---
title: "AUC integration methods"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{AUC integration methods}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
---
```{r knitr-setup, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
warning = FALSE,
comment = "#>"
)
```
```{r setup, include = FALSE}
library(PKNCA)
library(ggplot2)
auc_methods <- c("lin up/log down", "linear", "lin-log")
auc_types <- c("AUCinf", "AUClast", "AUCall")
comma_and <- function(x) {
paste(
paste(x[-length(x)], collapse = ", "),
ngettext(length(x) - 1, ", and", "and"),
x[length(x)]
)
}
```
# Integration methods for Area Under the Concentration-Time curve (AUC)
There are `r length(auc_methods)` methods for choosing how to integrate the AUC
between two concentration measurements in `PKNCA`. They are
`r comma_and(auc_methods)`. Additionally, there are `r length(auc_types)`
methods for extrapolating after the last concentration above the limit of
quantification. They are `r comma_and(auc_types)`. 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~:
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 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~:
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 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~:
1. If concentrations are both zero, interpolate as zero;
2. If concentrations are before T~max~, use linear interpolation;
3. If concentrations are after T~max~ (and before T~last~) and either concentration is zero, use linear interpolation; and
4. 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:
1. 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
2. Integrate linearly the triangle between T~last~ and the time of zero
concentration after T~last~.
## `"AUCinf"` extrapolation
AUCinf extrapolation requires estimation of a half-life. It extrapolates using the equation
$Extrap = \frac{C_{last}}{\lambda_z}$
# Examples
```{r setup-example, include=FALSE}
examples <-
list(
"Clast is above the LLOQ" =
data.frame(
conc = c(0, 1.8, 3, 2, 1, 0.5, 0.25),
time = 0:6
),
"Clast is below the LLOQ" =
data.frame(
conc = c(0, 1.8, 3, 2, 1, 0.5, 0),
time = 0:6
),
"Concentrations increase and decrease after T~max~ with zeros in the middle" =
data.frame(
conc = c(0, 1.8, 0, 0, 3, 2, 2.5, 0, 0, 2.5, 1, 0.5, 0),
time = 0:12
)
)
```
```{r example-figures, fig.cap = caption, echo = FALSE}
caption <- character()
for (current_example_nm in names(examples)) {
for (current_method in auc_methods) {
for (current_type in auc_types) {
caption <-
c(
caption,
sprintf(
"Example PK where %s; %s interpolation method; %s extrapolation method (AUC type); and keeping all BLQ values (not default BLQ handling)",
current_example_nm, current_method, current_type
)
)
current_example <- examples[[current_example_nm]]
# Handle BLQ
current_example_noblq <- clean.conc.blq(conc = current_example$conc, time = current_example$time)
current_example_noblq$BLQ <- "Above LLOQ"
current_example_points <-
merge(current_example, current_example_noblq, all = TRUE)
current_example_points$BLQ[is.na(current_example_points$BLQ)] <- "Below LLOQ"
# Calculate some NCA parameters required for interpolation and extrapolation
tlast <- pk.calc.tlast(conc = current_example$conc, time = current_example$time)
half_life <- pk.calc.half.life(conc = current_example$conc, time = current_example$time)
# Define the methods used for interpolation and extrapolation
current_example$method <-
factor(
PKNCA:::choose_interval_method(
conc = current_example$conc,
time = current_example$time,
tlast = tlast,
method = current_method,
auc.type = current_type,
options = list()
),
levels = c("zero", "linear", "log", "extrap_log")
)
stopifnot(!any(is.na(current_example$method)))
current_example_interpolate <-
data.frame(
time =
sort(c(
# make lines almost directly down at the transition after Tlast to
# zero, where applicable
current_example$time + sqrt(.Machine$double.eps),
seq(min(current_example$time), max(current_example$time) + diff(current_example$time)[1], length.out = 200)
))
)
current_example_interpolate$conc <-
interp.extrap.conc(
conc = current_example$conc,
time = current_example$time,
time.out = current_example_interpolate$time,
lambda.z = half_life$lambda.z,
method = current_method,
auc.type = current_type,
conc.blq = "keep"
)
current_example$time_after <- c(current_example$time[-1], Inf)
if (current_example$method[nrow(current_example)] == "zero") {
current_example_plot <- current_example
} else {
current_example_before_tlast <- current_example[current_example$time < tlast, ]
current_example_after_tlast <- current_example[current_example$time >= tlast, ][1, ]
current_example_after_tlast$method <- current_example$method[nrow(current_example)]
current_example_after_tlast$time_after <- Inf
current_example_plot <-
dplyr::bind_rows(
current_example_before_tlast,
current_example_after_tlast
)
}
print(
ggplot(current_example_points, aes(x = time, y = conc)) +
geom_rect(data = current_example_plot, aes(xmin = time, xmax = time_after, ymin = -Inf, ymax = Inf, fill = method), alpha = 0.2) +
geom_point(aes(shape = BLQ)) +
geom_line(data = current_example_interpolate) +
scale_colour_hue(drop = FALSE)
)
}
}
}
```