Heart rate (HR) and heart rate variability (HRV) are important quantitative markers of cardiovascular control, as regulated by the autonomic nervous system [1
]. It has long been understood that the healthy heart is influenced by multiple neural and hormonal inputs that result in variations of duration in the interbeat intervals (R-R intervals). The synergic interaction between the two branches of the autonomic nervous system to the heart has a major influence in regulating the cardiac dynamics and physiological mechanism of HRV. In particular, parasympathetic influences decrease the firing rate of pacemaker cells in the heart's sinus-atrial (SA) node, whereas sympathetic influences have the opposite effect [9
]. In cardiovascular physiology, it is known that lung volume tends to be correlated with variations in the timing of heart beat, or HRV. Typically, HR slows down during expiration and speeds up during inspiration. This phenomenon is known as the respiratory sinus arrhythmia (RSA) [28
]. RSA is primarily mediated by modulation of vagal outflow to the SA node. Quantification of RSA provides important information about some of the mechanisms involved in cardio-respiratory coupling [20
]. In clinical practice, RSA is often treated as an indirect and noninvasive measure of parasympathetic cardiac control [24
], even in the presence of paced breathing [29
], and may also be considered as a reliable indicator of cardiac dysfunction [26
]. A quite comprehensive review on RSA may be found in [10
A central goal in biomedical engineering applied to cardiovascular control is to develop quantitative measures and informative indices that can be extracted from physiological measurements. Specifically, a major challenge in cardiovascular engineering is to develop statistical models and apply signal processing tools to investigate various cardiovascular-respiratory functions [8
], such as HRV, RSA, and baroreflex.
In the literature, numerous methods have been proposed for quantitative HRV analysis [1
], including point process analysis [2
], frequency-domain analysis [7
], and nonlinear dynamics analysis [19
]. In [22
], RSA was defined as “the difference between the maximum HR rate after the onset of inspiratory flow and the immediately minimum HR"; whereas in [35
], RSA was calculated using the formula: 100 x (mean longest R-R - mean shortest R-R) / mean R-R interval. Saul and colleagues [27
] proposed a transfer function analysis approach for evaluating the RSA, which requires to directly model the SA node. In [4
], a bivariate autoregressive model was proposed to evaluate a time-varying index of RSA (within a temporal window). However, none of these RSA indices provide a truly instantaneous evaluation of the cardiorespiratory dynamics.
Several issues in RSA assessment from R-wave events have yet to be addressed [10
]. First, estimates of RSA have been derived from either HR or heart period data. The former is more commonly computed in clinical practice, whereas the latter would be preferred on biometric grounds, especially when the interest is in indexing parasympathetic control because of the relative linearity between vagal frequency and heart period. Second, the R-R interval series are unevenly spaced in time. Direct application to these data to spectral analysis is not appropriate and is usually solved by use of interpolating filters. In addition, longer heart periods may significantly decrease the Nyquist frequency under fast respiratory oscillation, giving rise to possible aliasing effects. Third, standard time-series analysis usually assumes that the data show at least weak sense stationarity, thus requiring particular care in choosing appropriate data segments for analysis, or requiring removal of nonstationary trends.
To address these issues, we investigate different probability models for human heart beat intervals with an adaptive point process filtering paradigm [3
], and illustrate the analysis with both synthetic data and electrocardiogram (ECG) and lung volume recordings from a previous study [31
] under an autonomic blockade assessment protocol. Furthermore, we extend the inverse Gaussian probability model to take into account the influence of respiration on HRV, based on which we derive an analytic form for computing the frequency response of RSA. Modeling accuracy is evaluated via goodness-of-fit tests, including the Kolmogorov-Smirnov (KS) test. To our knowledge, this is the first effort in the literature that investigates the dynamic RSA effect within a point process framework. In line with the most conventional guidelines [1
], our paradigm resolves the old conflict between heart period and HR, obviates the need of interpolation with the potential to solve possible aliasing problems, allows for instantaneous measures at virtually any time resolution offering (in contrast to the interpolated R-R interval values) a more rigorous frequency analysis, and overcomes nonstationarity issues associated with window-based estimation models.
The manuscript is organized as follows. Section II first introduces the point process framework, then presents several probability models of the heart beat dynamics, and finally proposes the extended bivariate probability model by inclusion of respiration (e.g., lung volume) measurements for the purpose of quantifying instantaneous RSA. In Section III, both synthetic and real experimental recordings are used to illustrate and validate the instantaneous RSA gain as computed by the novel point process algorithm. In addition, statistical tests are conducted on the autonomic blockade protocol to evaluate inter-subject statistical trends of RSA gain across different posture and pharmacological conditions. Finally, discussions and conclusion are given in Section IV.