ABSTRACT
Application of Linear-Order 1 Differential Equations: Drug Kinetics
A substance (e.g., a drug) placed in one compartment is eliminated from that compartment at a rate proportional to the quantity it contains, and this elimination moves it to a second compartment (such as blood) that originally does not contain the substance. The second compartment also eliminates the substance to an external sink and does so at a rate proportional to the quantity it contains. If D denotes the initial amount in the first compartment, and the elimination rate constants from each compartment are denoted k1 and k2, respectively, then the quantities in compartment 1
•
(denoted X) and compartment 2 (denoted Y) at any
time t are described by
dX
dt k X= − 1 X(0) = D (compartment 1)
dY
dt k X k Y= −1 2 Y(0) = 0 (compartment 2)
from which
so that
dY
dt k Y k De k t+ = −2 1 1 , a linear order 1 equation with
solution
Y k D
k k e ek t k t=
−
This illustrates a model that is commonly used to
describe the movement of a drug from some entry
site into and out of the blood.
