Almost everyone familiar with pharmaceuticals has heard a conversation like this before:

Scientist 1: “What are the pharmacokinetics of Drug X?”

Scientist 2: “Drug X follows a 1-compartment model in rats, but in monkeys it tends to have a distribution phase and seems to follow 2-compartment kinetics.”

Scientist 1: Thinks to himself/herself …’What does a compartment have to do with this! A compartment is something you find in a train!’

Compartments are an important concept in pharmacokinetics (and pharmacodynamics), but they are rarely explained to other scientists. Hopefully this post will demystify the idea of compartments and show you that the concept of compartments is simple.

To understand compartments, think about your heart for a minute. A human heart has 4 distinct chambers, each with a specific function. Blood, which has been depleted of oxygen returns through the veins to the right atrium. It is then transferred to the right ventricle. The right ventricle pumps the blood into the lungs and then the blood moves into the left atrium. Finally the blood moves into the left ventricle which pushes the blood through the arteries of the body to distribute the oxygenated blood to all of the organs and tissues of the body. Each chamber of the heart has a specific function, and there is a specific flow of blood involved. The following schematic depicts the 4 chambers of the heart along with the direction of blood flow.

As you can see, the blood has unidirectional flow from one chamber to the next. In other words, the blood does not move from the right ventricle back into the right atrium (at least it doesn’t happen with a normal, healthy heart!). If this makes sense to you, then you now understand the idea of compartments. In a very real way, the chambers of the heart are separate “compartments” that the blood passes through.

In pharmacokinetics we don’t use tangible “compartments” like the chambers of the heart. Instead we use theoretical, or imaginary “compartments”. If you were to draw a picture of all the organs and tissues of the body, each as a separate compartment, it would look something like this (image from dougneubauer.com):

Even this model is a bit simplistic for the body, are all muscles the same? What is the “Rest” of the body? Clearly, if we tried to identify every single different tissue in the body, we would have infinite “compartments” in our model. Pharmacokineticists like to simplify things significantly. Thus, instead of defining tangible compartments, we design theoretical compartments with *unique* names like 1, 2, 3, central, peripheral, etc. (I hope you noticed the sarcasm!). Then we draw arrows between these compartments to show how the drug travels from one compartment to the other. Here are 2 examples:

#### 1-Compartment Model

- Drug enters the central compartment (or compartment 1) from somewhere outside of the body.
- Drug then leaves the central compartment. This is analogous to the drug leaving the body.
- Drug recirculation does not occur (output line does not reconnect with input line).
- The 1-compartment model considers the entire body, and all of the organs and tissues to be one giant bucket.

#### 2-Compartment Model

- Drug enters the central compartment (or compartment 1) from somewhere outside of the body.
- Drug then leaves the central compartment by one of two paths:
- the peripheral compartment (also called compartment 2) or
- drug leaves the body

- Drug that is in the peripheral compartment can return to the central compartment.
- Drug recirculation occurs between the central and peripheral compartment, but once drug leaves the body, it does not re-enter the body.
- The 2-compartment model considers the entire body, and all of the organs and tissues to be two buckets, but all drug must leave the body through a single bucket.

In many ways the compartmental models are very similar to the heart chamber model. These models show movement from one “chamber” to another. The 2 key differences are that the pharmacokinetic models are not closed systems (drug is not recirculated from output to input); and pharmacokinetic models permit bi-directional movement (the heart chamber model only allows unidirectional movement).

Hopefully you now understand what is meant by compartmental models in pharmacokinetics. In essence, the number (1, 2, 3) refers to the number of circles drawn on the paper. Many may be asking * why* we use compartment models in pharmacokinetics. The brief answer is that the mathematical functions associated with compartment models seem to describe the observed data very well. It is for practical reasons, not physiologic reasons that we use compartmental models. I will leave the detailed explanation for another blog post.

its very useful to learn

Hi Nathan,

I would like to ask you what is the input for any PK/PD modeling. Is it the drug dose-conc-time data or any other data since we are trying to model the bioavailability of the drug in the system.

Anurag,

Input for PK/PD modeling includes concentration-time data and effect-time data (or response-time data). I’m unclear if your comment suggests that you are trying to model how different bioavailability values affect the pharmacologic response, or if you are trying to build a model to determine the bioavailability. If it is the former, you should use a PK/PD model. If it is the latter, then you should use a PK model (PD is not needed to calculate bioavailability).

Nathan

Nathan

what will be initial concentrationof drug ???

Aastha,

I’m not sure how to answer your question. The initial concentration of drug in the body is the amount of drug divided by the volume of distribution. But since the volume of distribution is not known

a priori, then the initial concentration must be extrapolated based on the route of administration and measured concentrations at later time points.Nathan

Hi Nathan,

I am wondering if it can be assumed that distribution, metabolism and excretion are dynamic processes that take place at the same time or if this depends on the model used.

Thanks a lot,

Emma

Emma,

That is exactly what happens. All three processes happen simultaneously and each is dynamic. The complex interplay between these processes is what we study in pharmacokinetics. It’s actually quite amazing that we can use (relatively) simple mathematical models to describe such complex relationships.

Nathan

Why do some drugs show one compartment models, and other drugs show two compartment models?

Rob,

Great question. Compartmental models are simply mathematical constructs that allow us to describe what we observed with respect to drug concentrations. The difference between 1- and 2-compartment models is that in a 2-compartment model, drug elimination appears to follow a 2-phase process. That could be due to distribution or elimination properties, but there will be a fast and a slow elimination process which shows up as a biexponential curve or a 2-compartment model. A 1-compartment model generally only exhibits a single elimination phase, thus it can be modeled with a single exponential curve or a 1-compartment model.

The physiologic reasons for 1- and 2-compartment models vary greatly and are poorly understood. Some people attribute 2-compartment models to tissue distribution, but most of those arguments lack solid data to prove the hypothesis. I generally avoid connecting physiology to specific compartmental models for this reason.

Nathan

Hello!

I am super confused about diff between Pharmacodynamic models and PK/PD modelling. Are they same or different? How is PD model different from dose response curve?

Thanks

Sagar,

Thank you for your message. These terms are often used interchangeably, but can mean different things. A pharmacodynamic model is a mathematical model that describes the time course of drug effects. The driving force can be a drug dose (e.g. dose-response), or it could be drug concentrations (e.g. concentration-response). A PK/PD model incorporates additional pharmacokinetic information such that the driving force is a drug dose which causes changes in drug concentrations which in turn influences drug effects. Thus the PK/PD model handles the translation of a dose to drug concentrations (PK portion of the model) and translation of concentrations to drug effects (PD portion of the model).

Nathan

What is open model and closed model?what is the difference?

Kavya,

I think you might be referring to a “closed-form” solution for a model. A “closed-form” simply means that there is an exact analytical solution for the equation. For example, a one compartment model can be written in the following ways:

Closed form: C = (Dose/V)*e^(-k*t)

Differential equations: dC/dt = (Dose/V)*(-k)

Sometimes you can convert a differential equation into the closed form, and sometimes you cannot. When you cannot convert the differential equation into a closed form, you have to use a numerical integrator to solve the equation.

Hope that helps!

Nathan