When performing non-compartmental analysis, the area under the concentration-time curve (AUC) is calculated to determine the total drug exposure over a period of time. Together with C_{max}, these two parameters are often used to define the systemic exposure of a drug for comparison purposes. For example, in bioequivalence trials, the entire statistical analysis is based on the comparison between formulations of AUC and C_{max}. While the mathematics involved in the calculation of AUC are simple, there are nuances to the methods that are often misunderstood. Hopefully I can review some of the key details here. You can also view my video on YouTube.

Although AUC can be calculated directly from primary PK parameters (CL and V), I will discuss only the numerical estimation of AUC using non-compartmental analysis techniques in this blog post.

### Linear Trapezoidal Method

The linear trapezoidal method uses linear interpolation between data points to calculate the AUC. This method is required by the OGD and FDA, and is the standard for bioequivalence trials. For a given time interval (t_{1} – t_{2}), the AUC can be calculated as follows:

In essence the first two terms calculate the average concentration over the time interval. The last piece (t_{1} – t_{2}) is the duration of time. So the linear method takes the average concentration (using linear methods) and applies it to the entire time interval. When you sum all of the intervals together, you will arrive at the total exposure from the first time point to the last. If you then divide the total AUC by the total time elapsed, you will arrive at the “average” concentration of drug in the body over the total time interval.

### Logarithmic Trapezoidal Method

The logarithmic trapezoidal method uses logarithmic interpolation between data points to calculate the AUC. This method is more accurate when concentrations are decreasing because drug elimination is exponential (which makes it linear on a logarithmic scale). For a given time interval (t_{1} – t_{2}), the AUC can be calculated as follows:

This method assumes that C_{1} > C_{2}. The fraction represents the logarithmic average of the two concentrations. Just as with the linear method, the average concentration is multiplied by the time interval.

### Linear-Log Trapezoidal Method

This is a combination of the first two methods and is also called “linear-up log-down”. When concentrations are increasing (as in the absorption phase), the linear trapezoidal method is used. When concentrations are decreasing (as in the elimination phase), the logarithmic trapezoidal method is used. This method is thought to be the most “accurate” because the linear method is the best approximation of drug absorption while logarithmic decline is best modeled by the logarithmic trapezoidal method during drug elimination.

### Why are there different methods?

The following figure demonstrates how the linear trapezoidal method overestimates the AUC during the elimination phase. The blue line represents mono-exponential decline of a drug. Samples were drawn at 16 and 20 hours. The red line represents the linear trapezoidal methods estimation of drug concentrations. As you can plainly see, the red line is higher than the blue line suggesting overestimation by the linear trapezoidal method.

The logarithmic trapezoidal method accurately estimates mono-exponential decline of drug concentrations. However, during an absorption phase, the logarithmic trapezoidal method can underestimate the exposure.

I hope you have a better understanding of how to calculate AUC using the different methods that are available. And I hope you understand the basis of these methods and the pitfalls and limitations of each.

Today’s pharmacokineticists and PK/PD modelers are under more pressure than ever to quickly and accurately characterize the safety and efficacy profiles of investigational drugs. They need the right tools to perform non-compartmental analysis (NCA), build pharmacometric models, and generate reports that communicate their findings.

Phoenix 7.0’s new features and enhancements are the direct result of user feedback we received to make the world’s most advanced PK/PD software package even better.

Watch this webinar to learn how Phoenix 7.0 helps you handle bigger datasets, perform lightning-fast NCA, and make gorgeous plots.

For IV bolus, what is the preferable calculation method? Since distribution phase typically fast, so the two methods would be similar? Thanks.

Qin,

I would recommend the log-linear method for IV bolus. This is because the concentrations follow a log-linear decline after administration.

Nathan

Do you know why there are Phoenix 64 NCA has 4 Calculation methods for non-sparse but only 2 methods for sparse sampling? For non-sparse there is both options Linear Log Trap and Linear Up Log Down? Are these in fact the same as you stated above?

The other 2 methods listed are for available for both sparse and non-sparse:

1. linear trap linear interp (is this what you refer to as linear trap?);

2. linear trap lin/log interp (is this fit with any of the three methods you described?)

Thanks, Sam

When sparse sampling is selected, only the linear trapezoidal methods are available (logarithmic methods are not available).

The definitions of each method are shown in the user guide for Phoenix WinNonlin on page 43 of the pdf document. You can locate the document by going to the Help menu of the software.

Dear all ! can anybody explain reasons for nonlinear PK?

Nonlinear PK occurs when the clearance is dependent upon the concentration of drug circulating in the bloodstream. This occurs when you have saturable clearance processes (usually enzyme-mediated).

Dear All,

Can anybody help me? For calculating Incremental AUC which Trapezoidal Rule should I apply (I mean Linear or Log )?

Thanks in advance.

What is the difference between the two approaches (Linear vs Log-lin) when the last sampled time point is BLQ? And in particular if we assume that BLQ can be replaced with “zero.”

Thanks,

M.

The log approach cannot be used with imputed concentrations of “zero” because ln(0) is undefined. That is why an imputation of zero for BLQs requires the use of the linear trapezoidal rule.

I have a question about Linear-Log Trapezoidal Method that I need clarity on:

You have described above that Linear-Log Trapezoidal Method as a combination of the first two methods (linear trapezoidal and log trapezoidal) and is also called “linear-up log-down”. Why then in Phoenix is there the “Linear log trapezoidal” and “Linear up log down” options if they are both the same?

Good question Carol. The “linear log” method uses the linear trapezoidal method up to Cmax and then uses the log trapezoidal method from Cmax to the last observation. The “linear up log down” method uses the linear trapezoidal method whenever the concentrations are increasing in an interval and the log trapezoidal method whenever the concentrations are decreasing in an interval. Thus, in most cases, they are identical, but if the concentration-time profile is a bit bumpy, they will be slightly different.

In pharmacokinetic study, all variables (Cmax, AUC) are calculated as mean with SD, in case of Tmax why median is calculated.

I have a question regarding sparse data. I am doing NCA using WinNonlin. Here, I have 5 replicates at each time point. Right now, WinNonlin gives an error when I put concentration of all 5 replicates at each time point (error 14062). It calculates the mean only after I select “Sparse” option.

However, as per my knowledge “Sparse” option is selected when there are very few replicates at each time point. Can I calculate the mean for all replicates at each time point without selecting “Sparse” option?

WinNonlin expects to see one sample at each time point when using the standard mode. Thus you get an error if you have 5 replicates. When the sparse option is used, WinNonlin calculates the mean concentration at each nominal timepoint (regardless of patient ID) and calculates the NCA parameters for a single mean profile. If you have replicates for each individual, you should calculate the mean concentration for each nominal time point yourself (using the summary statistic tool) and then feed the mean values into an NCA object.