# AUC Calculation with PKNCA

#### Bill Denney

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))

    ## Warning: A numeric `legend.position` argument in `theme()` was deprecated in ggplot2
    ## 3.5.0.
    ## ℹ Please use the `legend.position.inside` argument of `theme()` instead.
    ## This warning is displayed once every 8 hours.
    ## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    ## generated.

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, calculations requiring dose will return NA.

    kable(as.data.frame(results_obj))

subject | start | end | PPTESTCD | PPORRES | exclude
--- | --- | --- | --- | --- | ---
1 | 0 | 24 | auclast | 12.9965842 | NA
1 | 0 | 24 | aucall | 15.1965842 | 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.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 | NA
1 | 0 | 24 | aucinf.pred | 22.4947355 | 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 AUC~0-\(\infty\),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 treated like AUClast with start and end times separately selected. (In a future version of PKNCA, they will be more simply calculated using an AUCinterval.)

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, calculations requiring dose will return NA.

    kable(as.data.frame(results_obs_obj))

subject | start | end | PPTESTCD | PPORRES | exclude
--- | --- | --- | --- | --- | ---
1 | 0 | 2 | 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 | BLQ | 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, calculations requiring dose will return NA.

    as.data.frame(results_interp)

    ## # A tibble: 1 × 6
    ##   subject start   end PPTESTCD PPORRES exclude
    ##     <dbl> <dbl> <dbl> <chr>      <dbl> <chr>
    ## 1       1     0   1.5 auclast     2.56 <NA>