Question

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The Problem: Drugs in Series^[1]

One of the physician’s responsibilities is to give medicine dosage for a patient in an effective manner. Two mathematical techniques help physicians analyze the concentration of drug is the bloodstream of a patient: the exponential decay model (EDM) and the Geometric Series and its Formula (GSF). In this problem, we will analyze the situation where a drug is administered intravenously and that the concentration of the drug in the bloodstream jumps almost immediately to its highest level after a dose is administered.

Let Q(t) be the dose concentration of a drug at time t, and Q₀ represent the concentration just after the dose is administered intravenously.

According to the EDM, as the body eliminates the drug, the rate of decrease of concentration in the blood is proportional to the concentration itself. If Q₀ represents the concentration just after the dose is administered intravenously, explain why the amount of drug after t hours is given by

Q(t) = Q0e?kt

where k is a constant (depending on a property of the particular drug used).

Now suppose that at t = c, a second dose of the drug is given to the patient. The concentration of the drug in the bloodstream jumps almost immediately to its highest level Q(c) and then the concentration is diffused rapidly throughout the bloodstream over time. A picture of this is shown below.

Let Q(c^?) be the dose concentration just before the second dose is administered intravenously. Explain why

Q(c^?) = lim Q(t)

t?c^?

and find a value for Q(c^?) in terms of Q₀ and k.

Let Q(c⁺) be the dose concentration right after the second dose is administered intravenously. Write an expression for Q(c⁺) in terms of Q₀ and k.

Note that the concentration decays from Q(c⁺) according to the EDM, but with a slight twist.

The twist is that the initial concentration is at t = c instead of t = 0. Explain why

Q(t) = Q0(1 + e?kc)e?k(t?c)

when c ? t < 2c.

Repeat work similar to what you did in (b) to find a model for Q(t) when 2c ? t < 3c.

Write an sum that gives the concentration just before the nth dose of the drug, Q((n ? 1)c^?) and another sum that gives the concentration just after the nth dose of the drug, Q((n ? 1)c⁺)

Show that Q((n ? 1)c^?) is a geometric sum with r = e^?kc, and that 0 < r < 1. Use this fact to write a formula for Q((n ? 1)c^?).

Show that Q((n ? 1)c⁺) is a geometric sum with r = e^?kc, and that 0 < r < 1. Use this fact to write a formula for Q((n ? 1)c⁺).

Suppose a treatment for the patient is continued indefinitely.

Compute lim_n??Q((n ? 1)c^?). Consider the drug concentration a long time after teh initial dosel explain why this limit equals the long-term minimum concentration in the patient.

Compute lim_n??Q((n?1)c⁺), and explain why this limit equals the maximum concentration in the patient long-term.

How could these computations help a physician make clinical decisions

[1] Based on Annamalai, Chinnaraji. “Applications of exponential decay and geometric series in effective medicine dosage.” Advances in Bioscience and Biotechnology 1.01 (2010): 51.

Answer 1