Page 172 - Veterinary Toxicology, Basic and Clinical Principles, 3rd Edition

Toxicokinetics Chapter | 8 139

VetBooks.ir is described using mass balance equations that represent where V e and C e are the anatomic volume, and concentra-
Following a schematic design of the model, the model
tion in the extracellular space, V i and C i is the volume of
the intracellular space, and the concentration in the intra-
the changes in tissue concentrations over time. The limit-
ing factor in the exchange between compartments is an cellular space, and K t is the membrane permeability coef-
important consideration. In most models, the exchange is ficient for the membranes that separate the intracellular
either limited by diffusion through a membrane barrier and extracellular spaces.
(diffusion-limited), or it is limited by the amount of blood Further modifications are needed to describe metabo-
flow (flow-limited). Both types of exchanges may be lism or excretion. This can be done by adding a mass
present in a single PBPK model. Diffusion-limited removal term, R ex . The resulting equation for a flow-
exchanges are often associated with large, polar mole- limited tissue block is:
cules, and organs with small blood flow to mass ratios.
V t dC t C a C t
Flow-limited exchanges are most commonly associated 5 Q t R ex
with small, lipophilic compounds, and organs with rela- d t P t
tively small volume and large blood flow to mass ratios. The mathematical definition of R ex can be a simple or
The basic mass balance equation for a flow-limited as complex as needed to describe the important features
compartment is: of the process. Commonly used functions include simple
first-order exchanges, and Michaelis Menton equations
V t dC t
5 Q t ðC a C v Þ that describe processes that can become saturated. For
d t
example, if R ex is defined as the clearance of a specific
where Q t , V t , and C t are the blood flow, anatomic volume, organ the equation would be:
and concentration of the compound in the compartment,
and C a and C v are the concentrations of the compound in V t dC t C a C t
5 Q t C t Cl organ
the arterial and venous blood flowing into and out of the d t P t
tissue. It is assumed in flow-limited models that the com- where Cl organ is the clearance of the eliminating organ.
pound is in instantaneous equilibrium between the tissue Tissue blocks can be further refined by the addition of
and the blood, and that distribution in the compartment is modification terms to describe processes such as protein
homogeneous. This allows for the relationship between
binding, tissue binding, active transport, biliary excretion,
the venous blood concentration and the vascular space to
enterohepatic circulation, and metabolism. For example,
be defined according to the partition coefficient between
if R ex is governed by metabolism, and the appropriate
the tissue and blood:
parameter values of enzyme activity are known, it can
described using Michaelis Menton equations such as:
C t
C v 5
P t ðV m f 1 Þ
R ex 5
where P t is the tissue-to-blood partition coefficient.
ðK m 1 f 1 Þ
Therefore, the final mass balance equation is:
where V m is the maximum rate of metabolism, K m is the

V t dC t C a C t concentration at which the rate of metabolism is 50% of
5 Q t
maximum, and f 1 is the free concentration in the metabo-
d t P t
This equation can be used in models where all tissue lizing organ. The final step is to write the mass balance
compartments are simplified into a single compartment. equation for the central (venous blood) compartment. The
In contrast, membrane-limited exchanges do not assume input into this compartment is the combined venous blood
that tissue concentrations are in equilibrium with venous streams from the various tissue compartments. The rate of
blood concentrations. Instead, it uses subcompartments change in concentration in the central compartment is
representing extravascular and intracellular space. The described by:
assumption is that the vascular space is in equilibrium
V p dC p X
with the extracellular space. The mass balance equation is 5 Q t C v Q p C p
d t
then defined by rate of change in the extracellular space
per unit of time, and can be written as: where V p , Q p , and C p are the anatomic volume, the total
blood flow, and the concentration in the central compart-
V e dC e
5 Q t ðC a C e Þ K t ðC e C i Þ ment. C v represents the venous blood concentration from
d t
each tissue compartment. As with other compartments,
this compartment can be modified to incorporate pro-
0 1
V i dC i C e C t cesses such as protein binding, or partitioning into blood
5 K t @ A
cells.
d t P t