Mathematics 134 – Calculus for Physical and Life Sciences Discussion 3 – Drug Levels March 21, 2006

Background

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, 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.

Discussion Questions

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.

B)

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}.

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}.

(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.)

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 nth day. (Note these are two sequences!)

C) Now, let’s consider a general mathematical pattern. Let S_{n} be any sum of the form

(1)  S_{n} = a + ar + ar^{2} + ⋯ + ar^{n}

where a, r are two constants. (This is called a 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.

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.

2) Suppose |a| < 1. What happens in the limit as n → ∞ to a^{n+1}? What does that say about lim_{n→∞} S_{n} (1)?

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 ∞.

1) In the drug level problem in parts A and B, say in words what lim_{n→∞} Q_{n} and lim_{n→∞} P_{n} represent.

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.)

3) Suppose we want there to be ≥ 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.

Assignment

Group write-ups due in class on Monday, March 27.