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% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
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% Basically the Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
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\begin{document}

\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\\COLD PILL}




\markboth{Cold Pill}{Cold Pill}





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

\makeStitlePDFLaTex



\begin{abstract}
A model for the flow of a cold pill drug through the gastrointestinal compartment to the bloodstream compartment of a human subject is proposed. Students solve the system of differential equation model, use known parameter values, and plot solutions. Students also determine maximum amount of drug in the bloodstream and the time of such a peak level. Finally, students determine the rate at which the drug is absorbed into the bloodstream from the gastroinstestinal compartment in order to meet certain medical requirements, namely, set time for maximum amount of the drug in the bloodstream.
\end{abstract}




\section*{SCENARIO DESCRIPTION}

This modeling opportunity is based on an examination question, offered in quotes,  in a differential equations course given at the University of Delhi. We are grateful to our colleague DR Swarn Singh,  Mathematics, Sri Venkateswara College, University of Delhi, Delhi INDIA for sharing this material.

\begin{quote}
``Consider a model to describe the levels of the drug in a patient taking a single cold pill:


\begin{eqnarray}\label{eq:system}
\displaystyle{\frac{dx}{dt}} &=& - k_1 x\\ \nonumber
\displaystyle{\frac{dy}{dt}} &=&  k_1 x - k_2 y
\end{eqnarray}

                      
Here $k_1$ and $k_2$ describe rates at which the drugs move between the two sequential compartments (gastroinstestinal (GI)-tract and blood stream). At time $t$, $x = x(t)$ and $y = y(t)$ are the levels of the drug in Gi-tract and bloodstream, respectively. 

\begin{itemize}
\item [(a)] Find solution expressions for $x = x(t)$ and $y = y(t)$ which satisfy this pair of differential equations, when  $k_1$  and $k_2$ are not equal.''
\end{itemize}
\end{quote}

We add the following questions:

\begin{itemize}
\item [(b)] For parameter values   $k_1= 0.9$ 1/hr and $k_2= 0.6$ 1/hr with initial amounts of the drug being $x(0) = 300$ mg and $y(0) = 0$ mg plot the solutions.
\item [(c)]  For parameter values    $k_1= 0.9$ and $k_2= 0.6$ 1/hr with initial amounts of the drug being $x(0) = 300$ mg and $y(0) = 0$ mg  determine when the amount of drug peaks in the bloodstream and the amount of the drug in both the bloodstream and the GI-tract at this time.
\item [(d)] Pharmaceutical manufacturers have the capability of slowing down the rate at which the drug goes from the GI-tract into the bloodstream, i.e. altering $k_1$, through coating on the pill for example. This will not alter the natural rate at which the drug leaves the bloodstream or is consumed by the body.  Determine how sensitive the peak amount of drug in the bloodstream is, as well as the time of that peak amount, in terms of the value of  $k_1$. You may presume all other values are held to the same level, i.e. keep $k_2  = 0.6$ 1/hr, $x(0) = 300$ mg,   and $y(0) = 0$ mg.
\item [(e)] What value of $k_1$, hence what coating level on the ingested pill, should the pharmaceutical company select so that the  time of the maximum amount of drug in the bloodstream is 1.5 hr and the actual amount of drug in the bloodstream is 125 mg?
\end{itemize}

\teacher{
\section*{NOTES FOR TEACHERS}


We offer this modeling scenario to illustrate how one source idea from one teacher can reach across distance to stimulate a colleague and how collaboration can build quality teching material.

There are a number of ways to solve the 2 by 2 linear system (\ref{eq:system}) including (1) substitution the solution of the first equation in (\ref{eq:system}) for $x(t)$   into the second equation for $y(t)$ and solving the resulting differential equation, say, by integrating factor method; (2) using eigenvalue/eigenvector approach; or (3) using a computer algebra system such as Maple, SAGE, or Mathematica.

Activities (b) and (c) require a plot and an optimization of a function of time.

Activities (d) and (e) are meant to point out that manufacturers have choices in determining the rate constant $k_1$ which will delay or speed-up the movement of the drug from the GI-tract to the bloodstream and to actually compute the necessary rate constant, $k_1$ for a specific medical outcome.

We offer complete analyses and commentary in the Mathematica notebook 5-22-T-Mma-ColdPill-TeacherVersion.nb and the accompanying pdf file for those who do not have access to Mathematica.
}





\end{document}
