Page 112 - The Toxicology of Fishes

92 The Toxicology of Fishes

variable are then developed and used to estimate model parameter values for other sizes of fish or other
values of the environmental variable. For physiologically based models, prediction of toxicokinetic
behavior is accomplished by appropriate adjustments to model parameters such as organ and tissue
masses, blood flows to organs and tissues, and water flow across the gills.
The following text provides an overview of clearance concepts common to both compartmental and
physiologically based toxicokinetic modeling approaches. These two modeling approaches are then
described in detail. Both have advantages and disadvantages with respect to their utility and requirements
for development. Guidance on the application of each approach is given in separate summaries. A section
on noncompartmental modeling is also provided. Noncompartmental models are used extensively in
mammalian pharmacology and toxicology and offer important advantages when using sparse datasets.

Clearance Concepts
Toxicants taken up by fish may be eliminated by several pathways, including metabolism and excretion
into bile, urine, and respiratory water. In some cases, elimination rates are extremely slow and chemicals
are retained for days, weeks, or even months after the fish are transferred to a clean environment (Niimi
and Oliver, 1983). Alternatively, elimination pathways may clear >99% of an absorbed dose within
just a few hours. In either case, elimination may be characterized by an elimination clearance (CL).
The CL is defined as the rate of toxicant elimination (dX/dt) divided by the concentration in the
reference region (C):
CL = ( dX dt C (3.6)
)
/
and has units of flow (e.g., L/hr or mL/min). Normalization to body size (weight or surface area) may
also be appropriate, particularly when averaging values from fish of different sizes. The CL is the sum
of the individual clearances for all relevant elimination pathways; for example, the toxicant may be
simultaneously metabolized by the liver and excreted by the kidney and at the gills:
CL = CL + CL + CL b (3.7)
r
h
where the subscripts refer to hepatic (h), renal (r), and branchial (b) clearance.

Hepatic Clearance
CL is limited by the rate of blood flow to the liver. If blood perfusing the liver is completely cleared
h
of chemical, then CL is equal to hepatic blood flow. The CL may be less than hepatic blood flow,
h
h
however, if a chemical is poorly metabolized within the liver or if it is reversibly bound to plasma
proteins, impeding its transfer from the plasma into hepatocytes. A physiologically based model of
hepatic clearance may be developed by assuming that the liver is a well-stirred compartment (Figure
3.18). Toxicant concentrations in blood entering and leaving the liver are denoted as C (arterial con-
a
centration) and C (venous concentration), respectively. Blood flow is represented by Q, and –dX/dt is
v
the rate of toxicant elimination by the liver (the minus sign indicates that chemical is being lost from
the system). Metabolizing enzymes may be envisioned as a coating on the walls of the compartment.
For this well-stirred model, the rate of elimination of toxicant is:
/
Q
−dX dt = (C a − ) (3.8)
C v
If the right-hand side of this equation is multiplied by C /C , one obtains:
a
a
/
−dX dt = (C a − )C C a (3.9)
Q
a
C v
The ratio (C – C )/C is the hepatic extraction ratio (E). Substituting, the rate of elimination can be
a
v
a
written as: