\magnification=\magstep1
\centerline{Mathematics 136 -- Advanced Placement Calculus}
\centerline{Discussion -- Drug Levels}
\centerline{November 30, 2009}
\bigskip
\noindent
{\it Background}
\bigskip
The goal of today's discussion is to introduce some key ideas
relating to {\it finite and infinite series}, the topic of Chapter
8 in the text and the last (short) unit of the course, 
which we are now ready to begin.  We will do this
in the context of an important question in clinical medicine.  Namely,
if a patient is taking regular doses of some drug over an extended
period of time, {\it what is the
long-term level of the drug that will build up in the body?}  For
some very toxic drugs, this might limit the dosage that could be
given safely!

For instance, we will consider a drug called atenolol, which is
usually given in 50mg doses once a day to lower blood pressure. 

\bigskip
\noindent
{\it Discussion Questions}
\bigskip
\noindent
A)  The half-life of atenolol in the bloodstream is about 6.3 hours.
If a dose of $Q_0 = 50$ mg is given at the start of a 24 hour period, 
how much will be left after 24 hours have elapsed?  Call this amount
$P_1$, the amount left at the end of the first day.
\bigskip
\noindent
B)  
\item{1)}  Next, a second dose is administered.  For simplicity, let's
assume it is absorbed into the bloodstream immediately.
Call $Q_1$ the total amount present after the second dose  (the 
new, plus the remnants of the first).  Find $Q_1$.
Similarly, let $P_2$ be the amount left at the end of the second
day.  Find $P_2$.
\item{2)}  Next, at the start of the third day,
a third dose is administered.  
Call $Q_2$ the total amount present after the third dose  (the 
new, plus the remnants of the first and second).  Find $Q_2$.
Similarly, let $P_3$ be the amount left at the end of the third day
day.  Find $P_3$. 
\medskip
\item{} (If you aren't starting to see a general
pattern at this point, it may help to go a few days
farther into the process.)
\medskip
\item{3)}  
Say the patient is on a long-term treatment plan
for high blood pressure.  Continuing in the same way, find general 
formulas (using sums) for $Q_n = $ amount present after the $(n+1)$st
dose, and $P_n = $ amount present at the end of the $n$th day.
\bigskip
\noindent
C)  Now, let's consider a general mathematical pattern.
Let $S_n$ be any sum of the form
$$S_n = a + ar + ar^2 + \cdots + ar^n\leqno(1)$$
where $a,r$ are two constants.
(This is called a {\it finite geometric series} with first term $a$
and ratio $r$.)
Let's see how to write $S_n$ in ``closed form'' -- that is,
in an equivalent way but not including a summation of
lots of terms.  
\item{1)}  What happens if you take $(1 - r)S_n = S_n - rS_n$ using the 
formula in (1)? 
Solve the resulting equation to give a formula for $S_n$ in
terms of $a,r,n$, but with out a summation.
\item{2)}  Suppose $|a| < 1$.  What happens in the limit
as $n \to \infty$ to $a^{n+1}$?  What does that say about
$\lim_{n\to\infty} S_n$ (1)?
\bigskip
\noindent
D)  If we have a patient on a long-term drug regimen, say 
extending potentially over a period of many years, then
we are talking effectively about letting $n = $ number of 
days go to $\infty$.  
\item{1)}  In the drug level problem in parts A and B, say in words
what $\lim_{n\to\infty} Q_n$ and $\lim_{n\to\infty} P_n$ represent.
\item{2)}  Suppose it is not safe for the patient ever to have an atenolol
level $> 70$ mg.  Is the treatment regimen described here safe?
(Think about what your formula from part C says.)
\item{3)}  Suppose we want there to be $\ge 10$ mg of atenolol
present in the body at all times (even just before a new dose
is taken).  Is the patient getting enough atenolol so this
will be true ``in the long run''?  Explain.
\bigskip
\noindent
{\it Assignment}
\bigskip
Group write-ups due in class on Friday, December 4.
\bye
