How long before the dose is no longer expected to be active? What would we expect the box to say about taking more? If we were to continue with what the box says for 5 consecutive doses,

Question

Feeling congested? Pop some acid, it will clear you right up. Well no. don't really do that. but it is interesting that HCl is the active ingredient in nearly all decongestants, due to the property it has of compressing the blood vessels in your sinuses reducing swelling, and allowing you to both drain, and breathe easier. Now for the math part. One particular fast-acting decongestant contains 15mg HCl and you are supposed to take two every dose. It is known that you only need about 4.5mg in your system to be effective, and that the half life is expected to be about 150 minutes. How long before the dose is no longer expected to be active? What would we expect the box to say about taking more? If we were to continue with what the box says for 5 consecutive doses, how much will be in our system right before we would take the 6th dose?

Explain your process.

Answer 1

4.5 > 30(0.5)^t/150

How far apart are the 6 consecutive doses?

Answer 2

Hi Alyssa H.,

What malarky in the question! The HCl present in decongestants is merely present to balance the basic effects of the drug it is presented with: it makes the true decongestant moiety pH neutral and water-soluble. Pure HCl (quite dilute) might occasionally administered orally to establish proper low pH in the stomach, when a patient isn't secreting HCl well. However, it is then to be sipped through a straw and immediately swallowed and rinsed after, to minimize contact time with tooth enamel. Both excess strong acids and bases are generally to be avoided in your body.

However, back to your instructors artificial-reality scenario. If a half-life is quoted you, assume a first order (exponential) (A(t) = A(0)*exp(-kt)) decay in the active (HCl) substance level. You know how to go from a half-life value to an exponential rate constant, right? From there it is just a matter of following HCl level vs. time. You can do this by solving for remaining HCl just before each new dose is taken, boosting it by the administered dose, and then following the decay from the new level. There's no particularly more efficient way for you to calculate the end values, although you should and might note that the levels just before each "hit" do form a series which exponentially approaches what is called the steady-state minimum level. (In other words: there is a formula to calculate, but it would take more time to work it out than it would to just calculate by hand for this problem).

-- Cheers, -- Mr. d.