## Kinetics Of Drug Absorption

For all commonly used routes of administration except IV, the drug must dissolve in body fluids and diffuse through one or more membranes to enter the blood or plasma. Thus, all routes except IV are classed as extravascular routes, and absorption is defined as appearance of the drug in blood or plasma.

The most common extravascular route is oral. When a solution or a rapidly dissolving solid dosage form is given orally, the absorption process often obeys first-order kinetics. In these cases, absorption can be characterized by evaluating the absorption rate constant, ka, using plasma concentration versus time data.

The Method of "Residuals" ("Feathering" the Curve)

When absorption is first-order, the kinetic model may be written as shown in Scheme 3:

Scheme 3

where DG = drug at the absorption site (gut); DB = drug in the body; DE = eliminated drug; ka = first-order absorption rate constant, and kel = overall elimination rate constant.

The differential equations describing the rates of change of the three components of Scheme 3 are dDG dt

+kelDB

To determine ka from plasma concentration versus time data, it is necessary to integrate equation (30). This is best achieved through exponential expressions. First, integration of equation (29) gives

where DG is the initial amount of drug presented to the absorbing region of the gut. (DG = dose, if absorption is complete.)

Figure 8 Semilogarithmic plot of observed plasma concentrations (crosses) and "residuals" (circles) versus time for an orally administered drug absorbed by a first-order process.

Substituting equation (32) into equation (30) gives dDB = +kaD0c exp(-fcaf) - kelDB (33)

Integration of equation (33) may be accomplished with Laplace transforms (6,7). The result is

Do k

Thus, the amount of drug in the body following administration of an extravascular dose is a constant [(DjG ka)/(ka - kel)] multiplied by the difference between two exponential terms—one representing elimination [exp (-kelt)] and the other representing absorption

Dividing both sides of equation (34) by Vd yields an equation for plasma concentration versus time:

D0 k

Equation (35) describes the line in Figure 8, which is a semilogarithmic plot of Cp versus time for an orally administered drug absorbed by a first-order process. The plot begins as a rising curve and becomes a straight line with a negative slope after six hours. This behavior is the result of the biexponential nature of equation (35). Up to six hours, both the absorption process [exp (-kat)] and the elimination process [exp (-kelt)] influence the plasma concentration. After six hours, only the elimination process influences the plasma concentration.

This separation of the processes of absorption and elimination is the result of the difference in the values of ka and kei. If ka is much larger than kei (a good rule is that it must be at least five times larger), the second exponential term in equation (35) will approach zero much more rapidly than the first exponential term. And at large values of t, equation (35) will reduce to

Do k

where A is a constant term.

Converting to logarithms we obtain ln Cp =-kel t + ln A (37)

Thus after six hours the semilogarithmic plot of Cp versus time shown in Figure 8 becomes a straight line, and kel can be determined from the slope. Therefore, the overall elimination rate constant for a drug may be accurately determined from the "tail" of a semilogarithmic plot of plasma concentration versus time following extravascular administration if ka is at least five times larger than kel.

The value of ka can also be determined from plots like Figure 8 using the following logic:

In Figure 8 the curved line up to six hours is given by

The straight line after six hours and the extrapolated (dashed) line before six hours are given by

The difference (residual) between the curved line and the extrapolated (dashed) line up to six hours is given by

Converting to logs:

As shown in Figure 8, a semilogarithmic plot of residuals versus time is a straight line with a slope of —ka.

It should be noted that the intercepts (A) for both the extrapolated (dashed) line (equation 37) and the residuals line (equation 38) are the same and are equal to the constant in equation (35):

A is a function of the two rate constants (ka and kel), the apparent volume of distribution (Vd), and the amount of drug absorbed (DG). After ka and kel have been evaluated and A has been determined by extrapolation, a value for Vd can be calculated if it is assumed that DG is equal to the dose administered, i.e., absorption is 100% complete.

Example. Figure 8 is a plot of the data shown in Table 5. The extrapolated value of A is 11.8 mg/mL.

 Time (hr) Observed Cp (mg/mL) Extrapolated Cp (mg/mL) Residuals (
0 0