Equation 2.6.1. Rate of Change of X with Time
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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 µ&g/ml of a drug in plasma and at time 24 hours the concentration is 5 µ&g/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 2.6.1. 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 2.6.2. Rate of Elimination
Integration will allow us to convert this (Equation 2.6.2) and other differential equations to arrive at what we call integrated equations.
Giving:
Equation 2.6.3. 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 2.6.1 and Equation 2.6.3).
We will work with both differential and integrated equations during the year.
One example of this is the area under the plasma concentration time curve. Later we shall learn that this summation or integration process can be used to evaluate dosage forms, that is it can be used as a measure of performance.
Using integral calculus we can go in the reverse direction. In the section above we converted from the rate of change equation to an equation for X. We can also go further and get an area under the curve, which is a further integration.
Another example is the progression from distance, to speed (the rate of change of distance), to acceleration (the rate of change of speed).
Figure 2.6.1 Relationship between Rate and Integral
Copyright 2001-3 David W. A. Bourne (david@boomer.org)