% SIMIODESample.tex and simiode.cls for LaTeX2e
% Version 11 March 2013.
% 
% This class (simiode.cls) is a modification of that used by 
% the journal PRIMUS - Problems, Resources, and Issues in 
% Mathematics Undergraduate Studies. 
% 
% See http://www.tandfonline.com/loi/upri .
% 
% The class was developed by  Marc Frantz with the  suggestions of the 
% PRIMUS Editorial Board, especially John Thoo, and is much appreciated.
%
% This template "SIMIODEPaper.tex" is to be used with the 
% class file "simiode.cls".  You may rename the template.  
% The latest version of this class and template files 
% can be found at
% 
%                <http://www.SIMIODE.org> .
%
% For an explanation of using the class file "simiode.cls", 
% see the pdf ``Description and Illustration of Use: 
% A LaTeX Class File for SIMIODE'' found at  
% 
%                <http://www.SIMIODE.org> . 
%
% A "Quick Start Guide" appears below.
%
% For information on using TeX or LaTeX, go to %                 <http://www.tug.org>.

% This file must be processed with version LaTeX2e or higher.
%
\documentclass{simiode}

\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
%

\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding double spacing %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Manuscripts should be submitted doubled spaced.  This is 
% accomplished via the \baselinestretch command below.
% However, you may obtain an idea of how your article will 
% look in final form, so that you can better plan your 
% presentation.  To see your manuscript formatted very 
% similarly to the on-line published format of  SIMIODE, 
% comment the following line out by inserting a percent sign % (%) at its beginning.

\renewcommand{\baselinestretch}{1.3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding macros %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Any macros defined for your paper should be contained in 
% the top matter. Likewise, any environment definitions such 
% as \newtheorem or \newenvironment should also go in the 
% top matter. 
% 
% Do not \input your macros as separate files within your 
% final version because more files make it harder for the 
% editors and users to keep track of and/or modify and 
% append your paper.


\newcommand{\AmSLaTeX}{$\cal A$\kern-.1667em\lower.5ex\hbox{$\cal
M$}\kern-.125em $\cal S$-\LaTeX}
\newcommand{\BibTeX}{{\sc Bib}\TeX}
\newcommand{\scls}{{\tt simiode.cls}}
\newcommand{\stemp}{{\tt SIMIODEPaper.tex}}
\newcommand{\SI}{SIMIODE}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Top matter is above and 
% the beginning of the document below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\newpage

\title{\ifnum\TeacherEdition=1 TEACHER VERSION \footnote{DELETE THIS FOOTNOTE BEFORE SUBMISSION.  In the \TeX \, source code there is a toggle/option to produce a Teacher Version or a Student Version by ``carving'' out with  $\backslash$teacher\{\} that material which should only appear in the Teacher version. See the source code.} \else STUDENT VERSION \fi\\Name of Scenario}
\title{TEACHER VERSION\\PARACETAMOL ABSORPTION}


\markboth{Paracetamol Absorption}{Paracetamol Absorption}






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

\makeStitlePDFLaTex


\begin{abstract}
We offer drug absorption data on paracetamol, an analgesic pain reliever, for two different sets of patients, vegetarian and non-vegetarian. Students build a two compartment model for plasma and non-plasma (tissue) compartments of human patients and determine parameters in a complete model analysis.
\end{abstract}




\section*{SCENARIO DESCRIPTION}


Paracetamol is an analgesic (pain reliever) and this study\cite{PrescottEtAl1993} seeks to determine the effects of diet on the absorption (meaning the amount of the drug which gets into the plasma portion of the body from the non-plasma portion of the body through which it is ingested) of paracetamol from the non-plasma part of the body after it is ingested into the non-plasma part of the body through swallowing a solution  of `Coca Cola' in which the paracetamol was dissolved in the liquid. Two kinds of subjects were involved (a) vegetarian and (b) non-vegetarian.\cite{PrescottEtAl1993}

We will use data from  experiments on 10 patients reported in  \cite{PrescottEtAl1993} and attempt to model the amount of paracetamol in the non-plasma portion of the body, principally in the tissue  where the analgesic effect takes place.



\begin{figure}[!h]
\centering

\includegraphics[width=9cm]{5-20-ParacetamolAbsorption-TwoCompartmentModel.eps}


   
\parbox{10.8 true cm}{\caption{\label{fig:5-20-ParacetamolAbsorption-TwoCompartmentModel} Two compartment model for exchange of paracetamol between the plasma and non-plasma compartments of a human with urinary excretion.}}

\end{figure}

We use a two compartment model in Figure~\ref{fig:5-20-ParacetamolAbsorption-TwoCompartmentModel}  which is built from  understanding  the flow or kinetics of paracetamol absorption in the human body. $Q_A(t)$ is the concentration (mg/L) of paracetamol in the patient's non-plasma (or tissue) and $Q_B(t)$ is the concentration (mg/L) of paracetamol in the plasma (or bloodstream). $A$ is the non-plasma compartment, $B$ is the plasma compartment, $V_A$ and $V_B$ are the respective volumes of the compartments (in liters or L), $Q_A$ and $Q_B$  are the respective concentrations of the compartments (mg/L), and the $k$'s are the flow rates in units L/time, where time is in hours.  In our work we shall assume the ratio of non-plasma volume to plasma volume ($V_A/V_B$) is 7.4. Although, one could just leave $R =V_A/V_B$ as a parameter for the model as well or assume other values of $R$, say in the range 7 to 20 and compare models.


From \cite{WagnerEtAl1968} - The mass devoted to the plasma in a normal body is about 16.3\% while the mass devoted to the tissues is about 11.5\%. Thus $R = V_A/V_B =  16.3/11.5 =   1.42$  Aha the distinction is plasma to non-plasma and plasma to tissue ratio. 


\subsection*{Patients and Data Collection}

The purpose of the experiments in \cite{PrescottEtAl1993} was to study the effect of diet on the time for absorption of the parcetamol, for this is the time it will take for the drug to have its analgesic (pain killing) effect. Two different diets or regimens were studied, vegetarian and non-vegetarian. There were 10 patients in each diet group. Thus two sets of patient data were collected with the following protocol:

\begin{quote}
Following an overnight fast, 20 mg/kg of paracetamol \ldots  was taken, dissolved in 400 ml of `Coca Cola'. Fluid was withheld for the next 2 h, during which time the volunteers remained seated, and no food was allowed for 4 h after dosing. Venous blood was taken from an indwelling cannula at 0, 15, 30, 45, 60 and
90 min, and at 2, 3, 4, 5, 6, 7, 8, and 24 h, and urine was
collected  at 0, 4, 8, 12 and 24 h. Plasma and urine were stored frozen at -20$^{\circ}$ C until analyzed.\cite[p.~238]{PrescottEtAl1993}
\end{quote} 


\subsection*{Patient Data}

\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
\hspace{-.15 true cm} Time (h)  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 0.0 \hspace{-.15 true cm} &  \hspace{-.15 true cm} 0.25  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 0.5  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 0.75  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 1.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 1.5  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 2.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 3.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 4.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 5.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 6.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 7.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 8.0\\ \hline
\hspace{-.15 true cm} Paracetamol (mg/L)  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 0.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 4.5  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 8.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 9.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 10.5  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 11.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 11.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 9.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 7.5  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 5.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 4.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 3.0  \hspace{-.15 true cm} &  \hspace{-.15 true cm} 2.5\\ \hline
\end{tabular}
\parbox{8.4 true cm}{\caption{\label{table:PlasmaConcVeg} Time concentration of paracetamol in vegetarian diet patients, averaged over the 10 patients.}}
\end{table}

\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
\hspace{-.20 true cm}  Time (h) \hspace{-.20 true cm} &  \hspace{-.20 true cm}  0.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  0.25 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  0.5 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  0.75 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  1.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  1.5 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  2.0  \hspace{-.20 true cm} &  \hspace{-.20 true cm}  3.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  4.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  5.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  6.0  \hspace{-.20 true cm} &  \hspace{-.20 true cm}  7.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  8.0\\ \hline
\hspace{-.20 true cm}  Paracetamol (mg/L)   \hspace{-.20 true cm} &  \hspace{-.20 true cm}   0.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  11.25 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  13.5 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  15.25 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  13.75 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  13.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  12.0 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  8.75  \hspace{-.20 true cm} &  \hspace{-.20 true cm}  7.75 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  5.75 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  4.25 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  3.25 \hspace{-.20 true cm} &  \hspace{-.20 true cm}  2.0\\ \hline
\end{tabular}
\parbox{8.0 true cm}{\caption{\label{table:PlasmaConcNonVeg} Time concentration of paracetamol in non-vegetarian diet patients, averaged over the 10 patients..}}
\end{table}



\subsection*{Model Building Activities}

Each patient in the  study  \cite{PrescottEtAl1993} was given an initial dose of paracetamol,  $Q_A(0) = 20$  mg/kg of body mass, which ``is ingested into the non-plasma part of the body through swallowing a solution  of  `Coca Cola' in which the paracetamol was dissolved in the liquid.''\cite[p.~238]{PrescottEtAl1993}.

\begin{enumerate}
  \item Plot the data of paracetamol concentrations for vegetarian and non-vegetarian patients. Explain what you see.
  \item Build a model for the change in the concentration of paracetamol in both the non-plasma and plasma volumes, $Q_A(t)$ and $Q_B(t)$,  respectively of the body using  the information offered in Figure~\ref{fig:5-20-ParacetamolAbsorption-TwoCompartmentModel} for each situation, vegetarian and non-vegetarian diets. Use $Q_A(0) = 20$ mg/L and $Q_B(0) = 0$ mg/L.
  \item Solve your model for $Q_B(t)$, the concentration of paracetamol in the plasma or blood.
  \item Use the assumption that  the ratio of non-plasma volume to plasma volume ($V_A/V_B$) is 7.4 to eliminate variables.
  \item Using the data from both the vegetarian diet patients (Table~\ref{table:PlasmaConcVeg}) and then the non-vegetarian diet patients (Table~\ref{table:PlasmaConcNonVeg}) estimate all the parameters in the model you built.
  \item Plot the concentrations in plasma ($Q_B(t)$) of paracetamol in both the vegetarian and non-vegetarian patients cases and compare your models with the respective data sets.
  \item Use the results from your modeling with your determined parameters to compute and plot the total amount of paracetamol in the non-plasma (tissue) for each set of patients.
  \item From your analysis can you say there is any effect of diet (vegetarian or non-vegetarian) on absorption of paracetamol into the non-plasma (tissue) portion of the human body? If so what is that effect?  
  \item Write out a two paragraph description of what your modeling activity tells you with caveats and  concerns you have about the implications.
\end{enumerate}




\teacher{

\section*{NOTES FOR TEACHERS}

We enclose several analyses in a Mathematica notebook, 5-20-ParacetamolAbsorption-TeacherVersion.nb (and accompanying pdf file for those who do not use Mathematica) in which complete analysis of a reasonable two compartment flow model for both data sets, vegetarian and non-vegetarian. We believe the results we obtained say that the absorption rate for paracetamol, an analgesic pain reliever, is slower for those with a vegetarian diet  than those with a non-vegetarian diet. The paper \cite{PrescottEtAl1993} 
gives a great deal of background and physiological narrative and should interest students interested in this area of applications of differential equations.

We had difficulty in finding the ratio $R$ of the volume of non-plasma mass to plasma mass for humans, but our estimates suggested in the literature and common lore suggest that $R = V_A/V_B$ is about 7-12 range depending on the individual. Indeed, in our analysis we estimated, not only the parameter flow rates, but also the value of $R$ in an analysis for vegetarian and non-vegetarian patients and here we found $R$ to be 7.4 for vegetarians and 11.8 for non-vegetarians.  This is somewhat disturbing, but in fact, we do not know anything about these two pools. Indeed, the non-vegetarians could have been ``chunkier!''

Perhaps students in further research might ascertain a more reasonable range for $R$ and produce a slightly different, but same conclusion, model.
}

\begin{thebibliography}{99}

\bibitem{PrescottEtAl1993} Prescott, K. Yoovathaworn, K. Makrananda, R. Saivises, and K. Sriwatanakul. 1993. Impaired absorption of paracetamol in vegetarians. {\it British Journal of Clinical Pharmacology\/}. 36: 237-240. 



\bibitem{WagnerEtAl1968} Wagner, J. G., G. K. Aghajanian, and O. H. L. Bing. 1968.  Correlation of performance test scores with ``tissue concentration'' of lysergic acid diethylamide in human subjects. {\it Clinical Pharmacology and Therapeutics\/}. 9(5): 635-638.  

\end{thebibliography}



\end{document}
