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Pretomanid (Zou 2022)

Source: vignettes/articles/Zou_2022_pretomanid.Rmd
Zou_2022_pretomanid.Rmd
ui <- rxode2::rxode(readModelDb("Zou_2022_pretomanid"))
#> 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_ka_1, etaiov_ka_2, etaiov_ka_3, etaiov_ka_4, etaiov_mtt_1, etaiov_mtt_2, etaiov_mtt_3, etaiov_mtt_4
#> as a work-around try putting the mu-referenced expression on a simple line

Model and source

  • Citation: Zou Y, Nedelman J, Lombardi A, Pappas F, Karlsson MO, Svensson EM. Characterizing Absorption Properties of Dispersible Pretomanid Tablets Using Population Pharmacokinetic Modelling. Clin Pharmacokinet. 2022;61(11):1585-1593. doi:10.1007/s40262-022-01163-w
  • Description: One-compartment population PK model with one transit compartment and first-order absorption for pretomanid dispersible (pediatric) and marketed tablets given with a high-fat meal to healthy adults, with dose-dependent relative bioavailability and inter-occasion variability on absorption

Zou 2022 characterised a pediatric dispersible-tablet formulation (DTF) of pretomanid against the marketed 200 mg tablet (MF) in study CL-011, a phase 1 single-dose, four-period crossover in healthy adults. Each participant received 10, 50 and 200 mg (4 x 50 mg) DTF and 200 mg MF, separated by 7-day washouts. Only the panel dosed after an FDA standard high-fat breakfast was modelled, because pretomanid is labelled to be taken with food.

The starting point was the Salinger 2019 pooled model (also in this package as Salinger_2019_pretomanid). Zou 2022 kept its one-compartment disposition and its transit-plus-absorption-compartment layout, but reduced the transit chain from three compartments to one:

dose --> transit1 --> depot --> central --> eliminated
           ktr         ka         kel

with ktr = 1 / MTT, so the mean absorption time is MAT = 1/KA + MTT. As in the Salinger 2019 model, depot is the source control stream’s ABS compartment and sits downstream of the transit compartment; oral doses enter transit1, which is why the model file declares dosing <- "transit1".

Covariates in the final model:

  • body weight, allometric with fixed exponents 0.75 (CL) and 1 (V) around a 55 kg reference participant;
  • formulation, on KA only: the marketed tablet absorbs 1.65-fold faster than the dispersible tablet. Formulation was tested on bioavailability (ratio 1.00, 90% CI 0.87-1.14) and not retained;
  • dose, as a power function on relative bioavailability, F = (DOSE/200)^0.0822.

Interindividual variability is on CL and V (correlated). Inter-occasion variability, one occasion per crossover period, is on F, KA and MTT. The residual error is combined proportional and additive, and both components are multiplied by 3.46 for samples taken less than 10 h after the dose.

cat("ODE states:", paste(ui$state, collapse = " -> "), "\n")
#> ODE states: transit1 -> depot -> central
cat("Observation:", paste(ui$predDf$var, collapse = ", "), "\n")
#> Observation: Cc
stopifnot(identical(ui$state, c("transit1", "depot", "central")))

Population

The model was fitted to the fed panel of CL-011: 24 healthy adults (16 male, 8 female) contributing 1377 concentrations. Zou 2022 Table 1 (‘Fed’ column) reports a median age of 39.0 years (range 23-50), median weight 76.2 kg (64.4-117), median height 170 cm (155-193) and median BMI 27.0 kg/m^2 (22.6-31.5). The study was run at a single site in San Antonio, Texas. The same information is stored in the model’s population metadata:

str(readModelDb("Zou_2022_pretomanid")()$population, give.attr = FALSE)
#> Warning: some etas defaulted to non-mu referenced, possible parsing error: etaiov_fdepot_1, etaiov_fdepot_2, etaiov_fdepot_3, etaiov_fdepot_4, etaiov_ka_1, etaiov_ka_2, etaiov_ka_3, etaiov_ka_4, etaiov_mtt_1, etaiov_mtt_2, etaiov_mtt_3, etaiov_mtt_4
#> as a work-around try putting the mu-referenced expression on a simple line
#> List of 15
#>  $ species       : chr "human"
#>  $ n_subjects    : int 24
#>  $ n_studies     : int 1
#>  $ n_observations: int 1377
#>  $ age_range     : chr "23-50 years"
#>  $ age_median    : chr "39.0 years"
#>  $ weight_range  : chr "64.4-117 kg"
#>  $ weight_median : chr "76.2 kg"
#>  $ height_median : chr "170 cm (range 155-193)"
#>  $ bmi_median    : chr "27.0 kg/m^2 (range 22.6-31.5)"
#>  $ sex_female_pct: num 33.3
#>  $ disease_state : chr "healthy adult volunteers"
#>  $ dose_range    : chr "single oral doses of 10, 50 and 200 mg (4 x 50 mg) dispersible tablet and 200 mg marketed tablet, each after an"| __truncated__
#>  $ regions       : chr "United States (San Antonio, Texas)"
#>  $ notes         : chr "Phase 1 relative-bioavailability and food-effect study CL-011 (NCT04309656). The model uses the fed panel only "| __truncated__

Source trace

Every value comes from Zou 2022 Table 2 and the NONMEM control stream printed in Section C of the Electronic Supplementary Material (ESM). The control stream’s $THETA, $OMEGA and $SIGMA hold the final estimates: each one rounds to the corresponding Table 2 entry.

Quantity Value Source
lka (KA, DTF) log(0.396) 1/h Table 2 ‘Absorption rate of DTF’; ESM THETA(3)
e_form_mf_ka 1.65 Table 2 ‘Proportional effect (theta3) of MF on KA’; ESM THETA(5)
lmtt log(1.13) h Table 2 ‘Mean transit time’; ESM THETA(4)
lcl log(2.81) L/h Table 2 ‘Apparent clearance’; ESM THETA(1)
lvc log(68.0) L Table 2 ‘Apparent volume of distribution’; ESM THETA(2)
e_wt_cl, e_wt_vc 0.75, 1 (fixed) Table 2 weight-scaling rows; ESM $PK TVLCL / TVLV2
Weight reference 55 kg Table 2 footnote a; ESM LOG(WT/55)
lfdepot log(1) (fixed) Table 2 ‘Bioavailability (F)’
e_dose_fdepot 0.0822 Table 2 ‘Dose effect (theta4) on F’; ESM THETA(6)
etalcl, etalvc block 0.0622, 0.0111, 0.00751 ESM $OMEGA BLOCK(2); Table 2 CVs 24.9% and 8.67%
etaiov_fdepot_* 0.00561 ESM $OMEGA BLOCK(3) ‘IOV.F1’; Table 2 7.49%
etaiov_ka_* 0.284 ESM ‘IOV.KA’; Table 2 53.3%
etaiov_mtt_* 0.77 ESM ‘IOV.MATT’; Table 2 87.8%
propSd 0.0889 Table 2 ‘Proportional error’ 8.89%; ESM $SIGMA 0.0079
addSd 0.401 ng/mL Table 2 ‘Additive error’; ESM $SIGMA 0.161
e_tad_early_ruv 3.46 Table 2 ‘Time-varying error term (theta5)’; ESM THETA(7)
ktr = 1/MTT, dose into transit compartment – ESM $MODEL and $PK (K13 = KTR, K32 = KA)
F = (DOSE/200)^theta4 – Table 2; ESM TVLF1 = LOG(DOSE/200) * THETA(6)
ERRT = 3.46 when TAD < 10 h – Table 2 footnote d; ESM $ERROR

Table 2 footnote b states that the IIV, IOV and proportional-error entries are sqrt(variance) x 100, and the control-stream variances reproduce them: sqrt(0.0622) = 0.249, sqrt(0.00751) = 0.0867, sqrt(0.00561) = 0.0749, sqrt(0.284) = 0.533, sqrt(0.77) = 0.878, sqrt(0.0079) = 0.0889 and sqrt(0.161) = 0.401.

Typical-value checks against the paper

The paper states three derived typical values that follow from the parameters alone, so they are checked exactly.

p <- as.list(ui$theta)
mat_dtf <- 1 / exp(p$lka) + exp(p$lmtt)
mat_mf <- 1 / (exp(p$lka) * p$e_form_mf_ka) + exp(p$lmtt)
frel <- (c(10, 50) / 200)^p$e_dose_fdepot

typical <- data.frame(
  Quantity = c(
    "MAT, dispersible tablet (h)",
    "MAT, marketed tablet (h)",
    "MAT ratio DTF / MF (%)",
    "Relative F, 10 mg vs 200 mg (%)",
    "Relative F, 50 mg vs 200 mg (%)"
  ),
  Paper = c(3.7, 2.7, 137, 78, 89),
  Model = c(mat_dtf, mat_mf, 100 * mat_dtf / mat_mf, 100 * frel)
)
typical$Model <- signif(typical$Model, 3)
knitr::kable(
  typical,
  caption = "MAT from the Discussion and Results Section 3.2; relative bioavailability from Results Section 3.2."
)
MAT from the Discussion and Results Section 3.2; relative bioavailability from Results Section 3.2.
Quantity Paper Model
MAT, dispersible tablet (h) 3.7 3.66
MAT, marketed tablet (h) 2.7 2.66
MAT ratio DTF / MF (%) 137.0 137.00
Relative F, 10 mg vs 200 mg (%) 78.0 78.20
Relative F, 50 mg vs 200 mg (%) 89.0 89.20 
# Each paper value is printed to 2-3 significant figures; the model must
# reproduce it to that rounding.
stopifnot(
  abs(mat_dtf - 3.7) < 0.05,
  abs(mat_mf - 2.7) < 0.05,
  abs(100 * mat_dtf / mat_mf - 137) < 1,
  abs(100 * frel - c(78, 89)) < 0.5
)

Virtual cohort

A crossover cohort mirroring CL-011: every virtual participant receives all four treatments, one per period, with a 168-h interval between doses and the CL-011 sampling schedule (0-96 h after each dose). Treatments are assigned to periods in a fixed order (MF 200 mg, DTF 200 mg, DTF 50 mg, DTF 10 mg); with the same inter-occasion variance on every occasion and a half-life near 13 h, period order and carry-over do not affect the comparison. Body weight is log-normal around the fed-panel median of 76.2 kg, truncated to the observed 64.4-117 kg range.

n_sub <- 200
set.seed(20221163)
wt <- numeric(0)
while (length(wt) < n_sub) {
  draw <- rlnorm(2 * n_sub, log(76.2), 0.15)
  wt <- c(wt, draw[draw >= 64.4 & draw <= 117])
}
wt <- wt[seq_len(n_sub)]

treatments <- data.frame(
  OCC = 1:4,
  treatment = c("A: 200 mg MF", "B: 200 mg DTF", "C: 50 mg DTF", "D: 10 mg DTF"),
  DOSE_PRETOMANID_MG = c(200, 200, 50, 10),
  FORM_PRETOMANID_DT = c(0, 1, 1, 1)
)
period_h <- 168
sample_times <- c(0, 0.5, 1, 2, 3, 4, 6, 8, 12, 16, 24, 36, 48, 72, 96)

dose_rows <- tidyr::expand_grid(id = seq_len(n_sub), OCC = 1:4) |>
  dplyr::mutate(
    time = (OCC - 1) * period_h,
    evid = 1L,
    amt = treatments$DOSE_PRETOMANID_MG[OCC],
    cmt = "transit1"
  )
obs_rows <- tidyr::expand_grid(id = seq_len(n_sub), OCC = 1:4, tad = sample_times) |>
  dplyr::mutate(
    time = (OCC - 1) * period_h + tad,
    evid = 0L,
    amt = 0,
    cmt = "central"
  ) |>
  dplyr::select(-tad)

events <- dplyr::bind_rows(dose_rows, obs_rows) |>
  dplyr::left_join(treatments |> dplyr::select(-treatment), by = "OCC") |>
  dplyr::mutate(WT = wt[id]) |>
  dplyr::arrange(id, time, dplyr::desc(evid)) |>
  dplyr::select(id, time, evid, amt, cmt, dplyr::everything())

stopifnot(
  !anyNA(events),
  dplyr::n_distinct(events$id) == n_sub,
  sum(events$evid == 1) == 4 * n_sub
)

Simulation 

rxode2::rxSetSeed(20221163)
sim <- rxode2::rxSolve(ui, events, returnType = "data.frame")
# Recover the occasion and time after dose from the clock time: each dose
# opens a 168-h period, and every sample falls 0-96 h into its period.
sim <- sim |>
  dplyr::select(-dplyr::any_of(c("OCC", "DOSE_PRETOMANID_MG", "FORM_PRETOMANID_DT"))) |>
  dplyr::mutate(
    OCC = as.integer(floor(time / period_h)) + 1L,
    tad_h = time - (OCC - 1) * period_h
  ) |>
  dplyr::left_join(treatments, by = "OCC")
stopifnot(
  nrow(sim) == n_sub * 4 * length(sample_times),
  !anyNA(sim$Cc),
  all(sim$tad_h %in% sample_times)
)

Visual predictive check (replicates Figure 2)

Figure 2 of Zou 2022 shows the 10th, 50th and 90th percentiles of the fed-panel concentrations by treatment on a log scale. The simulated percentiles below include residual error (sim) and are the model’s counterpart of those bands.

vpc <- sim |>
  dplyr::filter(tad_h > 0) |>
  dplyr::group_by(treatment, tad_h) |>
  dplyr::summarise(
    p10 = quantile(sim, 0.10),
    p50 = quantile(sim, 0.50),
    p90 = quantile(sim, 0.90),
    .groups = "drop"
  )
ggplot(vpc, aes(tad_h)) +
  geom_ribbon(aes(ymin = p10, ymax = p90), fill = "steelblue", alpha = 0.3) +
  geom_line(aes(y = p50), colour = "steelblue4") +
  facet_wrap(~treatment) +
  scale_y_log10() +
  labs(
    x = "Time after dose (h)",
    y = "Pretomanid concentration (ng/mL)",
    caption = "Simulated median and 10th-90th percentiles; replicates Figure 2 of Zou 2022."
  ) +
  theme_bw()

Absorption-phase residual error

The residual SD is 3.46-fold larger for samples taken less than 10 h after the dose. At the 200 mg concentrations the proportional term dominates the additive one, so the ratio of the relative-residual SDs before and after 10 h recovers the multiplier.

ruv <- sim |>
  dplyr::filter(DOSE_PRETOMANID_MG == 200, tad_h > 0, Cc > 50) |>
  dplyr::mutate(rel_resid = (sim - Cc) / Cc, early = tad_h < 10) |>
  dplyr::group_by(early) |>
  dplyr::summarise(sd_rel = sd(rel_resid), n = dplyr::n(), .groups = "drop")
knitr::kable(ruv, digits = 4, caption = "Relative-residual SD before (TRUE) and after (FALSE) 10 h, 200 mg arms.")
Relative-residual SD before (TRUE) and after (FALSE) 10 h, 200 mg arms.
early sd_rel n
FALSE 0.0899 2621
TRUE 0.3135 2687 
ratio <- ruv$sd_rel[ruv$early] / ruv$sd_rel[!ruv$early]
cat("Early / late residual SD ratio:", signif(ratio, 3), "(model multiplier 3.46)\n")
#> Early / late residual SD ratio: 3.49 (model multiplier 3.46)
# Thousands of residuals per group put the SD ratio within a few percent of
# 3.46 (the additive term lowers it slightly); omitting the multiplier gives
# 1, and applying it to every sample also gives 1.
stopifnot(ratio > 3.0, ratio < 3.9)

Non-compartmental analysis

NCA uses the individual predictions (Cc, no residual error) on the CL-011 sampling schedule, per treatment. Reference values are the fed-panel geometric least-squares means in the ESM: Table D1 for treatments A and B, and Table D5 (primary analysis, ‘Test’ = fed) for treatments C and D. The simulated values are cohort medians, which for log-normally distributed exposure estimate the same quantity as a geometric mean.

conc <- sim |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::mutate(conc = pmax(Cc, 0)) |>
  dplyr::select(id, treatment, time = tad_h, conc)
dose <- events |>
  dplyr::filter(evid == 1) |>
  dplyr::left_join(treatments |> dplyr::select(OCC, treatment), by = "OCC") |>
  dplyr::transmute(id, treatment, time = 0, amt)

conc_obj <- PKNCA::PKNCAconc(conc, conc ~ time | treatment + id)
dose_obj <- PKNCA::PKNCAdose(dose, amt ~ time | treatment + id)
intervals <- data.frame(
  start = c(0, 0),
  end = c(96, Inf),
  cmax = c(TRUE, FALSE),
  tmax = c(TRUE, FALSE),
  auclast = c(TRUE, FALSE),
  aucinf.obs = c(FALSE, TRUE),
  half.life = c(FALSE, TRUE)
)
nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))
nca_long <- as.data.frame(nca_res$result) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "auclast", "aucinf.obs", "half.life"))
reference <- data.frame(
  treatment = c("A: 200 mg MF", "B: 200 mg DTF", "C: 50 mg DTF", "D: 10 mg DTF"),
  cmax = c(2120, 1820, 384, 75.3),
  auclast = c(53900, 54200, 12000, 1970),
  aucinf.obs = c(56100, 56700, 12400, 2030)
)
cmp <- nlmixr2lib::ncaComparisonTable(
  simulated = nca_long,
  reference = reference,
  by = "treatment",
  units = c(cmax = "ng/mL", auclast = "ng*h/mL", aucinf.obs = "ng*h/mL"),
  tolerance_pct = 20
)
knitr::kable(
  cmp,
  caption = paste(
    "Zou 2022 ESM Tables D1 and D5 (fed-panel geometric LS means) vs simulated",
    "cohort medians. AUC0-t is AUC over 0-96 h. * differs by more than 20%."
  )
)
Zou 2022 ESM Tables D1 and D5 (fed-panel geometric LS means) vs simulated cohort medians. AUC0-t is AUC over 0-96 h. * differs by more than 20%.
NCA parameter treatment Reference Simulated % diff
Cmax (ng/mL) A: 200 mg MF 2120 1620 -23.6%*
Cmax (ng/mL) B: 200 mg DTF 1820 1520 -16.4%
Cmax (ng/mL) C: 50 mg DTF 384 345 -10.2%
Cmax (ng/mL) D: 10 mg DTF 75.3 60.4 -19.8%
AUC0-∞ (obs) (ng*h/mL) A: 200 mg MF 56100 53400 -4.7%
AUC0-∞ (obs) (ng*h/mL) B: 200 mg DTF 56700 54800 -3.4%
AUC0-∞ (obs) (ng*h/mL) C: 50 mg DTF 12400 12100 -2.4%
AUC0-∞ (obs) (ng*h/mL) D: 10 mg DTF 2030 2150 +5.8%
AUClast (ng*h/mL) A: 200 mg MF 53900 51800 -3.9%
AUClast (ng*h/mL) B: 200 mg DTF 54200 52800 -2.6%
AUClast (ng*h/mL) C: 50 mg DTF 12000 11700 -2.6%
AUClast (ng*h/mL) D: 10 mg DTF 1970 2050 +4.1% 
pct <- nca_long |>
  dplyr::group_by(treatment, PPTESTCD) |>
  dplyr::summarise(sim = median(PPORRES), .groups = "drop") |>
  dplyr::inner_join(
    tidyr::pivot_longer(reference, -treatment, names_to = "PPTESTCD", values_to = "pub"),
    by = c("treatment", "PPTESTCD")
  ) |>
  dplyr::mutate(pct_diff = 100 * (sim / pub - 1))
auc_pct <- pct$pct_diff[pct$PPTESTCD != "cmax"]
cmax_pct <- pct$pct_diff[pct$PPTESTCD == "cmax"]
# AUC is set by F * dose / CL, so a mis-transcribed clearance, weight
# reference, dose exponent or unit moves every arm by tens of percent.
# Cmax is underpredicted in every arm, by about 10-25%, which is the peak
# underprediction the paper reports for its own model (Discussion), so it
# is gated on its centre with an envelope that admits cohort noise.
stopifnot(
  abs(median(auc_pct)) < 8,
  max(abs(auc_pct)) < 15,
  abs(median(cmax_pct)) < 25,
  max(abs(cmax_pct)) < 35
)

Simulated and observed AUCs agree across the 20-fold dose range. The largest AUC difference is at 10 mg, where the power model’s relative bioavailability of 78% sits slightly above the observed dose-normalised exposure (the observed 10 mg/200 mg AUC0-inf ratio is about 72%).

Cmax is underpredicted in every arm, most for the marketed 200 mg tablet and the 10 mg dispersible tablet (about 20-25% below the observed geometric means) and least at 50 mg. This reproduces a limitation of the published model rather than an encoding error: Zou 2022 reports that its own peak concentrations were underpredicted at 10 mg and for the 200 mg marketed tablet (Discussion and ESM Section B VPCs). The model was built to predict AUC, the exposure metric used for pediatric dose selection.

Assumptions and deviations

  • Fed conditions only. The model was fitted to the fed panel; it carries no food effect and should not be used to simulate fasted dosing. The fasted panel was analysed by NCA only, and the dose dependence of exposure reverses under fasting (NCA slope 0.824 versus 1.086 fed).
  • Weight reference. Clearance and volume are referenced to 55 kg, inherited from Salinger 2019, even though the fed panel weighed 64.4-117 kg. The typical values CL = 2.81 L/h and V = 68.0 L therefore describe a lighter participant than any enrolled.
  • Formulation indicator polarity. The source column DTF (1 = dispersible tablet) is carried as FORM_PRETOMANID_DT; the model forms the marketed-formulation indicator 1 - FORM_PRETOMANID_DT internally, as the control stream does.
  • Occasions. One occasion per crossover period (OCC 1-4). The ESM $OMEGA BLOCK(3) SAME blocks are encoded as independent per-occasion etas with occasions 2-4 fixed to the occasion-1 variances; the BLOCK(3) off-diagonals are zero in the source, so the etas are uncorrelated. Records with OCC outside 1-4 receive no inter-occasion variability.
  • Mu-referencing warning. rxode2 has no occasion level for random effects, so the per-occasion etas are combined through occasion indicators. rxode2 warns that these etas are not mu-referenced. The warning concerns estimation speed only; simulation is unaffected.
  • Time-varying residual error. The control stream switches on a TAD data column; the model uses rxode2’s tad(), the time since the most recent dose, which is the same quantity for single doses separated by washouts. NONMEM’s Y = IPRED + ERRT * (EPS(1) * IPRED + EPS(2)) with a diagonal $SIGMA is a combined error whose variances add (combined2()).
  • Unpublished covariance. The CL-V covariance (0.0111, correlation 0.51) is printed only in the ESM control stream, not in Table 2; the control stream’s other estimates all match Table 2, so it is taken as the final estimate.
  • NCA reference statistic. The ESM reports geometric least-squares means; the comparison uses simulated cohort medians.
  • Peak concentrations. Simulated Cmax is about 10-25% below the observed geometric means, as the source reports for its own model; AUC is reproduced within about 6%.
  • Virtual cohort. Body weight is log-normal (median 76.2 kg, log-SD 0.15) truncated to the fed-panel range; sex, age and height do not enter the model.