# Chapter 14

# Multiple IV Bolus Dose Administration

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## Cp_{max} and Cp_{min}

More useful equations can be derived from the general equation.

**Equation 14.6.1 Cp at time, t, after the nth IV Bolus Dose**

**General Equation**

These are equations to calculate the maximum and minimum plasma concentration after many
doses. That is as n approaches ∞, R^{n} (= e^{-n • kel • τ}) approaches 0 and with t = 0 or t = τ. These are the limits of the PLATEAU CONCENTRATIONS.

**Equation 14.6.2 Cp Immediately after Many Doses**

**Equation 14.6.3 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

What are the Cp_{max} and Cp_{min} values when the plateau values are reached

therefore

therefore the plasma concentration will fluctuate between 15.5 and 5.48 mg/liter
during each dosing interval when the plateau is reached.

### Accumulation Factor

In Equation 14.6.2 the ratio, Dose/V represents the initial concentration after the first dose. Thus the ratio between the highest, initial concentration at steady state, Cp_{max} and the highest concentration after the first dose, Cp^{0}_{1} can be expressed as the ratio.
**Equation 14.6.4 Accumulation Factor**

The accumulation factor describes how much drug accumulates during a multiple dosing regimen and gives a direct measure of how much higher the concentrations are during a dosing interval at steady state compared with the concentration during the first dosing interval.
It is interesting to note that if the dose is given every drug elimination half-life the accumulation factor is two since R equal to e^{-kel • τ} equals one half.

If dosing every half-life Cp_{max} is twice the Cp^{0}_{1} value and Cp_{min} is equal to the Cp^{0}_{1} value.

For the example above R is equal to 0.354 and the accumulation factor can be calculated.

To complete the example above we can calculate the plasma concentration at any time following multiple IV bolus administration (using Equation 14.5.11 on the previous page) AND we can calculate the Cp_{max} and Cp_{min} values (using Equation 14.6.2 and 14.6.3 above).

**Figure 14.6.1 Plot of Cp ***versus* Time showing Time to Approach

50% of Plateau during Multiple Dose Regimen

Click on the figure to view the interactive graph

### Time to Cp_{max}/Cp_{min}

Just as in the case of a continuous infusion it takes some time to get to the plateau where the concentrations vary between Cp_{max} and Cp_{min} during each dosing interval. As before, 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 again have a problem with an excessive time required to reach the plateau. Administration of a loading dose will achieve steady state concentrations rapidly.

Practice problems involving Cp_{max} and Cp_{min} at steady state after uniform multiple dose IV bolus doses.
Another practice problems involving Cp_{max} and Cp_{min} at steady state after uniform multiple dose IV bolus doses.
A third practice problems involving Cp_{max} and Cp_{min} 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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