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: t_{1/2} = 4 hr; IV dose 100 mg every 6 hours;
V = 10 liter

then

What are the Cp_{max} and Cp_{min} 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 Cp_{max} and Cp_{min} 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 Cp_{max} = 15.5 mg/liter

A suitable loading dose would be Cp_{max} * V

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

In general:-

The loading dose is Cp_{max} * V

And since

(see Equation XIV-17, page XIV-11)

**Equation XIV-19 Loading Dose**

or

Maintenance DOSE = Loading DOSE * (1 - R)

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 Cp_{max} =

Maintenance dose = Cp_{max} * 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**

Cp_{max} = = 34.3 mg/L

and

Cp_{min} = Cp_{max} * 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 = Cp_{max} * 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**

Practice problems involving Cp

This page was last modified: 10 July 2002

Copyright 2002 David W.A. Bourne