AUC integration methods"

In PKNCA: Perform Pharmacokinetic Non-Compartmental Analysis

knitr::opts_chunk$set(
  collapse = TRUE,
  warning = FALSE,
  comment = "#>"
)

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

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
      )
  )

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)
      )
    }
  }
}

PKNCA documentation built on June 22, 2024, 9:25 a.m.