# AUC integration methods

# Integration methods for Area Under the Concentration-Time curve (AUC)

There are 3 methods for choosing how to integrate the AUC between two concentration measurements in `PKNCA`. They are lin up/log down, linear and lin-log. Additionally, there are 3 methods for extrapolating after the last concentration above the limit of quantification. They are AUCinf, AUClast and AUCall. Other methods of calculating AUC (such as AUCtau and AUCint) are made with variants of these.

# Definitions and abbreviations

- AUC: Area under the concentration-time curve
- BLQ: Below the lower limit of quantification
- LLOQ: lower limit of quantification
- NCA: Noncompartmental analysis
- Profile: A set of concentration-time points for calculation
- Tlast: The last concentration above the limit of quantification within a profile
- Tmax: The time of the maximum concentration

# Description of methods of integrating between two concentrations before Tlast

Note that other NCA tools may not describe interpolation as zero. The zero-interpolation rules are used by PKNCA to assist with other methods used across the suite of tools for interpolation and data cleaning within PKNCA. The zero-interpolation rules could be swapped for linear trapezoidal rules with the same effects here.

## Linear up/logarithmic down (`"lin up/log down"`) interpolation

Linear up/logarithmic down interpolation is the most commonly used method for PK, and it is the default for `PKNCA`.

Linear up/logarithmic down interpolation is often used when an exogenous substance is dosed and measured, and when the elimination likely occurs by first-order elimination from the body.

Linear up/logarithmic down interpolation uses the following rules in order for each pair of concentrations through Tlast:

1. If concentrations are both zero, interpolate as zero;
2. If concentrations are decreasing and the second concentration is not zero, use logarithmic interpolation; and
3. If concentrations are decreasing before Tlast or increasing ever, use linear interpolation.

## Linear trapezoidal (`"linear"`) interpolation

Linear trapezoidal interpolation is often used when an endogenous substance is measured (and possibly dosed), and when the elimination may not occur by first-order elimination processes.

Linear trapezoidal interpolation uses the following rules in order for each pair of concentrations through Tlast:

1. If concentrations are both zero, interpolate as zero; and
2. Use linear interpolation for all other times (this could be the only rule).

## Linear to Tmax/logarithmic after Tmax (`"lin-log"`) interpolation

Linear to Tmax/logarithmic after Tmax interpolation is infrequently used. It uses the following rules in order for each pair of concentrations through Tlast:

1. If concentrations are both zero, interpolate as zero;
2. If concentrations are before Tmax, use linear interpolation;
3. If concentrations are after Tmax (and before Tlast) and either concentration is zero, use linear interpolation; and
4. If concentrations are after Tmax and neither is zero, use logarithmic interpolation.

# Description of methods of integrating between two concentrations after Tlast

## `"AUClast"` extrapolation

AUClast extrapolation after Tlast is the simplest. It is no extrapolation; the extrapolated AUC integral is zero.

## `"AUCall"` extrapolation

AUCall extrapolation after Tlast has two rules:

1. If the last concentration measured is above the limit of quantification (in other words, the last time is Tlast), then no extrapolation is done; otherwise
2. Integrate linearly the triangle between Tlast and the time of zero concentration after Tlast.

## `"AUCinf"` extrapolation

AUCinf extrapolation requires estimation of a half-life. It extrapolates using the equation

\(Extrap = \frac{C_{last}}{\lambda_z}\)

# Examples