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Theophylline (Rovei 1982)

Source: vignettes/articles/Rovei_1982_theophylline.Rmd
Rovei_1982_theophylline.Rmd

Model and source

  • Citation: Rovei V, Chanoine F, Strolin Benedetti M. Pharmacokinetics of theophylline: a dose-range study. Br J Clin Pharmacol. 1982;14(6):769-778. doi:10.1111/j.1365-2125.1982.tb02035.x
  • Description: One-compartment oral PK model for theophylline tablets (Rovei 1982): first-order absorption with lag time in healthy adult volunteers across single oral doses of 125-500 mg.
  • Article: Br J Clin Pharmacol 1982;14(6):769-778

Population

Eight healthy adult Caucasian volunteers (4 male, 4 female), age 22-35 years (mean 29 +/- SD 4), total body weight 48-77 kg (mean 62 +/- 9), lean body weight 33-65 kg (mean 50 +/- 11), all non-smokers on a xanthine-free diet with normal blood pressure (BP 120/80 mmHg), heart rate (65 +/- 8 bpm), hepatic function (SGOT 10 u/l, SGPT 10 +/- 1 u/l, alkaline phosphatase 28 +/- 5 u/l), and renal function (creatinine clearance 119 +/- 8 mL/min). Each subject received single oral 125, 250, 375 and 500 mg doses of theophylline tablets (Theodel) in a 4-period cross-over with at least a 2 week washout (Rovei 1982 Table 1, Methods page 770, Results page 772).

The same information is available programmatically via readModelDb("Rovei_1982_theophylline")$population.

Source trace

Per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Rovei_1982_theophylline.R. The table below collects them for review.

Equation / parameter Value Source location
llag (lag time) log(0.09) h Table 3 page 774: across-dose mean of t_lag column (0.11, 0.09, 0.09, 0.07 h at 125/250/375/500 mg). Paper text “tlag = lag-time (h)” page 771.
lka (absorption rate) log(1.73) 1/h Table 3 page 774: ka = ln(2)/t_abs with across-dose mean t_abs = 0.40 h (0.31, 0.44, 0.29, 0.55). Paper text “t_abs = half life of the absorption phase” page 771.
lcl (apparent CL) log(2.94) L/h Table 3 page 774: across-dose mean CL = 0.042 L/h/kg. Reference WT = 70 kg implies CL = 0.042 * 70 = 2.94 L/h.
lvc (apparent Vc) log(35.7) L Table 3 page 774: across-dose mean Vd = 0.51 L/kg. Reference WT = 70 kg implies Vc = 0.51 * 70 = 35.7 L.
etallag IIV log(1 + 0.50^2) Table 3 page 774: t_lag range 0-0.21 h across doses; CV approximated at 50%.
etalka IIV log(1 + 0.50^2) Table 3 page 774: t_abs range 0.07-1.37 h across doses; paper text page 773 “high intraindividual and interindividual variations of the rate of absorption”; CV approximated at 50%.
etalcl IIV log(1 + 0.25^2) Results page 773: CL = 0.041 +/- 0.007 to 0.043 +/- 0.012 L/h/kg across doses; CV ~17-28%, rounded to 25%.
etalvc IIV log(1 + 0.14^2) Results page 773: Vd = 0.46 +/- 0.07 to 0.54 +/- 0.07 L/kg across doses; CV ~13-15%, rounded to 14%.
propSd 0.10 (10%) Not reported in paper; default proportional residual error chosen for simulation. HPLC assay recovery 85 +/- 5% (Methods page 771-772) supports < 10% measurement CV.
Structure n/a One-compartment open model with first-order absorption and lag time, fit by Gauss-Newton iteration with 1/y^2 weighting (G-PHARM, Gomeni & Gomeni 1978). Pharmacokinetics page 771.
(WT/70)^1 scaling n/a Linear (exponent 1) body-weight scaling on CL and Vc to reproduce the paper’s per-kg parameterization.

Virtual cohort

Original observed data are not publicly available. The cohort below approximates the Rovei 1982 Table 1 demographics: 4 male / 4 female adults aged 22-35 years, body weight 48-77 kg (mean 62 +/- 9 kg), all subjects receiving each of the four dose levels in a balanced cross-over. The simulation expands the trial to 200 virtual subjects per dose to give a well-resolved median + percentile envelope.

set.seed(19821101) # paper accepted 1982-07-19; published Br J Clin Pharmacol 1982 issue 14(6)

n_per_dose <- 200L
doses <- c(125, 250, 375, 500)

make_cohort <- function(dose, n, id_offset = 0L) {
  tibble(
    id        = id_offset + seq_len(n),
    WT        = pmin(pmax(rnorm(n, mean = 62, sd = 9), 48), 77),
    treatment = factor(sprintf("%d mg", dose), levels = sprintf("%d mg", doses)),
    dose_mg   = dose
  )
}

cohort <- bind_rows(lapply(seq_along(doses), function(i) {
  make_cohort(dose = doses[i], n = n_per_dose, id_offset = (i - 1L) * n_per_dose)
}))
obs_times <- sort(unique(c(
  seq(0, 4, by = 0.1),                # dense around absorption / Cmax
  seq(4.5, 12, by = 0.5),
  seq(13, 48, by = 1)
)))

dose_rows <- cohort |>
  mutate(time = 0, amt = dose_mg, cmt = "depot", evid = 1L)

obs_rows <- cohort |>
  tidyr::crossing(time = obs_times) |>
  mutate(amt = 0, cmt = NA_character_, evid = 0L)

events <- bind_rows(dose_rows, obs_rows) |>
  select(id, time, amt, cmt, evid, WT, treatment, dose_mg) |>
  arrange(id, time, desc(evid))

stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))

Simulation 

mod <- rxode2::rxode2(readModelDb("Rovei_1982_theophylline"))
sim <- rxode2::rxSolve(mod, events = events, keep = c("WT", "treatment", "dose_mg"))

Typical-value (no IIV) simulation for the published-Cmax comparison:

mod_typ <- mod |> rxode2::zeroRe()
typ_cohort <- tibble(id = seq_along(doses), WT = 62, treatment = factor(sprintf("%d mg", doses), levels = sprintf("%d mg", doses)), dose_mg = doses)
typ_doses <- typ_cohort |> mutate(time = 0, amt = dose_mg, cmt = "depot", evid = 1L)
typ_obs   <- typ_cohort |> tidyr::crossing(time = obs_times) |> mutate(amt = 0, cmt = NA_character_, evid = 0L)
typ_events <- bind_rows(typ_doses, typ_obs) |>
  select(id, time, amt, cmt, evid, WT, treatment, dose_mg) |>
  arrange(id, time, desc(evid))
sim_typ <- rxode2::rxSolve(mod_typ, events = typ_events, keep = c("WT", "treatment", "dose_mg"))
#> ℹ omega/sigma items treated as zero: 'etallag', 'etalka', 'etalcl', 'etalvc'
#> Warning: multi-subject simulation without without 'omega'

Replicate published figures

Figure 1 – Mean plasma concentration vs. time (semi-log and linear)

Figure 1 of Rovei 1982 plots the mean plasma theophylline concentration vs. time across the 8 subjects for each of the four dose levels (semi-log on the top panel, linear on the bottom). The cohort summary below reproduces that shape, with the 5-50-95 percentile envelope across simulated subjects and the typical-value line overlaid.

sim_summary <- sim |>
  filter(!is.na(Cc), Cc > 0) |>
  group_by(time, treatment) |>
  summarise(
    Q05 = quantile(Cc, 0.05),
    Q50 = quantile(Cc, 0.50),
    Q95 = quantile(Cc, 0.95),
    .groups = "drop"
  )

ggplot(sim_summary, aes(time, Q50, colour = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.15, colour = NA) +
  geom_line(linewidth = 0.7) +
  scale_y_log10(limits = c(0.05, 100)) +
  scale_x_continuous(breaks = seq(0, 48, by = 12)) +
  labs(x = "Time (h)", y = "Plasma theophylline (mg/L), log scale",
    title = "Replicates Figure 1 (semi-log) of Rovei 1982",
    caption = paste("Median (line) and 5-95 percentile envelope (band) across", n_per_dose, "virtual subjects per dose."))
#> Warning: Removed 34 rows containing missing values or values outside the scale range
#> (`geom_ribbon()`).

ggplot(sim_summary |> filter(time <= 12), aes(time, Q50, colour = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.15, colour = NA) +
  geom_line(linewidth = 0.7) +
  labs(x = "Time (h)", y = "Plasma theophylline (mg/L)",
    title = "Replicates Figure 1 (linear) of Rovei 1982",
    caption = "Linear scale, 0-12 h window to highlight the absorption / Cmax phase.")

Figure 2 – Cmax and AUC linearity vs. dose

Figure 2 of Rovei 1982 shows that Cmax and AUC scale linearly with dose (per total body weight, mg/kg). The simulated typical-value Cmax across doses preserves this proportionality.

cmax_typ <- sim_typ |>
  filter(!is.na(Cc)) |>
  group_by(treatment, dose_mg, WT) |>
  summarise(Cmax = max(Cc), .groups = "drop") |>
  mutate(dose_per_kg = dose_mg / WT)

ggplot(cmax_typ, aes(dose_per_kg, Cmax)) +
  geom_point(size = 3) +
  geom_smooth(method = "lm", se = FALSE, formula = y ~ x) +
  labs(x = "Dose / total body weight (mg/kg)", y = "Typical Cmax (mg/L)",
    title = "Replicates Figure 2 (top-left panel, total body weight) of Rovei 1982",
    caption = "Typical-value (no IIV) simulation at WT = 62 kg, the population mean.")

PKNCA validation

Compute single-dose NCA parameters using PKNCA per dose group. The treatment grouping variable (treatment) is placed before id in the formula so per-dose summaries can be compared to the paper.

sim_nca <- sim |>
  filter(!is.na(Cc), Cc > 0) |>
  select(id, time, Cc, treatment)

conc_obj <- PKNCA::PKNCAconc(as.data.frame(sim_nca), Cc ~ time | treatment + id)

dose_df <- events |>
  filter(evid == 1) |>
  select(id, time, amt, treatment) |>
  as.data.frame()

dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id)

intervals <- data.frame(
  start      = 0,
  end        = 48,
  cmax       = TRUE,
  tmax       = TRUE,
  aucinf.obs = TRUE,
  half.life  = TRUE
)

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- suppressMessages(PKNCA::pk.nca(nca_data))
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
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#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
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#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
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#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
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#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
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#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.4) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.5) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.6) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.4) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.5) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
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#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
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#> (0.2) is not allowed
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#> (0.1) is not allowed
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#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
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#> (0.1) is not allowed
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#> (0.2) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.2) is not allowed
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#> Warning: Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (0.1) is not allowed
#> Warning: Requesting an AUC range starting (0) before the first measurement
#> (0.3) is not allowed
nca_summary <- summary(nca_res)
knitr::kable(nca_summary, caption = "Simulated NCA parameters by dose group (median +/- 5/95 percentile across virtual subjects).")
Simulated NCA parameters by dose group (median +/- 5/95 percentile across virtual subjects).
start end treatment N cmax tmax half.life aucinf.obs
0 48 125 mg 200 3.40 [21.6] 1.90 [0.800, 5.00] 8.67 [2.72] NC
0 48 250 mg 200 6.78 [17.4] 1.90 [0.700, 5.00] 8.46 [2.72] NC
0 48 375 mg 200 10.0 [18.8] 2.00 [0.800, 5.50] 8.97 [2.69] NC
0 48 500 mg 200 13.7 [19.1] 1.90 [0.900, 4.50] 8.75 [2.60] NC

Comparison against published NCA

Rovei 1982 Table 3 reports per-dose mean Cmax, tmax, AUC and elimination half-life across the 8 subjects. The side-by-side comparison below uses the typical-value (no IIV) simulation at the population mean weight (62 kg) so the comparison is against the paper’s reported population-mean point estimates.

nca_typ <- sim_typ |>
  filter(!is.na(Cc)) |>
  group_by(treatment, dose_mg) |>
  summarise(
    sim_Cmax_mgL = max(Cc),
    sim_tmax_h   = time[which.max(Cc)],
    .groups = "drop"
  )

paper_table3 <- tibble(
  treatment    = factor(sprintf("%d mg", doses), levels = sprintf("%d mg", doses)),
  paper_Cmax   = c(4.1, 8.0, 9.3, 15.1),
  paper_tmax   = c(1.6, 1.8, 1.6, 2.0),
  paper_AUC    = c(52, 106.1, 161, 210),
  paper_t_half = c(8.6, 8.1, NA, 8.8) # 375 mg t_half range only quoted as 7.0-11.3
)

cmp <- nca_typ |> left_join(paper_table3, by = "treatment")
knitr::kable(cmp, digits = 2,
  caption = "Simulated typical-value Cmax / Tmax (WT = 62 kg) vs. Rovei 1982 Table 3 mean Cmax / tmax / AUC / t_half across 8 subjects.")
Simulated typical-value Cmax / Tmax (WT = 62 kg) vs. Rovei 1982 Table 3 mean Cmax / tmax / AUC / t_half across 8 subjects.
treatment dose_mg sim_Cmax_mgL sim_tmax_h paper_Cmax paper_tmax paper_AUC paper_t_half
125 mg 125 3.39 1.9 4.1 1.6 52.0 8.6
250 mg 250 6.79 1.9 8.0 1.8 106.1 8.1
375 mg 375 10.18 1.9 9.3 1.6 161.0 NA
500 mg 500 13.58 1.9 15.1 2.0 210.0 8.8

The simulated typical-value Cmax is within ~20% of the paper’s Table 3 mean Cmax across all four doses (3.4 vs 4.1; 6.8 vs 8.0; 10.2 vs 9.3; 13.6 vs 15.1 mg/L). The slight under-prediction at low doses is consistent with the paper’s reported variability across 8 subjects (Table 3 ranges) and with the deterministic typical-value simulation not reproducing the per-subject distribution of Vd values that drive Cmax. Tmax (2 h simulated vs 1.6-2.0 h paper) and elimination half-life (8.4 h derived vs 8.1-8.8 h paper Table 3) match closely.

Assumptions and deviations

  • Source is a classical PK study, not a population PK fit. Rovei 1982 describes individual subject fits to a one-compartment open model with Gauss-Newton iteration, then summarizes the eight individual parameter estimates as means and ranges (Table 3) and means +/- SDs (Results page 773 narrative). The parameter values in ini() are the across-dose means; IIV omegas are approximated from the inter-subject SD/mean (CV) reported for each dose. These are not formal nonlinear-mixed-effect omega estimates.
  • Across-dose pooling. The paper concludes (page 773) that t_lag, t_abs, t_max, t_beta, CL, CL_R, Vd and F are not modified by dose (ANOVA at four doses, 8 subjects each), so the per-dose values from Table 3 are pooled to a single typical value for each parameter.
  • Reference body weight. The paper reports CL and Vd in L/h/kg and L/kg (per body weight). The model adopts a 70 kg reference for CL and Vc (a common adult reference) and applies linear (exponent = 1) WT scaling to reproduce the paper’s per-kg parameterization. At the population mean weight 62 kg, the model yields CL = 2.60 L/h and Vc = 31.6 L, equivalent to the paper’s 0.042 L/h/kg and 0.51 L/kg.
  • Bioavailability F = 1. The paper reports total urinary recovery (theophylline + 3-MX + 1-MU + 1,3-DMU) of 70-90% of the dose (Results pages 772-773) and concludes “the virtually complete recovery from urine of the given dose after each administration indicates the good absorption of the drug” (Discussion page 775). The CL reported in Table 3 is CL = F * Dose / AUC (page 771). The model assumes F = 1 throughout, so lcl is exposed as apparent oral clearance CL/F.
  • Residual error not reported. Rovei 1982 does not publish a residual error model. The vignette uses a 10% proportional residual error as a conservative simulation-only default. HPLC assay recovery was reported as 85 +/- 5% (Methods page 771-772), suggesting an analytical CV below 10% on top of any model misspecification.
  • IIV on absorption parameters approximate. t_abs ranged 0.07-1.37 h across subjects (Table 3, 500 mg dose); t_lag ranged 0-0.21 h. The paper describes “high intraindividual and interindividual variations of the rate of absorption” (page 773). Using a 50% CV for both etalka and etallag reproduces the order-of-magnitude variability observed but is not a formally fitted estimate.
  • Paper reports a “t_lag” half-life. Table 3’s row label reads “t_lag (h)” but the paper text on page 771 defines tlag = lag-time (h), not a half-life of a delay process. The model interprets the value as a lag time and applies it via lag(depot).
  • Paper does not report formal covariate models. No covariate effects on CL, Vc, ka or lag are claimed in the source (the paper’s ANOVA tested dose effects only). WT is included as a covariate solely to reproduce the per-kg parameterization; AGE, SEX, and other demographic variables are not used in the model.