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## Understanding Steady State Pharmacokinetics

“Steady state” is an important term in pharmacokinetics, but it can often seem a bit abstract and confusing to many. Here is how I define steady state:

When the rate of drug input is equal to the rate of drug elimination, steady state has been achieved.

Another way to think of this is imagine a carton of eggs in your kitchen. And imagine that when you use 2 eggs to make an omelette for breakfast. Someone in your house notices the empty spots in the carton of eggs and purchases 2 more eggs and places them in the carton. So when you wake up the next morning, the carton is full of eggs. If this process repeats itself over many days, it would appear that the eggs never change … there are always 12 eggs in the carton even though you use them for various meals and recipes. In this hypothetical scenario, the eggs are at steady state because the rate of elimination is equal to the rate of input.

The eggs represent individual drug molecules in the body. Using the eggs represents the variety of clearance mechanisms that eliminate drug molecules from circulation. And the replenishment of eggs represent taking new doses of medication.

### PK parameters associated with steady state

There are several special PK parameters associated with steady state kinetics. These parameters are not necessarily more important; however, they are useful because of the unique situation when drug input rate and elimination rate are equivalent. The first is the average plasma concentration at steady state, or Css. This parameter can be calculated based on the steady state definition where the rate of input is equal to the rate of elimination.

Rate of input: $\frac{F*Dose}{\tau}$, where τ is the dosing interval

Rate of elimination: $CL*C_p$

By solving for Cp, you get the following: $C_{ss}=C_p=\frac{F}{CL}*\frac{Dose}{\tau}$

As a further simplification, we know that there is a relationship between dose, clearance, and bioavailability shown by the following equation:

$CL=\frac{F*Dose}{AUC}$

By rearranging terms we can get the following:

$AUC=\frac{F*Dose}{CL}$

Then replacing terms from the equation for Css above with AUC, we get the following:

$C_{ss}=\frac{AUC}{\tau}$

Thus, the average concentration at steady state is simply the total exposure over 1 dosing interval divided by the time of the dosing interval. So while concentrations rise and fall during a dosing interval at steady state, the average concentration does not change. Furthermore, the only factors that control Css are the dose, the dosing interval, and the clearance. Assuming clearance cannot be altered by a clinician, the steady state levels of drug can be modulated using the dose and the dosing interval. Lower doses and longer intervals will result in lower Css values, while higher doses and shorter intervals will give higher Css values.

The situation is even simpler for intravenously administered drugs where you can directly calculate the Css using the following equation:

$C_{ss}=\frac{R_0}{CL}$ where R0 is the rate of drug input.

The Css is proportional to the infusion rate directly.

One additional trick you can use is the relationship between AUC for a dosing interval and AUC0-∞ after a single dose:

$AUC_{o-\tau}=AUC_{0-\infty}$

Based on this equality, if you calculate AUC0-∞ after a single dose, you can then predict the steady state concentrations for any dosing interval you choose by plugging it into the equation listed earlier for Css. This can also be extended to different dose levels if you assume dose proportionality.

### Time to reach steady state

The time to reach steady state is defined by the elimination half-life of the drug. After 1 half-life, you will have reached 50% of steady state. After 2 half-lives, you will have reached 75% of steady state, and after 3 half-lives you will have reached 87.5% of steady state. The rule of thumb is that steady state will be achieved after 5 half-lives (97% of steady state achieved).

If you have a drug with a long half life, you can achieve a target steady state level more quickly by using a loading dose. For example, perhaps you want to achieve a steady state of concentration (Css) of 10 ng/mL, which requires a 30 mg dose once daily. However, it will take 10 days to achieve steady state. Assuming dose proportionality, a 60 mg dose would achieve a Css of 20 ng/mL in 10 days also. But after a single dose of 60 mg, you would achieve 50% of steady state or 10 ng/mL (20 ng/mL * 50% = 10 ng/mL). Therefore you could achieve the target Css more quickly by administering one 60 mg dose followed by 30 mg doses to achieve steady state levels within 2 days. The 60 mg dose is called a loading dose, and the 30 mg dose is the maintenance dose (or dose to maintain steady state).

### Final thoughts

Steady state pharmacokinetics are important for chronically administered drug products. Understanding the basic concept that a drug reaches steady state once the rate of drug input and elimination are equivalent helps simplify the concept of steady state. Hopefully you have a clearer understanding of steady state and can utilize this information in your upcoming projects.

To learn about how we’ve improved Phoenix to make performing NCA and PK/PD modeling even easier, please watch this webinar I gave on the latest enhancements to Phoenix.

### About the author

By: Nathan Teuscher
Dr. Teuscher has been involved in clinical pharmacology and pharmacometrics work since 2002. He holds a PhD in Pharmaceutical Sciences from the University of Michigan and has held leadership roles at biotechnology companies, contract research organizations, and mid-sized pharmaceutical companies. Prior to joining Certara, Dr. Teuscher was an active consultant for companies and authored the Learn PKPD blog for many years. At Certara, Dr. Teuscher developed the software training department, led the software development of Phoenix, and now works as a pharmacometrics consultant. He specializes in developing fit-for-purpose models to support drug development efforts at all stages of clinical development. He has worked in multiple therapeutic areas including immunology, oncology, metabolic disorders, neurology, pulmonary, and more. Dr. Teuscher is passionate about helping scientists leverage data to aid in establishing the safety and efficacy of therapeutics.

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