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Cpmax and Cpmin

More useful equations can be derived from this general equation. These are equations to calculate the maximum and minimum plasma concentration after many doses. That is as n ---> ∞, Rn (= e-n • kel • τ) ---> 0 and with t = 0 or t = τ. These are the limits of the PLATEAU CONCENTRATIONS.

Equation 25.6.1 Cp Immediately after Many Doses

Equation 25.6.2 Cp Immediately before Many Doses

An example may be helpful: t1/2 = 4 hr; IV dose 100 mg every 6 hours; V = 10 liter

What are the Cpmax and Cpmin values when the plateau values are reached

therefore

therefore the plasma concentration will fluctuate between 15.5 and 5.4 mg/liter during each dosing interval when the plateau is reached.

We can now calculate the plasma concentration at any time following multiple IV bolus administration and we can calculate the Cpmax and Cpmin values.

Figure 25.6.3 Plot of Cp Versus Time showing Time to Approach 50% of Plateau during Multiple Dose Regimen

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

It can be shown that the time to reach a certain fraction of the plateau concentration is dependent on the drug elimination half-life only, much the same as for the approach to steady state during an IV infusion. Thus we may have a problem with an excessive time required to reach the plateau. Therefore we may want to determine a suitable loading dose to achieve steady state rapidly.

In the previous example Cpmax = 15.5 mg/liter

A suitable loading dose would be Cpmax * V

155 mg as a bolus would give Cp = 15.5 mg/liter, followed by 100 mg every 6 hours to maintain the Cpmax and Cpmin values at 15.5 and 5.5 mg/liter respectively.

In summary:

The IV bolus loading dose to quickly achieve a required drug concentration, Cpmax, can be calculated as Cpmax * V

Rewriting Equation 25.6.1 with Dose expressed more explicitly as the Maintenance Dose.

Equation 25.6.3 Cpmax

Equation 25.6.4 Loading Dose

or

Equation 25.6.5 Maintenance Dose

We can try another example of calculating a suitable dosing regimen.

Consider V = 25 liter; kel = 0.15 hr-1 for a particular drug and we need to keep the plasma concentration between 35 mg/liter (MTC) and 10 mg/liter (MEC).

What we need is the maintenance dose, the loading dose, and the dosing interval.

From Equations 25.6.1 and 25.6.2

Equation 25.6.6 R from Cpmax and Cpmin

therefore

taking the ln of both sides

- kel • τ = -1.252 = -0.15 • τ

τ (dosing interval) = 8.35 hr

A dosing interval of 8 hours would be more reasonable. Thus with τ = 8 hr and kel = 0.15 hr-1

R = e-kel • τ = e-8 x 0.15 = 0.301

From Equation 25.6.3

Equation 25.6.7 Maintenance Dose

Maintenance dose = 35 x 25 x (1 - 0.301) = 612 mg

Again a more realistic maintenance dose would be 600 mg every 8 hours.

To check this regimen

or

This regimen would be quite suitable as the maximum and minimum values are still within the limits suggested. All that remains is to calculate a suitable loading dose.

Loading dose = Cpmax • V = 35 x 25 = 875 mg either 875, 850 or 800 mg

This answer can be expressed graphically.

Figure 25.6.4 Plasma Concentration after Multiple IV Bolus Doses

Calculate a suitable multiple IV bolus dosage regimen for specified Cpmin and Cpmax

Patient (weight Kg) is to receive a drug by multiple IV bolus doses. For optimal treatment drug concentration should be kept between Cpmax mg/L and Cpmin mg/L. This drug has an apparent volume of distribution of L/Kg and an elimination rate constant of hr-1 (Clearance = L/kg.hr).

First calculate the dosing interval, tau.

Tau hour.

Enter a new (rounded) value for tau. hr.

The next step is to calculate a suitable maintenance and loading dose.

Start with a loading dose of mg as an IV Bolus and a maintenance dose of mg as an IV Bolus every hour.

Calculate Cpmin and Cpmax after multiple IV bolus doses

Patient (weight Kg) has received mg of a drug as an IV Bolus every hours. This drug has an apparent volume of distribution of L/Kg and an elimination rate constant of hr-1 (Clearance = L/kg.hr). Calculate the Cpmin and Cpmax at steady state.

Cpmin mg/L

Cpmax mg/L

Practice problems involving Cpmax and Cpmin at steady state after uniform multiple dose IV bolus doses. Another practice problems involving Cpmax and Cpmin at steady state after uniform multiple dose IV bolus doses. A third practice problems involving Cpmax and Cpmin at steady state after uniform multiple dose IV bolus doses.

Other practice problems involving the calculation of Cp at three times during a uniform dosing interval with Linear or Semi-log graphical answers or calculation of Cp at three times during a non-uniform dosing interval with Linear or Semi-log graphical answers

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

This file was last modified: Thursday 07 Aug 2003 at 11:55 AM