Mathematics

Problem

A pharmaceutical company models the amount

Q(t)

(in milligrams) of a drug in a patient’s bloodstream with the differential equation

\frac{dQ}{dt} = 0.4\,Q - 0.0008\,Q^2

, where

t

is measured in minutes. (a) Express the differential equation in logistic form and identify the carrying capacity. (b) Determine the time rate of change of

Q

when

Q = 300\,\text{mg}

. (c) The therapeutic window for the drug lies between

200\,\text{mg}

and

600\,\text{mg}

. For which values of

Q

in this range is the concentration increasing? Justify your answer mathematically. (d) Suppose the drug is effective only while

Q

is rising. Starting from an initial dose of

100\,\text{mg}

, determine whether the effectiveness is increasing or decreasing after

15\,\text{min}

if the concentration at that time has reached

240\,\text{mg}

. Explain your reasoning using the model parameters.

Intuitive explanation

This is a logistic growth model where the drug amount grows due to absorption but is limited by elimination and saturation effects. The key insight is recognizing that the $0.4Q$ term represents net growth while the $-0.0008Q^2$ term represents a self-limiting factor—as the drug concentration increases, elimination accelerates. You can find equilibrium points (where $\frac{dQ}{dt} = 0$ ) to understand the drug's long-term behavior, and analyze stability to predict whether the system approaches a safe steady state or dangerous accumulation.

Intuitive explanation

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