# AUC integration methods

Source:`vignettes/v23-auc-integration-methods.Rmd`

`v23-auc-integration-methods.Rmd`

## 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 AUC_{tau} and AUC_{int}) 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
- T
_{last}: The last concentration above the limit of quantification within a profile - T
_{max}: The time of the maximum concentration

## Description of methods of integrating between two concentrations
before T_{last}

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 T_{last}:

- If concentrations are both zero, interpolate as zero;
- If concentrations are decreasing and the second concentration is not zero, use logarithmic interpolation; and
- If concentrations are decreasing before T
_{last}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 T_{last}:

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

### Linear to T_{max}/logarithmic after T_{max}
(`"lin-log"`

) interpolation

Linear to T_{max}/logarithmic after T_{max}
interpolation is infrequently used. It uses the following rules in order
for each pair of concentrations through T_{last}:

- If concentrations are both zero, interpolate as zero;
- If concentrations are before T
_{max}, use linear interpolation; - If concentrations are after T
_{max}(and before T_{last}) and either concentration is zero, use linear interpolation; and - If concentrations are after T
_{max}and neither is zero, use logarithmic interpolation.

## Description of methods of integrating between two concentrations
after T_{last}

###
`"AUClast"`

extrapolation

AUClast extrapolation after T_{last} is the simplest. It is
no extrapolation; the extrapolated AUC integral is zero.

###
`"AUCall"`

extrapolation

AUCall extrapolation after T_{last} has two rules:

- If the last concentration measured is above the limit of
quantification (in other words, the last time is T
_{last}), then no extrapolation is done; otherwise - Integrate linearly the triangle between T
_{last}and the time of zero concentration after T_{last}.