**3.5. Pharmacokinetic Modeling**

**3.5.1. Modeling approaches**

There are three approaches that have been suggested for pharmacokinetic modeling, compartmental, physiological and model-independent.

The first is an empirical approach, which is based on a simple **compartmental
model**. These compartments have no strict physiological or anatomic basis.
The compartment can represent a body volume, or just as easily it could
represent a chemical state, for example a metabolite of the drug. Usually
this approach uses one or two compartments. Despite its simplistic nature,
many useful quantities can be derived using this approach and by comparing
predicted values with actual data.

The **physiological model** identifies the compartments with actual
body spaces. The model is a great deal more complex than the compartmental
models. Actual transfer and flow rates are employed together with experimentally
measured blood/tissue concentration ratios. This model can be used for predictions,
and is more adaptable to clinical therapy and to changing situations such
as alterations of flow rates due to conditions such as swelling, or fever.
Scale up is more easily from one species to another, which is convenient
since most of the required measurements must be performed in animals.

The **model** **independent approach** is the most recent, and
is purely mathematical. It avoids recourse to kinetic parameters which may
not be valid, and is a lot less complex. It is good for adsorption and elimination
rates, and clearances, but gives no specific information about how the drug
is distributed.

**3.5.2. One Compartment Open Model**

This approach models the entire body as a single compartment into which
drug is added by a rapid single dose, or **bolus**. It is assumed that
the drug concentration is uniform in the body compartment at all times and
is eliminated by a first order process that is described by a first order
rate constant kel. The model is shown in figure 3.5.1.

Figure 3.5.1. Basis of the one compartment open model

Analysis of this model relies on a simple mass balance .

This is a first order rate equation, which can be solved by a plot of ln C against time

Figure 3.5.2. Graphical treatment of plasma concentration with time to obtain kel and VD

The one compartment model fails to describe the actual drug disposition when, for example, a particular organ has a small, but strong affinity for a drug, which does not affect the overall plasma concentration, but which leads to toxicity on repeated doses. If this area is the site of drug action, the effect could continue after blood levels had subsided.

**3.5.2.1. One Compartment Model: Oral Administration**

When a drug is taken orally as a tablet, the drug has to dissolve and be absorbed by the gut. This is often a first order process, and this should be accounted for by a first order absorption term in the kinetic analysis, as shown in figure 3.5.3:

Figure 3.5.3. One compartment model with first order absorption

**3.5.3. Two compartment model**

Often drug does not distribute evenly amongst all the organs. To account for this a two compartment model is used in which drug disposition is biexponential. The drug is assumed to distribute into a second compartment but be eliminated from the first compartment only. This is obviously a simplification of the situation in the body, but it can give some data on rates in and out of specific organs. The plasma drug concentration initially declines quite rapidly due to elimination from the plasma and distribution into the second compartment, which can comprise several organs. This phase is called the a phase. Once equilibrium is reached, the plasma drug level declines more slowly due to elimination alone, in what is termed the phase. This is illustrated in figure 3.5.4.

Figure 3.5.4. Plasma-Tissue drug profile that obeys the two compartment model

In the two compartment model, again a rapid bolus of drug is assumed. Also a first order transfer is assumed between compartments and first order elimination from the blood compartment, with no elimination or metabolism in the tissue. The scheme is outlined in figure 3.5.5.

Figure 3.5.5. Two compartment open model

The kinetic equations for this case are as follows:

Integration with boundary conditions,

This represents the biexponential semilog concentration/time plot that is expected with the two compartment model and is shown in figure 3.5.6.

Figure 3.5.6.** **Method of fitting
data to biexponential curve

When experimental data is fitted to the semilog curve, a best fit straight line is generated from the long times points (phase) points, and the resulting slope is -. When line 2 is subtracted from the remaining short time data points, the slope of the resulting line 1 is -. This technique is known as curve stripping. From the intercepts I1 and I2 of these lines, we can obtain kinetic parameters as:

Comparing these equations with those for the one compartment model, we can see that drugs with a very shortphase, (rapid distribution) the model approaches that of the one compartment model.

**3.5.4. Constant IV Infusion: One Compartment Model**

Drugs are not always administered by a rapid bolus, as assumed in these
models. A frequent mode of drug administration is by a continuous intravenous
infusion, In this case a steady state of plasma drug concentration is achieved.
For the one compartment model, we assume a zero order infusion rate constant
k_{o}. The analysis follows:

**3.5.5. Constant IV Infusion: Two Compartment Model**

For the two compartment model and identical expression can be derived for steady state concentration in the body compartment:

The equivalent expression for the tissue compartment is derived as:

**3.5.6. Sustained Release Formulation: One compartment model**

Many of the drugs that one can purchase over the counter **(OTC drugs)**
are available is **sustained release** formulations. The vehicle in which
they are supplied has been modified so that not all of the drug is available
for absorption. This means that some fraction of the drug fi,
is available immediately, and some fraction f_{r},
[f_{r} = (1-f_{i})]
is released with first order kinetics, described by a constant k_{r} as illustrated in figure 3.5.7..

Figure 3.5.7. One Compartment model for a sustained release formulation

For the concentration of drug in the gut:

therefore:

For the concentration of drug in the body:

This reduces to the expression for oral administered dose with an absorption
term, equation (3.16) when f_{r} = 0 and f_{i}
= 1, and B = C_{B} V_{d}. Experimentally one can measure f_{r}, f_{i} and k_{r} *in vitro*
for a sustained release formulation and measure C_{B} (plasma concentration)
after administration in animal or human. These data are then fit to find
k_{el}, k_{a} and V_{d}. An example that appeared
in the literature was given by Wiegand and Taylor (1960) Biochemic. Pharmacol.
3: 256 for the drug 1-(o-methoxy propyl)-4-(g-methoxypropyl) piperazine
phosphate.

Figure 3.5.8. Comparison of sustained-release tablet and solution kinetics in six dogs. Initial dose is shown on the graph.