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Liposomal daunorubicin (Hempel 2003)

Source: vignettes/articles/Hempel_2003_daunorubicin_liposomal.Rmd
Hempel_2003_daunorubicin_liposomal.Rmd

Model and source 

mod_meta <- nlmixr2est::nlmixr(readModelDb("Hempel_2003_daunorubicin_liposomal"))$meta
#> ℹ parameter labels from comments will be replaced by 'label()'
  • Citation: Hempel G, Reinhardt D, Creutzig U, Boos J. Population pharmacokinetics of liposomal daunorubicin in children. Br J Clin Pharmacol. 2003;56(4):370-377. doi:10.1046/j.1365-2125.2003.01886.x
  • Description: One-compartment IV-infusion population PK model for total daunorubicin (free plus liposome-encapsulated) following liposomal daunorubicin (Daunoxome) in paediatric and adolescent oncology patients (Hempel 2003). Clearance and volume of distribution scale linearly with total body weight (CL = theta_CL * WT; V = theta_V * WT, i.e. the source paper’s per-kg parameterisation with allometric exponent fixed to 1 and no reference-weight normalisation). The final model (Table 2 model 15) retains inter-individual variability on CL (51% CV) and V (27% CV), inter-occasion variability on CL (16.7% CV) – documented but NOT encoded structurally here, per the Andrews 2017 / Brooks 2021 nlmixr2lib precedent for IOV without an operational occasion column – and a proportional residual error (22%). Distinct from Varatharajan_2016_daunorubicin (free daunorubicin + daunorubicinol metabolite in adult AML).
  • Article (DOI): https://doi.org/10.1046/j.1365-2125.2003.01886.x

This vignette validates the packaged Hempel_2003_daunorubicin_liposomal model – a one-compartment IV-infusion population PK model for total daunorubicin (free plus liposome-encapsulated) following liposomal daunorubicin (Daunoxome) in 24 paediatric and adolescent oncology patients – against the source publication’s Table 2 (population PK model development), Table 3 (BSA-normalised comparison with the prior literature), and the representative subject profile shown in Figure 5.

Population

Hempel 2003 pooled 214 plasma daunorubicin concentrations from 72 treatment cycles (mean 3 samples per cycle, 9 per patient) in 24 paediatric and adolescent patients enrolled across 16 German paediatric haematology / oncology centres in the AML-REZ 97 protocol of the German Society for Paediatric Oncology and Haematology (GPOH). Nineteen of 24 patients had relapsed acute myeloic leukaemia (AML); the remaining five had other relapsed malignancies (two acute lymphoblastic leukaemia, one osteosarcoma, one Ewing sarcoma, one multiple endocrine neoplasia). All patients had previously received conventional daunorubicin and / or doxorubicin. Per Table 1, median age was 15.4 years (range 2.84-23.2); median body weight 48.8 kg (range 14-76.5); median height 1.61 m (range 0.89-1.98); median body surface area 1.48 m^2 (range 0.58-1.98). Daunoxome was administered as a 1- to 2.5-hour IV infusion at 30 or 60 mg/m^2 on days 1 and 5 of each induction cycle (a pilot phase at 30 mg/m^2 was followed by escalation to 60 mg/m^2; four patients on the 60 mg/m^2 schedule intensified to days 1, 3, and 5). Total daunorubicin (free plus liposome-encapsulated) was quantified by capillary electrophoresis with laser-induced fluorescence detection following organic-solvent sample preparation that destroyed the liposomes; within-day accuracy / precision were 3.9-12.9% and 3.8-10.7% (n = 8), and the lower limit of quantification was 2 ug/L (= 2 ng/mL).

The same information is available programmatically via the model’s population metadata:

str(mod_meta$population)
#> List of 20
#>  $ species        : chr "human (paediatric and adolescent)"
#>  $ n_subjects     : int 24
#>  $ n_studies      : int 1
#>  $ age_range      : chr "2.84-23.2 years"
#>  $ age_median     : chr "15.4 years"
#>  $ weight_range   : chr "14-76.5 kg"
#>  $ weight_median  : chr "48.8 kg"
#>  $ height_range   : chr "0.89-1.98 m"
#>  $ height_median  : chr "1.61 m"
#>  $ bsa_range      : chr "0.58-1.98 m^2"
#>  $ bsa_median     : chr "1.48 m^2"
#>  $ bmi_range      : chr "12.6-23.5 kg/m^2"
#>  $ bmi_median     : chr "17.6 kg/m^2"
#>  $ sex_female_pct : num NA
#>  $ disease_state  : chr "Predominantly relapsed acute myeloic leukaemia (19/24); five other malignancies (two relapsed acute lymphoblast"| __truncated__
#>  $ dose_range     : chr "Liposomal daunorubicin (Daunoxome) 30-60 mg/m^2 as a 1- to 2.5-hour IV infusion on days 1 and 5 (induction); fo"| __truncated__
#>  $ regions        : chr "Germany (16 paediatric haematology/oncology centres enrolling in the AML-REZ 97 protocol of the German Society "| __truncated__
#>  $ sampling_window: chr "Plasma sampling suggested at end of infusion, 2-4 h, 4-8 h, 12-17 h, 20-28 h after first dose, predose and ~24/"| __truncated__
#>  $ assay          : chr "Total daunorubicin (free plus liposome-encapsulated; organic-solvent sample preparation destroys liposomes prio"| __truncated__
#>  $ notes          : chr "Patient demographics from Table 1. Sex split is not tabulated as a count; the paper notes 'boys are on average "| __truncated__

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Hempel_2003_daunorubicin_liposomal.R. The table below collects them in one place; values come from Hempel 2003 Table 2 model 15 (the final model) unless otherwise noted.

Parameter / equation Value Source location
lcl (Clearance per kg body weight) log(0.00641) L/h/kg Table 2 model 15 CL = 6.41 mL/h/kg; abstract identical
lvc (Volume of distribution per kg body weight) log(0.0654) L/kg Table 2 model 15 V = 65.4 mL/kg; abstract identical
etalcl (IIV on CL) 0.23133 Table 2 model 15 omega_CL 51% CV; log(1 + 0.51^2)
etalvc (IIV on V) 0.07033 Table 2 model 15 omega_V 27% CV; log(1 + 0.27^2)
propSd (proportional residual error) 0.22 Table 2 model 15 residual error 22%
cl <- exp(lcl + etalcl) * WT n/a Table 2 footnote *** (CL = q1 * Weight; V = q2 * Weight)
vc <- exp(lvc + etalvc) * WT n/a Table 2 footnote *** (CL = q1 * Weight; V = q2 * Weight)
d/dt(central) <- -kel * central n/a Methods (one-compartment ADVAN1 TRAN2); Results paragraph “A one-compartment model described the data adequately”
Cc <- central / vc * 1000 n/a Unit conversion mg/L -> ng/mL (= ug/L, the paper’s reported concentration unit)
Cc ~ prop(propSd) n/a Results paragraph “For the residual error, a proportional error model was chosen”

Inter-occasion variability on CL of 16.7% CV (Table 2 model 15) is explicitly noted in the source paper but is NOT encoded in the packaged model; see Assumptions and deviations.

Virtual cohort

The original observed daunorubicin concentrations are not publicly available. The virtual cohort below approximates the Hempel 2003 Table 1 demographics: 24 paediatric and adolescent subjects with body weight, height, and body surface area drawn to span the published ranges and medians. Each subject receives a single 60 mg/m^2 IV infusion of Daunoxome over 1.5 h (the cycle’s day-1 dose); sampling follows the schedule suggested in Methods (end of infusion, 2-4 h, 4-8 h, 12-17 h, 20-28 h post-infusion start) plus a 48-h late sample so PKNCA can estimate the terminal-phase parameters.

set.seed(20260604L)

n_subj <- 24L

# Body weight: log-normal centred on the reported median 48.8 kg with
# the SD chosen so the simulated range covers Table 1's 14-76.5 kg.
wt_draw <- function(n) {
  s <- exp(rnorm(n, mean = log(48.8), sd = log(76.5 / 14) / 4))
  pmin(pmax(s, 14), 76.5)
}

# Body height: log-normal centred on the reported median 1.61 m with
# the SD chosen so the simulated range covers Table 1's 0.89-1.98 m.
ht_draw <- function(n) {
  s <- exp(rnorm(n, mean = log(1.61), sd = log(1.98 / 0.89) / 4))
  pmin(pmax(s, 0.89), 1.98)
}

# Body surface area via Mosteller: BSA(m^2) = sqrt(WT(kg) * HT(cm) / 3600).
# HT here is in metres, so multiply by 100 inside the square root.
bsa_mosteller <- function(wt_kg, ht_m) {
  sqrt(wt_kg * ht_m * 100 / 3600)
}

cov_tab <- tibble::tibble(
  id   = seq_len(n_subj),
  WT   = wt_draw(n_subj),
  HT   = ht_draw(n_subj)
) |>
  mutate(BSA = bsa_mosteller(WT, HT))

# Sanity check: simulated medians close to Table 1 (within ~5%).
stopifnot(
  abs(median(cov_tab$WT)  / 48.8 - 1) < 0.15,
  abs(median(cov_tab$HT)  / 1.61 - 1) < 0.15,
  abs(median(cov_tab$BSA) / 1.48 - 1) < 0.15
)

# Dosing: 60 mg/m^2 over 1.5-hour IV infusion to central. amt in mg,
# rate in mg/h. Sampling: pre-dose, end-of-infusion, then the Methods
# schedule midpoints plus a 48-h tail for terminal-slope estimation.
infusion_h    <- 1.5
dose_per_bsa  <- 60                    # mg/m^2
sample_times  <- c(0, 1.5, 3, 6, 14, 24, 36, 48)

make_subject <- function(i) {
  row  <- cov_tab[i, ]
  amt  <- dose_per_bsa * row$BSA       # mg total
  rate <- amt / infusion_h             # mg/h
  doses <- tibble::tibble(
    id   = row$id,    time = 0,
    evid = 1L,        amt  = amt,
    rate = rate,      dv   = NA_real_
  )
  obs <- tibble::tibble(
    id   = row$id,    time = sample_times,
    evid = 0L,        amt  = NA_real_,
    rate = NA_real_,  dv   = NA_real_
  )
  bind_rows(doses, obs) |>
    mutate(
      WT  = row$WT,
      HT  = row$HT,
      BSA = row$BSA
    ) |>
    arrange(time, desc(evid))
}

events <- bind_rows(lapply(seq_len(n_subj), make_subject))

stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))

Simulation 

mod         <- readModelDb("Hempel_2003_daunorubicin_liposomal")
mod_typical <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'

sim_typical <- rxode2::rxSolve(
  object = mod_typical, events = events,
  keep   = c("WT", "HT", "BSA")
) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc'
#> Warning: multi-subject simulation without without 'omega'

sim_stoch <- rxode2::rxSolve(
  object = mod, events = events,
  keep   = c("WT", "HT", "BSA")
) |>
  as.data.frame()
#> ℹ parameter labels from comments will be replaced by 'label()'

Replicate published figures

Figure 5 – representative subject concentration-time profile

Hempel 2003 Figure 5 shows the measured plasma concentrations and individual model predictions (model 15) for one representative patient (ID 23) sampled across four administrations of Daunoxome. Plasma concentrations during the 1.5-h infusion rise to a peak of order ~10000 ng/mL at end of infusion, then decline mono-exponentially with an apparent terminal half-life of order 5-10 h. The figure below shows the typical-value (zeroRe) prediction for the median 48.8 kg, 1.48 m^2 virtual subject, alongside the cohort-wide stochastic spread.

typical_subject_id <- cov_tab |>
  arrange(abs(WT - 48.8)) |>
  slice(1) |>
  pull(id)

sim_typical_one <- sim_typical |>
  filter(id == typical_subject_id, time > 0)

ggplot() +
  geom_line(
    data = sim_stoch |> filter(time > 0),
    aes(time, Cc, group = id),
    colour = "gray70", alpha = 0.4
  ) +
  geom_line(
    data = sim_typical_one,
    aes(time, Cc),
    colour = "#1f77b4", linewidth = 1.1
  ) +
  geom_vline(xintercept = infusion_h, linetype = "dashed",
             colour = "gray40") +
  scale_y_log10() +
  labs(
    x = "Time after start of infusion (h)",
    y = "Total daunorubicin (ng/mL, log scale)",
    title = "Figure 5 (analogue) -- single-dose Daunoxome 60 mg/m^2",
    subtitle = paste(
      "Replicates Hempel 2003 Figure 5 schematic for a representative",
      "subject. Blue: typical-value (zeroRe) prediction for the",
      "median-weight (~48.8 kg) virtual subject. Gray:",
      "between-subject stochastic spread. Dashed line marks end of",
      "the 1.5-h infusion."
    )
  ) +
  theme_minimal()

Concentration-time profile by body-weight tertile

The model parameterises CL and V as a linear product of body weight (no reference-weight normalisation), so smaller patients have lower CL and V in absolute terms and the per-dose AUC and t1/2 are roughly invariant of weight in mass-normalised units. The plot below stratifies the virtual cohort by weight tertile to show this directly.

wt_breaks <- quantile(cov_tab$WT, probs = c(0, 1/3, 2/3, 1))
sim_stoch <- sim_stoch |>
  mutate(wt_tert = cut(WT, breaks = wt_breaks,
                       include.lowest = TRUE,
                       labels = c("Low (lightest 1/3)",
                                  "Mid",
                                  "High (heaviest 1/3)")))

sim_stoch |>
  filter(time > 0) |>
  group_by(time, wt_tert) |>
  summarise(
    q05 = quantile(Cc, 0.05, na.rm = TRUE),
    q50 = quantile(Cc, 0.50, na.rm = TRUE),
    q95 = quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time, q50, fill = wt_tert)) +
  geom_ribbon(aes(ymin = q05, ymax = q95), alpha = 0.2) +
  geom_line(aes(colour = wt_tert), linewidth = 0.9) +
  geom_vline(xintercept = infusion_h, linetype = "dashed",
             colour = "gray40") +
  scale_y_log10() +
  labs(
    x = "Time after start of infusion (h)",
    y = "Total daunorubicin (ng/mL, log scale)",
    colour = "Weight tertile",
    fill   = "Weight tertile",
    title  = "Concentration-time profile by body-weight tertile",
    subtitle = paste(
      "Median (line) and 5th-95th percentile (ribbon) of cohort",
      "stochastic simulation at 60 mg/m^2 IV infusion over 1.5 h."
    )
  ) +
  theme_minimal()

PKNCA validation

The Hempel 2003 paper does not tabulate NCA parameters per subject; however, Table 3 compares this study’s BSA-normalised V, CL, terminal half-life, and AUC at 60 mg/m^2 against four prior Daunoxome publications. PKNCA is used here to summarise Cmax, AUCinf, half-life, and clearance from the stochastic simulation, and the medians are compared against the Table 3 “this study” column.

sim_for_nca <- sim_stoch |>
  filter(!is.na(Cc), Cc > 0, time > 0) |>
  select(id, time, Cc) |>
  mutate(treatment = "60 mg/m^2") |>
  as.data.frame()

doses_for_nca <- events |>
  filter(evid == 1L) |>
  select(id, time, amt) |>
  mutate(treatment = "60 mg/m^2") |>
  as.data.frame()

conc_obj <- PKNCA::PKNCAconc(
  data    = sim_for_nca,
  formula = Cc ~ time | treatment + id,
  concu   = "ng/mL",
  timeu   = "hr"
)
dose_obj <- PKNCA::PKNCAdose(
  data    = doses_for_nca,
  formula = amt ~ time | treatment + id,
  doseu   = "mg"
)

intervals <- data.frame(
  start       = 0,
  end         = Inf,
  cmax        = TRUE,
  tmax        = TRUE,
  aucinf.obs  = TRUE,
  aucinf.pred = TRUE,
  half.life   = TRUE,
  clast.obs   = TRUE
)

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- suppressWarnings(PKNCA::pk.nca(nca_data))

knitr::kable(
  summary(nca_res),
  caption = paste(
    "Simulated NCA parameters (PKNCA) for a single 60 mg/m^2 Daunoxome",
    "IV infusion over 1.5 h in the 24-subject virtual cohort."
  )
)
Simulated NCA parameters (PKNCA) for a single 60 mg/m^2 Daunoxome IV infusion over 1.5 h in the 24-subject virtual cohort.
Interval Start Interval End treatment N Cmax (ng/mL) Tmax (hr) Clast (ng/mL) Half-life (hr) AUCinf,obs (hr*ng/mL) AUCinf,pred (hr*ng/mL)
0 Inf 60 mg/m^2 24 28000 [44.2] 1.50 [1.50, 1.50] 133 [4670] 7.51 [3.33] NC NC

Comparison against Table 3 (this-study column)

Hempel 2003 Table 3 reports, for the 60 mg/m^2 single-dose case (BSA-normalised):

  • V = 1.93 L/m^2
  • CL = 0.233 L/h/m^2
  • t1/2 = 5.66 h
  • AUC at 60 mg/m^2 = 231.3 mg.h/L (= 231300 ng.h/mL)

The model parameterises CL and V on body weight, not BSA. For the median paediatric subject (48.8 kg, 1.48 m^2, dose 88.8 mg), the typical-value (zeroRe) prediction is

  • CL_typ = 0.00641 L/h/kg * 48.8 kg = 0.313 L/h, equivalent to 0.313 / 1.48 = 0.211 L/h/m^2 (about 9% below Table 3’s 0.233 L/h/m^2)
  • V_typ = 0.0654 L/kg * 48.8 kg = 3.19 L, equivalent to 3.19 / 1.48 = 2.16 L/m^2 (about 12% above Table 3’s 1.93 L/m^2)
  • t1/2 = ln(2) * V_typ / CL_typ = 7.07 h (about 25% above Table 3’s 5.66 h)
  • AUC = 88.8 mg / 0.313 L/h = 283.5 mg.h/L (about 23% above Table 3’s 231.3 mg.h/L)

The differences between the model-15 (weight-based, this packaged implementation) and Table 3 (BSA-normalised individual NCA values for the same cohort) reflect the different parameterisations the source paper used for the two analyses: the per-BSA values in Table 3 were computed for comparison against the prior literature (which also used BSA-normalised reporting), whereas Table 2 model 15 is the per-weight final population model that achieved the lowest objective function (OF = 417). Both numbers are correct in their respective frames; the packaged model returns the weight-based model-15 predictions.

typ_wt   <- 48.8
typ_bsa  <- 1.48
typ_dose <- 60 * typ_bsa
typ_cl_lh   <- 0.00641 * typ_wt
typ_v_l     <- 0.0654  * typ_wt
typ_t12_h   <- log(2)  * typ_v_l / typ_cl_lh
typ_auc_mghl <- typ_dose / typ_cl_lh
typ_cl_per_bsa <- typ_cl_lh / typ_bsa
typ_v_per_bsa  <- typ_v_l   / typ_bsa

published <- tibble::tibble(
  parameter         = c("V (L/m^2)", "CL (L/h/m^2)",
                        "t1/2 (h)", "AUC at 60 mg/m^2 (mg.h/L)"),
  table_3_value     = c(1.93, 0.233, 5.66, 231.3),
  model_15_typical  = c(round(typ_v_per_bsa, 3),
                        round(typ_cl_per_bsa, 3),
                        round(typ_t12_h, 2),
                        round(typ_auc_mghl, 1))
) |>
  mutate(pct_diff = round(100 * (model_15_typical / table_3_value - 1), 1))

knitr::kable(
  published,
  caption = paste(
    "Hempel 2003 Table 3 (this-study column, BSA-normalised) vs",
    "packaged model 15 typical-value prediction at the median",
    "paediatric subject (48.8 kg, 1.48 m^2). Differences",
    "reflect the per-weight vs per-BSA parameterisation used for",
    "the two analyses; the packaged model implements model 15",
    "(the paper's chosen final population model)."
  )
)
Hempel 2003 Table 3 (this-study column, BSA-normalised) vs packaged model 15 typical-value prediction at the median paediatric subject (48.8 kg, 1.48 m^2). Differences reflect the per-weight vs per-BSA parameterisation used for the two analyses; the packaged model implements model 15 (the paper’s chosen final population model).
parameter table_3_value model_15_typical pct_diff
V (L/m^2) 1.930 2.156 11.7
CL (L/h/m^2) 0.233 0.211 -9.4
t1/2 (h) 5.660 7.070 24.9
AUC at 60 mg/m^2 (mg.h/L) 231.300 283.900 22.7

Assumptions and deviations

  • Inter-occasion variability (IOV) of 16.7% CV on CL is NOT encoded structurally. Hempel 2003 Table 2 model 15 retains an IOV component on CL alongside the IIV on CL (51% CV) and IIV on V (27% CV). The IOV captures within-subject variability in apparent clearance from one dosing occasion to the next (within a multi-cycle treatment course). nlmixr2lib has no canonical occasion-column convention, and the prevailing precedent for IOV without an operational OCC column is to omit it from the packaged model (Andrews 2017 tacrolimus, Brooks 2021 tacrolimus). Downstream users who want to reproduce the IOV can add an OCC indicator and a per-occasion eta in rxode2; the IOV variance on the log scale is log(1 + 0.167^2) = 0.0276. Population-level summaries (typical Cmax, typical AUC) are unaffected; only within-subject variability across cycles is missing.

  • Linear weight scaling with no reference-weight normalisation (per the source paper). Model 15 uses CL = theta_CL * WT and V = theta_V * WT directly, with theta_CL = 6.41 mL/h/kg and theta_V = 65.4 mL/kg reported per kg body weight. This is equivalent to a fixed allometric exponent of 1 with no reference weight; downstream comparison against models that use the more common allometric form (WT/70)^0.75 must account for the different functional shape.

  • Table 3 versus Table 2 model 15. The paper reports two parameterisations of the cohort: Table 2 model 15 (weight-based, the chosen final model, OF = 417) and Table 3 (per-BSA values calculated specifically for comparison against the prior literature that also reported per-BSA values). The packaged model implements Table 2 model 15; the Table 3 comparison values are approximately 20-25% different in t1/2 and AUC for the median paediatric subject because the per-BSA Table 3 values were computed for a different reporting frame, not because either set of numbers is wrong. See the comparison table above for the side-by-side breakdown.

  • Sex distribution not encoded. Hempel 2003 reports that boys were heavier on average (median 58.0 vs 37.5 kg) but found no significant sex difference in PK after accounting for weight; the paper does not tabulate the exact M/F count. The population$sex_female_pct field is left as NA and sex is not a covariate of the model.

  • Body weight time-fixed in the simulation. The paper does not explicitly state whether body weight was updated between dosing cycles. In the virtual cohort above weight is held constant across the 48-h observation window (a reasonable approximation given the short window). For multi-cycle simulations over weeks-to-months, users should consider supplying time-varying weight.

  • BSA computed via the Mosteller formula in the virtual cohort. The source paper does not state which BSA formula it used; the Mosteller formula (BSA = sqrt(WT * HT / 3600), HT in cm, WT in kg) is applied here to derive BSA from the simulated WT and HT distributions for the 60 mg/m^2 dose calculation. The model itself does NOT use BSA as a covariate – BSA appears only in the virtual cohort’s dose-amount computation.

  • **Concentration units (ng/mL) require an explicit *1000 scaling in model().** With dose in mg and vc in L, the ratio central / vc carries units of mg/L = 1000 ng/mL. The Cc output multiplies by 1000 to express concentrations in the paper’s reported ug/L units (numerically identical to ng/mL). checkModelConventions() may issue an info-level message flagging this dosing-vs-concentration magnitude mismatch; the scaling is intentional and documented in the model file.

  • Single-dose simulation only. This vignette simulates one cycle’s day-1 dose. The paper’s dosing schedule (Daunoxome on days 1 and 5 per cycle, four patients escalated to days 1, 3, 5) is mechanically straightforward to extend by adding further dose rows; the single-dose case is shown here as the validation pipeline.