# How does calculus relate to medicine?

May 20, 2015

One important area of application is to deciding drug dosages.

Suppose the dose of a drug is $Q$ milligrams (per pill) and that the patient is supposed to take the drug every $h$ hours. Futhermore, suppose the half-life of the drug (the amount of time for the amount of the drug to decay to 50% of the starting amount) in a person's bloodstream is $T$ hours (for simplicity, assume the drug enters the person's bloodstream instantaneously).

Let ${Q}_{n}$ be the amount of the drug in the body right after the ${n}^{\setminus m b \otimes \left\{t h\right\}}$ dose and let ${P}_{n}$ be the amount of the drug in the body right before the ${n}^{t h}$ dose so that ${Q}_{n} = {P}_{n} + Q$.

Let's seek a pattern: ${P}_{1} = 0$, ${Q}_{1} = 0 + Q = Q$, ${P}_{2} = Q \setminus \cdot {2}^{- \frac{h}{T}}$, ${Q}_{2} = Q \setminus \cdot {2}^{- \frac{h}{T}} + Q$, ${P}_{3} = \left(Q \setminus \cdot {2}^{- \frac{h}{T}} + Q\right) \setminus \cdot {2}^{- \frac{h}{T}} = Q \left({2}^{- 2 \frac{h}{T}} + {2}^{- \frac{h}{T}}\right)$, ${Q}_{3} = Q \left({2}^{- 2 \frac{h}{T}} + {2}^{- \frac{h}{T}}\right) + Q$, ${P}_{4} = \left(Q \left({2}^{- 2 \frac{h}{T}} + {2}^{- \frac{h}{T}}\right) + Q\right) \setminus \cdot {2}^{- \frac{h}{T}} = Q \left({2}^{- 3 \frac{h}{T}} + {2}^{- 2 \frac{h}{T}} + {2}^{- \frac{h}{T}}\right)$, ${Q}_{4} = Q \left({2}^{- 3 \frac{h}{T}} + {2}^{- 2 \frac{h}{T}} + {2}^{- \frac{h}{T}}\right) + Q$, etc...

The patterns indicate that ${P}_{n} = Q \setminus {\sum}_{k = 1}^{n - 1} {2}^{- \setminus \frac{k h}{T}}$ and ${Q}_{n} = Q \setminus {\sum}_{k = 0}^{n - 1} {2}^{- \setminus \frac{k h}{T}}$.

Here's the calculus-related part. As $n \to \setminus \infty$, it can be shown that ${P}_{n} \to \setminus \frac{Q}{{2}^{\frac{h}{T}} - 1}$ and ${Q}_{n} \to \setminus \frac{Q {2}^{\frac{h}{T}}}{{2}^{\frac{h}{T}} - 1}$.

What's the application to medicine? You want to choose $h$ and $Q$ so that $\setminus \frac{Q}{{2}^{\frac{h}{T}} - 1}$ is large enough to be effective in the patient's body and so that $\setminus \frac{Q {2}^{\frac{h}{T}}}{{2}^{\frac{h}{T}} - 1}$ is small enough that it is not dangerous to the patient.