Page 124 - The Toxicology of Fishes

104 The Toxicology of Fishes

from the central compartment. These equations may then be integrated to give X and X as functions
2
1
of time:
X 1 = [ X 0 (αβ )  k 21 − ) β e −β t −( k 21 − ) e −α t   (3.63)
− ] ( 
α
X 2 = [ X k 12 (αβ e ]( −β t − e ) (3.64)
−α
t
− )
0
where α and β are rate constants that are comprised of the model rate constants:
{ 2  }
12
 
.
 k
α= 05 k + k 21 + k 10 ) + ( 12 + k 21 + k 10 ) − 4k k (3.65)

( 12
21 10
{ 2  }
12
 
.
β= 05 k + k 21 + k 10 ) − ( 12 + k 21 + k 10 ) − 4k k (3.66)
 k

( 12
21 10
The total amount of toxicant in the body (X = X + X ) is:
1
2
X =   X (αβ  k + k − ) β e −β t −( 12 k − ) e −αt t   (3.67)
k +
21 α

21
− ) ( 12
0
The plasma concentration is X /V , or:
1
1
− ) ( 
C p =   X V (αβ  k − ) β e −β t −( k − ) e −α t   (3.68)
21 α
21
1
0
which is an equation of the form:
–αt
C = Ae + Be –βt (3.69)
p
This equation can be fit to the C ,t profile by graphical or nonlinear least-squares methods to obtain
p
estimates for A, α, B, and β. These values may then be used to calculate model parameters and other
parameters of interest (Gibaldi and Perrier, 1982):
V 1 = X 0 ( A B) (3.70)
+
+
k 21 = ( A + B ) ( A B) (3.71)
α
β
k 10 =αβ k 21 (3.72)
+ −
k 12 = αβ k 21 − k 10 (3.73)
1(
V 2 = V k 12 k 21) (3.74)
V ss = V + V 2 (3.75)
1
V k + k − β) ( k − β) = β
V β = ( 12 21 21 Vk 10 (3.76)
1
1
CL = k V = β β (3.77)
V
10 1
t 1/2,α = ln 2/α (3.78)
t 1/2,β = ln 2/β (3.79)
V and V are the steady-state and β-phase volumes of distribution. The former, when multiplied by C p
β
ss
at steady state (e.g., in a continuous waterborne exposure), gives the amount of toxicant in the organism.