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where
V
max
and
K
m
are the maximal velocity and Michaelis constant of the enzymatic
reaction. First-order metabolism is described by the equation:
dA
dt
met
(70)
=
V K C
,
t
f
vt
where
K
f
is the first-order rate constant. Both saturable and first-order metabolism may
occur simultaneously. For example, if metabolism by both saturable and first-order pro-
cesses is occurring in the liver, the mass balance equation for the liver would be:
V
dC
dt
V C
K
1
max
v
1
(71)
=
Q C
(
−
C
)
−
−
K C V
.
1
1
a
v
1
f
v
1 1
+
C
m
v
1
For inhalation exposure to volatile chemicals, equilibrium is established between the
chemical in the alveolar air space of the lung and the chemical in arterial blood. The
concentration of chemical in arterial blood (
C
a
) is described by the equation:
Q C
+
+
(
Q C
c
v
p i
C
(72)
)
,
=
a
Q
Q P
/
c
p
b
where
Q
p
is alveolar ventilation rate,
C
i
is the concentration in inhaled air, and
P
b
is the blood:air partition coefficient. If the chemical can be inhaled, it can also be
exhaled. The concentration of the chemical in exhaled alveolar air is the ratio of its
concentration in arterial blood and the blood:air partition coefficient,
C
a
/
P
b
.
Upon oral ingestion of a chemical (e.g., food, drinking water), the chemical may
be absorbed into the portal circulation or into the lymphatic system. For simplicity,
we assume first-order absorption into the portal circulation only. In this situation, the
chemical is delivered directly to the liver prior to distribution throughout the body.
This requires an input term in the mass balance equation for the liver and a mass bal-
ance equation that describes the rate of loss of chemical from the site of absorption
(stomach):
dA
dt
st
=
(73)
K A
st
,
a
V
dC
dt
V C
K
1
max
vt
(74)
=
Q C
(
−
C
)
+
K A
−
−
K C V
,
1
1
a
v
1
a
st
f
v
1 1
+
C
m
vt
where
K
a
is the oral absorption rate constant and
A
st
is the amount (mass) of chemi-
cal in the stomach (i.e., the site of absorption). Incorporation of dermal absorption is
more complex, in that a skin compartment must be added to the model (see
Krishnan
and Andersen, 1994
).
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