# Chapter 8

## Scheme or diagram

This model can be represented as:-

Figure 8.1.1 Representing Oral Administration, One Compartment Pharmacokinetic Model

Where Xg is the amount of drug to be absorbed, Xp is the amount of drug in the body, and ka is the first order absorption rate constant.

## Differential Equations

### Drug Amount Remaining to be Absorbed, Xg

The differential equation for Xg is shown in Equation 8.2.1

Equation 8.2.1 Differential Equation for Amount Remaining in the G-I Tract

This is similar to the equation for dCp/dt after an IV bolus administration.

Using Laplace transforms it is possible to derive the integrated equation.

Equation 8.2.2 Integrated Equation for Drug Amount Remaining in the G-I Tract available for Absorption

where F is the fraction of the dose which can be absorbed, the bioavailability.

We could therefore plot Xg (the amount remaining to be absorbed) versus time on semi-log graph paper and get a straight line with a slope representing ka, Figure 8.2.1.

Figure 8.2.1 Semi-log Plot of X(g) versus Time

And as a linear plot.

Figure 8.2.2 Linear Plot of X(g) versus Time

### Drug Amount in the Body, Xp

For Xp (= V • Cp) the amount of drug in the body, the differential equation is shown in Equation 8.2.3

Equation 8.2.3 Differential Equation for Amount of Drug in the Body

The first term, ka • Xg, represents absorption and the second term, kel • V • Cp, represents elimination

Even without integrating this equation we can get an idea of the plasma concentration time curve.

Shortly after the dose is administered ka • Xg is much larger than kel • V • Cp and the value of dCp/dt is positive, therefore the slope is positive and Cp will increase. With increasing time after the dose is administered, as Xg decreases, Cp is initially increasing, therefore there will be a time when ka • Xg will equal kel • V • Cp. At this time dCp/dt will be zero and there will be a peak in the plasma concentration. At even later times Xg will approach zero, and dCp/dt will become negative and Cp will decrease. It could be expected that the plasma concentration time curve will look like Figure 8.2.3.

Figure 8.2.3 Linear Plot of Cp versus Time after Oral Administration Showing Rise, Peak, and Fall in Cp

Click on the figure to view the interactive graph