## Repetitive IV Dosing

The plasma concentrations in Figure 12 can be calculated as follows: From equation (25) the plasma concentration at the end of the first dosing interval (T) is given by:

Immediately after the second dose is given, the plasma concentration will be:

and so on.

It is now helpful to define the parameter R as the fraction of the initial plasma concentration that remains at the end of any dosing interval; R is given by the following equation:

As was pointed out in section "Pharmacokinetics of Drug Eliminated by Simultaneous Metabolism and Excretion", when T = t1/2, R = 0.5. The plot in Figure 12 was constructed using these conditions; therefore, the plasma concentration at the end of each dosing interval is half the concentration at the beginning of the dosing interval. Equations (48) and (49) can be simplified to

CTi = CPiR

for the plasma concentration at the end of the first dosing interval, and

P2 P1 P1

for the plasma concentration at the beginning of the second dosing interval. The series can be carried further for more doses:

The plasma concentrations at the beginning and end of the nth dosing interval are given by the following power series:

Beginning = C°pi + C0i R + C°pi R2 + ... + C0i R"-1 (51)

Since R is always smaller than 1, Rn becomes smaller as n increases. For example, if R = 0.5, R10 = 0.001. Therefore, the high-power terms in equations (51) and (52) become negligible as n increases, and additional doses do not change the value of CQp or CTp significantly. This explains why the plasma concentrations reach a plateau instead of continuing to rise as more doses are given.

Hence, Cmax and Cmin (see Figure 12) are defined as the plasma concentrations at the beginning and end, respectively, of the nth dosing interval after the plateau has been reached (i.e., n approaches 1). When n = 1, equations (51) and (52) become

Thus, the maximum and minimum plasma concentrations on the plateau of a repetitive IV dosing regimen can be calculated if the dosing interval (T), the overall elimination rate constant (kei), and the initial plasma concentration (Cp) are known.

Example. A drug has a biologic half-life of four hours. Following an IV injection of 100 mg, Cp is found to be 10 mg/mL. Calculate Cmax and Cmin if the 100-mg IV dose is repeated every six hours until a plasma concentration plateau is reached.

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