McKay, Donald. "Front matter" Multimedia Environmental Models ...

Figure 8.10 Fish bioaccumulation processes expressed as concentration/rate constants and
fugacity/D values.
Apparently, the chemical passively diffuses into the fish along much the same route
as oxygen. In the laboratory, it is usual to expose a fish to a constant water concentration
for a period of time during which the concentration in the fish should rise
from zero to C F according to the integrated version of the differential equation with
C F initially zero and C W constant.
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C F = (k 1/k 2)C W[1 – exp(–k 2t)]
After prolonged exposure, when k 2t is large, i.e., >4, C F approaches (k 1/k 2)C W or
K FWC W where K FW is the bioconcentration factor. The fish is then placed in clean
water, and loss or clearance or depuration is followed, the corresponding equation
being
C F = C FO exp(–k 2t)
where C FO is the concentration at the start of clearance.
It is apparent that there are three parameters, k 1, k 2, and K FW or k 1/k 2, thus only
two can be defined independently. The most fundamental are k 1 (which is a kinetic
rate constant term quantifying the volume of water that the fish respires and from
which it removes chemical, divided by the volume of the fish) and K FW, which is a
thermodynamic term reflecting equilibrium partitioning. The loss rate constant k 2 is
best regarded as k 1/K FW.
The uptake and clearance half-times are both 0.693 K FW/k 1 or 0.693/k 2.
This equation can be expanded to include uptake from food with a rate constant
k A and food concentration C A, loss by metabolism with a rate constant k M, and loss
by egestion in feces with rate constant k E, namely,
dC F/dt = k 1C W + k AC A – C F(k 2 + k M + k E)