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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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%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, 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\\MULTIPLE DRUG DOSES}



\teacher{
    \keywords{drug, doses, Dirac, step function, Laplace Transform}

\tagwords{administration, inverse Laplace Transform, first-order, model, differential equation}
}


\markboth{Multiple Drug Doses}{Multiple Drug Doses}






\author{Brian Winkel, Director\\
SIMIODE\\
Cornwall NY \teacher{12518} USA\\
\teacher{Director@SIMIODE.org}
}

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\teacher{
\begin{abstract}
Two possible multiple dose drug administration regimens are offered for modeling.  A drug is to maintain a certain level (above a set minimum and below a set maximum) in the blood stream and one regimen involves bolus injections and the other involves steady drip flow over short periods of time.  One of the regimens cannot meet the constraints and a modification is required of the students.
\end{abstract}
}

\teacher{

\vspace*{-.1 true cm}
\listkeywords

\vspace*{-.9 true cm}
\listtags
}

\section*{STATEMENT}

You are to design an intravenous drug administration plan for a 24 hour span for a hospital patient. The patient has a port or shunt to their vein so medicine can be administered without further ``sticking.'' This medicine is to be administered in such a way that the amount of the drug in their blood stream must be between 2 and 4 mg throughout the 24 hour period. To go over 4 mg is to risk adverse effects and to go under 2 mg is to possibly not benefit from the drug.  The drug is absorbed into the tissues where it does its work at the rate of 50\% of the amount present in the bloodstream per hour.

There are two options:

\begin{enumerate}
\item bolus or plug injection of a set amount of the drug at fixed time intervals or 
\item continuous drip administration of the drug over set time intervals.  
 \end{enumerate}
 
We know that, ``In medicine, a bolus (from Latin bolus, ball) is the administration of a discrete amount of medication, drug or other compound in order to raise its concentration in blood to an effective level.''\cite{Wikia}  

Because of possible shock effect the drug can be administered in any amount for no more than 1/2 hour of each hour long time interval.

Here are the taskings for you:
\begin{enumerate}
\item  Build a mathematical model for (1) and (2) approaches, if possible, prescribing initial dosing and drug administration to comply with requirements. 
\item What criterion would you use to compare these two approaches?  Determine and demonstrate your optimal solution.
\item If you find one of these approaches is not possible, then modify the approach to make it possible and demonstrate that possibility.
\end{enumerate}



\begin{thebibliography}{99}

\bibitem{Wikia} Wikipedia contributors. Bolus (medicine). Wikipedia, The Free Encyclopedia. \href{https://en.wikipedia.org/w/index.php?title=Bolus\_(medicine)\&oldid=738637625}{https://en.wikipedia.org/w/index.php?title=Bolus\_(medicine)\&oldid=738637625}. Accessed 12 October 2016.


\end{thebibliography}
\teacher{
\section*{COMMENTS}


This is a great opportunity to apply Laplace Transform methods as both drug administration regimens use appropriate functions, namely,  Dirac delta functions and step functions.

In  Case 1 using a bolus or plug injection it is impossible to do this and meet the conditions.  One can reason that using the bolus approach of injecting a set amount, the same set amount, $a$, at fixed time intervals is not possible. For if the amount to be in the blood is to be between 2 and 4 mg all the time then the initial bolus must be at least $a = 2$ mg, which would then immediately drop below 2 mg level as time increases. So this means we must administer an initial bolus of more than 2 mg. Now if all future bolus administrations are to be the same size, i.e. greater than 2 we cannot have the amount of drug in the bloodstream both be in the region $[2, 4]$ mg BEFORE the administration bolus and AFTER the administration of the bolus. 

Full analysis and commentary is offered in the Mathematica notebook 7-10-T-Mma-MultipleDoses-TeacherVersion.nb and accompanying pdf file for those who do not read Mathematica files.

The essence of the analysis for Case 1 using a bolus or plug injection is that we can modify the bolus administration regimen  so as to administer a fixed amount, say 4 mg, initially, and then when half of that amount has been absorbed by the body tissue and we are down to 2 mg of the drug in the bloodstream we simply administer a  bolus of 2 mg, sending the total amount of drug in the bloodstream back up to 4 mg. Then we wait the same amount of time, in our case, 1.38692 hrs, and administer another bolus of size 2 mg, and so forth. Figure~\ref{fig:7-10-BolusDoseWorksB} illustrates this phenomenon. There are other combinations of initial dosage and follow-on constant dosing which will satisfy the constraints, but in each case we would need to compute the time between doses and the amount of the follow-on dose itself.


\begin{figure}[t]
\begin{center}
\includegraphics[width=10.6cm]{7-10-BolusDoseWorksB.png}



\parbox{10.7 true cm}{\caption{\label{fig:7-10-BolusDoseWorksB} Plot of modified bolus drug administration in which we initially administer 4 mg of the drug and at equal intervals of length 1.38692 hrs thereafter we administer an additional bolus of 2 mg of the drug.}}
\end{center}

\vspace*{-.5 true cm}
\end{figure}


Case 2 in which we apply a continuous drip administration of the drug at rate of  $a$ mg/hr over set time intervals of duration $dur$ as a percent of an hour has possible solutions and we depict these in Figure~\ref{fig:7-10-SteadyDoseWorksB}. 

\begin{figure}[h]
\begin{center}
\includegraphics[width=10.6cm]{7-10-SteadyDoseWorksB.png}



\parbox{10.3 true cm}{\caption{\label{fig:7-10-SteadyDoseWorksB} Plot of continuous drip administration of the drug at rate in mg/hr over set time intervals of duration as a percent of an hour.}}
\end{center}

\vspace*{-.5 true cm}
\end{figure}

In Figure~\ref{fig:7-10-SteadyDoseWorksB} we depict a large region of possible solutions which keep the amount of drug in the bloodstream between 2 and 4 mg. However, the constraint that we can only administer the drug for at most one half hour limits our application regimen to $a$ and $dur$ in the region marked allowable in the figure.


Tasking (2), ``What criterion would you use to compare these two approaches?  Determine and demonstrate your optimal solution.''  This is left to the individual student to design criterion on which to compare dosings.

}
 




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