Differential Equations

Drug Amount Remaining to be Absorbed, Xg

Equation 17.2.1 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 17.2.1 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 of -ka.

Drug Amount in the Body, Xp

Equation 17.2.3 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 Xg is much larger than V • Cp therefore the value of V • dCp/dt is positive, therefore the slope is positive and Cp will increase. With increasing time after the dose is administered, Xg will decrease, while initially Cp is increasing, therefore there will be a time when ka • Xg will equal kel • V • Cp. At this time V • dCp/dt will be zero and there will be a peak in the plasma concentration. At even later times Xg will approach zero, and V • dCp/dt will become negative and Cp will decrease. It could be expected that the plasma concentration time curve will look like Figure 17.2.1.

Figure 17.2.1 Linear Plot of Cp versus Time after Oral Administration Showing Rise, Peak, and Fall in Cp

Click on the figure to view the Java Applet window
Java Applet as a Semi-log Plot

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Copyright 2001-3 David W. A. Bourne (david@boomer.org)