Introduction

PKNCA considers two types of data grouping within data sets: the group and the interval. A group typically identifies a single subject given a single intervention type (a “treatment”) with a single analyte. An interval subsets a group by times within the group, and primary noncompartmental analysis (NCA) calculations are performed within an interval.

As a concrete example, consider the figure below shows the concentration-time profile of a study subject in a multiple-dose study. The group is all points in the figure, and the interval for the last day (144 to 168 hr) is the area with blue shading.

## Formula for concentration:
##  conc ~ time | treatment + ID
## With 1 subjects defined in the 'ID' column.
## Nominal time column is not specified.
## 
## First 6 rows of concentration data:
##    study treatment ID time      conc   analyte exclude volume duration
##  Study 1     Trt 1  1    0 0.0000000 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    1 0.6140526 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    2 0.8100022 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    4 0.8425422 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    6 0.7771994 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    8 0.7052469 Analyte 1    <NA>     NA        0

# Plot the concentration-time data and the interval
ggplot(d_conc_multi, aes(x=time, y=conc)) +
  geom_ribbon(data=d_conc_multi[d_conc_multi$time >= 144,],
              aes(ymax=conc, ymin=0),
              fill="skyblue") +
  geom_point() + geom_line() +
  scale_x_continuous(breaks=seq(0, 168, by=12)) +
  scale_y_continuous(limits=c(0, NA)) +
  labs(x="Time Since First Dose (hr)",
       y="Concentration\n(arbitrary units)")

intervals_manual <- data.frame(start=144, end=168, auclast=TRUE)
knitr::kable(intervals_manual)

start	end	auclast
144	168	TRUE

PKNCAdata(d_conc, intervals=intervals_manual)

## Formula for concentration:
##  conc ~ time | treatment + ID
## With 1 subjects defined in the 'ID' column.
## Nominal time column is not specified.
## 
## First 6 rows of concentration data:
##    study treatment ID time      conc   analyte exclude volume duration
##  Study 1     Trt 1  1    0 0.0000000 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    1 0.6140526 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    2 0.8100022 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    4 0.8425422 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    6 0.7771994 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    8 0.7052469 Analyte 1    <NA>     NA        0
## No dosing information.
## 
## With 1 rows of AUC specifications.
## No options are set differently than default.

Group Matching

Group matching occurs by matching all overlapping column names between the groups and the interval data.frame. (Note that grouping columns cannot be the word start, end, or share a name with an NCA parameter.)

Selecting the Subjects for an Interval

The groups for an interval prepare for summarization. Typically the groups will take a structure similar to the preferred summarization structure with groups nested in the logical method for summary. As an example, the group structure may be: study, treatment, day, analyte, and subject. The grouping names for an interval must be the same as or a subset of the grouping names used for the concentration data.

As the matching occurs with all available columns, the grouping columns names are only required to the level of specificity for the calculations desired. As an example, if you want AUC_inf,obs in subjects who received single doses and AUC_last on days 1 (0 to 24 hours) and 10 (216 to 240 hours) in subjects who received multiple doses, with treatment defined as “Drug 1 Single” or “Drug 1 Multiple”, the intervals could be defined as below.

intervals_manual <-
  data.frame(
    treatment=c("Drug 1 Single", "Drug 1 Multiple", "Drug 1 Multiple"),
    start=c(0, 0, 216),
    end=c(Inf, 24, 240),
    aucinf.obs=c(TRUE, FALSE, FALSE),
    auclast=c(FALSE, TRUE, TRUE)
  )
knitr::kable(intervals_manual)

treatment	start	end	aucinf.obs	auclast
Drug 1 Single	0	Inf	TRUE	FALSE
Drug 1 Multiple	0	24	FALSE	TRUE
Drug 1 Multiple	216	240	FALSE	TRUE

Intervals

Intervals are defined by data.frames with one row per interval, zero or more columns to match the groups from the PKNCAdata object, and one or more NCA parameters to calculate.

Selection of points within an interval occurs by choosing any point at or after the start and at or before the end.

To Infinity

The end of an interval may be infinity. An interval to infinity works the same as any other interval in that points are selected by being at or after the start and at or before the end of the interval. Selecting Inf or any value at or after the maximum time yields no difference in effect, but Inf is simpler when scripting to ensure that all points are selected.

## Formula for concentration:
##  conc ~ time | treatment + ID
## With 1 subjects defined in the 'ID' column.
## Nominal time column is not specified.
## 
## First 6 rows of concentration data:
##    study treatment ID time      conc   analyte exclude volume duration
##  Study 1     Trt 1  1    0 0.0000000 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    1 0.6140526 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    2 0.8100022 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    4 0.8425422 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    6 0.7771994 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    8 0.7052469 Analyte 1    <NA>     NA        0

# Use superposition to simulate multiple doses
ggplot(d_conc$data[d_conc$data$time <= 48,], aes(x=time, y=conc)) +
  geom_ribbon(data=d_conc$data,
              aes(ymax=conc, ymin=0),
              fill="skyblue") +
  geom_point() + geom_line() +
  scale_x_continuous(breaks=seq(0, 72, by=12)) +
  scale_y_continuous(limits=c(0, NA)) +
  labs(x="Time Since First Dose (hr)",
       y="Concentration\n(arbitrary units)")

intervals_manual <-
  data.frame(
    start=0,
    end=Inf,
    auclast=TRUE,
    aucinf.obs=TRUE
  )
print(intervals_manual)

##   start end auclast aucinf.obs
## 1     0 Inf    TRUE       TRUE

my.data <- PKNCAdata(d_conc, intervals=intervals_manual)

Multiple Intervals

More than one interval may be specified for the same subject or group of subjects by providing more than one row of interval specifications. In the figure below, the blue and green shaded regions indicate the first and second rows of the intervals, respectively.

## Formula for concentration:
##  conc ~ time | treatment + ID
## With 1 subjects defined in the 'ID' column.
## Nominal time column is not specified.
## 
## First 6 rows of concentration data:
##    study treatment ID time      conc   analyte exclude volume duration
##  Study 1     Trt 1  1    0 0.0000000 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    1 0.6140526 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    2 0.8100022 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    4 0.8425422 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    6 0.7771994 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    8 0.7052469 Analyte 1    <NA>     NA        0

# Plot the concentration-time data and the interval
ggplot(d_conc_multi, aes(x=time, y=conc)) +
  geom_ribbon(data=d_conc_multi[d_conc_multi$time <= 24,],
              aes(ymax=conc, ymin=0),
              fill="skyblue") +
  geom_ribbon(data=d_conc_multi[d_conc_multi$time >= 144,],
              aes(ymax=conc, ymin=0),
              fill="lightgreen") +
  geom_point() + geom_line() +
  scale_x_continuous(breaks=seq(0, 168, by=12)) +
  scale_y_continuous(limits=c(0, NA)) +
  labs(x="Time Since First Dose (hr)",
       y="Concentration\n(arbitrary units)")

intervals_manual <-
  data.frame(
    start=c(0, 144),
    end=c(24, 168),
    auclast=TRUE
  )
knitr::kable(intervals_manual)

start	end	auclast
0	24	TRUE
144	168	TRUE

my.data <- PKNCAdata(d_conc, intervals=intervals_manual)

Overlapping Intervals and Different Calculations by Interval

In some scenarios, multiple intervals may be needed where some intervals overlap. There is no issue with an interval specification that has two rows with overlapping times; the rows are considered separately. In the example below, the 0-24 interval is shared between both the first and second (shaded blue-green).

The example of overlapping intervals also illustrates that different calculations can be performed in different intervals. In this case, auclast is calculated in both intervals while aucinf.obs is only calculated in the 0-Inf interval.

## Formula for concentration:
##  conc ~ time | treatment + ID
## With 1 subjects defined in the 'ID' column.
## Nominal time column is not specified.
## 
## First 6 rows of concentration data:
##    study treatment ID time      conc   analyte exclude volume duration
##  Study 1     Trt 1  1    0 0.0000000 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    1 0.6140526 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    2 0.8100022 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    4 0.8425422 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    6 0.7771994 Analyte 1    <NA>     NA        0
##  Study 1     Trt 1  1    8 0.7052469 Analyte 1    <NA>     NA        0

# Use superposition to simulate multiple doses
ggplot(d_conc$data, aes(x=time, y=conc)) +
  geom_ribbon(data=d_conc$data,
              aes(ymax=conc, ymin=0),
              fill="lightgreen",
              alpha=0.5) +
  geom_ribbon(data=d_conc$data[d_conc$data$time <= 24,],
              aes(ymax=conc, ymin=0),
              fill="skyblue",
              alpha=0.5) +
  geom_point() + geom_line() +
  scale_x_continuous(breaks=seq(0, 168, by=12)) +
  scale_y_continuous(limits=c(0, NA)) +
  labs(x="Time Since First Dose (hr)",
       y="Concentration\n(arbitrary units)")

intervals_manual <-
  data.frame(
    start=0,
    end=c(24, Inf),
    auclast=TRUE,
    aucinf.obs=c(FALSE, TRUE)
  )
knitr::kable(intervals_manual)

start	end	auclast	aucinf.obs
0	24	TRUE	FALSE
0	Inf	TRUE	TRUE

my.data <- PKNCAdata(d_conc, intervals=intervals_manual)

Intervals with Duration

Some events have durations of times rather than instants in time associated with them. Two typical examples of duration data in NCA are intravenous infusions and urine or fecal sample collections. Inform PKNCA of durations with the duration argument to the PKNCAdose and PKNCAconc functions.

Durations data are selected based on both the beginning and ending of the duration existing within the interval.

Parameters Available for Calculation in an Interval

The following parameters are available in an interval. For more information about the parameter, see the documentation for the function.

Parameter Name	Parameter Description	Function for Calculation
adj.r.squared	The adjusted r^2 value of the half-life calculation	See the parameter name half.life
ae	The amount excreted (typically into urine or feces)	pk.calc.ae
aucall	The area under the concentration time curve from the beginning of the interval to the last concentration above the limit of quantification plus the triangle from that last concentration to 0 at the first concentration below the limit of quantification	pk.calc.auc.all
aucall.dn	Dose normalized aucall	pk.calc.dn
aucinf.obs	The area under the concentration time curve from the beginning of the interval to infinity with extrapolation to infinity from the observed Clast	pk.calc.auc.inf.obs
aucinf.obs.dn	Dose normalized aucinf.obs	pk.calc.dn
aucinf.pred	The area under the concentration time curve from the beginning of the interval to infinity with extrapolation to infinity from the predicted Clast	pk.calc.auc.inf.pred
aucinf.pred.dn	Dose normalized aucinf.pred	pk.calc.dn
aucint.all	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with the triangle from Tlast to the next point and zero thereafter (matching AUCall)	pk.calc.aucint.all
aucint.all.dose	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with the triangle from Tlast to the next point and zero thereafter (matching AUCall)	pk.calc.aucint.all
aucint.inf.obs	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with zeros (matching AUClast)	pk.calc.aucint.inf.obs
aucint.inf.obs.dose	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with zeros (matching AUClast)	pk.calc.aucint.inf.obs
aucint.inf.pred	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with the triangle from Tlast to the next point and zero thereafter (matching AUCall)	pk.calc.aucint.inf.pred
aucint.inf.pred.dose	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with the triangle from Tlast to the next point and zero thereafter (matching AUCall)	pk.calc.aucint.inf.pred
aucint.last	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with zeros (matching AUClast)	pk.calc.aucint.last
aucint.last.dose	The area under the concentration time curve in the interval extrapolating from Tlast to infinity with zeros (matching AUClast)	pk.calc.aucint.last
auclast	The area under the concentration time curve from the beginning of the interval to the last concentration above the limit of quantification	pk.calc.auc.last
auclast.dn	Dose normalized auclast	pk.calc.dn
aucpext.obs	Percent of the AUCinf that is extrapolated after Tlast calculated from the observed Clast	pk.calc.aucpext
aucpext.pred	Percent of the AUCinf that is extrapolated after Tlast calculated from the predicted Clast	pk.calc.aucpext
aumcall	The area under the concentration time moment curve from the beginning of the interval to the last concentration above the limit of quantification plus the moment of the triangle from that last concentration to 0 at the first concentration below the limit of quantification	pk.calc.aumc.all
aumcall.dn	Dose normalized aumcall	pk.calc.dn
aumcinf.obs	The area under the concentration time moment curve from the beginning of the interval to infinity with extrapolation to infinity from the observed Clast	pk.calc.aumc.inf.obs
aumcinf.obs.dn	Dose normalized aumcinf.obs	pk.calc.dn
aumcinf.pred	The area under the concentration time moment curve from the beginning of the interval to infinity with extrapolation to infinity from the predicted Clast	pk.calc.aumc.inf.pred
aumcinf.pred.dn	Dose normalized aumcinf.pred	pk.calc.dn
aumclast	The area under the concentration time moment curve from the beginning of the interval to the last concentration above the limit of quantification	pk.calc.aumc.last
aumclast.dn	Dose normalized aumclast	pk.calc.dn
cav	The average concentration during an interval	pk.calc.cav
cav.dn	Dose normalized cav	pk.calc.dn
ceoi	Concentration at the end of infusion	pk.calc.ceoi
cl.all	Clearance or observed oral clearance calculated with AUCall	pk.calc.cl
cl.last	Clearance or observed oral clearance calculated to Clast	pk.calc.cl
cl.obs	Clearance or observed oral clearance calculated with observed Clast	pk.calc.cl
cl.pred	Clearance or observed oral clearance calculated with predicted Clast	pk.calc.cl
clast.obs	The last concentration observed above the limit of quantification	pk.calc.clast.obs
clast.obs.dn	Dose normalized clast.obs	pk.calc.dn
clast.pred	The concentration at Tlast as predicted by the half-life	See the parameter name half.life
clast.pred.dn	Dose normalized clast.pred	pk.calc.dn
clr.last	The renal clearance calculated using AUClast	pk.calc.clr
clr.obs	The renal clearance calculated using AUCinf,obs	pk.calc.clr
clr.pred	The renal clearance calculated using AUCinf,pred	pk.calc.clr
cmax	Maximum observed concentration	pk.calc.cmax
cmax.dn	Dose normalized cmax	pk.calc.dn
cmin	Minimum observed concentration	pk.calc.cmin
cmin.dn	Dose normalized cmin	pk.calc.dn
ctrough	The trough (predose) concentration	pk.calc.ctrough
ctrough.dn	Dose normalized ctrough	pk.calc.dn
deg.fluc	Degree of fluctuation	pk.calc.deg.fluc
end	Ending time of the interval (potentially infinity)	(none)
f	Bioavailability or relative bioavailability	pk.calc.f
fe	The fraction of the dose excreted	pk.calc.fe
half.life	The (terminal) half-life	pk.calc.half.life
kel.iv.last	Elimination rate (as calculated from the intravenous MRTlast)	pk.calc.kel
kel.iv.obs	Elimination rate (as calculated from the intravenous MRTobs)	pk.calc.kel
kel.iv.pred	Elimination rate (as calculated from the intravenous MRTpred)	pk.calc.kel
kel.last	Elimination rate (as calculated from the MRT using AUClast)	pk.calc.kel
kel.obs	Elimination rate (as calculated from the MRT with observed Clast)	pk.calc.kel
kel.pred	Elimination rate (as calculated from the MRT with predicted Clast)	pk.calc.kel
lambda.z	The elimination rate of the terminal half-life	See the parameter name half.life
lambda.z.n.points	The number of points used for the calculation of half-life	See the parameter name half.life
lambda.z.time.first	The first time point used for the calculation of half-life	See the parameter name half.life
mrt.iv.last	The mean residence time to the last observed concentration above the LOQ correcting for dosing duration	pk.calc.mrt.iv
mrt.iv.obs	The mean residence time to infinity using observed Clast correcting for dosing duration	pk.calc.mrt.iv
mrt.iv.pred	The mean residence time to infinity using predicted Clast correcting for dosing duration	pk.calc.mrt.iv
mrt.last	The mean residence time to the last observed concentration above the LOQ	pk.calc.mrt
mrt.md.obs	The mean residence time with multiple dosing and nonlinear kinetics using observed Clast	pk.calc.mrt.md
mrt.md.pred	The mean residence time with multiple dosing and nonlinear kinetics using predicted Clast	pk.calc.mrt.md
mrt.obs	The mean residence time to infinity using observed Clast	pk.calc.mrt
mrt.pred	The mean residence time to infinity using predicted Clast	pk.calc.mrt
ptr	Peak-to-Trough ratio (fraction)	pk.calc.ptr
r.squared	The r^2 value of the half-life calculation	See the parameter name half.life
span.ratio	The ratio of the half-life to the duration used for half-life calculation	See the parameter name half.life
start	Starting time of the interval	(none)
swing	Swing relative to Cmin	pk.calc.swing
tfirst	Time of the first concentration above the limit of quantification	pk.calc.tfirst
thalf.eff.iv.last	The effective half-life (as determined from the intravenous MRTlast)	pk.calc.thalf.eff
thalf.eff.iv.obs	The effective half-life (as determined from the intravenous MRTobs)	pk.calc.thalf.eff
thalf.eff.iv.pred	The effective half-life (as determined from the intravenous MRTpred)	pk.calc.thalf.eff
thalf.eff.last	The effective half-life (as determined from the MRTlast)	pk.calc.thalf.eff
thalf.eff.obs	The effective half-life (as determined from the MRTobs)	pk.calc.thalf.eff
thalf.eff.pred	The effective half-life (as determined from the MRTpred)	pk.calc.thalf.eff
time_above	Time above a given concentration	pk.calc.time_above
tlag	Lag time	pk.calc.tlag
tlast	Time of the last concentration observed above the limit of quantification	pk.calc.tlast
tmax	Time of the maximum observed concentration	pk.calc.tmax
vd.obs	Apparent observed volume of distribution calculated with observed Clast	pk.calc.vd
vd.pred	Apparent observed volume of distribution calculated with predicted Clast	pk.calc.vd
vss.iv.last	The steady-state volume of distribution with intravenous infusion calculating through Tlast	pk.calc.vss
vss.iv.obs	The steady-state volume of distribution with intravenous infusion using observed Clast	pk.calc.vss
vss.iv.pred	The steady-state volume of distribution with intravenous infusion using predicted Clast	pk.calc.vss
vss.last	The steady-state volume of distribution calculating through Tlast	pk.calc.vss
vss.md.obs	The steady-state volume of distribution for nonlinear multiple-dose data using observed Clast	pk.calc.vss
vss.md.pred	The steady-state volume of distribution for nonlinear multiple-dose data using predicted Clast	pk.calc.vss
vss.obs	The steady-state volume of distribution using observed Clast	pk.calc.vss
vss.pred	The steady-state volume of distribution using predicted Clast	pk.calc.vss
vz.obs	The terminal volume of distribution using observed Clast	pk.calc.vz
vz.pred	The terminal volume of distribution using predicted Clast	pk.calc.vz

Selection of Calculation Intervals

Bill Denney