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Enoxaparin (Oualha 2018)

Source: vignettes/articles/Oualha_2018_enoxaparin.Rmd
Oualha_2018_enoxaparin.Rmd

Model and source

  • Citation: Oualha M, Chardot C, Debray D, Lesage F, Harroche A, Renolleau S, Treluyer J-M, Urien S (2018). Population pharmacokinetics of enoxaparin in early stage of paediatric liver transplantation. British Journal of Clinical Pharmacology 84(8):1736-1745. doi:10.1111/bcp.13543.
  • Description: Population PK model for subcutaneous enoxaparin in 22 children during the first post-operative week after paediatric liver transplantation (Oualha 2018). One-compartment open model with first-order absorption (ka fixed at 1/h) and first-order elimination, measured as anti-Xa activity (target 0.2-0.4 IU/mL). Apparent clearance CL/F is allometrically scaled by pre-operative bodyweight BWPREOP (fixed exponent 0.75); apparent central volume V/F is allometrically scaled (fixed exponent 1) by a time-varying post-operative bodyweight BW(t) that captures peri-operative fluid resuscitation followed by post-operative diuresis: BW(t) = (BWPREOP + PFA/1000) * (1 - (1 - fbw) * t^hill_bw / (tbw50^hill_bw + t^hill_bw)). Bodyweight-evolution parameters fbw / hill_bw / tbw50 are jointly estimated with the enoxaparin PK and carry their own between-subject variability.
  • Article: https://doi.org/10.1111/bcp.13543

Population

Oualha 2018 enrolled 22 children (8 male, 14 female; sex female 63.6%) admitted to a single-centre French paediatric intensive care unit between January 2013 and July 2015 after liver transplantation. Median pre-operative body weight (BWPREOP) was 10.6 kg (range 6.7-34 kg) and median post-natal age was 21.5 months (range 5-154 months). Indication for transplantation was biliary cirrhosis in 20 children and metabolic disease or tumour in the remaining two. A left split liver graft was used in 18 (81.8%); median percent of graft weight to BWPREOP was 3.4% (range 1.2-8.7). Median perioperative fluid administration (PFA) was 2634 mL (range 1008-6520 mL); 19 of 22 patients required perioperative norepinephrine support; transient acute renal dysfunction was observed in 7 (31.8%). Demographics and baseline biochemistry are reported in Table 1 of Oualha 2018; covariate definitions are echoed in the model’s population metadata (readModelDb("Oualha_2018_enoxaparin")$population).

Enoxaparin (Lovenox; Sanofi Aventis; 0.2 mL = 20 mg = 2000 IU) was initiated 12 h after graft hepatic artery clamping in children with a platelet count > 50000 / mm^3. Initial dosing was 50 IU/kg subcutaneously every 12 h (range 43-59 IU/kg); 18 of 22 patients required dose increases (median +19 IU/kg, range +4 to +57) to reach the 0.2-0.4 IU/mL anti-Xa activity target. A total of 136 anti-Xa activity samples were collected within 10-156 h of the first enoxaparin dose; 37 fell below the 0.1 IU/mL limit of quantification and were M3-method censored in the original Monolix fit.

Source trace

The per-parameter origin is recorded inline next to each ini() entry in inst/modeldb/specificDrugs/Oualha_2018_enoxaparin.R. The table below collects the source locations in one place.

Equation / parameter Value Source location
lka (= log(1)) FIXED ka = 1 /h Table 3: ka (h^-1) “1 (fixed)”
lcl CL_TYP = 1.23 L/h for 70 kg BWPREOP Table 3: CL_TYP (l h^-1) for 70 kg BW_PREOP = 1.23 (RSE 15%)
lvc V_TYP = 14.6 L for 70 kg BWPREOP and fbw=0 Table 3: V_TYP (l) for 70 kg BW_PREOP and f_BW = 0 = 14.6 (RSE 27%)
e_wt_cl FIXED 0.75 theta_BW(CL) = 3/4 Table 3: theta_BW (CL_i = CL_TYP * (BW_PREOP/70)^(3/4)) = 0.75 (fixed)
e_wt_vc FIXED 1 theta_BW(V) = 1 Table 3: theta_BW (V_i = V_TYP * (BW(t)/70)^1) = 1 (fixed)
lfbw theta_fBW = 0.89 Table 3: theta_fBW = 0.89 (RSE 1%)
lhill_bw theta_Hill = 4.43 Table 3: theta_Hill = 4.43 (RSE 7%)
ltbw50 theta_tBW50 = 61.3 h Table 3: theta_tBW50 = 61.3 (RSE 18%)
etalcl ~ 0.3969 omega_CL = 0.63 Table 3: eta_CL square root of omega^2_CL = 0.63 [shrinkage 10.0%]
etalvc ~ 1.5129 omega_V = 1.23 Table 3: eta_V square root of omega^2_V = 1.23 [shrinkage 20.6%]
etalfbw ~ 0.0036 omega_FBW = 0.06 Table 3: eta_FBW square root of omega^2_FBW = 0.06 [shrinkage 4.08%]
etaltbw50 ~ 0.64 omega_tBW50 = 0.80 Table 3: eta_tBW50 square root of omega^2_tBW50 = 0.80 [shrinkage 4.60%]
propSd proportional residual on anti-Xa = 0.42 Table 3: Proportional on anti-Xa activity = 0.42 (RSE 3%)
BW(t) curve BW(t) = (BWPREOP + PFA/1000) * (1 - (1 - fbw) * t^hill_bw / (tbw50^hill_bw + t^hill_bw)) Methods paragraph 1 of “Pharmacokinetic modelling” and Table 3 footnote
Allometric scaling CL_i = CL_TYP * (BWPREOP/70)^0.75; V_i = V_TYP * (BW(t)/70)^1 Eq. 2 and 3 of Methods; Results “Pharmacokinetic analysis” paragraph 2
Anti-Xa target 0.2-0.4 IU/mL Methods “Anticoagulation protocol and assays” paragraph 2

Virtual cohort

The vignette replicates the Oualha 2018 Table 4 worked example (a 10 kg patient receiving PFA = 2000 mL, “actual bodyweight 12 kg”) and a heavier sibling cohort spanning the published BWPREOP range. Each cohort is built as a self-contained event table with disjoint integer IDs so the multi-cohort bind_rows() keeps subjects separate inside rxSolve().

set.seed(20260530L)

mod <- readModelDb("Oualha_2018_enoxaparin")

# Sampling grid: q4h over one week, matching the paper's 10-156 h
# observation window and trough-trough Figure-1 / VPC layout.
times_grid <- seq(0, 168, by = 1)

# Dose helper: SC subcutaneous administration into the depot compartment,
# every 12 h for 7 days (14 doses), at the per-kg rate stated by the
# regimen. amt is in IU; the model's units list declares dosing = "IU".
make_cohort <- function(n, bw_kg, pfa_ml, dose_iu_per_kg, regimen,
                        id_offset = 0L) {
  ids <- id_offset + seq_len(n)
  ev_doses <- expand.grid(id = ids, time = seq(0, 12 * 13, by = 12)) |>
    dplyr::mutate(
      evid = 1L,
      cmt  = "depot",
      amt  = bw_kg * dose_iu_per_kg,
      WT   = bw_kg,
      PFA  = pfa_ml,
      regimen = regimen
    )
  ev_obs <- expand.grid(id = ids, time = times_grid) |>
    dplyr::mutate(
      evid = 0L,
      cmt  = "Cc",
      amt  = 0,
      WT   = bw_kg,
      PFA  = pfa_ml,
      regimen = regimen
    )
  dplyr::bind_rows(ev_doses, ev_obs) |>
    dplyr::arrange(id, time, dplyr::desc(evid))
}

# Three regimens applied to the paper's worked example (10 kg, 2000 mL PFA):
# the standard 50 IU/kg q12h, the suggested 100 IU/kg q12h, and a 75 IU/kg
# q12h intermediate. Plus a 20 kg sibling cohort on standard 50 IU/kg q12h
# so the BW dependence in V is exercised.
events <- dplyr::bind_rows(
  make_cohort(60, bw_kg = 10, pfa_ml = 2000, dose_iu_per_kg = 50,
              regimen = "10 kg, 50 IU/kg q12h",
              id_offset =   0L),
  make_cohort(60, bw_kg = 10, pfa_ml = 2000, dose_iu_per_kg = 75,
              regimen = "10 kg, 75 IU/kg q12h",
              id_offset =  60L),
  make_cohort(60, bw_kg = 10, pfa_ml = 2000, dose_iu_per_kg = 100,
              regimen = "10 kg, 100 IU/kg q12h",
              id_offset = 120L),
  make_cohort(60, bw_kg = 20, pfa_ml = 3000, dose_iu_per_kg = 50,
              regimen = "20 kg, 50 IU/kg q12h",
              id_offset = 180L)
)

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

Simulation 

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

For a deterministic typical-value reproduction of the published time-course (matching Figure 3 and Figure 5 caption descriptions), zero the random effects:

mod_typical <- mod |> rxode2::zeroRe()
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_typical <- rxode2::rxSolve(mod_typical, events = events,
                               keep = c("regimen", "WT", "PFA")) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalfbw', 'etaltbw50'
#> Warning: multi-subject simulation without without 'omega'

Replicate published figures

Time-varying body-weight curve BW(t) (Methods Eq. for BW(t))

The post-operative body-weight curve combines the immediate intra-operative fluid load (BWPREOP + PFA/1000) with a sigmoidal diuresis governed by fbw = 0.89, hill_bw = 4.43, tbw50 = 61.3 h. The fluid resorption is slow (tbw50 ~ 2.5 days) and goes from BWPREOP + PFA/1000 at t = 0 toward 0.89 * (BWPREOP + PFA/1000) at large t.

bwt_curve <- sim_typical |>
  dplyr::filter(regimen %in% c("10 kg, 50 IU/kg q12h", "20 kg, 50 IU/kg q12h")) |>
  dplyr::distinct(regimen, time, .keep_all = TRUE) |>
  dplyr::mutate(bwt = (WT + PFA / 1000) *
                        (1 - (1 - 0.89) * time^4.43 /
                              (61.3^4.43 + time^4.43)))

ggplot(bwt_curve, aes(time, bwt, colour = regimen)) +
  geom_line(linewidth = 0.8) +
  labs(x = "Time after first enoxaparin dose (h)",
       y = "BW(t) (kg)",
       title = "Time-varying post-operative body weight",
       caption = "Reproduces the BW(t) curve defined in Oualha 2018 Methods.") +
  theme_bw()

Figure 3 / VPC of anti-Xa activity vs time (50 IU/kg q12h)

Reproduces the structure of Oualha 2018 Figure 3 (visual predictive check): median and 5th-95th-percentile envelopes of simulated anti-Xa activity over the first week of treatment, stratified by regimen.

sim_summary <- sim |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::group_by(regimen, time) |>
  dplyr::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(sim_summary, aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.2) +
  geom_line(linewidth = 0.6) +
  geom_hline(yintercept = c(0.2, 0.4), linetype = "dashed", colour = "darkred") +
  facet_wrap(~regimen) +
  labs(x = "Time after first enoxaparin dose (h)",
       y = "Anti-Xa activity (IU/mL)",
       title = "Anti-Xa activity simulated VPCs by regimen",
       caption = paste("Dashed lines = target range 0.2-0.4 IU/mL.",
                       "Reproduces the structure of Oualha 2018 Figure 3.")) +
  theme_bw()

Typical anti-Xa profile (Figure 5 base trajectory; standard 50 IU/kg)

Oualha 2018 Figure 5A depicts the predicted anti-Xa trajectory after the standard 50 IU/kg q12h dosing for a 10 kg, 2000 mL PFA patient. The blue solid line in Figure 5A is the deterministic typical-value prediction (no BSV). Reproduced here without the Bayesian-adjustment overlay (the Bayesian overlay requires a per-subject observation at the 4-h time point and is not a property of the structural model).

sim_typical_one <- sim_typical |>
  dplyr::filter(regimen == "10 kg, 50 IU/kg q12h", id == 1)

ggplot(sim_typical_one, aes(time, Cc)) +
  geom_line(colour = "steelblue", linewidth = 0.9) +
  geom_hline(yintercept = c(0.2, 0.4), linetype = "dashed", colour = "darkred") +
  labs(x = "Time after first enoxaparin dose (h)",
       y = "Anti-Xa activity (IU/mL)",
       title = "Typical-value anti-Xa profile (10 kg, PFA 2000 mL, 50 IU/kg q12h)",
       caption = paste("Solid line = typical-value prediction (zeroRe).",
                       "Dashed lines = target range 0.2-0.4 IU/mL.",
                       "Standard regimen of 50 IU/kg q12h matches the",
                       "blue solid line of Oualha 2018 Figure 5A.")) +
  theme_bw()

PKNCA validation

The paper does not tabulate NCA parameters directly, but Table 4 reports the probability that the 4-h-post-dose anti-Xa activity falls in three ranges (<0.2, 0.2-0.4, >0.6 IU/mL) for the standard 50 IU/kg q12h regimen and a Bayesian-adjusted regimen, for a 10 kg / 2000 mL PFA patient. We instead report standard PKNCA steady-state metrics (Cmax,ss, Cmin,ss, AUC0-tau, Cav,ss) per regimen as a generic validation surface.

# Take the last steady-state dosing interval (the 13th dose at t = 156 h
# through t = 168 h) to avoid the initial accumulation transient and the
# slowly-rising end of the BW(t) curve at very late times. Anti-Xa
# concentrations at 1-h resolution within the interval are dense enough
# for Cmax / Cmin / AUC0-tau estimation.
tau <- 12

sim_nca <- sim |>
  dplyr::filter(!is.na(Cc),
                time >= 144, time <= 168) |>
  dplyr::select(id, time, Cc, regimen)

dose_df <- events |>
  dplyr::filter(evid == 1L, time >= 144) |>
  dplyr::select(id, time, amt, regimen)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | regimen + id,
                             concu = "IU/mL", timeu = "h")
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | regimen + id,
                             doseu = "IU")

intervals <- data.frame(
  start    = 156,
  end      = 168,
  cmax     = TRUE,
  tmax     = TRUE,
  cmin     = TRUE,
  auclast  = TRUE,
  cav      = TRUE
)

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- PKNCA::pk.nca(nca_data)
knitr::kable(summary(nca_res),
             caption = paste("Steady-state PKNCA (12-h interval 156-168 h):",
                             "Cmax,ss / Cmin,ss / AUC0-tau / Cav,ss per regimen."))
Steady-state PKNCA (12-h interval 156-168 h): Cmax,ss / Cmin,ss / AUC0-tau / Cav,ss per regimen.
Interval Start Interval End regimen N AUClast (h*IU/mL) Cmax (IU/mL) Cmin (IU/mL) Tmax (h) Cav (IU/mL)
156 168 10 kg, 100 IU/kg q12h 60 3.01 [73.0] 0.420 [74.1] 0.0599 [1560] 2.00 [1.00, 3.00] 0.251 [73.0]
156 168 10 kg, 50 IU/kg q12h 60 1.71 [67.0] 0.286 [69.1] 0.0134 [15700] 2.00 [1.00, 5.00] 0.143 [67.0]
156 168 10 kg, 75 IU/kg q12h 60 2.54 [63.0] 0.347 [67.7] 0.0597 [502] 2.00 [1.00, 3.00] 0.212 [63.0]
156 168 20 kg, 50 IU/kg q12h 60 2.15 [58.9] 0.276 [63.0] 0.0578 [550] 2.00 [1.00, 3.00] 0.179 [58.9]

Comparison against Table 4 of Oualha 2018

Table 4 reports, for a 10 kg / 2000 mL PFA patient on the standard 50 IU/kg q12h regimen, the probability that the 4-h post-dose anti-Xa activity falls in the < 0.2 IU/mL range. We approximate that probability from the simulated cohort and compare.

anti_xa_4h <- sim |>
  dplyr::filter(regimen %in% c("10 kg, 50 IU/kg q12h",
                               "10 kg, 100 IU/kg q12h"),
                time %in% c(4, 16, 28, 40)) |>
  dplyr::group_by(regimen, time) |>
  dplyr::summarise(
    pct_below_02 = mean(Cc < 0.2, na.rm = TRUE) * 100,
    pct_in_target = mean(Cc >= 0.2 & Cc <= 0.4, na.rm = TRUE) * 100,
    pct_above_06 = mean(Cc > 0.6, na.rm = TRUE) * 100,
    median_anti_xa = median(Cc, na.rm = TRUE),
    .groups = "drop"
  )
knitr::kable(anti_xa_4h, digits = 2,
             caption = paste("Probability of anti-Xa activity ranges at",
                             "4 h post-dose at the first four scheduled doses,",
                             "for the 10 kg / PFA = 2000 mL patient.",
                             "Compare to Oualha 2018 Table 4."))
Probability of anti-Xa activity ranges at 4 h post-dose at the first four scheduled doses, for the 10 kg / PFA = 2000 mL patient. Compare to Oualha 2018 Table 4.
regimen time pct_below_02 pct_in_target pct_above_06 median_anti_xa
10 kg, 100 IU/kg q12h 4 46.67 40.00 6.67 0.20
10 kg, 100 IU/kg q12h 16 38.33 41.67 13.33 0.27
10 kg, 100 IU/kg q12h 28 33.33 38.33 13.33 0.29
10 kg, 100 IU/kg q12h 40 28.33 43.33 13.33 0.30
10 kg, 50 IU/kg q12h 4 78.33 18.33 0.00 0.09
10 kg, 50 IU/kg q12h 16 68.33 26.67 0.00 0.13
10 kg, 50 IU/kg q12h 28 63.33 30.00 0.00 0.14
10 kg, 50 IU/kg q12h 40 63.33 30.00 0.00 0.15

The very high BSV on V (omega_V = 1.23) propagates into a wide distribution of simulated Cmax / Cmin / Cav at any time point, consistent with the paper’s central finding that 32% of children never reached the target range and all 22 experienced at least one sub-therapeutic exposure. Quantitative agreement with Table 4’s probabilities depends on the residual error realisation and the Bayesian-adjustment loop the paper applies after the first observed 4-h anti-Xa activity; the structural model alone reproduces the qualitative pattern but is not expected to match Table 4’s exact percentages without that empirical-Bayes update.

Assumptions and deviations

  • BW(t) joint estimation vs forward simulation. The paper jointly fits the BW(t) evolution against per-day measured body weights. In forward simulation the BW(t) curve is a deterministic output of the structural parameters and the per-subject etalfbw / etaltbw50 realisations; the paper’s additive (0.054 kg) and proportional (0.012) BW residuals reported in Table 3 belong to that joint-fit measurement-error model and are not loaded into the nlmixr2lib model file, since they have no role in forward simulation of anti-Xa exposure. A downstream user who wishes to fit the joint BW + anti-Xa data should add a second observation BW ~ add(0.054) + prop(0.012) and an extra bwt derived variable annotated as observable; this is out of scope for the standard model-library entry.
  • Below-LOQ handling. The original Monolix fit M3-method censored 37 of 136 anti-Xa observations below the 0.1 IU/mL limit of quantification. The packaged model does not encode any censoring; forward simulation produces the continuous (uncensored) anti-Xa trajectory. Users who want to reproduce the published M3 likelihood for re-fitting must add the censoring layer manually.
  • Bioavailability F. The paper does not estimate the absolute SC bioavailability of enoxaparin (the literature value is ~92%). The reported CL and V are therefore apparent CL/F and V/F; no lfdepot parameter is added to the model. Forward simulations assume the same apparent-clearance interpretation as the paper.
  • PFA-to-kg conversion (1 mL ~ 1 g). The intra-operative fluid volume PFA enters BW(t) as PFA/1000 (mL to kg) under the conventional fluid density-1 mapping. Packed-RBC (1.08 g/mL) or 5% albumin (1.02 g/mL) contributions to PFA give a small systematic underestimate of the resuscitation mass; this is well within the IIV on lfbw and ltbw50.
  • Time origin for BW(t). The model time t is interpreted as time since the first enoxaparin dose. The paper’s BW(t) curve is defined in the same reference frame (Methods, paragraph 1 of “Pharmacokinetic modelling”). Users who want to start simulation at a different post-operative time must shift the dosing event table accordingly.
  • Cohort representativeness of the vignette. The vignette uses 60 virtual subjects per regimen at fixed BWPREOP and PFA so the dose-finding comparison is interpretable; the original 22-subject cohort covers a wider BWPREOP / PFA range that the user can scan by varying the make_cohort() arguments.
  • Bayesian re-dosing loop. Oualha 2018 Figure 5 and Table 4 depict a Bayesian-updated dosing schedule that uses an observed 4-h anti-Xa value to choose the next dose. The vignette reproduces the structural a-priori forward simulation only; the Bayesian re-dosing loop is application-layer logic outside the structural model and is not encoded in the model file.