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<TITLE>The dynamical model</TITLE>
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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">
&nbsp;</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">
&nbsp;</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>
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