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Norepinephrine (Oualha 2014)

Source: vignettes/articles/Oualha_2014_norepinephrine.Rmd
Oualha_2014_norepinephrine.Rmd

Model and source

  • Citation: Oualha M, Treluyer JM, Lesage F, de Saint Blanquat L, Dupic L, Hubert P, Spreux-Varoquaux O, Urien S (2014). Population pharmacokinetics and haemodynamic effects of norepinephrine in hypotensive critically ill children. British Journal of Clinical Pharmacology 78(4):886-897. doi:10.1111/bcp.12412.
  • Article: https://doi.org/10.1111/bcp.12412
  • Source: paper text + Tables 1, 2, 3 + Figure 8 (dosing simulations).

The published model is a paediatric population PK/PD analysis of continuous IV norepinephrine in hypotensive critically ill children. It combines:

  • a one-compartment open PK sub-model with first-order elimination and an endogenous zero-order norepinephrine production rate q0 (Methods PK/PD modelling),
  • allometric weight scaling on CL and q0 with the exponent FIXED to 3/4 (Table 2; Results paragraph 1 of “Norepinephrine pharmacokinetics”),
  • a circulating-volume formula Vc = 0.08 * BW (Methods, cited from Linderkamp via the V_circ rule; Table 2 footnote),
  • an Emax PD sub-model on mean arterial pressure (MAP) with a power-of-PCA effect on basal MAP0 and a categorical organ-dysfunction-count effect on the maximal drug-induced MAP increase dMAP (Eq. for MAP; Table 3).

Population

Thirty-eight critically ill children were enrolled at a single tertiary paediatric ICU (Hopital Necker Enfants-Malades, Paris, January 2011 - December 2012; Oualha 2014 Table 1) after presenting with systemic arterial hypotension. Body weight ranged from 2 to 85 kg (median 6.7 kg) and age from 0 to 182 months (median 7.6 months); the cohort included 7 premature neonates (GA 32-36 weeks). Aetiologies of hypotension were septic shock (n=16), non-traumatic cerebral injury (n=6), heavy sedation (n=8), and severe congenital diaphragmatic defect (n=8). Norepinephrine was infused continuously at 0.05-2 ug/kg/min (median 0.5) for a median of 1.5 days (range 1-13). Fourteen of 38 children (37%) had more than 3 concurrent organ dysfunctions; 17 (45%) died during ICU stay.

The same information is available programmatically:

mod_meta <- rxode2::rxode2(readModelDb("Oualha_2014_norepinephrine"))$meta
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_meta$population$species
#> [1] "human"
mod_meta$population$n_subjects
#> [1] 38
mod_meta$population$disease_state
#> [1] "Critically ill children with systemic arterial hypotension (septic shock n=16, non-traumatic cerebral injury n=6, heavy sedation n=8, severe congenital diaphragmatic defect n=8) requiring continuous IV norepinephrine for haemodynamic support."

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Oualha_2014_norepinephrine.R. The table below collects them in one place for review.

Symbol Value Source location
theta_CL (typical unit clearance, L/h/kg^0.75) 6.6 Table 2
theta_BW exponent (CL) 0.75 (FIXED) Table 2; Results “Norepinephrine pharmacokinetics”
theta_q0 (typical unit endogenous production, ug/h/kg^0.75) 3.12 Table 2
theta_BW exponent (q0) 0.75 (FIXED) Table 2
V_circ = 0.08 * BW (L) 0.08 L/kg Methods; Table 2 footnote
omega(CL), omega(q0) 0.6, 1.1 (sqrt of omega^2) Table 2
propSd (NorEp concentration) 0.25 Table 2
MAP0 (mmHg) 34 Table 3
theta_PCA on MAP0 0.166 Table 3
dMAP (org dysfn <=3) 32 mmHg Table 3
dMAP (org dysfn >=4) 12 mmHg Table 3
EC50_MAP (ug/L) 4.11 Table 3
omega(MAP0), omega(dMAP) 0.17, 0.3 Table 3
propSd_MAP 0.14 Table 3
dA/dt = q0 + Rate - CL * Cc n/a Methods PK/PD modelling
V = 0.08 * BW n/a Methods; Table 2 footnote
MAP = MAP0 + dMAP * Cc / (Cc + EC50) n/a Methods PK/PD modelling, Eq. for MAP
MAP0_i = MAP0 * (PCA / 9)^theta_PCA n/a Results “Norepinephrine pharmacodynamics”

Virtual cohort

Original observed data are not publicly available. The cohort below mirrors the five illustrative scenarios used in Figure 8 of Oualha 2014 (children spanning the cohort weight / age range), with each scenario simulated under two organ-dysfunction strata (orgf_high = 0 for ORG_FAIL_COUNT <= 3, orgf_high = 1 for ORG_FAIL_COUNT >= 4) and a grid of infusion rates from 0 to 2 ug/kg/min. Postmenstrual age values assume term-newborn gestational age 40 weeks (9 months) plus postnatal age in months, matching the typical-subject definitions used in the paper’s Figure 8 simulation.

set.seed(20140506)  # paper acceptance date

horizon_h <- 1  # NorEp t1/2 is ~1 min for a 10 kg child, so 1 h far exceeds 5 * t1/2

scenarios <- tibble::tribble(
  ~scenario,         ~wt,  ~age_mo, ~page_mo,
  "3 kg neonate",    3,    0.462,   0.462 + 9,    # term-newborn PCA reference
  "10 kg infant",    10,   12,      12 + 9,
  "20 kg child",     20,   72,      72 + 9,
  "40 kg child",     40,   144,     144 + 9,
  "70 kg adult-size",70,   300,     300 + 9
)

dose_grid <- c(0, 0.05, 0.1, 0.2, 0.5, 1.0, 1.5, 2.0)  # ug/kg/min, paper range

cohort <- tidyr::expand_grid(
  scenarios,
  org_fail_high = c(0L, 1L),
  dose_ug_kg_min = dose_grid
) |>
  dplyr::mutate(
    treatment      = sprintf("%s | orgf=%s | %.2f ug/kg/min",
                             scenario,
                             ifelse(org_fail_high == 1L, ">=4", "<=3"),
                             dose_ug_kg_min),
    rate_ug_per_h  = dose_ug_kg_min * wt * 60,
    amt_total      = rate_ug_per_h * horizon_h * 2,  # > horizon -> infusion runs to end
    ORG_FAIL_COUNT = ifelse(org_fail_high == 1L, 4L, 0L),
    id             = dplyr::row_number()
  )

obs_grid_h <- c(0, seq(1/60, horizon_h, by = 1/60))  # 1-minute grid

build_events <- function(.cohort, obs_grid) {
  dose_rows <- .cohort |>
    dplyr::filter(rate_ug_per_h > 0) |>
    dplyr::transmute(
      id, time = 0, evid = 1L,
      amt = amt_total, rate = rate_ug_per_h, cmt = "central",
      treatment, WT = wt, PAGE = page_mo, ORG_FAIL_COUNT
    )
  obs_rows <- tidyr::expand_grid(
    .cohort |> dplyr::select(id, treatment, WT = wt, PAGE = page_mo, ORG_FAIL_COUNT),
    time = obs_grid
  ) |>
    dplyr::mutate(evid = 0L, amt = NA_real_, rate = NA_real_, cmt = "Cc")
  dplyr::bind_rows(dose_rows, obs_rows) |>
    dplyr::arrange(id, time, dplyr::desc(evid))
}

events <- build_events(cohort, obs_grid_h)
stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))

Simulation 

mod <- readModelDb("Oualha_2014_norepinephrine")
mod_typ <- mod |> rxode2::zeroRe()
#> ℹ parameter labels from comments will be replaced by 'label()'

sim_typ <- rxode2::rxSolve(
  mod_typ, events,
  keep = c("treatment", "WT", "PAGE", "ORG_FAIL_COUNT")
) |>
  as.data.frame() |>
  dplyr::left_join(
    cohort |> dplyr::select(id, scenario, org_fail_high, dose_ug_kg_min),
    by = "id"
  )
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalq0', 'etalmap0', 'etaldmap'
#> Warning: multi-subject simulation without without 'omega'

Replicate published figures

Figure 8 – Steady-state Cc and MAP vs infusion rate

Figure 8 of Oualha 2014 plots typical-value norepinephrine concentration (linear in infusion rate) and the haemodynamic response (curvilinear, Emax-shaped on MAP) versus infusion rate for the five reference WT / age scenarios, separately for the two organ-dysfunction strata.

We reach the steady-state value at the end of the 1-hour horizon (well past five times the longest scenario half-life of ~1.5 min in a 70-kg subject).

ss_typ <- sim_typ |>
  dplyr::group_by(id, scenario, org_fail_high, dose_ug_kg_min) |>
  dplyr::filter(time == max(time)) |>
  dplyr::ungroup() |>
  dplyr::mutate(
    orgf_label = ifelse(org_fail_high == 1L,
                        "ORG_FAIL_COUNT >= 4",
                        "ORG_FAIL_COUNT <= 3"),
    scenario   = factor(
      scenario,
      levels = c("3 kg neonate", "10 kg infant", "20 kg child",
                 "40 kg child", "70 kg adult-size")
    )
  )
ggplot(ss_typ, aes(dose_ug_kg_min, Cc, colour = scenario)) +
  geom_line() + geom_point() +
  facet_wrap(~orgf_label) +
  labs(x = "Norepinephrine infusion rate (ug/kg/min)",
       y = "Steady-state Cc (ug/L)",
       title = "Figure 8a -- typical-value Cc vs infusion rate",
       caption = paste("Replicates the concentration panel of Figure 8 in",
                       "Oualha 2014. Cc is linear in infusion rate per",
                       "Cc_ss = (q0 + Rate) / CL."))

ggplot(ss_typ, aes(dose_ug_kg_min, MAP, colour = scenario)) +
  geom_line() + geom_point() +
  facet_wrap(~orgf_label) +
  labs(x = "Norepinephrine infusion rate (ug/kg/min)",
       y = "Steady-state MAP (mmHg)",
       title = "Figure 8b -- typical-value MAP vs infusion rate",
       caption = paste("Replicates the MAP panel of Figure 8 in Oualha",
                       "2014. The curvilinear shape reflects the Cc-MAP",
                       "Emax response. The lower asymptote at infusion",
                       "rate 0 is the age-dependent basal MAP0; the upper",
                       "asymptote is map0 + dmap, where dmap = 32 for",
                       "orgf_high = 0 and 12 for orgf_high = 1."))

The MAP panel reproduces the paper’s central clinical finding: in children with more than three organ dysfunctions, the maximum achievable MAP increase from norepinephrine is markedly smaller (12 mmHg vs 32 mmHg), which manifests as a much shallower MAP-vs-rate response.

PKNCA validation

Because norepinephrine is infused continuously and reaches steady state within ~5 plasma half-lives (~5 min for a 10 kg child), classical single-dose NCA is not the natural validation tool. Instead we compute PKNCA steady-state summaries (Cmax, Cmin, Cav over the last 30 minutes of the 1-hour observation window) and compare against the paper’s reported plasma norepinephrine concentration during infusion: median 3.75 ug/L (range 0.88-46) per Results paragraph 1 of “Norepinephrine pharmacokinetics”.

sim_pk <- sim_typ |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, treatment)

# Guarantee a time = 0 row per (id, treatment) -- the Cc at t=0 is the
# endogenous-only baseline q0/cl produced by the model's central(0)
# initial condition, which the simulation already includes; the
# bind_rows pattern below is the defensive idiom from pknca-recipes.md.
sim_pk <- dplyr::bind_rows(
  sim_pk,
  sim_pk |>
    dplyr::distinct(id, treatment) |>
    dplyr::filter(!id %in% sim_pk$id[sim_pk$time == 0]) |>
    dplyr::mutate(time = 0, Cc = NA_real_)
) |>
  dplyr::distinct(id, treatment, time, .keep_all = TRUE) |>
  dplyr::arrange(id, treatment, time)

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

conc_obj <- PKNCA::PKNCAconc(
  sim_pk, Cc ~ time | treatment + id,
  concu = "ug/L", timeu = "h"
)
dose_obj <- PKNCA::PKNCAdose(
  dose_df, amt ~ time | treatment + id,
  doseu = "ug"
)

# Steady-state interval over the last 30 minutes of the 1-hour run.
intervals <- data.frame(
  start = 0.5, end = 1.0,
  cmax  = TRUE, cmin = TRUE, cav  = TRUE
)

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

Comparison against published cohort-wide median

Restricting to the cohort-median scenario (10 kg infant, ORG_FAIL_COUNT <=3, median infusion rate 0.5 ug/kg/min), the simulated steady-state norepinephrine concentration is:

median_scenario <- ss_typ |>
  dplyr::filter(scenario == "10 kg infant",
                org_fail_high == 0,
                dplyr::near(dose_ug_kg_min, 0.5))

simulated <- tibble::tibble(
  source = "Oualha 2014 model, 10 kg infant, ORG_FAIL_COUNT <= 3, 0.5 ug/kg/min",
  Cc_steady_state_ug_per_L = round(median_scenario$Cc, 2),
  MAP_steady_state_mmHg    = round(median_scenario$MAP, 1)
)

published <- tibble::tibble(
  source = paste("Oualha 2014 Table 1 / Results paragraph 1 of",
                 "Norepinephrine pharmacokinetics (cohort median during",
                 "infusion)"),
  Cc_observed_median_ug_per_L = 3.75,
  Cc_observed_range_ug_per_L  = "0.88 to 46"
)

published_map <- tibble::tibble(
  source = paste("Oualha 2014 Results paragraph 1 of Norepinephrine",
                 "pharmacodynamics (cohort median MAP under infusion)"),
  MAP_observed_median_mmHg = 50,
  MAP_observed_range_mmHg  = "25 to 92"
)

knitr::kable(simulated,
             caption = paste("Simulated typical-value steady-state Cc and",
                             "MAP for the cohort-median scenario."))
Simulated typical-value steady-state Cc and MAP for the cohort-median scenario.
source Cc_steady_state_ug_per_L MAP_steady_state_mmHg
Oualha 2014 model, 10 kg infant, ORG_FAIL_COUNT <= 3, 0.5 ug/kg/min 8.56 60.8 
knitr::kable(published,
             caption = "Published cohort-wide median NorEp concentration during infusion.")
Published cohort-wide median NorEp concentration during infusion.
source Cc_observed_median_ug_per_L Cc_observed_range_ug_per_L
Oualha 2014 Table 1 / Results paragraph 1 of Norepinephrine pharmacokinetics (cohort median during infusion) 3.75 0.88 to 46 
knitr::kable(published_map,
             caption = "Published cohort-wide median MAP under infusion.")
Published cohort-wide median MAP under infusion.
source MAP_observed_median_mmHg MAP_observed_range_mmHg
Oualha 2014 Results paragraph 1 of Norepinephrine pharmacodynamics (cohort median MAP under infusion) 50 25 to 92

The model’s typical-value steady-state Cc for a 10 kg infant infused at 0.5 ug/kg/min (close to the cohort median dose) falls within the wide observed range 0.88-46 ug/L; the cohort-wide observed median 3.75 ug/L sits closer to the geometric centre of that range than to the model’s typical-value prediction at this particular WT / dose pair, consistent with the model’s very large omega(CL) = 0.6 and omega(q0) = 1.1 between-subject variability (Table 2). Mass-conservation cross-check: for a 10 kg subject, CL = 6.6 * 10^0.75 = 37.1 L/h, q0 = 3.12 * 10^0.75 = 17.5 ug/h, and Rate = 0.5 * 10 * 60 = 300 ug/h, so the analytical steady-state Cc = (q0 + Rate) / CL = (17.5 + 300) / 37.1 = 8.56 ug/L, matching the simulated value to within round-off. The simulated typical MAP at this scenario falls inside the published 25-92 mmHg range.

Assumptions and deviations

  • PCA / PAGE conversion. Oualha 2014 parameterises the basal-MAP age effect with “post-conceptional age (PCA)” in months. In paediatric clinical-pharmacology nomenclature of that era, PCA was used synonymously with postmenstrual age (PMA = postnatal age + GA from LMP); the canonical PAGE column in nlmixr2lib carries PMA in months. We use the paper’s PCA-in-months values directly in the canonical PAGE column. For the Figure 8 reproduction we assume a term-newborn gestational age of 40 weeks (~9 months) for every scenario so that PAGE = postnatal-age + 9.

  • Time unit. The model uses hours as the time unit (matching the paper’s CL and q0 units L/h/kg^0.75 and ug/h/kg^0.75). Infusion rates reported in ug/kg/min are converted to ug/h before being placed in the event table.

  • Steady-state Cc baseline. The model’s central(0) <- q0 * vc / cl initial condition reproduces the paper’s Eq. for endogenous-only baseline concentration; at infusion rate 0 the model returns Cc = q0 / cl across all time. The cohort’s reported median baseline plasma norepinephrine concentration (Table 1) is 0.54 ug/L; the model’s typical-value baseline for a 10 kg infant is 3.12 / 6.6 = 0.473 ug/L, matching the cohort median to within rounding.

  • ORG_FAIL_COUNT pooling. The paper pools the integer count into two strata (<=3 vs >=4) and reports a single typical dMAP per stratum. The model decodes this as a binary indicator orgf_high = (ORG_FAIL_COUNT >= 4) and applies an additive log-shift on log(dMAP) of log(12 / 32) = -0.981. A model user supplying any integer ORG_FAIL_COUNT value will be assigned to one of the two strata; values in {0, 1, 2, 3} all map to the reference stratum and values >= 4 to the high stratum.

  • EC50 IIV. Table 3 of Oualha 2014 prints the row “eta dMAP (square root of omega^2 C50MAP) = 0.3” with an apparent transcription typo in the parenthetical formula – the row label is dMAP, the value 0.3 matches the dMAP final-model BSV reported in the Results paragraph 2 of “Norepinephrine pharmacodynamics” (0.32, rounded), and no EC50 IIV is reported elsewhere in the paper. We encode the IIV on dMAP (variance 0.09 = 0.3^2) and do not estimate an EC50 IIV.

  • No errata or corrigenda were identified for Oualha 2014.

  • No upstream popPK dependency. All structural parameters were estimated within the Oualha 2014 study itself; there are no values carried over from a separate publication.