22 Dose-Aware Interpolation and Extrapolation

22.1 Why dose timing matters for interpolation

Standard concentration interpolation (e.g. interp.extrap.conc()) assumes a smooth concentration-time curve with no discontinuities. But when a dose is given at or near the interpolation target time, the correct value depends critically on whether you are asking about the concentration before or after that dose event.

interp.extrap.conc.dose() is the dose-aware version. It accounts for:

Whether the route is intravascular (bolus causes instantaneous jump) or extravascular (no instantaneous change)
Whether you want the concentration before (out.after = FALSE) or after (out.after = TRUE) the dose
Whether the dose is an infusion with a finite duration

22.2 The key argument: `out.after`

`out.after`	Meaning
`FALSE` (default)	Return the concentration just before the dose event
`TRUE`	Return the concentration just after the dose event

out.after has a meaningful effect only at the instant of an IV bolus dose (route.dose = "intravascular" with duration.dose = 0), where the concentration jumps instantaneously: FALSE gives the value from the data before the dose (the trough) and TRUE gives the value from the data after the dose. For extravascular doses — and for infusions, which have duration.dose > 0 — there is no instantaneous jump, so out.after makes no difference.

22.3 Example data: single oral dose

# Single-dose oral profile (absorption + elimination)
conc <- c(0.0, 2.5, 5.0, 4.2, 2.8, 1.5, 0.8)
time <- c(0,   1,   2,   3,   5,   7,   10 )

d <- data.frame(time = time, conc = conc)

ggplot(d, aes(x = time, y = conc)) +
  geom_line(colour = "steelblue") +
  geom_point(size = 3, colour = "steelblue") +
  labs(title = "Example single-dose oral profile",
       x = "Time (h)", y = "Concentration (mg/L)") +
  theme_minimal()

22.4 Basic interpolation (no dose at target time)

When the target time is between observed samples and does not coincide with a dose, the result equals standard log-linear interpolation.

# t = 4 h falls between t = 3 and t = 5
interp.extrap.conc.dose(
  conc       = conc,
  time       = time,
  time.dose  = 0,          # dose was given at t=0
  route.dose = "extravascular",
  time.out   = 4
)

[1] 3.429286
attr(,"Method")
[1] "Interpolation"

Compare to dose-unaware interpolation:

interp.extrap.conc(conc = conc, time = time, time.out = 4)

[1] 3.429286

They agree when no dose occurs at the target time.

22.5 At the dose time: oral (extravascular)

For an oral dose, concentration does not change instantaneously at the dose time — absorption is gradual. So out.after does not change the interpolated value at the dose time itself.

# Oral: t=0, before dose
interp.extrap.conc.dose(
  conc       = conc, time = time,
  time.dose  = 0, route.dose = "extravascular",
  time.out   = 0, out.after  = FALSE
)

[1] 0
attr(,"Method")
[1] "Observed concentration"

# Oral: t=0, after dose (same — no jump)
interp.extrap.conc.dose(
  conc       = conc, time = time,
  time.dose  = 0, route.dose = "extravascular",
  time.out   = 0, out.after  = TRUE
)

[1] 0
attr(,"Method")
[1] "Interpolation"

22.6 At the dose time: IV bolus (intravascular)

For an IV bolus — indicated by route.dose = "intravascular" together with duration.dose = 0 — the dose causes an instantaneous concentration jump. out.after = FALSE returns the pre-dose (trough) concentration; out.after = TRUE returns the post-dose concentration.

# Two IV bolus doses (t=0 and t=6); the t=6 sample is the pre-dose trough
conc_iv <- c(10,  8,  6,  4,  2.5, 9,  6)
time_iv <- c(0,   1,  2,  4,  6,   7,  9)

# Just before the second dose at t=6: the observed trough
interp.extrap.conc.dose(
  conc       = conc_iv, time = time_iv,
  time.dose  = c(0, 6), route.dose = "intravascular", duration.dose = 0,
  time.out   = 6, out.after  = FALSE
)

[1] 2.5
attr(,"Method")
[1] "Observed concentration"

# Just after the second dose at t=6: the post-dose concentration
interp.extrap.conc.dose(
  conc       = conc_iv, time = time_iv,
  time.dose  = c(0, 6), route.dose = "intravascular", duration.dose = 0,
  time.out   = 6, out.after  = TRUE
)

[1] 11.0227
attr(,"Method")
[1] "Immediately after an IV bolus with a concentration next"

The pre-dose value is the measured trough (2.5). The post-dose value (about 11.0) is estimated the same way as pk.calc.c0(): log-slope back-extrapolation from the concentrations measured after the dose.

22.7 Multiple dose times

The IV bolus example above already passed two dose times — time.dose accepts a vector whenever multiple doses were given during the sampling window. Here is the same two-dose scenario with oral dosing.

# Oral doses at t=0 and t=6
conc_md <- c(0, 2.5, 5.0, 4.2, 2.8, 4.5, 6.0, 5.1, 3.5, 2.0)
time_md <- c(0, 1,   2,   3,   5,   6,   7,   8,   10,  12 )

# Concentration just before the second dose at t=6
interp.extrap.conc.dose(
  conc       = conc_md, time = time_md,
  time.dose  = c(0, 6), route.dose = "extravascular",
  time.out   = 6, out.after = FALSE
)

[1] 4.5
attr(,"Method")
[1] "Observed concentration"

# Concentration just after the second dose at t=6
interp.extrap.conc.dose(
  conc       = conc_md, time = time_md,
  time.dose  = c(0, 6), route.dose = "extravascular",
  time.out   = 6, out.after = TRUE
)

[1] 4.5
attr(,"Method")
[1] "Interpolation"

For extravascular doses, before and after are the same (no instantaneous jump).

22.8 IV infusion: dose duration

For IV infusions, supply duration.dose (the infusion duration in the same time units).

# 30-minute (0.5h) infusion starting at t=0
conc_inf <- c(0, 3,  5,  4.5, 3,  2,  1  )
time_inf <- c(0, 0.25, 0.5, 1, 2, 4, 8  )

# During the infusion, between the samples at t=0.25 and t=0.5
interp.extrap.conc.dose(
  conc          = conc_inf, time = time_inf,
  time.dose     = 0,
  route.dose    = "intravascular",
  duration.dose = 0.5,
  time.out      = 0.4, out.after = FALSE
)

[1] 4.2
attr(,"Method")
[1] "Interpolation"

Because duration.dose > 0, the dose is treated as an infusion rather than a bolus: there is no instantaneous jump at t = 0, and concentrations during the infusion are interpolated between the observed samples (here 4.2, between the 3 and 5 measurements).

22.9 Extrapolating beyond the last sample

Returning to the single oral-dose profile from the top of the page (last sample at t = 10): extrapolation past the last measured concentration requires an elimination rate. Pass lambda.z (forwarded via ... to the extrapolation functions); without it, the result is NA.

# t=14 is beyond the last sample (t=10); without lambda.z the result is NA
interp.extrap.conc.dose(
  conc       = conc, time = time,
  time.dose  = 0, route.dose = "extravascular",
  time.out   = 14
)

[1] NA
attr(,"Method")
[1] "Extrapolation"

# With lambda.z, extrapolation follows clast * exp(-lambda.z * (t - tlast))
interp.extrap.conc.dose(
  conc       = conc, time = time,
  time.dose  = 0, route.dose = "extravascular",
  time.out   = 14, lambda.z = 0.25
)

[1] 0.2943036
attr(,"Method")
[1] "Extrapolation"

22.10 How PKNCA uses `interp.extrap.conc.dose()` internally

Within pk.nca(), the interval AUC and AUMC parameters (aucint.* and aumcint.*) call this function to estimate the concentration at the start and end of their window whenever dosing data are given (≥ 0.13.0); every aucint/aumcint parameter is dose-aware, and the former .dose variants are retired. You normally never call it directly. The rules that apply inside pk.nca() are:

Only the window boundaries are estimated, and always as the concentration just before any dose at that time (out.after = FALSE). A concentration measured at the same time as a dose counts as measured before it.
The profile ends at the next dose. Concentrations measured after the first dose at or after end are not integrated and are not used to interpolate or extrapolate back into the window.
A dose inside the window is integrated across using the concentrations measured on either side of it; no concentration is imputed at that dose.
The route and duration of each dose are used, so an IV bolus inside the window is handled as a bolus, not as an extravascular dose.
Without dosing data, pk.nca() uses interp.extrap.conc() instead, and the PPANMETH column reports “Interpolation: not dose-aware (no dosing data)”.

Because the window start is the concentration before the dose, an aucint.* window that starts at an IV bolus with no sample at that time starts from the pre-dose concentration rather than from a back-extrapolated C0. For IV bolus profiles, use the aucivint.* parameters, which back-extrapolate C0:

# Two IV bolus doses (t = 0 and t = 12) with no sample at either dose time
d_iv2 <- data.frame(
  subject = "A",
  time = c(0.5, 1, 2, 4, 6, 12.5, 13, 14, 16, 18, 24),
  conc = c(9, 8, 6, 4, 2.8, 10, 9, 7, 5, 3.6, 1.5)
)
d_iv2_dose <- data.frame(subject = "A", time = c(0, 12), dose = 100)

o_conc_iv2 <- PKNCAconc(d_iv2, conc ~ time | subject)
o_dose_iv2 <- PKNCAdose(d_iv2_dose, dose ~ time | subject,
                        route = "intravascular")

# The dosing interval after the second dose
o_nca_iv2 <- pk.nca(PKNCAdata(
  o_conc_iv2, o_dose_iv2,
  intervals = data.frame(start = 12, end = 24,
                         aucint.last = TRUE, aucivint.last = TRUE, c0 = TRUE)
))

as.data.frame(o_nca_iv2) |>
  filter(PPTESTCD %in% c("aucint.last", "aucivint.last", "c0")) |>
  select(start, end, PPTESTCD, PPORRES)

# A tibble: 3 × 4
  start   end PPTESTCD      PPORRES
  <dbl> <dbl> <chr>           <dbl>
1    12    24 aucint.last      50.0
2    12    24 c0               11.1
3    12    24 aucivint.last    52.8

aucint.last starts from the concentration estimated just before the 12-hour dose and rises linearly to the 12.5-hour sample, so it is lower than aucivint.last, which starts from the back-extrapolated c0. The AUC chapter shows the effect of the next-dose boundary on an oral profile, and the IV dosing chapter covers the auciv* family.

Understanding these rules helps diagnose unusual NCA results when dosing and sampling times coincide — for example, when an interval boundary such as end = tau coincides with the next dose (the trough sample time). Every combination of dose events and observations, and the method used for each, is tabulated in the complete methods table of the dose-aware interpolation vignette, and Selection of Calculation Intervals explains how interval boundaries are chosen.

Several edge cases changed in PKNCA 0.13.0: interpolation no longer fails when the concentrations around a dose are all BLQ; the AUCall extrapolation finds the first BLQ measurement after Tlast instead of stopping at the last measured concentration; and a concentration estimated before a dose is extrapolated from the Clast of the profile before that dose rather than from the Clast of the interval being calculated.

22.11 Argument reference

Argument	Required	Meaning
`conc`	Yes	Observed concentrations
`time`	Yes	Observation times (same units as `time.dose`)
`time.dose`	Yes	Time(s) at which doses were given
`route.dose`	No (default `"extravascular"`)	`"extravascular"` or `"intravascular"`; scalar or one value per dose
`duration.dose`	No (default `NA`)	Dose duration: `0` marks an IV bolus, a positive value an infusion; scalar or one value per dose
`time.out`	Yes	Target time at which to interpolate/extrapolate
`out.after`	No (default `FALSE`)	`FALSE` = before dose; `TRUE` = after dose (only meaningful at the instant of an IV bolus)
`options`	No	PKNCA options list (controls AUC method / interpolation rule)

The table is abridged: interp.extrap.conc.dose() also accepts conc.blq and conc.na (BLQ and NA concentration handling), check (input validation), and ... (passed through to the interpolation/extrapolation functions, e.g. lambda.z).

pkgdown reference: interp.extrap.conc.dose() · interp.extrap.conc() · pk.calc.c0() · pk.calc.aucint() · pk.calc.auciv() · PKNCAconc() · PKNCAdose() · PKNCAdata() · pk.nca()

--- title: "Dose-Aware Interpolation and Extrapolation" --- ```{r setup, include=FALSE} library(PKNCA) library(dplyr) library(ggplot2) conflicted::conflicts_prefer(dplyr::filter, dplyr::select, .quiet = TRUE) ``` ## Why dose timing matters for interpolation Standard concentration interpolation (e.g. `interp.extrap.conc()`) assumes a smooth concentration-time curve with no discontinuities. But when a dose is given **at** or **near** the interpolation target time, the correct value depends critically on whether you are asking about the concentration *before* or *after* that dose event. **`interp.extrap.conc.dose()`** is the dose-aware version. It accounts for: - Whether the route is intravascular (bolus causes instantaneous jump) or extravascular (no instantaneous change) - Whether you want the concentration before (`out.after = FALSE`) or after (`out.after = TRUE`) the dose - Whether the dose is an infusion with a finite duration --- ## The key argument: `out.after` | `out.after` | Meaning | |---|---| | `FALSE` (default) | Return the concentration just **before** the dose event | | `TRUE` | Return the concentration just **after** the dose event | `out.after` has a meaningful effect only at the instant of an IV bolus dose (`route.dose = "intravascular"` with `duration.dose = 0`), where the concentration jumps instantaneously: `FALSE` gives the value from the data before the dose (the trough) and `TRUE` gives the value from the data after the dose. For extravascular doses — and for infusions, which have `duration.dose > 0` — there is no instantaneous jump, so `out.after` makes no difference. --- ## Example data: single oral dose ```{r} # Single-dose oral profile (absorption + elimination) conc <- c(0.0, 2.5, 5.0, 4.2, 2.8, 1.5, 0.8) time <- c(0, 1, 2, 3, 5, 7, 10 ) d <- data.frame(time = time, conc = conc) ggplot(d, aes(x = time, y = conc)) + geom_line(colour = "steelblue") + geom_point(size = 3, colour = "steelblue") + labs(title = "Example single-dose oral profile", x = "Time (h)", y = "Concentration (mg/L)") + theme_minimal() ``` --- ## Basic interpolation (no dose at target time) When the target time is **between** observed samples and does not coincide with a dose, the result equals standard log-linear interpolation. ```{r} # t = 4 h falls between t = 3 and t = 5 interp.extrap.conc.dose( conc = conc, time = time, time.dose = 0, # dose was given at t=0 route.dose = "extravascular", time.out = 4 ) ``` Compare to dose-unaware interpolation: ```{r} interp.extrap.conc(conc = conc, time = time, time.out = 4) ``` They agree when no dose occurs at the target time. --- ## At the dose time: oral (extravascular) For an oral dose, concentration does not change instantaneously at the dose time — absorption is gradual. So `out.after` does not change the interpolated value at the dose time itself. ```{r} # Oral: t=0, before dose interp.extrap.conc.dose( conc = conc, time = time, time.dose = 0, route.dose = "extravascular", time.out = 0, out.after = FALSE ) ``` ```{r} # Oral: t=0, after dose (same — no jump) interp.extrap.conc.dose( conc = conc, time = time, time.dose = 0, route.dose = "extravascular", time.out = 0, out.after = TRUE ) ``` --- ## At the dose time: IV bolus (intravascular) For an IV bolus — indicated by `route.dose = "intravascular"` together with `duration.dose = 0` — the dose causes an instantaneous concentration jump. `out.after = FALSE` returns the pre-dose (trough) concentration; `out.after = TRUE` returns the post-dose concentration. ```{r} # Two IV bolus doses (t=0 and t=6); the t=6 sample is the pre-dose trough conc_iv <- c(10, 8, 6, 4, 2.5, 9, 6) time_iv <- c(0, 1, 2, 4, 6, 7, 9) # Just before the second dose at t=6: the observed trough interp.extrap.conc.dose( conc = conc_iv, time = time_iv, time.dose = c(0, 6), route.dose = "intravascular", duration.dose = 0, time.out = 6, out.after = FALSE ) ``` ```{r} # Just after the second dose at t=6: the post-dose concentration interp.extrap.conc.dose( conc = conc_iv, time = time_iv, time.dose = c(0, 6), route.dose = "intravascular", duration.dose = 0, time.out = 6, out.after = TRUE ) ``` The pre-dose value is the measured trough (2.5). The post-dose value (about 11.0) is estimated the same way as `pk.calc.c0()`: log-slope back-extrapolation from the concentrations measured after the dose. --- ## Multiple dose times The IV bolus example above already passed two dose times — `time.dose` accepts a vector whenever multiple doses were given during the sampling window. Here is the same two-dose scenario with oral dosing. ```{r} # Oral doses at t=0 and t=6 conc_md <- c(0, 2.5, 5.0, 4.2, 2.8, 4.5, 6.0, 5.1, 3.5, 2.0) time_md <- c(0, 1, 2, 3, 5, 6, 7, 8, 10, 12 ) # Concentration just before the second dose at t=6 interp.extrap.conc.dose( conc = conc_md, time = time_md, time.dose = c(0, 6), route.dose = "extravascular", time.out = 6, out.after = FALSE ) ``` ```{r} # Concentration just after the second dose at t=6 interp.extrap.conc.dose( conc = conc_md, time = time_md, time.dose = c(0, 6), route.dose = "extravascular", time.out = 6, out.after = TRUE ) ``` For extravascular doses, before and after are the same (no instantaneous jump). --- ## IV infusion: dose duration For IV infusions, supply `duration.dose` (the infusion duration in the same time units). ```{r} # 30-minute (0.5h) infusion starting at t=0 conc_inf <- c(0, 3, 5, 4.5, 3, 2, 1 ) time_inf <- c(0, 0.25, 0.5, 1, 2, 4, 8 ) # During the infusion, between the samples at t=0.25 and t=0.5 interp.extrap.conc.dose( conc = conc_inf, time = time_inf, time.dose = 0, route.dose = "intravascular", duration.dose = 0.5, time.out = 0.4, out.after = FALSE ) ``` Because `duration.dose > 0`, the dose is treated as an infusion rather than a bolus: there is no instantaneous jump at t = 0, and concentrations during the infusion are interpolated between the observed samples (here 4.2, between the 3 and 5 measurements). --- ## Extrapolating beyond the last sample Returning to the single oral-dose profile from the top of the page (last sample at t = 10): extrapolation past the last measured concentration requires an elimination rate. Pass `lambda.z` (forwarded via `...` to the extrapolation functions); without it, the result is `NA`. ```{r} # t=14 is beyond the last sample (t=10); without lambda.z the result is NA interp.extrap.conc.dose( conc = conc, time = time, time.dose = 0, route.dose = "extravascular", time.out = 14 ) ``` ```{r} # With lambda.z, extrapolation follows clast * exp(-lambda.z * (t - tlast)) interp.extrap.conc.dose( conc = conc, time = time, time.dose = 0, route.dose = "extravascular", time.out = 14, lambda.z = 0.25 ) ``` --- ## How PKNCA uses `interp.extrap.conc.dose()` internally Within `pk.nca()`, the interval AUC and AUMC parameters (`aucint.*` and `aumcint.*`) call this function to estimate the concentration at the `start` and `end` of their window whenever dosing data are given (≥ 0.13.0); every `aucint`/`aumcint` parameter is dose-aware, and the former `.dose` variants are retired. You normally never call it directly. The rules that apply inside `pk.nca()` are: - **Only the window boundaries are estimated**, and always as the concentration just before any dose at that time (`out.after = FALSE`). A concentration measured at the same time as a dose counts as measured before it. - **The profile ends at the next dose.** Concentrations measured after the first dose at or after `end` are not integrated and are not used to interpolate or extrapolate back into the window. - **A dose inside the window is integrated across** using the concentrations measured on either side of it; no concentration is imputed at that dose. - **The route and duration of each dose are used**, so an IV bolus inside the window is handled as a bolus, not as an extravascular dose. - **Without dosing data**, `pk.nca()` uses `interp.extrap.conc()` instead, and the `PPANMETH` column reports "Interpolation: not dose-aware (no dosing data)". Because the window start is the concentration *before* the dose, an `aucint.*` window that starts at an IV bolus with no sample at that time starts from the pre-dose concentration rather than from a back-extrapolated C0. For IV bolus profiles, use the `aucivint.*` parameters, which back-extrapolate C0: ```{r} # Two IV bolus doses (t = 0 and t = 12) with no sample at either dose time d_iv2 <- data.frame( subject = "A", time = c(0.5, 1, 2, 4, 6, 12.5, 13, 14, 16, 18, 24), conc = c(9, 8, 6, 4, 2.8, 10, 9, 7, 5, 3.6, 1.5) ) d_iv2_dose <- data.frame(subject = "A", time = c(0, 12), dose = 100) o_conc_iv2 <- PKNCAconc(d_iv2, conc ~ time | subject) o_dose_iv2 <- PKNCAdose(d_iv2_dose, dose ~ time | subject, route = "intravascular") # The dosing interval after the second dose o_nca_iv2 <- pk.nca(PKNCAdata( o_conc_iv2, o_dose_iv2, intervals = data.frame(start = 12, end = 24, aucint.last = TRUE, aucivint.last = TRUE, c0 = TRUE) )) as.data.frame(o_nca_iv2) |> filter(PPTESTCD %in% c("aucint.last", "aucivint.last", "c0")) |> select(start, end, PPTESTCD, PPORRES) ``` `aucint.last` starts from the concentration estimated just before the 12-hour dose and rises linearly to the 12.5-hour sample, so it is lower than `aucivint.last`, which starts from the back-extrapolated `c0`. The [AUC chapter](auc-methods.qmd#why-dose-awareness-matters) shows the effect of the next-dose boundary on an oral profile, and the [IV dosing chapter](intravascular.qmd) covers the `auciv*` family. Understanding these rules helps diagnose unusual NCA results when dosing and sampling times coincide — for example, when an interval boundary such as `end = tau` coincides with the next dose (the trough sample time). Every combination of dose events and observations, and the method used for each, is tabulated in the [complete methods table](https://humanpred.github.io/pknca/articles/v21-methods-for-dose-aware-interpolation-and-extrapolation.html#appendix-complete-methods-table) of the dose-aware interpolation vignette, and [Selection of Calculation Intervals](https://humanpred.github.io/pknca/articles/v03-selection-of-calculation-intervals.html) explains how interval boundaries are chosen. Several edge cases changed in PKNCA 0.13.0: interpolation no longer fails when the concentrations around a dose are all BLQ; the `AUCall` extrapolation finds the first BLQ measurement after Tlast instead of stopping at the last measured concentration; and a concentration estimated before a dose is extrapolated from the Clast of the profile before that dose rather than from the Clast of the interval being calculated. --- ## Argument reference | Argument | Required | Meaning | |---|---|---| | `conc` | Yes | Observed concentrations | | `time` | Yes | Observation times (same units as `time.dose`) | | `time.dose` | Yes | Time(s) at which doses were given | | `route.dose` | No (default `"extravascular"`) | `"extravascular"` or `"intravascular"`; scalar or one value per dose | | `duration.dose` | No (default `NA`) | Dose duration: `0` marks an IV bolus, a positive value an infusion; scalar or one value per dose | | `time.out` | Yes | Target time at which to interpolate/extrapolate | | `out.after` | No (default `FALSE`) | `FALSE` = before dose; `TRUE` = after dose (only meaningful at the instant of an IV bolus) | | `options` | No | PKNCA options list (controls AUC method / interpolation rule) | The table is abridged: `interp.extrap.conc.dose()` also accepts `conc.blq` and `conc.na` (BLQ and `NA` concentration handling), `check` (input validation), and `...` (passed through to the interpolation/extrapolation functions, e.g. `lambda.z`). --- ::: {.callout-note icon=false appearance="minimal"} **pkgdown reference:** [interp.extrap.conc.dose()](https://humanpred.github.io/pknca/reference/interp.extrap.conc.html) · [interp.extrap.conc()](https://humanpred.github.io/pknca/reference/interp.extrap.conc.html) · [pk.calc.c0()](https://humanpred.github.io/pknca/reference/pk.calc.c0.html) · [pk.calc.aucint()](https://humanpred.github.io/pknca/reference/pk.calc.auxcint.html) · [pk.calc.auciv()](https://humanpred.github.io/pknca/reference/pk.calc.auxciv.html) · [PKNCAconc()](https://humanpred.github.io/pknca/reference/PKNCAconc.html) · [PKNCAdose()](https://humanpred.github.io/pknca/reference/PKNCAdose.html) · [PKNCAdata()](https://humanpred.github.io/pknca/reference/PKNCAdata.html) · [pk.nca()](https://humanpred.github.io/pknca/reference/pk.nca.html) :::