Vol. 283, Issue 5, H1873-H1886, November 2002
Poincaré plot interpretation using a physiological
model of HRV based on a network of oscillators
Michael
Brennan1,
Marimuthu
Palaniswami1, and
Peter
Kamen2
1 Department of Electrical and Electronic
Engineering, University of Melbourne, Parkville, Victoria 3010; and
2 Department of Cardiology, Austin Hospital, Melbourne,
Victoria 3084, Australia
 |
ABSTRACT |
In this paper, we develop a physiological
oscillator model of which the output mimics the shape of the R-R
interval Poincaré plot. To validate the model, simulations of
various nervous conditions are compared with heart rate variability
(HRV) data obtained from subjects under each prescribed condition. For
a variety of sympathovagal balances, our model generates Poincaré
plots that undergo alterations strongly resembling those of actual R-R
intervals. By exploiting the oscillator basis of our model, we detail
the way that low- and high-frequency modulation of the sinus node
translates into R-R interval Poincaré plot shape by way of
simulations and analytic results. With the use of our model, we
establish that the length and width of a Poincaré plot are a
weighted combination of low- and high-frequency power. This provides a
theoretical link between frequency-domain spectral analysis techniques
and time-domain Poincaré plot analysis. We ascertain the degree
to which these principles apply to real R-R intervals by testing the
mathematical relationships on a set of data and establish that the
principles are clearly evident in actual HRV records.
heart rate variability; quantitative beat-to-beat analysis
| |
INTRODUCTION |
THE STUDY OF HEART
RATE variability (HRV) centers on the analysis of beat-to-beat
fluctuations in heart rate. The series of time intervals between
heartbeats, referred to as R-R intervals, are measured over a period of
anywhere from 10 min to 24 h (15). Attention has
focused on HRV as a method of quantifying cardiac autonomic function.
In this study, we present new results in developing a novel
mathematical model that describes the interactions between the
sympathetic and the parasympathetic nervous systems and heart rate
fluctuations over a short-term period of 5-10 min. Whereas our
model is based on standard and already accepted physiological principles, the mathematical formulation permits in-depth numerical and
analytic investigations yielding valuable insight into clinical R-R
interval analysis techniques.
Standard analysis techniques commonly estimate the levels of
sympathetic and parasympathetic activity from the variability in the
R-R intervals. Our attention has focused on two specific HRV analysis
techniques. The first is the frequency domain spectral analysis of R-R
intervals (2, 4, 6, 14, 20). R-R interval Poincaré
plot analysis is the second technique, which is a newer nonlinear
method (8-10, 21, 22). To date, R-R interval Poincaré plot analysis has not been clearly related to a
physiological model of HRV. The main objective of our model is to
provide insight into the significance of Poincaré plot morphology
and not to accurately reproduce the complex autonomic activity of any
particular individual.
Our model emulates the differing varieties of Poincaré plot
patterns seen in subjects over a range of sympathovagal balances. In
addition, the model provides a unique link between spectral analysis
techniques and the emerging analysis techniques that rely on the shape
and/or other morphological properties of the Poincaré plot.
Analytic results on the "lengths" and "widths" of the
Poincaré plots generated by our model are developed. Simulations are employed to confirm the analytic results on the model. However, the
model does not necessarily represent the full range of autonomic activity. Therefore, to evaluate the validity and scope of the model
and analysis, we provide results by using a set of data from actual subjects.
| |
GLOSSARY |
Model
| HR |
Mean heart rate, Hz
|
| t |
Time, s
|
| k |
Beat number
|
| tk |
Time of kth beat, s
|
 |
Mean interbeat interval, s
|
| Cs |
Sympathetic coupling constant, Hz
|
| Cp |
Parasympathetic coupling constant, Hz
|
 s |
Frequency of sympathetic modulation, rad/s
|
| p |
Frequency of parasympathetic modulation, rad/s
|
| s(t) |
Sympathetic activation
|
| p(t) |
Parasympathetic activation
|
| m(t) |
Modulation function, Hz
|
| x(t) |
IPFM output spike train, Hz
|
| y(t) |
IPFM integration process output
|
Analysis
| N |
Number of sinusoids
|
| Cn |
Coupling constant for sinusoid n, Hz
|
 n |
Phase of sinusoid n, radians
|
 k |
Deviation time of beat k from regular spike train, s
|
| n |
Frequency of sinusoid n, rad/s
|
| RRk |
Interbeat interval, s
|
 RRk |
Delta interbeat interval, s
|
| L |
Length of Poincaré plot, s
|
| W |
Width of Poincaré plot, s
|
HRV indexes
| LF |
Low-frequency power, 1/s2
|
| HF |
High-frequency power, 1/s2
|
| SDRR |
Standard deviation of interbeat intervals, s
|
| SDSD |
Standard deviation of successive differences of interbeat intervals, s
|
| |
HRV ANALYSIS |
It is well known that perturbations to autonomic activity, such as
respiratory sinus arrhythmia and vasomotor oscillations, cause
corresponding fluctuations in heart rate (2, 17). HRV analysis seeks to determine the autonomic activity from heart rate
variability. Spectral analysis is the standard technique used to
determine the presence of respiratory sinus arrhythmia and vasomotor
oscillations (17, 18). This is accomplished by dividing
the spectrum into low- (0.04-0.15 Hz) and high- (0.15-0.4 Hz)
frequency bands, known as the LF and HF bands, effectively distinguishing between rapid respiratory modulator activity and slow
vasomotor modulation of heart rate (see Fig.
1A). HF power is supposedly a
pure measure of parasympathetic activity, and LF power is reflective of
sympathetic modulation and parasympathetic tone, although it is
sometimes considered to reflect sympathetic tone (6). In
this study, spectral estimates are given by the autoregressive
parametric technique by using the modified covariance method
(12) for the smooth spectrum and easy identification of
the spectral peaks.
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Fig. 1.
A: heart reat variability (HRV) spectrum.
Respiratory component near 0.3 Hz and the vasomotor component near 0.1 Hz are clearly present. HF, high frequency; LF, low frequency.
B: Poincaré plot of the same data. Length and width
are shown graphically on plot.
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|
The Poincaré plot is a scatterplot of the current R-R
interval plotted against the preceding R-R interval. Poincaré
plot analysis is a quantitative visual technique, whereby the shape of
the plot is categorized into functional classes (21, 22). The plot provides summary information as well as detailed beat-to-beat information on the behavior of the heart (9). Points above the line of identity indicate R-R intervals that are longer than the
preceding R-R interval, and points below the line of identity indicate
a shorter R-R interval than the previous. Accordingly, the dispersion
of points perpendicular to the line of identity (the "width")
reflects the level of short-term variability. This dispersion can be
quantified by the standard deviation of the distances the points lie
from the line of identity. This measure is equivalent to the standard
deviation of the successive differences of the R-R intervals [standard
deviation of successive differences (SDSD) or root-mean-square of
successive differences (RMSSD)] (10). The standard
deviation of points along the line of identity (the "length")
reflects the standard deviation of the R-R intervals (SDRR).
Figure 1B details these quantitative measures of
Poincaré plot shape. Poincaré plots appear under different
names in the literature: scatter plots, first return maps, and Lorenz
plots being prominent terms. A distinct advantage of Poincaré
plots is their ability to identify beat-to-beat cycles and patterns in
data that are difficult to identify with spectral analysis (21,
22).
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PHYSIOLOGICAL HRV MODEL BASED ON INTERACTING OSCILLATORS |
In this section, we develop a model by using a coupled network of
oscillators, each representing a specific facet of the baroreflex and
autonomic nervous system. The architecture of the network and the
coupling are shown in Fig. 2. The
coupling constants Cs and
Cp denote the level at which the corresponding
oscillator modulates the sinus node oscillator, where s is sympathetic
and p is parasympathetic. For the purpose of clarity, we define the respiratory oscillator as the parasympathetic oscillator.
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Fig. 2.
Three coupled oscillators representing the cardiac
control system. Cs, sympathetic coupling
constant; Cp, parasympathetic coupling
constant.
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|
Sympathetic oscillator.
The sympathetic oscillator (s) represents the combined LF power of the
HRV spectrum, which includes vasomotor activity. It is governed by
Eq. 1
|
(1)
|
where s represents the level of sympathetic activation.
Sympathetic activity occurs on a slow time scale-altering heart rate over a long duration (2, 16, 17). Accordingly
s is assigned a small value, producing slow waves of
more than 10 s duration.
It is generally accepted that low levels of sympathetic activity will
result in slow oscillations of sympathetic nerve activity entrained to
the vasomotor oscillations. However, as the level of sympathetic
activity increases, these oscillations are damped and the fluctuations
disappear such that under intense sympathetic drive, the heart rate
becomes metronomic in its regularity. This damped effect can be
achieved by taking s 
0 or by reducing the coupling
between the sympathetic oscillator and the sinus oscillator by taking
Cs 0.
Parasympathetic respiratory oscillator.
This oscillator is intended to represent short-term activity
impinging on the sinus node via the parasympathetic nervous system. Respiratory oscillations affect both the sympathetic and
parasympathetic nervous systems; however, because of the slow response
time of the sympathetic system, these rapid oscillations are mediated purely by the parasympathetic system (2, 16, 17). The
effects of respiration are described by the parasympathetic respiratory oscillator (p), which is governed by Eq. 2,
|
(2)
|
where p represents the level of parasympathetic respiratory
activation. This oscillator has a value of p larger than
s, typically at the modeled respiration frequency.
Sinus oscillator.
The sinus node oscillator is based on the formulation of the well-known
integral pulse frequency modulation (IPFM) model. It is a useful
description of how cardiac events are modulated by autonomic nervous
activity, and its suitability for modeling the sinus node has been
discussed by a number of researchers (1, 3, 7). The IPFM
model generates heartbeats by integrating an input signal until it
reaches a preset threshold of unity. At this point, a pulse is produced
and the integrator is reset to zero. See Fig.
3. The mathematical representation is
given in Eq. 3.
![1=<LIM><OP>∫</OP><LL>t<SUB>k</SUB></LL><UL>t<SUB>k+1</SUB></UL></LIM>[HR+m(t)]d<IT>t</IT>](/content/vol283/issue5/fulltext/H1873/img003.gif)
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(3)
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The signal m(t) is the input signal
representing autonomic activity, and tk is the
time of the kth R wave. When the input signal is
zero, the IPFM model generates heartbeats with an interval equal to
= 1/HR, where HR is a
variable parameter that represents mean heart rate. It is equal to the
actual frequency of heartbeats in the absence of any modulatory
autonomic nervous activity. The input signal m(t)
represents the effects of modulatory autonomic nervous input and is
defined by Eq. 4. If the input signal is positive, then
heartbeats are generated at a faster rate, whereas a negative input
signal causes heartbeats to be generated at a slower rate. The function
x(t) represents the series of pulses representing
the heartbeats generated by the model, whereas
y(t) designates the integrator's output as a
function of time.
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Fig. 3.
Integral pulse frequency modulation (IPFM) model. Input
signal HR + m(t) is integrated
until the integrator output y(t) reaches the
threshold of unity. At this point a pulse is produced in the output
signal x(t) and the integrator is reset.
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We have formulated the modulation of the sinus oscillator by the
sympathetic and parasympathetic oscillators as described by Eq. 4
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(4)
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As a result, the sinus oscillator beats at a base rate of
HR Hertz, which is increased or decreased in an additive
linear fashion by sympathetic and parasympathetic respiratory
modulation. For the modulating frequencies to appear unaliased in the
beat sequence, the mean beat frequency HR should be higher
than the highest modulating frequency component p
|
(5)
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The coupling constants Cs and
Cp reflect the levels of sympathetic and
parasympathetic modulation of the sinus node, which is not equivalent
to the tonic (mean) levels of sympathetic and parasympathetic activity.
The tonic autonomic influences are included in the parameter
HR, which is a combined function of sympathetic and
parasympathetic activity, hormonal responses, and various parameters of
the individual such as blood pressure. Accordingly HR is a
function of the intrinsic heart rate HR0 and the
tonic influences of the autonomic system commonly referred to as the sympathovagal balance (5). Whereas the exact nature of
sympathovagal balance is not completely understood, this concept has
been formalized by the following model, HR = HR0 × m × n,
due to Rosenblueth and Simeone (16) and Katona et al.
(11) in which m > 1 is the net sympathetic
influence and n < 1 is the net parasympathetic influence.
It is still being debated whether there exist any reliable connections
between the tonic influences m and n and the
levels of modulation Cs and
Cp; however, it is often observed that heart rate and HRV are inversely related.
Accordingly, we model HR, Cs, and
Cp as free variables so that it is possible to
investigate sympathetic and parasympathetic interactions with
Cs and Cp as functions of
HR or sympathovagal balance. Note that
Cs and Cp should be
chosen such that HR + m(t) is
strictly positive.
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CONVERGENCE BETWEEN MODEL AND ACTUAL HRV DATA |
In this section, we demonstrate that our model displays the
features of real R-R intervals under various induced autonomic balances. We consider the following conditions: complete autonomic blockade, parasympathetic blockade, and normal
sympathetic-parasympathetic balance. Poincaré plots of the
model's output are compared with plots of actual R-R intervals
obtained from patients under the prescribed autonomic perturbations.
The model's simulated autonomic balance is adjusted by varying the
coupling constants, which alters the levels at which the oscillators
influence the sinus oscillator. For all simulations, except where
otherwise mentioned, the following constants were used
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(6)
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HR corresponds to an R-R interval of 850 ms. The period
of the sympathetic oscillator is set to ~40 s, and the
parasympathetic oscillator is set to a period of ~3 s. Such a LF was
used for the sympathetic oscillator, because it needs to account for
the combined power of the LF and very low-frequency (VLF) bands to achieve a high degree of similarity between the plots from simulated and actual data.
Complete autonomic blockade.
First, we consider the model's output in the absence of coupling, a
state that is easily simulated with Cs and
Cp taking on very small values. Consider Fig.
4B for which the coupling
constants were Cs = Cp = 0.01. The Poincaré plot appears
as a single dense point termed a "tight cluster." Because of the
low coupling, there is very little variation in
m(t), and subsequently the sinus oscillator beats
at a constant frequency of HR Hertz. Accordingly, the R-R intervals varied little from the constant value 1/HR
seconds. The behavior of a denerved heart, such as found in the case of a transplant patient as in Fig. 4A, is mimicked. Figure
4C shows the power spectra of Fig. 4, A and
B. It is seen that neither the transplant patient nor the
model has any significant spectral power in either the LF or HF bands.
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Fig. 4.
Complete autonomic blockade. A: Poincaré
plot of subject with complete autonomic denervation (transplant
patient) with mean R-R interval of 800 ms. B: sinus
oscillator with low coupling to sympathetic and parasympathetic
oscillators (Cs = Cp = 0.01). C: power spectra of
A and B.
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Unopposed sympathetic activity: parasympathetic blockade.
This scenario is simulated by a high degree of coupling between the
sympathetic oscillator and the sinus oscillator, whereas a low coupling
level is used for the parasympathetic respiratory oscillator.
Accordingly, the coupling constants take the values Cs = 0.21 and
Cp = 0. The model's output, viewed as a
Poincaré plot in Fig.
5B, is a slender closed loop
oriented along the line y = x and is suggestive of a
"cigar" due to its shape. No variability is present other than the
motion around the loop, a direct result of excluding the
parasympathetic respiratory oscillator.
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Fig. 5.
A: Poincaré plot of a subject that has
been given atropine to block parasympathetic activity. B:
model-generated R-R intervals when the sympathetic oscillator is
coupled (Cs = 0.21 and
Cp = 0). C: additive Gaussian
noise with a standard deviation of 10 ms. D: power spectra
of A (top), B (middle), and
C (bottom).
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The plot in Fig. 5A is of a healthy subject who has been
infused with atropine. The variability witnessed in this plot is therefore largely a product of sympathetic activity. The total lack of
any short-term variability in our model's output prevents a clear
comparison to this subject except at the most qualitative level. Figure
5C shows the effect of artificially adding a small amount of
short-term variability to the model's output by adding zero mean
Gaussian noise with a standard deviation of 10 ms to the simulated
intervals. A Poincaré plot very similar to actual observed
cigar-shaped plots is observed. Real-life physiological systems usually
do contain some level of spontaneous random variability that is best
modeled as noise, particularly at this level. The model's output
resembles R-R intervals recorded from patients with degraded
parasympathetic nervous control, such as patients with heart failure
(10). The length of the cigar is directly proportional to
the amplitude of the sympathetic modulation of the sinus oscillator.
The power spectrum of the atropine-infused subject in Fig.
5A is shown in Fig. 5D, top. The
spectrum is seen to consist of a substantial level of LF power and very
little HF power. Figure 5D, middle, shows the
power spectra of the model-generated R-R intervals. The single peak in
the LF band is the effect of the sympathetic oscillator with a coupling
intensity of 0.21. Finally, Fig. 5D, bottom,
shows the power spectrum of the model-generated R-R intervals with
added noise. The noise adds a constant level across all frequencies to
the power spectrum, and therefore, its presence does not overly alter
the shape of the spectrum.
Sympathetic: parasympathetic balance.
In this scenario, levels of parasympathetic respiratory activity are
introduced. This is simulated by way of a small coupling intensity for
the parasympathetic respiratory oscillator in addition to a high level
of sympathetic coupling. Figure
6B shows the model's R-R
interval output for the coupling constants
Cs = 0.3 and Cp = 0.05. A large degree of variability emerges in the model's output in
which the parasympathetic oscillator is responsible for the flanging
effect or widening of the cigar shape into a "comet." Comparing
Fig. 6, A and B, shows how closely the simulated
R-R intervals resemble a Poincaré plot of a subject at rest
breathing quietly in the supine position. Increasing the
parasympathetic respiratory oscillator's coupling intensity increases
the width of the comet and consequently the level of short-term
variability in the R-R intervals. Figure 6C demonstrates
this effect with Cp taking on the value of 0.1. The width of the comet is also dependent on the frequency of the
parasympathetic oscillator in an intuitive manner: larger values of
p yield wider comets because short-term variability is
increased.
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Fig. 6.
Balanced sympathetic and parasympathetic activity.
A: R-R intervals obtained from a subject who is lying supine
and at rest. B: model's simulated output for the coupling
constants Cs = 0.3 and
Cp = 0.05. C: effect of
increasing Cp to 0.1. D: power
spectra of A (top), B
(middle), and C (bottom).
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The power spectrum of the supine subject of Fig. 6A is
presented in Fig. 6D, top. A substantial level of
both LF and HF power is displayed. Figure 6D,
middle, shows the power spectrum of the model-generated R-R
intervals. The two peaks produced by the sympathetic and the
parasympathetic oscillators with coupling intensities 0.3 and 0.05 are
clearly shown. Figure 6D, bottom, displays the power spectrum of Fig. 6C, which has an increased value of
0.1 for the parasympathetic oscillator's coupling.
A significant difference exists between the Poincaré plots
of the model-generated R-R intervals and the R-R intervals obtained clinically: the density of the points in the simulated cases are skewed
toward the lower left corner of the plot, whereas actual R-R intervals
are more centrally distributed. The core of this discrepancy lies in
the highly periodic nature of the oscillators. Fluctuations produced by
the actual autonomic nervous system are not pure sinusoidal signals.
Instead they resemble a random walk, which obtains low and high R-R
interval lengths occasionally, while usually fluctuating about a mean
value without deviating widely. It is important to observe that the
lengths and widths of Fig. 6, A and B, are
roughly the same. Our model shows it is this feature that corresponds
to the balance of LF and HF power being similar, not the dispersion of
the points within the Poincaré plot.
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MATHEMATICAL ANALYSIS OF HRV MODEL |
This section develops a mathematical analysis used to investigate
the length and width of the Poincaré plots generated from the HRV
model developed in the previous sections. Because the model is a
simplification of actual HRV mechanisms, these results will not apply
to real HRV data in an exact sense. However, the results provide clear
insight into the manner in which Poincaré plot descriptors vary
as sympathetic and parasympathetic modulation levels are varied.
Specifically, we characterize the theoretical dependency between LF and
HF modulators and the shape of an R-R interval Poincaré plot
generated by our model.
In accomplishing this analysis, we require an explicit solution to the
R-R interval series. The remainder of this section derives this result.
By defining the time of the initial beat to be the origin
t0 = 0, the defining equation for the IPFM
oscillator (Eq. 3) can be expressed nonrecursively as
![<LIM><OP>∫</OP><LL>0</LL><UL>t<SUB>k</SUB></UL></LIM> [HR+m(t)]d<IT>t=k</IT>](/content/vol283/issue5/fulltext/H1873/img010.gif)
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(7)
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where m(t) is the modulating signal
(3). In our model m(t) consists of
two frequency components. It turns out to be just as easy to work with
N frequency components, so we consider
m(t) = 
Cncos(nt + n) with n < 2
HR for all n, i.e., slow modulation. The
defining equation becomes
|
(8)
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We have also divided through by HR and expressed
= 1/HR to make the equations simpler.
After integratation, the general relationship
![t<SUB>k</SUB>+<A><AC>I</AC><AC>&cjs1171;</AC></A> <LIM><OP>∑</OP><LL>n=1</LL><UL>N</UL></LIM> <FR><NU>C<SUB>n</SUB></NU><DE>&ohgr;<SUB>n</SUB></DE></FR>[sin (<IT>&ohgr;<SUB>n</SUB>t<SUB>k</SUB>+&phgr;<SUB>n</SUB></IT>)<IT>−</IT>sin (<IT>&phgr;<SUB>n</SUB></IT>)]<IT>=k<A><AC>I</AC><AC>&cjs1171;</AC></A></IT>](/content/vol283/issue5/fulltext/H1873/img014.gif)
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(9)
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is obtained. Performing the substitution tk = k
+ k, as per De Boer et al. (3),
the following nonlinear relationship for k is
obtained
![&dgr;<SUB>k</SUB>=−<IT><A><AC>I</AC><AC>&cjs1171;</AC></A> </IT><LIM><OP>∑</OP><LL><IT>n=</IT>1</LL><UL><IT>N</IT></UL></LIM> <FR><NU><IT>C<SUB>n</SUB></IT></NU><DE><IT>&ohgr;<SUB>n</SUB></IT></DE></FR> [sin (<IT>&ohgr;<SUB>n</SUB>k<A><AC>I</AC><AC>&cjs1171;</AC></A>+&phgr;<SUB>n</SUB>+&ohgr;<SUB>n</SUB>&dgr;<SUB>k</SUB></IT>)<IT>−</IT>sin (<IT>&phgr;<SUB>n</SUB></IT>)]](/content/vol283/issue5/fulltext/H1873/img016.gif)
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(10)
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The k terms represent the amount each
beat deviates from the regular pulse train tk = k. Equation 10 can be
linearized about k = 0 provided
nk is small for
all n 
[1...N]. If the event times are
close to a regular pulse train (k < /2) and the modulation frequencies are less
than the mean beat frequency (n < 2HR), it is obvious that
nk < 1. Hence for
a large class of practical pulse trains, including R-R intervals, a
linear analysis is an accurate approximation. Linearizing about
k = 0, we obtain
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(11)
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Solving for k gives the final expression
for the beat times
![t<SUB>k</SUB>=k<A><AC>I</AC><AC>&cjs1171;</AC></A>+&dgr;<SUB>k</SUB>=k<A><AC>I</AC><AC>&cjs1171;</AC></A>+<FR><NU>−<IT><A><AC>I</AC><AC>&cjs1171;</AC></A> </IT><LIM><OP>∑</OP><LL><IT>n=</IT>1</LL><UL><IT>N</IT></UL></LIM> <FR><NU><IT>C<SUB>n</SUB></IT></NU><DE><IT>&ohgr;<SUB>n</SUB></IT></DE></FR> [sin (<IT>&ohgr;<SUB>n</SUB>k<A><AC>I</AC><AC>&cjs1171;</AC></A>+&phgr;<SUB>n</SUB></IT>)<IT>−</IT>sin (<IT>&phgr;<SUB>n</SUB></IT>)]</NU><DE>1<IT>−<A><AC>I</AC><AC>&cjs1171;</AC></A> </IT><LIM><OP>∑</OP><LL><IT>n=</IT>1</LL><UL><IT>N</IT></UL></LIM> C<SUB><IT>n</IT></SUB> cos (<IT>&ohgr;<SUB>n</SUB>k<A><AC>I</AC><AC>&cjs1171;</AC></A>+&phgr;<SUB>n</SUB></IT>)</DE></FR>](/content/vol283/issue5/fulltext/H1873/img019.gif)
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(12)
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The R-R intervals are RRk = tk + 1 
tk. For our model, N = 2 and
C1 = Cs,
C2 = Cp,
1 = s, 2 = p and
1 = 2 = 0. In this case,
Eq. 12 provides us with an accurate approximation to the R-R
interval series generated by our HRV model. This result holds so long
as the intervals are approximately regular and the modulation is slow.
This is generally the case for R-R intervals. However, for subjects
with very large HRV, the assumption that the intervals are
approximately regular may be somewhat inaccurate. For the
assumption k < /2 to be compromised, an R-R interval would
have to deviate from the mean R-R interval by
an amount greater than / 
0.32.
Length of Poincaré plot main cloud.
In this section we develop an approximation to the length of a
Poincaré plot, depicted in Fig. 1B, as a function of
the HRV model's coupling constants Cs and
Cp. Researchers, who are dealing with noisy
data, often employ the SDNN as a measure of Poincaré length
(10, 19). For the purposes of this section, in which sequences lacking random variability are analyzed, it is simpler to
define the length to be the distance between the extreme right- and
left-most points of the Poincaré plot. The agreement between these two measures is a simple scaling by a constant. Thus length (L) is defined as the difference between the largest and
smallest R-R intervals as shown.
Analytically deriving the maximum and minimum of the R-R interval
series from Eq. 12 is not straightforward; fortunately, these quantities can be approximated. By employing the standard approximation (1 + z)
1 = 1 z for z < 1, Eq. 12 can be
approximated as
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(13)
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Expanding the brackets, combining sums, and using standard
trigonometric identities, it is possible to express Eq. 13
as a sum of sinusoids
![−&phgr;<SUB>m</SUB>]+sin [(<IT>&ohgr;<SUB>n</SUB>+&ohgr;<SUB>m</SUB></IT>)<IT>k<A><AC>I</AC><AC>&cjs1171;</AC></A>+&phgr;<SUB>n</SUB>+&phgr;<SUB>m</SUB></IT>]}](/content/vol283/issue5/fulltext/H1873/img025.gif)
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(14)
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Hence, the R-R interval series, RRk = + k
k1, is
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(15)
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Assuming the maximum values of the time-varying sinusoids
(those dependent on k) of Eq. 15 are eventually
sampled simultaneously at some point in time, an approximation to the
upper limit of the length is obtained by replacing the sinusoids with
the value 1. This approach gives the maximum length obtainable, a
figure that is strictly an upper bound, yet also serves as an
approximation to the true length L for modulation
frequencies significantly less than the mean beat frequency. This is by
virtue of having sampled frequently enough to examine arbitrarily close
to the upper bound at some point in time. The upper bound on
L is then twice the sum of the amplitudes of the
frequency components described in Eq. 15. As
Cn < 1, L is largely determined
by the first-order terms. Equation 16 is the first-order
approximation to length. It is noted that this quantity is no longer
the strict upper bound on L due to discarding the higher
order contributions; however, it remains an approximation to the true
length.
|
(16)
|
Therefore, Poincaré plots obtained from our HRV model have a
length approximated by
|
(17)
|
The actual (true) Poincaré plot length as a function of the
HRV model's coupling constants Cs and
Cp over the range 0.0-0.15 is shown in Fig.
8A (obtained via simulations). Length appears to be
dependent on Cs and Cp in
an almost identical manner and to behave linearly, in agreement with
this analysis. Figure 8C compares true length to the
approximation to length given by Eq. 17. For
Cs + Cp < 1, the approximation is in excellent agreement with the true
length. As Cs + Cp
increases, second-order influences begin to become significant due to
nonlinear effects becoming prominent, as expected from the analysis.
The approximately identical manner that the coupling constants control
the length can be explained by noting that sin(x) x when x < 1 and for low modulation
frequencies p < 2HR. Accordingly,
Eq. 17 behaves as
|
(18)
|
These results state that HF and LF modulations affect L
in equivalent manners for slow modulation and in a linear fashion for
small coupling intensities. Under these conditions, length reflects
neither the HF nor the LF modulations more significantly than the
other. Thus, for practical purposes, length may be considered a measure
of total modulation and is akin to the total power of the modulating signal.
Width of the Poincaré plot main cloud.
The width of the main cloud of an R-R interval Poincaré plot
characterizes the dispersion of points about the line of identity. Common measures of the width are the SDSD and the RMSSD of the R-R
intervals (10, 19). As for the length of the model-based Poincaré plot, the lack of any random component is
exploited, and the width is defined to be the distance between the
extremities as depicted in Fig. 1B. Thus the width is
as Fig. 7 details. This expression
involves the "delta" R-R intervals,
RRk = RRk RRk1, which are also known as the successive
differences of the R-R intervals. They are given by
![+2<A><AC>I</AC><AC>&cjs1171;</AC></A><SUP>2</SUP> <LIM><OP>∑</OP><LL>n=1</LL><UL>N</UL></LIM> <LIM><OP>∑</OP><LL>m=1</LL><UL>N</UL></LIM> <FR><NU>C<SUB>n</SUB>C<SUB>m</SUB></NU><DE>&ohgr;<SUB>n</SUB></DE></FR><FENCE><AR><R><C> sin<SUP>2</SUP> <FENCE><FR><NU>(<IT>&ohgr;<SUB>n</SUB>−&ohgr;<SUB>m</SUB></IT>)<IT><A><AC>I</AC><AC>&cjs1171;</AC></A></IT></NU><DE>2</DE></FR></FENCE> sin [(<IT>&ohgr;<SUB>n</SUB>−&ohgr;<SUB>m</SUB></IT>)<IT>k<A><AC>I</AC><AC>&cjs1171;</AC></A>+</IT>(<IT>&phgr;<SUB>n</SUB>−&phgr;<SUB>m</SUB></IT>)<IT>−</IT>(<IT>&ohgr;<SUB>n</SUB>−&ohgr;<SUB>m</SUB></IT>)<IT><A><AC>I</AC><AC>&cjs1171;</AC></A></IT>]</C></R><R><C>+sin<SUP>2</SUP> <FENCE><FR><NU>(<IT>&ohgr;<SUB>n</SUB>+&ohgr;<SUB>m</SUB></IT>)<IT><A><AC>I</AC><AC>&cjs1171;</AC></A></IT></NU><DE>2</DE></FR></FENCE> sin [(<IT>&ohgr;<SUB>n</SUB>+&ohgr;<SUB>m</SUB></IT>)<IT>k<A><AC>I</AC><AC>&cjs1171;</AC></A>+</IT>(<IT>&phgr;<SUB>n</SUB>+&phgr;<SUB>m</SUB></IT>)<IT>−</IT>(<IT>&ohgr;<SUB>n</SUB>+&ohgr;<SUB>m</SUB></IT>)<IT><A><AC>I</AC><AC>&cjs1171;</AC></A></IT>]</C></R></AR></FENCE>](/content/vol283/issue5/fulltext/H1873/img033.gif)
|
(19)
|
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Fig. 7.
Poincaré width, W, is measured as the
largest difference between consecutive intervals, multiplied by the
square root of 2. See text for additional information.
|
|
As can be seen from Eq. 19, the RR intervals posses
no direct current component, which is expected due to the zero average. Similar frequency content is present as for the length, except for being phase shifted and being multiplied by an extra
sin( · ) term leading to the squared coefficient. An
approximation to W is determined by taking an upper bound
for W (by replacing the time-varying sinusoids with unity)
and retaining only first-order terms as detailed in the calculations
for length
|
(20)
|
For our HRV model, this expression is
|
(21)
|
Figure 8B details how
true width varies as the coupling parameters are varied over the range
0.0-0.15. A comparison of Eq. 21 to the true width is
given in Fig. 8D. It is seen that the approximation to
W is accurate when Cs + Cp < 1 but deviates widely as
Cs + Cp becomes
large, due mainly to second-order influences becoming prominent. It can
be seen from Fig. 8B that the level of HF modulation, Cp, is the dominant parameter controlling width.
This property is clearly seen from the analysis, especially for small
modulation frequencies (s < 2HR) as
Eq. 21 behaves approximately as
|
(22)
|
Roughly speaking, the width of a Poincaré plot is a function
of the weighted sum of the LF and HF amplitudes, where each amplitude
is weighted by the respective angular frequency. Accordingly, HF
components contribute to the width in larger amounts, and
LF components contribute at minor, yet still significant levels. As will be explained later, Poincaré plot width should correlate highly with HF power and correlate at small levels with LF power.
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Fig. 8.
Plots of width and length of Poincaré plot main
clouds as the two coupling parameters are varied over the range
0.0-0.15. A and B: length and width,
respectively, obtained from simulated R-R intervals. C and
D: how analytic approximations to length (solid line
compared with Eq. 17, dotted line) and width (solid line
compared with Eq. 12, dotted line) compare.
|
|
Poincaré plot morphological properties for the HRV model.
As the previous sections have shown, the correspondence between the HRV
model's parameters and the Poincaré plot's shape can be
accurately approximated by a linear transformation for small coupling
intensities
|
(23)
|
The significance of this result is that the morphology of a
Poincaré plot encodes the amplitudes of the modulation signal, allowing recovery of the amplitudes for signals composed of two known
frequency components.
|
(24)
|
For our model, it is theoretically possible to estimate similar
characteristics to HRV spectral analysis, such as LF power, HF power,
and HF/LF ratios, from the shape of the Poincaré plot by
assigning appropriate values to the constants s and
p. This is in addition to investigating the detailed
beat-to-beat characteristics of HRV data. It should be noted that this
property only applies exactly for modulation signals composed of only
two frequency components. How well the correspondence generalizes to
actual HRV data is dependent on how well the HRV spectrum is
approximated by two dominant peaks.
| |
GENERALIZATION TO REAL HRV DATA |
At this point it is interesting to consider how well the results
of the previous section apply to actual data obtained from subjects
under various autonomic conditions. The results are not expected to
apply completely because they stem from a model of a discrete spectrum,
but the principles identified by the analysis should be evident.
Data set acquisition.
We employ the data set of a previous study (9) because it
contains subjects over a wide range of autonomic conditions. The data
set consists of 10 healthy subjects (5 female, 5 male) aged between 20 and 40 yr (30.2 ± 7.2 means ± SD). Each subject underwent
four autonomic purtubations: 1) baseline study with subjects
in the supine position in a quiet environment; 2) 70° head-up tilt, which increases sympathetic activity and decreases parasympathetic activity; 3) atropine infusion, which
markedly decreases parasympathetic nervous system activity; and
4) transdermal scopolomine, which increases parasympathetic
nervous activity. In all, 40 records were collected, each containing
1,024 R-R intervals.
Data set analysis.
For each data set, the length and width of the Poincaré plot and
the LF and HF power were calculated. The length was calculated by
L = 2SDRR, and the width by W = 
SDSD, as can be derived from simple geometry. The LF and HF
parameters were calculated by using the autoagressive technique with
the modified covariance technique (12). The bands were
LF = 0.04-0.15 Hz and HF = 0.15-0.4 Hz. The length
and width of the Poincaré plot were then derived from the LF and
HF power by using Eq. 23 with
Cs/HR2 = 
and
Cp/HR2 = 
. The coupling constants need to be divided by
HR twice, once as HRV spectral analysis techniques assume
that the modulation signal in Eq. 3 is dimensionless
(1-3) and again as we need to multiply by the mean
beat interval to normalize the discrete spectra units to those of the
continuous spectrum (3). The derived length and width are
compared with the actual length and width by plotting them against each
other as scatterplot. The value of HR is calculated as the
inverse of the average R-R interval. The choice of suitable values for
s and p is akin to the choice of the LF
and HF bands. The midfrequencies of the bands is the most appropriate
choice, i.e., s = 2(0.1) and
p = 2(0.28) rad/s.
Figure 9A displays the derived
length on the vertical axis and the actual length on the horizontal
axis. The points do reflect the line of identity; however, there exists
a fair amount of variability, which indicates that that Eq. 23 does not hold entirely. The goodness of fit to the line of
identity can be quantified by the correlation coefficient. Figure
9A has a correlation coefficient of 0.94, indicating that
that Eq. 23 holds reasonably well in determining the actual
length. Equation 23 has a tendency to underestimate the
actual length, which is partially explained by noting that the length
is a measure of all the modulation, yet LF and HF measure only the
power from 0.04 Hz upwards, ignoring the VLF band. The derived width
versus the actual width is plotted on Fig. 9B. A very good
fit with a correlation coefficient of 0.97 occurs. The superior
performance of Eq. 23, when predicting the width of a Poincaré plot, can be explained by noting that ignoring the VLF power will not adversely affect the width as it is dominated by HF
power.
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Fig. 9.
Comparisons of derived parameters versus actual values.
Correlation coefficients are 0.94 (A, length from LF and
HF), 0.97 (B, width from LF and HF), 0.81 (C, LF
from length and width), and 0.94 (D, HF from length and
width).
|
|
The same analysis is now repeated for the reverse situation. Starting
with the length and width of a Poincaré plot, we derive the LF
and HF power by using Eq. 24 with = Cs/HR2 and = Cp/HR2. The derived
values of LF and HF are compared with the actual LF and HF values
calculated by spectral analysis. Figure 9C displays the
actual LF power versus the derived LF power. A correlation coefficient
of 0.81 indicates a reasonable fit, and it is clear that the main trend
of the relationship between LF power and length and width expressed by
Eq. 24 holds. Figure 9D compares
derived HF power with actual HF power. A correlation coefficient of
0.93 indicates that Eq. 24 explains the
dependency of HF on the length and width very well.
These results clearly show that the principles identified from
Eqs. 23 and 24 are indeed present for actual HRV
data. The fact that a discrete spectrum consisting of only two
components can explain so much about the relationships among LF, HF,
length, and width of a Poincaré plot is remarkable.
Poincaré plot morphology for real data.
The results of the previous sections imply that the width is a measure
of short-term variability and the length is a measure of total
variability. This result has consequences for the correlations between
frequency domain indexes and Poincaré plot indexes. Attempting to
correlate LF power with Poincaré length (or equivalent SDNN measures) will explain only part of the variations in Poincaré length. Substantial portions of the variations are due to the codependency with HF power and will appear as uncorrelated noise. In
data sets where significant variations in both LF and HF power are
present, our model predicts that Poincaré length will correlate reasonably well with both LF and HF power; however, it will correlate highly with neither due to the variations introduced by the other. For
Poincaré width, the dependencies on HF power are stronger than
those of LF power. A strong correlation is expected when comparing HF
power to Poincaré width, because the variations due to LF power
will be small. LF power should correlate with Poincaré width,
albeit at low levels, because LF power does influence the width, but
the variations present due to HF power are large and reduce the
correlation coefficient markedly.
Many of these results have already been shown experimentally.
Specifically, our findings corroborate the findings of Otzenberger et
al. (13), who found that SDNN (Poincaré length)
correlated with both LF and HF power and RMSSD (Poincaré width)
correlated with HF power and, to a lesser extent, LF power. Tullppo et
al. (19), who investigated HRV and exercise, also present
experimental results that agree: SDNN correlated almost equally with HF
(Pearson's correlation coefficient: r = 0.75) and LF
(r = 0.72) power, and RMSSD correlated highly with HF power
(r = 0.97) and to a lesser yet significant extent with LF
power (r = 0.65).
In conclusion, we develop a new mathematical model with a network of
oscillators. For the first time, Poincaré plots are generated
from the model and compared with Poincaré plots generated from
subjects under various autonomic conditions. Now one can clearly
understand how various autonomic regimes appear on the Poincaré
plot through the use of the model.
Traditionally, researchers have identified length and width of
Poincaré plots with LF and HF powers, respectively, of the HRV
signal. However, with the use of our model, we establish that the
length and width are not separately related but are a weighted combination of LF and HF power. This investigation provides a theoretical link between frequency domain spectral analysis techniques and time domain Poincaré plot analysis.
To determine the degree to which our results generalize to actual HRV
data, we applied the model-based formulas to a set of clinical data.
The results indicate that the formulas do identify clear trends in the
relationships between the spectral components and Poincaré length
and width. This gives definitive evidence that for HRV data, the length
is a display of total modulation and the width indicates predominately
short-term modulation. In summary, this study provides clear
mathematical insight into the nature of observed data.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: M. Palaniswami, Dept. of Electrical & Electronic Engineering, The Univ. of
Melbourne, Victoria 3010, Australia (E-mail:
swami{at}ee.mu.oz.au).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
10.1152/ajpheart.00405.2000
Received 5 May 2000; accepted in final form 22 May 2002.
| |
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Am J Physiol Heart Circ Physiol 283(5):H1873-H1886
0363-6135/02 $5.00
Copyright © 2002 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
![<IT>+&ohgr;<SUB>n</SUB>&dgr;<SUB>k</SUB></IT> cos (<IT>&ohgr;<SUB>n</SUB>k<A><AC>I</AC><AC>&cjs1171;</AC></A>+&phgr;<SUB>n</SUB></IT>)]](/content/vol283/issue5/fulltext/H1873/img018.gif)
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