# A Grapefruit A Day… brings the doctor to stay?

Note: although this is a post about grapefruits and their effect on drugs, it is *not* a literal model fit to real data, it is a *qualitative* model to explain the type of effect that might be seen. Do not use this post to make specific medical decisions.

I had a conversation with my mother in law recently that encouraged me to look at qualitative models of drug metabolism. The conversation was about grapefruit and its moderately well known effect on certain medicines. Actually I think this is less well known than it should be. It’s not just Statins for cholesterol control, but a very large list of drugs whose metabolism is slowed by the presence of grapefruit enzymes.

One of the issues is that an orally administered drug has a kinetic cycle in which it is first absorbed, and then metabolized away. The timing of the drug doses, whether they are given once, twice, three, or four times per day for example is related to how quickly the drug disappears from your bloodstream. You might imagine that most of the drug is gone after say 4 hours so that you are supposed to take the drug every 4 hours and keep some level of the drug in your system throughout the day. However, grapefruit enzymes can last for several days, and so the standard dosing of the drug doesn’t disappear as quickly in the presence of grapefruit, and the concentration of the drug can be much higher. A braindead simple model for drug metabolism is:

$$\frac{dy}{dt} = – \frac{1}{k}y$$

Where $$y$$ is the concentration of the drug in your bloodstream and $$k$$ is the number of hours it takes to reduce the drug concentration by a factor $$1/e$$. The solution to this differential equation is an exponential decay function $$y = Y_0 e^{-t/k}$$.

Now we can express this relationship in a non-dimensional form. Replace $$y=Y_0 y’, t = k t’$$. The equation becomes:

$$\frac{Y_0}{k} \frac{dy’}{dt’} = – \frac{Y_0}{k}y’ \rightarrow \frac{dy’}{dt’} = – y’$$

In other words by measuring time in units of $$k$$ and measuring the drug concentration in units of the initial concentration after the drug is absorbed $$Y_0$$ the equation becomes simpler.

Now let’s modify the equation in the presence of a grapefruit enzyme at non-dimensional concentration $$C$$. We’ll measure $$C$$ as a fraction of the concentration needed to double the metabolism time for the drug, and we’ll make a simple qualitative assumption about the effect that $$C$$ has on the drug metabolism:

$$\frac{dy’}{dt’} = – \frac{1}{1+C}y$$. Now when $$C=1$$ the drug is being metabolized at rate $$1/2$$. If $$C=10$$ the drug is being metabolized at rate $$1/11$$ and when $$C=0$$ the drug is being metabolized at rate $$1$$ as expected from the previous model.

If we further assume that the metabolism of $$C$$ follows the same kind of law as the metabolism of the drug in the absence of $$C$$ with a time constant $$s$$ which is bigger than 1 so that the grapefruit takes much longer to disappear than the un-modified drug, we have the following equations:

$$C(t) = C_0e^{-t’/s}$$ and solving the differential equation using Maxima for $$y’$$ we have:

$$y'(t) = e^{s (\log(C_0+1) – \log(C_0 + e^{t/s}))}$$

Which is not quite equal to substituting $$C$$ directly into the solution for $$y’$$ in the absence of $$C$$:

$$y'(t) \ne e^{-\frac{t}{1+C(t)}}$$ but you can think of this to get the idea that the decay rate of the drug $$y’$$ is a time varying function which is linked to the concentration and decay rate of grapefruit enzyme.

Now let’s assume that grapefruit decays on a time scale that is s=4 times longer than the unmodified drug time (this is reasonable for many oral drugs that decay in a fraction of a day, whereas grapefruit can stay around for 2 or 3 days). Look at the time courses *of the drug* for initial grapefruit concentrations 0,1,3,9 (green, yellow, orange, red):

If you look at say time 2, the green curve is where the pharmacologists who designed the drug thought the concentration would be, and with even 1 unit of grapefruit, it’s about 2 times as large. With 9 units of grapefruit it’s perhaps 8 times as large as expected! What is a “unit” of grapefruit? It’s not clear to me how much grapefruit you need to eat, and it seems as if it’s dependent on the individual’s physiology and the particular drug that’s been ingested, but it’s clear that even 1 grapefruit can make a noticeable difference for some conditions, as there are case studies of hospitalizations related to this topic where the doses of grapefruit are in the vicinity of 2 or 3 fruits, or a liter or so of juice.