Page 128 - The Toxicology of Fishes

108 The Toxicology of Fishes

Q i Q i
VASCULAR
SPACE
C art C vi
EXTRAVASCULAR
SPACE

FIGURE 3.23 Schematic representation of a compartment exhibiting flow-limited kinetics. Symbols: Q i , tissue blood flow
rate; C art , chemical concentration in arterial blood; C vi , chemical concentration in venous blood draining the tissue.

A relatively simple tissue description can be developed if it is assumed that the rate of chemical
exchange between blood and tissue is fast relative to the blood perfusion rate (Figure 3.23). Under these
circumstances, chemical partitioning proceeds to equilibrium with the result that C is determined by
vi
the chemical concentration in the tissue and the relative affinity of the compound for the tissue and for
blood, denoted as P , the equilibrium tissue–blood partition coefficient:
i
C vi = C P i (3.84)
i
Substituting Equation 3.84 into Equation 3.83, one obtains:

dX dt = ( C P I) (3.85)
Q C art −
i
i
i
or equivalently:
VdC dt = ( C P i) (3.86)
Q C art −
i
i
i
i
where C is the chemical concentration in the tissue, and V is the tissue volume. This type of chemical
i
i
distribution is said to be flow limited because the rate of chemical accumulation is controlled by the
blood flow rate.
Chemical distribution is said to be diffusion limited when the rate of chemical diffusion across a
biological membrane limits chemical flux between blood and tissues. Under these circumstances, the
tissue must be divided into two subcompartments (Figure 3.24). Chemical flux between subcompartments
is then modeled using the Fick relationship (see Equation 3.1). In practice, chemical permeability (i.e.,
DP /h) and the surface area for diffusion are difficult to determine. The permeability-area (PA) product
mw
is therefore frequently treated as a first-order transport parameter (k ), the value of which is determined
i
from exposure data. If a diffusion limitation exists at the capillary endothelium, the relevant subcom-
partments are the vascular and extravascular tissue spaces. Alternatively, if the diffusion limitation exists
at the cellular membrane, the relevant subcompartments are the extracellular and intracellular spaces.
In either case, the mathematical treatment is identical, but different volumes must be assigned to represent
each subcompartment.
For illustration, it is assumed here that the diffusion limitation exists at the capillary endothelium.
Each subcompartment is considered to be a well-mixed phase; therefore, venous blood exiting the tissue
has the same chemical concentration as that of the vascular subcompartment space. With these assump-
tions, a mass-balance equation for the extravascular space (ei) may written as:
VdC dt = ( C ei) (3.87)
k C vi −
i
i
ei
and for the vascular space (bi) may written as:
dt = ( C vi) − ( C ei)
k C vi −
Q C art −
bi
VdC bi i i (3.88)
The chemical concentration in the whole tissue may then be calculated from the volume-weighted
contributions of the vascular and extravascular spaces: