Periodic Dosing with Exponential Decay
by Mike Martin

This page allows you to analyze a model for drug delivery.   Assume a uniform dosage and that dose administration results in the recipient's drug concentration increasing by co units.   N doses are administered at uniform intervals of T time units.   Furthermore, the body eliminates the drug according to an exponential decay model where 1/e is removed in τ time units; equivalently, we sometimes enter this with k = 1/τ.   In a given dosage interval, let c(t) be the concentration and let t = 0 when dosed, then the concentration obeys the differential equation  

c ' (t) = - c (t) / τ

or, equivalently,
c ' (t) = - k c (t)

with the initial condition   c (0) = co.   This initial-value problem has the unique solution on that interval of  

c (t) = co e - t / τ

or, equivalently,
c (t) = co e - k t

The toxic concentration, ctoxic, may be entered and is displayed as a dashed red line.   The therapeutic concentration, ctherapeutic, may be entered and is displayed as a dashed blue line.   The limiting values over time of the drug concentration (both max and min) are displayed as dashed purple lines.   Enter the value of the parameters below, then press Evaluate.

co =
T =
N =
τ =
ctoxic =
ctherapeutic =


Jones, D.S. and B.D. Sleeman Differential Equations & Mathematical Biology. Chapman & Hall/CRC, Boca Raton, FL, 2002.

Mazumdar, J. An Introduction to Mathematical Physiology & Biology. Cambridge University Press, Cambridge, UK, 1999.