AUC Calculation with PKNCA

Area Under the Curve (AUC) is a commonly-used metric for assessing exposure to a drug. Many variants of AUC exist, and the information below will assist in determining both the appropriate AUC and how to calculate it.

Preparation

For the below examples, the following data will be used.

suppressPackageStartupMessages({
  library(PKNCA)
  library(dplyr)
  library(cowplot)
  library(knitr)
  library(ggplot2)
})
scale_colour_discrete <- scale_colour_hue
scale_fill_discrete <- scale_fill_hue

my_conc <- data.frame(conc=c(0, 2.5, 3, 2, 1.5, 1.2, 1.1, 0, 0),
                      time=c(0:5, 8, 12, 24),
                      subject=1)
my_conc$BLQ <- my_conc$conc == 0
my_conc$measured <- TRUE
ggplot(my_conc,
       aes(x=time,
           y=conc,
           shape=BLQ,
           group=subject)) +
  geom_line() +
  geom_point(size=4) +
  scale_x_continuous(breaks=my_conc$time) +
  theme(legend.position=c(0.8, 0.8))

For the purpose of illustration, NCA parameters will also be calculated for each of the AUC types below. Note that in the results, more parameters are returned than were requested. The additional parameters are the set of parameters required to calculate the requested parameters.

conc_obj <- PKNCAconc(my_conc, conc~time|subject)
data_obj <- PKNCAdata(data.conc=conc_obj,
                      intervals=data.frame(start=0,
                                           end=24,
                                           aucall=TRUE,
                                           auclast=TRUE,
                                           aucinf.pred=TRUE,
                                           aucinf.obs=TRUE))
results_obj <- pk.nca(data_obj)
kable(as.data.frame(results_obj))
subject start end PPTESTCD PPORRES PPANMETH exclude
1 0 24 auclast 12.9965842 AUC: lin up/log down NA
1 0 24 aucall 15.1965842 AUC: lin up/log down NA
1 0 24 tmax 2.0000000 NA
1 0 24 tlast 8.0000000 NA
1 0 24 clast.obs 1.1000000 NA
1 0 24 lambda.z 0.1075592 NA
1 0 24 r.squared 0.7580245 NA
1 0 24 adj.r.squared 0.6370368 NA
1 0 24 lambda.z.corrxy -0.8706460 NA
1 0 24 lambda.z.time.first 3.0000000 NA
1 0 24 lambda.z.time.last 8.0000000 NA
1 0 24 lambda.z.n.points 4.0000000 NA
1 0 24 clast.pred 1.0216136 NA
1 0 24 half.life 6.4443313 NA
1 0 24 span.ratio 0.7758757 NA
1 0 24 aucinf.obs 23.2235095 AUC: lin up/log down NA
1 0 24 aucinf.pred 22.4947355 AUC: lin up/log down NA

AUC to the Last Value Above the Limit of Quantification (AUClast)

AUC0-last calculates the AUC from time 0 to the last value above the limit of quantification, tlast (within PKNCA, this is the last value above 0). In the figure below, AUC0-last integrates the shaded region. Integration after tlast is 0.

tlast <- pk.calc.tlast(conc=my_conc$conc,
                       time=my_conc$time)
tlast
## [1] 8
my_conc$include_auclast <- my_conc$time <= tlast
ggplot(my_conc,
       aes(x=time,
           y=conc,
           shape=BLQ,
           group=subject)) +
  geom_ribbon(data=my_conc[my_conc$include_auclast,],
              aes(ymin=0, ymax=conc),
              fill="lightblue") +
  geom_line() +
  geom_point(size=4) +
  scale_x_continuous(breaks=my_conc$time) +
  theme(legend.position=c(0.8, 0.8))

AUCall

AUCall starts with AUC0-last and then integrates from tlast to the first point after tlast with a linear interpolation to zero. From the second point after tlast to \(\infty\) is considered zero.

first_after_tlast <- my_conc$time[my_conc$time > tlast][1]
first_after_tlast
## [1] 12
my_conc$include_aucall <- my_conc$time <= first_after_tlast
ggplot(my_conc,
       aes(x=time,
           y=conc,
           shape=BLQ,
           group=subject)) +
  geom_ribbon(data=my_conc[my_conc$include_aucall,],
              aes(ymin=0, ymax=conc),
              fill="lightblue") +
  geom_line() +
  geom_point(size=4) +
  scale_x_continuous(breaks=my_conc$time) +
  theme(legend.position=c(0.8, 0.8))

AUC to Infinity (AUC\(\infty\))

AUC0-\(\infty\) is commonly used for single-dose data. It calculates the AUC0-last and then extrapolates to \(\infty\) using the estimated half-life. Two starting points are used to estimate from tlast to \(\infty\), the observed or half-life predicted concentration at tlast (clast.obs and clast.pred).

The two figures below illustrate the integration with AUC0-\(\infty\),obs and AUC0-\(\infty\),pred. The difference between the two figures is most evident at time=8 where there is a discontinuity in integration at tlast due to using clast.pred after that point and clast.obs before that point. (To illustrate the integration differences, BLQ indicator shapes have been removed. BLQ is handled identically to previous figures.)

# Add one row to illustrate extrapolation beyond observed data
my_conc <-
  rbind(my_conc,
        data.frame(conc=NA,
                   time=36,
                   subject=1,
                   BLQ=NA,
                   measured=FALSE,
                   include_auclast=FALSE,
                   include_aucall=FALSE))
# Extrapolate concentrations for aucinf.obs
my_conc$conc_aucinf.obs <- my_conc$conc
my_conc$conc_aucinf.obs[my_conc$BLQ | is.na(my_conc$BLQ)] <-
  interp.extrap.conc(conc=my_conc$conc,
                     time=my_conc$time,
                     time.out=my_conc$time[my_conc$BLQ | is.na(my_conc$BLQ)],
                     lambda.z=as.data.frame(results_obj)$PPORRES[as.data.frame(results_obj)$PPTESTCD %in% "lambda.z"])

# Extrapolate concentrations for aucinf.pred
my_conc$conc_aucinf.pred <- my_conc$conc
my_conc$conc_aucinf.pred[my_conc$BLQ | is.na(my_conc$BLQ)] <-
  interp.extrap.conc(conc=my_conc$conc,
                     time=my_conc$time,
                     time.out=my_conc$time[my_conc$BLQ | is.na(my_conc$BLQ)],
                     lambda.z=as.data.frame(results_obj)$PPORRES[as.data.frame(results_obj)$PPTESTCD %in% "lambda.z"],
                     clast=as.data.frame(results_obj)$PPORRES[as.data.frame(results_obj)$PPTESTCD %in% "clast.pred"])
my_conc$conc_aucinf.pred[my_conc$time == tlast] <-
  as.data.frame(results_obj)$PPORRES[as.data.frame(results_obj)$PPTESTCD %in% "clast.pred"]
ggplot(my_conc[!is.na(my_conc$conc),],
       aes(x=time,
           y=conc,
           #shape=BLQ,
           group=subject)) +
  geom_ribbon(data=my_conc,
              aes(ymin=0, ymax=conc_aucinf.obs),
              fill="lightblue") +
  geom_line() +
  #geom_point(size=2) +
  scale_x_continuous(breaks=my_conc$time) +
  theme(legend.position=c(0.8, 0.8)) +
  labs(title="Extrapolation using AUCinf,obs")

ggplot(my_conc[!is.na(my_conc$conc),],
       aes(x=time,
           y=conc,
           #shape=BLQ,
           group=subject)) +
  geom_ribbon(
    data=arrange(
      bind_rows(mutate(my_conc,
                       before=FALSE),
                mutate(filter(my_conc, time == tlast),
                       conc_aucinf.pred=conc,
                       before=TRUE)),
      time, desc(before)),
    aes(ymin=0,
        ymax=conc_aucinf.pred),
    fill="lightblue") +
  geom_line() +
  #geom_point(size=2) +
  scale_x_continuous(breaks=my_conc$time) +
  theme(legend.position=c(0.8, 0.8)) +
  labs(title="Extrapolation using AUCinf,pred")

Partial AUCs

Partial AUCs integrate part of the area within a time range of interest. Partial AUCs are often of interest to assess bioequivalence with more detail than AUC0-\(\infty\) or AUC0-last may indicate. Within PKNCA, partial AUCs are calculated with the interval AUC (aucint) family of parameters: aucint.last, aucint.all, aucint.inf.obs, and aucint.inf.pred integrate over the full interval from start to end while handling the region after tlast the same way as auclast, aucall, aucinf.obs, and aucinf.pred, respectively.

Request an aucint parameter like any other interval parameter with the start and end of the interval defining the range of integration. When the interval boundaries do not match observed time points, concentrations at the boundaries are automatically interpolated (or extrapolated, according to the method of the specific parameter), so no manual data preparation is required. In the example below, the concentration at the end time of 1.5 is not in the observed data, and it is interpolated automatically during the calculation.

When dosing data are given, the aucint parameters are dose-aware: the profile being integrated ends at the first dose at or after the end of the interval, so a concentration measured after that dose is never interpolated back into the interval. Only the start and end of the interval are estimated; a dose within the interval is integrated across using the concentrations measured on either side of it. The method (PPANMETH) column reports whether the calculation was dose-aware and how the region after tlast was handled – with the half-life, with AUCall (when aucint.inf.obs or aucint.inf.pred was requested but the half-life is not estimable), or not at all when the interval ends at or before tlast.

data_aucint_obj <-
  PKNCAdata(conc_obj,
            intervals=data.frame(start=0, end=c(2, 1.5), aucint.last=TRUE))
results_aucint_obj <- pk.nca(data_aucint_obj)
kable(as.data.frame(results_aucint_obj))
subject start end PPTESTCD PPORRES PPANMETH exclude
1 0 2.0 aucint.last 4.0000 AUC: lin up/log down. Interpolation: not dose-aware (no dosing data). Extrapolation: none NA
1 0 1.5 aucint.last 2.5625 AUC: lin up/log down. Interpolation: not dose-aware (no dosing data). Extrapolation: none NA

The area under the first moment curve (AUMC) has matching interval parameters (aumcint.last, aumcint.all, and the other aumcint* parameters), as do the intravenous C0 back-extrapolated AUMC parameters (aumciv*), in the development version of PKNCA.