# Multiple IV Bolus Dose Administration

## Development of General Equation

We can now consider a general equation which could describe the plasma concentration at any time after multiple IV bolus drug administration.

Figure 14.5.1 Drug Concentration after Multiple Doses Showing Accumulation

Concentration at the end of the first dosing interval

Equation 14.5.1 Cp after the First Dose

where

This gives the plasma concentrations at the end of first interval, where τ is the dosing interval in hours.

Concentration at the start of the second interval

Equation 14.5.2 Cp at the Start of the Second Interval

Concentration at the end of the second dose interval

Equation 14.5.3 Cp at the End of the Second Interval

and so on.

It will help if we define the parameter, R = e-kel • τ, which is the fraction of the initial plasma concentration remaining at the end of the dosing interval.

Then

Equation 14.5.4 Cp at the End of the Second Interval

Equation 14.5.5 Cp at the Start of the Third Interval

this is a geometric series with each term R times the preceding term.

Equation 14.5.6 Cp at the Start of the nth Interval

Equation 14.5.7 Cp at the End of the nth Interval

these two sums are sums of geometric series and they can be simplified to give

Equation 14.5.8 Cp at the Start of the nth Interval

Equation 14.5.9 Cp at the End of the nth Interval

Starting with the first equation (Equation 14.5.8) we can calculate drug concentration in blood or plasma at any time following uniform multiple IV bolus administration.

Equation 14.5.10 Cp at time,t, after the nth IV Bolus Dose

where here t = time since the last dose.

Equation 14.5.11 Cp at time, t, after the nth IV Bolus Dose
General Equation