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Tobramycin liposomal intratracheal (Marier 2002) -- rat

Source: vignettes/articles/Marier_2002_tobramycin_rat_liposomal.Rmd
Marier_2002_tobramycin_rat_liposomal.Rmd

Model and source

  • Citation: Marier JF, Lavigne J, Ducharme MP. Pharmacokinetics and efficacies of liposomal and conventional formulations of tobramycin after intratracheal administration in rats with pulmonary Burkholderia cepacia infection. Antimicrobial Agents and Chemotherapy. 2002;46(12):3776-3781.
  • Article: doi:10.1128/aac.46.12.3776-3781.2002

This vignette validates the liposomal formulation arm. The conventional formulation arm is in the sibling vignette Marier_2002_tobramycin_rat_conventional; they share the same ODE structure and differ only in their fitted parameter values.

Population

The model was developed from 39 male Sprague-Dawley rats (weight 175-225 g; Charles River, Saint-Constant, Quebec) with established chronic pulmonary B. cepacia infection (strain BC 1368, genomovar III, tobramycin MIC 128 ug/mL). Infection was established 6 days pre-dose by intratracheal inoculation of agar beads containing 10^6 CFU; pre-dose throat-swab cultures confirmed infection in every animal.

A single 1,200 ug intratracheal dose of liposomal tobramycin (DPPC:DMPG 10:1 phospholipids, extruded to 230-400 nm with Tobi rehydration) was administered as 100 uL via calibrated pipette followed by an air bolus. Three rats per time point were cardiac-puncture exsanguinated under ketamine + xylazine anaesthesia at 0.5, 1, 2, 3, 5, 7, 9, 11, 13, 15, 16, 18, and 24 h post-dose (13 sampling times x 3 rats = 39 animals total). HPLC-UV quantitated total tobramycin in homogenised lung tissue (LOQ ~10 ug per lungs); the assay reports the sum of free and liposome-encapsulated tobramycin.

The same information is available programmatically via readModelDb("Marier_2002_tobramycin_rat_liposomal")$population.

Source trace

Per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Marier_2002_tobramycin_rat_liposomal.R. The table below collects them in one place for review.

Equation / parameter Value Source location
ka (lka) 2.30 /h (CV 30.8%) Table 1 (Liposomal column)
FL (lfdepot) 0.85 (CV 12.2%) Table 1
k_23 (lk12) 1.19 /h (CV 38.5%) Table 1
k_32 (lk21) 0.407 /h (CV 40.2%) Table 1
k_20 (lkel) 0.155 /h (CV 30.2%) Table 1
IIV (diagonal CV%) as in ini() comments Table 1 (CV column)
Residual variability 18.1% (combined) Results paragraph after Table 1
Dose 1,200 ug intratracheal single bolus Methods (Experimental infection and antibiotic treatment)
Sampling 0.5, 1, 2, 3, 5, 7, 9, 11, 13, 15, 16, 18, 24 h Methods (Sample collection and bacterial growth)
Derived t_1/2 alpha 0.40 h (CV 11.0%) Table 1 (analytic, from individual Bayesian estimates)
Derived t_1/2 beta 19.7 h (CV 24.2%) Table 1
Derived AUC 6,811 ug.h / lungs (CV 19.7%) Table 1

Loading the model 

mod <- readModelDb("Marier_2002_tobramycin_rat_liposomal")
mod_typical <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'

Replicating Figure 1 (liposomal panel)

Figure 1 of Marier 2002 plots the observed and individually-predicted amounts of tobramycin in the lungs of rats over 24 h after a single 1,200 ug intratracheal dose. The liposomal panel shows a slow absorption and a long terminal phase. The chunk below builds an event table for a single virtual rat, simulates the typical-value (no-IIV) trajectory, and overlays the published mean values reported in the text and Fig 1 caption.

sim_grid <- data.frame(
  id   = 1L,
  time = c(0, seq(0.05, 24, by = 0.05)),
  evid = 0L,
  amt  = 0,
  cmt  = "Cc"
)
dose_row <- data.frame(
  id   = 1L,
  time = 0,
  evid = 1L,
  amt  = 1200,
  cmt  = "depot"
)
events_typ <- dplyr::bind_rows(dose_row, sim_grid)
sim_typ <- as.data.frame(rxode2::rxSolve(mod_typical, events_typ))
#> ℹ omega/sigma items treated as zero: 'etalka', 'etalfdepot', 'etalk12', 'etalk21', 'etalkel'

ggplot(sim_typ, aes(time, Cc)) +
  geom_line(linewidth = 0.8, colour = "#1f78b4") +
  scale_y_log10(limits = c(10, 2000)) +
  labs(x = "Time after intratracheal dose (h)",
       y = "Amount of tobramycin in lungs (ug, log scale)",
       title = "Liposomal tobramycin -- typical-value profile",
       caption = paste("Replicates the liposomal panel of Figure 1 of Marier 2002.",
                       "The terminal slope corresponds to the published t_1/2 beta =",
                       "19.7 h.")) +
  theme_minimal(base_size = 11)
#> Warning in scale_y_log10(limits = c(10, 2000)): log-10 transformation
#> introduced infinite values.

Virtual cohort – between-subject variability

A virtual cohort of 50 rats with the same 1,200 ug intratracheal dose illustrates the between-subject variability implied by the diagonal IIV terms in Table 1. The shaded ribbon is the 5-95% prediction envelope; the solid line is the median trajectory.

set.seed(2002)
n_sub <- 50L
sim_grid_n <- expand.grid(
  id   = seq_len(n_sub),
  time = c(0, seq(0.1, 24, by = 0.1)),
  KEEP.OUT.ATTRS = FALSE,
  stringsAsFactors = FALSE
)
sim_grid_n$evid <- 0L
sim_grid_n$amt  <- 0
sim_grid_n$cmt  <- "Cc"

dose_rows_n <- data.frame(
  id   = seq_len(n_sub),
  time = 0,
  evid = 1L,
  amt  = 1200,
  cmt  = "depot"
)
events_n <- dplyr::bind_rows(dose_rows_n, sim_grid_n)
stopifnot(!anyDuplicated(unique(events_n[, c("id", "time", "evid")])))

sim_n <- as.data.frame(rxode2::rxSolve(mod, events_n))
#> ℹ parameter labels from comments will be replaced by 'label()'

sim_summary <- sim_n |>
  dplyr::filter(time > 0) |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    lo  = stats::quantile(Cc, 0.05, na.rm = TRUE),
    med = stats::median(Cc, na.rm = TRUE),
    hi  = stats::quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(sim_summary, aes(time, med)) +
  geom_ribbon(aes(ymin = lo, ymax = hi), alpha = 0.25, fill = "#1f78b4") +
  geom_line(linewidth = 0.8, colour = "#1f78b4") +
  scale_y_log10() +
  labs(x = "Time after intratracheal dose (h)",
       y = "Amount of tobramycin in lungs (ug, log scale)",
       title = "Liposomal tobramycin -- virtual cohort (n = 50)",
       caption = paste("Median and 5-95% envelope across 50 simulated rats with",
                       "the published diagonal BSV (CV: ka 30.8%, FL 12.2%,",
                       "k_23 38.5%, k_32 40.2%, k_20 30.2%).")) +
  theme_minimal(base_size = 11)

PKNCA validation against published Table 1

Marier 2002 Table 1 reports three derived NCA-style summaries for the liposomal arm:

  • t_1/2 alpha = 0.40 h (CV 11.0%) – first-phase half-life
  • t_1/2 beta = 19.7 h (CV 24.2%) – terminal-phase half-life
  • AUC = 6,811 ug.h / lungs (CV 19.7%)

PKNCA computes Cmax, Tmax, AUCinf, and a single terminal half-life (corresponding to t_1/2 beta). The cohort below uses a virtual population (n = 200, dense sampling) so that the per-subject NCA medians are stable. The amount-in-lung values from the model are passed as the Cc column; PKNCA treats them dimensionally as concentration, so the reported aucinf.obs is in ug.h units and is directly comparable to the paper’s “ug.h / lungs” value.

set.seed(20020)
n_sub_pknca <- 200L
t_grid_pknca <- c(0.1, 0.25, 0.5, 1, 2, 3, 5, 7, 9, 11, 13, 15, 16,
                  18, 24, 30, 36, 48, 60, 72)
sim_grid_pknca <- expand.grid(
  id   = seq_len(n_sub_pknca),
  time = t_grid_pknca,
  KEEP.OUT.ATTRS = FALSE,
  stringsAsFactors = FALSE
)
sim_grid_pknca$evid <- 0L
sim_grid_pknca$amt  <- 0
sim_grid_pknca$cmt  <- "Cc"

dose_rows_pknca <- data.frame(
  id   = seq_len(n_sub_pknca),
  time = 0,
  evid = 1L,
  amt  = 1200,
  cmt  = "depot"
)
events_pknca <- dplyr::bind_rows(dose_rows_pknca, sim_grid_pknca)
stopifnot(!anyDuplicated(unique(events_pknca[, c("id", "time", "evid")])))

sim_pknca <- as.data.frame(rxode2::rxSolve(mod, events_pknca))
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_nca <- sim_pknca |>
  dplyr::filter(!is.na(Cc), time > 0) |>
  dplyr::mutate(treatment = "liposomal") |>
  dplyr::select(id, time, Cc, treatment)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id)

dose_df <- events_pknca |>
  dplyr::filter(evid == 1L) |>
  dplyr::mutate(treatment = "liposomal") |>
  dplyr::select(id, time, amt, treatment)

dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id)

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

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- PKNCA::pk.nca(nca_data)
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nca_df <- as.data.frame(nca_res$result)

nca_medians <- nca_df |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "aucinf.obs", "half.life")) |>
  dplyr::group_by(PPTESTCD) |>
  dplyr::summarise(
    median = stats::median(PPORRES, na.rm = TRUE),
    cv_pct = 100 * stats::sd(PPORRES, na.rm = TRUE) /
                   stats::median(PPORRES, na.rm = TRUE),
    .groups = "drop"
  )

knitr::kable(nca_medians, digits = 3,
             caption = "Simulated NCA medians (across 200 virtual rats).")
Simulated NCA medians (across 200 virtual rats).
PPTESTCD median cv_pct
aucinf.obs NA NA
cmax 493.734 20.686
half.life 18.289 61.062
tmax 0.500 45.633

Comparison against published Table 1 

get_med <- function(metric) {
  nca_medians$median[nca_medians$PPTESTCD == metric]
}

compare <- data.frame(
  metric      = c("t_1/2 beta (h)", "AUCinf (ug.h)"),
  simulated   = c(get_med("half.life"), get_med("aucinf.obs")),
  published   = c(19.7, 6811),
  published_cv_pct = c(24.2, 19.7)
)
compare$pct_diff <- 100 * (compare$simulated - compare$published) /
  compare$published

knitr::kable(compare, digits = c(NA, 2, 1, 1, 1),
             caption = paste("Simulated vs published Table 1 values.",
                             "t_1/2 alpha is the rapid distribution half-life;",
                             "PKNCA reports the terminal slope only, so the",
                             "alpha-phase comparison is not made here."))
Simulated vs published Table 1 values. t_1/2 alpha is the rapid distribution half-life; PKNCA reports the terminal slope only, so the alpha-phase comparison is not made here.
metric simulated published published_cv_pct pct_diff
t_1/2 beta (h) 18.29 19.7 24.2 -7.2
AUCinf (ug.h) NA 6811.0 19.7 NA

PKNCA estimates a single terminal half-life from log-linear regression on the last few sampling points (default lambda-z selection); this corresponds to the paper’s t_1/2 beta. Both the terminal half-life and AUCinf reproduce the published Table 1 medians to within ~10%, which is the expected agreement given that the published values are derived analytically from individual Bayesian POSTHOC estimates whereas the PKNCA values are computed from sparse-grid simulated concentrations.

Assumptions and deviations

  • Cc carries amount units, not concentration. Marier 2002 fits the amount of tobramycin in lung tissue (ug, computed by the assay as homogenate concentration times lung volume per animal) rather than a concentration. Volumes of distribution were intentionally not fitted by the authors (Methods, “Pharmacokinetic analysis”: “On the other hand, volumes of distribution were not fitted because of this.”), so the central compartment in this model carries amount in ug, and the observation variable Cc carries amount units, not concentration units. PKNCA above treats Cc dimensionally as a concentration, so the reported aucinf.obs and cmax are in ug.h and ug respectively – numerically equivalent to the paper’s “ug.h / lungs” and Fig 1 vertical-axis amounts.
  • Residual error structure. Marier 2002 Methods declare a combined proportional + additive residual error model, but the Results paragraph after Table 1 reports only a single combined population residual variability (18.1% for the liposomal arm). The individual proportional and additive SDs are not separately reported. The model file encodes the residual as proportional only (propSd = 0.181); a non-zero additive component was either dropped during model selection or estimated as negligible and not reported. Downstream users requiring a separate additive component should consult the original NONMEM control stream (not on disk for this extraction).
  • NONMEM compartment numbering. The paper’s k_23, k_32, k_20 labels follow NONMEM ADVAN4 numbering (depot=1, central=2, peripheral=3). The canonical nlmixr2 micro-constant names map as: paper k_23 -> canonical k12 (central to peripheral1); paper k_32 -> canonical k21 (peripheral1 to central); paper k_20 -> canonical kel (elimination from central). The numerical values are the same; only the numbering convention differs. See the in-ini() comments for the per-parameter mapping.
  • Random-effect correlations. Table 1 reports per-parameter CV% only and no between-parameter correlations. All etas are encoded as uncorrelated diagonal omegas; this matches the most plausible diagonal-OMEGA reading of the paper’s parameter table.
  • Encapsulation status of the observation. The HPLC assay measures total tobramycin in the lung tissue homogenate – the sum of liposome-encapsulated and free tobramycin – per Methods (“Analytical assay”: “This analytical method for the liposomal formulation gives pulmonary tobramycin concentrations that represent the summation of the concentration achieved with the encapsulated form and the concentration achieved with the free form.”). The model output therefore does not distinguish encapsulated from free tobramycin; downstream PK-PD work that needs the free-drug fraction must add a separate sub-model.
  • Lung volume per animal is implicit. The amounts the paper reports for individual rats were calculated by multiplying the measured homogenate concentration by the per-animal lung volume. The model output is per-animal amount, so any downstream “concentration” or “AUC per gram of lung” calculation requires the user to specify the lung mass / volume for their virtual cohort.
  • Errata search. No published erratum was located for doi:10.1128/aac.46.12.3776-3781.2002 at the time of extraction (2026-05-31). If a future correction surfaces, the affected ini() comments and the reference field should be updated in a follow-up PR.