t
C
C0
C = C0 e-kt
t
C
CS
CE
t
C
CS
CE
MATH 164, Spring 2001 Due Date: Name(s):
Honors Project 0: Drug Dosage
Background
Two facts will be important in this project.
First, the concentration of a drug in the bloodstream from a single dose
decreases with time. (Can you see why?) In general, the rate of decrease is
proportional to the amount present: thus if C = C(t) is concentration (in
mg/mL) as a function of time t (in hours), then
dC
dt
= -kC
where k > 0 is a constant, known as the elimination constant of the drug. It follows that
C = C0e-kt
where C0 is the initial concentration, or
C = C0
1
2
t
tH
where tH is the half-life of the drug.
Second, for most drugs there is an effective concentration level CE above
which C must stay if the drug is to be effective, and a safe concentration
level CS > CE below which C must stay if the drug is to be safe.
The Problems
1. You are working with a drug whose elimination constant k is known. (In general, what might k depend
on? How might you determine k in practice?) Design a schedule for long-term administration of the drug
that will:
a) get the concentration to an effective level as soon as possible,
b) keep the concentration below a known safe level CS, and
c) keep the concentration above a known effective level CE.
You may assume that upon administration the drug immediately becomes completely mixed in the bloodstream
at its maximum concentration level. (Is this assumption reasonable?) Hint: Over the long-term,
concentration levels should look something like this:
Two types of administration schedule you will want to avoid are illustrated below:
C
CS
CE
t t
C
CS
CE
C
CS
CE
t
P1
P2 P3
R1 R2
t
C
CS
CE
P1 P2
P3
R2R1
What is wrong with these schedules, and what led to the problems?
2. So far, your computations have been quite general in nature. Find data for a real drug1
, and report the
administration schedule you designed in concrete terms.
Two sequences of interest in understanding the long-term effects of drugs are the (sequence of) peak
concentrations {Pn} and the (sequence of) residual concentrations {Rn}. The peak concentration Pn is
the concentration of the drug immediately after the nth dose, and the residual concentration Rn is the
concentration of the drug immediately before the n + 1st dose.
3. Assuming that a dose of a drug that raises the bloodstream concentration level by C0 is taken at regular
time intervals t0, that the elimination constant k of the drug is known, and that upon administration the
drug immediately becomes completely mixed in the bloodstream at its maximum concentration level:
a) find closed form expressions for Pn and Rn,
b) find P = limn Pn,
c) describe in pharmaceutical terms the significance of P,
d) find R = limn Rn, and
e) describe in pharmaceutical terms the significance of R.
1Consult the Physicians Desk Reference, aka "the PDR".