11 Superposition

11.1 What superposition does

Superposition predicts steady-state concentration-time profiles from a single-dose profile by linearly summing lagged copies of the single-dose curve. This is the NCA equivalent of multi-dose simulation, and it requires no compartmental model.

The core assumption is linear, time-invariant pharmacokinetics: the drug does not accumulate non-linearly, and clearance and volume do not change over time.

flowchart LR
    A["Single dose\nprofile (0→∞)"] --> B["Shift by τ,\nshift by 2τ, ..."]
    B --> C["Sum contributions\nat each time t"]
    C --> D["Predicted SS\nprofile (0→τ)"]

11.2 Basic superposition

# Use Theoph subject 1 as single-dose reference
subj1 <- d_conc |> filter(subject == "1")
actual_dose <- d_dose |> filter(subject == "1") |> pull(dose)

o_conc_1 <- PKNCAconc(subj1, conc ~ time | subject)

# Predict steady-state profile with τ = 24 h, show 3 dosing intervals
ss_profile <- superposition(
  o_conc_1,
  tau         = 24,           # dosing interval (h)
  dose.input  = actual_dose,  # dose used to generate the single-dose data
  dose.amount = actual_dose,  # dose to simulate at SS (same here)
  n.tau       = 3,            # simulate 3 intervals
  check.blq   = FALSE         # Theoph has non-zero first sample
)

head(ss_profile, 12)

# A tibble: 12 × 3
   subject  conc   time
   <ord>   <dbl>  <dbl>
 1 1        5.12  0    
 2 1        7.17  0.25 
 3 1        8.54  0.370
 4 1       10.8   0.57 
 5 1       14.7   1.12 
 6 1       13.6   2.02 
 7 1       12.2   3.82 
 8 1       11.8   5.1  
 9 1       10.6   7.03 
10 1        9.72  9.05 
11 1        8.38 12.1  
12 1        4.71 24

11.3 Visualising the approach to steady state

With n.tau = Inf, superposition runs until the profile converges to steady state (within steady.state.tol).

ss_converged <- superposition(
  o_conc_1,
  tau         = 24,
  dose.input  = actual_dose,
  dose.amount = actual_dose,
  n.tau       = Inf,
  check.blq   = FALSE
)

ggplot(ss_converged, aes(x = time, y = conc)) +
  geom_line(colour = "steelblue") +
  geom_point(size = 2, colour = "steelblue") +
  labs(title = "Superposition: predicted steady-state profile (τ = 24 h)",
       x = "Time (h)", y = "Predicted SS concentration (mg/L)") +
  theme_minimal()

11.4 Changing dose at steady state

To predict the effect of a dose change, set dose.amount to the new dose while keeping dose.input as the original dose. PKNCA scales the concentrations proportionally.

ss_half_dose <- superposition(
  o_conc_1,
  tau         = 24,
  dose.input  = actual_dose,
  dose.amount = actual_dose / 2,   # half dose
  n.tau       = Inf,
  check.blq   = FALSE
)

bind_rows(
  ss_converged   |> mutate(regimen = "Full dose"),
  ss_half_dose   |> mutate(regimen = "Half dose")
) |>
  ggplot(aes(x = time, y = conc, colour = regimen)) +
  geom_line() + geom_point(size = 2) +
  labs(title = "Dose scaling via superposition",
       x = "Time (h)", y = "Predicted SS concentration (mg/L)") +
  scale_colour_manual(values = c("Full dose" = "steelblue", "Half dose" = "coral")) +
  theme_minimal()

11.5 Changing the dosing interval (τ)

ss_8h <- superposition(
  o_conc_1,
  tau         = 8,
  dose.input  = actual_dose,
  dose.amount = actual_dose,
  n.tau       = Inf,
  check.blq   = FALSE
)

ggplot(ss_8h, aes(x = time, y = conc)) +
  geom_line(colour = "firebrick") +
  geom_point(size = 2, colour = "firebrick") +
  labs(title = "Superposition with τ = 8 h (more frequent dosing)",
       x = "Time within interval (h)", y = "Predicted SS concentration (mg/L)") +
  theme_minimal()

11.6 Extracting accumulation ratio

The accumulation ratio (Rac) is Cmax at steady state divided by Cmax after the first dose:

cmax_sd <- max(subj1$conc, na.rm = TRUE)
cmax_ss <- max(ss_converged$conc, na.rm = TRUE)
rac      <- cmax_ss / cmax_sd
cat("Accumulation ratio (Rac):", round(rac, 2), "\n")

Accumulation ratio (Rac): 1.44

cat("Single-dose Cmax:", round(cmax_sd, 2), "mg/L\n")

Single-dose Cmax: 10.5 mg/L

cat("Steady-state Cmax:", round(cmax_ss, 2), "mg/L\n")

Steady-state Cmax: 15.1 mg/L

11.7 Key arguments reference

Argument	Required	Meaning
`tau`	Yes	Dosing interval (same time units as data)
`dose.input`	When `dose.amount` ≠ NULL	Dose used to generate the observed profile
`dose.amount`	When `dose.input` ≠ NULL	Dose to simulate (for dose scaling)
`n.tau`	No (default `Inf`)	Number of intervals to simulate before stopping
`steady.state.tol`	No (default 0.001)	Convergence tolerance for `n.tau = Inf`
`check.blq`	No (default `TRUE`)	Require first concentration = 0; set `FALSE` if first sample is non-zero
`auc.type`	No (default `"AUCinf"`)	How to extrapolate the single-dose tail: `"AUCinf"` or `"AUClast"`
`additional.times`	No	Extra timepoints to include in the output

Limitation: Superposition assumes linear PK. It should not be used for drugs with non-linear kinetics (e.g., Michaelis-Menten elimination, time-varying clearance, or saturable protein binding).

pkgdown reference: PKNCAconc() · superposition()

--- title: "Superposition" --- ```{r setup, include=FALSE} library(PKNCA) library(dplyr) library(ggplot2) conflicted::conflicts_prefer(dplyr::filter, dplyr::select, .quiet = TRUE) d_conc <- as.data.frame(Theoph) |> rename(time = Time, subject = Subject) d_dose <- Theoph |> as.data.frame() |> group_by(Subject) |> summarise(dose = Dose[1] * Wt[1], .groups = "drop") |> rename(subject = Subject) |> mutate(time = 0) ``` ## What superposition does **Superposition** predicts steady-state concentration-time profiles from a single-dose profile by linearly summing lagged copies of the single-dose curve. This is the NCA equivalent of multi-dose simulation, and it requires no compartmental model. The core assumption is **linear, time-invariant pharmacokinetics**: the drug does not accumulate non-linearly, and clearance and volume do not change over time. ```{mermaid} flowchart LR A["Single dose\nprofile (0→∞)"] --> B["Shift by τ,\nshift by 2τ, ..."] B --> C["Sum contributions\nat each time t"] C --> D["Predicted SS\nprofile (0→τ)"] ``` --- ## Basic superposition ```{r} # Use Theoph subject 1 as single-dose reference subj1 <- d_conc |> filter(subject == "1") actual_dose <- d_dose |> filter(subject == "1") |> pull(dose) o_conc_1 <- PKNCAconc(subj1, conc ~ time | subject) # Predict steady-state profile with τ = 24 h, show 3 dosing intervals ss_profile <- superposition( o_conc_1, tau = 24, # dosing interval (h) dose.input = actual_dose, # dose used to generate the single-dose data dose.amount = actual_dose, # dose to simulate at SS (same here) n.tau = 3, # simulate 3 intervals check.blq = FALSE # Theoph has non-zero first sample ) head(ss_profile, 12) ``` --- ## Visualising the approach to steady state With `n.tau = Inf`, superposition runs until the profile converges to steady state (within `steady.state.tol`). ```{r} ss_converged <- superposition( o_conc_1, tau = 24, dose.input = actual_dose, dose.amount = actual_dose, n.tau = Inf, check.blq = FALSE ) ggplot(ss_converged, aes(x = time, y = conc)) + geom_line(colour = "steelblue") + geom_point(size = 2, colour = "steelblue") + labs(title = "Superposition: predicted steady-state profile (τ = 24 h)", x = "Time (h)", y = "Predicted SS concentration (mg/L)") + theme_minimal() ``` --- ## Changing dose at steady state To predict the effect of a dose change, set `dose.amount` to the new dose while keeping `dose.input` as the original dose. PKNCA scales the concentrations proportionally. ```{r} ss_half_dose <- superposition( o_conc_1, tau = 24, dose.input = actual_dose, dose.amount = actual_dose / 2, # half dose n.tau = Inf, check.blq = FALSE ) bind_rows( ss_converged |> mutate(regimen = "Full dose"), ss_half_dose |> mutate(regimen = "Half dose") ) |> ggplot(aes(x = time, y = conc, colour = regimen)) + geom_line() + geom_point(size = 2) + labs(title = "Dose scaling via superposition", x = "Time (h)", y = "Predicted SS concentration (mg/L)") + scale_colour_manual(values = c("Full dose" = "steelblue", "Half dose" = "coral")) + theme_minimal() ``` --- ## Changing the dosing interval (τ) ```{r} ss_8h <- superposition( o_conc_1, tau = 8, dose.input = actual_dose, dose.amount = actual_dose, n.tau = Inf, check.blq = FALSE ) ggplot(ss_8h, aes(x = time, y = conc)) + geom_line(colour = "firebrick") + geom_point(size = 2, colour = "firebrick") + labs(title = "Superposition with τ = 8 h (more frequent dosing)", x = "Time within interval (h)", y = "Predicted SS concentration (mg/L)") + theme_minimal() ``` --- ## Extracting accumulation ratio The **accumulation ratio (Rac)** is Cmax at steady state divided by Cmax after the first dose: ```{r} cmax_sd <- max(subj1$conc, na.rm = TRUE) cmax_ss <- max(ss_converged$conc, na.rm = TRUE) rac <- cmax_ss / cmax_sd cat("Accumulation ratio (Rac):", round(rac, 2), "\n") cat("Single-dose Cmax:", round(cmax_sd, 2), "mg/L\n") cat("Steady-state Cmax:", round(cmax_ss, 2), "mg/L\n") ``` --- ## Key arguments reference | Argument | Required | Meaning | |---|---|---| | `tau` | Yes | Dosing interval (same time units as data) | | `dose.input` | When `dose.amount` ≠ NULL | Dose used to generate the observed profile | | `dose.amount` | When `dose.input` ≠ NULL | Dose to simulate (for dose scaling) | | `n.tau` | No (default `Inf`) | Number of intervals to simulate before stopping | | `steady.state.tol` | No (default 0.001) | Convergence tolerance for `n.tau = Inf` | | `check.blq` | No (default `TRUE`) | Require first concentration = 0; set `FALSE` if first sample is non-zero | | `auc.type` | No (default `"AUCinf"`) | How to extrapolate the single-dose tail: `"AUCinf"` or `"AUClast"` | | `additional.times` | No | Extra timepoints to include in the output | > **Limitation:** Superposition assumes linear PK. It should not be used for drugs with non-linear kinetics (e.g., Michaelis-Menten elimination, time-varying clearance, or saturable protein binding). --- ::: {.callout-note icon=false appearance="minimal"} **pkgdown reference:** [PKNCAconc()](https://humanpred.github.io/pknca/reference/PKNCAconc.html) · [superposition()](https://humanpred.github.io/pknca/reference/superposition.html) :::