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\else STUDENT VERSION \fi\\
Two Compartment Pharmacokinetic Model with \\First-Order Drug Interactions}



% left page running head, then right
\markboth{Digoxin Elimination}{Digoxin Elimination}

\author{
Therese Shelton\\
Southwestern University\\
Georgetown TX 78626 USA\\
shelton@southwestern.ed\\ \\
 Beulah Agyemang-Barimah\\
 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 concentration of digoxin eliminated from the human body at a rate proportional to the concentration. This is a ``first-order reaction'' in the language of pharmacokinetics -- the study of how drugs move in the body. This activity can be used to introduce compartmentalized models for a $2\times 2$ system of linear 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*{STATEMENT}
% 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}.
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%   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.

The human body eliminates some drugs, particularly digoxin, with transfer rates that depend on which part of the body we consider. \\

 % ==========================================================

% ==========================================================
\section{SCENARIO DESCRIPTION - DIGOXIN ELIMINATION} 
\subsection*{Activity 1: First Thoughts}
{\indent } (1.1)  Typically when a drug is administered to an individual, the concentration of the drug $C(t)$ in the body changes over time.  The drug is gradually eliminated from the body. The concentration is usually amount per unit volume, though we will see an alternative idea of concentration. A first-order drug reaction is one in which the rate of change is proportional to the concentration of the drug in the body.

Sometimes it is useful to think of how drugs distribute throughout the various parts of the body, particularly tissues (like fat and muscle where proteins might bind to the drug) and the plasma.  Even though there are many tissues throughout the body, and they are all different, we can make a simplification of treating some as a single compartment. Two main compartments of interest are the ``tissue compartment'' and the ``plasma compartment.'' 

{\noindent }(1.2)  Draw a diagram with two rectangles, each labeled to represent the drug concentration in the theoretical compartments. 
Include arrows on the diagram, and label each with an appropriate rate constant to indicate first-order drug reactions  (that is, in which the transfers of the drug occur at rates proportional to the concentration)  as follows. 

\begin{itemize}
\item From the plasma compartment, the drug can move out of the body entirely, with proportionality constant $k_{10}$, and the drug can move into the tissue compartment, with proportionality constant $k_{12}$. 
\item From the tissue compartment, the drug can move back into the plasma compartment with proportionality constant $k_{21}$.
\end{itemize}

 \teacher{  {\em Answer:} 
 \begin{figure}[htb]
    \includegraphics[scale=.7]{DigoxinDiagram2Crp.pdf} 
   \centering
   \caption{Diagram for Digoxin.}
    \label{fig:diag1}
 \end{figure} % reference as  \ref{fig:diag1}
 }
 
{\noindent }(1.3) Now write a system of differential equations relating the rates of change of the concentrations of the drug in the plasma, $C_p$, and in the tissues, $C_t$. Keep in mind all constants are positive.   
\begin{equation}\label{eq:GenDEs}
\begin{split}
       \frac{dC_p}{dt} &=  (\hspace{1 in}) C_p +  (\hspace{1 in}) C_t\\
       \frac{dC_t}{dt}  &= \hspace{2.8 in}
\end{split}
\end{equation}

 \teacher{  {\em Answer:}   
\begin{equation}\label{xx}
\begin{split}
      \frac{dC_p}{dt} &= -(k_{12} + k_{10}) \; C_p(t) + k_{21} \; C_t(t) \\
      \frac{dC_t}{dt} &=  \;  \;  k_{21}  \; C_p(t) -k_{12}  \; C_t(t) 
\end{split}
\end{equation}
}


%% - - - - - - 
\subsection*{Activity 2: Types of Patients and Model Parameter Values}  
The way a drug interacts with the body is highly affected by certain health conditions. Kidneys are responsible for a great deal of drug elimination, so a person with renal failure has a lower rate of overall elimination.  A person without conditions that require special consideration has ``normal parameters'' and a person with reduced kidney function has ``renal failure parameters.''  A great deal of research that is beyond the scope of our assignments is needed to properly treat different patients, so we use published results. According to \cite{Shargel}, Table \ref{table:1} gives some pharmacokinetic parameters of {\em digoxin}, a drug used to treat congestive heart failure. Here, $h^{-1}$ means ``per hour'' and ``per kg'' means ``per kg of body weight.''  

\begin{table}[h!]
\centering
  \begin{tabular} {| l   | c  | c   | c   | }  
 \hline
Parameter  & Unit & Set A & Set B \\
\hline  
$k_{10}$  & $h^{-1}$   & 0.04  &  0.18 \\  \hline 
$k_{12}$  & $h^{-1}$   & 0.45  &  1.02 \\ \hline 
$k_{21}$  & $h^{-1}$   & 0.11  &  0.15 \\ \hline 
$V_p $ per kg  & L/kg & 0.73  &  0.78 \\ \hline  
 \end{tabular}
 \caption{Parameter Sets of Digoxin for Subjects with Different Renal Functions.} 
\label{table:1} % reference as  \ref{table:1}
 \end{table} 
    
Which set of parameters, A or B, in Table \ref{table:1} could be used to treat a patient with renal failure? Explain, and briefly mention specific parameters.  

 \teacher{  {\em Answer:} 
   Parameter set A is for a patient with reduced renal function. 
Someone with less functional kidneys would experience a slower rate   $k_{10}$ of elimination of the drug from the body. (Other answers are possible.) 
}

%% - - - - - - 
\subsection*{ Activity 3: A Specific Model}  
Consider the following pharmacokinetic model for a specific patient.
\begin{eqnarray}\label{eq:DEs1}
\frac{dC_p}{dt} &=& -1.20 C_p(t) + 0.15   C_t(t) \\ \nonumber
\frac{dC_t}{dt} &=& 1.02   C_p(t) - 0.15   C_t(t) 
\end{eqnarray}
{\noindent }(3.1)    Determine the values for the parameters 
$k_{10}, k_{12}$, and $k_{21}$
in (\ref{eq:DEs1}) to represent a two compartment model as in Activity 1 for digoxin. 

 \teacher{  {\em Answer:} 
  This represents parameter set B for the subject with functional kidneys.  \\}

{\noindent }(3.2)   The concentration of the drug in each of the two compartments depends upon time, and they are interdependent upon each other.  Find an expression for $\frac{dC_t}{dC_p},$  and interpret this as a rate of change. We will visualize this on the phase plane. 

 \teacher{  {\em Answer:}  
 $\frac{-(k_{12} + k_{10}) \; C_p(t) + k_{21} \; C_t(t)}{ k_{21}  \; C_p(t) -k_{12}  \; C_t(t) }   \; =  \;  \frac{-1.20 \; C_p + 0.15  \; C_t}{  \; 1.02  \; C_p - 0.15  \; C_t}$ represents the rate of change of the concentration of digoxin in the tissues as a function of the concentration in the plasma compartment.  \\}

{\noindent }(3.3)   A drug administered intravenously as a ``bolus dose'', meaning over a short period of time, distributes so quickly throughout the plasma that it is modeled as being instantaneous. 
The ``volume of the plasma compartment'' is $V_p$ and is considered to be constant for the time interval of interest. 
The initial amount of the drug given in the intraveneous bolus dose is $D_0$. 
A concentration can be expressed as an amount per unit volume, or as amount per kg body weight.  
State initial conditions in general. State conditions specifically for (\ref{eq:DEs1})  to treat a 70-kg patient with an initial dose of 3.6 micrograms (mcg) of digoxin per kg of body weight, based on the information provided.  (It can be shown that this dosage is actually too high by current standards.) 

 \teacher{  {\em Answer:} \begin{equation}\label{eq:DEs1IC} 
 C_p(0)=\frac{D_0}{V_p}=4.61 {\rm mcg/L},  \hspace{.1in} C_t(0)=0.
 \end{equation}
}


%% - - - - - - 
\subsection*{ Activity 4: Phase Plane Analysis}

 We can use our first order $2\times 2$ system of autonomous (independent of time) differential equations 
as a vector field and analyze  the {\em phase plane}. 

{\noindent }(4.1)    Use technology to create a well-labeled vector field for (\ref{eq:DEs1}, \ref{eq:DEs1IC}).  Begin with a scale so that drug concentrations in either compartment might be as high as 5 mcg/L, micrograms per litre. You may adjust this scale if needed. 

{\noindent }(4.2)      A ``$C_p$ nullcline'' is a curve on which  $C_p$ does not change. Determine the $C_p$ nullcline, solving for $C_t$ if possible. Sketch on the phase plane using a dashed line. 

{\noindent }(4.3)     Complete the inequality condition that describes when the drug has a net transfer {\em out of} the plasma compartment:  $C_p$  \underline{\hspace{.3 in} }  0. 
Determine this condition for  (\ref{eq:DEs1}).  
Which side of the $C_p$ nullcline represents a net transfer of drug out of the plasma compartment? 

{\noindent }(4.4)     Determine the ``$C_t$ nullcline''. Determine the $C_t$ nullcline, solving for $C_t$ if possible.  Sketch on the phase plane using a dot-dashed line. 

{\noindent }(4.5)     Complete the inequality condition that describes when the drug has a net transfer {\em into}  the tissue compartment:  $C_t$  \underline{\hspace{.3 in} }  0.   
Determine this condition for  (\ref{eq:DEs1}).  
Which side of the $C_t$ nullcline represents a net transfer of drug into the tissue compartment? 


{\noindent }(4.6)     These nullclines break the first quadrant into three regions. 
 In one region, the vectors all point up and to the left.  Remembering that 
 $\frac{dC_p}{dC_t} = ( \frac{dC_p}{dt} ) / (\frac{dC_t}{dt}), $ this means that 
 $C_p$ is \underline{\hspace{.5 in} }  (increasing or decreasing?) and that 
 $C_t$ is \underline{\hspace{.5 in} }.  
 Describe this in terms of the net drug transfer into/out of  the two compartments.


{\noindent }(4.7)      Give the specific initial conditions for an initial drug concentration of 4.6 mcg/L. 
Sketch a solution curve on the phase plane using these initial conditions. Indicate a direction on the curve. Estimate the coordinates for the highest point on the curve. 


\teacher{  {\em Answers:} 
 Our comments to the instructor here include answers to later questions
The $C_p$-nullcline is  $C_t = (k_{12} + k_{10} )/k_{21} C_p$, and the  
$C_t$-nullcline is  $C_t = k_{12}/k_{21} C_p$. Digoxin has a net transfer out of the plasma compartment when $C_p'(t) < 0 \Leftrightarrow  C_t < (k_{12}+k_{10})/k_{21} C_p;$ we never cross the $C_p$-nullcline for our parameter sets. 
Drug is transferring into the tissue compartment when the ratio of tissue concentration to plasma  concentration, $C_t/C_p$, exceeds the ratio $k_{21}/k_{12}$, which is the rate at which drug transfers from the tissue compartment to the plasma compartment, divided by the rate at which drug transfers from the plasma compartment to the tissue compartment.  Note that one could lead with this sort of description to have students develop the differential equation model in (\ref{eq:GenDEs}), as an alternative to using the diagram approach. 


\begin{figure}[htb]
    \centering
    \includegraphics[scale=.45]{DigoxinPhasePlane1Crpd.pdf} 
    \includegraphics[scale=.45]{DigPhasePlaneZoomLegendSideCrp.pdf} 
    \caption{Normal Parameters: Phase Portrait (left), Focus on Maximal Tissue Concentration (right).}
    \label{fig:phasePlanes} 
 \end{figure}  % reference as  \ref{fig:phasePlanes}


Figure \ref{fig:phasePlanes} shows the phase portrait for the specific digoxin situation in  (\ref{eq:DEs1}, \ref{eq:DEs1IC}) with $C_t$ plotted against $C_p$.  On the left graph, we see unscaled vectors of $\frac{dC_t}{dC_p}$ in a direction field, nullclines differentiated as dashed and dot-dashed, a point to mark the initial value, and a solution curve as a parametric function \{$C_p(t),C_t(t)$\}. Because the concentration in the plasma is always decreasing, we move on the parametric curve from right to left as time increases. More specifically, our solution begins on the lower right of the phase portraits, progresses up and to the left until the time of maximum tissue concentration, and then the curve shifts down. The slopes of the $C_p$-nullcline,  $C_t = 8.0 C_p$, and 
the $C_t$-nullcline,  $C_t = 6.8 C_p$, are visualized in Figure \ref{fig:phasePlanes}. 

 
On the right graph in Figure \ref{fig:phasePlanes}, we see a zoomed-in version that helps distinguish the nullclines, with a point to mark the maximal tissue concentration. The scaled direction vectors on the right graph indicate where the rate of change of tissue concentration with respect to plasma concentration is greater. The solution curve appears smooth on the zoomed-in graph, even though the turn appears quite sharp on the zoomed-out graph; $\frac{dC_t}{dC_p}$ is well-defined. 

 The highest point on the trajectory in each view indicates the maximum tissue concentration.  This occurs when the rate of change of the tissue concentration is zero, i.e., where the parametric curve crosses the $C_t$-nullcline. This transition separates the  {\em distribution phase}, 
 with a rapid decline of drug from the plasma compartment and a rapid increase into the tissue compartment, from the  {\em elimination phase} with a slower decline from the plasma compartment.  
The nullclines divide the phase portrait into regions; vectors are similar within a region but differ across regions. For instance, vectors point down and to the left between the two nullclines in Figure \ref{fig:phasePlanes}, as both plasma concentration and tissue concentration decrease, corresponding to negative values of $\frac{dC_p}{dt}$ and $\frac{dC_t}{dt}$ in the elimination phase. 

 
The system is mathematically tractable for solution via an eigensystem, decoupling, or Laplace Transforms.  Our students practice solution techniques in other assignments and use a computer algebra system in this activity to allow them to focus on the pharmacokinetic content. Our students will have seen vector fields for systems of differential equations, but not nullclines.  
}


\subsection*{ Activity 5: Meaning of the Solution of the IVP -  as functions of time}

{\noindent }(5.1)       Use a DE solver, and record the results: Solve the initial value problem with (\ref{eq:DEs1}) and initial conditions 
\begin{equation} \label{eq:ICs1}
%\begin{align}
C_p(0) = 4.616, 
C_t(0) = 0.000
%\end{align}
\end{equation} 
 
{\noindent }(5.2)        Consider the magnitude (absolute value) of the coefficients within the exponential functions.  Let $\alpha$ be the larger magnitude and $\beta$ the smaller:  
$\alpha$ = \underline{\hspace{.5 in} }, 
$\beta$ = \underline{\hspace{.5 in} }, 

 \teacher{  {\em Answers:}   
The digoxin levels are described by (\ref{eq:Solns1}) with $\alpha=-1.3297, \beta=-0.02031.$
\begin{equation}\label{eq:Solns1}
\begin{split}
      C_p(t) &= \; 4.1582 e^{-1.3297 t}  +   0.4571 e^{-0.02031 t} \\
      C_t(t) &= -3.5953 e^{-1.3297 t}   +  3.5953   e^{-0.02031  t}  
\end{split}
\end{equation}
}

{\noindent }(5.3)        Graph the concentration (also known as the ``level'') of drug in the plasma for 24 hours after the IV bolus dose.  The graph of $C_p(t)$ is called the ``drug plasma level-time curve."  Graph the drug tissue level-time curve on the same set of axes.
Describe the behavior of the curve in calculus terms.

{\noindent }(5.4)        In a two-compartment first-order-reaction pharmacokinetic model from an IV injection, the drug declines rapidly from the plasma compartment in a period of time called the {\em distribution phase}, followed by a slower decline in the {\em elimination phase}.  \\
Indicate the phases on the drug plasma level-time curve. (An approximate distinction will suffice.) \\
Consider the coefficients within the exponential functions. Why is the distribution phase also referred to as the {\em alpha phase}?

{\noindent }(5.5)        Graph the drug tissue level-time curve. 
Describe the behavior of the curve in calculus terms.  

{\noindent }(5.6)        Graph the drug plasma level-time curve  on the same set of axes as  the drug tissue level-time curve. Label each axis, including units, and include a title. 

{\noindent }(5.7)        Label the maximum on the drug tissue level-time curve with a large dot.  \\
From the graph, estimate the time at which this occurs, the maximum tissue concentration, and the corresponding concentration in the plasma. Record and label the values here, with units:  \\
\centerline{$t_{max} =$       \underline{\hspace{.3 in} }  \hspace{.2 in}    
$C_t(t_{max}) =$       \underline{\hspace{.3 in} }  \hspace{.2 in}      
$C_p(t_{max}) =$      \underline{\hspace{.3 in} }  \hspace{.2 in} }     

{\noindent }(5.8)        Use the solutions and/or the IVPs to calculate these values to 4 decimal places. \\

 \teacher{  {\em Answers and comments:}  
\begin{figure}[htb]
   \centering
    \includegraphics[scale=.6]{NormConcVSTime.pdf} 
   \caption{Drug Level-Time Curves with Normal Parameters.}
    \label{fig:NormDrugLevelCurves}
 \end{figure} 
Figure \ref{fig:NormDrugLevelCurves} shows the drug level-time curves and points to mark maximal tissue concentration and the corresponding plasma concentration. At first, the drug distributes quickly to the tissues, so the tissue concentration increases rapidly during the alpha phase.  Tissue concentration has its greatest value of 3.32 mcg/L at  $t=3.2$ hours, with corresponding plasma concentration 0.49 mcg/L. After this time, the tissue concentration decreases, which marks the elimination phase.  Our students often estimate the maximal tissue concentration to the left of its accurate placement, perhaps focusing on the greater curvature. 
 The bi-exponential form of the solutions in  (\ref{eq:Solns1}) corresponds beautifully to the description of the distribution and elimination phases.  
Those who wish to connect more to solution techniques can have the students discuss the superposition of solutions using the eigenvalues of this linear system, as well as the stable nature  of the isolated critical point at (0,0) as indicated by the negative values for $\alpha$ and $\beta$.
}


%% +++++++++++++++++++++


%% - - - - - - - - 
\subsection*{ Activity 6: Solution as a parametric function}
{\noindent }      Use technology to plot the curve formed by the parametric expression \{$C_p(t), C_t(t)$\} and the vector field.  
Label the point of maximum tissue concentration with a large dot. 

 \teacher{  {\em Answers: } See the discussion for Activity 4.    }



%% +++++++++++++++++++++
\subsection*{ Activity 7: Model 2} 

Consider the following pharmacokinetic model
\begin{eqnarray}\label{eq:DEs2}
\frac{dC_p}{dt} &=& -0.49 C_p(t) + 0.11   C_t(t) \\ \nonumber
\frac{dC_t}{dt} &=&   0.45   C_p(t) - 0.11   C_t(t) 
\end{eqnarray}

\begin{equation}\label{eq:ICs2}
C_p(0) = 4.932, \,\,
C_t(0) = 0.000
\end{equation}

{\noindent }(7.1)         Determine the values for the parameters 
$k_{10}, k_{12}, k_{21}$
in (\ref{eq:DEs2}) to represent a two compartment model as in  (\ref{eq:GenDEs}).  

{\noindent }(7.2)          Use technology to solve the initial value problem  for  (\ref{eq:DEs2},  \ref{eq:ICs2}).  Record the results:   

{\noindent }(7.3)         Use the solutions and/or the IVPs to calculate these values to 4 decimal places. 

\centerline{$t_{max} =$       \underline{\hspace{.3 in} }  \hspace{.2 in}    
$C_t(t_{max}) =$       \underline{\hspace{.3 in} }  \hspace{.2 in}      
$C_p(t_{max}) =$      \underline{\hspace{.3 in} }  \hspace{.2 in}}      

{\noindent }(7.4)          Plot the parametric drug-level curve and the vector field.  Label each axis, including units, and include a title. 
Label the point of maximum tissue concentration with a large dot. 

{\noindent }(7.5)          Plot both the drug plasma level-time curve and the drug tissue level-time curve on the same set of axes here. Label each axis, including units, and include a title. 
Label the point of maximum tissue concentration with a large dot. 

{\noindent }(7.6)          Compare Model 1 and Model 2.  
Use these questions to help formulate your response, but other comments are also appropriate. 
Does one have a longer distribution phase? 
Which has the higher maximum tissue concentration? 
Does that take longer to achieve in one model? 
Which one might be used for a patient with renal failure? \\



 \teacher{  {\em Answers: } 

Figure \ref{fig:diagBothParam} shows both solution trajectories; ``RF" indicates ``renal failure." 
Here, time increases from right to left; time values cannot be determined from the graph. 
Figure \ref{fig:diagBothNonParam} shows the drug-level curves for each patient. Here, time increases from left to right. 

\begin{figure}[htb]
   \centering
    \includegraphics[scale=.8]{DigParamBothCrp.pdf} 
   \caption{Parametric Representation of Sensitivity to Parameters.}
    \label{fig:diagBothParam}
 \end{figure} 
 \begin{figure}[htb]
   \centering
    \includegraphics[scale=.9]{BothConcVSTimeCrp.pdf}
   \caption{Drug-Level Curve Representation of Sensitivity to Parameters.}
    \label{fig:diagBothNonParam}
 \end{figure} 

Although the dose is the same for each patient, the initial plasma concentration is higher for the subject with renal failure. The distribution phase for each patient appears on Figure \ref{fig:diagBothParam} to the right of each maximal tissue concentration; on Figure \ref{fig:diagBothNonParam}, this appears to the left of each maximal tissue concentration. 
The patient with renal failure experiences a longer time until maximal tissue concentration, as seen in Figure \ref{fig:diagBothNonParam}. This is also captured with the smaller distribution rate constant, $\alpha=0.5926$ in the analytic solution. 
In Figure \ref{fig:diagBothParam}, the slopes of each portion of the solution curve on either side of the maximal tissue concentration, are apparently greater for the patient with renal failure. 
Drug transfers into the tissue compartment when the ratio $C_t/C_p$, exceeds $k_{21}/k_{12}$, which is 0.15 for the subject with unimpaired kidneys and 0.24 in the subject with less functional kidneys. Similar comparisons can be made regarding the elimination phase. 
All of the comparisons help explain the significantly higher tissue concentration for the subject with renal failure at the end of the first twenty-four hours.


There are many quantities of pharmaceutical interest in drug models, including rates per minute of  various quantities, such as the amount in mcg of drug eliminated by the body, number of mL of body fluid cleared of drug, and fraction of drug eliminated \cite{Shargel}, to name a few. A third variable can account for the concentration of drug that is eliminated from the body, extending the analysis to a $3\times 3$ system of ordinary differential equations. 
We could determine the amount of drug per kg body weight of the subject, or scale the drug amount in any compartment by the body mass of the individual to obtain the actual amount of the drug in mcg. 

In the healthcare profession, specific values of effective and non-toxic drug concentrations are of enormous importance and define the {\em therapeutic window}. This can be expressed in terms of the drug concentrations. According to \cite{Digoxin-Elderly}, digoxin has a narrow therapeutic range, and subjects over sixty-five years of age, in particular, can experience potentially fatal toxicity.  
Currently recommended plasma concentrations of digoxin are between 0.5 and 1.0 mcg/L to treat congestive heart failure in the elderly -- half the level that was previously recommended\cite{Digoxin-Elderly}. 
Elderly patients generally have decreased renal function, by as much as 50\%,  as a result of the aging process. This and other factors make older patients more sensitive to drug dosing. 
However, digoxin is so widely used to manage congestive heart failure, 
which occurs in about 80\% of the elderly, that it is one of
the most frequently prescribed medications for those over sixty-five.

In our implementation from \cite{Shargel}, the plasma concentrations are mostly between 2.5 and 4.5 mcg/L -- up to 450\% of current recommendations.  The initial plasma concentrations in our model for the patient with less functional kidneys is almost five times the recommended level. 
This demonstrates the power of mathematical modeling by virtually investigating a situation that could have toxic effects in reality.

}



 


\begin{thebibliography}{98}


\bibitem{Digoxin-Elderly}
Currie, G. M., J. M. Wheat and H. Kia.  2011. 
Pharmacokinetic Considerations for Digoxin in Older People.
{\em The Open Cardiovascular Medicine Journal}. 5: 130-135. 
{\tt http://doi.org/10.2174/1874192401105010130}. 
 Accessed 12 August 2018.

\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 ODEs, the last of which is a $2 \times 2$ system. 
Two styles of the student version for aspirin are provided: one with just the questions, and a fuller style with student learning outcomes and a small self-assessment. 
The modules can be used independently from one another or in the sequence indicated. The level of inquiry-oriented learning progress through the sequence of activities. We define inquiry-oriented learning broadly to be any type of pedagogy that is active, student-centered, and helps students gain confidence in doing and creating mathematics on their own. 
The aspirin and caffeine 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. 
Students are guided through a discussion of nullclines, beginning with the definition, so no prior knowledge is needed. 


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.  
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{PK-Concepts}
Rowland, M. and T. Tozer  2011. 
{\it Clinical Pharmacokinetics and Pharmacodynamics:  Concepts and Applications, 4th ed\/}. Philadelphia: 
Lippincott Williams \& Wilkins.


\bibitem{Shelton}
Shelton, T., T. Laurent and B. Agyemang-Barimah.  To appear. Pharmacokinetic Models for Active Learning of Differential Equations.
{\it PRIMUS.} 29(7): 742-767.
\href{https://www.tandfonline.com/doi/abs/10.1080/10511970.2018.1484398}{https://www.tandfonline.com/doi/abs/10.1080/10511970.2018.1484398}. 

\end{thebibliography}
}





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