PHAR 4634 - Chapter 14 Page 8     ### Cpmax and Cpmin equations

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 ---> and t = 0 or t = t. These are the limits of the PLATEAU CONCENTRATIONS.   Equation XIV-17 Cp Immediately after Many Doses

and   Equation XIV-18 Cp Immediately before Many Doses

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

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

kel = = 0.17 hr-1

R = e-kel * = e-0.17 x 6 = 0.35

therefore and 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 XIV-11, Plot of Cp Versus Time showing Time to Approach 50% of Plateau during Multiple Dose Regimen

Using a JAVA aware browser you can create your own version of Figure XIV-11.

Plasma Concentration versus Time Plots

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

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 general:-

And since (see Equation XIV-17, page XIV-11) or

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.

Since therefore Also

R = e-kel * = 0.2857

then

- kel * = -1.2528 or = 8.35 hour; the dosing interval.

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

R = e-kel * = e-8 x 0.15 = 0.3012

If we use Cpmax = Maintenance dose = Cpmax * V * (1 -R) = 35 x 25 x (1 -0.3012) = 611 mg

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

To check

Cpmax = = 34.3 mg/L

and

Cpmin = Cpmax * R = 10.3 mg/L

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 XIV-11. 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. 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     