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# What is a Half-life?

One of the most incorrectly used but most often quoted parameters is half life. The symbol for half-life is t1/2. The reason for the multitude of errors with half-life is that in most cases, the assumptions required to use half-life correctly are rarely valid. And when the assumptions are false, the derived interpretations are also incorrect. Here are some common statements that use half-life:

• If you double the dose, you will increase the duration of effect by 1 half-life.
• Because the drug has a long half-life, you don’t need to dose very frequently.
• Half-life is a primary pharmacokinetic parameter.

The first two statements are true under specific conditions. The third statement is never true! To understand half-life, and what it means, we need to talk about where it comes from. If you remember the discussion on primary pharmacokinetic parameters, these parameters depend on the physiology of the body and the physiochemical properties of the drug. These primary parameters are volume of distribution, clearance, bioavailability, and absorption rate. By combining these primary parameters, we can generate secondary pharmacokinetic parameters. One of the secondary pharmacokinetic parameters is the elimination rate (kel), which is calculated as:
$k_{el}=\frac{CL}{V}$
The rate at which drug is eliminated from the body is in the units of inverse time (e.g. 1/h). To understand how this relates to half-life requires a bit more math.
$C(t)=\frac{Dose}{V}*e^{\frac{-CL}{V}*t}$
substituting $k_{el}$for $\frac{CL}{V}$gives
$C(t)=\frac{Dose}{V}*e^{-k_{el}*t}$
At time = 0, $C(0) = \frac{Dose}{V}$.
Using this equation, we can determine when half of the drug has left the body, which is also known as the half-life of the drug. To do this we are looking for a time = t1/2 in which
$C(t_{1/2})=\frac{1}{2}*\frac{Dose}{V}$
$C(t_{1/2})=\frac{1}{2}*\frac{Dose}{V}=\frac{Dose}{V}*e^{-k_{el}*t_{1/2}}$
$frac{1}{2}=e^{-k_{el}*t_{1/2}}$
$ln(\frac{1}{2})=-k_{el}*t_{1/2}$
$ln(1)-ln(2)=-k_{el}*t_{1/2}$
$-ln(1)+ln(2)=k_{el}*t_{1/2}$
$ln(2)-ln(1)=k_{el}*t_{1/2}$
$ln(2)=k_{el}*t_{1/2}$
$t_{1/2}=\frac{ln(2)}{k_{el}}$

This final equation shows the relationship between t1/2 and the elimination rate constant. The definition of half-life (t1/2) is the time required for the concentration to fall to 50% of its current value. This was derived by assuming a 1-compartment model and linear elimination. Therefore, the estimation of half-life by using kel is only valid when the drug follows first order elimination from a single compartment. As you can see, half-life is clearly not a primary pharmacokinetic parameter. And the only way to derive it is to assume a specific compartmental model with first-order elimination.

But, if these conditions are valid, then the half-life parameter is useful to determine how the time the drug will be present in the body. Using the following table, you can see how much drug will be left after each half-life.

Half-life % Drug Remaining % Drug Eliminated
1 50% 50%
2 25% 75%
3 12.5% 87.5%
4 6.25% 93.75%
5 3.125% 96.875%

Although half-life can make your life simple as a pharmacokineticist, you must remember that these assumptions are only valid when the drug follows a single compartment model with first-order elimination. If these assumptions are not true, then the interpretations derived from the half-life parameter (like % drug remaining) will be inaccurate.

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.

### 筆者について

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.