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\centerline{Mathematics 134 -- Calculus for Physical and Life Sciences}
\centerline{Discussion 3 -- Drug Levels}
\centerline{March 21, 2006}
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\noindent
{\it Background}
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The goal of today's discussion is to show how sequences and 
sums of sequences (= series) can arise in a very practical
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. 

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{\it Discussion Questions}
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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.
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\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 $P_3$. 
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\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.)
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\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.
(Note these are two sequences!)
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\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)?
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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.
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{\it Assignment}
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Group write-ups due in class on Monday, March 27.
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