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Pharmacokinetics - First Order Rate Equation

Derivation by David McAuley, Pharm D.

concentration time curve
By definition, first order process:
deltaC/deltaT proportional C
[The rate of  change in concentration versus time is directly proportional to the concentration 'C' at any point along the curve].  Note: if you graph the log of the concentration versus time, a linear graph will result.

Using the concept of variation in relation to proportionality you get:
deltaC/deltaT = kC
(The proportionality is replaced by the constant 'k')

'k' is known as the coefficient of variation - "direct proportionality constant". Also,
because we are dealing with an elimination process (concentration is decreasing), a
negative sign is added:
deltaC/deltaT = - kC
 or rearranging:
deltaC/C = - k deltaT

Next, we will consider a particular time and use the the short hand seen in calculus "with respect to".
d[C]/C = -kdt

Next, using a definite integral and integrating from C -->Co  and T --> 0
1st order

Result:
lnC - lnCo = -k[ t - 0]   or  
lnC  - lnCo =  -kt   or   lnC =  lnCo -kt

Raising everything to the inverse log (base e) you get:
C = Co e-kt

Application

Additional background information can be found here.
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