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# Nonlinear PK: What Does That Mean?

You may come across a phrase like the following and wonder what it means: “… this drug exhibits nonlinear pharmacokinetics …”. An example of a drug that has nonlinear pharmacokinetics (PK) is erythropoietin or EPO. You may have heard about EPO in the context of sports because it is a performance enhancing drug (PED). EPO regulates red blood cell production, and it is used therapeutically to stimulate bone marrow to produce more red blood cells. Patient who undergo chemotherapy often receive EPO to help boost red blood cell production to replace the cells lost during the chemotherapy regimen. Athletes use EPO to gain a competitive advantage by increasing the oxygen transport capacity in their body and thus increasing endurance. EPO is a common PED for cyclists, runners, and other endurance sports.

But the use of EPO has nothing to do with the nonlinear PK that it exhibits when administered to humans. Nonlinear PK means that increases in drug exposure are not linearly related to increases in administered doses. For a drug with linear PK, we would expect that a 2-fold increase in dose would result in a 2-fold increase in drug exposure. When the dose of EPO is doubled from 1 μg/kg to 2 μg/kg, the exposure increases more than 2-fold. The reason for the non-linearity is related to the clearance of the drug from the body. The following equation shows the relationship between Dose, AUC (exposure) and CL (clearance):

$AUC=\frac{1}{CL} * Dose$

If clearance is constant, then increases in Dose are directly proportional to increases in AUC. This is why the term “constant clearance” is often substituted for the term “linear PK”. Both describe the same set of conditions. If clearance is not changing, then exposure increases linearly with Dose. Nonlinear PK occurs when clearance is not constant (i.e. clearance changes with dose). Most often, the non-linearity is due to a saturation of a clearance mechanism. This can happen if the mechanism of clearance is dependent upon an enzyme system with fixed capacity. Once the amount of drug exceeds the capacity of the enzyme to metabolize it, you begin to see non-linearity.

One common way to represent this phenomenon is to use a Michaelis-Menton kinetic model, like the one shown below:

$CL=\frac{V_{max} * C_{drug}}{K_M + C_{drug}}$

Clearance is a function of the concentration of drug which changes over time. Vmax represents the maximum elimination rate of the system. KM is the drug concentration that produces 50% of the maximal elimination rate of the system. Cdrug is the concentration of the drug at a specific point in time (note that it changes over time as drug is eliminated). At very low concentrations when Cdrug is much less than KM, the clearance can be estimated as follows:

$CL=\frac{V_{max} * C_{drug}}{K_M} = \frac{V_{max}}{K_M} * C_{drug}$

Thus, at concentrations below KM clearance increases linearly with concentration. Then at very high concentrations when Cdrug is much greater than KM, the clearance can be estimated as follows:

$CL=\frac{V_{max} * C_{drug}}{C_{drug}} = V_{max}$

At high concentrations, the capacity of the metabolizing system becomes saturated and no metabolism beyond Vmax can occur. It is important to appreciate that all drugs exhibit nonlinear PK at high doses. Some drugs exhibit nonlinear PK in the therapeutic range used to treat patients, while other drugs do not exhibit non-linearity until doses exceed the therapeutic window by several orders of magnitude.

When you hear the term nonlinear PK in the future, you can just say to yourself “aha, the drug exhibits saturable clearance at high doses.”You can also be certain that at high doses, exposure increases faster than dose. Nonlinear PK is an interesting area of research that is easily understood once you break it down into the component parts.

Read this case study to learn Certara scientists used Phoenix NLME to build a physiologic PK model to define the relationship between systemic and hepatic exposure of an orphan drug in patients with and without liver impairment.

### 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.