## Extrapolating AUC to Infinity

Area under the curve or AUC is a pharmacokinetic statistic used to describe the total exposure to a drug. More specifically, it is the time-averaged concentration of drug circulating in the body fluid analyzed (normally plasma, blood or serum). Standard calculation of AUC involves using non-compartmental techniques to calculate the AUC from time 0 to the last measurable concentration. This is called AUC0-t and represents the observed exposure to a drug. But what happens after the last measurable concentration? How much drug is “left” in the body? And what happens to it?

The total AUC or AUC0-∞ is the area under the curve from time 0 extrapolated to infinite time. This parameter is calculated using the following equation:

$AUC_{0-infty}=AUC_{o-t}+AUC_{t-infty}$

I described previously how to calculate AUC0-t. Thus we only have to calculate AUCt-∞ to complete the equation listed above. The extrapolation of AUC to infinity requires several assumptions. The first assumption is that after the last measurable concentration, drug declines in mono-exponential fashion. The second assumption is that the elimination rate constant of that mono-exponential decline is accurately estimated by the terminal elimination rate constant observed with the measurable data. The final assumption is that no other processes besides elimination are involved. In general, these assumptions are valid. At low concentrations, drug usually declines in mono-exponential fashion. And the terminal elimination rate constant does not change over time or with different concentrations of circulating drug. And finally, other processes such as absorption and distribution do not play a significant role in the terminal phase of the pharmacokinetic profile.

If those assumptions are valid, we can then treat the extrapolated portion of the AUC similar to an IV bolus dose. If an IV bolus dose is administered, and the drug follows mono-exponential decline, the AUC can be calculated as the following:

$AUC_{IV bolus} = frac{C_0}{k_{el}}$

Using this same equation, we can substitute the numbers we have and get the following:

$AUC_{t-infty} = frac{C_{last}}{k_{el}}$

This extrapolated AUC is then added to the observed AUC to give a value for total AUC. Another useful metric is to calculate the fraction of the total AUC that is due to the extrapolated AUC. This can be calculated using the following equation:

$% extrapolated = frac{AUC_{t-infty}}{AUC_{0-infty}}$

If the % extrapolated is greater than 20%, than the total AUC may be unreliable. The unreliability of the data is not due to a calculation error. Instead it indicates that more sampling is needed for an accurate estimate of the elimination rate constant and the observed area under the curve.

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.