# Pain Management

While I did get a tooth pulled yesterday, I was back at work and talking to students again in class.  I really can’t complain as things feel as good as I could hope for; however, I am still sore enough that I did take some Ibuprofen throughout the day.  The doctor had given me a prescription for 600mg of Ibuprofen to be taken every 6 to 8 hours as needed.  Following these instructions seem to work well, and I would not suggest changing a prescribed dosage; however, it is in my curious nature to wonder why this dosage was chosen and how much would of the Ibuprofen would be in your bloodstream at any given point in time.

Therefore, let’s model the amount of Ibuprofen in my bloodstream.  First, we should note that if I were to take a single dose, my body will metabolize the medication in an exponential manner.  That is, the more there is in my system, the more will be metabolized.  In particular, if A is the amount in my system in milligrams, t is time in hours, then we get that dA/dt=rA, where r is a constant and dA/dt is the rate of change of amount with respect to time.  Now, if we solve this differential equation by separation of variables, we get A=A0ert where A0 is the initial amount taken.  Using the fact that the half-life of Ibuprofen is approximately 2 hours (1), we can find that the constant r=ln(.5)/2≈-.3466 per hour.

Now, if we assume that we take the Ibuprofen continuously, that is as if it was given through an IV drip so that we received 600mg every 6 hours, we would find that we could model this situation with the equation dA/dt=rA+600/6=-.3466A+100.  This would give us a solution of A=288.5147 + 311.483e(-0.3466 t).  The plot of this is given below.

We can see that we would start off with the 600mg dosage, however, the body would be able to metabolize the Ibuprofen faster than it would enter your body.  Hence the amount in your body would decay until you reached a constant amount of approximately 288.5mg of Ibuprofen in your body.  That is the effectiveness of the painkiller would decrease over time until reaching a steady state where it would until you ceased taking the medicine.

However, medicine is not given intravenously at home.  Therefore, while a continuous model can give us some insight, it is not accurate.  If we change the situation so that 600mg is taken every 6 hours and none is administered between doses, then we get a piecewise differential equation that looks like

If we solve this for differential equation we get

If we graph this, we get

Note that this model show a much larger variation in the amount of medication over time.  We have a minimum after the first dose of approximately 75mg with subsequent minimums after dosages increasing until approximately 85.7mg.  This will give a maximum amount of Ibuprofen after a dosage of approximately 685.7mg.  Looking at this model, we can see that there would be a significant difference in efficiency at the time of dosage as compared to just prior to the next dose, which is what you would expect to experience.

While I would hope for a more constant level of pain relief while recovering from my pulled tooth, I still find it worthwhile to know how the dosages are determined and what to expect.  As a note compared to other medications, the half-life and dosage amount will change what level of medication would be maintained, but we would end up with a similar qualitative outlook.

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