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\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\DOG DRUGS}


\markboth{Dog Drugs}{Dog Drugs}

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

\makeStitlePDFLaTex


\begin{abstract}
We offer a problem to determine the necessary drug administration in order to keep a dog
sedated with specific information on half-life for an exponentially decaying presence of the drug in the body.
\end{abstract}

\section*{SCENARIO DESCRIPTION}

In \cite[p.~166, Exercise 6]{Riggs1963} the following problem is posed.


\begin{quote}

In the dog, an intravenous dose of 30 mg of pentobarbital sodium per		
kilogram of body weight will usually produce surgical anesthesia. Also in		
the dog, pentobarbital has a biological half-life of about 4.5 hours, due almost entirely to metabolism.	

You anesthetize a 14-kg dog with the above dose of pentobarbital. Two		
hours later the anesthesia is obviously beginning to lighten and you want		
to restore the original depth of anesthesia. How many milligrams of pentobarbital sodium should you inject?		
\end{quote}


\teacher{
\section*{NOTES FOR TEACHERS}

\subsection*{ Preliminaries}

This is a little problem in which half-life or decay rate plays a role, but it is one in which a differential equation model for exponential decay  also plays a role.

The problem as stated can be offered anywhere along the development of students' abilities to model with differential equations or solve differential equations. Students can be asked to look up notions, e.g., half-life, on the web and build their modeling concept and relate the rate of reaction to the half-life. Or students could be introduced to the exponential decay model and led through the derivation below which relates half-life to decay constant and then left to their own devices to actually come up with the dose of the drug to administer in order to accomplish a return to full sedation levels as required by the problem.

If the notions of half-life and decay rate in a first order differential equation (\ref{eq:Exp}), 

\begin{equation}\label{eq:Exp}
P'(t) = -kP(t)\, , \quad P(0) = P_0\, ,	 
\end{equation}
which models the amount in mg of pentobarbital sodium, $P(t)$, in the dog's body at time $t$ in hours have been introduced, then one can proceed with relating half-life to decay rate. Here $k > 0$ is called the {\it decay rate\/} and has units 1/hr.

If not then that relationship needs to be established, and it is often the case that after solving (\ref{eq:Exp}) students can do so by simply solving (\ref{eq:Half}) for $T$, namely how long ($T$) does it take the amount of pentobarbital sodium to drop to half of its original amount at any time with any starting amount, $P(t)$?

\begin{equation}\label{eq:Half}
P(t+T) = 0.5 P(t) \, .
\end{equation}

The solution to (\ref{eq:Exp}) is obtained by separation of variables or a computer algebra system and is found to be

\begin{equation}\label{eq:solved}
P(t) = P_0 e^{-k\, t} \, .
\end{equation}
Thus, substituting $P(t)$ from (\ref{eq:solved}) into (\ref{eq:Half}) we have

\begin{equation}\label{eq:Param}
P_0 e^{-k\, t}e^{-k\, T}  = P_0 e^{-k\, (t+ T)} = P(t+T) =  0.5 P(t) = 0.5 P_0 e^{-k\, t}\, .	
\end{equation}

From here we can divide through by $P_0 e^{-k\, t}$ to obtain $e^{-k\,T} = 0.5$. Notice that the original amount of the drug, $P_0$, disappears from the expressions and so this affirms that the notion of half-life does NOT depend on amount of drug present at any time. Now, taking logarithms of both sides yields $-k T = \ln(0.5)$ which leads to a relationship between the half life or length of time until half of the drug disappears from the body (presumably by metabolism), namely $T_{\rm half}  = T$, and the rate constant $k$.

\begin{equation}\label{eq:half}
T_{\rm half} = \frac{\ln(.05)}{-k} = \frac{\ln(2)}{k} \quad {\rm or} \quad k = \frac{\ln(2)}{T_{\rm half}} \, . 
\end{equation}


A reasonable assumption in our situation is that a 14 kg dog was administered 30 mg of pentobarbital
sodium per kilogram of dog mass to keep it fully sedated, i.e. the total amount of pentobarbital
sodium in the dog's body at the start of the time frame can be presumed to be (14 kg)$\cdot$(30 mg/kg) = 420 mg. If the half-life, $T_{\rm half}$, of pentobarbital sodium is 4.5 hours then we know from above that the decay constant for pentobarbital sodium is $k = \ln(2)/(4.5) = 0.154033$ and thus a full description of the amount of pentobarbital sodium in the dog's body t hours after the start of the experiment is given by
 \begin{equation}\label{eq:half}
P(t) = 420 e^{-0.154033\, t}\, .	
\end{equation}
Now we need to know just how much pentobarbital sodium is in the dog's body after 2 hours so we can then tell how much of the drug we have to administer to get the total back up to 420 mg, which will be enough to sedate the dog again.
We see that at time $t = 2$ hours into the experiment there is $P(2) = 420e^{-0.1540332}  = 308.644$ mg pentobarbital sodium in the dog's body. Thus we have to administer a dose of $420-308.644 = 111.356$ mg of pentobarbital sodium to get the amount of the drug in the body back to where the dog is sedated.

\subsection*{Timing}

This problem can be issued in class witih minimal discussion of 5-10 minutes and/or assigned as a homework problem. Of course, should this activity be used to introduce the  separation of variable technique for solution of the differential equation  then more time needs to be allotted.

\subsection*{Final Comment}

From this discussion it is clear that there may be a number of approaches and which one the students take will involve their level of mathematical abilities with regard to differential equations. The teacher can offer strong guidance or permit the students to discover the mathematics with some gently fed information. Either way, we have a nice application of differential equations in the area of anesthesiology.

}
\begin{thebibliography}{99}


\bibitem{Riggs1963} Riggs, D.~S. 1963.  {\it  The Mathematical Approach to Physiological Problems\/}. Cambridge MA:  The M.I.T. Press.  



\end{thebibliography}




\end{document}
 
 



}
