Amount of the drug with respect to time

I have been discussing with a friend on an exercise question in a book. We couldn’t get our heads around them.

The questions are as follows:

1.) When a patient receives an injection, the amount of the injected substance in the body immediately goes up. Comment on whether the function that describes the amount of the drug with respect to time is a continuous function.

2.) When a person takes a pill, the amount of the drug contained in the pill is immediately inside the body. Comment on whether the function that describes the
amount of the drug in the blood stream with respect to time is a continuous

To me, the second one is obvious. It is a continuous function. The body has to process drug (which takes around 30 min, perhaps), and in the small intestine drug will slowly move in to the bloodstream.

But the first question is tricky. The drug is in the syringe. More specifically, it is in the barrel of the syringe. A person pushes the plunger towards the needle. The plunger’s head gets closer and closer to the needle, and as the head gets closer to the needle, the liquid moves moves into the bloodstream. But the drug (or the liquid) is not instantly in the bloodstream. It does not happen “in the blink of an eye”, so to say. So, the second case is also continuous. What would you say?

I would interpret the question the opposite way. A drug injected into the bloodstream is almost immediately available for all practical purposes. Acute effects can start to show up in seconds.

Pills are generally not readily active by nature and sometimes by design. Effects cannot begin until absorption takes place and that often depends on the capsule or outer coating being digested and the drug can only then start to be absorbed. Some orally administered drugs are designed specifically to be time-released as well.

I am not a pharmacist nor a medical doctor but I do have some background in academic psychopharmacology. I may be misinterpreting the questions because it still isn’t obvious to me what they are asking but my best guess is that the questions are asking which route of administration is most direct and controllable and that is intravenous rather than oral.

Isn’t this a homework question?

It sounds like a question that they are trying to “keep simple” and expecting the simple answer. Just like all the questions that say “ignore air resistance.”

No, it is not. We have been going through the book to find some useful exercise problems.

I took an Intro to Logic class, where nearly every exercise problem in the book could be parsed and re-parsed with “gotchas” like that. I always went to the prof’s office to argue with him over all the interesting cases, of which there were plenty. He must have been bored teaching the same old class year after year, because he really got a kick out of all that.

One of the questions on the final exam was something like:
“Name the fallacy: ‘Mary claims there is no global warming, but that’s nonsense because everyone knows that Mary is an alcoholic and drunk all the time’.”

I called it the “Ad feminem” fallacy.

What kind of ‘injection’? SubQ, IM, I/V?

What kind of pill? Is it an ‘ER’ (Extended release) derivative?

A subq injection will most closely imitate continuous delivery.
IM is second
I/V is as far from continuous as you can get (depending on the drug used)
Where did you find these questions? I hope they came after a lengthy setup as to answer my questions.

I am not sure where the question is. I wonder if I am missing something that makes it more of a question.

Of course it is a continuous function.

IV push, the fastest way of getting a medicine into the bloodstream, still pushes medicine into an IV tubing over a second or so and from there is flushed in through the tubing with some extra normal saline, another push of a second or so. If the range is hitting the bloodstream then the function goes from zero to its maximum over one to two seconds. The function is differentiable at all points; there is no hole and no “jump”.

Now of course that might not be the answer that the writer of the question was looking for as they may never have pushed an IV med. They may be imagining that a push is instantly in the bloodstream and wanting an answer that demonstrates understanding that a function that jumps vertically is not differentiable at that point and not continuous at that point. I would be sure on a test to explain that I got that concept as I answered why it was actually was a continuous function.

The functions of when the medication hits various end targets, of how it is redistributed into different body compartments, and how it is eliminated, are also a bit more complicated and interesting and creates shivers recalling pharmacology class … and all continuous functions.

Each individual molecule of drug crosses the arbitrary ‘inside the body’ line at a slightly different time. If you zoom in on your continuous function far enough you see it is actually a series of tiny jumps, one for each molecule. There are so many molecules that continuity is a very useful and close approximation, but at the barest level, the function is discontinuous.

Good try but no. The width of the entry point is such that many thousands are “crossing the line” at the same time, such that a fraction of each one is across at any moment. Moment A a total of so many thousands are crossing. Any way even if they somehow went one by one in single file you’d still have 0.000001 of a molecule across, .000002 … Now if you want to take it to an extreme barest and accept the concept of Planck seconds as the smallest possible unit of time, then every function that describes reality is discontinuous. But unlikely they are going there.

I agree with your math & physics, but disagree with your philosophical conclusion.

Functions are math ideals. The real world is discrete. Everything is discontinuous under sufficient conceptual magnification.

To me the first question to be answered is what level of detail the discussion is aimed at. It’s really the only interesting question as it determines the answer to all the rest.

IANA medical anything, but it seems pretty obvious that both the OP’s delivery routes are continuous at the gross scale of whole bodies and pharmacological effects. Though clearly the effective dose derivative with respect to time is much steeper for an IV than for a time release pill.

So the question only gets interesting when we can home in to the scale they’re thinking about. Whatever that may be.

Is this a math question or a medical question? It changes the answer a great deal, it seems. “Continuity” in math is well defined. In medicine, it may mean something different, or be less rigorously defined.


How long does it take to get from “introduced into the body” to “reasonably uniformly dispersed throughout the body”?

The question is nonsense. Whoever drafted it does not understand continuity. What is the function here? If it is the number of molecules in the bloodstream that is an integer and no non-constant integer-valued function can ever be continuous. Of course, on a sufficiently large scale they can appear continuous. But without defining the scale, the question is meaningless.

And let’s not even mention whether or not the drug reactions are governed by zero, first, second, or even third order pharmacological kinetics!

[sub]oops, I mentioned it[/sub]

That may be so if concentration is high enough, or the opening is wide enough, but those are both assumptions.

Furthermore, metabolism may be modelled by half-life but it, too, is a discontinuous function. The drug molecule has either been deactivated by its enzyme (/ receptor / whatever) or it hasn’t. There’s no “half steps” at all here. Again, it’s an assumption to assume that not a single molecule of the drug is metabolized before the injection is complete; at which point, the function becomes discontinuous anyway.

The amount of drug as a function of time can be modeled very well by a discontinuous function. It could also be modeled by a continuous function with a very large slope in that vicinity. And ultimately, all we really care about are the models of the process, not the actual process itself.

It would be a very very very very very uncommon IV push circumstance in which the concentration was dilute enough that molecules went in one by one and no IV entry port is so narrow (can’t be as then resistance impedes flow).

Hari’s point is however most cogent: scale of analysis needs to be defined.

Population growth is typically modeled by a logistic equation, which is generally considered continuous over time. But of course population actually grows by one individual being born at a time. It is deciding which level of analysis is in play that matters … and even how one defines a continuous function. I went with the “is differentiable” at all points (or at at least “at least one point”), but I am clearly wrong.

QtM, just don’t mention them three times and we should be okay.

Assume the body is “well-stirred”?