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Indacaterol, glycopyrronium and mometasone furoate (Bartels 2021)

Source: vignettes/articles/Bartels_2021_indacaterol_glycopyrronium_mometasone.Rmd
Bartels_2021_indacaterol_glycopyrronium_mometasone.Rmd

Model and source

Bartels 2021 developed one population PK model per component of the indacaterol/glycopyrronium/mometasone furoate (IND/GLY/MF) inhaled triple therapy, fitted to asthma patients who received IND/MF or IND/GLY/MF via the Breezhaler device. The three models are independent and ship as three files that share this article:

  • Bartels_2021_indacaterol: Two-compartment population PK model with sequential zero-order/first-order absorption for inhaled indacaterol in adults and adolescents with asthma receiving the indacaterol/mometasone furoate (IND/MF) or indacaterol/glycopyrronium/mometasone furoate (IND/GLY/MF) fixed-dose combinations via the Breezhaler device (PALLADIUM and IRIDIUM Phase III studies), with estimated allometric body-weight exponents on CL/F and Vc/F, fixed allometric exponents on Q/F and Vp/F, a Japanese-ethnicity effect on Vc/F and an IRIDIUM study effect on Vc/F (Bartels 2021)
  • Bartels_2021_glycopyrronium: Two-compartment population PK model with bolus input for inhaled glycopyrronium in adults and adolescents with asthma receiving the indacaterol/glycopyrronium/mometasone furoate (IND/GLY/MF) fixed-dose combination or glycopyrronium monotherapy via the Breezhaler device (IRIDIUM Phase III study), with fixed allometric body-weight exponents on all clearance and volume terms and grouped-race (Japanese, other) effects on Vc/F (Bartels 2021)
  • Bartels_2021_mometasoneFuroate: Two-compartment population PK model with mixed (simultaneous) zero-order/first-order absorption for inhaled mometasone furoate in adults and adolescents with asthma receiving the indacaterol/mometasone furoate (IND/MF) or indacaterol/glycopyrronium/mometasone furoate (IND/GLY/MF) fixed-dose combinations, or MF monotherapy, via the Breezhaler device (PALLADIUM, IRIDIUM and E2201 studies), with estimated allometric body-weight exponents on CL/F and Vc/F, baseline FEV1 on CL/F and Vc/F, and formulation (IND/GLY/MF; medium-dose MF strength) and IRIDIUM-study effects on Vc/F and relative bioavailability (Bartels 2021). The MF Twisthaler monotherapy comparator arms are not covered: their Vc/F, F and Vp/F formulation effects are not reported.
  • Citation: Bartels C, Jain M, Yu J, Tillmann HC, Vaidya S. Population Pharmacokinetic Analysis of Indacaterol/Glycopyrronium/Mometasone Furoate After Administration of Combination Therapies Using the Breezhaler Device in Patients with Asthma. Eur J Drug Metab Pharmacokinet. 2021;46(4):489-506. doi:10.1007/s13318-021-00689-x
  • Article (open access): https://doi.org/10.1007/s13318-021-00689-x
  • Supplement (Online Resources 1-4): https://static-content.springer.com/esm/art%3A10.1007%2Fs13318-021-00689-x/MediaObjects/13318_2021_689_MOESM1_ESM.pdf

No correction notice for the article was registered with CrossRef as of 2026-09-28.

Population

The analysis pooled 698 patients with asthma from two Phase III studies, PALLADIUM (NCT02554786, n = 273) and IRIDIUM (NCT02571777, n = 249), and the Phase II Breezhaler-versus-Twisthaler device-bridging study E2201 (NCT01555151, n = 176) (Bartels 2021 Table 4). 398 patients (57%) were female. The grouped-race covariate had 523 Caucasian/White, 107 Japanese and 68 other patients. Study means were 44.6-52.7 years of age (range 11-79), 74.7-82 kg of body weight (range 33.6-156 kg), 1.9-2.1 L baseline FEV1 and 84-96 mL/min/1.73 m2 eGFR. The Phase III studies sampled sparsely up to 1 h after the dose on Days 30 and 84/86. E2201 took six MF samples over 24 h.

Each drug was fitted to the studies that gave it:

  • Indacaterol, 150 ug once daily, in PALLADIUM and IRIDIUM, as IND/MF 150/160 or 150/320 ug or IND/GLY/MF 150/50/80 or 150/50/160 ug.
  • Glycopyrronium, 50 ug once daily, in IRIDIUM, as IND/GLY/MF.
  • Mometasone furoate in all three studies, as the two FDCs, as MF alone via Breezhaler (80 or 320 ug, E2201), or as MF alone via Twisthaler (200-800 ug per day, PALLADIUM and E2201).

The population element of each model holds the same information, for example readModelDb("Bartels_2021_indacaterol")()$population.

Source trace

Every ini() value carries an in-file comment naming its source. Unless stated otherwise, the values come from Bartels 2021 Table 5, which gives the final-model estimates for all three drugs. Monolix reports random effects as SDs, so each variance below is SD squared and each CL/F-Vc/F covariance is r x SD_CL x SD_Vc.

Parameter IND GLY MF Source
lcl (CL/F, L/h) log(54) log(89) log(210) Table 5
lvc (Vc/F, L) log(600) log(440) log(1800) Table 5
lq (Q/F, L/h) log(380) log(350) log(250) Table 5
lvp (Vp/F, L) log(5700) fixed(log(1300)) log(3700) Table 5 (GLY fixed from the COPD model, Sect. 2.3)
lka (Ka, 1/h) fixed(log(50)) - log(2.4) Table 5
ld1 (zero-order duration, h) log(0.055) - fixed(log(0.01)) Table 5
logitffo (logit of 1 - Fr) qlogis(1 - 0.33) - qlogis(1 - 0.37) Table 5 ‘Fr’
e_wt_cl 0.28 fixed(0.75) 0.34 Table 5
e_wt_vc 0.43 fixed(1) 0.33 Table 5
e_wt_q, e_wt_vp fixed(0.75), fixed(1) fixed(0.75), fixed(1) fixed(0.75), fixed(1) Table 5
e_race_japanese_vc -0.29 -0.65 - Table 5
e_race_other_vc not reported -0.063 - Sect. 3.4 text (GLY)
e_study_iridium_vc 0.25 - 0.19 Table 5
e_fev1_cl, e_fev1_vc - - -0.17, -0.23 Table 5
e_form_mf_indglymf_vc - - -0.32 Table 5
e_form_mf_indglymf_f - - fixed(2) Sect. 2.3 factor 0.5, applied as a dose divisor
e_form_mf_medium_f - - 0.18 Table 5
etalcl + etalvc c(0.2304, 0.098496, 0.1444) c(0.1521, 0.10335, 0.2809) c(0.2401, 0.07644, 0.1521) Table 5 SDs and correlations
etalq 0.0225 - 0.1521 Table 5
etalvp 1.69 - 4 Table 5
etalogitffo 1.44 - 0.0484 Table 5 ‘BSV on Fr’
propSd 0.24 0.34 0.36 Table 5
Covariate form: power of WT/75, FEV1/2; exp(theta x indicator) Sect. 2.4 Eqs. 1-2, Table 3 reference values
Two-compartment disposition, first-order elimination Sect. 3.4
IND: sequential zero-order (fraction Fr over D) then first-order Sect. 3.2, 3.4
GLY: bolus input to the central compartment Sect. 3.2
MF: simultaneous zero-order (Fr over D) and first-order Sect. 3.2, 3.4

Dosing the models

The IND and MF models split each inhalation between a zero-order input to the central compartment and a first-order input from the depot. Every inhaled dose is therefore two records with the same amt at the same time: one on cmt = "central" with rate = -2, so rxode2 applies the modelled duration d1, and one on cmt = "depot". The model’s f() statements divide the dose between the two records. Doses are the nominal ug of each component (MF as the nominal ug of the FDC strength). The GLY model takes one bolus record on cmt = "central". Concentrations are returned in pg/mL.

rxode2 cannot combine ss = 2 with a modelled f(), so steady state cannot be imposed with ss = 1 / ss = 2 pairs on these split doses. The article reaches steady state with a 60-day once-daily run-in instead. The IND terminal half-life is about 90 h, and 14 days leaves IND about 8% short of steady state.

n_run <- 60 # once-daily doses before the evaluated interval
t_last <- (n_run - 1) * 24 # time of the last dose

# One dose-plus-observation event table per subject. `subj` has one row per
# subject: an `id`, every covariate, the eta columns and a `variant` label.
make_events <- function(subj, amt, split, obs_tad) {
  dose_t <- (seq_len(n_run) - 1) * 24
  doses <- if (split) {
    tidyr::expand_grid(id = subj$id, time = dose_t, cmt = c("central", "depot")) |>
      dplyr::mutate(rate = ifelse(cmt == "central", -2, 0))
  } else {
    tidyr::expand_grid(id = subj$id, time = dose_t, cmt = "central") |>
      dplyr::mutate(rate = 0)
  }
  doses <- dplyr::mutate(doses, evid = 1L, amt = amt)
  obs <- tidyr::expand_grid(id = subj$id, time = t_last + obs_tad) |>
    dplyr::mutate(cmt = "central", rate = 0, evid = 0L, amt = 0)
  dplyr::bind_rows(doses, obs) |>
    dplyr::left_join(subj, by = "id") |>
    dplyr::arrange(id, time, dplyr::desc(evid))
}

# Draw subject-level random effects in R. Solving the zeroRe() model with these
# columns in the data gives each virtual subject the same etas in every
# covariate variant, and makes the cohort identical on every machine.
draw_etas <- function(mod, n) {
  om <- rxode2::rxode2(mod)$omega
  z <- matrix(stats::rnorm(n * ncol(om)), n) %*% chol(om)
  colnames(z) <- colnames(om)
  as.data.frame(z)
}

# PKNCA run separately for each treatment. One call over all treatments
# spends most of its time subsetting the full data set for every subject, and
# took more than 5 min for the covariate cohort below; per-treatment calls take
# about 2 s each. Returns the long `$result` data frame.
nca_one <- function(d, intervals) {
  dose <- d |> dplyr::distinct(treatment, id) |> dplyr::mutate(tad = 0, amt = 1)
  res <- PKNCA::pk.nca(PKNCA::PKNCAdata(
    PKNCA::PKNCAconc(d, Cc ~ tad | treatment + id),
    PKNCA::PKNCAdose(dose, amt ~ tad | treatment + id),
    intervals = intervals
  ))
  as.data.frame(res$result)
}
nca_by_treatment <- function(conc, intervals) {
  dplyr::bind_rows(lapply(split(conc, conc$treatment), nca_one, intervals = intervals))
}

mod_ind <- rxode2::rxode2(readModelDb("Bartels_2021_indacaterol"))
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_gly <- rxode2::rxode2(readModelDb("Bartels_2021_glycopyrronium"))
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_mf <- rxode2::rxode2(readModelDb("Bartels_2021_mometasoneFuroate"))
#> ℹ parameter labels from comments will be replaced by 'label()'

Typical-value profiles (Figure 4)

Figure 4 of Bartels 2021 shows the steady-state typical profile of each drug under each FDC. The IND curves of all four FDCs coincide, as do the GLY curves of both IND/GLY/MF strengths. The MF curves differ by product and strength. The maintainers digitised the curves from the vector graphic in the article PDF. The coordinates are exact, so the only error is the width of the plotted line, about 1% on the log axis. The figure’s IND and MF curves match the IRIDIUM study effect (the only study that gave all four FDCs), so the typical patient below is a 75 kg Caucasian from IRIDIUM with a baseline FEV1 of 2 L.

fig4_digitised <- tibble::tribble(
  ~drug, ~product, ~tad, ~conc,
  "IND", "IND/GLY/MF 150/50/160", 0.5, 243.1,
  "IND", "IND/GLY/MF 150/50/160", 1, 206.6,
  "IND", "IND/GLY/MF 150/50/160", 2, 161.2,
  "IND", "IND/GLY/MF 150/50/160", 4, 124.7,
  "IND", "IND/GLY/MF 150/50/160", 6, 113.8,
  "IND", "IND/GLY/MF 150/50/160", 8, 110.8,
  "IND", "IND/GLY/MF 150/50/160", 12, 101.5,
  "IND", "IND/GLY/MF 150/50/160", 16, 95.6,
  "IND", "IND/GLY/MF 150/50/160", 20, 91.9,
  "IND", "IND/GLY/MF 150/50/160", 23.9, 89.8,
  "GLY", "IND/GLY/MF 150/50/160", 0.5, 84.5,
  "GLY", "IND/GLY/MF 150/50/160", 1, 57.6,
  "GLY", "IND/GLY/MF 150/50/160", 2, 37.9,
  "GLY", "IND/GLY/MF 150/50/160", 4, 29.2,
  "GLY", "IND/GLY/MF 150/50/160", 6, 26.2,
  "GLY", "IND/GLY/MF 150/50/160", 8, 23.9,
  "GLY", "IND/GLY/MF 150/50/160", 12, 20.0,
  "GLY", "IND/GLY/MF 150/50/160", 16, 16.9,
  "GLY", "IND/GLY/MF 150/50/160", 20, 14.2,
  "GLY", "IND/GLY/MF 150/50/160", 23.9, 11.9,
  "MF", "IND/GLY/MF 150/50/160", 0.5, 171.6,
  "MF", "IND/GLY/MF 150/50/160", 1, 175.5,
  "MF", "IND/GLY/MF 150/50/160", 2, 142.4,
  "MF", "IND/GLY/MF 150/50/160", 4, 97.6,
  "MF", "IND/GLY/MF 150/50/160", 6, 73.9,
  "MF", "IND/GLY/MF 150/50/160", 8, 59.0,
  "MF", "IND/GLY/MF 150/50/160", 12, 43.2,
  "MF", "IND/GLY/MF 150/50/160", 16, 33.5,
  "MF", "IND/GLY/MF 150/50/160", 20, 28.8,
  "MF", "IND/GLY/MF 150/50/160", 23.9, 24.9,
  "MF", "IND/MF 150/320", 0.5, 140.2,
  "MF", "IND/MF 150/320", 1, 147.7,
  "MF", "IND/MF 150/320", 2, 130.1,
  "MF", "IND/MF 150/320", 4, 92.9,
  "MF", "IND/MF 150/320", 6, 75.9,
  "MF", "IND/MF 150/320", 12, 46.8,
  "MF", "IND/MF 150/320", 16, 37.7,
  "MF", "IND/MF 150/320", 20, 32.4,
  "MF", "IND/MF 150/320", 23.9, 28.1,
  "MF", "IND/GLY/MF 150/50/80", 0.5, 98.8,
  "MF", "IND/GLY/MF 150/50/80", 1, 105.8,
  "MF", "IND/GLY/MF 150/50/80", 2, 85.6,
  "MF", "IND/GLY/MF 150/50/80", 4, 57.3,
  "MF", "IND/GLY/MF 150/50/80", 6, 44.8,
  "MF", "IND/GLY/MF 150/50/80", 8, 35.5,
  "MF", "IND/GLY/MF 150/50/80", 12, 25.9,
  "MF", "IND/GLY/MF 150/50/80", 16, 20.1,
  "MF", "IND/GLY/MF 150/50/80", 20, 17.3,
  "MF", "IND/GLY/MF 150/50/80", 23.9, 15.0,
  "MF", "IND/MF 150/160", 0.5, 84.3,
  "MF", "IND/MF 150/160", 2, 78.1,
  "MF", "IND/MF 150/160", 4, 57.1,
  "MF", "IND/MF 150/160", 8, 38.1,
  "MF", "IND/MF 150/160", 12, 28.7
)
# Digitised curve maxima (concentration only; the time of a sub-hour peak is
# not resolvable at the plotted line width).
fig4_peaks <- tibble::tribble(
  ~drug, ~product, ~cmax_fig,
  "IND", "IND/GLY/MF 150/50/160", 278.7,
  "GLY", "IND/GLY/MF 150/50/160", 129.7,
  "MF", "IND/GLY/MF 150/50/160", 176.9,
  "MF", "IND/MF 150/320", 148.2,
  "MF", "IND/GLY/MF 150/50/80", 106.3,
  "MF", "IND/MF 150/160", 88.4
)
# MF products: nominal dose and the two formulation indicators.
mf_products <- tibble::tribble(
  ~product, ~amt, ~FORM_MF_INDGLYMF, ~FORM_MF_MEDIUM,
  "IND/MF 150/320", 320, 0, 0,
  "IND/GLY/MF 150/50/160", 160, 1, 0,
  "IND/MF 150/160", 160, 0, 1,
  "IND/GLY/MF 150/50/80", 80, 1, 1
)
tad_fine <- sort(unique(round(c(seq(0.005, 1, by = 0.005), seq(1, 24, by = 0.05)), 6)))

typ_one <- function(mod, amt, split, cov, drug, product) {
  ev <- make_events(dplyr::bind_cols(tibble::tibble(id = 1L), cov), amt, split, tad_fine)
  rxode2::rxSolve(rxode2::zeroRe(mod), events = ev, returnType = "data.frame") |>
    dplyr::transmute(drug = drug, product = product, tad = round(time - t_last, 6), Cc)
}

typ <- dplyr::bind_rows(
  typ_one(mod_ind, 150, TRUE, tibble::tibble(WT = 75, RACE_JAPANESE = 0, STUDY_IRIDIUM = 1),
          "IND", "IND/GLY/MF 150/50/160"),
  typ_one(mod_gly, 50, FALSE, tibble::tibble(WT = 75, RACE_JAPANESE = 0, RACE_OTHER = 0),
          "GLY", "IND/GLY/MF 150/50/160"),
  dplyr::bind_rows(lapply(seq_len(nrow(mf_products)), function(i) {
    p <- mf_products[i, ]
    typ_one(mod_mf, p$amt, TRUE,
            tibble::tibble(WT = 75, FEV1 = 2, STUDY_IRIDIUM = 1,
                           FORM_MF_INDGLYMF = p$FORM_MF_INDGLYMF, FORM_MF_MEDIUM = p$FORM_MF_MEDIUM),
            "MF", p$product)
  }))
)
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'

fig4_check <- fig4_digitised |>
  dplyr::inner_join(typ, by = c("drug", "product", "tad"), relationship = "one-to-one") |>
  dplyr::mutate(pct_diff = 100 * (Cc / conc - 1))
peak_check <- typ |>
  dplyr::group_by(drug, product) |>
  dplyr::summarise(cmax_model = max(Cc), tmax_model = tad[which.max(Cc)], .groups = "drop") |>
  dplyr::inner_join(fig4_peaks, by = c("drug", "product")) |>
  dplyr::mutate(pct_diff = 100 * (cmax_model / cmax_fig - 1))
stopifnot(nrow(fig4_check) == nrow(fig4_digitised), nrow(peak_check) == nrow(fig4_peaks))

fig4_check |>
  dplyr::group_by(drug, product) |>
  dplyr::summarise(
    n_points = dplyr::n(),
    `max abs % diff, tad <= 6 h` = max(abs(pct_diff[tad <= 6])),
    `max abs % diff, tad > 6 h` = max(abs(pct_diff[tad > 6])),
    .groups = "drop"
  ) |>
  dplyr::left_join(
    peak_check |> dplyr::select(drug, product, `Cmax % diff` = pct_diff), by = c("drug", "product")
  ) |>
  knitr::kable(digits = 1, caption = "Typical-value model vs. digitised Figure 4 curves.")
Typical-value model vs. digitised Figure 4 curves.
drug product n_points max abs % diff, tad <= 6 h max abs % diff, tad > 6 h Cmax % diff
GLY IND/GLY/MF 150/50/160 10 3.3 4.6 -4.1
IND IND/GLY/MF 150/50/160 10 2.1 6.4 0.1
MF IND/GLY/MF 150/50/160 10 6.9 19.6 3.2
MF IND/GLY/MF 150/50/80 10 6.8 18.8 2.8
MF IND/MF 150/160 5 5.1 3.3 0.8
MF IND/MF 150/320 9 7.9 14.6 0.5

The IND and GLY curves reproduce Figure 4 to within 7% at every digitised point, and within 5% at the peak. The MF curves match through the absorption and early distribution phase and at the peak. From about 8 h onward the model runs up to about 20% above the figure for three of the four products (see the second assertion block and the Assumptions section).

# Deterministic comparison (typical values, no random numbers). Measured maxima:
# IND 6.4%, GLY 4.6% (all points); peaks 4.1% (GLY), MF tad <= 6 h 7.9%.
stopifnot(
  max(abs(fig4_check$pct_diff[fig4_check$drug != "MF"])) < 9,
  max(abs(peak_check$pct_diff)) < 6,
  max(abs(fig4_check$pct_diff[fig4_check$drug == "MF" & fig4_check$tad <= 6])) < 11
)
# Known deviation, kept visible rather than hidden: after 6 h the MF points sit
# above the figure, by up to 15-20% for three of the four products (measured
# maximum 19.6%). Guard only against a gross error such as a dropped
# peripheral compartment.
stopifnot(max(abs(fig4_check$pct_diff[fig4_check$drug == "MF" & fig4_check$tad > 6])) < 30)

Covariate simulations (Section 3.6)

Section 3.6 of Bartels 2021 quantifies the covariate effects by simulating steady-state profiles for 200 virtual patients per covariate variant, with BSV and without residual error. It samples every 6 min over 24 h and compares population-mean AUC0-24h, Cmax and Ctrough against a 75 kg reference patient. The simulation below repeats that design. The same 200 sets of random effects are used in every variant, so each ratio compares like with like. The paired design, together with R’s own random-number generator, makes the cohort identical on every machine.

set.seed(20210522)
n_sub <- 200 # per variant (Bartels 2021 Sect. 2.6)
etas <- list(
  IND = draw_etas(mod_ind, n_sub),
  GLY = draw_etas(mod_gly, n_sub),
  MF = draw_etas(mod_mf, n_sub)
)

variants <- tibble::tribble(
  ~drug, ~variant, ~amt, ~WT, ~RACE_JAPANESE, ~RACE_OTHER, ~STUDY_IRIDIUM, ~FEV1, ~FORM_MF_INDGLYMF, ~FORM_MF_MEDIUM,
  "IND", "reference (PALLADIUM)", 150, 75, 0, 0, 0, NA, NA, NA,
  "IND", "55 kg", 150, 55, 0, 0, 0, NA, NA, NA,
  "IND", "105 kg", 150, 105, 0, 0, 0, NA, NA, NA,
  "IND", "Japanese", 150, 75, 1, 0, 0, NA, NA, NA,
  "IND", "IRIDIUM", 150, 75, 0, 0, 1, NA, NA, NA,
  "GLY", "reference", 50, 75, 0, 0, NA, NA, NA, NA,
  "GLY", "55 kg", 50, 55, 0, 0, NA, NA, NA, NA,
  "GLY", "105 kg", 50, 105, 0, 0, NA, NA, NA, NA,
  "GLY", "Japanese", 50, 75, 1, 0, NA, NA, NA, NA,
  "GLY", "other race", 50, 75, 0, 1, NA, NA, NA, NA,
  "MF", "reference (IND/MF 150/320, PALLADIUM)", 320, 75, NA, NA, 0, 2, 0, 0,
  "MF", "55 kg", 320, 55, NA, NA, 0, 2, 0, 0,
  "MF", "105 kg", 320, 105, NA, NA, 0, 2, 0, 0,
  "MF", "FEV1 1.2 L", 320, 75, NA, NA, 0, 1.2, 0, 0,
  "MF", "FEV1 3 L", 320, 75, NA, NA, 0, 3, 0, 0,
  "MF", "IND/MF 150/320", 320, 75, NA, NA, 1, 2, 0, 0,
  "MF", "IND/GLY/MF 150/50/160", 160, 75, NA, NA, 1, 2, 1, 0,
  "MF", "IND/MF 150/160", 160, 75, NA, NA, 1, 2, 0, 1,
  "MF", "IND/GLY/MF 150/50/80", 80, 75, NA, NA, 1, 2, 1, 1
) |>
  dplyr::mutate(vidx = dplyr::row_number())

tad_grid <- round(seq(0.1, 24, by = 0.1), 6) # 6-min grid of Bartels 2021 Sect. 2.6

build_variant <- function(v) {
  cov_cols <- c(IND = list(c("WT", "RACE_JAPANESE", "STUDY_IRIDIUM")),
                GLY = list(c("WT", "RACE_JAPANESE", "RACE_OTHER")),
                MF = list(c("WT", "FEV1", "STUDY_IRIDIUM", "FORM_MF_INDGLYMF", "FORM_MF_MEDIUM")))[[v$drug]]
  subj <- dplyr::bind_cols(
    tibble::tibble(id = (v$vidx - 1L) * n_sub + seq_len(n_sub), variant = v$variant),
    v[rep(1, n_sub), cov_cols],
    etas[[v$drug]]
  )
  make_events(subj, v$amt, split = v$drug != "GLY", obs_tad = tad_grid)
}

solve_drug <- function(drug, mod) {
  vv <- variants[variants$drug == drug, ]
  ev <- dplyr::bind_rows(lapply(split(vv, seq_len(nrow(vv))), build_variant))
  stopifnot(!anyDuplicated(unique(ev[, c("id", "time", "evid", "cmt")])))
  rxode2::rxSolve(rxode2::zeroRe(mod), events = ev, keep = "variant",
                  returnType = "data.frame", maxsteps = 1e6) |>
    dplyr::transmute(drug = drug, id, variant, tad = round(time - t_last, 6), Cc)
}

sim_cov <- dplyr::bind_rows(
  solve_drug("IND", mod_ind),
  solve_drug("GLY", mod_gly),
  solve_drug("MF", mod_mf)
)
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'
#> Warning: multi-subject simulation without without 'omega'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc'
#> Warning: multi-subject simulation without without 'omega'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'
#> Warning: multi-subject simulation without without 'omega'
stopifnot(!anyNA(sim_cov$Cc), all(sim_cov$Cc > 0))

PKNCA over the steady-state interval

At steady state the pre-dose concentration equals the concentration 24 h after the dose. Each profile therefore gets a time-zero row that carries its 24 h value. This anchors AUC0-24 and Ctrough without an observation at the dosing instant, where a GLY bolus would already have raised the level.

nca_in <- dplyr::bind_rows(
  sim_cov,
  sim_cov |> dplyr::filter(abs(tad - 24) < 1e-8) |> dplyr::mutate(tad = 0)
) |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::mutate(treatment = paste(drug, variant, sep = ": ")) |>
  dplyr::arrange(treatment, id, tad)

nca_cov <- nca_by_treatment(
  nca_in,
  intervals = data.frame(start = 0, end = 24, cmax = TRUE, cmin = TRUE, auclast = TRUE)
)

nca_means <- nca_cov |>
  dplyr::filter(PPTESTCD %in% c("cmax", "cmin", "auclast")) |>
  dplyr::group_by(treatment, PPTESTCD) |>
  dplyr::summarise(mean = mean(PPORRES), .groups = "drop") |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = mean)
stopifnot(nrow(nca_means) == nrow(variants))

Comparison against the published covariate effects

Each published figure is the percent difference of a population mean from the reference variant, rounded to whole percent. The IND reference is PALLADIUM, as in Section 3.6. The MF formulation comparisons use IRIDIUM, the only study that gave all four FDCs.

claims <- tibble::tribble(
  ~drug, ~variant, ~reference, ~metric, ~published,
  "IND", "55 kg", "reference (PALLADIUM)", "auclast", 10,
  "IND", "105 kg", "reference (PALLADIUM)", "auclast", -10,
  "IND", "55 kg", "reference (PALLADIUM)", "cmax", 12,
  "IND", "105 kg", "reference (PALLADIUM)", "cmax", -12,
  "IND", "Japanese", "reference (PALLADIUM)", "cmax", 20,
  "IND", "Japanese", "reference (PALLADIUM)", "auclast", 0,
  "IND", "IRIDIUM", "reference (PALLADIUM)", "auclast", 0,
  "GLY", "55 kg", "reference", "auclast", 26,
  "GLY", "105 kg", "reference", "auclast", -22,
  "GLY", "55 kg", "reference", "cmax", 33,
  "GLY", "105 kg", "reference", "cmax", -27,
  "GLY", "Japanese", "reference", "cmax", 60,
  "GLY", "other race", "reference", "cmax", 5,
  "GLY", "Japanese", "reference", "cmin", -8,
  "GLY", "Japanese", "reference", "auclast", 0,
  "MF", "55 kg", "reference (IND/MF 150/320, PALLADIUM)", "auclast", 11,
  "MF", "105 kg", "reference (IND/MF 150/320, PALLADIUM)", "auclast", -11,
  "MF", "FEV1 1.2 L", "reference (IND/MF 150/320, PALLADIUM)", "auclast", -8,
  "MF", "FEV1 3 L", "reference (IND/MF 150/320, PALLADIUM)", "auclast", 7,
  "MF", "55 kg", "reference (IND/MF 150/320, PALLADIUM)", "cmax", 11,
  "MF", "105 kg", "reference (IND/MF 150/320, PALLADIUM)", "cmax", -11,
  "MF", "FEV1 1.2 L", "reference (IND/MF 150/320, PALLADIUM)", "cmax", -10,
  "MF", "FEV1 3 L", "reference (IND/MF 150/320, PALLADIUM)", "cmax", 9,
  "MF", "reference (IND/MF 150/320, PALLADIUM)", "IND/MF 150/320", "cmax", 13,
  "MF", "IND/GLY/MF 150/50/160", "IND/MF 150/320", "auclast", 0,
  "MF", "IND/GLY/MF 150/50/80", "IND/MF 150/160", "auclast", 0
)

mean_of <- function(drug, variant, metric) {
  v <- nca_means[[metric]][nca_means$treatment == paste(drug, variant, sep = ": ")]
  if (length(v) != 1L) stop("no unique NCA row for ", drug, ": ", variant)
  v
}
claims$simulated <- vapply(seq_len(nrow(claims)), function(i) {
  c_i <- claims[i, ]
  100 * (mean_of(c_i$drug, c_i$variant, c_i$metric) / mean_of(c_i$drug, c_i$reference, c_i$metric) - 1)
}, numeric(1))
claims$difference <- claims$simulated - claims$published

claims |>
  dplyr::mutate(metric = dplyr::recode(metric, auclast = "AUC0-24", cmax = "Cmax", cmin = "Ctrough")) |>
  dplyr::rename(
    Drug = drug, Variant = variant, `Compared with` = reference, Metric = metric,
    `Published (%)` = published, `Simulated (%)` = simulated, `Difference (points)` = difference
  ) |>
  knitr::kable(digits = 1, caption = "Covariate effects on population-mean exposure, Bartels 2021 Section 3.6 vs. simulation.")
Covariate effects on population-mean exposure, Bartels 2021 Section 3.6 vs. simulation.
Drug Variant Compared with Metric Published (%) Simulated (%) Difference (points)
IND 55 kg reference (PALLADIUM) AUC0-24 10 10.9 0.9
IND 105 kg reference (PALLADIUM) AUC0-24 -10 -11.0 -1.0
IND 55 kg reference (PALLADIUM) Cmax 12 12.8 0.8
IND 105 kg reference (PALLADIUM) Cmax -12 -12.4 -0.4
IND Japanese reference (PALLADIUM) Cmax 20 21.7 1.7
IND Japanese reference (PALLADIUM) AUC0-24 0 0.0 0.0
IND IRIDIUM reference (PALLADIUM) AUC0-24 0 -0.1 -0.1
GLY 55 kg reference AUC0-24 26 26.1 0.1
GLY 105 kg reference AUC0-24 -22 -22.2 -0.2
GLY 55 kg reference Cmax 33 33.4 0.4
GLY 105 kg reference Cmax -27 -26.8 0.2
GLY Japanese reference Cmax 60 61.0 1.0
GLY other race reference Cmax 5 4.9 -0.1
GLY Japanese reference Ctrough -8 -7.6 0.4
GLY Japanese reference AUC0-24 0 -0.9 -0.9
MF 55 kg reference (IND/MF 150/320, PALLADIUM) AUC0-24 11 11.6 0.6
MF 105 kg reference (IND/MF 150/320, PALLADIUM) AUC0-24 -11 -11.3 -0.3
MF FEV1 1.2 L reference (IND/MF 150/320, PALLADIUM) AUC0-24 -8 -8.1 -0.1
MF FEV1 3 L reference (IND/MF 150/320, PALLADIUM) AUC0-24 7 7.0 0.0
MF 55 kg reference (IND/MF 150/320, PALLADIUM) Cmax 11 11.2 0.2
MF 105 kg reference (IND/MF 150/320, PALLADIUM) Cmax -11 -11.0 0.0
MF FEV1 1.2 L reference (IND/MF 150/320, PALLADIUM) Cmax -10 -9.9 0.1
MF FEV1 3 L reference (IND/MF 150/320, PALLADIUM) Cmax 9 8.5 -0.5
MF reference (IND/MF 150/320, PALLADIUM) IND/MF 150/320 Cmax 13 12.3 -0.7
MF IND/GLY/MF 150/50/160 IND/MF 150/320 AUC0-24 0 -0.1 -0.1
MF IND/GLY/MF 150/50/80 IND/MF 150/160 AUC0-24 0 -0.1 -0.1 
# The cohort is fixed by set.seed() and the zeroRe() solve, so this is
# deterministic. Measured maximum |difference| 1.7 points (IND Japanese Cmax).
# The published values are themselves 200-subject Monte-Carlo means rounded to
# whole percent, so the bound leaves room for their own sampling error.
stopifnot(nrow(claims) == 26, !anyNA(claims$simulated), max(abs(claims$difference)) < 4)

Section 4.2 also gives an absolute value: the simulated mean GLY Cmax in Japanese patients was 208.8 pg/mL for IND/GLY/MF 150/50/160 ug.

gly_jap_cmax <- mean_of("GLY", "Japanese", "cmax")
gly_jap_cmax
#> [1] 203.9151
# Measured 203.9 (-2.3%); the paper's own figure is a 200-subject Monte-Carlo
# mean with its own sampling error.
stopifnot(abs(gly_jap_cmax / 208.8 - 1) < 0.08)

Variability bands (Figure 4) 

band <- sim_cov |>
  dplyr::filter(
    (drug == "IND" & variant == "IRIDIUM") |
      (drug == "GLY" & variant == "reference") |
      (drug == "MF" & variant %in% mf_products$product)
  ) |>
  dplyr::mutate(product = dplyr::if_else(drug == "MF", variant, "IND/GLY/MF 150/50/160")) |>
  dplyr::group_by(drug, product, tad) |>
  dplyr::summarise(Q05 = quantile(Cc, 0.05), Q95 = quantile(Cc, 0.95), .groups = "drop")

ggplot() +
  geom_ribbon(data = band, aes(tad, ymin = Q05, ymax = Q95), fill = "grey80") +
  geom_line(data = typ, aes(tad, Cc)) +
  geom_point(data = fig4_digitised, aes(tad, conc), colour = "firebrick", size = 1.5) +
  facet_wrap(~ drug + product, scales = "free_y") +
  scale_y_log10() +
  labs(
    x = "Time after dose (h)", y = "Plasma concentration (pg/mL)",
    title = "Steady-state typical profiles with 5th-95th percentile BSV band",
    caption = paste0(
      "Replicates Figure 4 of Bartels 2021. Line: typical value; band: 200 virtual\n",
      "75 kg patients with BSV; red points: digitised from the published figure."
    )
  )

Observed Day 86 concentrations (Online Resource 2)

Online Resource 2 lists the observed mean Ctrough and Cmax on Day 86 for each FDC arm. As in Figure 1, Cmax is the largest sample up to 1 h after the dose. For IND the 2-min sample is excluded, and for GLY it is included (IRIDIUM sampled at -25 min, 2 min, 15 min and 1 h). Ctrough is the -25 min pre-dose sample. The comparison below simulates 200 virtual patients per arm. Body weight is drawn from each study’s Table 4 mean and SD, truncated to the study range. The grouped-race indicators are drawn at each study’s Table 4 frequency. Proportional residual error is added in R, and values below the LLOQ are set to the LLOQ, as the paper’s summary statistics were.

set.seed(86)
studies <- tibble::tribble(
  ~study, ~wt_mean, ~wt_sd, ~wt_min, ~wt_max, ~p_jap, ~p_oth,
  "PALLADIUM", 79.5, 19, 33.6, 156, 21 / 273, 51 / 273,
  "IRIDIUM", 82, 17.6, 44, 136, 13 / 249, 17 / 249
)
obs_arms <- tibble::tribble(
  ~drug, ~study, ~treatment, ~amt, ~cmin, ~cmax,
  "IND", "PALLADIUM", "IND PALLADIUM IND/MF 150/320", 150, 106.8, 302.2,
  "IND", "PALLADIUM", "IND PALLADIUM IND/MF 150/160", 150, 94.0, 340.3,
  "IND", "IRIDIUM", "IND IRIDIUM IND/GLY/MF 150/50/160", 150, 96.6, 275.3,
  "IND", "IRIDIUM", "IND IRIDIUM IND/GLY/MF 150/50/80", 150, 101.0, 285.9,
  "IND", "IRIDIUM", "IND IRIDIUM IND/MF 150/320", 150, 84.7, 261.6,
  "IND", "IRIDIUM", "IND IRIDIUM IND/MF 150/160", 150, 87.0, 265.4,
  "GLY", "IRIDIUM", "GLY IRIDIUM IND/GLY/MF 150/50/160", 50, 13.1, 140.6,
  "GLY", "IRIDIUM", "GLY IRIDIUM IND/GLY/MF 150/50/80", 50, 12.3, 157.7
) |>
  dplyr::mutate(aidx = dplyr::row_number())

rtnorm <- function(n, mean, sd, lo, hi) {
  x <- stats::rnorm(n, mean, sd)
  while (any(out <- x < lo | x > hi)) x[out] <- stats::rnorm(sum(out), mean, sd)
  x
}

build_arm <- function(a) {
  s <- studies[studies$study == a$study, ]
  grp <- sample(c("cau", "jap", "oth"), n_sub, replace = TRUE,
                prob = c(1 - s$p_jap - s$p_oth, s$p_jap, s$p_oth))
  subj <- tibble::tibble(
    id = 10000L + (a$aidx - 1L) * n_sub + seq_len(n_sub),
    treatment = a$treatment,
    WT = rtnorm(n_sub, s$wt_mean, s$wt_sd, s$wt_min, s$wt_max),
    RACE_JAPANESE = as.integer(grp == "jap"),
    RACE_OTHER = as.integer(grp == "oth"),
    STUDY_IRIDIUM = as.integer(a$study == "IRIDIUM")
  )
  subj <- dplyr::bind_cols(subj, draw_etas(if (a$drug == "IND") mod_ind else mod_gly, n_sub))
  if (a$drug == "IND") subj$RACE_OTHER <- NULL else subj$STUDY_IRIDIUM <- NULL
  # -25 min before the Day-86 dose is 23 h 35 min after the previous dose.
  samp <- if (a$drug == "IND") c(-25 / 60, 0.25, 1) else c(-25 / 60, 2 / 60, 0.25, 1)
  make_events(subj, a$amt, split = a$drug == "IND", obs_tad = samp)
}

solve_arms <- function(drug, mod, prop_sd, lloq) {
  aa <- obs_arms[obs_arms$drug == drug, ]
  ev <- dplyr::bind_rows(lapply(split(aa, seq_len(nrow(aa))), build_arm))
  rxode2::rxSolve(rxode2::zeroRe(mod), events = ev, keep = "treatment",
                  returnType = "data.frame", maxsteps = 1e6) |>
    dplyr::mutate(
      tad = pmax(round(time - t_last, 6), 0), # the pre-dose sample anchors the interval at 0
      Cc = pmax(Cc * (1 + prop_sd * stats::rnorm(dplyr::n())), lloq)
    ) |>
    dplyr::select(id, treatment, tad, Cc)
}

sim_obs <- dplyr::bind_rows(
  solve_arms("IND", mod_ind, 0.24, 5),
  solve_arms("GLY", mod_gly, 0.34, 0.25)
)
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalogitffo'
#> Warning: multi-subject simulation without without 'omega'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc'
#> Warning: multi-subject simulation without without 'omega'

nca_obs <- nca_by_treatment(
  dplyr::filter(sim_obs, !is.na(Cc)),
  intervals = data.frame(start = 0, end = 1, cmax = TRUE, cmin = TRUE)
)

# The paper reports arithmetic means; ncaComparisonTable() would otherwise
# summarise the simulated subjects by their median.
obs_means <- nca_obs |>
  dplyr::filter(PPTESTCD %in% c("cmax", "cmin")) |>
  dplyr::group_by(treatment, PPTESTCD) |>
  dplyr::summarise(PPORRES = mean(PPORRES), .groups = "drop")

cmp_obs <- nlmixr2lib::ncaComparisonTable(
  simulated = obs_means,
  reference = dplyr::select(obs_arms, treatment, cmax, cmin),
  by = "treatment",
  units = c(cmax = "pg/mL", cmin = "pg/mL"),
  tolerance_pct = 20
)
knitr::kable(cmp_obs, caption = paste(
  "Simulated vs. observed Day 86 mean concentrations (Online Resource 2).",
  "Cmin over the sparse 0-1 h samples is the pre-dose trough. * differs by >20%."
))
Simulated vs. observed Day 86 mean concentrations (Online Resource 2). Cmin over the sparse 0-1 h samples is the pre-dose trough. * differs by >20%.
NCA parameter treatment Reference Simulated % diff
Cmax (pg/mL) IND PALLADIUM IND/MF 150/320 302 339 +12.2%
Cmax (pg/mL) IND PALLADIUM IND/MF 150/160 340 340 -0.2%
Cmax (pg/mL) IND IRIDIUM IND/GLY/MF 150/50/160 275 279 +1.2%
Cmax (pg/mL) IND IRIDIUM IND/GLY/MF 150/50/80 286 294 +2.8%
Cmax (pg/mL) IND IRIDIUM IND/MF 150/320 262 291 +11.1%
Cmax (pg/mL) IND IRIDIUM IND/MF 150/160 265 291 +9.6%
Cmax (pg/mL) GLY IRIDIUM IND/GLY/MF 150/50/160 141 138 -2.0%
Cmax (pg/mL) GLY IRIDIUM IND/GLY/MF 150/50/80 158 153 -3.3%
Cmin (pg/mL) IND PALLADIUM IND/MF 150/320 107 94.1 -11.9%
Cmin (pg/mL) IND PALLADIUM IND/MF 150/160 94 90.7 -3.5%
Cmin (pg/mL) IND IRIDIUM IND/GLY/MF 150/50/160 96.6 93.8 -2.9%
Cmin (pg/mL) IND IRIDIUM IND/GLY/MF 150/50/80 101 99 -2.0%
Cmin (pg/mL) IND IRIDIUM IND/MF 150/320 84.7 100 +18.5%
Cmin (pg/mL) IND IRIDIUM IND/MF 150/160 87 94.1 +8.1%
Cmin (pg/mL) GLY IRIDIUM IND/GLY/MF 150/50/160 13.1 12.5 -4.3%
Cmin (pg/mL) GLY IRIDIUM IND/GLY/MF 150/50/80 12.3 13.9 +13.2%

The simulated means sit close to the observed ones. The largest differences are in the IRIDIUM IND/MF arms, whose observed IND Cmax was 4-9% and Ctrough 10-16% below those of the IRIDIUM IND/GLY/MF arms. The model has no IND formulation effect (Sect. 3.4), so it gives all four IRIDIUM arms the same exposure, and the observed spread between arms stays in the residual.

obs_pct <- obs_means |>
  dplyr::inner_join(
    obs_arms |> dplyr::select(treatment, cmax, cmin) |>
      tidyr::pivot_longer(c(cmax, cmin), names_to = "PPTESTCD", values_to = "published"),
    by = c("treatment", "PPTESTCD")
  ) |>
  dplyr::mutate(pct = 100 * (PPORRES / published - 1))
stopifnot(nrow(obs_pct) == 16)
obs_pct |>
  dplyr::group_by(PPTESTCD) |>
  dplyr::summarise(median_pct = median(pct), max_abs_pct = max(abs(pct)), .groups = "drop") |>
  knitr::kable(digits = 1, caption = "Summary of the simulated-vs-observed differences.")
Summary of the simulated-vs-observed differences.
PPTESTCD median_pct max_abs_pct
cmax 2.0 12.2
cmin -2.5 18.5 
# Observed data sit behind this comparison, and the covariate distributions
# are reconstructed from summaries, so the gate is on the centre and a wide
# envelope. Measured: median +2.0% (Cmax) and -2.5% (Ctrough), largest single
# row 18.5% (IND IRIDIUM IND/MF 150/320 trough). A mis-transcribed CL/F, Vc/F,
# dose or unit moves every row by tens of percent.
stopifnot(abs(median(obs_pct$pct)) < 8, max(abs(obs_pct$pct)) < 30)

Assumptions and deviations

  • Three models, one article. Bartels 2021 fitted a separate model to each drug, so each ships as its own file.
  • Split dose records. Each IND or MF inhalation is two records of the same amt: one on central with rate = -2 and one on depot (see “Dosing the models”). The Monolix absorption macros are not reproduced literally; the split reproduces them.
  • IND absorption is sequential. Section 3.2 describes “a short zero-order absorption of a fraction of the drug followed by a rapid first-order absorption of the rest”, so the first-order fraction starts when the zero-order input ends (alag(depot) <- d1). With Ka fixed at 50 1/h the lag moves the profile by only a few minutes. For MF the paper describes the first-order input as “overlaid” on the zero-order one, so both start at the dose.
  • Fr is a dose split, not a time fraction. Taking Fr as the zero-order share of the absorption time, D / (D + 1/Ka), gives 0.73 for IND and 0.02 for MF, not the tabulated 0.33 and 0.37. The canonical logitffo holds the first-order share, 1 - Fr.
  • BSV on Fr is on the logit scale. Section 2.3 says that BSV was modelled with “multiplicative exponential random effects”. Applied to IND’s Fr (0.33, SD 1.2), a log-normal effect would give about 18% of patients a zero-order fraction above 1, which is impossible. The maintainers therefore used the logit-normal distribution that Monolix uses for fractions for both drugs. Because logit(1 - Fr) = -logit(Fr), the variance on logitffo equals the published variance on logit(Fr).
  • MF dose-normalisation factor. Section 2.3 describes “a fixed multiplicative factor on the bioavailability … 2.5 in Twisthaler, 1.0 as part of IND/MF FDC …, and 0.5 as part of IND/GLY/MF FDC”. Read as an F multiplier, IND/GLY/MF 150/50/160 ug would give 160 x 0.5 = 80 ug-equivalents against 320 for IND/MF 150/320 ug. That is a four-fold exposure difference, but Section 3.6 reports equal AUC0-24h and Figure 4 shows overlapping curves. Read as a divisor of the nominal dose, the factors match the Table 2 pairs (80/160 ug in IND/GLY/MF = 160/320 ug in IND/MF = 400/800 ug via Twisthaler), and the model gives the equal AUCs shown above. The model therefore carries a fixed relative bioavailability of 1/0.5 = 2 for IND/GLY/MF.
  • MF Twisthaler monotherapy is not covered. The final MF model has Vc/F and F effects for MF via Twisthaler and a Vp/F effect for the E2201 Twisthaler puff strengths, but the paper does not report them (“Formulation effects for monotherapies … are not shown”, Table 5 legend). The file covers the Breezhaler products only. The reported Twisthaler AUCs (25% and 37% below the high- and medium-dose FDCs) together imply a Twisthaler F effect of about log(0.75), but the Vc/F and Vp/F effects cannot be recovered.
  • The IND grouped-race level ‘other’ is not encoded. Section 3.4 lists grouped race (Caucasian/White, Japanese, other) on IND Vc/F, but only the Japanese coefficient is reported. Patients of the ‘other’ group are simulated as the Caucasian reference; Section 3.6 puts their Cmax only 5% above Caucasian patients. The GLY ‘other’ coefficient (-0.063) is reported in the Section 3.4 text and is included.
  • GLY ‘other’ race uses the RACE_OTHER canonical for the residual level of the paper’s grouped-race covariate. It pools non-Japanese Asian, Black, Native American and other-race patients.
  • Residual error is proportional only. Section 2.3 used a combined error model but dropped the additive term when it tended to zero. Table 5 lists only the proportional term for all three final models.
  • Figure 4 study. The Figure 4 IND curve matches the IRIDIUM study effect to 3% and would sit 17% higher at Cmax under PALLADIUM. IRIDIUM is also the only study that gave all four FDCs, so the typical-value replication uses IRIDIUM.
  • MF late-phase deviation from Figure 4. The MF typical curves match Figure 4 through 6 h (within 8%) and at Cmax (within 4%), but run up to 15-20% above the figure from about 8 h to the trough for three of the four products. The late phase of a two-compartment model depends on Q/F and Vp/F, and the MF Vp/F carries a very large BSV (SD 2). Figure 4 may show a median rather than a strict typical-value simulation, or use slightly different reference covariates. The parameters were not tuned to close the gap.
  • Covariate claims are population means. Section 3.6 reports percent differences of 200-patient population means. The simulation reproduces them with a paired 200-patient cohort. The largest gap is 1.7 percentage points (IND Japanese Cmax, +21.7% simulated against +20% published).
  • Covariate distributions for the Day-86 comparison were drawn from the Table 4 means, SDs and ranges of each study (normal, truncated). The source gives only these summaries.
  • Steady state by run-in. rxode2 cannot combine ss = 2 with a modelled f(), so steady state is reached by 60 once-daily doses.