Enhancing the deceleration capacity index of heart rate by modified-phase-rectified signal averaging

Abstract

Deceleration capacity (DC) of heart rate is a novel indicator of autonomic nervous system (ANS) activity. In this paper, we proposed a modified DC index based on improved phase-rectified signal averaging (PRSA) algorithm. Sinusoidal analysis is applied to elucidate the rationality of the improved PRSA. Then the validity of the modified DC is verified by the databases of chronic heart failure (CHF) patients and control group. Both the conventional and modified DCs are significantly lower in CHF patients than that in the control group (2.12 ± 2.98 vs. 6.34 ± 1.92 ms, P < 0.0001 and 5.45 ± 2.48 vs. 10.64 ± 1.76 ms, P < 0.0001, respectively). And the modified DC provides higher accuracy in distinguishing CHF than the conventional one (87.4 vs. 82.1%). The results indicate that the suggested technique enhances the performance of PRSA and improves the efficiency of DC in assessing ANS activity in CHF patients.

Appendix

A theoretical derivation is presented in the appendix for calculating the number of pseudo anchor points in one period of a discrete sinusoid.

The discrete sinusoid is given as $x_{n} = \sin \left( {2\pi n{\frac{a}{b}} + \delta } \right)$, n = 1, …, N. The sample points categorized to pseudo anchor points follow two criteria:

• (1) The sample points locate on the decreasing edge;

• (2) Each point is larger than the former one.

The two criteria lead to a relationship among the index of the sample points, the normalized frequency and the initial phase. Phases of the points follow criterion (1) correspond to:

$$ {\frac{1}{2}}\,\pi + 2k\pi < 2\pi n{\frac{a}{b}} + \delta < {\frac{3}{2}}\pi + 2k\pi . $$

(7)

It can be further derived as:

$$ {\frac{b\pi + 4kb\pi - 2\delta b}{4a\pi }} < n < {\frac{3b\pi + 4kb\pi - 2\delta b}{4a\pi }} $$

(8)

and the phases of the points follow criterion (2) correspond to:

$$ \sin \left( {2\pi n{\frac{a}{b}} + \delta } \right) > \sin \left( {2\pi (n - 1){\frac{a}{b}} + \delta } \right). $$

(9)

It can be simplified as:

$$ 2\cos \left( {2\pi n{\frac{a}{b}} - {\frac{a}{b}}\pi + \delta } \right)\sin \left( {{\frac{a}{b}}\pi } \right) > 0. $$

(10)

Since a and b are positive integers and a < b (because b is the period of the sinusoid), the range of ${\frac{a}{b}}\pi$ is limited in (0, π) and thus $\sin \left( {{\frac{a}{b}}\pi } \right) > 0.$ So $\cos \left( {2\pi n{\frac{a}{b}} - {\frac{a}{b}}\pi + \delta } \right) > 0.$ The range of the phase is given as:

$$ \begin{gathered} 2k\pi < 2\pi n{\frac{a}{b}} - {\frac{a}{b}}\pi + \delta < {\frac{\pi }{2}} + 2k\pi \hfill \\ {\text{or}} \hfill \\ {\frac{3}{2}}\pi + 2k\pi < 2\pi n{\frac{a}{b}} - {\frac{a}{b}}\pi + \delta < 2\pi + 2k\pi . \hfill \\ \end{gathered} $$

(11)

The relationship among n and a, b, δ is obtained as:

$$ \begin{gathered} {\frac{2kb\pi + a\pi - \delta b}{2a\pi }} < n < {\frac{4kb\pi + b\pi + 2a\pi - 2\delta b}{4a\pi }} \hfill \\ {\text{or}} \hfill \\ {\frac{4kb\pi + 3b\pi + 2a\pi - 2\delta b}{4a\pi }} < n < {\frac{2kb\pi + 2b\pi + a\pi - \delta b}{2a\pi }} \hfill \\ \end{gathered} $$

(12)

In both (8) and (12), k ranges from 0 to a−1. In conclusion, when the index of the sample point, the normalized frequency, and the initial phase satisfy with (8) and (12), this sample point is a pseudo anchor point.