Page 145 - The Toxicology of Fishes

Toxicokinetics in Fishes 125

dX/dt = –CLC p (3.114)

where X represents the amount of chemical in the body, CL is the plasma clearance of the chemical,
and C is the plasma concentration. From this relationship, the amount of chemical eliminated over a
p
small time interval, dt, is:
dX = –CLC dt (3.115)
p

At the time (t) that dX is eliminated, dX has spent t time in the body. The total time in residence for
all the molecules that make up X is the sum of the products of all the dX multiplied by the corresponding
residence times; that is, the integral of tdX:

∞
∫ −CLC tdt (3.116)
p
0
The total amount introduced into the body is the total amount eliminated, which may be expressed as:

∞
∫ −CLC dt (3.117)
p
0
The mean residence time is the total time in residence divided by the total amount introduced into the
body:

∞ ∞
∫ − CLC tdt ∫ Ctdt
p
p
MRT = 0 ∞ = 0 ∞ = AUUMC (3.118)
AUC
∫ − CLC dt ∫ Cdt
p
p
0 0
where AUC is the area under the plasma concentration–time profile, and AUMC is the area under the
moment curve, which is the integral of the product of plasma concentration and time.
After intravascular bolus administration of chemical, the MRT can be used to calculate the V using
ss
the relationship:
V = MRT CL (3.119)
ss
where:
CL = dose/AUC (3.120)

Even when the plasma concentration–time profile is complex and irregular, it is generally possible to
calculate AUC and AUMC, provided the profile is completely characterized.
To obtain the total AUC, the area beyond the last measured plasma concentration must be estimated.
This may be accomplished using the relationship:

−∞ = z λ (3.121)
AUC tlast, C p last,
where λ is –2.3 times the slope of the log-linear portion. Although the plasma concentration–time profile
z
can be highly irregular, it has to have a terminal log-linear phase to provide a λ value for the extrapolation.
z
In the absence of a log-linear phase, it is necessary to have a sufficient number of samples so the measured
concentrations fully outline the AUC. In that case, a CL value may be obtained but a value for V cannot
ss
be determined.