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)
})

## Warning: package 'dplyr' was built under R version 3.4.3

## Warning: package 'cowplot' was built under R version 3.4.3

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$BLOQ <- my_conc$conc == 0
my_conc$measured <- TRUE

ggplot(my_conc,
       aes(x=time,
           y=conc,
           shape=BLOQ,
           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)

## No dose information provided, assuming default dosing information.

kable(results_obj$result)

end	subject	PPTESTCD	PPORRES	exclude
24	1	auclast	12.9965842	NA
24	1	aucall	15.1965842	NA
24	1	tmax	2.0000000	NA
24	1	tlast	8.0000000	NA
24	1	clast.obs	1.1000000	NA
24	1	lambda.z	0.1075592	NA
24	1	r.squared	0.7580245	NA
24	1	adj.r.squared	0.6370368	NA
24	1	lambda.z.time.first	3.0000000	NA
24	1	lambda.z.n.points	4.0000000	NA
24	1	clast.pred	1.0216136	NA
24	1	half.life	6.4443313	NA
24	1	span.ratio	0.7758757	NA
24	1	aucinf.obs	23.2235095	NA
24	1	aucinf.pred	22.4947355	NA

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

AUC_0-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, AUC_0-last integrate 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=BLOQ,
           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))

AUC_all

AUC_all starts with AUC_0-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=BLOQ,
           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_$\infty$)

AUC_0-$\infty$ is commonly used for single-dose data. It calculates the AUC_0-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 AUC_{0-$\infty$,obs} and AUC~0-$\infty$,pred$. The difference between the tow 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, BLOQ indicator shapes have been removed. BLOQ 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,
                   BLOQ=NA,
                   measured=FALSE,
                   include_auclast=FALSE,
                   include_aucall=FALSE))
# Extrapolate concentrations for aucinf.obs
my_conc$conc_aucinf.obs <- my_conc$conc
my_conc$conc_aucinf.obs[my_conc$BLOQ | is.na(my_conc$BLOQ)] <-
  interp.extrap.conc(conc=my_conc$conc,
                     time=my_conc$time,
                     time.out=my_conc$time[my_conc$BLOQ | is.na(my_conc$BLOQ)],
                     lambda.z=results_obj$result$PPORRES[results_obj$result$PPTESTCD %in% "lambda.z"])

# Extrapolate concentrations for aucinf.pred
my_conc$conc_aucinf.pred <- my_conc$conc
my_conc$conc_aucinf.pred[my_conc$BLOQ | is.na(my_conc$BLOQ)] <-
  interp.extrap.conc(conc=my_conc$conc,
                     time=my_conc$time,
                     time.out=my_conc$time[my_conc$BLOQ | is.na(my_conc$BLOQ)],
                     lambda.z=results_obj$result$PPORRES[results_obj$result$PPTESTCD %in% "lambda.z"],
                     clast=results_obj$result$PPORRES[results_obj$result$PPTESTCD %in% "clast.pred"])
my_conc$conc_aucinf.pred[my_conc$time == tlast] <-
  results_obj$result$PPORRES[results_obj$result$PPTESTCD %in% "clast.pred"]

ggplot(my_conc[!is.na(my_conc$conc),],
       aes(x=time,
           y=conc,
           #shape=BLOQ,
           group=subject)) +
  geom_ribbon(data=my_conc,
              aes(ymin=0, ymax=conc_aucinf.obs),
              fill="lightblue") +
  geom_line() +
  #geom_point(size=2) +
  scale_x_continuous(breaks=my_conc$time) +
  theme(legend.position=c(0.8, 0.8)) +
  labs(title="Extrapolation using AUCinf,obs")

ggplot(my_conc[!is.na(my_conc$conc),],
       aes(x=time,
           y=conc,
           #shape=BLOQ,
           group=subject)) +
  geom_ribbon(
    data=arrange(
      bind_rows(mutate(my_conc,
                       before=FALSE),
                mutate(filter(my_conc, time == tlast),
                       conc_aucinf.pred=conc,
                       before=TRUE)),
      time, desc(before)),
    aes(ymin=0,
        ymax=conc_aucinf.pred),
    fill="lightblue") +
  geom_line() +
  #geom_point(size=2) +
  scale_x_continuous(breaks=my_conc$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 AUC_0-$\infty$ or AUC_0-last may indicate. Within PKNCA, partial AUCs are treated like AUC_last with start and end times separately selected. (In a future version of PKNCA, they will be more simply calculated using an AUC_interval.)

When the starting and ending times are observed within the data, partial AUCs can be calculated using the parameter auclast as illustrated below.

# Interpolation not required
data_obs_obj <- PKNCAdata(conc_obj, intervals=data.frame(start=0, end=2, auclast=TRUE))
results_obs_obj <- pk.nca(data_obs_obj)

## No dose information provided, assuming default dosing information.

kable(results_obs_obj$result)

start	end	subject	PPTESTCD	PPORRES	exclude
0	2	1	auclast	4	NA

When the starting and ending times are not observed within the data or when samples are below the limit of quantification, concentrations must be interpolated and added to the dataset before calculation as illustrated below.

# Interpolation required
my_conc_interp <-
  arrange(
    bind_rows(
      my_conc,
      data.frame(conc=interp.extrap.conc(conc=my_conc$conc, time=my_conc$time, time.out=1.5),
                 time=1.5,
                 subject=1)),
    time)
kable(my_conc_interp)

conc	time	subject	BLOQ	measured	include_auclast	include_aucall	conc_aucinf.obs	conc_aucinf.pred
0.00	0.0	1	TRUE	TRUE	TRUE	TRUE	0.0000000	0.0000000
2.50	1.0	1	FALSE	TRUE	TRUE	TRUE	2.5000000	2.5000000
2.75	1.5	1	NA	NA	NA	NA	NA	NA
3.00	2.0	1	FALSE	TRUE	TRUE	TRUE	3.0000000	3.0000000
2.00	3.0	1	FALSE	TRUE	TRUE	TRUE	2.0000000	2.0000000
1.50	4.0	1	FALSE	TRUE	TRUE	TRUE	1.5000000	1.5000000
1.20	5.0	1	FALSE	TRUE	TRUE	TRUE	1.2000000	1.2000000
1.10	8.0	1	FALSE	TRUE	TRUE	TRUE	1.1000000	1.0216136
0.00	12.0	1	TRUE	TRUE	FALSE	TRUE	0.7153906	0.6644116
0.00	24.0	1	TRUE	TRUE	FALSE	FALSE	0.1967862	0.1827632
NA	36.0	1	NA	FALSE	FALSE	FALSE	0.0541310	0.0502736

conc_interp_obj <- PKNCAconc(my_conc_interp, conc~time|subject)
data_interp_obj <- PKNCAdata(conc_interp_obj, intervals=data.frame(start=0, end=1.5, auclast=TRUE))
results_interp <- pk.nca(data_interp_obj)

## No dose information provided, assuming default dosing information.

results_interp$result

##   start end subject PPTESTCD PPORRES exclude
## 1     0 1.5       1  auclast  2.5625    <NA>

AUC Calculation with PKNCA

Bill Denney

2018-01-02

Preparation

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

AUC_all

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

Partial AUCs

AUC Calculation with PKNCA

Bill Denney

2018-01-02

Preparation

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

AUCall

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

Partial AUCs

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

AUC_all

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