ABSTRACT
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 X D
dY dt k X k Y Y D
= − =
= − =
( ) ( )
( ) (
compartment 1
compartment 2)
from which
X De k t= − 1
so that
dY dt k Y k De
with solution
Y k Dk k e e
−
⎛
⎝
⎜
⎞
⎠
⎟
−
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.
