# Understanding the Pharmacokinetics of Repeated Doses

Most prescription drugs are administered repeatedly for a limited duration (for acute illnesses) or for an extended period of time (for chronic conditions). As such, it is important to understand the pharmacokinetic behavior of drugs when they are administered according to repeat-dose regimens.

### Single-Dose PK Behavior

For the purpose of this discussion, we will use single-dose IV bolus administration as the starting point. Let’s assume we administer 500 mg of a drug into a hypothetical volume of 10 liters. Since the full IV bolus dose is administered at once, we will see an initial concentration of 50 mg/L (500 mg / 10 L of volume = 50 mg/L). The human body has evolved multiple mechanisms (hepatic, renal, etc.) that allow for elimination of drugs and other substances. As a result, immediately following the initial bolus dose, drug concentrations will begin to decline. We frequently describe this decline in terms of *half-life* (the time required for concentration to decline by 50%). If our hypothetical drug has a half-life of 3 hours we would observe a concentration of 25 mg/L at three hours post-dose, a concentration of 12.5 mg/L at six hours post-dose, a concentration of 6.25 mg at 9 hours post-dose, etc. If no further doses are administered, the concentration will continue to decline by an additional 50% every 3 hours.

### Superposition of Repeated Doses

What if a second 500 mg dose is given at the 6-hour mark? For most drugs, the concentrations produced by this second dose would be comparable to concentrations produced by the first dose. However, at the 6-hour time point we still have 12.5 mg/L of drug remaining from the first dose, as described above. So, immediately following the second dose, we will have 12.5 mg/L remaining from the original dose, plus an additional 50 mg/L resulting from the dose that was just administered. This gives a combined *observed concentration* of 62.5 mg/L. Each additional 50 mg dose will produce concentrations comparable to the very first dose. However, the concentration we *observe *will be the sum of concentrations remaining from each prior dose combined with concentrations from the most recent dose. For example, a third dose delivered at the 12-hour time point will also produce an initial concentration of 50 mg/L. The concentration remaining from the second dose would be 12.5 mg/L, and the concentration remaining from the first dose would be 3.125 mg/L. If we combine all of these concentrations, our observed concentration would be 50 mg/L + 12.5 mg/L + 3.125 mg/L ≈ 65.6 mg/L (an additive combination of concentrations from the first, second, and third doses).

The process of adding concentrations from multiple doses to determine the observed concentrations is often referred to as the *principle of superposition*. The pattern of superposition described above assumes that each dose behaves approximately the same despite rising concentrations. It is important to note that simple, additive superposition is approximately true for very many drugs. However, superposition of exposures can become more complicated when the PK behavior of each dose changes as concentrations rise (e.g., drugs with saturable clearance, such as phenytoin). For the remainder of this discussion, we will stick with the simple, additive superposition scenario.

### Reaching Steady State

Successive doses will result in increasing concentrations of the drug in the body until a plateau is reached. This plateau is called *steady state*. At steady state, the amount of drug administered on each dosing occasion is matched by an equivalent amount of drug leaving the body between each dose (rate in = rate out). At steady state, concentrations will rise and fall according to a repeating pattern as long as we continue to administer drug at the same dose level and with the same time period between doses. This repeated time period of dosing is often called the *dosing interval* and is abbreviated using the Greek letter tau (τ). Drug accumulation and attainment of steady state does not require IV bolus dosing. It is possible to observe a similar pattern of accumulation and attainment of steady state for virtually any route of administration.

For most drugs, it takes roughly 5 half-lives to reach an approximate steady state. It follows that, the time to steady state during a repeat-dose regimen is dictated by the half-life of the drug. Intuitively we might think that increasing the dose or giving doses more frequently would accelerate attainment of steady state. However, neither of these changes will alter the speed at which steady state is achieved. Changing the dose or dosing interval *will affect the concentrations achieved at steady state, but not the time required to achieve steady state*. Some drugs have quite prolonged half-lives (days to weeks, or longer). For drugs with extended half-lives we may not be able to wait the necessary 5 half-lives to reach a desired steady state concentration. When time is crucial (such as antibiotic use for critical-care patients), there is a method to achieve steady state more rapidly: the *loading dose*. A loading dose is an initial dose (or series of doses) intended to quickly achieve desired concentrations. A loading dose typically won’t achieve steady state on its own (that would take 5 half-lives). However, once the desired concentrations are achieved with a loading dose, a repeat-dose (maintenance) regimen can be started at a lower dose level. If calculated correctly, this new maintenance regimen will maintain stable steady state exposure for the remaining duration of repeated drug administration.

### AUC and Accumulation

Following a single dose, we can easily calculate the area under the concentration-time curve from zero to infinity (AUC_{0-∞}) as a measure of overall drug exposure. During repeat-dose administration we often calculate the AUC during a steady state dosing interval (AUC_{0-τ}) as a measure of overall drug exposure. Interestingly, if clearance remains constant for a drug (no change in CL with increasing concentrations), AUC_{0-τ} at steady state will be identical to the AUC_{0-∞} following single-dose administration. This equivalence assumes that the dose level administered as a single dose is identical to the dose level administered in the repeat-dose regimen. This AUC equivalence proves useful in terms of calculating PK parameters. For example, following a single IV bolus dose, we can calculate CL using the following expression: CL = Dose/ AUC_{0-∞}. AUC equivalence also allows us to estimate CL using steady state AUC_{0-τ} : CL = Dose/ AUC_{0-τ}. The latter clearance estimate is frequently termed steady state clearance (CL,ss).

### Conclusions

Most drugs will deviate from the ideal accumulation pattern described above to some extent. However, understanding the principle of superposition allows for reasonable predictions of repeat-dose PK behavior for a very large number of drugs. This becomes particularly useful when progressing from single-dose studies to repeat-dose studies for the first time during drug development. This knowledge also enables us to design repeat-dose regimens that efficiently and reliably achieve desired concentrations within a clinically-acceptable time frame.

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