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Treosulfan and EBDM in rat plasma and brain (Danielak 2020)

Source: vignettes/articles/Danielak_2020_treosulfan_rat.Rmd
Danielak_2020_treosulfan_rat.Rmd

Model and source

  • Citation: Danielak D, Romanski M, Kasprzyk A, Tezyk A, Glowka F. Population pharmacokinetic approach for evaluation of treosulfan and its active monoepoxide disposition in plasma and brain on the basis of a rat model. Pharmacol Rep. 2020;72(5):1297-1309. doi:10.1007/s43440-020-00115-0. Correction: Pharmacol Rep. 2020;72:1443. doi:10.1007/s43440-020-00144-9 (units of the first-order rate constants corrected from L/h to 1/h; no values changed)
  • Description: Preclinical (rat). Joint parent-metabolite population PK model for treosulfan (TREO) and its active monoepoxide (S,S)-1,2-epoxybutane-3,4-diol-4-methanesulfonate (EBDM) in plasma and brain of Wistar rats after a single 500 mg/kg intraperitoneal dose (Danielak 2020). First-order absorption into a one-compartment TREO plasma model; irreversible first-order conversion of TREO to EBDM (fixed rate constants in plasma and in brain); one-compartment EBDM plasma model; bidirectional blood-brain barrier transport of TREO and EBDM parameterised by an influx clearance and the influx/efflux clearance ratio; a second (deep) brain compartment for TREO. All clearances and volumes are apparent (divided by F) and per kg body weight. Male sex lowers TREO plasma clearance by 14.6%. IIV on ka and TREO CL; proportional residual error fixed to 15% (plasma) and 20% (brain).
  • Article: https://doi.org/10.1007/s43440-020-00115-0 (open access)
  • Correction: https://doi.org/10.1007/s43440-020-00144-9

Treosulfan is a prodrug: it converts non-enzymatically (pH- and temperature-dependent) to the active monoepoxide EBDM, and then to the diepoxide DEB. Danielak et al. fitted a joint parent-metabolite model to plasma and brain concentrations of treosulfan (TREO) and EBDM in Wistar rats to quantify blood-brain barrier (BBB) penetration.

Population

96 ten-week-old Wistar rats (48 male, 48 female; mean body weight 306 +/- 25 g for males and 188 +/- 15 g for females) received a single intraperitoneal dose of 500 mg/kg treosulfan (1797 umol/kg). The design was destructive (one animal per sampling time): plasma and brain were collected predose and at 0.25, 0.5, 1, 2, 4, 6 and 24 h. One female with an outlying 0.25 h plasma concentration was excluded, and the 24 h samples (all below the limit of quantitation) were dropped (Methods “Animals” and “Sample collection”, Fig. 2; Results first paragraph).

The same information is available programmatically via readModelDb("Danielak_2020_treosulfan_rat")()$population.

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Danielak_2020_treosulfan_rat.R. All clearances and volumes are apparent (divided by the unknown intraperitoneal bioavailability F) and per kg body weight.

Equation / parameter Value Source location
lka (k12) log(5.12) 1/h Table 1
lcl (CL1/F, females) log(0.419) L/h/kg Table 1
lvc (V2/F) fixed log(1.03) L/kg Table 1
lkmet (k23, TREO to EBDM in plasma) fixed log(0.451) 1/h Table 1; Methods assumption (4)
lkmet_brain (k45, TREO to EBDM in brain) fixed log(0.271) 1/h Table 1; Methods assumption (5), log10(kf) = -7.479 + 0.960 pH at pH 7.2
lcl_ebdm (CL2/F) log(6.24) L/h/kg Table 1
lvc_ebdm (V3/F) fixed log(0.914) L/kg Table 1
lclin (Q1/F, TREO BBB influx clearance) log(0.0233) L/h/kg Table 1; Methods assumption (7)
lkp_brain (BBB1 = CLin/CLout, TREO) log(0.120) Table 1
lv_brain_extravascular (V4/F) fixed log(6.56e-3) L/kg Table 1; Results “Prediction of treosulfan concentrations in brain”
lq_brain_deep (Q2/F) log(0.304) L/h/kg Table 1
lv_brain_deep (V6/F) fixed log(0.364) L/kg Table 1
lclin_ebdm (Q3/F, EBDM BBB influx clearance) log(0.0122) L/h/kg Table 1
lkp_brain_ebdm (BBB2 = CLin/CLout, EBDM) log(0.317) Table 1
lv_brain_extravascular_ebdm (V5/F) fixed log(6.56e-3) L/kg Table 1
e_sex_cl (COV CL-MALE) -0.146 Table 1 and footnote a
etalka 0.38941 (69% CV) Table 1, footnote b
etalcl 0.00995 (10% CV) Table 1, footnote b
propSd, propSd_ebdm fixed 0.15 Table 1; Methods “Population pharmacokinetic analysis”
propSd_Cbrain, propSd_Cbrain_ebdm fixed 0.20 Table 1; Methods
Structure (six compartments, arrows) n/a Figure 3
clef = clin / kp_brain n/a Methods assumption (7): BBB = CLin / CLout
cl = tvcl * (1 + e_sex_cl * (1 - SEXF)) n/a Table 1 footnote a

Deterministic checks on the typical animal

The typical male and female are solved without random effects on a log-spaced early grid (absorption is fast, k12 = 5.12 1/h) out to 72 h so that the AUCs are essentially complete.

mod <- readModelDb("Danielak_2020_treosulfan_rat")
mod_typ <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'
dose <- 1797 # umol/kg; 500 mg/kg / 278.29 g/mol (Methods)

grid <- sort(unique(c(0, 10^seq(-3, log10(72), length.out = 600))))
ev_typ <- dplyr::bind_rows(
  data.frame(id = 1L, time = 0, amt = dose, evid = 1L, cmt = "depot",
             dvid = NA_integer_, SEXF = 1),
  data.frame(id = 1L, time = grid, amt = 0, evid = 0L, cmt = "central",
             dvid = 1L, SEXF = 1),
  data.frame(id = 2L, time = 0, amt = dose, evid = 1L, cmt = "depot",
             dvid = NA_integer_, SEXF = 0),
  data.frame(id = 2L, time = grid, amt = 0, evid = 0L, cmt = "central",
             dvid = 1L, SEXF = 0)
)
sim_typ <- rxode2::rxSolve(mod_typ, events = ev_typ, rtol = 1e-10, atol = 1e-12,
                           maxsteps = 1e6, returnType = "data.frame")
#> ℹ omega/sigma items treated as zero: 'etalka', 'etalcl'
#> Warning: multi-subject simulation without without 'omega'
sim_typ$sex <- ifelse(sim_typ$id == 1, "Female", "Male")

trap <- function(t, y) sum(diff(t) * (head(y, -1) + tail(y, -1)) / 2)
auc_typ <- sim_typ |>
  dplyr::group_by(sex) |>
  dplyr::summarise(
    cl = cl[1], cl_ebdm = cl_ebdm[1],
    auc_treo = trap(time, Cc),
    auc_ebdm = trap(time, Cc_ebdm),
    auc_brain = trap(time, Cbrain),
    auc_brain_ebdm = trap(time, Cbrain_ebdm),
    c025 = Cc[which.min(abs(time - 0.25))],
    .groups = "drop"
  ) |>
  dplyr::mutate(
    recovered_pct = 100 * (cl * auc_treo + cl_ebdm * auc_ebdm) / dose,
    ratio_treo = auc_brain / auc_treo,
    ratio_ebdm = auc_brain_ebdm / auc_ebdm
  )
knitr::kable(auc_typ, digits = 4,
             caption = "Typical-animal exposures (AUC in umol*h/L, CL in L/h/kg).")
Typical-animal exposures (AUC in umol*h/L, CL in L/h/kg).
sex cl cl_ebdm auc_treo auc_ebdm auc_brain auc_brain_ebdm c025 recovered_pct ratio_treo ratio_ebdm
Female 0.4190 6.24 2033.518 151.4520 241.8083 59.1800 1108.299 100.0058 0.1189 0.3908
Male 0.3578 6.24 2184.750 162.7154 259.7914 63.5812 1118.068 100.0058 0.1189 0.3908

Three checks follow, all on deterministic quantities.

  1. Mass balance. Every molecule of treosulfan leaves the body either unchanged through CL1/F or, after conversion, as EBDM through CL2/F (the brain compartments have no elimination of their own). Therefore CL1 * AUC(TREO, plasma) + CL2 * AUC(EBDM, plasma) = Dose. A wrong conversion or transfer term breaks this identity.
  2. Sex effect. The typical male CL1/F must be 0.419 * (1 - 0.146) = 0.358 L/h/kg, the male median shown in Figure 6 of the paper.
  3. Brain penetration of treosulfan. Integrating the brain mass balance to infinity gives AUC(brain) / AUC(plasma) = CLin / (CLout + k45 * V4) = 0.1189; the value is independent of sex and is close to BBB1 = 0.120 because conversion in the small brain volume is slow compared with efflux.
kin_ratio <- 0.0233 / (0.0233 / 0.120 + 0.271 * 6.56e-3)
stopifnot(
  nrow(auc_typ) == 2,
  all(abs(auc_typ$recovered_pct - 100) < 0.5),
  abs(auc_typ$cl[auc_typ$sex == "Male"] - 0.419 * (1 - 0.146)) < 1e-6,
  abs(auc_typ$cl[auc_typ$sex == "Female"] - 0.419) < 1e-6,
  all(abs(auc_typ$ratio_treo / kin_ratio - 1) < 0.01)
)

The paper that first analysed these same animals by naive pooling reported a brain-to-plasma AUC ratio of about 0.1 for treosulfan (Danielak 2020 Discussion, citing Romanski et al.); the model gives 0.119. EBDM penetrates better (typical AUC ratio 0.391), consistent with BBB2 = 0.317 plus local formation of EBDM from treosulfan inside the brain.

The mean plasma treosulfan concentration observed 0.25 h after dosing was 1147 uM (Results first paragraph, excluding the outlier); the typical-animal prediction at 0.25 h is 1108 uM (female) and 1118 uM (male).

# The observed mean is one sample per animal at a single time; the gate only
# catches a dose, unit or volume error, which would move it by >2-fold.
stopifnot(all(abs(auc_typ$c025 / 1147 - 1) < 0.25))

Virtual cohort and simulation

Original data are not available. The cohort below is 100 males and 100 females dosed with 1797 umol/kg into the peritoneal depot, observed on the paper’s sampling grid (0.25-6 h) plus a dense grid for the NCA. The model has four error endpoints (Cc, Cc_ebdm, Cbrain, Cbrain_ebdm), so observation rows carry dvid = 1; every concentration is still returned as a column.

set.seed(2020)
n_per_sex <- 100
obs_times <- sort(unique(c(0, 0.05, 0.1, 0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4, 5, 6,
                           8, 10, 12, 16, 20, 24)))
make_cohort <- function(n, sexf, id_offset) {
  ids <- id_offset + seq_len(n)
  dplyr::bind_rows(
    data.frame(id = ids, time = 0, amt = dose, evid = 1L, cmt = "depot",
               dvid = NA_integer_),
    tidyr::expand_grid(id = ids, time = obs_times) |>
      dplyr::mutate(amt = 0, evid = 0L, cmt = "central", dvid = 1L)
  ) |>
    dplyr::mutate(SEXF = sexf, sex = ifelse(sexf == 1, "Female", "Male")) |>
    dplyr::arrange(id, time, dplyr::desc(evid))
}
events <- dplyr::bind_rows(
  make_cohort(n_per_sex, 1, 0L),
  make_cohort(n_per_sex, 0, n_per_sex)
)
stopifnot(!anyDuplicated(events[, c("id", "time", "evid")]))

rxode2::rxSetSeed(2020)
sim <- rxode2::rxSolve(mod, events = events, keep = "sex",
                       returnType = "data.frame")
#> ℹ parameter labels from comments will be replaced by 'label()'

Replicate Figure 5 (VPC) 

# Replicates Figure 5 of Danielak 2020: 5th, 50th and 95th percentiles of the
# four measured analytes over the 0.25-6 h sampling window.
sim |>
  dplyr::filter(time >= 0.25, time <= 6) |>
  dplyr::select(id, time, sex, Cc, Cc_ebdm, Cbrain, Cbrain_ebdm) |>
  tidyr::pivot_longer(c(Cc, Cc_ebdm, Cbrain, Cbrain_ebdm),
                      names_to = "analyte", values_to = "conc") |>
  dplyr::mutate(analyte = factor(
    analyte,
    levels = c("Cc", "Cc_ebdm", "Cbrain", "Cbrain_ebdm"),
    labels = c("Treosulfan, plasma", "EBDM, plasma",
               "Treosulfan, brain", "EBDM, brain")
  )) |>
  dplyr::group_by(analyte, time) |>
  dplyr::summarise(
    Q05 = quantile(conc, 0.05), Q50 = median(conc), Q95 = quantile(conc, 0.95),
    .groups = "drop"
  ) |>
  ggplot(aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.3) +
  geom_line() +
  facet_wrap(~analyte, scales = "free_y") +
  scale_y_log10() +
  labs(x = "Time after dose (h)", y = "Concentration (uM)",
       caption = "Replicates Figure 5 of Danielak 2020 (90% prediction interval, median).")

Replicate Figure 6 (clearance by sex) 

cl_ind <- sim |>
  dplyr::distinct(id, sex, cl)
ggplot(cl_ind, aes(sex, cl)) +
  geom_boxplot() +
  labs(x = "Sex", y = "CL1/F (L/h/kg)",
       caption = "Replicates Figure 6 of Danielak 2020 (individual CL1/F by sex).")

cl_med <- cl_ind |>
  dplyr::group_by(sex) |>
  dplyr::summarise(median_cl = median(cl), .groups = "drop")
knitr::kable(cl_med, digits = 3)
sex median_cl
Female 0.420
Male 0.357 
# Figure 6 medians: about 0.358 (male) and 0.419 (female). With a 10% CV the
# median of 100 draws is within ~3% of the typical value; 8% still catches
# a sign error on e_sex_cl (0.48 vs 0.36 L/h/kg).
stopifnot(
  abs(cl_med$median_cl[cl_med$sex == "Male"] / 0.358 - 1) < 0.08,
  abs(cl_med$median_cl[cl_med$sex == "Female"] / 0.419 - 1) < 0.08
)

PKNCA validation

Plasma treosulfan NCA by sex. The paper does not report NCA values, so there is no published table to compare against; the checks above serve as the quantitative validation.

sim_nca <- sim |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, sex)
sim_nca <- dplyr::bind_rows(
  sim_nca,
  sim_nca |> dplyr::distinct(id, sex) |> dplyr::mutate(time = 0, Cc = 0)
) |>
  dplyr::distinct(id, sex, time, .keep_all = TRUE) |>
  dplyr::arrange(id, time)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | sex + id)
dose_df <- events |>
  dplyr::filter(evid == 1) |>
  dplyr::select(id, time, amt, sex)
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | sex + id)
intervals <- data.frame(start = 0, end = Inf, cmax = TRUE, tmax = TRUE,
                        aucinf.obs = TRUE, half.life = TRUE)
nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj,
                                          intervals = intervals))
nca_sum <- as.data.frame(nca_res$result) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "aucinf.obs", "half.life")) |>
  dplyr::group_by(sex, PPTESTCD) |>
  dplyr::summarise(median = median(PPORRES, na.rm = TRUE), .groups = "drop") |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = median)
nca_sum |>
  dplyr::rename(
    "Sex" = sex,
    "Cmax (uM)" = cmax,
    "Tmax (h)" = tmax,
    "AUC0-inf (uM*h)" = aucinf.obs,
    "t1/2 (h)" = half.life
  ) |>
  knitr::kable(digits = 2, caption = "Median simulated plasma treosulfan NCA by sex.")
Median simulated plasma treosulfan NCA by sex.
Sex AUC0-inf (uM*h) Cmax (uM) t1/2 (h) Tmax (h)
Female 2012.83 1188.40 2.18 0.5
Male 2169.22 1225.87 2.18 0.5 

# AUC0-inf does not depend on ka, and IIV on CL1/F is only 10% CV, so the
# cohort median must sit close to the typical-animal AUC computed above. A
# dose, unit or clearance error moves it by far more than 10%.
auc_m <- nca_sum$aucinf.obs[nca_sum$sex == "Male"]
auc_f <- nca_sum$aucinf.obs[nca_sum$sex == "Female"]
stopifnot(
  length(auc_m) == 1, length(auc_f) == 1,
  abs(auc_f / auc_typ$auc_treo[auc_typ$sex == "Female"] - 1) < 0.1,
  abs(auc_m / auc_typ$auc_treo[auc_typ$sex == "Male"] - 1) < 0.1
)

Assumptions and deviations

  • Abstract vs. Table 1 for male clearance. The abstract gives the male CL1/F as 0.273 L/h/kg, which is 0.419 - 0.146 (the coefficient subtracted as an absolute value). Table 1 footnote a defines the effect as fractional, CL1-MALE/F = CL1/F + CL1/F * COV, giving 0.358 L/h/kg, and the male median in Figure 6 is 0.358. The model follows the footnote and Figure 6.
  • Units of the rate constants. The original online version printed the first-order rate constants in L/h; the Correction (doi:10.1007/s43440-020-00144-9) changes the units to 1/h without changing any value. The open-access article carries the corrected units.
  • Q1/F and Q3/F are the BBB influx clearances. Methods assumption (7) parameterises BBB transport by the influx clearance CLin and the ratio BBB = CLin / CLout; Figure 3 labels the plasma-brain arrows Q1/F, BBB1 and Q3/F, BBB2. The model therefore uses clin = Q/F and clef = clin / BBB.
  • Brain treosulfan converts to EBDM only in the central brain compartment (k45), and there is no other brain elimination (Figure 3; Results “Blood- brain transport of treosulfan and EBDM”).
  • Brain observations are the central brain compartments (V4/F for treosulfan, V5/F for EBDM), per Results “Prediction of treosulfan concentrations in brain”. The deep brain compartment (V6/F) is exposed as Cbrain_deep but was not observed.
  • Per-kg dosing. The paper specified the dose as 1797 umol/kg for every animal, so all volumes and clearances are per kg and no body-weight covariate is needed. Supply doses in umol/kg.
  • Bioavailability. All parameters are apparent (/F); F was not identifiable and is implicitly 1.
  • Residual error was fixed to the assay precision (15% plasma, 20% brain) because a one-animal-per-sample design cannot identify it.