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# The Superposition Principle

The superposition principle has nothing to do with a super-hero; however, you might be perceived as a hero if you can explain the principle to others. The superposition principle is a mathematical concept that helps us analyze concentration-time data. While it may seem complicated, it is actually nothing more than addition!

The superposition principle states that under linear conditions (ie, constant clearance) the total concentration of drug in the body is the sum of the remaining concentrations from each administered dose at that point in time when a measurement is made.

The superposition principle assumes that subsequent dosing events will not be impeded or affected by drug that is already circulating in the blood stream. From a pharmacokinetic point of view, the drug in the body doesn’t know anything about the drug that is outside of the body waiting to be absorbed. Therefore, each dose can be considered as an independent event, and the sum of all these dosing events provides the aggregate concentration of drug in circulation.

For example, let’s assume we dose a heparin every 8 hours for thromboembolism prophylaxis. Imagine the doses are administered at 6 am (Dose 1), 2 pm (Dose 2), and 10 pm (Dose 3). If we want to determine the concentration of heparin at 6 pm, we could use the superposition principle in the following way: $C_{total}=C_{\text{Dose 1}}(\text{t=12 h}) + C_{\text{Dose 2}}(\text{t=4 h}) + C_{\text{Dose 3}}(\text{t=0 h})$ $C_{\text{Dose 1}}(\text{t=12 h})=\frac{Dose}{V}*e^{-k*\text{12 h}}$ $C_{\text{Dose 2}}(\text{t=4 h})=\frac{Dose}{V}*e^{-k*\text{4 h}}$ $C_{\text{Dose 3}}(\text{t=0 h})=0$

Using this logic, the concentration-time curve of any dosing regimen can be generated simply by calculating the Ctotal using the concentration from each individual dosing event. You can even generalize this equation for a 1-compartment system with intravenous administration as the following: $Cn(t^{\prime})=C_1(t)+C_1(t)*[\frac{1-e^{-(n-1)\beta \tau}}{1-e^{-\beta \tau}}]*e^{-\beta t^{\prime}}$

where t’ = time after administration of the nth dose, t = time after administration of the first dose, n = the number of doses, β = elimination rate constant, and τ = the dosage interval.

Now you can be the super-hero and explain the superposition principle to your colleagues.

Today, drug development is carried out in human subjects and animals. However, as computing power and the number of sophisticated technology platforms grow exponentially, and our knowledge of human health and disease increases, the virtualization of clinical research and development will grow steadily. Read this article to learn more.

### 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.