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\begin{document}

\newpage
\title{TEACHER VERSION\\LIPOPROTEIN METABOLISM}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION   \else STUDENT VERSION \fi\\LIPOPROTEIN METABOLISM}


\author{Brian Winkel, Director
SIMIODE\\
Chardon OH 44024 USA\\
BrianWinkel@simiode.org}

\markboth{Lipoprotein Metabolism}{Lipoprotein Metabolism}



\makeStitlePDFLaTex

\begin{abstract}
Data from a study on the amounts of  low-density-lipoprotein (LDL), a form of cholesterol, in blood plasma  is presented. Students  build, validate, and use a compartment model of the kinetic exchange of the LDL between the extravascular body tissue and the blood plasma. 

\end{abstract}





\section*{SCENARIO DESCRIPTION}


The background for this modeling scenario comes from \cite[pp.~208-217]{Anderson1983} and appears in a section entitled The Basic Estimation Problem in which the author describes a number of parameter estimation techniques using   ``\ldots the data which is based on baboon low-density-lipoprotein (LDL) data taken over a 10-day period in  laboratory animal studies relating to the material in \cite{BliheimerEtAl1975}.'' The data (see Table~\ref{table:data}) is in time, days, with amount of a tracer compound in Compartment 1 - Blood Plasma while ``\ldots the experiment consists of a bolus injection of a unit amount of tracer into Compartment 1 -  Blood Plasma at time $t = 0$, and then subsequent sampling of Compartment 1 at times $t$ in days.  Compartment 2 is the extravascular or non-blood plasma space of the body.  The single model exit ($a_{02} = 0$) is used.'' The tracer (${}^{\rm 125}$I radioactive labeled LDL) is a normalized dose of size 1 unit to begin at time $t = 0$.

\begin{quote}
The kinetic parameters for LDL turnover were calculated
using methods originally described by Matthews \cite{Matthews1957} and
by Nosslin \cite{NosslinEtAl1964} and recently adapted to LDL turnover
studies by Langer \cite{LangerEtAl1972}.  Curve fitting was performed
using the SAAM 25 Computer Program to derive the two
exponential components of the plasma die-away curve.''\cite[p. 1422]{BliheimerEtAl1975}
\end{quote}


In Table~\ref{table:data} we see data on a tracer for low-density-lipoprotein (LDL) after an initial intravenous injection. Again, this is ``\ldots based on baboon low-density-lipoprotein data taken over a 10-day period in laboratory animal studies relating to the material in \cite{BliheimerEtAl1975}.''  We will attempt to model this using the two compartment model pictured in Figure~\ref{fig:5-12-LipoproteinMetabolismDiagram}.

\begin{table}[h]\label{table:data}
\centering

\begin{tabular}{|c|c|} \hline \hline
\multicolumn{1}{|c|}{Time (days)} & \multicolumn{1}{|c|}{Tracer in Units in A}\\ \hline \hline
0.0 & 1.0000\\ \hline 
0.5& 0.4610\\ \hline 
1.0 & 0.2590\\ \hline
1.5  & 0.1700\\ \hline
2.0  & 0.1210\\ \hline
3.0  & 0.0722\\ \hline
4.0  & 0.0451\\ \hline
5.0  & 0.0319\\ \hline
6.0  & 0.0240\\ \hline
7.0  & 0.0182\\ \hline
8.0  & 0.0141\\ \hline
9.0  & 0.0100\\ \hline
\!\!10.0 & 0.0094\\ \hline
\end{tabular}

\parbox{8.9 true cm}{\caption{\label{table:data}  Data on amount of low-density-lipoprotein (LDL) tracer over time in days found in plasma compartment of baboon after initial intravenous injection. \cite[p.~211]{Anderson1983})}}

\end{table}


\begin{figure}[h]
\centering


\includegraphics[width=13cm]{5-12-LipoproteinMetabolismDiagram}

\vspace*{.2 true cm}
\parbox{9.9 true cm}{\caption{\label{fig:5-12-LipoproteinMetabolismDiagram}  Diagram (from \cite[p.~103]{Anderson1983}) of two compartment model for the study of low-density-lipoprotein (LDL) kinetics in a baboon.}}
\end{figure}




\subsection*{Activity 1}


Using the notion of ``simple change in something,'' in this case concentration of LDL in each compartment, we can produce the system of differential equations in (\ref{eq:LDLEQ}). 

\begin{itemize}
\item [a)] Explain how we can develop the differential equations   (\ref{eq:LDLEQ}) from the diagram in Figure~\ref{fig:5-12-LipoproteinMetabolismDiagram}which shows the flow of LDL between the plasma compartment and the non-plasma compartment of the body. 
\end{itemize}
 
 If we are going to  model this phenomenon as offered in Figure~\ref{fig:5-12-LipoproteinMetabolismDiagram} then we need to identify variables and relationships between variables.
\begin{itemize}
\vspace*{-.2 true cm}  \item[] \hspace*{-.44 true cm} $x(t)$ - amount (unknown units) of LDL in the body plasma.
\vspace*{-.2 true cm} \item[] \hspace*{-.4 true cm} $y(t)$ - amount (unknown units) of LDL in the  extravascular space. 
\vspace*{-.2 true cm} \item[] \hspace*{-.4 true cm} $V_x$ - volume of the body plasma.
\vspace*{-.2 true cm} \item[] \hspace*{-.4 true cm} $V_y$ - volume of the  extravascular space.
\item[] \hspace*{-.4 true cm} $a_{21}$ - rate constant (1/day) for the flow of LDL from  plasma to the extravascular space.
\vspace*{-.2 true cm} \item[] \hspace*{-.4 true cm} $a_{12}$ - rate constant (1/day) for the flow of LDL from the  extravascular space to plasma. 
\vspace*{-.2 true cm} \item[] \hspace*{-.4 true cm} $a_{01}$ - rate constant (1/day) for the metabolism and excretion  of LDL in  plasma.
\vspace*{-.2 true cm} \item[] \hspace*{-.45 true cm} $a_{02}$ - rate constant (1/day) for the metabolism and excretion  of LDL in  extravascular.
\end{itemize}

\noindent Here is the resulting differential equations model we ask you to defend.

\vspace*{-.7 true cm}
\begin{eqnarray} \label{eq:LDLEQ}
x'(t) &=&   -a_{21}x(t) - a_{01} x(t) + a_{12} y(t)   \\ \nonumber
y'(t)&=&  a_{21} x(t) - a_{12} y(t)  - a_{02} y(t  \, .
\end{eqnarray}

Notice that the volumes of  parts of our model, $V_x$ and $V_y$ are not given to us. However, we do know the initial concentration of LDL in the plasma compartment, i.e. $x(0) = 1$, an unknown unit, but we presume there is no LDL in the rest of the extravascular space, so $y(0) = 0$ is used. Furthermore, recall that  ``single exit'' ($a_{02} = 0$) model is assumed \cite[p.210]{Anderson1983}. 






\subsection*{Activity 2}

We seek to estimate the  parameters $a_{21}$, $a_{12}$, and $a_{01}$,  each with units $1/{\rm day}$  if we are to fully understand this process.

\begin{itemize}
  \item [a)] Perform the necessary analysis to determine the parameters  $a_{21}$, $a_{12}$, and $a_{01}$,    using the data in Table~\ref{table:data}.
  \end{itemize} 

So how do  we estimate these parameters?  We use Mathematica to get closed form solutions for $x(t)$ and $y(t)$, each of which is about 5 lines of complex algebraic expressions in the familiar format of solutions to such systems, i.e. sums of exponentials with messy terms involving  $a_{21}$, $a_{12}$, and $a_{01}$.  If we place the  values   of the amount of LDL tracer in the plasma at each time $t$ into an array, calling it $X$,  with time as the first coordinate and the amount of LDL tracer in the plasma at the corresponding time as the second coordinate we have  13 (from observations) time-units data pairs. From these we  form the sum of square errors 
(\ref{eq:SSE})  between the model $x(t)$ values and the observed values for amount (units) of LDL tracer in the plasma. Call the latter $O_i, i = 1, 2, \ldots, 13$. 

\begin{equation}\label{eq:SSE}
{\rm SSE}(a_{21}, a_{12},  a_{01}) = \sum_{i=1}^7 (x(t_i) - O_i)^2
\end{equation}


 

Now with Mathematica's powerful {\tt FindMinimum} command we can determine the values of the parameters $a_{21}$, $a_{12}$, and $a_{01}$, which minimize this ${\rm SSE}(a_{21}, a_{12},  a_{01})$ function, 

\vspace*{.15 true cm}
\centerline{\tt FindMinimum[SSE[$a_{21}$, $a_{12}$,  $a_{01}$], \{$a_{21}$, 1\}, \{$a_{12}$, 1\}, \{$a_{01}$, 1\}]\, .}
\vspace*{.15 true cm}

\begin{itemize}
 \item [b)] Use the resulting parameters in the model (\ref{eq:LDLEQ}) and determine a function for $x(t)$, the amount  of LDL in the body plasma or bloodstream at time $t$ minutes.
 \item [b)] Plot the model function, $x(t)$, with the data on observed levels of LDL found in Table~\ref{table:data} and compare them. 
 \item [c)] How much time will it take to get the LDL levels down to 5\% in the blood plasma compartment of the original amount injected according to this two compartment model?  
 \item [d)] Use your parameters in the model (\ref{eq:LDLEQ}) and determine a function for $y(t)$, the amount of LDL in the extravascular space, i.e. the extravascular space  at time $t$ minutes.
 \item [e)] Determine the time at which the  amount of LDL in the extravascular space, i.e. $y(t)$,  peaks and the time at which the concentration of LDL in the extravascular space will fall to  5\% of that maximum value. 
\end{itemize}




\subsection*{Activity 3}

 Vary one of the parameters, say  $a_{21}$ the rate at which the tracer goes from the bloodstream to the extravascular region of the body and see how sensitive the mode is to slight or large changes in the parameter. Discuss what you see and how it makes sense in your understanding of the physiology involved.

\teacher{
\section*{NOTES FOR TEACHERS}


This could be a first time through for a modeling scenario in which a linear model is built (notice we ask students to justify a presented model from a diagram, not build a model from scratch). Moreover, we suggest the approach in using the analytic solution with the parameters involved to do a minimization of the sum of square errors analysis to obtain the best parameters.  This entire activity would take one or two periods and an outside writing assignment with a significant write-up of both the mathematics and the modeling work. I have used this sort of modeling activity before studying systems of differential  equations  and I have used it in the midst of studying systems. Either way, modeling activities can serve to motivate the study.  I have found that once we have done a few models with parameter estimation, students pick up on this quickly and see the big picture as well as the use of  Mathematica quite easily. This is not a semester project type problem, rather students enjoy seeing their differential equations modeling skills pay off in yet another area just as they study the mathematics. 

It is important to motivate the study of differential equations. The appropriate and historical way I see to do this is through applications in which the differential equations arise naturally as a model for the phenomena. Furthermore, I believe it is important, not to dwell on all the by-hand techniques of the past, but to use technology such as  Mathematica or {\it Maple} so the students can see the structures (however complicated they may be) of the solutions and  analyze the data. This LDL study illustrates just how this comes together, I believe.


We have placed   development and modeling results in a Mathematica file at the SIMIODE web site, 5-12-T-Mma-LipoproteinMetabolism-TeacherVersion.nb, as well as in a pdf file 5-12-T-Mma-LipoproteinMetabolism-TeacherVersion.pdf for those who do not use Mathematica.


Figure~\ref{fig:5-12-LipoproteinMetabolismPlot} shows a convincing plot from the above Mathematica file that the two compartment model does a fine job of modeling the data. This is a very convincing plot for students to validate the model and the estimated parameters. 


\begin{figure}[h]
\centering


\includegraphics[width=13cm]{5-12-LipoproteinMetabolismPlot}

\vspace*{.2 true cm}
\parbox{11.9 true cm}{\caption{\label{fig:5-12-LipoproteinMetabolismPlot}  Plot of actual data from Table~\ref{table:data} with model superimposed over the data.}}
\end{figure}
}

\begin{thebibliography}{99}

\bibitem{Anderson1983} Anderson, D.~H. 1983. {\it Compartmental Modeling and Tracer Kinetics. Lecture Notes in Biomathematics, Number 50\/}.  New York:  Springer-Verlag.

\bibitem{BliheimerEtAl1975} Billheimer, D.~W., J.~L.~Goldstein, S.~M.~Grundy, and M.~S.~Brown. 1975.  Reduction in Cholestorol and Low Density Lipoprotein Synthesis after Portacaval Shunt Surgery in a Patient with Homozygous Familial Hypercholesterolomemia. {\it The Journal of Clinical Investigation\/} 56: 1420-1430. 

\bibitem{LangerEtAl1972} Langer, T., W.~Strober, and R.~I.~Levy. 1972. The
metabolism of low density lipoprotein in familial type
II hyperlipoproteinemia. {\it J.~Clin.~Invest\/}. 51: 1528-1536.


\bibitem{Matthews1957} Matthews, C.~M.~E. 1957. The theory of tracer experiments
with iodine$^{\rm 131}$-labeled plasma proteins. {\it Phys.~Med.~Biol\/}. 2: 36-53.

\bibitem{NosslinEtAl1964} Nosslin, B., J.~E.~Olsson, and H.~Link. 1964. Metabolic Studies on ${}^{\rm 125}$I-labelled Beta-trace Protein, with Special  Reference to Synthesis within the Central Nervous system.  In Appendix. Anderson, D.~H.~(Ed.) {\it Metabolism of Human Gamma Globulin\/}. Oxford UK: Scientific Publications,
Ltd. pp.~103-120.

\end{thebibliography}






\end{document}

