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

In patients with heart failure and disordered intracardiac conduction of activation, doctors implant a biventricular 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.

Responses of physiological variables to changes in pacemaker settings fit well to a parabola in the vicinity of the optimum [6,7].Curve-fitting to calculate a clinical optimum has the advantage of permitting interpolation to settings which were not directly tested, and also avoids the problem that simply "picking the highest" leads to illusory optima, illusory increments in physiology from optimization, and illusory changes in optimum over time [8].
In clinical practice physiological measurements con-tain noise which can sometimes be substantial in comparison to the underlying signal, and which prevents the true underlying optimum being identified precisely even with curve fitting (Figure 1).The impact of such noise can be reduced by averaging multiple measurements, but this consumes resources such as time in a clinical environment or battery power if the measurements are conducted by the implanted device.It is therefore important to be able to calculate the precision of an optimum so that resource usage can be planned to be appropriate to achieve the clinically-required precision.
As well as needing to know how many replicates are required, doctors planning an optimization protocol also need guidance regarding what range of settings to cover during testing, and how coarsely or finely.Widening the range or making finer-grained measurements (i.e. at more closely-spaced intervals) increase the cost of the optimization process, and so are only justified if there is a clinically-valuable increase in precision of the optimum.
Finally, doctors caring for patients from day to day need to know the uncertainty of the optimum obtained this clinical optimization process.Without this knowledge, it is impossible to interpret apparent differences in optimum within an individual patient between one assessment method and another, or over time [8,9] or after an operation or a heart attack.
This paper derives a simple formula for the uncertainty of the parabolically-defined optimal setting of a pacemaker, and presents practical implications for protocol design and for medical practice.

Physiological Measurements with Noise
The pacemaker setting may be adjusted over a wide range of values.Within individual patients, clinical information provides a priori a constrained range which contains all biologically plausible locations of the optimum for that patient.During the optimization procedure, the clinician acquires a series of pairs of values of the pacemaker setting and the corresponding physiological measurement.The first value, the pacemaker setting, has negligible uncertainty because it is programmed digitally.However the second value, the measured physiological variable, is not perfectly reproducible and therefore has an element of uncertainty.Some choices of physiological variable, such as blood pressure, permit automatic acquisition, while others, such as Doppler velocity-time integral, typically require human involvement for each measurement.In practice the protocol is often to make more than one replicate measurement at each setting, and then to summarize the data for each setting by the mean of the raw measurements at that setting.Let the optimization protocol try S different values x i of the pacemaker setting, evenly spaced at intervals of x.Let the averaged physiological measurements representing each setting be denoted y i , each of which is the mean of R replicate raw measurements.The observed data y i during clinical testing are composed of an underlying value y und,i and an error component ε R .
of y und,i and normally distributed, with its standard deviation for a single raw measurement being 1   and, by the central limit theorem, the standard deviation for the error ε R of the average of R replicates being

The Optimization Process
The best-fit parabola, Ax i 2 + Bx i + C, to the observed data is defined by the standard least-squares method, to minimize the squared deviation between the observations y i and the parabola.The clinical optimum from that fitted parabola is -B/2A, which will differ from the true optimum Opt True by an amount whose standard deviation can be calculated as follows.The best-fit parabola is defined by having A, B and C values that minimize the squared error   This is achieved by setting all the partial derivatives of F with respect to A, B and C to zero: These can be solved for A, B and C as follows:

The Clinical Optimum
The clinician will choose as optimal the pacemaker setting which corresponds to the middle of the fitted parab- . In terms of the clinical data x i and y i , For simplicity, and without loss of generality, a coordinate system for x can be chosen that makes 0 the centre of the range of settings tested during optimization.This symmetrical arrangement of x i values provides the following convenient identities: Applying these substitutions permits the clinical optimum to be described as follows: If y i is augmented by any constant k, the numerator is unchanged because . The denominator is also unchanged because it is augmented by Therefore without loss of generality the observed values y i may be defined in terms of the underlying quadratic curvature coefficient A und , the true optimum x opt , and the noise component ε R as follows:

Expression for the Clinical Optimum
Applying these within Equation (1) gives:

Imprecision of the Clinical Optimum
It can be seen from this that the clinically observed optimum ˆopt x differs from the underlying optimum opt x because of two types of error, which affect the numerator and denominator respectively.The component in the numerator is a simple additive error.The impact of the denominator error, however, scales with opt x .In clinical situations where the errors are large in comparison with the degree of curvature that is manifested, the entire denominator of this formula falls near (or beyond) zero and behavior of the expression becomes strongly nonlinear.In such a situation there is almost no useful information about the underlying optimum available from the acquired data.Clinically this can be considered likely whenever the information content (or intraclass correlation coefficient) is low [8].
In most situations of well-designed optimization protocols, however, the error is not large compared with the curvature of the signal, which makes it possible to provide a closed-form expression for the imprecision of the optimum.
To do this, the variance of the numerator is first shown to be which can be approximated with 1 + z as long as the magnitude of z is well below 1.Thus as long as we can make the following assumption: and as long as the numerator and denominator errors are not large, a linear approximation can be used for the variance of ˆopt x as follows: The standard error of the clinical optimum, being the square root of this, is therefore

Contributory Factors to Imprecision of the Clinical Optimum
In general the optimization protocol may have multiple replicates, i.e.R  1, in order to reduce the effect of noise.
1 R   so that in terms of the fundamental biological characteristics, the standard error of the clinical optimum is This formulation highlights the 4 contributory factors to imprecision of the optimum clearly.First, the imprecision of the clinical optimum falls with the square root of , the number of individual physiological measurements made.Since making additional measurements either consumes scarce time in a clinical environment, or battery power if conducted automatically by the pacemaker, having knowledge of this tradeoff between number of measurements and imprecision may be helpful.

S R 
Second, by observing that the second term is almost the reciprocal of the width of the range of settings tested, , it can be seen that the wider the range tested, the more precise the optimum.In practice this is limited by loss of fit to the parabola for settings far from the optimum.However, within the range over which behaviour is parabolic, this analysis suggests it is desirable to cover a wide range rather than to focus exclusively on the very close vicinity of the optimum.

 
1 S  Third, biology sets an lower limit on 1 und A   .While measurement error can be reduced by choosing measurement techniques with smaller instrument noise, eventually almost all the variability between replicates is genuine biological variation between heart beats, which places a lower boundary on 1   .Meanwhile A und is a manifestation of dependence of physiology upon changes in pacemaker setting, and is determined by the patient's own biological characteristics.For ideal measurement modalities, where the equipment contributes no noise, all the variability is biological, and may well be similar across different modalities.For example, measures of pressure and flow, although fundamentally different and having distinct units, may change proportionally with the same constant of proportionality in response to both signal (change in pacemaker setting) and noise (spontaneous biological variability).Thus The final term can be neglected if the optimum is very close to the centre of the tested range, i.e.
. However, it rises as opt x rises.For rapid interpretation by non-mathematicians it may be useful to develop a dimensionless variable E, representing how far the true optimum is away from the centre of the tested range, running from 0 when opt x = 0, to 1 when Reworking Equation (3) to use this, and introducing x   , the width of the range of settings tested, gives: This formula is only valid when the condition described above in (2) is satisfied.That condition, the bare minimum number of raw measurements needed before the standard error of the optimum can be validly estimated, can be approximated in this simplified way: If exceeds the above formula by (for example) more than 5-fold, the uncertainty of the optimum is identified reliably using Equation (4).If exceeds by only a small margin, Equation (4) will underestimate the observed scatter between repeat optimizations.Finally, if RS does not even exceed the right hand side, the dataset is so poor in information content that it should be discarded, and the protocol redesigned.

R S 
In practice the doctor does not know the underlying values of  The expressions can be applied to any unit of "pacemaker setting", and any unit of "physiological response".Typically pacemaker settings (atrioventricular and interventricular delay) are expressed in either milliseconds or seconds.Physiological response may be based on flow, for which suitable units might be true flow rates (e.g.ml/min, L/min) or expressed per beat (ml/beat), or as a peak instantaneous flow rate, or as an average velocity (cm/min or m/min) or peak velocity, or velocity-time integral (cm/beat).Alternatively physiological response may be pressure (e.g.mmHg) as a systolic, mean, or pulse pressure, or a value derived from pressure such as intraventricular peak first derivative of pressure (dp/dt max ).In principle uncalibrated physiological response variables and even those of unknown physical unit may be used, as long as it is reasonable to believe they vary approximately linearly with cardiac performance.
Any physical units may be used to express scatter 1   and width W but, once they are decided, the units that must be used to express curvature und A are [scatter]/ [width] 2 .

Simplified Expression for Rapid Appreciation
In practice the number of settings tested is usually fairly large, e.g. S  6, and E does not exceed 1.Thus to explain the implications of the formula to a non-mathematician protocol designer, it may be sufficient to use the following simplified approximation.First check that  , otherwise redesign the experiment, with more replicates or with a variable which has a smaller The left factor contains 3 variables known precisely from the experimental design.The right factor contains 3 variables that must be estimated from observations in the patient. is almost double, at 0.010 s.For optima at the edge of the tested spectrum, is almost 4-fold higher than at the middle of the range, i.e. ≈ 0.019 s.

 ˆopt SE x 
Pacemakers only permit quantized values to be programmed, for example in steps of 0.010 s.For optima lying near the centre of the tested range in that study, the optimization procedure can be seen to be sufficient to identify the optimum for clinical purposes.However, for optima at the edge of the tested range, the protocol would need to be adjusted to maintain that level of optimization precision.The most generically effective step to achieve this would be to conduct more replicate measurements.

Example 2. Interventricular Delay Optimization
In the same study, [10] optimization of interventricular delay was also examined.There were again S = 6 settings tested, and R = 6 replicate measurements.The width of the spectrum of settings covered was W = 0.120 s.Observed measurement scatter 1 mmHg.Curvature was measured to be A und = 67 mmHg•s -2 .The bare minimum number of measurements needed Step 1.To exclude grossly inadequate optimizations, check that In this case, 15 >> 0.08, which is very satisfactory.
Step 2. Calculate standard error of optimum Step 1.To exclude grossly inadequate optimizations, check that In this case, 15 >> 0.08, which is very satisfactory.
Step 2. Calculate standard error of optimum for an optimum lying in the middle of the tested range, Equation (4) provides > ~0.118 s.For optima at the edge of the tested spectrum, is 4-fold higher, i.e. > ~0.422 ms.These lower limits on the standard error are so large that the 95% confidence intervals (1.96 standard errors) dwarf the entire spectrum of clinically plausible settings.

SE x
This analysis shows that interventricular delay optimization conducted in this way in such patients is unreliable.Interventricular delay optimization protocols recommending a few (or even just one) measurement at each setting will never work.Biological variability has an ineradicable lower limit which can only be overcome by repetitions so numerous as to be unrealistic for manual methods such as echocardiography, and challenging even for automatable haemodynamic approaches.
The curious silence of the world literature on the elementary question of blinded test-retest reproducibility of interventricular delay optimization is a dog that didn't bark [11].

Examples of Application in Protocol Design
A doctor designing an optimization protocol can use this formula to ensure that the protocol delivers a clinicallysatisfactory degree of precision.Suppose the protocol needs to deliver optimization within 0.010 s on 95% of occasions, i.e.

SE x
needs to be 0.005 s.Rewriting Equation (4), we see that the number of replicate measurements required is: 1 5 (7).
A simplified form of the relationship, for easy appreciation by non-mathematicians, is shown below: Three of the variables are handled easily.The protocol designer can choose how many settings to test (S, e.g.6), and the width covered (W, e.g.0.160 s) and can arrange from physiological knowledge that the optimum will lie in the middle 50% of the spectrum of tested settings, i.e.E< 1/2.
The value of


for each proposed optimization variable.This requires only a few minutes per subject.
Table 1 shows how this ratio, and other variables, affect the number of replicates needed, R, to achieve the desired optimization precision.
Although mathematically the tested range W, the required precision , and the scatter-to-curvature ratio The many haemodynamic variables that are affected by AV delay appear to be initially change proportionally [12] which suggests that A und may be approximately the same proportion of the baseline value of each variable.The protocol designer desiring small has by far the greatest range of possible values, and the design process should therefore focus on ensuring that this is small, if the optimization protocol is to be practical.


Although it is convenient to describe optimization responses with a parabola, the true underlying shape may   In practice the number of tested settings, width of spectrum, desired precision, and extremeness, all have a limited range of realistic values (as shown).In contrast, curvature and scatter have a wide range of possible values and, although rarely formally quantified in research reports, have enormous impact on the precision of the optimization process.This table can be used for physiological response measures that have any measurement unit (not only mmHg) since it is the relative size of scatter versus curvature, rather than their absolute values, that is important.For example, if "mmHg" was replaced by "% of value at reference setting", the table could without any other alteration be used for any variable such as echo-Doppler velocity-time integral, uncalibrated pressure signal, bioimpedance-derived stroke volume, or any marker of pulsatility in the peripheral circulation.
be more complex.For example, where intrinsic (or even just fusion) conduction becomes active as AV delay is lengthened, the data points may deviate upwards from the parabolic trend, and so fitting to a parabola may bias the fitted AV delay optimum toward higher values.Nevertheless, the principles in this manuscript would still hold true.Moreover, in the close vicinity of the optimum, which is relevant to precise optimization, curvature may be closer to parabolic.Merely applying formulae will not make optima more precise.Only selecting a physiological variable of suitably low 1 Â  ratio, and taking time to conduct enough measurements (shown in Equation ( 7)), can improve precision.Lack of interest in scatter and curvature in a doctor conducting optimization is as uninspiring as lack of interest in wings and engines in an aircraft pilot.

Conclusions
The practical implications may be summarized from inspection of each factor in Equation ( 6) in turn.To obtain precise clinical optima:  most importantly, either commit enough resources to obtain enough measurements, or do not embark on optimization;  cover a wide range of pacemaker settings while remaining within the parabolic region of the response curve;  choose a physiological variable with as narrow a random variability (in relation to its sensitivity to pacemaker setting change) as possible; and  if possible, design the spectrum of settings tested so that the true optimum will lie near its middle rather than at an extreme.Clinicians treating patients and scientists designing and conducting studies have not had simple, quick methods to establish the uncertainty in planned or actual optimization procedures.As a result, patients may be undergoing apparent optimization procedures that are in fact worsening the programming of their pacemaker.Moreover without methods for recognizing unreliable optimizations, clinicians finding the optimum appearing to change every 1 or 2 years may feel compelled to carry out worthless optimization procedures more frequently, [8] which wastes clinical resources and may be harmful to patients.
The results in this paper permit easy quantification of uncertainty of the optimum of a cardiac resynchronization therapy pacemaker.They help patients gain the best physiological benefit, help doctors design protocols that deliver efficient and reliable optimization, and assist in distinguishing genuine change in patient physiology over time, versus random noise.An unexamined optimization is not worth doing.

Figure 1 .
Figure 1.Sketch showing how noise in the measured data creates uncertainty in the optimum.Note: If there is enough time to conduct many optimizations, they can be analyzed separately to provide separate estimates of the optimum, so the uncertainty in the optimum can be observed.The upper panel sketches 3 optimization datasets in one simulated patient.All 3 datasets are plotted on the same graph and labeled "1" to "3".The lower panel shows the 3 individual interpolated optima obtained by curve fitting with a parabola.This paper provides a method of calculating the uncertainty in a clinically-obtained optimum, without having to conduct several independent optimizations.
lower limit within an individual patient which cannot be improved upon by the clinician.

Figure 2
summarizes all the relevant variables in a format convenient for visual appreciation.It shows the steps necessary to gauge the uncertainty of the optimum.

Existing Protocols 3 . 6 . 1 . 1 ˆ
Example 1. Atrioventricular Delay OptimizationIn a published study[10] of clinical optimization in 15 patients, there were S = 6 atrioventricular delay settings tested, and R = 6 replicate measurements.The width of the spectrum of settings covered was W = 0.200 − 0.040 s = 0.160 s.The observed scatter between individual replicate measurements was = 3.9 mmHg.The ob- served curvature was Â = 1194 mmHg•s -2 .With these observed data, 9, which RS comfortably exceeded.For an optimum lying near the middle of the tested range, optima half-way to the edge of the tested spectrum, which RS falls far below.This signals that application of Equation (4) will likely substantially underestimate the true standard error of the optimum.Despite therefore being only a crude lower estimate,

Figure 2 .
Figure 2. Illustration of the 6 variables that affect precision of the optimum.Note: Three are set by the design of the protocol and are therefore under the control of the clinician (within reason) and three are measured experimentally and depend on biology and on the physiological variable chosen for monitoring.Step 1 highlights the necessity for performing enough replicate measurements before attempting to calculate a standard error.
smallest proportion of its baseline value.Â  each have the same impact on the number of replicates required, in clinical practice