Load the data

## It is always a good idea to look at the data
knitr::kable(head(datasets::Theoph))

The columns that we will be interested in for our analysis are conc, Time, and Subject in the concentration data.

## By default it is groupedData; convert it to a data frame for use
conc_obj <- PKNCAconc(as.data.frame(datasets::Theoph), conc~Time|Subject)

Compute the Superposition from Single-Dose Data to Steady-State

With a simple call, we can have the estimated steady-state concentration for each subject. At minimum, the time between dosing (tau) must be provided.

steady_state <- superposition(conc_obj, tau=24)

## Error in superposition.numeric(conc = nested_data$data_conc[[idx]]$conc, : The first concentration must be 0 (and not NA).  To change this set check.blq=FALSE.

The error noting that the first concentration must be zero is due to the fact that superposition usually occurs with single-dose data. If the first concentration is nonzero, the data are not likely to be single-dose (or a data error should be fixed). Let’s find the offending data.

knitr::kable(subset(datasets::Theoph, Time == 0 & conc > 0),
             caption="Nonzero predose measurements",
             row.names=FALSE)

For this example, we will assume that these were errors, correct them to zero, and recalculate.

## Correct nonzero concentrations at time 0 to be BLQ.
theoph_corrected <- as.data.frame(datasets::Theoph)
theoph_corrected$conc[theoph_corrected$Time == 0] <- 0
conc_obj_corrected <- PKNCAconc(theoph_corrected, conc~Time|Subject)

## Calculate the new steady-state concentrations with 24 hour dosing
steady_state <- superposition(conc_obj_corrected, tau=24)
knitr::kable(head(steady_state, n=14),
             caption="Superposition at steady-state")

The output is a tbl_df, tbl, data.frame including all the grouping factors as columns, a column for concentration, and a column for time. Time point selection ensures that the beginning and end of the interval are included and that every measured time that contributes to the interval is included. The points at the beginning and end of the interval are very similar; they are within a tolerance of 0.001 as defined by the steady.state.tol argument to superposition.

Nonstandard Superposition Computations

## Calculate the unsteady-state concentrations with 24 hour dosing
unsteady_state <- superposition(conc_obj_corrected, tau=24, n.tau=2)
knitr::kable(head(unsteady_state, n=14),
             caption="Superposition before steady-state")

## Calculate the new steady-state concentrations with 24 hour dosing
complex_interval_steady_state <- superposition(conc_obj_corrected, tau=24, dose.times=c(0, 2, 4))
knitr::kable(head(complex_interval_steady_state, n=10),
             caption="Superposition at steady-state with complex dosing")

ggplot(complex_interval_steady_state,
       aes(y=conc, x=time, colour=Subject)) +
  geom_line()

up_to_steady_state <- superposition(conc_obj_corrected,
                                    tau=4*24,
                                    n.tau=1,
                                    dose.times=seq(0, 3*24, by=12))

ggplot(up_to_steady_state, aes(x=time, y=conc, colour=Subject)) +
  geom_line()

Time Point Selection and Addition

Superposition is often used to estimate NCA parameters with nonparametric methods. To ensure that estimated parameters are as accurate as possible (especially \(C_{max}\)), each dose has every post-dose time point included. Specifically, each dose will have the following times:

• 0 (zero) and tau,
• The time of each dose (the dose.times argument)
• Every value from the time column of the data modulo tau (shifting the time for each measurement to be within the dosing interval) repeated for each dose, and
• each time from the additional.times argument.

How the number of time points increases can be seen by comparing the time points for subject 1 in the steady-state single dosing and the complex dosing examples above.

steady_state$time[steady_state$Subject == 1]

##  [1]  0.00  0.25  0.37  0.57  1.12  2.02  3.82  5.10  7.03  9.05 12.12 24.00

sum(steady_state$Subject == 1)

## [1] 12

complex_interval_steady_state$time[complex_interval_steady_state$Subject == 1]

##  [1]  0.00  0.25  0.37  0.57  1.12  2.00  2.02  2.25  2.37  2.57  3.12  3.82
## [13]  4.00  4.02  4.25  4.37  4.57  5.10  5.12  5.82  6.02  7.03  7.10  7.82
## [25]  9.03  9.05  9.10 11.03 11.05 12.12 13.05 14.12 16.12 24.00

sum(complex_interval_steady_state$Subject == 1)

## [1] 34

Interpolation and Extrapolation Methods

The interpolation and extrapolation methods align with those used for calculating the AUC. By default, interpolation uses the PKNCA.options selection for "auc.method" and extrapolation follows the curve of \(AUC_{inf}\). These can be modified with the interp.method and extrap.method arguments.