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.