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Tolvaptan (Lanke 2019)

Source: vignettes/articles/Lanke_2019_tolvaptan.Rmd
Lanke_2019_tolvaptan.Rmd

Model and source

  • Citation: Lanke S, Shoaf SE. (2019). Population Pharmacokinetic Analyses and Model Validation of Tolvaptan in Subjects With Autosomal Dominant Polycystic Kidney Disease. J Clin Pharmacol 59(5):763-770. doi:10.1002/jcph.1370
  • Description: One-compartment population PK model for oral tolvaptan in adults with autosomal dominant polycystic kidney disease (Lanke 2019): zero-order absorption directly into the central compartment (duration D1), first-order elimination, and a dose-dependent relative bioavailability F1 = Dosemax / (Dose50 + Dose) that falls by about 37 percent from 15 mg to 120 mg per administration. eGFR (power), body weight (power) and CYP3A inhibitor coadministration (fractional change) act on CL/F; body weight (power) acts on Vd/F. Correlated IIV on CL/F, Vd/F and D1, independent IIV on Dose50 and Dosemax, and combined additive plus proportional residual error.
  • Article: https://doi.org/10.1002/jcph.1370 (open access, PMC6590359)

Lanke 2019 developed a population PK model of oral tolvaptan, the vasopressin V2-receptor antagonist approved to slow kidney function decline in autosomal dominant polycystic kidney disease (ADPKD), and validated it externally against a separate phase 3b trial. The structural model is one compartment with zero-order absorption (duration D1) directly into the central compartment and first-order elimination. The distinguishing feature is a relative bioavailability that falls with the amount administered,

F1=DosemaxDose50+DoseF1 = \frac{Dose_{max}}{Dose_{50} + Dose}

(Lanke 2019 Equation 4), which the authors attribute to the low solubility of tolvaptan. eGFR, body weight and CYP3A inhibitor coadministration act on CL/F, and body weight acts on Vd/F.

Population

The model was developed from 7335 plasma concentrations in 1091 adults with ADPKD pooled from five phase 2 trials and the phase 3b open-label extension NCT01214421 (Lanke 2019 Methods). Median age was 42 years (range 18-62), median weight 79.5 kg (42.3-164.5), median BMI 25.9 kg/m^2 and median CKD-EPI eGFR 69.4 mL/min/1.73 m^2 (13.62-137.37) (Table 1); 565 of the 1091 subjects were female and 18 were enrolled in Japan (Table 2). Of the 1073 subjects enrolled outside Japan, 1024 were White, 14 Black, 9 Asian, 1 Hispanic and 25 other. Doses ranged from single 15-120 mg doses to split-dose regimens of 15/15 to 90/30 mg, the morning dose at about 8 AM and the second dose 8-9 h later. 131 subjects had at least one record of CYP3A inhibitor coadministration (3.46 percent of observations). External validation used 678 subjects and 3012 observations from the randomized withdrawal trial NCT02160145 at 60/30 or 90/30 mg.

The same information is available programmatically via readModelDb("Lanke_2019_tolvaptan")$population.

Source trace

Element Value Source location
Structure: one compartment, zero-order input into central, first-order elimination – Results, Base Model; Equation 5
lcl (CL/F) 12.6 L/h Table 3, Final Model
lvc (Vd/F) 110 L Table 3, Final Model
ld1 (D1) 0.58 h Table 3, Final Model
lfdepot_dose50 (Dose50) 166 mg Table 3, Final Model; Abstract
lfdepot_dosemax (Dosemax) 182 mg Table 3, Final Model; Abstract
F1 = Dosemax / (Dose50 + Dose) – Equation 4
e_crcl_cl (eGFR power on CL/F) 0.52 Table 3; normalised to median 69.4 (Table 1, Equation 2)
e_wt_cl (WT power on CL/F) -0.53 Table 3; normalised to median 79.5 kg (Table 1, Equation 1)
e_cyp3a4_inh_cl (Inh on CL/F) -0.23 Table 3; linear form of Equation 3
e_wt_vc (WT power on Vd/F) 0.23 Table 3; Equation 1
etalcl, etalvc, etald1 block 0.40; -0.25, 0.49; 1.13, -1.51, 5.67 Table 3, Interindividual variability
etalfdepot_dose50 0.086 Table 3
etalfdepot_dosemax 0.019 Table 3
addSd sqrt(5.3) = 2.30 ng/mL Table 3, Residual variability (variance)
propSd sqrt(0.18) = 0.424 Table 3, Residual variability (variance)

The IIV entries are variances: sqrt(0.49) = 70 percent and sqrt(5.67) = 238 percent reproduce the Abstract’s IIV for Vd/F and D1, and sqrt(0.40) = 63 percent is the Abstract’s 64 percent for CL/F to rounding.

mod <- readModelDb("Lanke_2019_tolvaptan")
ui <- rxode2::rxode2(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_typ <- rxode2::zeroRe(ui)

Dose-dependent bioavailability

The Abstract states that F1 decreased by 36 percent as the dose rose from 15 mg to 120 mg, and the Results state that F1 equals 1 at the lowest dose of 15 mg. Both follow from the Table 3 values. The function decreases monotonically and has no minimum, so the Table 3 footnote’s description of Dosemax (“the highest dose at which F1 is lowest”) does not describe it; likewise F1 reaches 0.5 at 2 x 182 - 166 = 198 mg rather than at Dose50.

f1 <- function(dose) 182 / (166 + dose)
f1_tab <- data.frame(dose_mg = c(15, 30, 45, 60, 90, 120)) |>
  dplyr::mutate(F1 = f1(dose_mg))
knitr::kable(f1_tab, digits = 3, caption = "Typical relative bioavailability by administered amount.")
Typical relative bioavailability by administered amount.
dose_mg F1
15 1.006
30 0.929
45 0.863
60 0.805
90 0.711
120 0.636 

stopifnot(
  abs(f1(15) - 1) < 0.01,
  # Abstract: 'F1 decreased by 36%' from 15 to 120 mg
  abs((1 - f1(120) / f1(15)) * 100 - 36) < 1
)
data.frame(dose = seq(0, 240, by = 1)) |>
  dplyr::mutate(F1 = f1(dose)) |>
  ggplot(aes(dose, F1)) +
  geom_line() +
  geom_point(data = f1_tab, aes(dose_mg, F1)) +
  labs(x = "Tolvaptan amount per administration (mg)", y = "Relative bioavailability F1") +
  theme_bw()
Typical relative bioavailability F1 versus the administered amount (Lanke 2019 Equation 4).

Typical relative bioavailability F1 versus the administered amount (Lanke 2019 Equation 4).

The same ratio must come out of the packaged model. Typical-value single doses of 15, 30, 60 and 120 mg are simulated in a subject at the covariate medians, and the PKNCA dose-normalised AUC is compared with the closed form F1(dose) x dose / CL.

sd_doses <- c(15, 30, 60, 120)
sd_times <- c(seq(0, 2, by = 0.02), seq(2.1, 24, by = 0.1), seq(24.5, 200, by = 0.5))
sd_events <- dplyr::bind_rows(
  data.frame(id = seq_along(sd_doses), time = 0, evid = 1, amt = sd_doses, rate = -2, cmt = "central"),
  tidyr::expand_grid(id = seq_along(sd_doses), time = sd_times) |>
    dplyr::mutate(evid = 0, amt = 0, rate = 0, cmt = "central")
) |>
  dplyr::arrange(id, time, dplyr::desc(evid)) |>
  dplyr::mutate(
    dose_group = paste(sd_doses[id], "mg"),
    DOSE_TOLVAPTAN_MG = sd_doses[id],
    WT = 79.5, CRCL = 69.4, CONMED_CYP3A4_INH = 0
  )

sd_sim <- as.data.frame(rxode2::rxSolve(mod_typ, sd_events, returnType = "data.frame",
                                        keep = "dose_group"))
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etald1', 'etalfdepot_dose50', 'etalfdepot_dosemax'
#> Warning: multi-subject simulation without without 'omega'

sd_conc <- PKNCA::PKNCAconc(
  dplyr::filter(sd_sim, !is.na(Cc)) |> dplyr::select(id, time, Cc, dose_group),
  Cc ~ time | dose_group + id, concu = "ng/mL", timeu = "h"
)
sd_dose <- PKNCA::PKNCAdose(
  dplyr::filter(sd_events, evid == 1) |> dplyr::select(id, time, amt, dose_group),
  amt ~ time | dose_group + id, doseu = "mg"
)
sd_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(
  sd_conc, sd_dose,
  intervals = data.frame(start = 0, end = Inf, cmax = TRUE, tmax = TRUE,
                         aucinf.obs = TRUE, half.life = TRUE)
))
sd_auc <- as.data.frame(sd_res) |>
  dplyr::filter(PPTESTCD == "aucinf.obs") |>
  dplyr::mutate(
    dose = sd_doses[id],
    closed_form = 1000 * f1(dose) * dose / 12.6,
    pct_diff = 100 * (PPORRES - closed_form) / closed_form,
    auc_per_mg = PPORRES / dose
  )
knitr::kable(
  sd_auc |>
    dplyr::select(dose_group, PPORRES, closed_form, pct_diff, auc_per_mg) |>
    dplyr::rename(
      "Dose" = dose_group, "AUC0-inf simulated (ng*h/mL)" = PPORRES,
      "AUC0-inf closed form (ng*h/mL)" = closed_form,
      "Difference (percent)" = pct_diff, "AUC per mg" = auc_per_mg
    ),
  digits = 2, caption = "Typical-value single-dose AUC0-inf from PKNCA against F1 x Dose / CL."
)
Typical-value single-dose AUC0-inf from PKNCA against F1 x Dose / CL.
Dose AUC0-inf simulated (ng*h/mL) AUC0-inf closed form (ng*h/mL) Difference (percent) AUC per mg
120 mg 6060.60 6060.61 0 50.51
15 mg 1197.05 1197.05 0 79.80
30 mg 2210.88 2210.88 0 73.70
60 mg 3834.81 3834.81 0 63.91 

# Same parameters on both sides, so only quadrature error separates them.
stopifnot(all(abs(sd_auc$pct_diff) < 1))
auc_ratio <- with(sd_auc, auc_per_mg[dose == 120] / auc_per_mg[dose == 15])
stopifnot(abs((1 - auc_ratio) * 100 - 36) < 1.5)

The dose-normalised AUC at 120 mg is 63.3 percent of that at 15 mg, the 36 percent decrease the Abstract reports.

Covariate effects (replicates Figure 1)

The Results give the covariate effects at the 5th and 95th percentiles of eGFR (32 and 110 mL/min/1.73 m^2) and body weight (56 and 113 kg) relative to the median subject; the Discussion gives the effect of inhibitor coadministration. These are the bars of Lanke 2019 Figure 1.

The typical CL/F and Vd/F are read from a typical-value solve of the packaged model for each covariate scenario, so the check exercises the model() equations rather than a re-typed formula.

scen <- tibble::tribble(
  ~scenario, ~WT, ~CRCL, ~CONMED_CYP3A4_INH,
  "median", 79.5, 69.4, 0,
  "eGFR 32", 79.5, 32, 0,
  "eGFR 110", 79.5, 110, 0,
  "WT 56 kg", 56, 69.4, 0,
  "WT 113 kg", 113, 69.4, 0,
  "CYP3A inhibitor", 79.5, 69.4, 1
) |>
  dplyr::mutate(id = dplyr::row_number())
scen_events <- dplyr::bind_rows(
  dplyr::mutate(scen, time = 0, evid = 1, amt = 15, rate = -2),
  dplyr::mutate(scen, time = 1, evid = 0, amt = 0, rate = 0)
) |>
  dplyr::mutate(cmt = "central", DOSE_TOLVAPTAN_MG = 15) |>
  dplyr::select(id, time, evid, amt, rate, cmt, DOSE_TOLVAPTAN_MG, WT, CRCL, CONMED_CYP3A4_INH, scenario) |>
  dplyr::arrange(id, time)
scen_par <- as.data.frame(rxode2::rxSolve(mod_typ, scen_events, returnType = "data.frame",
                                          keep = "scenario")) |>
  dplyr::distinct(scenario, cl, vc)
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etald1', 'etalfdepot_dose50', 'etalfdepot_dosemax'
#> Warning: multi-subject simulation without without 'omega'
ref <- dplyr::filter(scen_par, scenario == "median")
stopifnot(abs(ref$cl - 12.6) < 1e-6, abs(ref$vc - 110) < 1e-6)

cov_tab <- tibble::tribble(
  ~parameter, ~covariate, ~published_pct,
  "CL/F", "eGFR 32", -33,
  "CL/F", "eGFR 110", 27,
  "CL/F", "WT 56 kg", 20,
  "CL/F", "WT 113 kg", -17,
  "CL/F", "CYP3A inhibitor", -23,
  "Vd/F", "WT 56 kg", -8,
  "Vd/F", "WT 113 kg", 8
) |>
  dplyr::left_join(scen_par, by = c("covariate" = "scenario")) |>
  dplyr::mutate(
    value = ifelse(parameter == "CL/F", cl / ref$cl, vc / ref$vc),
    model_pct = 100 * (value - 1)
  )

knitr::kable(
  cov_tab |>
    dplyr::select(parameter, covariate, model_pct, published_pct) |>
    dplyr::rename(
      "Parameter" = parameter, "Covariate value" = covariate,
      "Model change (percent)" = model_pct, "Published change (percent)" = published_pct
    ),
  digits = 1,
  caption = "Covariate effects against the Lanke 2019 Results and Discussion."
)
Covariate effects against the Lanke 2019 Results and Discussion.
Parameter Covariate value Model change (percent) Published change (percent)
CL/F eGFR 32 -33.1 -33
CL/F eGFR 110 27.1 27
CL/F WT 56 kg 20.4 20
CL/F WT 113 kg -17.0 -17
CL/F CYP3A inhibitor -23.0 -23
Vd/F WT 56 kg -7.7 -8
Vd/F WT 113 kg 8.4 8 
stopifnot(all(abs(cov_tab$model_pct - cov_tab$published_pct) < 1))
cov_tab |>
  dplyr::mutate(label = paste(parameter, covariate)) |>
  ggplot(aes(x = model_pct, y = reorder(label, model_pct))) +
  geom_col(width = 0.6, orientation = "y") +
  geom_vline(xintercept = c(-20, 25), linetype = "dashed") +
  labs(x = "Change from the median subject (percent)", y = NULL) +
  theme_bw()
Replicates Figure 1 of Lanke 2019: change in CL/F and Vd/F at the 5th and 95th covariate percentiles. Dashed lines mark 80 and 125 percent.

Replicates Figure 1 of Lanke 2019: change in CL/F and Vd/F at the 5th and 95th covariate percentiles. Dashed lines mark 80 and 125 percent.

The Discussion also gives CL/F of 14.7 L/h at 56 kg and 10.2 L/h at 113 kg against “the typical value of 12.3 L/h”. The model gives 15.2 and 10.5 L/h from the Table 3 value of 12.6 L/h; the ratios to the typical value agree to within 1 percent, and only the typical value differs (see Assumptions and deviations).

Virtual cohort

97 percent of the development observations, and 93.7 percent of the validation observations, came from the 60/30 and 90/30 mg split-dose regimens, so the stochastic simulation uses those two arms with 200 subjects each. Body weight is log-normal around the median of 79.5 kg with the dispersion implied by the published 5th and 95th percentiles (56 and 113 kg); eGFR is normal around 69.4 with the dispersion implied by its percentiles (32 and 110), truncated to the observed range. CYP3A inhibitor coadministration is drawn per subject with the 3.46 percent observation-level frequency of Table 2 and held constant within a subject.

rxode2::rxSetSeed(20190501)
n_per_arm <- 200
regimens <- tibble::tribble(
  ~regimen, ~am_mg, ~pm_mg,
  "60/30 mg", 60, 30,
  "90/30 mg", 90, 30
)
cohort <- tidyr::expand_grid(regimen = regimens$regimen, subj = seq_len(n_per_arm)) |>
  dplyr::mutate(
    id = dplyr::row_number(),
    WT = pmin(pmax(79.5 * exp(rnorm(dplyr::n(), 0, log(113 / 56) / (2 * 1.645))), 42.3), 164.5),
    CRCL = pmin(pmax(rnorm(dplyr::n(), 69.4, (110 - 32) / (2 * 1.645)), 13.62), 137.37),
    CONMED_CYP3A4_INH = rbinom(dplyr::n(), 1, 0.0346)
  ) |>
  dplyr::left_join(regimens, by = "regimen")

knitr::kable(
  cohort |>
    dplyr::group_by(regimen) |>
    dplyr::summarise(
      n = dplyr::n(), WT_median = median(WT), CRCL_median = median(CRCL),
      CRCL_p05 = quantile(CRCL, 0.05), CRCL_p95 = quantile(CRCL, 0.95),
      inh_pct = 100 * mean(CONMED_CYP3A4_INH)
    ),
  digits = 1, caption = "Virtual cohort summary."
)
Virtual cohort summary.
regimen n WT_median CRCL_median CRCL_p05 CRCL_p95 inh_pct
60/30 mg 200 81.5 65.3 27.9 105.3 3
90/30 mg 200 80.1 68.7 32.2 102.6 4

Simulation

Fourteen days of split dosing: the morning dose at 0 h of each day and the afternoon dose 8 h later, both as modelled-duration zero-order input (rate = -2). DOSE_TOLVAPTAN_MG carries the amount of each administration, so the morning and afternoon doses get different bioavailabilities. Day 14 (312-336 h) is sampled densely.

n_days <- 14
ss_start <- (n_days - 1) * 24
ss_end <- n_days * 24
dose_rows <- cohort |>
  tidyr::expand_grid(day = seq_len(n_days) - 1, slot = c("am", "pm")) |>
  dplyr::mutate(
    time = 24 * day + ifelse(slot == "am", 0, 8),
    amt = ifelse(slot == "am", am_mg, pm_mg),
    evid = 1, rate = -2, cmt = "central"
  )
obs_rows <- cohort |>
  tidyr::expand_grid(time = c(seq(ss_start, ss_start + 2, by = 0.05),
                              seq(ss_start + 2.1, ss_start + 10, by = 0.05),
                              seq(ss_start + 10.1, ss_end, by = 0.1))) |>
  dplyr::mutate(evid = 0, amt = 0, rate = 0, cmt = "central")
events <- dplyr::bind_rows(dose_rows, obs_rows) |>
  dplyr::arrange(id, time, dplyr::desc(evid)) |>
  dplyr::group_by(id) |>
  dplyr::mutate(DOSE_TOLVAPTAN_MG = ifelse(evid == 1, amt, NA_real_)) |>
  tidyr::fill(DOSE_TOLVAPTAN_MG, .direction = "down") |>
  dplyr::ungroup() |>
  dplyr::select(id, time, evid, amt, rate, cmt, DOSE_TOLVAPTAN_MG, WT, CRCL,
                CONMED_CYP3A4_INH, regimen, am_mg, pm_mg)

sim <- as.data.frame(rxode2::rxSolve(
  ui, events, returnType = "data.frame",
  keep = c("regimen", "am_mg", "pm_mg")
))
sim |>
  dplyr::filter(time >= ss_start) |>
  dplyr::mutate(tad_day = time - ss_start) |>
  dplyr::group_by(regimen, tad_day) |>
  dplyr::summarise(
    p05 = quantile(Cc, 0.05), p50 = median(Cc), p95 = quantile(Cc, 0.95),
    .groups = "drop"
  ) |>
  ggplot(aes(tad_day, p50)) +
  geom_ribbon(aes(ymin = p05, ymax = p95), alpha = 0.25) +
  geom_line() +
  facet_wrap(~regimen) +
  scale_y_log10() +
  labs(x = "Time after the day-14 morning dose (h)", y = "Tolvaptan (ng/mL)") +
  theme_bw()
Simulated day-14 tolvaptan concentrations by regimen: median and 5th-95th percentile band of the individual predictions (compare the external-validation VPC of Lanke 2019 Figure 2).

Simulated day-14 tolvaptan concentrations by regimen: median and 5th-95th percentile band of the individual predictions (compare the external-validation VPC of Lanke 2019 Figure 2).

PKNCA validation

Lanke 2019 reports no NCA summary for its simulations, so the steady-state NCA is checked against the closed form it must satisfy instead. For any subject at steady state the AUC over one day equals the bioavailable daily amount divided by clearance, AUC0-24 = 1000 x (F1(AM) x AM + F1(PM) x PM) / CL, whatever the absorption duration. Both sides use the same drawn parameters, so the check fails on a mis-applied bioavailability (for example the daily total instead of the per-administration amount, or a bioavailability that is not evaluated on its own dose record).

nca_conc <- sim |>
  dplyr::filter(!is.na(Cc), time >= ss_start) |>
  dplyr::select(id, time, Cc, regimen)
nca_dose <- events |>
  dplyr::filter(evid == 1, time >= ss_start) |>
  dplyr::select(id, time, amt, regimen)

conc_obj <- PKNCA::PKNCAconc(nca_conc, Cc ~ time | regimen + id, concu = "ng/mL", timeu = "h")
dose_obj <- PKNCA::PKNCAdose(nca_dose, amt ~ time | regimen + id, doseu = "mg")
intervals <- data.frame(start = ss_start, end = ss_end, cmax = TRUE, tmax = TRUE,
                        cmin = TRUE, auclast = TRUE)
nca_res <- suppressMessages(PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)))
knitr::kable(summary(nca_res), caption = "Simulated day-14 NCA per regimen (one dosing day, 24 h).")
Simulated day-14 NCA per regimen (one dosing day, 24 h).
Interval Start Interval End regimen N AUClast (h*ng/mL) Cmax (ng/mL) Cmin (ng/mL) Tmax (h)
312 336 60/30 mg 200 6510 [92.8] 446 [79.0] 76.1 [691] 2.18 [0.0500, 23.7]
312 336 90/30 mg 200 8670 [82.5] 615 [65.4] 107 [472] 0.550 [0.0500, 23.1] 
indiv <- sim |>
  dplyr::filter(time == ss_start) |>
  dplyr::distinct(id, regimen, am_mg, pm_mg, cl, fdepot_dose50, fdepot_dosemax, d1)
chk <- as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD == "auclast") |>
  dplyr::select(id, regimen, auc_sim = PPORRES) |>
  dplyr::left_join(indiv, by = c("id", "regimen")) |>
  dplyr::mutate(
    auc_closed = 1000 * (am_mg * fdepot_dosemax / (fdepot_dose50 + am_mg) +
                           pm_mg * fdepot_dosemax / (fdepot_dose50 + pm_mg)) / cl,
    pct_diff = 100 * (auc_sim - auc_closed) / auc_closed
  )
knitr::kable(
  chk |>
    dplyr::group_by(regimen) |>
    dplyr::summarise(
      median_auc_sim = median(auc_sim), median_auc_closed = median(auc_closed),
      median_pct_diff = median(pct_diff), p90_abs_pct_diff = quantile(abs(pct_diff), 0.9)
    ) |>
    dplyr::rename(
      "Regimen" = regimen, "Median AUC0-24 simulated (ng*h/mL)" = median_auc_sim,
      "Median AUC0-24 closed form (ng*h/mL)" = median_auc_closed,
      "Median difference (percent)" = median_pct_diff,
      "90th percentile absolute difference (percent)" = p90_abs_pct_diff
    ),
  digits = 2, caption = "Steady-state AUC0-24 from PKNCA against the per-subject closed form."
)
Steady-state AUC0-24 from PKNCA against the per-subject closed form.
Regimen Median AUC0-24 simulated (ng*h/mL) Median AUC0-24 closed form (ng*h/mL) Median difference (percent) 90th percentile absolute difference (percent)
60/30 mg 6833.32 6833.95 -0.01 0.06
90/30 mg 9186.58 9186.93 0.00 0.16 

# Centre and robust envelope only: subjects with a very long D1 (IIV variance
# 5.67) are still approaching steady state on day 14.
stopifnot(
  abs(median(chk$pct_diff)) < 1,
  quantile(abs(chk$pct_diff), 0.9) < 5
)

A typical subject at the covariate medians is checked the same way with no random effects.

typ_events <- events |>
  dplyr::filter(id %in% c(1, n_per_arm + 1)) |>
  dplyr::mutate(WT = 79.5, CRCL = 69.4, CONMED_CYP3A4_INH = 0)
typ_sim <- as.data.frame(rxode2::rxSolve(mod_typ, typ_events, returnType = "data.frame",
                                         keep = c("regimen", "am_mg", "pm_mg")))
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etald1', 'etalfdepot_dose50', 'etalfdepot_dosemax'
#> Warning: multi-subject simulation without without 'omega'
typ_chk <- typ_sim |>
  dplyr::filter(time >= ss_start) |>
  dplyr::group_by(regimen, am_mg, pm_mg) |>
  dplyr::summarise(
    auc_sim = sum(diff(time) * (head(Cc, -1) + tail(Cc, -1)) / 2),
    .groups = "drop"
  ) |>
  dplyr::mutate(
    auc_closed = 1000 * (am_mg * f1(am_mg) + pm_mg * f1(pm_mg)) / 12.6,
    pct_diff = 100 * (auc_sim - auc_closed) / auc_closed
  )
knitr::kable(typ_chk, digits = 2, caption = "Typical-subject steady-state AUC0-24.")
Typical-subject steady-state AUC0-24.
regimen am_mg pm_mg auc_sim auc_closed pct_diff
60/30 mg 60 30 6045.37 6045.69 -0.01
90/30 mg 90 30 7288.61 7289.01 -0.01 
stopifnot(all(abs(typ_chk$pct_diff) < 0.5))

Comparison against published NCA

Lanke 2019 reports no Cmax, AUC or half-life values for tolvaptan, so there is no published NCA table to compare against. The quantitative checks above replace it: the 36 percent fall in F1 from the Abstract, the covariate effects from the Results and Discussion, and the closed-form steady-state exposure.

Assumptions and deviations

  • Dose50 = 166 mg, not 160 mg. The Results prose gives Dose50 as 160 mg; Table 3 and the Abstract give 166 mg. 166 mg is used because it makes F1 equal to 1 at 15 mg (182 / 181 = 1.006), as the Results state, and reproduces the Abstract’s 36 percent decrease from 15 to 120 mg (36.7 percent); 160 mg gives 1.04 and 37.5 percent.
  • Residual error read as variances. Table 3 prints Additive 5.3 and Proportional 0.18 without saying whether they are variances or standard deviations, and no control stream is published. They are read as NONMEM SIGMA variances (addSd = sqrt(5.3) = 2.30 ng/mL, propSd = sqrt(0.18) = 0.424), consistent with the IIV block of the same table, which is provably on the variance scale. This was a maintainer ruling; it affects only the spread of simulated observations, not Cc.
  • Per-administration dose in F1. Equation 4 does not say whether Dose is the amount of each administration or the daily total. The model uses the amount of each administration, carried in DOSE_TOLVAPTAN_MG, because the paper anchors F1 to the lowest single dose of 15 mg, which is also the smallest administration in any split regimen.
  • New parameter names. lfdepot_dose50 and lfdepot_dosemax were registered for the two dose-scale constants (maintainers’ ruling), following the lcl_dose50 grammar of the dose-dependent clearance family. The paper’s own glosses for them (“the dose at which F1 is 50%”, “the highest dose at which F1 is lowest”) do not describe Equation 4.
  • Typical CL/F in the Discussion. The Discussion computes the body-weight examples (14.7 and 10.2 L/h) from a typical CL/F of 12.3 L/h, whereas Table 3 gives 12.6 L/h. The model uses Table 3; the ratios are the same.
  • F1 is relative. Only oral data were modelled, so F1 is relative to a 15 mg administration and CL/F and Vd/F are apparent values.
  • Food not modelled. Food intake was not recorded in the phase 3b trials, and the paper notes that a high-fat meal raised Cmax by 15-96 percent; there is no food effect in the model.
  • Virtual cohort. Body weight and eGFR distributions are reconstructed from the published medians and 5th/95th percentiles; CYP3A inhibitor use is a per-subject indicator drawn at the observation-level frequency, whereas in the source data it changed record by record. The CYP3A4 inducer indicator, age, BMI, sex and Japan site were screened but are not in the final model.