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Pediatric nintedanib PK and FVC exposure-response (Hartmann 2026)

Source: vignettes/articles/Hartmann_2026_nintedanib.Rmd
Hartmann_2026_nintedanib.Rmd

Model and source

This paper contributes three models to the library, matching how the authors built them: one pediatric population PK model and two independent exposure-response models for the two FVC-derived efficacy endpoints.

pk    <- readModelDb("Hartmann_2026_nintedanib")
erpp  <- readModelDb("Hartmann_2026_nintedanib_fvcpp")
erz   <- readModelDb("Hartmann_2026_nintedanib_fvcz")
  • Citation: Hartmann S, Chan Kwong A, Ribbing J, Gahlemann M, Korell J. Population Pharmacokinetics and Exposure-Response Model-Based Bayesian Extrapolation of FVC-Based Efficacy Endpoints From Adults to Pediatric Patients Receiving Nintedanib. CPT Pharmacometrics Syst Pharmacol. 2026;15(1):e70135. doi:10.1002/psp4.70135. PMCID PMC12823301. Structural parameters, covariate coefficients and variance terms are the final estimates in Table S5 of Data S2 and in the final pediatric popPK NONMEM control stream reproduced in Data S1. The adult nintedanib population PK model that supplied the frequentist prior is a separate publication; a different adult nintedanib popPK analysis is packaged as modellib(‘Schmid_2017_nintedanib’).
  • Article: https://doi.org/10.1002/psp4.70135
  • Supplement (Data S1, NONMEM control streams): https://doi.org/10.1002/psp4.70135
  • PMC record: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12823301/

Nintedanib is an oral triple angiokinase inhibitor that slows the decline in forced vital capacity (FVC) in adults with idiopathic pulmonary fibrosis, systemic-sclerosis-associated interstitial lung disease and other chronic fibrosing ILDs with a progressive phenotype. The pediatric phase 3 trial InPedILD and its open-label extension InPedILD-ON studied it in children and adolescents 6 to less than 18 years old. Because 53 patients cannot support a stand-alone analysis, the authors used Bayesian partial extrapolation: previously developed adult popPK and FVC exposure-response meta-models supplied frequentist priors through the NONMEM NWPRI functionality, and only the genuinely pediatric quantities were released from those priors and estimated from the pediatric data alone.

The library’s other nintedanib model, modellib("Schmid_2017_nintedanib"), is a different adult analysis (NSCLC + IPF, parent plus the BIBF 1202 metabolite). It is not the adult prior used here, though its apparent clearance of 897 L/h at 71.5 kg is reassuringly close to the 910 L/h at 75 kg estimated below.

Population 

pop <- attr(pk, "population")
str(pop, max.level = 1, give.attr = FALSE)
#>  NULL

The PK analysis set is 44 patients contributing 446 nintedanib plasma concentrations (Hartmann 2026 Table S4); the exposure-response analysis set is all 53 enrolled patients contributing 517 FVC observations (Table 1). Mean body weight was 42.9 kg (SD 17.8) in the PK set, 27.2 kg (SD 10.3) in the 6 to less than 12 year group and 50.3 kg (SD 15.7) in the 12 to less than 18 year group. The cohort was 77% Caucasian with no Chinese, Korean, Indian or Japanese patients, which is why the ethnicity coefficients in the PK model are carried entirely by the adult prior and why every ethnicity covariate of the adult FVC models was dropped from the pediatric FVC models.

Dosing was assigned from body weight (Hartmann 2026 Table S1) and re-assigned as body weight changed:

dose_bands <- tibble::tribble(
  ~band, ~wt_lo, ~wt_hi,   ~dose_bid_mg, ~reduced_dose_bid_mg,
  1L,    13.5,   23.0,     50,           25,
  2L,    23.0,   33.5,     75,           50,
  3L,    33.5,   57.5,     100,          75,
  4L,    57.5,   Inf,      150,          100
)
dose_bands |>
  dplyr::mutate(`Weight range (kg)` = ifelse(
    is.finite(wt_hi),
    sprintf("%.1f to < %.1f", wt_lo, wt_hi),
    sprintf(">= %.1f", wt_lo)
  )) |>
  dplyr::select(
    `Weight bin` = band,
    `Weight range (kg)`,
    `Dose (BID, mg)` = dose_bid_mg,
    `Dose after one-step reduction (BID, mg)` = reduced_dose_bid_mg
  ) |>
  knitr::kable(caption = "Hartmann 2026 Table S1. Patients below 13.5 kg were excluded from the trial.")
Hartmann 2026 Table S1. Patients below 13.5 kg were excluded from the trial.
Weight bin Weight range (kg) Dose (BID, mg) Dose after one-step reduction (BID, mg)
1 13.5 to < 23.0 50 25
2 23.0 to < 33.5 75 50
3 33.5 to < 57.5 100 75
4 >= 57.5 150 100

Source trace

Every parameter in the three model files carries an in-file comment naming its source location. The table below is the reviewer-facing summary.

tibble::tribble(
  ~Model, ~Quantity, ~Value, ~`Source location`,
  "popPK", "CL/F at 75 kg",                "909.629 L/h",  "Data S1 popPK $THETA 1; Table S5 'Apparent clearance (CL/F)' = 910",
  "popPK", "V/F at 75 kg",                 "10696.3 L",    "Data S1 popPK $THETA 2; Table S5 'Apparent volume (V/F)' = 1.07E+04",
  "popPK", "ka",                           "2.73232 /h",   "Data S1 popPK $THETA 3; Table S5 'Absorption rate (ka)' = 2.73",
  "popPK", "Lag time",                     "0.71714 h",    "Data S1 popPK $THETA 4; Table S5 'Lag time (tlag)' = 0.717",
  "popPK", "Frel reference",               "1 (fixed)",    "Data S1 popPK $PK: TVF1 = 1 * F1RACEREG * F1DIAG6 * F1LDHBL",
  "popPK", "WT exponent on CL/F, V/F",     "0.75, 1 (fixed)", "Data S1 popPK $PK WTCL/WTV; Table S5 footnotes a, b; Table S2 footnote a",
  "popPK", "Other ethnicities on Frel",    "+0.3289",      "Data S1 popPK $THETA 5; Table S5 = 0.329 fraction change",
  "popPK", "Korean on Frel",               "-0.143797",    "Data S1 popPK $THETA 6; Table S5 = -0.144 fraction change",
  "popPK", "SSc-ILD on Frel",              "-0.138784",    "Data S1 popPK $THETA 7; Table S5 = -0.139 fraction change",
  "popPK", "LDH on Frel (centred 206 U/L)","0.00155711 per U/L", "Data S1 popPK $THETA 8 / F1LDHBL; Table S5 = 0.00156",
  "popPK", "Pediatric age on IOV Frel",    "-0.0525537 per year", "Data S1 popPK $THETA 9 / IOVPEDAGE; Table S5 = -0.0526",
  "popPK", "IIV V/F, ka, Frel",            "0.0952477, 1.56911, 0.173479", "Data S1 popPK $OMEGA 1-3; Table S5 CV column = sqrt of each",
  "popPK", "IOV Frel (6 occasions)",       "0.0961255",    "Data S1 popPK $OMEGA BLOCK(1) + 5 x SAME; Table S5 'IOV Frel' CV = 0.310",
  "popPK", "Residual (log-additive) SD",   "0.391754",     "Data S1 popPK $SIGMA = 0.153471; Table S5 'RUV' CV = 0.392",
  "popPK", "Child-Pugh A on Frel",         "2.15 (fixed assumption)", "Section 2.3.3 'a 115% higher nintedanib bioavailability was assumed'",
  "FVC %pred", "Baseline",                 "54.9 %",       "Table 2 'Baseline'",
  "FVC %pred", "Slope (adult)",            "-4.74 %/year", "Table 2 'Slope', footnote a 'Supported by adult priors'",
  "FVC %pred", "Pediatric change in slope","+2.28 %/year", "Table 2 'Pediatric change in slope', footnote b",
  "FVC %pred", "Emax",                     "4.17 %/year",  "Table 2 'Rate of change ... at maximum drug effect (Emax)'",
  "FVC %pred", "EC50",                     "8.05 nM",      "Table 2 'EC50'; repeated in Section 3.4",
  "FVC %pred", "IIV baseline / slope / RUV","CV 0.401 / SD 5.55 / CV 0.468", "Table 2, unit column gives the scale of each",
  "FVC %pred", "Additive residual SD",     "3.26 %",       "Table 2 'Add. RUV'",
  "FVC Z",     "Baseline",                 "-3.49 Z",      "Table 3 'Baseline'",
  "FVC Z",     "Slope (adult)",            "-0.308 Z/year","Table 3 'Slope', footnote b 'Supported by adult priors'",
  "FVC Z",     "Pediatric change in slope","+0.133 Z/year","Table 3 'Pediatric change in slope', footnote a",
  "FVC Z",     "Emax",                     "0.292 Z/year", "Table 3 'Rate of change in FVC Z-score at maximum drug effect (Emax)'",
  "FVC Z",     "EC50",                     "8.12 nM",      "Table 3 'EC50'; repeated in Section 3.5",
  "FVC Z",     "IIV baseline / slope / RUV","SD 1.84 / SD 0.379 / CV 0.464", "Table 3, unit column gives the scale of each",
  "FVC Z",     "Additive residual SD",     "0.276 Z",      "Table 3 'Add. RUV'"
) |>
  knitr::kable()
Model Quantity Value Source location
popPK CL/F at 75 kg 909.629 L/h Data S1 popPK $THETA 1; Table S5 ‘Apparent clearance (CL/F)’ = 910
popPK V/F at 75 kg 10696.3 L Data S1 popPK $THETA 2; Table S5 ‘Apparent volume (V/F)’ = 1.07E+04
popPK ka 2.73232 /h Data S1 popPK $THETA 3; Table S5 ‘Absorption rate (ka)’ = 2.73
popPK Lag time 0.71714 h Data S1 popPK $THETA 4; Table S5 ‘Lag time (tlag)’ = 0.717
popPK Frel reference 1 (fixed) Data S1 popPK $PK: TVF1 = 1 * F1RACEREG * F1DIAG6 * F1LDHBL
popPK WT exponent on CL/F, V/F 0.75, 1 (fixed) Data S1 popPK $PK WTCL/WTV; Table S5 footnotes a, b; Table S2 footnote a
popPK Other ethnicities on Frel +0.3289 Data S1 popPK $THETA 5; Table S5 = 0.329 fraction change
popPK Korean on Frel -0.143797 Data S1 popPK $THETA 6; Table S5 = -0.144 fraction change
popPK SSc-ILD on Frel -0.138784 Data S1 popPK $THETA 7; Table S5 = -0.139 fraction change
popPK LDH on Frel (centred 206 U/L) 0.00155711 per U/L Data S1 popPK $THETA 8 / F1LDHBL; Table S5 = 0.00156
popPK Pediatric age on IOV Frel -0.0525537 per year Data S1 popPK $THETA 9 / IOVPEDAGE; Table S5 = -0.0526
popPK IIV V/F, ka, Frel 0.0952477, 1.56911, 0.173479 Data S1 popPK $OMEGA 1-3; Table S5 CV column = sqrt of each
popPK IOV Frel (6 occasions) 0.0961255 Data S1 popPK $OMEGA BLOCK(1) + 5 x SAME; Table S5 ‘IOV Frel’ CV = 0.310
popPK Residual (log-additive) SD 0.391754 Data S1 popPK $SIGMA = 0.153471; Table S5 ‘RUV’ CV = 0.392
popPK Child-Pugh A on Frel 2.15 (fixed assumption) Section 2.3.3 ‘a 115% higher nintedanib bioavailability was assumed’
FVC %pred Baseline 54.9 % Table 2 ‘Baseline’
FVC %pred Slope (adult) -4.74 %/year Table 2 ‘Slope’, footnote a ‘Supported by adult priors’
FVC %pred Pediatric change in slope +2.28 %/year Table 2 ‘Pediatric change in slope’, footnote b
FVC %pred Emax 4.17 %/year Table 2 ‘Rate of change … at maximum drug effect (Emax)’
FVC %pred EC50 8.05 nM Table 2 ‘EC50’; repeated in Section 3.4
FVC %pred IIV baseline / slope / RUV CV 0.401 / SD 5.55 / CV 0.468 Table 2, unit column gives the scale of each
FVC %pred Additive residual SD 3.26 % Table 2 ‘Add. RUV’
FVC Z Baseline -3.49 Z Table 3 ‘Baseline’
FVC Z Slope (adult) -0.308 Z/year Table 3 ‘Slope’, footnote b ‘Supported by adult priors’
FVC Z Pediatric change in slope +0.133 Z/year Table 3 ‘Pediatric change in slope’, footnote a
FVC Z Emax 0.292 Z/year Table 3 ‘Rate of change in FVC Z-score at maximum drug effect (Emax)’
FVC Z EC50 8.12 nM Table 3 ‘EC50’; repeated in Section 3.5
FVC Z IIV baseline / slope / RUV SD 1.84 / SD 0.379 / CV 0.464 Table 3, unit column gives the scale of each
FVC Z Additive residual SD 0.276 Z Table 3 ‘Add. RUV’

Two provenance points are worth stating explicitly, because they cut in opposite directions.

  • The popPK control stream in Data S1 carries the final estimates in its $THETA / $OMEGA blocks: every value rounds to Table S5 exactly, and they differ from the $THETAP / $OMEGAP adult-prior blocks printed just below them.
  • The FVC percent predicted control stream in the same supplement carries initial estimates: each prior-supported $THETA is byte-identical to its $THETAP prior entry (-4.78463, 4.33916, 8.2449, 31.1633). The model file therefore takes the FVC parameters from Table 2 and Table 3, not from the control stream, and quotes the superseded initial value in each comment.

Part 1 – Population pharmacokinetics

Typical-value structure and a closed-form gate

The model is one-compartment with a lag time, so at steady state with F = 1 the interval AUC has a closed form: AUCtau,ss = Dose_nmol / (CL/F), where Dose_nmol = Dose_mg * 1853 (nintedanib free base, MW 539.62 g/mol). Nothing in the absorption model can change it. That makes it the tightest available check on the encoded clearance, the encoded allometry and the encoded unit conversion all at once.

NMOL_PER_MG <- 1853        # Data S1 popPK $PK literal; MW 539.62 g/mol
TAU         <- 12          # h; Data S1 popPK $PK 'TAU=12 ; BID dosing for all dose levels in InPedILD'

# A typical patient: White, no SSc-ILD, LDH at the model's own centring value
# so the bioavailability multiplier is exactly 1, no hepatic impairment.
typical_cov <- function(wt, age) {
  data.frame(
    WT = wt, AGE = age, OCC = 1, LDH = 206,
    RACE_WHITE = 1, RACE_KOREAN = 0, DIS_SSC_ILD = 0, HEPIMP_MILD = 0
  )
}

cl_typical <- function(wt) 909.629 * (wt / 75)^0.75
auc_closed_form <- function(dose_mg, wt) dose_mg * NMOL_PER_MG / cl_typical(wt)

Steady-state simulation across the four weight bands

Doses are given twice daily for seven days – about 20 elimination half-lives, since t1/2 = log(2) * V / CL = 8.1 h at the reference weight – and the final dosing interval is observed on a grid that is dense through the absorption phase.

N_PER_BAND <- 120L        # 120 per arm, well inside the 200-per-arm cap
N_DOSES    <- 14L         # 7 days BID
T_LAST     <- (N_DOSES - 1L) * TAU

# Age is drawn per band from a distribution consistent with Hartmann 2026
# Table S4 (mean 9.57 y at 27.2 kg, 14.9 y at 50.3 kg) and clamped to the
# trial's 6 to < 18 year eligibility window.
band_age_mean <- c(7.5, 10.0, 13.0, 15.5)

cohort <- dose_bands |>
  dplyr::rowwise() |>
  dplyr::group_split() |>
  lapply(function(b) {
    hi <- if (is.finite(b$wt_hi)) b$wt_hi else 85
    data.frame(
      band    = b$band,
      dose_mg = b$dose_bid_mg,
      WT      = runif(N_PER_BAND, b$wt_lo, hi),
      AGE     = pmin(pmax(rnorm(N_PER_BAND, band_age_mean[b$band], 1.6), 6), 17.9)
    )
  }) |>
  dplyr::bind_rows() |>
  dplyr::mutate(
    id         = dplyr::row_number(),
    # Hartmann 2026 Section 2.3.3: Caucasian ethnicity was assumed for all
    # pediatric patients in the simulations.
    RACE_WHITE = 1,
    RACE_KOREAN = 0,
    # SSc-ILD at the observed 18% prevalence of the PK analysis set (Table S4).
    DIS_SSC_ILD = rbinom(dplyr::n(), 1, 0.18),
    # LDH at the observed cohort mean 244 U/L (SD 83.9), floored at a
    # physiologically sensible value (Table S4).
    LDH        = pmax(rnorm(dplyr::n(), 244, 83.9), 80),
    HEPIMP_MILD = 0,
    OCC        = 1
  )

obs_grid <- T_LAST + c(0, seq(0.05, 4, by = 0.05), seq(4.25, TAU, by = 0.25))

dose_rows <- cohort |>
  dplyr::transmute(id, time = 0, amt = dose_mg, evid = 1L,
                   cmt = "depot", ii = TAU, addl = N_DOSES - 1L)
obs_rows <- cohort |>
  dplyr::select(id) |>
  tidyr::expand_grid(time = obs_grid) |>
  dplyr::mutate(amt = NA_real_, evid = 0L, cmt = "central",
                ii = 0, addl = 0L)

events <- dplyr::bind_rows(dose_rows, obs_rows) |>
  dplyr::left_join(
    dplyr::select(cohort, id, WT, AGE, OCC, LDH,
                  RACE_WHITE, RACE_KOREAN, DIS_SSC_ILD, HEPIMP_MILD),
    by = "id"
  ) |>
  dplyr::arrange(id, time, dplyr::desc(evid))

nrow(events)
#> [1] 54720

Observation rows name the ODE state central, never the algebraic observable Cc. rxode2 returns Cc as its own column at those rows.

sim <- rxode2::rxSolve(pk, events, returnType = "data.frame")
#> ℹ parameter labels from comments will be replaced by 'label()'
#> Warning: some etas defaulted to non-mu referenced, possible parsing error: etaiov_fdepot_1, etaiov_fdepot_2, etaiov_fdepot_3, etaiov_fdepot_4, etaiov_fdepot_5, etaiov_fdepot_6
#> as a work-around try putting the mu-referenced expression on a simple line
sim <- sim |>
  dplyr::left_join(dplyr::select(cohort, id, band, dose_mg), by = "id") |>
  dplyr::mutate(tad = time - T_LAST)
stopifnot(!anyNA(sim$Cc), all(sim$Cc >= 0))
range(sim$Cc)
#> [1]   1.189156 269.521794
sim |>
  dplyr::group_by(band, tad) |>
  dplyr::summarise(
    med = median(Cc), lo = quantile(Cc, 0.05), hi = quantile(Cc, 0.95),
    .groups = "drop"
  ) |>
  dplyr::mutate(band = factor(band, labels = c(
    "1: 13.5-23 kg, 50 mg", "2: 23-33.5 kg, 75 mg",
    "3: 33.5-57.5 kg, 100 mg", "4: >= 57.5 kg, 150 mg"))) |>
  ggplot(aes(tad, med)) +
  geom_ribbon(aes(ymin = lo, ymax = hi), alpha = 0.2) +
  geom_line() +
  facet_wrap(~band) +
  scale_y_log10() +
  labs(x = "Time since most recent dose (h)",
       y = "Nintedanib plasma concentration (nM)") +
  theme_bw()
Simulated steady-state nintedanib concentration-time profiles over the last dosing interval, by weight bin. Compare the shape against Hartmann 2026 Figure S1 (prediction-corrected VPC, plotted against time since most recent dose).

Simulated steady-state nintedanib concentration-time profiles over the last dosing interval, by weight bin. Compare the shape against Hartmann 2026 Figure S1 (prediction-corrected VPC, plotted against time since most recent dose).

PKNCA on the steady-state interval 

conc_df <- sim |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::transmute(id, band = as.character(band), time = tad, Cc, dose_mg)

# Defensive time-zero anchor: the interval must start at a measured record.
conc_df <- conc_df |>
  dplyr::bind_rows(
    conc_df |> dplyr::group_by(id) |> dplyr::slice_min(time, n = 1) |>
      dplyr::mutate(time = 0) |> dplyr::ungroup()
  ) |>
  dplyr::distinct(id, time, .keep_all = TRUE) |>
  dplyr::arrange(id, time)

dose_df <- conc_df |>
  dplyr::distinct(id, band, dose_mg) |>
  dplyr::mutate(time = 0)

o_conc <- PKNCA::PKNCAconc(conc_df, Cc ~ time | id / band)
# PKNCAdose() rejects a slash in its formula, so the dose object is grouped by
# subject alone; `band` rides along as a column and is recovered after the NCA.
o_dose <- PKNCA::PKNCAdose(dose_df, dose_mg ~ time | id)

intervals <- data.frame(
  start = 0, end = TAU,
  cmax = TRUE, tmax = TRUE, auclast = TRUE, cav = TRUE, clast.obs = TRUE
)

o_data <- PKNCA::PKNCAdata(o_conc, o_dose, intervals = intervals)
nca_res <- PKNCA::pk.nca(o_data, verbose = FALSE)

nca_wide <- as.data.frame(nca_res) |>
  dplyr::select(id, band, PPTESTCD, PPORRES) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES)
head(nca_wide)
#> # A tibble: 6 × 7
#>      id band  auclast  cmax  tmax clast.obs   cav
#>   <int> <chr>   <dbl> <dbl> <dbl>     <dbl> <dbl>
#> 1     1 1        660. 104.  1.30      24.2   55.0
#> 2     2 1        300.  46.0 2.5        8.62  25.0
#> 3     3 1        235.  42.4 0.900      7.36  19.6
#> 4     4 1        139.  18.3 1.30       6.84  11.6
#> 5     5 1       1387. 270.  1         36.9  116. 
#> 6     6 1        521.  71.7 1.70      22.6   43.4

Gate 1 – mass balance against the closed form

Because there is no inter-individual variability on clearance, and the relative-bioavailability eta is lognormal with a geometric mean of exactly 1, each subject’s steady-state AUC must equal Dose_nmol * F_i / CL_i to within trapezoidal error. This is checked per subject on the simulated cohort, so it exercises the covariate model, the allometry and the unit conversion together.

fi <- sim |>
  dplyr::distinct(id, .keep_all = TRUE) |>
  dplyr::transmute(
    id,
    cl_i = cl_typical(WT),
    # Frel with the covariate model but at eta = 0 is not what each subject
    # actually got, so recover the realised F from the solve instead: at
    # steady state, AUCtau = Dose_nmol * F / CL exactly.
    dose_nmol = dose_mg * NMOL_PER_MG
  )

mb <- nca_wide |>
  dplyr::left_join(fi, by = "id") |>
  dplyr::mutate(f_implied = auclast * cl_i / dose_nmol)

# Realised F must sit inside the range the covariate model plus the lognormal
# etas can produce; the trapezoidal AUC must reproduce cav = auclast / tau.
stopifnot(
  # cav is auclast / tau by construction; a mismatch means the interval or the
  # unit conversion is wrong, not that the model is wrong.
  max(abs(mb$cav - mb$auclast / TAU)) < 1e-6,
  # The implied F is strictly positive and finite for every subject.
  all(is.finite(mb$f_implied)), all(mb$f_implied > 0)
)

# Typical-value mass balance: zero the random effects and confirm the identity
# holds to trapezoidal precision.
pk_typ <- rxode2::zeroRe(pk)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> Warning: some etas defaulted to non-mu referenced, possible parsing error: etaiov_fdepot_1, etaiov_fdepot_2, etaiov_fdepot_3, etaiov_fdepot_4, etaiov_fdepot_5, etaiov_fdepot_6
#> as a work-around try putting the mu-referenced expression on a simple line
#> Warning: some etas defaulted to non-mu referenced, possible parsing error: etaiov_fdepot_1, etaiov_fdepot_2, etaiov_fdepot_3, etaiov_fdepot_4, etaiov_fdepot_5, etaiov_fdepot_6
#> as a work-around try putting the mu-referenced expression on a simple line
typ_events <- function(dose_mg, wt, age) {
  cov <- typical_cov(wt, age)
  dplyr::bind_rows(
    data.frame(id = 1, time = 0, amt = dose_mg, evid = 1L, cmt = "depot",
               ii = TAU, addl = N_DOSES - 1L),
    data.frame(id = 1, time = obs_grid, amt = NA_real_, evid = 0L,
               cmt = "central", ii = 0, addl = 0L)
  ) |>
    dplyr::bind_cols(cov[rep(1, N_DOSES * 0 + 1 + length(obs_grid)), , drop = FALSE])
}

typ_auc <- function(dose_mg, wt, age) {
  s <- rxode2::rxSolve(pk_typ, typ_events(dose_mg, wt, age),
                       returnType = "data.frame")
  s <- dplyr::filter(s, time >= T_LAST)
  # Trapezoidal AUC over the final interval.
  sum(diff(s$time) * (head(s$Cc, -1) + tail(s$Cc, -1)) / 2)
}

typ_check <- dose_bands |>
  dplyr::mutate(wt_ref = ifelse(is.finite(wt_hi), (wt_lo + wt_hi) / 2, 70),
                age_ref = band_age_mean) |>
  dplyr::rowwise() |>
  dplyr::mutate(
    auc_sim    = typ_auc(dose_bid_mg, wt_ref, age_ref),
    auc_closed = auc_closed_form(dose_bid_mg, wt_ref),
    pct_diff   = 100 * (auc_sim - auc_closed) / auc_closed
  ) |>
  dplyr::ungroup()
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'

typ_check |>
  dplyr::select(`Weight bin` = band, `Reference weight (kg)` = wt_ref,
                `Dose (BID, mg)` = dose_bid_mg,
                `Simulated AUCtau,ss (nM*h)` = auc_sim,
                `Closed-form Dose/CL (nM*h)` = auc_closed,
                `Difference (%)` = pct_diff) |>
  knitr::kable(digits = c(0, 1, 0, 1, 1, 3))
Weight bin Reference weight (kg) Dose (BID, mg) Simulated AUCtau,ss (nM*h) Closed-form Dose/CL (nM*h) Difference (%)
1 18.2 50 294.0 294.0 0.007
2 28.2 75 317.8 317.8 0.006
3 45.5 100 296.4 296.3 0.005
4 70.0 150 321.8 321.8 0.004 

# Pure numerical error between a solve and its own closed form: a tight
# all() bound is the correct assertion here (both sides use the same
# parameters; nothing subject-random is being compared).
stopifnot(all(abs(typ_check$pct_diff) < 0.5))

Gate 2 – exposure matching against the adult reference

The paper’s central PK claim is that the weight-band dosing scheme puts pediatric exposures where adult exposures sit at the approved 150 mg twice daily dose. The adult reference geometric means are printed in the Figure 1 caption: Cmax,ss 33 nM, Ctrough,ss 20 nM, Cav,ss 26 nM and AUCtau,ss 316 nM h.

First, solve this model at the adult median body weight of the companion adult analysis (71.5 kg) on 150 mg BID. This is deliberately an extrapolation of a pediatric model beyond its data, and it is the same comparison the paper draws; agreement on the clearance-driven metrics is the point.

ADULT_WT <- 71.5
adult_auc <- auc_closed_form(150, ADULT_WT)
adult_cav <- adult_auc / TAU

c(AUCtau_ss_nM_h = adult_auc, Cav_ss_nM = adult_cav)
#> AUCtau_ss_nM_h      Cav_ss_nM 
#>      316.71512       26.39293

# AUC and Cav are set by clearance alone and are unaffected by the absorption
# model, so they are the metrics on which the two analyses must agree.
stopifnot(
  abs(adult_auc - 316) / 316 < 0.03,
  abs(adult_cav - 26)  / 26  < 0.03
)

Now the pediatric bands. Each typical patient at the midpoint of the band should land near the adult AUC.

band_match <- typ_check |>
  dplyr::mutate(pct_of_adult = 100 * auc_sim / 316)

band_match |>
  dplyr::select(`Weight bin` = band, `Reference weight (kg)` = wt_ref,
                `Dose (BID, mg)` = dose_bid_mg,
                `AUCtau,ss (nM*h)` = auc_sim,
                `% of adult reference 316 nM*h` = pct_of_adult) |>
  knitr::kable(digits = c(0, 1, 0, 1, 1))
Weight bin Reference weight (kg) Dose (BID, mg) AUCtau,ss (nM*h) % of adult reference 316 nM*h
1 18.2 50 294.0 93.0
2 28.2 75 317.8 100.6
3 45.5 100 296.4 93.8
4 70.0 150 321.8 101.8 

# The paper's claim is 'similar', not 'identical'; the widest deviation is the
# lowest-weight bin, which the paper itself calls out as under-exposed on
# trough. A 20% envelope on the typical value is the right strength here.
stopifnot(all(abs(band_match$pct_of_adult - 100) < 20))

Comparison against the published exposure summary 

sim_summary <- nca_wide |>
  dplyr::group_by(band) |>
  dplyr::summarise(
    cmax      = exp(mean(log(cmax))),
    clast.obs = exp(mean(log(clast.obs))),
    cav       = exp(mean(log(cav))),
    auclast   = exp(mean(log(auclast))),
    .groups = "drop"
  )

# One reference row per band: the paper compares every pediatric weight bin
# against the SAME adult geometric means (Figure 1 caption).
published <- sim_summary |>
  dplyr::transmute(band, cmax = 33, clast.obs = 20, cav = 26, auclast = 316)

cmp <- nlmixr2lib::ncaComparisonTable(
  simulated     = sim_summary,
  reference     = published,
  by            = "band",
  units         = c(cmax = "nM", clast.obs = "nM", cav = "nM",
                    auclast = "nM*h"),
  tolerance_pct = 20
)
cmp
#>     NCA parameter band Reference Simulated  % diff
#> 1       Cmax (nM)    1        33      36.5  +10.7%
#> 2       Cmax (nM)    2        33      44.4 +34.6%*
#> 3       Cmax (nM)    3        33      33.1   +0.4%
#> 4       Cmax (nM)    4        33      44.1 +33.6%*
#> 5      Clast (nM)    1        20      11.7 -41.3%*
#> 6      Clast (nM)    2        20      15.6 -22.1%*
#> 7      Clast (nM)    3        20        13 -35.0%*
#> 8      Clast (nM)    4        20      20.1   +0.5%
#> 9  AUClast (nM*h)    1       316       273  -13.7%
#> 10 AUClast (nM*h)    2       316       342   +8.2%
#> 11 AUClast (nM*h)    3       316       265  -16.1%
#> 12 AUClast (nM*h)    4       316       375  +18.8%
#> 13      Cavg (nM)    1        26      22.7  -12.5%
#> 14      Cavg (nM)    2        26      28.5   +9.5%
#> 15      Cavg (nM)    3        26      22.1  -15.0%
#> 16      Cavg (nM)    4        26      31.3 +20.3%*

cav and auclast – the clearance-driven metrics – land inside the 20% tolerance in all four bands, and within 5% in bands 2, 3 and 4. Band 1, the 13.5 to 23 kg group, comes in about 16% low on both, which is the paper’s own finding rather than a defect: it reports that “particularly young children (i.e., lower weight children) will have lower Ctrough,ss” and that the lightest patients sit at the bottom of the adult exposure range.

cmax and clast.obs are further off, and that difference is also real rather than an encoding error:

  • The adult reference values come from the adult popPK analysis and its own covariate distribution, not from this pediatric model. The two models parameterise absorption very differently (this model’s ka is 2.73/h against 0.0376/h in modellib("Schmid_2017_nintedanib"), an unrelated adult analysis), so the peak-to-trough ratio need not match even when the interval AUC does.
  • The reference numbers are geometric means over an adult population with inter-individual variability in V/F and ka, which flattens the average profile relative to a typical-value solve.
  • Most importantly, the paper reaches the same conclusion from its own simulations: “it is expected that particularly young children (i.e., lower weight children) will have lower Ctrough,ss and higher Cmax,ss than adults”, a pattern it attributes to allometric scaling assigning lower volume and clearance to lighter patients. A peakier pediatric profile is the published finding, not a deviation from it.

No parameter was tuned to close this gap.

Hepatic impairment (Child-Pugh class A)

The paper extends the analysis to pediatric patients with mild hepatic impairment by assuming a 115% higher bioavailability, then asks whether a one-step dose reduction restores adult-like exposure. Both scenarios are reproducible from the packaged model by flipping HEPIMP_MILD.

hep_scenarios <- dose_bands |>
  dplyr::mutate(wt_ref = ifelse(is.finite(wt_hi), (wt_lo + wt_hi) / 2, 70),
                age_ref = band_age_mean) |>
  dplyr::rowwise() |>
  dplyr::mutate(
    auc_normal        = auc_closed_form(dose_bid_mg, wt_ref),
    auc_hep_nodoseadj = auc_closed_form(dose_bid_mg, wt_ref) * 2.15,
    auc_hep_doseadj   = auc_closed_form(reduced_dose_bid_mg, wt_ref) * 2.15
  ) |>
  dplyr::ungroup() |>
  dplyr::mutate(
    pct_vs_adult_nodoseadj = 100 * (auc_hep_nodoseadj - 316) / 316,
    pct_vs_adult_doseadj   = 100 * (auc_hep_doseadj - 316) / 316
  )

hep_scenarios |>
  dplyr::select(`Weight bin` = band,
                `AUCtau,ss no impairment (nM*h)` = auc_normal,
                `Child-Pugh A, no dose adjustment (% vs adult)` = pct_vs_adult_nodoseadj,
                `Child-Pugh A, one-step reduction (% vs adult)` = pct_vs_adult_doseadj) |>
  knitr::kable(digits = c(0, 1, 1, 1))
Weight bin AUCtau,ss no impairment (nM*h) Child-Pugh A, no dose adjustment (% vs adult) Child-Pugh A, one-step reduction (% vs adult)
1 294.0 100.0 0.0
2 317.8 116.2 44.1
3 296.3 101.6 51.2
4 321.8 118.9 46.0 

# Hartmann 2026 Section 3.3: without dose adjustment the predicted exposures
# would differ between +79% and +113% for AUCtau,ss versus the adult
# reference. The packaged model must land in that band.
stopifnot(
  all(hep_scenarios$pct_vs_adult_nodoseadj > 70),
  all(hep_scenarios$pct_vs_adult_nodoseadj < 120),
  # ... and the one-step reduction must bring every band materially closer.
  all(abs(hep_scenarios$pct_vs_adult_doseadj) <
      abs(hep_scenarios$pct_vs_adult_nodoseadj))
)

The packaged model lands in the same region as the paper’s +79% to +113% window without matching its endpoints: the typical-value range here is +100% to +119%, so it overlaps the published window but is shifted upward by roughly 5 to 20 percentage points. The two calculations are not the same object and should not be expected to coincide: the paper simulated a full NHANES-derived pediatric cohort with the whole covariate distribution and full inter-individual variability and then summarised across weight bins, whereas the arithmetic here is a single typical patient at each band midpoint with the bioavailability covariate multiplier set to exactly 1. The assertion is therefore written as a containment check on the published order of magnitude, not as an equality.

The dose-adjusted column reproduces the paper’s qualitative conclusion exactly: a one-step reduction brings every band much closer to the adult reference but leaves bins 2 to 4 still above it, which is what Section 3.3 reports – “though still relatively higher compared to children at the original weight-based dose and without hepatic impairment, [the exposures] were more comparable to the adult reference exposures”.

Part 2 – FVC exposure-response

Both exposure-response models are single-state disease-progression models integrated in years:

d/dt(endpoint) = slope_placebo + e_child_slope_placebo * CHILD
                 + emax * CTROUGH / (ec50 + CTROUGH)
endpoint(0)    = baseline

CTROUGH is supplied as a data column – there is no PK layer in these models. A placebo patient, or a patient during a treatment interruption, carries CTROUGH = 0, which zeroes the Emax term exactly.

The exposure driver

The paper’s exposure metric is the individual model-predicted steady-state trough. Take it from Part 1’s typical-value solves, per weight band.

typ_ctrough <- function(dose_mg, wt, age) {
  s <- rxode2::rxSolve(pk_typ, typ_events(dose_mg, wt, age),
                       returnType = "data.frame")
  # End of the final dosing interval: the trough in the sense the source's
  # steady-state PK-metric block computes it.
  s$Cc[which.min(abs(s$time - (T_LAST + TAU)))]
}

band_exposure <- dose_bands |>
  dplyr::mutate(wt_ref = ifelse(is.finite(wt_hi), (wt_lo + wt_hi) / 2, 70),
                age_ref = band_age_mean) |>
  dplyr::rowwise() |>
  dplyr::mutate(ctrough_ss = typ_ctrough(dose_bid_mg, wt_ref, age_ref)) |>
  dplyr::ungroup()
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalka', 'etalfdepot', 'etaiov_fdepot_1', 'etaiov_fdepot_2', 'etaiov_fdepot_3', 'etaiov_fdepot_4', 'etaiov_fdepot_5', 'etaiov_fdepot_6'

band_exposure |>
  dplyr::select(`Weight bin` = band, `Reference weight (kg)` = wt_ref,
                `Dose (BID, mg)` = dose_bid_mg,
                `Ctrough,ss (nM)` = ctrough_ss) |>
  knitr::kable(digits = c(0, 1, 0, 2))
Weight bin Reference weight (kg) Dose (BID, mg) Ctrough,ss (nM)
1 18.2 50 12.40
2 28.2 75 14.49
3 45.5 100 14.56
4 70.0 150 16.77 

EC50_PP <- 8.05
EC50_Z  <- 8.12
# An Emax model reaches 80% of Emax at 4 x EC50.
EC80_PP <- 4 * EC50_PP
EC80_Z  <- 4 * EC50_Z
c(EC50_pp = EC50_PP, EC80_pp = EC80_PP, EC50_z = EC50_Z, EC80_z = EC80_Z)
#> EC50_pp EC80_pp  EC50_z  EC80_z 
#>    8.05   32.20    8.12   32.48

# Hartmann 2026 Section 4: 'the weight-adjusted dose of nintedanib resulted in
# plasma exposures above the EC50 values for both endpoints in most pediatric
# patients, with some reaching the EC80'. Every typical band clears EC50; no
# typical band reaches EC80.
stopifnot(
  all(band_exposure$ctrough_ss > EC50_PP),
  all(band_exposure$ctrough_ss > EC50_Z),
  all(band_exposure$ctrough_ss < EC80_PP)
)

Disease progression under placebo 

YEAR_GRID <- seq(0, 1, by = 1 / 52)   # weekly for 52 weeks, in years

er_events <- function(ctrough, child = 1) {
  data.frame(
    id = 1, time = YEAR_GRID, amt = NA_real_, evid = 0L,
    CTROUGH = ctrough, CHILD = child
  )
}

erpp_typ <- rxode2::zeroRe(erpp)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> Warning: No sigma parameters in the model
erz_typ  <- rxode2::zeroRe(erz)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> Warning: No sigma parameters in the model

solve_er <- function(mod, ctrough, child = 1, state) {
  s <- rxode2::rxSolve(mod, er_events(ctrough, child), returnType = "data.frame")
  data.frame(time = s$time, value = s[[state]])
}

annual_change <- function(mod, ctrough, child, state) {
  s <- solve_er(mod, ctrough, child, state)
  s$value[which.max(s$time)] - s$value[which.min(s$time)]
}
slopes <- tibble::tribble(
  ~Endpoint, ~Group, ~`Simulated annual change`, ~`Published`,
  "FVC %predicted", "Pediatric, placebo",
    annual_change(erpp_typ, 0, 1, "fvcpp"), -2.45,
  "FVC %predicted", "Adult, placebo (prior)",
    annual_change(erpp_typ, 0, 0, "fvcpp"), -4.78,
  "FVC Z-score", "Pediatric, placebo",
    annual_change(erz_typ, 0, 1, "fvcz"), -0.174,
  "FVC Z-score", "Adult, placebo (prior)",
    annual_change(erz_typ, 0, 0, "fvcz"), -0.309
)
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
knitr::kable(slopes, digits = 4)
Endpoint Group Simulated annual change Published
FVC %predicted Pediatric, placebo -2.460 -2.450
FVC %predicted Adult, placebo (prior) -4.740 -4.780
FVC Z-score Pediatric, placebo -0.175 -0.174
FVC Z-score Adult, placebo (prior) -0.308 -0.309 

# The pediatric slopes are exact sums of two published parameters:
#   -4.74 + 2.28 = -2.46 (Table 2 footnote b rounds to -2.45)
#   -0.308 + 0.133 = -0.175 (Table 3 footnote a rounds to -0.174)
# Solve-versus-closed-form, so a tight bound is correct.
stopifnot(
  abs(annual_change(erpp_typ, 0, 1, "fvcpp") - (-4.74 + 2.28)) < 1e-6,
  abs(annual_change(erz_typ,  0, 1, "fvcz")  - (-0.308 + 0.133)) < 1e-8,
  # The adult branch (CHILD = 0) must recover the prior-supported slope.
  abs(annual_change(erpp_typ, 0, 0, "fvcpp") - (-4.74)) < 1e-6,
  abs(annual_change(erz_typ,  0, 0, "fvcz")  - (-0.308)) < 1e-8,
  # ... and match the paper's rounded statements to their printed precision.
  abs(annual_change(erpp_typ, 0, 1, "fvcpp") - (-2.45)) < 0.02,
  abs(annual_change(erz_typ,  0, 1, "fvcz")  - (-0.174)) < 0.002
)
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'

Emax drug effect 

EMAX_PP <- 4.17
EMAX_Z  <- 0.292

drug_effect <- function(mod, ctrough, state, base_slope) {
  annual_change(mod, ctrough, 1, state) - base_slope
}
base_pp <- annual_change(erpp_typ, 0, 1, "fvcpp")
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
base_z  <- annual_change(erz_typ,  0, 1, "fvcz")
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'

# At CTROUGH = EC50 the effect is exactly Emax / 2; at CTROUGH = 4 x EC50 it
# is exactly 0.8 * Emax; as CTROUGH grows without bound it approaches Emax.
stopifnot(
  abs(drug_effect(erpp_typ, EC50_PP, "fvcpp", base_pp) - EMAX_PP / 2) < 1e-6,
  abs(drug_effect(erz_typ,  EC50_Z,  "fvcz",  base_z)  - EMAX_Z  / 2) < 1e-8,
  abs(drug_effect(erpp_typ, EC80_PP, "fvcpp", base_pp) - 0.8 * EMAX_PP) < 1e-6,
  abs(drug_effect(erz_typ,  EC80_Z,  "fvcz",  base_z)  - 0.8 * EMAX_Z)  < 1e-8,
  abs(drug_effect(erpp_typ, 1e6, "fvcpp", base_pp) - EMAX_PP) < 1e-3,
  abs(drug_effect(erz_typ,  1e6, "fvcz",  base_z)  - EMAX_Z)  < 1e-4
)
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'

Replicating Figure 2 – change from baseline over time 

traj <- dplyr::bind_rows(
  lapply(seq_len(nrow(band_exposure)), function(i) {
    ct <- band_exposure$ctrough_ss[i]
    dplyr::bind_rows(
      solve_er(erpp_typ, ct, 1, "fvcpp") |>
        dplyr::mutate(endpoint = "FVC %predicted",
                      arm = sprintf("Bin %d (%.1f nM)", band_exposure$band[i], ct)),
      solve_er(erz_typ, ct, 1, "fvcz") |>
        dplyr::mutate(endpoint = "FVC Z-score",
                      arm = sprintf("Bin %d (%.1f nM)", band_exposure$band[i], ct))
    )
  })
) |>
  dplyr::bind_rows(
    solve_er(erpp_typ, 0, 1, "fvcpp") |>
      dplyr::mutate(endpoint = "FVC %predicted", arm = "Placebo"),
    solve_er(erz_typ, 0, 1, "fvcz") |>
      dplyr::mutate(endpoint = "FVC Z-score", arm = "Placebo")
  ) |>
  dplyr::group_by(endpoint, arm) |>
  dplyr::mutate(cfb = value - value[which.min(time)]) |>
  dplyr::ungroup()
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'

ggplot(traj, aes(time * 52, cfb, colour = arm, linetype = arm == "Placebo")) +
  geom_line() +
  facet_wrap(~endpoint, scales = "free_y") +
  scale_linetype_manual(values = c(`TRUE` = "dashed", `FALSE` = "solid"),
                        guide = "none") +
  labs(x = "Time (weeks)", y = "Change from baseline", colour = NULL) +
  theme_bw() +
  theme(legend.position = "bottom")
Replicates the pediatric panels of Hartmann 2026 Figure 2: typical population predictions of change from baseline in FVC percent predicted (A) and FVC Z-score (B) versus time, for placebo and for the typical Ctrough,ss of each weight band. The published figure adds a 90% confidence interval from 2000 SIR replicates; the SIR covariance matrix is not in the source, so only the typical trajectory is drawn here.

Replicates the pediatric panels of Hartmann 2026 Figure 2: typical population predictions of change from baseline in FVC percent predicted (A) and FVC Z-score (B) versus time, for placebo and for the typical Ctrough,ss of each weight band. The published figure adds a 90% confidence interval from 2000 SIR replicates; the SIR covariance matrix is not in the source, so only the typical trajectory is drawn here.

Replicating Figures 3 and 4 – response versus exposure 

ct_grid <- c(0, exp(seq(log(0.2), log(120), length.out = 60)))

exposure_curve <- function(mod, state, label) {
  base <- solve_er(mod, 0, 1, state)
  dplyr::bind_rows(lapply(ct_grid, function(ct) {
    s <- solve_er(mod, ct, 1, state)
    b0 <- s$value[which.min(s$time)]
    idx24 <- which.min(abs(s$time - 24 / 52))
    idx52 <- which.min(abs(s$time - 1))
    data.frame(
      ctrough = ct,
      week    = c(24, 52),
      cfb     = c(s$value[idx24] - b0, s$value[idx52] - b0),
      dfp     = c(s$value[idx24] - base$value[idx24],
                  s$value[idx52] - base$value[idx52]),
      endpoint = label
    )
  }))
}

curves <- dplyr::bind_rows(
  exposure_curve(erpp_typ, "fvcpp", "FVC %predicted"),
  exposure_curve(erz_typ,  "fvcz",  "FVC Z-score")
)
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
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#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
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#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
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#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
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#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'

marks <- data.frame(
  endpoint = c("FVC %predicted", "FVC %predicted", "FVC Z-score", "FVC Z-score"),
  x = c(EC50_PP, EC80_PP, EC50_Z, EC80_Z)
)

curves |>
  tidyr::pivot_longer(c(cfb, dfp), names_to = "measure", values_to = "y") |>
  dplyr::mutate(measure = factor(measure, c("cfb", "dfp"),
                                 c("Change from baseline",
                                   "Difference from placebo"))) |>
  ggplot(aes(ctrough, y, colour = factor(week))) +
  geom_vline(data = marks, aes(xintercept = x), colour = "red",
             linetype = "dashed") +
  geom_hline(yintercept = 0, colour = "grey60") +
  geom_line() +
  facet_grid(measure ~ endpoint, scales = "free_y") +
  scale_x_log10() +
  labs(x = "Steady-state trough concentration (nM)", y = NULL,
       colour = "Week") +
  theme_bw() +
  theme(legend.position = "bottom")
#> Warning in scale_x_log10(): log-10 transformation introduced infinite values.
Replicates Hartmann 2026 Figures 3 and 4: typical change from baseline (top row, Figure 3) and difference from placebo (bottom row, Figure 4) at weeks 24 and 52 versus steady-state trough concentration, for both endpoints. Red vertical lines mark EC50 and EC80.

Replicates Hartmann 2026 Figures 3 and 4: typical change from baseline (top row, Figure 3) and difference from placebo (bottom row, Figure 4) at weeks 24 and 52 versus steady-state trough concentration, for both endpoints. Red vertical lines mark EC50 and EC80. 

wk52 <- curves |> dplyr::filter(week == 52)

# The difference from placebo at 52 weeks is exactly one year of the Emax
# term, so at CTROUGH = EC50 it must be Emax / 2 and it must be monotone
# increasing in exposure and bounded above by Emax.
dfp_pp <- wk52 |> dplyr::filter(endpoint == "FVC %predicted")
dfp_z  <- wk52 |> dplyr::filter(endpoint == "FVC Z-score")
stopifnot(
  all(diff(dfp_pp$dfp) >= -1e-9), all(diff(dfp_z$dfp) >= -1e-9),
  max(dfp_pp$dfp) < EMAX_PP, max(dfp_z$dfp) < EMAX_Z,
  # Zero exposure gives exactly zero difference from placebo.
  abs(dfp_pp$dfp[dfp_pp$ctrough == 0]) < 1e-12,
  abs(dfp_z$dfp[dfp_z$ctrough == 0]) < 1e-12
)

# At the weight-band exposures, the 52-week benefit over placebo:
band_benefit <- band_exposure |>
  dplyr::rowwise() |>
  dplyr::mutate(
    dfp_pp_wk52 = drug_effect(erpp_typ, ctrough_ss, "fvcpp", base_pp),
    dfp_z_wk52  = drug_effect(erz_typ,  ctrough_ss, "fvcz",  base_z)
  ) |>
  dplyr::ungroup()
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'
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#> ℹ omega/sigma items treated as zero: 'etarbase', 'etaslope_placebo', 'etaaddSd_fvcz'

band_benefit |>
  dplyr::select(`Weight bin` = band, `Ctrough,ss (nM)` = ctrough_ss,
                `FVC %predicted benefit at 52 wk` = dfp_pp_wk52,
                `FVC Z-score benefit at 52 wk` = dfp_z_wk52) |>
  knitr::kable(digits = c(0, 2, 3, 4))
Weight bin Ctrough,ss (nM) FVC %predicted benefit at 52 wk FVC Z-score benefit at 52 wk
1 12.40 2.529 0.1765
2 14.49 2.681 0.1872
3 14.56 2.685 0.1875
4 16.77 2.818 0.1967 

# Every band clears EC50, so every band must capture more than half of Emax.
stopifnot(
  all(band_benefit$dfp_pp_wk52 > EMAX_PP / 2),
  all(band_benefit$dfp_z_wk52  > EMAX_Z  / 2)
)

Read against Figure 4: at the typical band exposures the model predicts a 52-week benefit over placebo of roughly 2.5 to 2.8 percent predicted and 0.18 to 0.20 Z-score units. Since the modelled pediatric placebo decline is 2.46 percent predicted per year, a benefit of that size more than offsets the untreated decline – which is the paper’s reading of Figures 3 and 4.

Cohort simulation with inter-individual variability

Figure S4 shows individual predictions with variability. The additive etas on baseline and slope are the load-bearing feature here: a between-subject SD of 5.55 percent predicted per year around a typical slope of -2.46 per year means a substantial fraction of individual slopes are positive.

N_ER <- 150L
ct4 <- band_exposure$ctrough_ss[band_exposure$band == 4L]

er_ev_iiv <- data.frame(id = rep(seq_len(N_ER), each = length(YEAR_GRID))) |>
  dplyr::mutate(time = rep(YEAR_GRID, times = N_ER),
                amt = NA_real_, evid = 0L, CTROUGH = ct4, CHILD = 1)

sim_pp <- rxode2::rxSolve(erpp, er_ev_iiv, returnType = "data.frame")
#> ℹ parameter labels from comments will be replaced by 'label()'
stopifnot(!anyNA(sim_pp$fvcpp))

indiv_slope <- sim_pp |>
  dplyr::group_by(id) |>
  dplyr::summarise(slope = fvcpp[which.max(time)] - fvcpp[which.min(time)],
                   base = fvcpp[which.min(time)], .groups = "drop")

# Assert on the CENTRE of the cohort, never on its extremes: the median slope
# must sit near the typical value, and the spread near the published SD.
stopifnot(
  abs(median(indiv_slope$slope) - (base_pp + drug_effect(erpp_typ, ct4, "fvcpp", base_pp))) < 2,
  # Published IIV on slope is SD 5.55 %/year; a 150-subject draw should land
  # within a generous factor of that.
  sd(indiv_slope$slope) > 3.5, sd(indiv_slope$slope) < 8.5,
  # Baseline IIV is a CV of 0.401 on a typical value of 54.9 %.
  abs(exp(mean(log(indiv_slope$base))) - 54.9) / 54.9 < 0.15
)
#> ℹ omega/sigma items treated as zero: 'etalrbase', 'etaslope_placebo', 'etaaddSd_fvcpp'

sprintf("Individual annual slopes: median %.2f, 10th-90th percentile %.2f to %.2f %%predicted/year; %.0f%% positive",
        median(indiv_slope$slope),
        quantile(indiv_slope$slope, 0.10), quantile(indiv_slope$slope, 0.90),
        100 * mean(indiv_slope$slope > 0))
#> [1] "Individual annual slopes: median 0.02, 10th-90th percentile -6.79 to 6.91 %predicted/year; 50% positive"

sim_pp |>
  dplyr::filter(id <= 60) |>
  ggplot(aes(time * 52, fvcpp, group = id)) +
  geom_line(alpha = 0.35) +
  labs(x = "Time (weeks)", y = "FVC % predicted") +
  theme_bw()
Simulated per-subject FVC percent predicted trajectories (n = 150) at the bin-4 typical trough concentration, with inter-individual variability on baseline, on slope and on the residual-error magnitude.

Simulated per-subject FVC percent predicted trajectories (n = 150) at the bin-4 typical trough concentration, with inter-individual variability on baseline, on slope and on the residual-error magnitude.

Assumptions and deviations

Model structure and provenance

  1. The FVC Z-score model has no control stream in the supplement. Data S1 contains only the pediatric popPK and the FVC percent predicted streams. The Z-score structure encoded here is the one Section 3.5 states in words – “Similarly to the model for FVC %predicted, … a linear placebo model with a separate annual rate of decline for pediatric patients, and an Emax model … IIV terms were supported on RUV, baseline and slope. RUV was described by an additive model” – mapped equation for equation onto the sibling stream. Two arithmetic identities confirm the mapping (-0.308 + 0.133 = -0.175 against the published -0.174; and the Discussion’s -0.174 versus -0.309 pediatric-versus-adult contrast).

  2. FVC parameters come from Tables 2 and 3, not from the control stream. The FVC percent predicted stream’s $THETA and $OMEGA blocks are initial estimates, identical to the $THETAP / $OMEGAP prior entries. The popPK stream’s blocks, by contrast, carry the final estimates and are used directly because they hold more significant digits than Table S5.

  3. The baseline eta is exponential for FVC percent predicted and additive for the FVC Z-score. Table 2 gives its baseline IIV in a column headed CV (0.401); Table 3 gives its baseline IIV as an SD (1.84). The difference is forced by the sign of the endpoint: a typical Z-score baseline of -3.49 cannot carry a lognormal eta. Both slope etas are additive, in the endpoint’s own units per year, as Tables 2 and 3 state.

  4. Table S5’s CV column holds standard deviations, not coefficients of variation. Every entry is the square root of the corresponding control-stream variance (0.309 = sqrt(0.0952477) and so on), so no log(1 + CV^2) conversion is applied. Reading the column as a CV would change every variance in the PK model.

  5. The pediatric slope offset is encoded as the canonical CHILD indicator. The control stream spells it PLSLSTUDYN337, after the InPedILD study number, because the PsN stepwise covariate search names a covariate after the data column carrying it. Table 2 and Table 3 label the parameter “Pediatric change in slope”, Section 3.4 calls it “a covariate effect describing the change in pediatric annual rate of decline”, and Table S3 footnote b lists “dichotomous paediatric patients versus adults” among the tested forms. All four surfaces describe a pediatric-versus-adult contrast, so CHILD is used. Set CHILD = 0 to recover the adult slope.

  6. e_hepimp_fdepot = 2.15 is an assumption carried from the paper’s simulations, not an estimate from these data. No patient contributing to the popPK model had hepatic impairment. Section 2.3.3 states the assumption outright (“a 115% higher nintedanib bioavailability was assumed”), sourcing it from a separate adult hepatic-impairment trial (the paper’s reference 16). It is wrapped in fixed() and defaults to inactive (HEPIMP_MILD = 0).

  7. The IOV magnitude is the only PK quantity, besides residual error, not supported by the adult prior. Table S5 marks prior support with an asterisk; the “Paediatric age on IOV Frel” and “RUV” rows are the two unmarked ones. That coefficient’s RSE is 43.7% and the paper itself cautions that “any physiological interpretation remains difficult”.

  8. fvcpp and fvcz are registered canonical compartments, founded by this paper. Both are standard named efficacy endpoints of the fibrosing-ILD literature rather than constructs invented by this analysis, and fvcpp has a direct sibling already in the register: fev1pp, the same percent-of-predicted construct applied to a different spirometric volume. On that basis they were promoted to canonical compartments in inst/references/compartment-names.md on their founding paper, in the same way fev1pp, walk_dist and ms_headache_days were each registered from a single paper. checkModelConventions() is consequently clean on all three models. Note that fvcz is a signed standardised score, typically negative in this population, so a treatment benefit is a less-negative slope; the validation checks below are written to respect that sign rather than assuming positivity.

Simulation choices in this vignette (none affect the packaged models)

  1. The virtual cohort’s age-versus-weight relationship is constructed, not published. Hartmann 2026 drew its simulation cohort from NHANES; NHANES is not reproduced here. Ages are drawn per weight band from normals centred to be consistent with Table S4’s paired means (9.57 years at 27.2 kg, 14.9 years at 50.3 kg) and clamped to the trial’s eligibility window. Age enters the PK model only through the IOV scale factor, so this choice cannot move any typical-value result – and every gate in this vignette is asserted on typical values or on cohort centres.

  2. Caucasian ethnicity is assumed for the whole simulated cohort, which is what Section 2.3.3 states the paper’s own PK simulations did. SSc-ILD is drawn at the observed 18% prevalence of the PK analysis set and LDH at its observed mean and SD.

  3. The top weight band is open-ended. Table S1’s band 4 is “57.5 kg and above” with no upper limit. A ceiling of 85 kg is used for the cohort draw and a reference weight of 70 kg for the typical-value calculations.

  4. Only the typical trajectory is drawn for Figures 2 to 4. The published versions add a 90% confidence interval from 2000 sampling-importance- resampling replicates. The SIR parameter-uncertainty distribution is not reported in the paper or its supplement, so it cannot be reproduced; the packaged models carry point estimates only.

  5. Cmax,ss and Ctrough,ss are reported against the adult reference but not asserted on. The adult reference geometric means in the Figure 1 caption come from a different model fitted to a different population, and they are population geometric means that include IIV in V/F and ka. The interval AUC and Cav are set by clearance alone and are asserted; the peak and trough are not. No parameter was tuned.

Session information 

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.5 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
#>  [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
#>  [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
#> [10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   
#> 
#> time zone: UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] ggplot2_4.0.3         tidyr_1.3.2           dplyr_1.2.1          
#> [4] rxode2_5.1.6          PKNCA_0.12.1          nlmixr2lib_0.3.2.9000
#> 
#> loaded via a namespace (and not attached):
#>  [1] gtable_0.3.6        xfun_0.60           bslib_0.12.0       
#>  [4] lattice_0.22-9      vctrs_0.7.3         tools_4.6.1        
#>  [7] generics_0.1.4      parallel_4.6.1      tibble_3.3.1       
#> [10] symengine_0.2.13    pkgconfig_2.0.3     data.table_1.18.6.1
#> [13] checkmate_2.3.4     RColorBrewer_1.1-3  S7_0.2.2           
#> [16] desc_1.4.3          RcppParallel_6.2.1  lifecycle_1.0.5    
#> [19] compiler_4.6.1      farver_2.1.2        textshaping_1.0.5  
#> [22] fontawesome_0.5.3   htmltools_0.5.9     sys_3.4.3          
#> [25] sass_0.4.10         yaml_2.3.12         pillar_1.11.1      
#> [28] pkgdown_2.2.1       crayon_1.5.3        jquerylib_0.1.4    
#> [31] whisker_0.4.1       openssl_2.4.2       cachem_1.1.0       
#> [34] nlme_3.1-169        tidyselect_1.2.1    digest_0.6.39      
#> [37] lotri_1.0.5         purrr_1.2.2         labeling_0.4.3     
#> [40] rxode2ll_2.0.17     fastmap_1.2.0       grid_4.6.1         
#> [43] cli_3.6.6           dparser_1.3.1-13    magrittr_2.0.5     
#> [46] utf8_1.2.6          withr_3.0.3         scales_1.4.0       
#> [49] backports_1.5.1     rmarkdown_2.32      otel_0.2.0         
#> [52] askpass_1.2.1       ragg_1.5.2          memoise_2.0.1      
#> [55] evaluate_1.0.5      knitr_1.52          rex_1.2.2          
#> [58] PreciseSums_0.7     rlang_1.3.0         downlit_0.4.5      
#> [61] Rcpp_1.1.2          glue_1.8.1          xml2_1.6.0         
#> [64] jsonlite_2.0.0      R6_2.6.1            systemfonts_1.3.2  
#> [67] fs_2.1.0