## Calculus

### Differential Calculus

Pharmacokinetics is the study of drug disposition (movement) in the body and thus differential calculus is an important stepping stone in the development of many of the equations used. Although I won't stress the derivation of equations in this course, some of the derivations will be covered and an understanding of the derivations will assist in the use of the equations.

Differential calculus is involved with the study of rates of processes. The calculus part comes in when we look at these processes in detail, that is, during small time intervals.

We may say that at time zero a patient has a concentration of 25 ug/ml of a drug in plasma and at time 24 hours the concentration is 5 ug/ml. That may be interesting in its self, but it doesn't give us any idea of the concentration between 0 and 24 hours, or after 24 hours. Using differential calculus we are able to develop equations to look at the process during the small time intervals that make up the total time interval of 0 to 24 hours. Then we can calculate concentrations at any time after the dose is given.

In many cases the rate of elimination of a drug can be described as being dependent on or proportional to the amount of drug remaining to be eliminated. That is, the process obeys first order kinetics. Thus:-

Equation II-2. Rate of Change of X with Time

where k is a proportionality constant we call a rate constant and X is the amount remaining to be eliminated.

Equation II-3. Rate of Elimination

Integration will allow us to convert this (Equation 2) and other differential equations to arrive at what we call integrated equations.

Giving:

Equation II-4. Integrated Equation. X versus Time

This is the resulting integrated equation. We will talk more of these equations later in the semester.

What we have done is convert the rate equation for X into an equation for X versus time. (Compare Equation II-2 and Equation II-4).

We will work with both differential and integrated equations during the semester.