Drug Targeting Organ-Specific Strategies

340 13 Pharmacokinetic/Pharmacodynamic Modelling in Drug Targeting
Usually, the differential equations are written in a different form, by relating the rate of
transport from a compartment to the quantity of the drug in that compartment and a rate
constant (dimension: time -1 ), as depicted in Figure 13.2. and formulated as follows.
dA 1 = R1 + k 21 · A 2 – k 12 · A 1 – k 10 · A 1
dt
(13.2)
where A 1 is the quantity of the drug in compartment 1, k 12 and k 21 are distribution rate
constants and k 10 is the elimination rate constant.
Comparing Eq. 13.1 and 13.2, it follows that a rate constant k xy is equal to CL xy / V x.
From the assumption that there is no net transport between two compartments if the concentrations
in both compartments are equal, it follows that
k 21 · V 2 = k 12 · V 1 (= CL 21 = CL 12).
Eq. 13.1 and 13.2 are mathematically equivalent, and thus may be used arbitrarily without
affecting the modelling results. However, Eq. 13.1 (and Figure 13.1) is preferred since it reflects
better the mechanistic basis, as drug transport is governed by drug concentration, both
for passive diffusion according to Fick’s Law, and for carrier-mediated transport. In the case
of the latter, the terms referring to the rate of transport from compartment x to compartment
y,
CL xy · C x
should be replaced by their Michaelis–Menten equivalent
Vmaxxy · Cx Kmxy + Cx (13.3)
(13.4)
where Vmax xy is the maximum transport rate between compartments x and y, and Km xy is
the Michaelis–Menten constant of the transport between x and y.
An example of the use of Michaelis–Menten kinetics in a compartmental model is given in
the model of Stella and Himmelstein [5], depicted in Figure 13.3.
13.2.4.2 Physiologically-based Pharmacokinetic (PB-PK) Models
These are relatively complex models describing drug transport between blood and a series of
physiological and/or anatomical entities, for example, organs, tissues, or cells [15–20]. PB-PK
models are characterized by a relative large number of parameters. In many cases, several of
these can be estimated from physiology or anatomy (for example, blood flow and volumes),
others may be obtained from in vitro experiments (for example, partition coefficients between
water and tissue), or by experiments in isolated tissues (for example, binding and metabolism
in isolated liver cells or slices; see Chapter 12). In principle, PB-PK models are well
adapted to take into account the extracellular and/or intracellular events in the disposition of
the targeting device.
The number of compartments in a physiologically-based pharmacokinetic model may vary
between two (in drug targeting: a target compartment and a non-target compartment) and 10
or more, depending on the desired degree of differentiation. The more compartments, the
greater the ability of the model to define the true behaviour of the drug. However, the in-