Non-Compartmental Analysis (NCA): The Basics of PK/PD Statistics, Part 1

Crucial in understanding the safety and efficacy of a drug are how it percolates through the body, or pharmacokinetics (PK), and how the body responds to its presence, or pharmacodynamics (PD). As with any aspect of drug development, testing PK and PD requires collecting data, and, as with all data, interpreting that data requires statistics. Of course, that’s where we come in! We’re going to take a look at the statistics behind PK/PD analysis – with a particular focus on the non-compartmental analysis, or NCA.

First, let’s examine how drugs typically behave in the human body, so we can understand what we need to study. As we mentioned, PK is all about how the drug moves through the body, from being introduced to being excreted.

Our goal is to understand how the concentration of the drug at its site of action changes. It is, however, often prohibitively invasive to measure concentrations at a site of action – such as the brain or heart, for example – so many PK studies focus on the concentration of the drug in the blood (properly the blood plasma, but the terms are used interchangeably) as a proxy. This is typically measured by taking blood samples from subjects at regular intervals following administration, and measuring the concentration of the drug in the sample.

We often break the movement of a drug in the body into four phases, denoted using the initialism ADME:

Absorption: This is the phase in which the drug enters the bloodstream. The speed at which this happens depends on the method of administration: if the drug is given as an IV bolus (drug injected directly into the bloodstream), the entire dose is absorbed instantly. At the other end of the scale, if the drug is given orally or topically (i.e. in a cream), then the dose of the drug can take from minutes to hours to be fully absorbed.

Distribution  After the drug enters the bloodstream, it is distributed through the body, moving from the blood into other tissues and organs until, eventually reaching the target site. For example, the paracetamol from an oral headache tablet is absorbed through the small intestine, and must then be distributed to the central nervous system and brain to have its pain-killing effect.

Metabolism & Excretion (together, Elimination): This phase encapsulates how the drug leaves the bloodstream by two main mechanisms. The first is by metabolism, where the molecular structure of the drug is broken down by mechanisms in the body. The second is by excretion, where the drug is filtered out by, for example, the kidneys so it can leave via the urinary system.

Metabolism and Excretion are typically considered together as the elimination phase since we often aren’t worried about how the drug leaves the body, only how quickly it does.

Note that, while we refer to these as phases, they do not happen neatly one after another. The phases overlap and often occur simultaneously – some of the dose will be metabolised as soon as it enters the body, for example. The phases, instead, refer to when each of these processes is dominant: changes in drug concentration soon after administration are mainly due to absorption, while those some time after administration are mainly due to elimination.

So, the essence of PK is to understand how the concentration of the drug in the body changes over time. The main goal of a PK study is, therefore, to determine the concentration-time curve of the drug, which, as the name suggests, models the concentration of the drug in the blood over time. There are several PK parameters this helps us establish:

C_max: This is the maximum concentration of the drug observed after administration – the peak of the concentration-time curve. This marks the end of the absorption phase, and is particularly of note when considering the safety of the drug: it is important to check whether the maximum concentration of the drug exceeds a level which could be toxic.

T_max: The time after administration at which C_max is observed, and, therefore, the time at which the absorption phase ends. This duration will depend strongly on the administration method: T_max for an IV bolus will be 0, since the maximal blood concentration occurs immediately, while T_max for an oral dose will be much later.

Area Under the Curve (AUC): This is typically measured between two time points, e.g. AUC_0-t where t is an arbitrary time point, and is, as the name suggests, the area under the concentration-time curve between those time points. This represents the exposure to the drug over that time, meaning AUC_0-∞ represents the total exposure to the drug. This is also important when considering the safety of the drug, since we want to ensure that the total exposure isn’t toxic.

Half-life (T_1/2): The half-life of a drug characterises the elimination phase, and gives the time for the blood plasma concentration of the drug to decrease by half.

These properties are easy to determine from the concentration-time curve, and can be used to characterise the PK properties of the drug. Many regulators, however, call for properties which require a bit more interpretation. Specifically, they also require:

Volume of Distribution (V_d): This is the total volume into which the drug distributes. If the V_d is low, most of the drug remains in the blood plasma rather than distributing into body tissue.

Clearance (C_L): This is another way of characterising the elimination phase. While the half-life focuses on the concentration of the drug, the clearance is instead the volume of plasma which is cleared of the drug per unit time.

So, we know what metrics we need, and how to get at them – the concentration-time curve. How do we go from a series of blood concentration data points to determining what we want to find out?

One method is the non-compartmental analysis (NCA). This is a simple approach, but it can still provide useful results.

An NCA doesn’t model the concentration-time curve. Instead, the data collected in the study is used directly to calculate the PK parameters. For example, C_max is given simply by the maximum concentration observed in any of the collected data points, with T_max given by the associated time. This can prove problematic, however, if the measurements around T_max aren’t dense enough as the true C_max can easily be missed.

This is more likely to be an issue when the drug is quickly absorbed. If a drug is fully absorbed one minute after an intra-muscular injection, for example, it might be difficult to take blood samples quickly enough to fully characterise the concentration-time curve around T_max.

To find the AUC, we assume that we can linearly interpolate between the datapoints. This gives us a series of trapezia, the area of which can be added up to give us the AUC. This also relies on the density of the data – the linear interpolation will not be a good approximation to the true concentration-time curve if the data is too sparse.

This is one of the disadvantages of an NCA: it requires a high data density to give accurate results. It is also not a predictive methodology. Since we don’t fit a model to the data, we have no way of predicting if our results would change if the initial conditions were different.

For example, if we were to, say, halve the administered dose, would T_max change? Using an NCA, we would have to collect a fresh batch of data using that halved dose to answer the question.

On the other hand, the big benefit of an NCA is that it’s quick and easy to implement. Since we’re essentially just reading numbers off a graph, we don’t need to worry about solving complex differential equations, as in methods we’ll talk about in the next part. An NCA can also often provide most PK information regulators look for. This means that NCA is a good option for the early stages of development when large time and resource investments in analyses are less desirable.

So, an NCA can be useful – particularly thanks to its simplicity – but it also has drawbacks which can limit its utility for later stages of drug development. In the next part of this series, we’ll take a look at a more sophisticated method of modelling PK/PD data: the compartmental analysis.

To find out more about how Quantics can support your PK/PD analysis, visit https://www.quantics.co.uk/biostatistics-services/pk-pd-statistics/