Noisy Biological Data, and Implications for Quantitative Optimization of Biventricular Pacemakers (Cardiac Resynchronization Therapy)

Author(s)
Darrel P. Francis

ABSTRACT

In patients with heart failure and disordered intracardiac conduction of activation, doctors implant a biven- tricular pacemaker (“cardiac resynchronization therapy”, CRT) to allow adjustment of the relative timings of activation of parts of the heart. The process of selecting the pacemaker timings that maximize cardiac function is called “optimization”. Although optimization—more than any other clinical assessment—needs to be precise, it is not yet conventional to report the standard error of the optimum alongside its value in clinical practice, nor even in research, because no method is available to calculate precision from one optimization dataset. Moreover, as long as the determinants of precision remain unknown, they will remain unconsidered, preventing candidate haemodynamic variables from being screened for suitability for use in optimization. This manuscript derives algebraically a clinically-applicable method to calculate the precision of the optimum value of x arising from fitting noisy biological measurements of y (such as blood flow or pressure) obtained at a series of known values of x (such as atrioventricular or interventricular delay) to a quadratic curve. A formula for uncertainty in the optimum value of x is obtained, in terms of the amount of scatter (irreproducibility) of y, the intensity of its curvature with respect to x, the width of the range and number of values of x tested, the number of replicate measurements made at each value of x, and the position of the optimum within the tested range. The ratio of scatter to curvature is found to be the overwhelming practical determinant of precision of the optimum. The new formulae have three uses. First, they are a basic science for anyone desiring time-efficient, reliable optimization protocols. Second, asking for the precision of every reported optimum may expose optimization methods whose precision is unacceptable. Third, evaluating precision quantitatively will help clinicians decide whether an apparent change in optimum between successive visits is real and not just noise.

In patients with heart failure and disordered intracardiac conduction of activation, doctors implant a biven- tricular pacemaker (“cardiac resynchronization therapy”, CRT) to allow adjustment of the relative timings of activation of parts of the heart. The process of selecting the pacemaker timings that maximize cardiac function is called “optimization”. Although optimization—more than any other clinical assessment—needs to be precise, it is not yet conventional to report the standard error of the optimum alongside its value in clinical practice, nor even in research, because no method is available to calculate precision from one optimization dataset. Moreover, as long as the determinants of precision remain unknown, they will remain unconsidered, preventing candidate haemodynamic variables from being screened for suitability for use in optimization. This manuscript derives algebraically a clinically-applicable method to calculate the precision of the optimum value of x arising from fitting noisy biological measurements of y (such as blood flow or pressure) obtained at a series of known values of x (such as atrioventricular or interventricular delay) to a quadratic curve. A formula for uncertainty in the optimum value of x is obtained, in terms of the amount of scatter (irreproducibility) of y, the intensity of its curvature with respect to x, the width of the range and number of values of x tested, the number of replicate measurements made at each value of x, and the position of the optimum within the tested range. The ratio of scatter to curvature is found to be the overwhelming practical determinant of precision of the optimum. The new formulae have three uses. First, they are a basic science for anyone desiring time-efficient, reliable optimization protocols. Second, asking for the precision of every reported optimum may expose optimization methods whose precision is unacceptable. Third, evaluating precision quantitatively will help clinicians decide whether an apparent change in optimum between successive visits is real and not just noise.

Cite this paper

nullD. Francis, "Precision of a Parabolic Optimum Calculated from Noisy Biological Data, and Implications for Quantitative Optimization of Biventricular Pacemakers (Cardiac Resynchronization Therapy),"*Applied Mathematics*, Vol. 2 No. 12, 2011, pp. 1497-1506. doi: 10.4236/am.2011.212212.

nullD. Francis, "Precision of a Parabolic Optimum Calculated from Noisy Biological Data, and Implications for Quantitative Optimization of Biventricular Pacemakers (Cardiac Resynchronization Therapy),"

References

[1] W. T. Abraham, W. G. Fisher and A. L. Smith, “Cardiac Resynchronization in Chronic Heart Failure,” New England Journal of Medicine, Vol. 346, 2002, pp. 1845-1853. doi:10.1056/NEJMoa013168

[2] A. Kyriacou, P. A. Pabari, and D. P. Francis, “Cardiac Resynchronization Therapy Is Certainly Cardiac Therapy, But How Much Resynchronization and How Much Atrioventricular Delay Optimization?” Heart Failure Reviews, 2011, Article in Press. doi:10.1007/s10741-011-9271-1

[3] J. G. Cleland, J. C. Daubert, E. Erdmann, N. Freemantle, D. Gras, L. Kappenberger and L. Tavazzi, “Cardiac Resynchronization-Heart Failure (CARE-HF) Study Investigators. The Effect of Cardiac Resynchronization on Morbidity and Mortality in Heart Failure,” New England Journal of Medicine, Vol. 352, No. 15, 2005, pp. 1539-1549. doi:10.1056/NEJMoa050496

[4] A. Auricchio, C. Stellbrink, S. Sack, M. Block, J. Vogt, P. Bakker, C. Huth, F. Sch?ndube, U. Wolfhard, D. B?cker, O. Krahnefeld and H. Kirkels, “Pacing Therapies In Congestive Heart Failure (PATH-CHF) Study Group. Long-Term Clinical Effect of Hemodynamically Optimized Cardiac Resynchronization Therapy in Patients With Heart Failure and Ven-Tricular Conduction Delay,” Journal of the American College of Cardiology, Vol. 39, No. 12, 2002, pp. 2026-2033. doi:10.1016/S0735-1097(02)01895-8

[5] M. D. Bogaard, P. A. Doevendans, G. E. Leenders, P. Loh, R. N. Hauer, H. van Wessel and M. Meine, “Can Optimization of Pacing Settings Compensate for a Non-Optimal Left Ventricular Pacing Site?” Europace, Vol. 12, No. 9, 2010, pp. 1262-1269. doi:10.1093/europace/euq167

[6] I. E. van Geldorp, T. Delhaas, B. Hermans, K. Vernooy, B. Broers, J. Klimusina, F. Regoli, F. F. Faletra, T. Moccetti, B. Gerritse, R. Cornelussen, J. Settels, H. J. Crijns, A. Auricchio and F. W. Prinzen, “Comparison of a Non-Invasive Arterial Pulse Contour Technique and Echo Doppler Aorta Velocity-Time Integral on Stroke Volume Changes in Optimization of Cardiac Resynchronization Therapy,” Europace, Vol. 13, No. 1, 2011, pp. 87-95. doi:10.1093/europace/euq348

[7] Z. I. Whinnett, J. E. Davies, G. Nott, K. Willson, C. H. Manisty, N. S. Peters, P. Kanagaratnam, D. W. Davies, A. D. Hughes, J. Mayet and D. P. Francis, “Efficiency, Reproducibility and Agreement of Five Different Hemodynamic Measures for Optimization of Cardiac Resynchronization Therapy,” International Journal of Cardiology, Vol. 129, No. 2, 2008, pp. 216-226. doi:10.1016/j.ijcard.2007.08.004

[8] P. A. Pabari, K. Willson, B. Stegemann, I. E. van Geldorp, A. Kyriacou, M. Moraldo, J. Mayet, A. D. Hughes and D. P. Francis, “When Is an Optimization Not an Optimization? Evaluation of Clinical Impli-Cations of Information Content (Signal-To-Noise Ratio) in Optimization of Cardiac Resynchronization Therapy, and How to Measure and Maximize It,” Heart Failure Reviews, Vol. 16, No. 3, 2011, pp. 277-290. doi:10.1007/s10741-010-9203-5

[9] W. T. Abraham, D. Gras, C. M. Yu, L. Guzzo and M. S. Gupta, “Rationale and Design of A Randomized Clinical Trial to Assess the Safety and Efficacy of Frequent Optimization of Cardiac Resynchronization Therapy: The Frequent Optimization Study Using the Quickopt Method (FREEDOM) Trial,” American Heart Journal, Vol. 159, No. 6, 2010, pp. 944-948. doi:10.1016/j.ahj.2010.02.034

[10] Z. I. Whinnett, J. E. R. Davies, K. Willson, C. H. Manisty, A. C. Chow, R. A. Foale, D. W. Davies, A. D. Hughes, J. Mayet and D. P. Francis, “Haemodynamic Effects of Changes in AV and VV Delay in Cardiac Resynchronization Therapy Show a Consistent Pattern: Analysis of Shape, Magnitude and Relative Importance of AV and VV Delay,” Heart, Vol. 92, 2006, pp. 1628-1634. doi:10.1136/hrt.2005.080721

[11] A. C. Doyle, “The Curious Incident of the Dog in the Night-Time,” Strand Magazine, Vol. 4, No. 4, 1892, pp. 645-660. http://www.archive.org/download/StrandMagazine24/ Strand24_text.pdf

[12] C. H. Manisty, A. Al-Hussaini, B. Unsworth, R. Baruah, P. A. Pabari, J. Mayet, A. D. Hughes, Z. I. Whinnett and D. P. Francis, “The Acute Effects of Changes to AV Delay on Blood Pressure and Stroke Volume: Potential Implications for Design of Pacemaker Optimization Protocols,” Circulation: Arrhythmia and Electrophysiology, Article in Press.

[1] W. T. Abraham, W. G. Fisher and A. L. Smith, “Cardiac Resynchronization in Chronic Heart Failure,” New England Journal of Medicine, Vol. 346, 2002, pp. 1845-1853. doi:10.1056/NEJMoa013168

[2] A. Kyriacou, P. A. Pabari, and D. P. Francis, “Cardiac Resynchronization Therapy Is Certainly Cardiac Therapy, But How Much Resynchronization and How Much Atrioventricular Delay Optimization?” Heart Failure Reviews, 2011, Article in Press. doi:10.1007/s10741-011-9271-1

[3] J. G. Cleland, J. C. Daubert, E. Erdmann, N. Freemantle, D. Gras, L. Kappenberger and L. Tavazzi, “Cardiac Resynchronization-Heart Failure (CARE-HF) Study Investigators. The Effect of Cardiac Resynchronization on Morbidity and Mortality in Heart Failure,” New England Journal of Medicine, Vol. 352, No. 15, 2005, pp. 1539-1549. doi:10.1056/NEJMoa050496

[4] A. Auricchio, C. Stellbrink, S. Sack, M. Block, J. Vogt, P. Bakker, C. Huth, F. Sch?ndube, U. Wolfhard, D. B?cker, O. Krahnefeld and H. Kirkels, “Pacing Therapies In Congestive Heart Failure (PATH-CHF) Study Group. Long-Term Clinical Effect of Hemodynamically Optimized Cardiac Resynchronization Therapy in Patients With Heart Failure and Ven-Tricular Conduction Delay,” Journal of the American College of Cardiology, Vol. 39, No. 12, 2002, pp. 2026-2033. doi:10.1016/S0735-1097(02)01895-8

[5] M. D. Bogaard, P. A. Doevendans, G. E. Leenders, P. Loh, R. N. Hauer, H. van Wessel and M. Meine, “Can Optimization of Pacing Settings Compensate for a Non-Optimal Left Ventricular Pacing Site?” Europace, Vol. 12, No. 9, 2010, pp. 1262-1269. doi:10.1093/europace/euq167

[6] I. E. van Geldorp, T. Delhaas, B. Hermans, K. Vernooy, B. Broers, J. Klimusina, F. Regoli, F. F. Faletra, T. Moccetti, B. Gerritse, R. Cornelussen, J. Settels, H. J. Crijns, A. Auricchio and F. W. Prinzen, “Comparison of a Non-Invasive Arterial Pulse Contour Technique and Echo Doppler Aorta Velocity-Time Integral on Stroke Volume Changes in Optimization of Cardiac Resynchronization Therapy,” Europace, Vol. 13, No. 1, 2011, pp. 87-95. doi:10.1093/europace/euq348

[7] Z. I. Whinnett, J. E. Davies, G. Nott, K. Willson, C. H. Manisty, N. S. Peters, P. Kanagaratnam, D. W. Davies, A. D. Hughes, J. Mayet and D. P. Francis, “Efficiency, Reproducibility and Agreement of Five Different Hemodynamic Measures for Optimization of Cardiac Resynchronization Therapy,” International Journal of Cardiology, Vol. 129, No. 2, 2008, pp. 216-226. doi:10.1016/j.ijcard.2007.08.004

[8] P. A. Pabari, K. Willson, B. Stegemann, I. E. van Geldorp, A. Kyriacou, M. Moraldo, J. Mayet, A. D. Hughes and D. P. Francis, “When Is an Optimization Not an Optimization? Evaluation of Clinical Impli-Cations of Information Content (Signal-To-Noise Ratio) in Optimization of Cardiac Resynchronization Therapy, and How to Measure and Maximize It,” Heart Failure Reviews, Vol. 16, No. 3, 2011, pp. 277-290. doi:10.1007/s10741-010-9203-5

[9] W. T. Abraham, D. Gras, C. M. Yu, L. Guzzo and M. S. Gupta, “Rationale and Design of A Randomized Clinical Trial to Assess the Safety and Efficacy of Frequent Optimization of Cardiac Resynchronization Therapy: The Frequent Optimization Study Using the Quickopt Method (FREEDOM) Trial,” American Heart Journal, Vol. 159, No. 6, 2010, pp. 944-948. doi:10.1016/j.ahj.2010.02.034

[10] Z. I. Whinnett, J. E. R. Davies, K. Willson, C. H. Manisty, A. C. Chow, R. A. Foale, D. W. Davies, A. D. Hughes, J. Mayet and D. P. Francis, “Haemodynamic Effects of Changes in AV and VV Delay in Cardiac Resynchronization Therapy Show a Consistent Pattern: Analysis of Shape, Magnitude and Relative Importance of AV and VV Delay,” Heart, Vol. 92, 2006, pp. 1628-1634. doi:10.1136/hrt.2005.080721

[11] A. C. Doyle, “The Curious Incident of the Dog in the Night-Time,” Strand Magazine, Vol. 4, No. 4, 1892, pp. 645-660. http://www.archive.org/download/StrandMagazine24/ Strand24_text.pdf

[12] C. H. Manisty, A. Al-Hussaini, B. Unsworth, R. Baruah, P. A. Pabari, J. Mayet, A. D. Hughes, Z. I. Whinnett and D. P. Francis, “The Acute Effects of Changes to AV Delay on Blood Pressure and Stroke Volume: Potential Implications for Design of Pacemaker Optimization Protocols,” Circulation: Arrhythmia and Electrophysiology, Article in Press.