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),"

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