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/* Century Schoolbook font is very similar to Computer Modern Math: cmmi */
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ecgsyn-1.0.0/paper/footnode.html
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!--Converted with jLaTeX2HTML 2002 (1.62) JA patch-1.4
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patched version by: Kenshi Muto, Debian Project.
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LaTeX2HTML 2002 (1.62),
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
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<HEAD>
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<TITLE>Footnotes</TITLE>
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<META NAME="description" CONTENT="Footnotes">
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<BODY >
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<DL>
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<DT><A NAME="foot31">...<IMG
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WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img6.png"
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ALT="$^{2,3}$"></A><A
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HREF="index.shtml#tex2html1"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/share/latex2html/icons/footnote.png"></SUP></A></DT>
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<DD><IMG
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WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img5.png"
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ALT="$^1$">Department of Engineering Science,
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University of Oxford, Oxford OX1 3PJ.
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<PRE>.
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</PRE>
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</DD>
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<DT><A NAME="foot32">...</A><A
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HREF="index.shtml#tex2html2"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/share/latex2html/icons/footnote.png"></SUP></A></DT>
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<DD><IMG
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WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$^2$">Mathematical Institute, University of Oxford, Oxford OX1 3LB.
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<PRE>.
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</PRE>
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</DD>
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<DT><A NAME="foot33">...</A><A
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HREF="index.shtml#tex2html3"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/share/latex2html/icons/footnote.png"></SUP></A></DT>
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<DD><IMG
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WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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ALT="$^3$">Centre for the Analysis of Time Series, London School
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of Economics, London WC2A 2AE.
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<PRE>.
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</PRE>
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</DD>
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<DT><A NAME="foot34">...</A><A
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HREF="index.shtml#tex2html4"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/share/latex2html/icons/footnote.png"></SUP></A></DT>
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<DD>E-mail: mcsharry@robots.ox.ac.uk
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<PRE>.
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</PRE>
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</DD>
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<html>
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<head><title>Index of /static/published-projects/ecgsyn/1.0.0/paper/</title></head>
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<body>
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<h1>Index of /static/published-projects/ecgsyn/1.0.0/paper/</h1><hr><pre><a href="../">../</a>
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<a href="ecgpqrst.css">ecgpqrst.css</a> 12-Apr-2019 17:35 891
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<a href="footnode.html">footnode.html</a> 12-Apr-2019 17:35 2394
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<a href="img1.png">img1.png</a> 12-Apr-2019 17:35 446
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<TITLE>A dynamical model for generating synthetic electrocardiogram signals</TITLE>
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<H1 ALIGN="CENTER">A dynamical model for generating synthetic electrocardiogram signals</H1>
|
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<P ALIGN="CENTER"><STRONG>Patrick E. McSharry<IMG
|
||||
WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
|
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ALT="$^{1,2}$">, Gari Clifford<IMG
|
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WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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||||
SRC="img5.png"
|
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ALT="$^1$">,
|
||||
Lionel Tarassenko<IMG
|
||||
WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img5.png"
|
||||
ALT="$^1$"> and Leonard A. Smith<IMG
|
||||
WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img6.png"
|
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ALT="$^{2,3}$">
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<A NAME="tex2html1"
|
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HREF="footnode.html#foot31"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/share/latex2html/icons/footnote.png"></SUP></A>
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<A NAME="tex2html2"
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HREF="footnode.html#foot32"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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<A NAME="tex2html4"
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HREF="footnode.html#foot34"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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||||
<center><table bgcolor="lightblue" width="90%">
|
||||
<tr><td>
|
||||
<p>
|
||||
This article originally appeared in <em>IEEE Transactions on Biomedical
|
||||
Engineering</em>, <b>50</b>(3):289-294; March 2003. Please cite this
|
||||
publication when referencing this material.
|
||||
<p>
|
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(C)2002 IEEE. Personal use of this material is permitted. However, permission
|
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to reprint/republish this material for advertising or promotional purposes or
|
||||
for creating new collective works for resale or redistribution to servers or
|
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lists, or to reuse any copyrighted component of this work in other works must
|
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be obtained from the IEEE. This material is presented to ensure timely
|
||||
dissemination of scholarly and technical work. Copyright and all rights therein
|
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are retained by authors or by other copyright holders. All persons copying this
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information are expected to adhere to the terms and constraints invoked by each
|
||||
author's copyright. In most cases, these works may not be reposted without the
|
||||
explicit permission of the copyright holder.
|
||||
<p>
|
||||
Software that implements the model described in this paper is freely available
|
||||
<a href="../">here</a>.
|
||||
</td></tr>
|
||||
</table></center>
|
||||
|
||||
<H3>Abstract:</H3>
|
||||
<DIV>
|
||||
A dynamical model based on three coupled ordinary differential equations
|
||||
is introduced which is capable of generating realistic synthetic
|
||||
electrocardiogram (ECG) signals. The operator can specify the mean and
|
||||
standard deviation of the heart rate, the morphology of the PQRST
|
||||
cycle and the power spectrum of the RR tachogram.
|
||||
In particular, both Respiratory Sinus Arrhythmia at the high frequencies
|
||||
(HF) and Mayer waves at the low frequencies (LF)
|
||||
together with the LF/HF ratio are incorporated in the model.
|
||||
Much of the beat-to-beat variation in morphology and timing of the human
|
||||
ECG, including QT dispersion and R-peak amplitude modulation
|
||||
are shown to result. This model may be employed to assess biomedical
|
||||
signal processing techniques which are used to compute clinical statistics
|
||||
from the ECG.
|
||||
</DIV>
|
||||
<P>
|
||||
|
||||
<BR>
|
||||
<IMG
|
||||
WIDTH="555" HEIGHT="51" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="\begin{keywords}
|
||||
Dynamical model, synthetic ECG, QRS morphology, Respiratory si...
|
||||
... Heart rate variability, RR tachogram, RR-interval, QT-interval.
|
||||
\end{keywords}">
|
||||
<BR>
|
||||
<BR><HR>
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"></A>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html18"
|
||||
HREF="node1.html">Introduction</A>
|
||||
<LI><A NAME="tex2html19"
|
||||
HREF="node2.html">ECG morphology</A>
|
||||
<LI><A NAME="tex2html20"
|
||||
HREF="node3.html">Heart rate variability</A>
|
||||
<LI><A NAME="tex2html21"
|
||||
HREF="node4.html">The dynamical model</A>
|
||||
<LI><A NAME="tex2html22"
|
||||
HREF="node5.html">Results</A>
|
||||
<LI><A NAME="tex2html23"
|
||||
HREF="node6.html">Conclusions</A>
|
||||
<LI><A NAME="tex2html24"
|
||||
HREF="node7.html">Acknowledgements</A>
|
||||
<LI><A NAME="tex2html25"
|
||||
HREF="node8.html">Bibliography</A>
|
||||
<LI><A NAME="tex2html26"
|
||||
HREF="node9.html">About this document ...</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
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<BR><HR>
|
||||
<ADDRESS>
|
||||
|
||||
2003-10-08
|
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</HTML>
|
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169
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
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* with significant contributions from:
|
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
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<HTML>
|
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|
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<TITLE>Introduction</TITLE>
|
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|
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|
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||||
<H1><A NAME="SECTION00010000000000000000">
|
||||
Introduction</A>
|
||||
</H1>
|
||||
The
|
||||
electrocardiogram (ECG) is a time-varying signal reflecting the
|
||||
ionic current flow which causes the cardiac fibres to contract and
|
||||
subsequently relax.
|
||||
The surface ECG is obtained by recording the potential difference between
|
||||
two electrodes placed on the surface of the skin.
|
||||
A single normal cycle of the ECG represents the successive atrial
|
||||
depolarisation/repolarisation and ventricular depolarisation/repolarisation
|
||||
which occurs with every heart beat. These can be approximately associated
|
||||
with the peaks and troughs of the ECG waveform labelled P,Q,R,S and T as
|
||||
shown in Fig. <A HREF="node1.html#f:garipqrst">1</A>.
|
||||
|
||||
Extracting useful clinical information from the real (noisy) ECG requires
|
||||
reliable signal processing techniques [<A
|
||||
HREF="node8.html#goldberger77">1</A>].
|
||||
These include R-peak detection [<A
|
||||
HREF="node8.html#pan85">2</A>,<A
|
||||
HREF="node8.html#kaplan91">3</A>],
|
||||
QT-interval detection [<A
|
||||
HREF="node8.html#davey99">4</A>]
|
||||
and the derivation of heart rate and respiration rate
|
||||
from the ECG [<A
|
||||
HREF="node8.html#moody85">5</A>,<A
|
||||
HREF="node8.html#moody86">6</A>].
|
||||
The RR-interval is the time between
|
||||
successive R-peaks, the inverse of this time interval gives the
|
||||
instantaneous heart rate.
|
||||
A series of RR-intervals is known as a RR tachogram and variability of
|
||||
these RR-intervals reveals important information about the physiological state
|
||||
of the subject [<A
|
||||
HREF="node8.html#malik95">7</A>].
|
||||
At present, new biomedical signal processing algorithms are usually evaluated
|
||||
by applying them to ECGs in a large database such as the Physionet
|
||||
database [<A
|
||||
HREF="node8.html#physionet">8</A>].
|
||||
While this gives the operator an indication of the accuracy of a given
|
||||
algorithm when applied to real data, it is difficult to infer how the
|
||||
performance would vary in different clinical settings with a range of
|
||||
noise levels and sampling frequencies.
|
||||
Having access to realistic artificial ECG signals may
|
||||
facilitate this evaluation.
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:garipqrst"></A><A NAME="53"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
||||
Morphology of a mean PQRST-complex of an ECG recorded
|
||||
from a normal human.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="279" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=garipqrst.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
This paper presents a model for generating a synthetic ECG signal with
|
||||
realistic PQRST morphology and prescribed heart rate dynamics.
|
||||
The aim of this model is to provide a
|
||||
standard realistic ECG signal with known characteristics,
|
||||
which can be generated with specific statistics
|
||||
such as the mean and standard deviation of the heart rate
|
||||
and frequency-domain characteristics of heart rate variability (HRV),
|
||||
such as the LF/HF ratio, defined as the ratio of power between 0.015 and
|
||||
0.15 Hz and 0.15 and 0.4 Hz in the RR tachogram [<A
|
||||
HREF="node8.html#malik95">7</A>].
|
||||
By generating a signal which represents a <I>typical</I> human ECG, this
|
||||
facilitates a comparison of different signal processing techniques.
|
||||
A synthetic ECG can be generated with different sampling frequencies and
|
||||
different noise levels in order to establish the performance of
|
||||
a given technique. This performance can be presented, for example,
|
||||
as the number of true positives, false positives, true negatives and false
|
||||
negatives for each test. Such performance assessment could be used as
|
||||
a ``standard'' and would enable clinicians to ascertain which biomedical
|
||||
signal processing techniques were best for a given application.
|
||||
|
||||
The layout of this paper is as follows; section <A HREF="node2.html#s:morphology">II</A>
|
||||
summarises the physiological mechanisms underlying the cardiac cycle and
|
||||
reviews the morphological variability which is reflected in the ECG signal.
|
||||
A brief review of HRV is presented in
|
||||
section <A HREF="node3.html#s:hrv">III</A>. The dynamical model is introduced in
|
||||
section <A HREF="node4.html#s:model">IV</A> and investigated in section <A HREF="node5.html#s:results">V</A>.
|
||||
Section <A HREF="node6.html#s:conclusions">VI</A> concludes and discusses extensions to
|
||||
the model which may be useful for simulating specific disorders.
|
||||
|
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<H1><A NAME="SECTION00020000000000000000"></A>
|
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<A NAME="s:morphology"></A>
|
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<BR>
|
||||
ECG morphology
|
||||
</H1>
|
||||
Each beat of the heart can be observed as a series of deflections away from
|
||||
the baseline on the ECG. These deflections reflect the time
|
||||
evolution of electrical activity in the heart which initiates
|
||||
muscle contraction. A single sinus (normal) cycle of the ECG,
|
||||
corresponding to one
|
||||
heart beat, is traditionally labelled with the letters P,Q,R,S and T
|
||||
on each of its turning points (Fig. <A HREF="node1.html#f:garipqrst">1</A>).
|
||||
The ECG may be divided into the following sections:
|
||||
|
||||
<UL>
|
||||
<LI>P-wave: a small low-voltage deflection away from the baseline
|
||||
caused by the depolarisation of the atria prior to atrial contraction as the
|
||||
activation (depolarisation) wave-front propagates
|
||||
from the SA node through the atria.
|
||||
<BR>
|
||||
|
||||
</LI>
|
||||
<LI>PQ-interval:
|
||||
the time between the beginning of atrial
|
||||
depolarisation and the beginning of ventricular depolarisation.
|
||||
<BR>
|
||||
|
||||
</LI>
|
||||
<LI>QRS-complex: the largest-amplitude portion of the ECG, caused by
|
||||
currents generated when the ventricles depolarise prior to their
|
||||
contraction. Although atrial repolarisation occurs before ventricular
|
||||
depolarisation, the latter waveform (i.e. the QRS-complex) is of much greater
|
||||
amplitude and atrial repolarisation is therefore not seen on the ECG.
|
||||
<BR>
|
||||
|
||||
</LI>
|
||||
<LI>QT-interval: the time between the onset of ventricular
|
||||
depolarisation and the end of ventricular repolarisation.
|
||||
Clinical studies have demonstrated that the QT-interval
|
||||
increases linearly as the RR-interval increases [<A
|
||||
HREF="node8.html#davey99">4</A>].
|
||||
Prolonged QT-interval may be associated with delayed ventricular
|
||||
repolarisation which may cause ventricular tachyarrhythmias
|
||||
leading to sudden cardiac death [<A
|
||||
HREF="node8.html#schwartz78">9</A>].
|
||||
|
||||
</LI>
|
||||
<LI>ST-interval: the time between the end of S-wave and the
|
||||
beginning of T-wave. Significantly elevated or depressed amplitudes away
|
||||
from the baseline are often associated with cardiac illness.
|
||||
|
||||
</LI>
|
||||
<LI>T-wave: ventricular repolarisation, whereby the cardiac muscle
|
||||
is prepared for the next cycle of the ECG.
|
||||
<BR>
|
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|
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|
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|
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Heart rate variability
|
||||
</H1>
|
||||
Analysis of variations in the instantaneous heart rate time series
|
||||
using the beat-to-beat RR-intervals (the RR tachogram) is known as
|
||||
Heart Rate Variability (HRV) analysis [<A
|
||||
HREF="node8.html#malik95">7</A>,<A
|
||||
HREF="node8.html#eursoccard96">10</A>].
|
||||
HRV analysis has been shown to provide an assessment of cardiovascular
|
||||
disease [<A
|
||||
HREF="node8.html#crawford99">11</A>].
|
||||
The heart rate may be increased by slow acting sympathetic activity
|
||||
or decreased by fast acting parasympathetic (vagal) activity.
|
||||
The balance between the effects of the sympathetic and parasympathetic
|
||||
systems, the two opposite acting branches of the autonomic nervous system,
|
||||
is referred to as the sympathovagal balance and
|
||||
is believed to be reflected in the beat-to-beat changes of the
|
||||
cardiac cycle [<A
|
||||
HREF="node8.html#malik95">7</A>].
|
||||
The heart rate is given by the reciprocal of the RR-interval in units
|
||||
of beats per minute.
|
||||
Spectral analysis of the RR tachogram is typically used to
|
||||
estimate the effect of the sympathetic and parasympathetic modulation
|
||||
of the RR-intervals. The two main frequency bands of interest
|
||||
are referred to as the Low-Frequency (LF) band (0.04 to 0.15 Hz)
|
||||
and the High-Frequency (HF) band (0.15 to 0.4 Hz) [<A
|
||||
HREF="node8.html#eursoccard96">10</A>].
|
||||
Sympathetic tone is believed to influence the LF component whereas
|
||||
both sympathetic and parasympathetic activity have an effect on the HF
|
||||
component [<A
|
||||
HREF="node8.html#malik95">7</A>]. The ratio of the power contained in
|
||||
the LF and HF components has been used as a measure of the sympathovagal
|
||||
balance [<A
|
||||
HREF="node8.html#malik95">7</A>,<A
|
||||
HREF="node8.html#eursoccard96">10</A>].
|
||||
|
||||
Respiratory Sinus Arrhythmia (RSA) [<A
|
||||
HREF="node8.html#hales1733">12</A>,<A
|
||||
HREF="node8.html#ludwig1847">13</A>]
|
||||
is the name given to the oscillation in the RR tachogram due to
|
||||
parasympathetic activity which is synchronous with the respiratory cycle.
|
||||
The RSA oscillation manifests itself as a
|
||||
peak in the HF band of the spectrum. For example, 15 breaths per minute
|
||||
corresponds to a 4 second oscillation with a peak in the power spectrum at
|
||||
0.25 Hz. A second peak is often found in the LF
|
||||
band of the spectrum at approximately 0.1 Hz. While the cause of this
|
||||
10 second rhythm is strongly debated, one possible explanation is that it
|
||||
may be due to baroreflex regulation which creates the so-called
|
||||
<I>Mayer waves</I> in the blood pressure signal [<A
|
||||
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|
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SRC="file:/usr/share/latex2html/icons/up.png"></A>
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HREF="node3.html">
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SRC="file:/usr/share/latex2html/icons/prev.png"></A>
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<BR>
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<B> Next:</B> <A NAME="tex2html66"
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HREF="node5.html">Results</A>
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<B> Up:</B> <A NAME="tex2html64"
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HREF="index.shtml">A dynamical model for</A>
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<B> Previous:</B> <A NAME="tex2html58"
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HREF="node3.html">Heart rate variability</A>
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<BR>
|
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<BR>
|
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<!--End of Navigation Panel-->
|
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|
||||
<H1><A NAME="SECTION00040000000000000000"></A>
|
||||
<A NAME="s:model"></A>
|
||||
<BR>
|
||||
The dynamical model
|
||||
</H1>
|
||||
The model generates a trajectory in a three-dimensional state space
|
||||
with co-ordinates <IMG
|
||||
WIDTH="56" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img1.png"
|
||||
ALT="$(x,y,z)$">. Quasi-periodicity of the ECG is reflected
|
||||
by the movement of the trajectory around an attracting limit cycle of
|
||||
unit radius in the <IMG
|
||||
WIDTH="41" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$(x,y)$">-plane.
|
||||
Each revolution on this circle corresponds to one RR-interval or heart beat.
|
||||
Inter-beat variation in the ECG is reproduced using the
|
||||
motion of the trajectory in the <IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$z$">-direction.
|
||||
Distinct points on the ECG, such as the P,Q,R,S and T are described by
|
||||
<I>events</I> corresponding to negative and positive attractors/repellors
|
||||
in the <IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$z$">-direction. These events are placed at fixed angles along the
|
||||
unit circle given by <IMG
|
||||
WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img13.png"
|
||||
ALT="$\theta_P$">, <IMG
|
||||
WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img14.png"
|
||||
ALT="$\theta_Q$">,<IMG
|
||||
WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="$\theta_R$">,<IMG
|
||||
WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img16.png"
|
||||
ALT="$\theta_S$"> and
|
||||
<IMG
|
||||
WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
ALT="$\theta_T$"> (see Fig. <A HREF="node4.html#f:pqrst3d">2</A>). When the trajectory
|
||||
approaches one of these events, it is pushed upwards or downwards
|
||||
away from the limit cycle, and then as it moves away it is pulled back
|
||||
towards the limit cycle.
|
||||
|
||||
The dynamical equations of motion are given by a set of three ordinary
|
||||
differential equations
|
||||
<BR>
|
||||
<DIV ALIGN="CENTER"><A NAME="e:pqrst"></A>
|
||||
<!-- MATH
|
||||
\begin{eqnarray}
|
||||
{\dot x} &=& \alpha x - \omega y, \nonumber \\
|
||||
{\dot y} &=& \alpha y + \omega x, \nonumber \\
|
||||
{\dot z} &=& - \!\!\!\!\!\! \sum_{i \in \{P,Q,R,S,T\}} \!\!\!\!\!\!
|
||||
a_i \Delta \theta_i
|
||||
\exp(-\Delta \theta_i^2 / 2 b_i^2) - (z - z_0),
|
||||
\end{eqnarray}
|
||||
-->
|
||||
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
|
||||
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
|
||||
WIDTH="13" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img18.png"
|
||||
ALT="$\displaystyle {\dot x}$"></TD>
|
||||
<TD ALIGN="CENTER" NOWRAP><IMG
|
||||
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$\textstyle =$"></TD>
|
||||
<TD ALIGN="LEFT" NOWRAP><IMG
|
||||
WIDTH="66" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$\displaystyle \alpha x - \omega y,$"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
</TD></TR>
|
||||
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
|
||||
WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$\displaystyle {\dot y}$"></TD>
|
||||
<TD ALIGN="CENTER" NOWRAP><IMG
|
||||
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$\textstyle =$"></TD>
|
||||
<TD ALIGN="LEFT" NOWRAP><IMG
|
||||
WIDTH="66" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img22.png"
|
||||
ALT="$\displaystyle \alpha y + \omega x,$"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
</TD></TR>
|
||||
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
|
||||
WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img23.png"
|
||||
ALT="$\displaystyle {\dot z}$"></TD>
|
||||
<TD ALIGN="CENTER" NOWRAP><IMG
|
||||
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$\textstyle =$"></TD>
|
||||
<TD ALIGN="LEFT" NOWRAP><IMG
|
||||
WIDTH="304" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img24.png"
|
||||
ALT="$\displaystyle - \!\!\!\!\!\! \sum_{i \in \{P,Q,R,S,T\}} \!\!\!\!\!\!
|
||||
a_i \Delta \theta_i
|
||||
\exp(-\Delta \theta_i^2 / 2 b_i^2) - (z - z_0),$"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(1)</TD></TR>
|
||||
</TABLE></DIV>
|
||||
<BR CLEAR="ALL"><P></P>
|
||||
where <!-- MATH
|
||||
$\alpha = 1 - \sqrt{x^2 + y^2}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="130" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$\alpha = 1 - \sqrt{x^2 + y^2}$">,
|
||||
<!-- MATH
|
||||
$\Delta \theta_i = (\theta - \theta_i) \ {\rm mod} \ 2 \pi$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="163" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img26.png"
|
||||
ALT="$\Delta \theta_i = (\theta - \theta_i) \ {\rm mod} \ 2 \pi$">,
|
||||
<!-- MATH
|
||||
$\theta = {\rm atan2}(y,x)$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="109" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img27.png"
|
||||
ALT="$\theta = {\rm atan2}(y,x)$"> and <IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img28.png"
|
||||
ALT="$\omega$"> is the angular velocity
|
||||
of the trajectory as it moves around the limit cycle.
|
||||
Baseline wander was introduced by coupling the baseline value <IMG
|
||||
WIDTH="19" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img29.png"
|
||||
ALT="$z_0$">
|
||||
in (<A HREF="node4.html#e:pqrst">1</A>) to the respiratory frequency <IMG
|
||||
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$f_2$"> using
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
z_0(t) = A \sin(2 \pi f_2 t),
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="e:baseline"></A><IMG
|
||||
WIDTH="142" HEIGHT="28" BORDER="0"
|
||||
SRC="img31.png"
|
||||
ALT="\begin{displaymath}
|
||||
z_0(t) = A \sin(2 \pi f_2 t),
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(2)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
where <IMG
|
||||
WIDTH="66" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$A = 0.15$"> mV.
|
||||
|
||||
These equations of motion given by (<A HREF="node4.html#e:pqrst">1</A>) were integrated
|
||||
numerically using a fourth order
|
||||
Runge-Kutta method [<A
|
||||
HREF="node8.html#press92">15</A>] with a fixed time step <!-- MATH
|
||||
$\Delta t = 1/f_s$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="75" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img33.png"
|
||||
ALT="$\Delta t = 1/f_s$">
|
||||
where <IMG
|
||||
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$f_s$"> is the sampling frequency.
|
||||
|
||||
Visual analysis of a section of typical ECG from a normal subject
|
||||
was used to suggest suitable times (and therefore angles <IMG
|
||||
WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$\theta_i$">)
|
||||
and values of <IMG
|
||||
WIDTH="18" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img36.png"
|
||||
ALT="$a_i$"> and <IMG
|
||||
WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img37.png"
|
||||
ALT="$b_i$"> for the PQRST points.
|
||||
The times and angles are specified relative to
|
||||
the position of the R-peak as shown in Table <A HREF="node4.html#t:pqrst">I</A>.
|
||||
|
||||
A trajectory generated by equation (<A HREF="node4.html#e:pqrst">1</A>) in three-dimensions
|
||||
corresponding to <IMG
|
||||
WIDTH="56" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img1.png"
|
||||
ALT="$(x,y,z)$"> is illustrated in Fig. <A HREF="node4.html#f:pqrst3d">2</A>.
|
||||
This demonstrates how the
|
||||
positions of the events <IMG
|
||||
WIDTH="89" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img38.png"
|
||||
ALT="$P,Q,R,S,T$"> act on the trajectory in the
|
||||
<IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$z$">-direction as it precesses around the unit circle in the <IMG
|
||||
WIDTH="41" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$(x,y)$">-plane.
|
||||
The <IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$z$"> variable from the three-dimensional system (<A HREF="node4.html#e:pqrst">1</A>)
|
||||
yields a synthetic ECG with realistic PQRST morphology
|
||||
(Fig. <A HREF="node4.html#f:pqrstcomplex">3</A>). The similarity between the synthetic
|
||||
ECG and the real ECG may be seen by comparing Fig. <A HREF="node4.html#f:pqrstcomplex">3</A>
|
||||
with Fig. <A HREF="node1.html#f:garipqrst">1</A>. Note that noise has not been added
|
||||
to the model at this point.
|
||||
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
|
||||
<DIV ALIGN="CENTER">
|
||||
<A NAME="286"></A>
|
||||
<TABLE CELLPADDING=3 BORDER="1">
|
||||
<CAPTION><STRONG>Table I:</STRONG>
|
||||
Parameters of the ECG model given by (<A HREF="node4.html#e:pqrst">1</A>)</CAPTION>
|
||||
<TR><TD ALIGN="LEFT">Index (i)</TD>
|
||||
<TD ALIGN="LEFT">P</TD>
|
||||
<TD ALIGN="LEFT">Q</TD>
|
||||
<TD ALIGN="LEFT">R</TD>
|
||||
<TD ALIGN="LEFT">S</TD>
|
||||
<TD ALIGN="LEFT">T</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT">Time (secs)</TD>
|
||||
<TD ALIGN="LEFT">-0.2</TD>
|
||||
<TD ALIGN="LEFT">-0.05</TD>
|
||||
<TD ALIGN="LEFT">0</TD>
|
||||
<TD ALIGN="LEFT">0.05</TD>
|
||||
<TD ALIGN="LEFT">0.3</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><IMG
|
||||
WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$\theta_i$"> (radians)</TD>
|
||||
<TD ALIGN="LEFT"><!-- MATH
|
||||
$-\frac{1}{3}\pi$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img39.png"
|
||||
ALT="$-\frac{1}{3}\pi$"></TD>
|
||||
<TD ALIGN="LEFT"><!-- MATH
|
||||
$-\frac{1}{12}\pi$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img40.png"
|
||||
ALT="$-\frac{1}{12}\pi$"></TD>
|
||||
<TD ALIGN="LEFT">0</TD>
|
||||
<TD ALIGN="LEFT"><!-- MATH
|
||||
$\frac{1}{12}\pi$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$\frac{1}{12}\pi$"></TD>
|
||||
<TD ALIGN="LEFT"><!-- MATH
|
||||
$\frac{1}{2}\pi$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img42.png"
|
||||
ALT="$\frac{1}{2}\pi$"></TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><IMG
|
||||
WIDTH="18" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img36.png"
|
||||
ALT="$a_i$"></TD>
|
||||
<TD ALIGN="LEFT">1.2</TD>
|
||||
<TD ALIGN="LEFT">-5.0</TD>
|
||||
<TD ALIGN="LEFT">30.0</TD>
|
||||
<TD ALIGN="LEFT">-7.5</TD>
|
||||
<TD ALIGN="LEFT">0.75</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><IMG
|
||||
WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img37.png"
|
||||
ALT="$b_i$"></TD>
|
||||
<TD ALIGN="LEFT">0.25</TD>
|
||||
<TD ALIGN="LEFT">0.1</TD>
|
||||
<TD ALIGN="LEFT">0.1</TD>
|
||||
<TD ALIGN="LEFT">0.1</TD>
|
||||
<TD ALIGN="LEFT">0.4</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
<A NAME="t:pqrst"></A>
|
||||
|
||||
</DIV>
|
||||
</DIV>
|
||||
<BR>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:pqrst3d"></A><A NAME="288"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
|
||||
A typical trajectory generated by the dynamical model
|
||||
(<A HREF="node4.html#e:pqrst">1</A>) in the three-dimensional space given by <IMG
|
||||
WIDTH="56" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img1.png"
|
||||
ALT="$(x,y,z)$">. The dashed
|
||||
line reflects the limit cycle of unit radius while the small circles show the
|
||||
positions of the P,Q,R,S,T events.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="351" HEIGHT="271" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=pqrst3d.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:pqrstcomplex"></A><A NAME="128"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
|
||||
Morphology of one PQRST-complex of the ECG.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="275" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=pqrstcomplex.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
By contrasting the dynamical model (<A HREF="node4.html#e:pqrst">1</A>) with the mechanisms
|
||||
underlying the cardiac cycle, it is obvious that the time required to
|
||||
complete one lap of the limit cycle is equal to the RR-interval
|
||||
of the synthetic ECG signal. Variations in the length of the RR-intervals
|
||||
can be incorporated by varying the angular velocity <IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img28.png"
|
||||
ALT="$\omega$">.
|
||||
|
||||
The effects of both RSA and Mayer waves in the power spectrum
|
||||
<IMG
|
||||
WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$S(f)$"> of the RR-intervals are incorporated
|
||||
by generating RR-intervals which have a bimodal power spectrum
|
||||
consisting of the sum of two Gaussian distributions,
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
S(f) = \frac{\sigma_1^2}{\sqrt{2 \pi c_1^2}}
|
||||
\exp \left( \frac{(f - f_1)^2}{2 c_1^2} \right)
|
||||
+ \frac{\sigma_2^2}{\sqrt{2 \pi c_2^2}}
|
||||
\exp \left( \frac{(f - f_2)^2}{2 c_2^2} \right),
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="e:Sf"></A><IMG
|
||||
WIDTH="423" HEIGHT="48" BORDER="0"
|
||||
SRC="img45.png"
|
||||
ALT="\begin{displaymath}
|
||||
S(f) = \frac{\sigma_1^2}{\sqrt{2 \pi c_1^2}}
|
||||
\exp \left( ...
|
||||
...c_2^2}}
|
||||
\exp \left( \frac{(f - f_2)^2}{2 c_2^2} \right),
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(3)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
with means <IMG
|
||||
WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$f_1,f_2$"> and standard
|
||||
deviations <IMG
|
||||
WIDTH="39" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img47.png"
|
||||
ALT="$c_1,c_2$">. Power in the LF and HF bands are given by
|
||||
<IMG
|
||||
WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$\sigma_1^2$"> and <IMG
|
||||
WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img49.png"
|
||||
ALT="$\sigma_2^2$"> respectively whereas the variance
|
||||
equals the total area <!-- MATH
|
||||
$\sigma^2 = \sigma^2_1+\sigma^2_2$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="95" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$\sigma^2 = \sigma^2_1+\sigma^2_2$">,
|
||||
yielding an LF/HF ratio of <!-- MATH
|
||||
$\sigma^2_1/\sigma^2_2$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$\sigma^2_1/\sigma^2_2$">.
|
||||
Fig. <A HREF="node4.html#f:Sf">4</A> shows the power spectrum <IMG
|
||||
WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$S(f)$"> given
|
||||
by <IMG
|
||||
WIDTH="61" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img52.png"
|
||||
ALT="$f_1 = 0.1$">,
|
||||
<IMG
|
||||
WIDTH="69" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img53.png"
|
||||
ALT="$f_2 = 0.25$">, <IMG
|
||||
WIDTH="68" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$c_1 = 0.01$">, <IMG
|
||||
WIDTH="68" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img55.png"
|
||||
ALT="$c_2 = 0.01$"> and <!-- MATH
|
||||
$\sigma^2_1/\sigma^2_2 = 0.5$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img56.png"
|
||||
ALT="$\sigma^2_1/\sigma^2_2 = 0.5$">.
|
||||
The Gaussian frequency distribution is motivated by the
|
||||
typical power spectrum of a real RR tachogram [<A
|
||||
HREF="node8.html#malik95">7</A>].
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:Sf"></A><A NAME="145"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
|
||||
Power spectrum <IMG
|
||||
WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$S(f)$"> of the RR-interval process
|
||||
with a LF/HF ratio of <!-- MATH
|
||||
$\sigma_1^2/\sigma_2^2 = 0.5$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img3.png"
|
||||
ALT="$\sigma _1^2/\sigma _2^2 = 0.5$">.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="275" BORDER="0"
|
||||
SRC="img57.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=Sf.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
A RR-interval time series <IMG
|
||||
WIDTH="34" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img58.png"
|
||||
ALT="$T(t)$"> with power spectrum <IMG
|
||||
WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$S(f)$"> is generated by
|
||||
taking the inverse Fourier transform of a sequence of complex numbers with
|
||||
amplitudes <IMG
|
||||
WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img59.png"
|
||||
ALT="$\sqrt{S(f)}$"> and phases which are randomly
|
||||
distributed between 0 and <IMG
|
||||
WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$2 \pi$">.
|
||||
By multiplying this time series by an appropriate
|
||||
scaling constant and adding an offset value, the resulting time series can be
|
||||
given any required mean and standard deviation.
|
||||
Suppose that <IMG
|
||||
WIDTH="34" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img58.png"
|
||||
ALT="$T(t)$"> represents the time series generated by the RR-process
|
||||
with power spectrum <IMG
|
||||
WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$S(f)$">. The time-dependent
|
||||
angular velocity <IMG
|
||||
WIDTH="33" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img61.png"
|
||||
ALT="$\omega(t)$"> of motion around the limit cycle is then given
|
||||
by
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
\omega(t) = \frac{2 \pi}{T(t)}.
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
|
||||
WIDTH="88" HEIGHT="42" BORDER="0"
|
||||
SRC="img62.png"
|
||||
ALT="\begin{displaymath}
|
||||
\omega(t) = \frac{2 \pi}{T(t)}.
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(4)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
In this way the series of RR-intervals of the resultant
|
||||
synthetic ECG will also have a power spectrum equal to <IMG
|
||||
WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$S(f)$">; this will be
|
||||
demonstrated in the next section.
|
||||
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
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<A NAME="tex2html65"
|
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HREF="node5.html">
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SRC="file:/usr/share/latex2html/icons/up.png"></A>
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<A NAME="tex2html57"
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<B> Next:</B> <A NAME="tex2html66"
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<B> Up:</B> <A NAME="tex2html64"
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HREF="index.shtml">A dynamical model for</A>
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<B> Previous:</B> <A NAME="tex2html58"
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HREF="node3.html">Heart rate variability</A>
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<ADDRESS>
|
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|
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2003-10-08
|
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</ADDRESS>
|
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</BODY>
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</HTML>
|
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248
ecgsyn-1.0.0/paper/node5.html
Normal file
248
ecgsyn-1.0.0/paper/node5.html
Normal file
@@ -0,0 +1,248 @@
|
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
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|
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<!--Converted with jLaTeX2HTML 2002 (1.62) JA patch-1.4
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patched version by: Kenshi Muto, Debian Project.
|
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LaTeX2HTML 2002 (1.62),
|
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
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* with significant contributions from:
|
||||
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
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<HTML>
|
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<HEAD>
|
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<TITLE>Results</TITLE>
|
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<META NAME="description" CONTENT="Results">
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<META NAME="keywords" CONTENT="ecgpqrst">
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<LINK REL="STYLESHEET" HREF="ecgpqrst.css">
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<LINK REL="next" HREF="node6.html">
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<LINK REL="previous" HREF="node4.html">
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<LINK REL="up" HREF="index.shtml">
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<LINK REL="next" HREF="node6.html">
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<BODY >
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<!--Navigation Panel-->
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<A NAME="tex2html75"
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HREF="node6.html">
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SRC="file:/usr/share/latex2html/icons/prev.png"></A>
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<B> Next:</B> <A NAME="tex2html76"
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HREF="node6.html">Conclusions</A>
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<B> Up:</B> <A NAME="tex2html74"
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HREF="index.shtml">A dynamical model for</A>
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<B> Previous:</B> <A NAME="tex2html68"
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HREF="node4.html">The dynamical model</A>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
|
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|
||||
<H1><A NAME="SECTION00050000000000000000"></A>
|
||||
<A NAME="s:results"></A>
|
||||
<BR>
|
||||
Results
|
||||
</H1>
|
||||
|
||||
The synthetic ECG (Fig. <A HREF="node5.html#f:ecgsynthetic">5</A>) illustrates
|
||||
the modulation of the QRS-complex due to RSA.
|
||||
Observational uncertainty is incorporated by adding normally
|
||||
distributed measurement errors with mean zero
|
||||
and standard deviation 0.025 mV (Fig. <A HREF="node5.html#f:ecgcomparison">6</A>a),
|
||||
yielding a similar signal to a segment of real ECG from a normal human
|
||||
(Fig. <A HREF="node5.html#f:ecgcomparison">6</A>b).
|
||||
|
||||
In order to illustrate the
|
||||
dynamics of the RR-intervals obtained from this synthetic ECG, peak detection
|
||||
was used to identify the times of the R-peaks.
|
||||
In the noise-free case, a simple algorithm which looks for local maxima
|
||||
within a small window is sufficient. For ECGs with noise and artefacts it
|
||||
may be necessary to use more complicated methods [<A
|
||||
HREF="node8.html#pan85">2</A>,<A
|
||||
HREF="node8.html#kaplan91">3</A>].
|
||||
A comparison between the continuous process with power spectrum <IMG
|
||||
WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$S(f)$">
|
||||
given by (<A HREF="node4.html#e:Sf">3</A>) and the piecewise constant
|
||||
reconstruction of the RR-process obtained from the R-peak detection
|
||||
(Fig. <A HREF="node5.html#f:rrinout">7</A>) illustrates the measurement errors that
|
||||
arise when computing heart rate variability statistics from
|
||||
RR-intervals.
|
||||
|
||||
The RR-intervals (Fig. <A HREF="node5.html#f:rrsynthetic">8</A>a) and corresponding
|
||||
instantaneous heart rate (Fig. <A HREF="node5.html#f:rrsynthetic">8</A>b)
|
||||
in units of beats per minute (bpm)
|
||||
for a mean of 60 bpm and standard deviation of 5 bpm
|
||||
display variability due to both RSA and Mayer waves.
|
||||
A spectral estimation technique
|
||||
for unevenly sampled time series, the Lomb periodogram
|
||||
[<A
|
||||
HREF="node8.html#press92">15</A>,<A
|
||||
HREF="node8.html#laguna98">16</A>], was used to calculate the power
|
||||
spectrum (Fig. <A HREF="node5.html#f:rrsynthetic">8</A>c) from the RR tachogram,
|
||||
derived from 5 minutes of data as recommended
|
||||
by [<A
|
||||
HREF="node8.html#malik95">7</A>,<A
|
||||
HREF="node8.html#eursoccard96">10</A>].
|
||||
Despite the loss of information in going from the continuous process to the
|
||||
piecewise constant reconstruction, a comparison between Fig. <A HREF="node4.html#f:Sf">4</A> and
|
||||
Fig. <A HREF="node5.html#f:rrsynthetic">8</A>c illustrates that it is still
|
||||
possible to obtain a reasonable estimate of the power spectrum.
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:ecgsynthetic"></A><A NAME="168"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
|
||||
ECG generated by dynamical model: (a) 10 seconds and (b) 50 seconds.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="275" BORDER="0"
|
||||
SRC="img63.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=ecgsynthetic.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:ecgcomparison"></A><A NAME="173"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6:</STRONG>
|
||||
Comparison between (a) synthetic ECG with additive normally
|
||||
distributed measurement errors and (b) real ECG signal from a normal human.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="275" BORDER="0"
|
||||
SRC="img64.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=ecgcomparison.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:rrinout"></A><A NAME="296"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7:</STRONG>
|
||||
Reconstruction of RR-process from R-peak detection:
|
||||
the underlying RR-process generated using (<A HREF="node4.html#e:Sf">3</A>) (black line)
|
||||
and the RR-interval time series obtained using R-peak
|
||||
detection of the synthetic ECG (grey line).</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="275" BORDER="0"
|
||||
SRC="img65.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=rrinout.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:rrsynthetic"></A><A NAME="298"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8:</STRONG>
|
||||
Analysis of RR-intervals from R-peak detection of the ECG signal
|
||||
generated by the dynamical model (<A HREF="node4.html#e:pqrst">1</A>) with mean heart rate 60 bpm
|
||||
and standard deviation 5 bpm: (a) RR-intervals,
|
||||
(b) instantaneous heart rate and (c) power spectrum of the RR-intervals.
|
||||
Note the two active frequencies belonging to RSA (0.25 Hz) and Mayer
|
||||
waves (0.1 Hz).</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="272" BORDER="0"
|
||||
SRC="img66.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=rrsynthetic.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
An increase in the RR-interval implies that the trajectory has more time to
|
||||
get pushed into the peak and trough given by the R and S events.
|
||||
This is reflected by the strong correlation between the RR-intervals and the
|
||||
RS-amplitude as shown in Fig. <A HREF="node5.html#f:rsrr">9</A>. A technique for deriving a measure
|
||||
of the rate of respiration
|
||||
from the ECG has been proposed [<A
|
||||
HREF="node8.html#moody85">5</A>,<A
|
||||
HREF="node8.html#moody86">6</A>].
|
||||
This ECG-derived respiratory signal (EDR) is of clinical use in
|
||||
situations where the ECG, but not respiration, is recorded.
|
||||
The synthetic ECG provides a means of testing the robustness of such
|
||||
techniques against noise and the effects of different sampling frequencies.
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:rsrr"></A><A NAME="190"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9:</STRONG>
|
||||
RS-amplitudes versus RR-intervals for the synthetic ECG.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="351" HEIGHT="275" BORDER="0"
|
||||
SRC="img67.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=rsrr.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
As a consequence of constructing the model with a variable angular frequency
|
||||
<IMG
|
||||
WIDTH="33" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img61.png"
|
||||
ALT="$\omega(t)$">, the time taken to
|
||||
move from the Q event to the T event, known as the QT-interval, varies with
|
||||
the RR-interval on a beat-to-beat basis.
|
||||
The relationship between the QT-interval and the
|
||||
RR-interval is linear as shown in Fig. <A HREF="node5.html#f:qtrr">10</A>.
|
||||
Such a linear relationship has been reported for real ECGs and
|
||||
has been used to calculate a corrected QT-interval [<A
|
||||
HREF="node8.html#davey99">4</A>].
|
||||
It is interesting that this relationship is a direct consequence of the
|
||||
model. Furthermore it may be possible to use the model to assess how much of
|
||||
the variation in the QT-interval is due to RR-interval variability so that
|
||||
this effect can be factored out.
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="f:qtrr"></A><A NAME="197"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10:</STRONG>
|
||||
QT-intervals versus RR-intervals for the synthetic ECG.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="352" HEIGHT="275" BORDER="0"
|
||||
SRC="img68.png"
|
||||
ALT="\begin{figure}
|
||||
\centerline{\psfig{file=qtrr.eps,width=7.75cm}}
|
||||
\end{figure}"></TD></TR>
|
||||
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<H1><A NAME="SECTION00060000000000000000"></A>
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<A NAME="s:conclusions"></A>
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<BR>
|
||||
Conclusions
|
||||
</H1>
|
||||
A new dynamical model has been introduced which is capable of replicating
|
||||
many of the important features of the human ECG.
|
||||
Moreover, many of the morphological changes observed in the human ECG
|
||||
manifest as a consequence of the geometrical structure of the model.
|
||||
Model parameters may be
|
||||
chosen to generate different morphologies for the PQRST-complex.
|
||||
The power spectrum of the RR-intervals can be selected <I>a priori</I> and used
|
||||
to drive the ECG generator. This allows the operator to prescribe specific
|
||||
characteristics of the heart rate dynamics such as the mean and standard
|
||||
deviation of the heart rate and
|
||||
spectral properties such as the LF/HF ratio.
|
||||
In addition the average morphology can be controlled by specifying the
|
||||
positions of the P,Q,R,S and T events and the magnitude of their effect
|
||||
on the ECG.
|
||||
|
||||
Having access to a realistic ECG provides a benchmark for testing
|
||||
numerous biomedical signal processing techniques. In order to establish
|
||||
the operational properties of these techniques in a clinical setting,
|
||||
it is important to know
|
||||
how they perform for different noise levels and sampling frequencies.
|
||||
|
||||
A number of applications and simple extensions of the model are possible:
|
||||
|
||||
(i) By fitting (see [<A
|
||||
HREF="node8.html#mcsharry99a">17</A>]) the model
|
||||
to the morphology of a particular subject's ECG
|
||||
and the power spectrum of their RR-intervals,
|
||||
a database of realistic ECGs could be created.
|
||||
This database could be employed for statistical hypothesis testing.
|
||||
Furthermore, it may be possible derive a corrected QT-interval which is
|
||||
independent of the heart rate.
|
||||
|
||||
(ii) The synthetic ECG could be used to assess the effectiveness of
|
||||
different techniques for noise and artefact removal.
|
||||
These could be evaluated by adding noise and/or artefact onto the
|
||||
synthetic signal and then comparing the original with the processed signal.
|
||||
|
||||
(iii) Abnormal morphological changes with time could be introduced
|
||||
by using a parameter to control the position of any of the P,Q,R,S or T events.
|
||||
This extension would be particularly useful for testing techniques
|
||||
which aim to detect ST depression or elevation by decreasing or
|
||||
increasing the <IMG
|
||||
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$z$">-position of the T wave over time.
|
||||
Similarly QT prolongation could be replicated by moving the T point away
|
||||
from the Q point in the <IMG
|
||||
WIDTH="41" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$(x,y)$"> plane (increasing
|
||||
<!-- MATH
|
||||
$\theta_T - \theta_Q$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="59" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img69.png"
|
||||
ALT="$\theta_T - \theta_Q$">).
|
||||
|
||||
(iv) The model could be used to produce multi-lead ECG signals by
|
||||
introducing a measurement function which maps from the
|
||||
<IMG
|
||||
WIDTH="56" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img1.png"
|
||||
ALT="$(x,y,z)$"> model space to the ECG signal: <IMG
|
||||
WIDTH="94" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img70.png"
|
||||
ALT="$s = h(x,y,z)$">. Different lead
|
||||
configurations and modulations due to respiration and movement of the
|
||||
cardiac axis could be modelled using time-dependent functions for <IMG
|
||||
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img71.png"
|
||||
ALT="$h$">.
|
||||
|
||||
(v) Abnormal beats, such as ectopics, can be simulated by
|
||||
modifying the position of the R-peak for one cycle of the dynamics.
|
||||
|
||||
The new model presented here reflects a data-driven approach to modelling the
|
||||
electrical activity of the heart. Key physiological features have been
|
||||
incorporated using motion of a trajectory throughout a three-dimensional state
|
||||
space. The quasi-periodicity of the cardiac cycle is represented by
|
||||
attraction towards a limit cycle.
|
||||
The model produces QT-intervals and R-peak height variation (RSA)
|
||||
which vary linearly with the RR-intervals
|
||||
as has been found in real ECGs [<A
|
||||
HREF="node8.html#moody86">6</A>,<A
|
||||
HREF="node8.html#davey99">4</A>].
|
||||
It is hoped that this model will provide a valuable tool
|
||||
for testing biomedical signal processing algorithms applied to ECG signals
|
||||
with different sampling frequencies and
|
||||
levels of noise and/or movement artefact.
|
||||
|
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<H1><A NAME="SECTION00070000000000000000">
|
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Acknowledgements</A>
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</H1>
|
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This research was supported by EPSRC grant GR/N02641 and Oxford BioSignals.
|
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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|
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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|
||||
<H2><A NAME="SECTION00080000000000000000">
|
||||
Bibliography</A>
|
||||
</H2><DL COMPACT><DD><P></P><DT><A NAME="goldberger77">1</A>
|
||||
<DD>
|
||||
A. L. Goldberger and E. Goldberger,
|
||||
<BR><EM>Clinical Electrocardiography</EM>,
|
||||
<BR>Mosby, St. Louis, 1977.
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||||
|
||||
<P></P><DT><A NAME="pan85">2</A>
|
||||
<DD>
|
||||
J. Pan and W. J. Tompkins,
|
||||
<BR>``A real-time QRS detection algorithm,''
|
||||
<BR><EM>IEEE Trans. Biomed. Eng.</EM>, vol. 32, no. 3, pp. 220-236, 1985.
|
||||
|
||||
<P></P><DT><A NAME="kaplan91">3</A>
|
||||
<DD>
|
||||
D. T. Kaplan,
|
||||
<BR>``Simultaneous QRS detection and feature extraction using simple
|
||||
matched filter basis functions,''
|
||||
<BR>in <EM>Computers in Cardiology</EM>. 1991, pp. 503-506, IEEE Computer
|
||||
Society Press.
|
||||
|
||||
<P></P><DT><A NAME="davey99">4</A>
|
||||
<DD>
|
||||
P. Davey,
|
||||
<BR>``A new physiological method for heart rate correction of the QT
|
||||
interval,''
|
||||
<BR><EM>Heart</EM>, vol. 82, pp. 183-186, 1999.
|
||||
|
||||
<P></P><DT><A NAME="moody85">5</A>
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||||
<DD>
|
||||
G. B. Moody, R. G. Mark, A. Zoccola, and S. Mantero,
|
||||
<BR>``Derivation of respiratory signals from multi-lead ecgs,''
|
||||
<BR><EM>Computers in Cardiology</EM>, vol. 12, pp. 113-116, 1985.
|
||||
|
||||
<P></P><DT><A NAME="moody86">6</A>
|
||||
<DD>
|
||||
G. B. Moody, R. G. Mark, M. A. Bump, J. S. Weinstein, A. D. Berman, J. E.
|
||||
Mietus, and A. L. Goldberger,
|
||||
<BR>``Clinical validation of the ecg-derived respiration (edr)
|
||||
technique,''
|
||||
<BR><EM>Computers in Cardiology</EM>, vol. 13, pp. 507-510, 1986.
|
||||
|
||||
<P></P><DT><A NAME="malik95">7</A>
|
||||
<DD>
|
||||
M. Malik and A. J. Camm,
|
||||
<BR><EM>Heart Rate Variability</EM>,
|
||||
<BR>Futura Publishing, Armonk, NY, 1995.
|
||||
|
||||
<P></P><DT><A NAME="physionet">8</A>
|
||||
<DD>
|
||||
A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. Ch. Ivanov,
|
||||
R. G. Mark, J. E. Mietus, G. B. Moody, C. K. Peng, and H. E. Stanley,
|
||||
<BR>``Physiobank, physiotoolkit, and physionet: Components of a new
|
||||
research resource for complex physiologic signals,''
|
||||
<BR><EM>Circulations</EM>, vol. 101, no. 23, pp. e215-e220, 2000.
|
||||
|
||||
<P></P><DT><A NAME="schwartz78">9</A>
|
||||
<DD>
|
||||
P. J. Schwartz and Wolf S.,
|
||||
<BR>``Q-T interval as predictor of sudden death in patients with
|
||||
myocardial infarction,''
|
||||
<BR><EM>Circulation</EM>, vol. 57, pp. 1074-1077, 1978.
|
||||
|
||||
<P></P><DT><A NAME="eursoccard96">10</A>
|
||||
<DD>
|
||||
Task Force of the European Society of Cardiology, the North American
|
||||
Society of Pacing, and Electrophysiology,
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<BR>
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<IMG
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WIDTH="553" HEIGHT="148" ALIGN="BOTTOM" BORDER="0"
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SRC="img72.png"
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ALT="\begin{biography}{Patrick E. McSharry} was born in Leitrim, Ireland in 1972.
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H...
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...cal modelling,
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forecasting and biomedical signal processing.
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\end{biography}">
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<BR>
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<BR>
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<IMG
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WIDTH="554" HEIGHT="148" ALIGN="BOTTOM" BORDER="0"
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SRC="img73.png"
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ALT="\begin{biography}{Gari D. Clifford}
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was born in Alton, Hampshire, UK in 1971.
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...
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...sis, biomedical signal processing
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and mathematical modelling.
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\end{biography}">
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<BR>
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<BR>
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<IMG
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WIDTH="556" HEIGHT="208" ALIGN="BOTTOM" BORDER="0"
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SRC="img74.png"
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ALT="\begin{biography}{Lionel Tarassenko} MA, DPhil, FREng, FIEE was born in
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Paris ...
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...d
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to a Fellowship of the Royal Academy of Engineering in 2000.
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\end{biography}">
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<BR>
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<BR>
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<IMG
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WIDTH="554" HEIGHT="152" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="\begin{biography}{Leonard A. Smith} received a BS in Physics,
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Mathematics and ...
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... the definition of noise; and nonlinear time series
|
||||
analysis.
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\end{biography}">
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||||
<BR><BR><HR>
|
||||
<ADDRESS>
|
||||
|
||||
2003-10-08
|
||||
</ADDRESS>
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||||
</BODY>
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LaTeX2HTML 2002 (1.62),
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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|
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
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<HTML>
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<TITLE>About this document ...</TITLE>
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<H1><A NAME="SECTION00090000000000000000">
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About this document ...</A>
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</H1>
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<STRONG>A dynamical model for generating synthetic electrocardiogram signals</STRONG><P>
|
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This document was generated using the
|
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<A HREF="http://www.latex2html.org/"><STRONG>LaTeX</STRONG>2<tt>HTML</tt></A> translator Version 2002 (1.62)
|
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<P>
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Copyright © 1993, 1994, 1995, 1996,
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<A HREF="http://cbl.leeds.ac.uk/nikos/personal.html">Nikos Drakos</A>,
|
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Computer Based Learning Unit, University of Leeds.
|
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<BR>
|
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Copyright © 1997, 1998, 1999,
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<A HREF="http://www.maths.mq.edu.au/~ross/">Ross Moore</A>,
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Mathematics Department, Macquarie University, Sydney.
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<P>
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The command line arguments were: <BR>
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<STRONG>latex2html</STRONG> <TT>ecgpqrst.tex ecgpqrst.html</TT>
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<P>
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The translation was initiated by on 2003-10-08
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<ADDRESS>
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2003-10-08
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