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\title{\bf Automatically Finding the Average Output of a Steadily Beating Ventricle}
\author{\large Alexander Yeh\thanks{Supported by the National Institute of Health
through grant R01-LM04493 from the National Library of Medicine and
grant R01-HL33041 from the National Heart, Lung, and Blood Institute.}\\ MIT Laboratory for Computer Science\\ 545 Technology Sq., NE43-413\\ Cambridge, MA 02139 USA} 
\date{}
\begin{document}
%to get variable names to be tighter in appearance (so `HR' does not look like `H times R').
\newcommand{\Bi}{\mbox{\it Bi}}
\newcommand{\Bo}{\mbox{\it Bo}}
\newcommand{\PI}{\mbox{\it Pi}} %Not to be mistaken with \Pi or \pi
\newcommand{\Po}{\mbox{\it Po}}
\newcommand{\HR}{\mbox{\it HR}}
\newcommand{\Vd}{\mbox{\it Vd}}
\newcommand{\Vs}{\mbox{\it Vs}}

\newcommand{\Md}{\mbox{\it Md}}
\newcommand{\Ms}{\mbox{\it Ms}}
\newcommand{\SV}{\mbox{\it SV}}

\maketitle
%200word limit
\paragraph{ABSTRACT:}
The repetitive behavior of a device or system can be described in two
ways: a detailed description of one iteration of the behavior, or a
summary description of the behavior over many repetitions.  In this
paper, an implemented program called AIS is applied to a model of a
steadily beating ventricle (part of the heart). AIS transforms the first
type of description into the second type. The output consists of the
symbolic average rates of change in parameter values and how those rates
would be different if various constants and functions had been
different. AIS's results are compared to results in the literature.

\paragraph{INTRODUCTION:}
A program called AIS (short for Analyzer of Iterated Sequences) has been
developed and applied to a model of a steadily beating ventricle.  When
given a state-description of a system and a sequence of actions or
transformations on that state, AIS symbolically finds some of the
time-averaged effects of continually iterating that sequence. The
specific effects found at present include 1) the symbolic average rate
of change in parameters, and 2) an assessment of how those rates of
change would be different with different values for various constants
and functions (sensitivity analysis).  The sequences handled by AIS are
ones which have the following ``constancy'': the sequence always repeats
the same actions in the same order and each occurrence of a given action
changes the parameters by the same amounts. Such an iterated sequence of
actions is exemplified by the beat cycle of a heart at steady-state.
Effects to be found include the average rate at which blood enters the
heart and how increasing that entering blood's pressure affects that
rate.

A motivation for finding such effects is that many periodic sub-systems
iterate at such a fast rate that the other parts of a system respond
only to the behavior of such a sub-system $\beta$ averaged over many
iterations.  Then a steady-state model for the entire system would only
require a description of $\beta$'s averaged behavior; $\beta$ can be
modeled as a constant iteration of the same sequence of parameter value
changes.  In this paper, the sub-system and system combination is the
heart and the human circulatory system.

Some other approaches of finding the behaviors of a continually
iterating sequence have combined qualitative simulation with cycle
detection \cite{Bobrow}.  For complicated systems (such as the heart),
these simulations predict many possible sequences of actions besides the
actual sequence. {\em Aggregation} \cite{Weld86} and {\em comparative
analysis} \cite{Weld88a} are useful after isolating the actual sequence.

Another approach \cite{Sacks88b} takes in a system description that
consists of a single set of differential equations that are always
applicable.  Creating such a description may often be hard, such as when
doing so for a heart.

The next two sections summarize the methodology behind AIS. More details
are given in \cite{Yeh90-2}. Afterwards is a section giving the results
of AIS for a ventricle. The paper ends with a summary.

\paragraph{METHOD (AIS Input):} \label{s:AIS-input}
An input description consists of three parts: the parameters which
describe the system state, static conditions on those parameters, and
the sequence of actions (transformations) that gets iterated.

Parameters are divided by the model-builder into 4 types. The first
3 types are classified by how a parameter behaves as the sequence of
actions is iterated:
\begin{enumerate}
 \item {\em Constant parameters} do not change in value at all.

  \item {\em Periodic parameters} change in value, but the sequence of
values repeats exactly with each sequence iteration.

  \item {\em Accumulating parameters} monotonically change in value with
each sequence iteration.
\end{enumerate}
In general, parameters are represented by symbols. The constant
parameter type also includes numbers and arbitrary functions of
expressions of constant parameters. The fourth parameter ``type'' has
only one parameter: the rate at which the sequence of actions is
iterated. At present, the rate must be expressed as a constant parameter
that is a symbol or number.

The second part of the input is a set of static conditions between
constant parameters.  These conditions are inequalities between numbers
and expressions made up of constant parameters. The expressions can have
algebraic and the more common transcendental functions. Also permissible
are (partial) derivatives of constant parameters which are arbitrary
functions. 

The last part of the input gives the sequence of actions
(transformations) that is iterated. The sequence is partitioned into
{\em phases} so that 1) every part of a sequence is put in exactly one
phase, and 2) during each phase, every parameter is either monotonically
non-decreasing or non-increasing in value.

For each phase, the input description needs to supply an expression for
every parameter that changes in value during that phase. For a periodic
parameter, the corresponding expression gives that parameter's value at
the end of the phase.\footnote{Due to the requirements on choosing
phases, a periodic parameter's value at a phase's beginning and the
preceding phase's end is the same.  And because the sequence iterates,
the last phase in the sequence is also deemed to ``precede'' the
1st phase.} For an accumulating parameter, the expression gives the
change in that parameter's value each time that phase occurs. An
expression may have algebraic and the more common transcendental
functions. The expression's arguments can consist of constant
parameters, periodic parameters' values at the beginning or end of that
phase, and/or accumulating parameters' change in values\footnote{Only
the change in value can be referred to because it stays the same from one
iteration of the sequence to the next. The actual value changes with
each iteration of the sequence.} each time that phase occurs.

The limitations on describing parameter changes are to assure that each
occurrence of a phase alters the parameters by the same constant amount.
Without some restrictions on how phases alter parameters, it will be
hard to impossible for AIS to determine the effects of steadily
iterating the sequence of actions.

Each phase also has a list of the conditions that are inequalities
between expressions and numbers.

\paragraph{METHOD (AIS Output):}\label{s:AIS-output}
AIS takes the input equations, solves them and checks for inconsistencies
(using the Bounder system \cite{Sacks88b}). Bounder is also used to find
the numeric bounds mentioned below.

Then, to derive the average rate of change in an accumulating parameter
$a$, AIS locates the change in that parameter's value during each phase
of a sequence, adds all those changes together, and then multiplies the
sum by the rate of cycle repetition. Next AIS finds numeric bounds on
this rate.

After deriving an average rate for $a$, AIS can observe how that rate
would be different if any one constant symbol or function were
different. For each symbol, AIS takes the first two (symbolic)
derivatives of the rate with respect to that symbol and obtains numeric
bounds on those derivatives.  Each constant symbol is considered to be
independent of all other symbols.

At present, AIS also tries to plot a ``qualitative'' graph of the rate
versus each constant symbol. The first derivative described above
provides slope information and the second provides convexity
information. AIS makes the assumption that the rate versus constant
function is smooth (differentiable). If the second derivative can be
both more or less than zero, AIS gives up. Otherwise, depending on how
the second derivative is bounded by zero and on how the first
derivative's bounds relate to zero, AIS determines which of the
following shapes the curve may possibly have:
\begin{picture}(10,10)\thicklines\put(1,10){\line(1,-1){10}}\end{picture} ,
\begin{picture}(10,10)\thicklines\put(1,5){\line(1,0){10}}\end{picture} ,
\begin{picture}(10,10)\thicklines\put(1,1){\line(1,1){10}}\end{picture} ,
\begin{picture}(10,10)\thicklines\put(9,9){\oval(18,18)[bl]}\end{picture} ,
\begin{picture}(20,10)\thicklines\put(9,9){\oval(17,17)[b]}\end{picture} ,
\begin{picture}(10,10)\thicklines\put(1,9){\oval(18,18)[br]}\end{picture} ,
\begin{picture}(10,10)\thicklines\put(9,1){\oval(18,18)[tl]}\end{picture} ,
\begin{picture}(20,10)\thicklines\put(9,1){\oval(17,17)[t]}\end{picture} and/or
\begin{picture}(10,10)\thicklines\put(1,1){\oval(18,18)[tr]}\end{picture} .
For example, if the first derivative is $<0$ and the second is $=0$
(such as when the rate is $-3x$ and the symbol is $x$), then the curve shape is
\mbox{\begin{picture}(10,10)\thicklines\put(1,10){\line(1,-1){10}}\end{picture} .}

Afterwards, AIS derives the effects of functions having different values
by performing some symbolic substitution \& subtraction and some
expression value bounding to observe how the rates would be different if
a function were larger in value.

\paragraph{RESULTS for Ventricle Model:}
This section describes the current version of AIS running on a model of
the beating of the left ventricle.\footnote{The description is based on
various texts and articles \cite{Ross85,Sato86} \cite[Ch.~13: Mechanisms
of Cardiac Contraction and Relaxation]{Braunwald} and makes many
assumptions.  One assumption is that blood is an incompressible fluid
without inertia.} The ventricle is a chamber with two one-way valves:
one valve lets in blood from the lungs at a pressure of $\PI$, and the
other valve lets out blood going to the rest of the body at a pressure
of $\Po$. The chamber consists of muscle which can either relax or
contract.  When relaxed ({\em diastole}), the ventricle volume ($V$)
versus pressure ($P$) curve ($\Vd [P]$) is roughly as shown in
Figure~\ref{f:heart-curves}a (the $P$ and $V$ axes are interchanged from
their usual positions). When contracted ({\em systole}), the $V$ versus
$P$ curve ($\Vs [P,\HR ]$) is roughly as shown in
Figure~\ref{f:heart-curves}b. The symbol $\HR $ appears because with
$\Vs $, $V$ decreases as the rate at which the ventricle contracts and
relaxes increases. This rate is known as the heart rate ($\HR $).
Figure~\ref{f:heart-curves}c shows with a dashed line the $V$ versus $P$
path that ventricle takes as it contracts and relaxes once (a beat
sequence): 1) The ventricle contracts, but no blood moves. So, $V$ stays
the same while $P$ increases to $\Po $. Move from $a$ to $b$ in the
diagram. 2) The ventricle continues contracting, but now, blood ejects
out the output valve.  $P$ stays the same while $V$ decreases to $\Vs
[\Po ,\HR ]$.  Move from $b$ to $c$. 3) The ventricle now starts to
relax and the blood stops moving.  $V$ becomes constant as $P$ decreases
to $\PI $. Go from $c$ to $d$. 4) The ventricle continues relaxing,
but now blood enters from the input valve. $P$ stays the same while $V$
increases to $\Vd [\PI ]$. Return from $d$ to $a$.

\begin{figure}
\center
\begin{picture}(80,70)(-20,-10)
\thinlines \put(0,0){\vector(1,0){50}} \put(0,0){\vector(0,1){50}}
\put(52,0){\makebox(0,0)[l]{$P$}} \put(-3,50){\makebox(0,0)[tr]{$V$}}
\put(25,55){\makebox(0,0)[b]{a) Diastole}}
\thicklines \put(30,25){\oval(40,40)[tl]} \put(34,45){\makebox(0,0)[tl]{$\Vd [P]$}}
\end{picture}
\begin{picture}(90,70)(-30,-10)
\thinlines \put(0,0){\vector(1,0){50}} \put(0,0){\vector(0,1){50}}
\put(52,0){\makebox(0,0)[l]{$P$}} \put(-3,50){\makebox(0,0)[tr]{$V$}}
\put(25,55){\makebox(0,0)[b]{b) Systole}}
\thicklines \put(10,25){\oval(40,40)[br]} \put(34,25){\makebox(0,0)[tl]{$\Vs [P,\HR ]$}}
\end{picture}
\begin{picture}(80,80)(-30,-10)
\thinlines \put(0,0){\vector(1,0){50}} \put(0,0){\vector(0,1){50}}
\put(52,0){\makebox(0,0)[l]{$P$}} \put(-3,50){\makebox(0,0)[tr]{$V$}}
\put(13,15){\dashbox{2}(14,20)}
\put(10,15){\makebox(0,0)[r]{$d$}}\put(29,13){\makebox(0,0)[tl]{$c$}}
\put(30,35){\makebox(0,0)[l]{$b$}}\put(11,36){\makebox(0,0)[br]{$a$}}
\put(13,0){\line(0,-1){3}} \put(27,0){\line(0,-1){3}}
\put(13,-4){\makebox(0,0)[tr]{$\PI $}} \put(27,-4){\makebox(0,0)[tl]{$\Po $}}
\put(0,15){\line(-1,0){3}} \put(0,35){\line(-1,0){3}}
\put(-5,15){\makebox(0,0)[r]{$\Vs [\Po ,\HR ]$}} \put(-5,35){\makebox(0,0)[r]{$\Vd [\PI ]$}}
\put(25,55){\makebox(0,0)[b]{c) Beat Path}}
\thicklines \put(10,25){\oval(40,40)[br]}\put(30,25){\oval(40,40)[tl]}
\end{picture}
\caption{Curves for a Left Ventricle}\label{f:heart-curves} 
\end{figure}

The input to AIS has the following: The symbol $\HR $ gives the rate at
which the ventricle beat sequence repeats. The constants are $\PI $, $\Po $,
$\Vd [\PI ]$ and $\Vs [\Po ,\HR ]$.\footnote{$\PI $ and $\Po $ are assumed to be
constant during the ventricle beats. These assumptions then force
$\Vd [\PI ]$ and $\Vs [\Po ,\HR ]$ to be also constant during the beats.} The
periodic parameters are $P$ and $V$. The accumulating parameters are the
amount of work done by the blood in moving through the ventricle ($W$),
and the amount of blood that has entered the ventricle ($\Bi $) and left
the ventricle ($\Bo $). The static conditions on the constants are:
\begin{center}
$\PI < \Po $,\hspace{1em}$\Vd [\PI ] > \Vs [\Po ,\HR ]$,\hspace{1em}$0\le \Vd [\PI ]$,\\
$0\le \Vs [\Po ,\HR ]$,\hspace{1em}$0<d(\Vd [\PI ])/d(\PI )$,\\ 
$0>d^2(\Vd [\PI ])/d(\PI )^2$, $0<\partial (\Vs [\Po ,\HR ])/\partial (\Po )$, $0<\partial ^2(\Vs [\Po ,\HR ])/\partial (\Po )^2$, $0>\partial (\Vs [\Po ,\HR ])/\partial (\HR )$.
\end{center}
Most of the conditions help describe the shape of $\Vd [\PI ]$ and $\Vs
[\Po ,\HR ]$. There are four phases in the sequence. Each phase has a
name, condition(s), and equation(s) for value changes. The notation is:
$X_b$ stands for parameter $X$'s value at the beginning of a phase,
$X_e$ for the value at the end, and $X_c$ for $X$'s change in value when
the phase occurs. In order, the phases are:
\begin{enumerate}
 \item Isovolumetric Contraction: $0\le V$,\hspace{1em}$P_e=\Po $.

 \item Ejection: $0\le V_b$,\hspace{1em}$0\le V_e$,\hspace{1em}$V_e=\Vs [\Po ,\HR ]$,\\$W_c=-P\cdot \Bo _c$,\hspace{1em}\mbox{$\Bo _c=V_b-V_e$}.

 \item Isovolumetric Relaxation: $0\le V$,\hspace{1em}$P_e=\PI $.

 \item Filling: $0\le V_b$,\hspace{1em}$0\le V_e$,\hspace{1em}$V_e=\Vd [\PI ]$,\\$W_c=P\cdot \Bi _c$,\hspace{1em}\mbox{$\Bi _c=V_e-V_b$}.
\end{enumerate}

After solving these phases' equations, AIS discovers the following
average rates of change for the accumulating parameters and bounds on
those rates:
\begin{eqnarray}
dW/dt&=&((\PI \cdot (\Vd [\PI ]-\Vs [\Po ,\HR ]))+\nonumber\\
&&(-\Po \cdot (\Vd [\PI ]-\Vs [\Po ,\HR ])))\cdot \HR\nonumber \\
d(\Bi )/dt&=& \HR \cdot (\Vd [\PI ]-\Vs [\Po ,\HR ]) >0 \label{eq:normal-CO}
\end{eqnarray}
Also, $d(\Bo )/dt = d(\Bi )/dt$. One can show that $dW/dt<0$, but the
bounding mechanism misses this.

After finding the rates, AIS derives and bounds the first two
derivatives of those rates with respect to each constant symbol, and
tries to give the shape of the curve of each rate versus each constant.
For $d(\Bi )/dt$, its first derivative with respect to $\HR $ is $>0$, but
no bounds are found for the second derivative. No curve shape is deduced.
With respect to the constant $\PI $, the first derivative is $>0$ but the
second is $<0$.  Assuming smoothness, AIS deduces a
\begin{picture}(10,10)\thicklines\put(9,1){\oval(18,18)[tl]}\end{picture}
shape for $d(\Bi )/dt$ versus $\PI $. This shape is consistent with the
Frank-Starling mechanism \cite[p.~212]{Ross85}. With respect to $\Po $,
both derivatives are $<0$, so the curve has a
\begin{picture}(10,10)\thicklines\put(1,1){\oval(18,18)[tr]}\end{picture}
shape. These results also apply to $d(\Bo )/dt$. As a check on the
ventricle model, these rate shape results are compared to experimental
results.  The results for $\PI $ and $\Po $ agree \cite{Sagawa67} in that
the corresponding AIS and experiment curves have the same general shapes
(signs of the first and second derivatives are the same). For $\HR $, the
AIS and experimental results are incomparable because the latter came
from intact systems where changing $\HR $ can change $\PI $ and $\Po $.

For the rate $dW/dt$, the only bound AIS can derive is that this
rate's second derivative with respect to either $\PI $ or $\Po $ is $>0$.
So for $dW/dt$ versus either $\PI $ and $\Po $, the possible curve shapes
are
\begin{picture}(10,10)\thicklines\put(9,9){\oval(18,18)[bl]}\end{picture},
\begin{picture}(20,10)\thicklines\put(9,9){\oval(17,17)[b]}\end{picture} or 
\begin{picture}(10,10)\thicklines\put(1,9){\oval(18,18)[br]}\end{picture}.

As for the $\Vd $ and $\Vs $ functions, AIS deduces that if $\Vd $ were
larger, both $d(\Bi )/dt$ and $d(\Bo )/dt$ would be also. But if $\Vs $
were larger, these rates would be smaller.  These results agree with the
description in \cite{Sato86}.

When modeling a circulatory system that has been averaged over many
heart beats and is in a steady-state, such as done in
\cite{Greenway82,LNCJ86,Sato86}, most of the system's mechanics can be
modeled by using direct current electrical circuit analogies, such as
[pressure drop] = \mbox{[resistance]$\cdot$[flow]}. Too complicated to
be modeled this way is the part of the mechanics that relates the $\PI
$, $\Po $, $\HR $, $\Vs $, and $\Vd $ for each ventricle to the rate at
which blood flows through that ventricle ($d(\Bi )/dt=d(\Bo )/dt$).
Current modeling efforts either directly use empirically derived
relationships (like \cite{Sagawa67}) or derive the needed equations by
hand from an AIS-input-like description (done in \cite{Sato86}). AIS can
perform the latter derivations automatically:
equation~(\ref{eq:normal-CO}) found by AIS for $d(\Bi )/dt$ provides the
desired relationship for the left ventricle.  The right ventricle is
similar.  Actually, to use this relationship numerically, one must be
more specific about the $\Vs $ and $\Vd $ curves, such as specifying
that $\Vd [x]=\log x$.

Other than needing more specific curve shapes, the AIS $d(\Bi )/dt$
equation is similar to the equations derived by others. The differences
are caused by modeling with slightly different sets of assumptions and
beliefs on what relationships exist and are important.

Sagawa \cite{Sagawa67} experimentally measured the effects of different
$\PI $ and $\Po $ values on the flow of blood ($d(\Bi )/dt=d(\Bo )/dt$) through
the left ventricles of dogs. The results were numerically fitted to a
relationship (curve) of the following form (translated to the notation
used in this paper): $d(\Bi )/dt =$
\begin{eqnarray*}
&&K_1\cdot (\PI -\PI _0)\cdot\\
&&(1-\exp [-(1-\Po /\Po _{max})/(K_2\cdot (\PI -\PI _0))]),
\end{eqnarray*}
where $K_1$, $\PI _0$, $\Po _{max}$ and $K_2$ are constants. This result
agrees with AIS's result in that both have a positive first derivative
for rate $d(\Bi )/dt$ with respect to $\PI $ and a negative first and second
derivative for that rate with respect to $\Po $.  The major difference
between this result and AIS's result is that this result does not
consider the effects of $\HR $ at all.  This omission is not surprising
given that $\HR $'s effects were never tested in the experiments. Another
difference is that with Sagawa, the minimum $\PI $ and maximum $\Po $ needed
to keep $d(\Bi )/dt$ above zero are given by the simple thresholds $\PI _0$
and $\Po _{max}$ respectively. With the AIS result, the minimum $\PI $ is a
more complex function of $\Po $, and similarly with the maximum $\Po $ and
$\PI $. A possible reason for this difference is that Sagawa determined
the effects of $\PI $ and $\Po $ on $d(\Bi )/dt$ separately in the experiments
and then combined the resulting equations. A third difference is that
the effects of $\PI $ have been linearized somewhat to simplify the
relationship: the actual data in the reference indicates that at large
values of $\PI $, $d(\Bi )/dt$ starts to increase sub-linearly with respect
to $\PI $, which agrees with AIS's result rather than the equation fitted
in the reference.

Sato and associates \cite{Sato86} have built a simultaneous equation
model of the cardiovascular system at steady-state. The model was built
to show the effects of heart failure (the heart muscle gets weaker or
less elastic) and to help find the optimum drug dosages for heart
failure therapies. Among the equations are the ones that give $d(\Bi )/dt$
for each ventricle (as before, the $d(\Bo )/dt$ equations are equivalent).
These equations have the form (translated to the notation used in this
paper):
\begin{displaymath}
d(\Bi )/dt = K\cdot \ln (\PI -\PI _0)-H\cdot \Po +M,
\end{displaymath}
where $K$, $\PI _0$, $H$ and $M$ are constants.  As mentioned above, these
$d(\Bi )/dt$ equations were derived by essentially carrying out what AIS
does by hand. The shape of the $d(\Bi )/dt$ versus $\PI $ curve from these
equations is
\begin{picture}(10,10)\thicklines\put(9,1){\oval(18,18)[tl]}\end{picture}
, which is the same shape as the one given by the AIS results. A
difference between these equations and the ones found by AIS are due to
Sato {\em et  al.} having more specific forms for the $\Vd $ and $\Vs $
functions:
\begin{eqnarray*}
\Vd [\PI ]&=&(K\cdot \ln (\PI -\PI _0)+\Md )/\HR \\
\Vs [\Po ,\HR ]&=&(H\cdot \Po -\Ms )/\HR ,
\end{eqnarray*}
where $M=\Md +\Ms$, so their equations have those more specific forms in
place of the $\Vd $ and $\Vs $ functions.  Also, their $\Vs $ function has
been linearized with respect to $\Po $, so the resulting $d(\Bi )/dt$ versus
$\Po $ curve has a
\begin{picture}(10,10)\thicklines\put(1,10){\line(1,-1){10}}\end{picture}
shape instead of the
\begin{picture}(10,10)\thicklines\put(1,1){\oval(18,18)[tr]}\end{picture}
shape found by AIS.  Another difference is that their
versions of the $\Vd $ and $\Vs $ functions are proportional to $1/\HR $, so
their $d(\Bi )/dt$ is independent of $\HR $ instead of increasing with
increases in $\HR $. In experiments on intact circulatory systems at rest,
this independence does hold for a wide range of $\HR $ values
\cite[p.~222,~294]{Ross85}. However, the latter source attributes the
constancy of $d(\Bi )/dt$ as $\HR $ increases to a decline in $\PI $. In
\cite{Sato86}, as $\HR $ increases, $\PI $ stays the same while $\Vd $ and
$\Vs $ decrease.

Another simultaneous equation model of the cardiovascular system at
steady-state was built earlier by Greenway \cite{Greenway82}. This model
was built to show the effects of a multitude of drugs on the
cardiovascular system. Greenway uses some of the same relationships
given to AIS as input. But a relationship\footnote{$d(\Bi )/dt$ as a
function of $\HR $, $\PI $, $\Po $, $\Vd $ and $\Vs $.} comparable to
equation~(\ref{eq:normal-CO}) is never explicitly derived. Instead, $\Po $
is solved out by the addition of the parameters for the arterial
capacitance and the body's resistance to blood flow. However, by noting
that $d(\Bi )/dt=\HR \cdot \SV$, where $\SV$ is the stroke volume, one can
rearrange Greenway's equations into one comparable to
equation~(\ref{eq:normal-CO}) to derive (translated to the notation used
in this paper) the following: $d(\Bi )/dt=$
\begin{displaymath}
\HR \cdot ((\PI +K\cdot F_A)\cdot C_{DV}-\Po /E_{max}-V_D),
\end{displaymath}
where $K$, $F_A$ and $V_D$ are constants, and $C_{DV}$ is a ``constant''
that decreases as $\PI $ gets very large. This equation matches the ones
produced by AIS and Sato {\em et al.} in that all three predict a
\begin{picture}(10,10)\thicklines\put(9,1){\oval(18,18)[tl]}\end{picture}
shape for the $d(\Bi )/dt$ versus $\PI $ curve ($C_{DV}$ decreases in size
as $\PI $ increases). And like with Sato and associates, this equation has
more specific forms in the place of the $\Vd $ and $\Vs $ functions in AIS's
result. In this equation:
\begin{eqnarray*}
\Vd [\PI ]&=&(\PI +K\cdot F_A)\cdot C_{DV}\\
\Vs [\Po ,\HR ]&=&\Po /E_{max}+V_D.
\end{eqnarray*}
Also like Sato {\em et al.} and unlike AIS's result, the $\Vs $ function
in this equation has been linearized with respect to $\Po $, so the
resulting $d(\Bi )/dt$ versus $\Po $ curve also has a
\begin{picture}(10,10)\thicklines\put(1,10){\line(1,-1){10}}\end{picture}
shape. On the other hand, this equation predicts that $d(\Bi )/dt$ will
increase as $HR$ increases, which is what the AIS result predicts but
not Sato {\em et al.}'s result.  A difference between this equation and
the ones given by both Sato {\em et al.} and AIS is that this one has
some terms to account for the affects of the atria (the $K\cdot F_A$
term) while the other two do not (they were given models that assume
that the atrial effects are either negligible or can be folded into the
expressions for the ventricles).  Also, unlike the AIS result, $\Vs $ in
this equation is independent of $\HR $.

This comparison of AIS's results to existing steady-state ventricle
models shows that the former are fairly similar to the latter.
Furthermore, the existing differences are due to different assumptions
being made about the ventricles, not to deficiencies in AIS itself.
Two major differences between AIS's results and the existing models are
that the latter have more specific relationships for the volume versus
pressure curves than the former and that these more specific curves are
also more linearized. In addition, in two of the existing ventricular
models, the blood flow rate is independent of the heart rate, which is
often not true, especially during exercise or other times of increased
venous return \cite[p.~414]{Braunwald} \cite[p.~222]{Ross85}.

\paragraph{SUMMARY:} 
A program called AIS has been applied to a description of a ventricle.
AIS takes in a description of the parameter changes in a heart-beat
sequence and finds the symbolic average rate of change for various
parameters. These rates form a steady-state model of a ventricle, and
this model is similar to existing ones. The differences that exist can
be traced back to differing assumptions one can make about a ventricle.
A major difference is that two of the existing models have the
shortcoming of letting the blood flow rate be independent of the heart
rate.

\paragraph{ACKNOWLEDGMENTS:} 
Various members of the the lab's CDMG group have aided this research,
especially Peter Szolovits, Jon Doyle, Elisha Sacks and William Long.

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