# Medication in the Bloodstream [on hold]

A.
With the cold season quickly approaching, you might want to know how cold medicine works. Pills for colds contain a decongestant to relieve stuffiness and an antihistamine to stop the sneezing and to dry up a runny nose. The pill dissolves and releases the medications into the gastrointestinal (GI) tract. The medications diffuse from there into the blood, and the bloodstream takes each medication to the site where it has a therapeutic effect. Both medications are gradually cleared from the blood by the kidneys and the liver.

The flow of a medication through the body can be modeled by treating the parts of the body as compartments, and then tracking the medication as it enters and leaves each compartment. A typical cold medication leaves one compartment (e.g., the GI tract) and moves into another (such as the bloodstream) at a rate proportional to the amount present in the first compartment. The constant of proportionality depends upon the medication, the compartment, and the age and general health of the individual.

Flow of medication
GI tract Blood

Suppose that there are A units of antihistamine in the GI tract at time 0 and that x(t)is the number of units remaining at any later time t. Time will be measured in hours. We assume that no medication enters the GI tract after the initial dose.
What is the initial value problem that models the amount of antihistamine in the GI tract?
(1)
Assume that the blood compartment includes the tissues where the medication does its work. The level y(t) of antihistamine in the blood builds up from zero but then falls as the kidneys and liver do their job of clearing foreign substances from the blood.

What is the initial value problem that models the amount of antihistamine in the blood?
(2)

Put (1) and (2) together and we have a mathematical model for the flow of a single dose
of A units of medication through the GI tract and blood compartments.
(3)
Describe a procedure that you can use to solve the system of equations.
Solve the system of equations.

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