## Why are PK Parameters Lognormally Distributed?

The statistical analysis of pharmacokinetic parameters is often overlooked and not always well understood. The disconnect between the pharmacokineticist and the biostatistician can often be a huge stumbling block that prevents the appropriate analysis of PK parameters. While I cannot solve all the disagreements between pharmacokineticists and biostatisticians in a single blog post, I hope to address a single point of contention between the two disciplines.

Most pharmacokineticists will be familiar with the following statement:

PK parameters are lognormally distributed.

A biostatistician will then follow that statement with this question:

Why do you say that? How do you know that the distribution is log-normal?

At this point in the conversation, there is either a long silence, or lots of hand-waving. The disconnect between the two scientists is born.

To understand the reason for the statement about the distribution of PK parameters, we must first understand the difference between a normal and a log-normal distribution.

 Normal Distribution Lognormal Distribution

These two images explain why PK parameters are lognormally distributed. A normal distribution reaches from – infinity to +infinity and is centered on the arithmetic mean value of the population. In contrast, a lognormal distribution reaches from 0 to +infinity and is centered on the geometric mean of the population. The difference between the distributions is the limit on the left side. In one case (lognormal) it is bound by zero. In the other case (normal) it is not bound at all. In real life there are physiologic limits to PK parameters. For example, the volume of distribution cannot be negative, or even zero. Yet it can be infinitely large. Thus the lognormal distribution, with the truncated left side and the long arm to the right is more representative of the expected distribution of the PK parameter volume of distribution than the normal distribution would be. Similar arguments can be made for clearance.

A normal distribution would require that the probability of an estimate is equally likely above or below the mean. As noted above, on the lower end there is a limit of zero. Thus the requirement for equal probability on either side of the mean is not possible for a PK parameter such as volume of distribution.

Thus PK parameters are log-normally distributed because they are limited by physiologic constraints that are not normally distributed. This limitation requires us to use a different set of distributional assumptions. It cannot be proven using a mathematical theorem, but it makes physiologic sense!

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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.