Derivation of the activity index

We may transform any sequence of interbeat intervals into an instantaneous heart rate signal $H(t)$, and derive a corresponding trajectory in the 3-d space defined by $\overline{H}_{N}(t)$, $P(t)$, and $S(t)$ (Figure 1).

Figure 1: Derivation of the activity index.

In general, one would expect to find that all three variables increase with activity, and that the times of minimum activity would occur when the trajectory most closely approaches the point indicated in Figure 1 as the ``ideal resting ECG''. This point, $\{H_{r}, P_{r}, S_{r}\}$, does not coincide with the origin, since extremely low mean heart rate measurements are likely to be erroneous; for this reason, it has been placed at $H_{r} = 40$ beats per minute (bpm), with $P_{r} = S_{r} = 0$. An activity index may be defined as the Euclidean distance of the trajectory from the ``ideal resting ECG'':

$$A(t) = c + \sqrt{a_{1}(\overline{H}_{N}(t) - {H}_{r})^2 + a_{2}S(t)^{2} + a_3 P(t)}$$ (6)

where the scaling constants may be determined empirically (for this study, $a_{1} = 1$, $a_{2} = 10$, and $a_{3} = 100$), and $c$ is a correction term for very low heart rate measurements:

$$c = \left\{ \begin{array}{ll} 25 \, {\rm bpm} - \overline{H... ...(t) < 25$\ bpm} \\ 0 & \mbox{otherwise} \end{array} \right.$$ (7)

In unusual cases, $P(t)$ may be large throughout a recording. This can occur in the context of sustained atrial fibrillation, or if the recording quality is uniformly poor, resulting in many detector errors. In such cases, random fluctuations in $P(t)$ might have a dominant influence on $A(t)$. In HRV studies, such recordings are typically excluded, so that this behavior is not usually a problem. It may be avoided if necessary by placing an upper bound on $P(t)$ before determining $A(t)$.