Differential and Integrated equation

The differential equation for V*Cp is then:

Equation VI-1. Differential Equation During an IV Infusion.

This is the differential equation during the infusion period and it can be integrated to give:

Equation VI-2. Integrated Equation for Cp versus Time

Javascript Calculators using Equation VI-2

Calculate Cp Given k0, kel and V at time t

Enter your own values into each field
 k0 (zero order mass/time) kel (first order reciprocal time) V (volume) t (time) Cp (mass/volume) is:
Calculate k0 required to give Cp at time t

Enter your own values into each field
 Desired Cp (mass/volume) kel (first order reciprocal time) V (volume) t (time) k0 (mass/time) is:

You may notice that Equation VI-2 for Cp is quite similar to Equation V-8 we had before for the cumulative amount of drug excreted into urine. As you might expect the plot of Cp would be similar in shape.

Figure VI-1. Linear Plot of Cp versus Time During a Continuous Infusion

Using a JAVA aware browser you can create your own version of Figure VI-1.

Plasma Concentration versus Time Plots

If we continue the infusion indefinitely then we will approach a steady state plasma concentration when the rate of infusion will be equal to the rate of elimination.

This is because the rate of infusion will be constant whereas the rate of elimination will increase as the plasma concentration increases. At steady state the two rates become equal. We can determine the steady state concentration from the differential equation by setting the rate of change of Cp, i.e. dCp/dt = 0.

Then

therefore

Equation VI-3. Steady State Concentration after Continuous Infusion

This could also be calculated from the integrated equation by setting e- kel * t = 0 at t = .

We can now calculate the infusion rate necessary to produce some desired steady state plasma level.

For Example:

A desired steady state plasma level of theophylline maybe 15 mg/L. The average half-life of theophylline is about 4 hr and the apparent volume of distribution is about 25 liter. What infusion rate is necessary?

First, kel = 0.693/4 = 0.17 hr-1

then k0 = kel * V * Cp = 0.17 * 25 * 15 = 63.8 mg/hr

We would probably use an infusion of 60 mg/hr which would produce a Cpss value given:

Javascript Calculators using Equation VI-3

Calculate Cpss Given k0, kel and V

Enter your own values into each field
 k0 (zero order mass/time) kel (first order reciprocal time) V (volume) Cpss (mass/volume) is:
Calculate k0 required to give Cpss

Enter your own values into each field
 Desired Cpss (mass/volume) kel (first order reciprocal time) V (volume) k0 (mass/time) is:

For practice try calculating concentrations or required infusion rates. Compare your answers with the computer! These problems includes calculation of drug concentration or required infusion rates during an IV infusion or at steady state.
This page was last modified: 25 May 2002

Copyright 2002 David W.A. Bourne