% SIMIODE-TeacherStudent-Template.tex and simiode.cls for LaTeX2e
% Version 22 November 2016.
%
% This template is to be used with the class file ``simiode.cls".
%
% The latest version of this class and template files
% can be found at
%
% in the Author section.
%
% A ``Quick Start Guide'' appears below.
%
% This file must be processed with version LaTeX2e or higher.
% Appreciation to Mark Frantz, IUPUI; John Thoo, Yuba College; and Eric Sullivan, Carroll College.
%
\documentclass{simiode}
%
% 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.
%
% 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 only 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.
%
%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.
%
% Authors can prepare a full Teacher Version using the notions immediately below to create a pdf for review, but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
%
%*********** Toggle the following line to turn the teacher version on (1) or off (0). ***********
%
\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
%
% The \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.
% Rename the pdf after you compile each final version.
%
\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{url}
%ADDED
% makes list spacing much better. A Alexander
\newenvironment{my_itemize}{
\begin{itemize}
\setlength{\itemsep}{1pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}}{\end{itemize}
}
%
\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.
%
% Top matter is above and the beginning of the document below.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
%\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\\Drug Model for Aspirin Absorption}
% left page running head, then right
\markboth{Aspirin Absorption}{Aspirin Absorption}
\author{
Therese Shelton\\
Southwestern University\\
Georgetown TX 78626 USA\\
shelton@southwestern.edu\\ \\
Beulah Agyemang-Barimah\\
Graduate Program in Computational Biology\\
Cornell University\\
Ithaca NY 14850 USA\\
bagyemangbarimah@gmail.com\\ \\
Theresa Laurent\\
St. Louis College of Pharmacy\\
St. Louis MO 63110-1088 USA\\
Theresa.Laurent@stlcop.edu\\
}
\makeStitlePDFLaTex
\begin{abstract}
We model the amount of aspirin absorbed by the human body at a constant rate. This is a ``zero-order reaction'' in the language of pharmacokinetics -- the study of how drugs move in the body. This simple activity can be used to introduce differential equations and it can be used to introduce the study of drugs in the body to students who know some differential equations.
%Offer an abstract which describes what the students will do and learn, including technical descriptions of mathematics, area of application, and outcomes expected for the scenario. This should help the teacher decide if the material is suitable for a given class. This will appear in Teacher Version only.
\end{abstract}
\section*{SCENARIO DESCRIPTION}
% This is the STATEMENT of the scenario for the student, with pointers to data sets, supplementary files, videos, etc. for student consumption. It should have clear statements, instructions, hints, necessary materials and possible reference to original source material, but not to material which ``gives away'' a solution. E.g., if there is historical material cited then that should be given in a set of references \cite{Hutchinson1978}.
% (deleted info on naming conventions)
% This STATEMENT section should be the first section in the Teacher Version and then followed by the COMMENTS section in which the teacher shares with colleagues issues and possible solution strategies surrounding this modeling scenario.
How does the human body absorb some drugs, particularly aspirin? That is the question for our study.
\section{ASPIRIN ABSORPTION}
\subsection*{Activity 1: First Thoughts on Modeling Drug Absorption}
{\noindent }(1.1) Typically when a drug is administered to an individual, the amount of the drug in mg, A(t), in the body changes over time in minutes.
{\em Pharmacokinetics} is ``the science of the kinetics of drug absorption, distribution, and elimination (i.e. metabolism and excretion.)''\cite[p.~4]{Shargel} Write an equation that corresponds to a {\bf constant release} of the drug from an ingested tablet into the body over time.
\begin{equation} \frac{dA}{dt} = \end{equation}
{\noindent }(1.2) Identify the following for your equation, or write ``none'':
independent variable(s); dependent variable(s); constant(s); parameter(s).
{\noindent }(1.3) For a drug that is released into the body at a constant rate, would you expect the amount of drug in the body to increase, stay the same, or decrease with time, at least for a while? What value would you expect for the initial amount of drug in the body?
\subsection*{Activity 2: A General Model }
{\noindent }(2.1)
%Drugs might be administered to people or other animals in a variety of forms, including, but not limited to, a fast acting powder, tablet, and rectal suppository.
One general model of drug amount in the body is given by
\begin{equation}
\frac{dA}{dt} = k. \label{de1}
\end{equation}
Use your mathematical background: $\frac{dA}{dt}$ represents the
\underline { \hspace {.2 in} } % rate of change
\underline { \hspace {.05 in} }
\underline { \hspace {.2 in} }
of the amount of drug in the body over time, in units of \underline { \hspace {.25 in} } .
{\noindent }(2.2) A ``zero-order reaction'' in pharmacokinetics is exemplified by \eqref{de1}.
Classify this differential equation using mathematical terms. Determine a general solution.
\subsection*{Activity 3: ASA Model and Specific ASA Situation.}
ASA stands for {\it acetylsalicylic acid\/}, which can be used to treat pain and other conditions.
It is the primary ingredient in Bayer\textregistered Aspirin\textsuperscript{TM}. Note that Aspirin is trademarked in some countries. Other ASA variants include BC\textregistered Powder and Excedrin\textregistered.)
A patient swallows a tablet that contains 325 mg of ASA. A specific model of drug amount in this case is given below from \cite{Shargel}.
\begin{equation}
A(t) = 0.86 t - 0.04 \label{fcn1}
\end{equation}
{\noindent }(3.1) Is this a zero-order reaction?
{\noindent }(3.2) The tablet takes a while to dissolve. What are the smallest and largest amounts of ASA in the body? At what times do these occur?
Determine a realistic time interval for \eqref{fcn1} to be in effect and graph the resulting realistic $ A(t)$.
{\noindent }(3.3) Give the differential equation form of \eqref{fcn1} and provide a realistic initial condition.
{\noindent }(3.4) Create a ``phase plane'', which in the context of our differential equation, is a graph with $A(t)$ on the horizontal axis and the rate of change of the drug amount $\frac{dA}{dt}$ on the vertical axis.
{\noindent }(3.5)
What about the phase plane might indicate that the reaction is ``zero-order''?
\begin{thebibliography}{98}
\bibitem{Shargel}
Shargel, L. and A. Yu 2016. {\it Applied Biopharmaceutics and Pharmacokinetics. 7th ed.} New York: McGraw Hill.
\end{thebibliography}
\teacher{
\section*{NOTES FOR TEACHERS}
%This is a set of COMMENTS about the scenario, for the teacher only. It might include a ``solution'' or a number of solution strategies. However, the main narrative should be about how to use the scenario in teaching, e.g., what are prerequisites, how long might one devote to it, what resources for students could be expected to be used, what issues which cause students difficult or ease should be expected to come up, discussion of technology use, pointers to other files, etc. This set of COMMENTS might have its own pointers to data sets, supplementary files, videos, etc. for student and teacher use, the material for students could be something the teacher might use with discretion.
This modeling scenario is one of a set of activities, each dealing with a different drug over time in the human body: (1) amount of aspirin absorbed, (2) concentration of caffeine eliminated, and (3) concentration of digoxin in two compartments of the body.
Teachers can see \cite{Shelton} for a discussion and brief analysis of these first order linear ordinary differential equations, the last of which is a $2 \times 2$ system.
Two styles of the student version for aspirin absorption are provided: one with just the questions, and a fuller style with student learning outcomes and a small self-assessment.
The scenarios (1) - (3) can be used independently from one another or in the sequence indicated. The level of inquiry-oriented learning progresses through the sequence of activities.
The aspirin scenarios include a phase plane; the pair of activities allows a visual distinction between ``zero-order reaction'' and ``first-order reaction'' in pharmacokinetics.
The digoxin module has the students develop a compartmental diagram from which they write the system of differential equations, and the analysis is much fuller.
All activities engage students in an interplay between mathematics and pharmacokinetics.
We had students work in groups and focused on their written explanations.
We have implemented the activities in a modeling class in which some students had differential equations recently, others a long time ago, and some not at all. The activity was a good way to level the playing field. We also used it in a differential equations class. We found that the style of questions provided sufficient guidance that students were mostly able to work independently from the instructor.
The instructor will note that we ask the students for a differential equation in Aspirin Activity (1.1) and then give the answer as a general model in Activity (2.1). Adding spacing to the questions makes this less obvious. In this scaffolded assignment, we have found that students benefit from the opportunity to self-check.
\begin{thebibliography}{99}
\bibitem{Shelton}
Shelton, T., T.~Laurent, and B.~Agyemang-Barimah. To appear. Pharmacokinetic Models for Active Learning of Differential Equations.
{\it PRIMUS.} DOI 10.1080/10511970.2018.1484398.
\href{https://www.tandfonline.com/doi/full/10.1080/10511970.2018.1484398}{https://www.tandfonline.com/doi/full/10.1080/10511970.2018.1484398}.
\end{thebibliography}
}
\end{document}