I should really know the answer to this question, so I feel a little silly asking, but here goes.
For a lot of drugs, medications, etc. I often see a ‘half life’ listed, i.e. the amount of time it takes for the substance to be drawn down to half of its original concentration. This is usually just listed as a number. For a first order reaction (exponential curve) that makes sense. Reaction rate is proportional to the amount of the substance:
dC/dt = - rC,
so it breaks down at a constant rate, and the half life is independent of the initial concentration, i.e. constant:
t1/2 = ln 2 / r.
Assuming that the breakdown of the compound is catalyzed by an enzyme, though, (and really oversimplifying) it seems to me that the rate limiting step should be more like a Michaelis-Menten reaction:
dC/dt = -vC / (k + C),
or equivalently dt/dC = k/vC + 1/v.
In this case the half life should be a linear function of the initial concentration:
t1/2 = k ln2/v + 2 C0 / v.
In this model the initial slope (V/k) is the same as r in the exponential model, so the two models are:
t1/2 = ln 2 / r (exponential)
t1/2 = ln 2 / r + 2 C0 / v (Michaelis-Menten).
In other words, it seems to me that medications and other compounds which are broken down enzymatically in the body, shouldn’t have a constant half-life: the half life should be dependent on the initial concentration. So why do you sometimes see drugs and so forth listed with a constant ‘half life’, and how is that a meaningful concept? Sorry if this is a silly question, I’m sure I must be missing something here.
For substrate concentrations well below the Michaelis constant, you can show – expand the denominator in powers of C/k – that MM kinetics are well-approximated by first-order kinetics.
And why would you want to give meds that approach overloading the body’s ability to dispose of them? You would expect that to be the regime of toxic side-effects.
Additionally, it seems not especially likely that there is only one pathway for degradation, so the actual kinetics may be extremely complicated, and the half-life just quoted to give you a rough idea of the overall rate of decay, which will be only crudely exponential.
What you are asking about is pharmacokinetics, the study of how drugs are cleared in the body… and one of the most HATED classes in pharmacy school.
You are welcome to go to the link to read all about it, but to quickly answer you question, with most drugs we can assume they operate as a zero order reaction and just give a half life independent on concentration. The reason for this is that most drugs saturate the enzymes that metabolize them, so the rate limited step is the enzyme, not the concentration of the drug. However, there are a few drugs that are first order, or second order, and there are complicated equations to figure out their half life… but, most of these drugs are IV only, and normally given in hospitals.
In all hospitals, one of the jobs of the pharmacists working there is to dose these drugs so they maintain the correct level. In some of the larger hospitals, they have a PK pharmacist who’s sole job is drug dosing. Most of these drugs are strong antibiotics like Vancomycin and Aminoglycosides, but there are other common drugs that have to be monitored like Digoxin, Theophylline, and many anti seizure and immune suppressive drugs (among others).
So yes, there are drugs where the half life varies, many of them in fact, but most of the drugs that the public comes in contact with, including all OTCs, we can assume they are zero order, and give a simple half life.
Hirka thanks for your response. This was exactly my question. Let’s say you are operating in the saturated region of the MM curve (i.e., C is high relative to K, so dC/dt = approximately constant, i.e. reaction rate is approximately zero order). If the substance is breaking down at a constant rate, though, then the half life shouldn’t be independent of concentration, right? Let’s say for example that your body can process 0.1 mg of some compound an hour, then the amount of time it takes for half the substance to break down is going to be greater if you have more of the substance there.
Wouldn’t a constant half life only correspond to a first-order reaction, rather than a zero-order or second-order reaction? Or a MM curve, for that matter (which would be approximately first-order at low C, like Carl notes, and zero-order at high C, as you note).
We’re not just a bag of enzymes. Processing, elimination, and target sites are usually separate. IIRC in a lot of cases uptake from the blood to the liver or kidneys is the rate-limiting step, which is where you get the zero-order kinetics.
Oops, you’re correct, we assume a first order reaction for most drugs… What can I say, it was a late night after working a full day… and I did say it was the most hated class.
This of course is in the simple models, a non-compartmental model. There are more complicated models, but for most drugs the simple way is good enough for most uses.
I hate PK! I’m having nightmares just thinking of it.
Oh, good point. If uptake of the compound is the rate-limiting step, and if it’s happening by passive transport, then (it seems to me) it should be first-order, which would explain why you can get a constant half life.
