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Diamorphine (Cai 2025)

Source: vignettes/articles/Cai_2025_diamorphine.Rmd
Cai_2025_diamorphine.Rmd

Model and source

  • Citation: Cai L, Zhai J, Ji B, Han F, Niu T, Wang L, Wang J. Intranasal diamorphine population pharmacokinetics modeling and simulation in pediatric breakthrough pain. CPT Pharmacometrics Syst Pharmacol. 2025;14(3):435-447. doi:10.1002/psp4.13186.
  • Description: Integrated four-compartment parent-plus-two-metabolite population PK model for diamorphine (heroin) and its sequential deacetylation products 6-monoacetylmorphine (6-MAM) and morphine, after intramuscular or intranasal diamorphine in adult male heroin users (Cai 2025). Diamorphine reaches a one-compartment central pool through a common first-order absorption rate constant shared by both routes; the intramuscular route is the reference (F = 1) and the intranasal route carries an estimated relative bioavailability of about 52% on a logit-normal scale. Diamorphine and 6-MAM have no elimination other than sequential conversion (the paper discards their renal elimination and treats both as fully converted to morphine), so each is described by a single first-order rate constant; the 6-MAM central volume is set equal to the diamorphine central volume because it is not identifiable. Morphine follows a two-compartment disposition with its own central volume and clearance. All states are molar amounts, so the sequential 1:1 deacetylation transfers amounts directly. Every parameter is standardised to 70 kg by theory-based allometry with fixed exponents (0.75 for clearance, 1 for volumes, -0.25 for first-order rate constants), and morphine clearance additionally carries a sigmoid postmenstrual-age maturation function fixed from Holford 2012 (TM50 = 58.1 weeks, Hill = 3.58) that allows extrapolation from the adult fit to children.
  • Article: https://doi.org/10.1002/psp4.13186
  • Supplement (Mlxtran source of the final model, Supplementary Material S1, and the covariate equations, Supplementary Material S2): served with the open-access record as PSP4-14-435-s001.txt and PSP4-14-435-s003.docx via https://www.ebi.ac.uk/europepmc/webservices/rest/PMC11919270/supplementaryFiles

Cai 2025 pools published diamorphine, 6-monoacetylmorphine (6-MAM) and morphine concentrations from two adult studies into a single integrated four-compartment model, then extrapolates it to children in order to test the empirical 0.1 mg/kg intranasal paediatric dose. The structure is a linear chain: diamorphine is absorbed from either an intramuscular or an intranasal depot into a one-compartment central pool, converts to 6-MAM, which converts to morphine, which follows a two-compartment disposition and is the only species with a clearance.

Two features of the encoding are worth stating up front because they are what make the chain work:

  • Everything is molar. The paper converted every concentration from ng/mL to nM so that the two sequential deacetylations transfer amounts 1:1 with no molecular-weight ratio on any arrow. States here are nmol and observations nmol/L, so a dose in milligrams must be converted before use.
  • Diamorphine and 6-MAM have no elimination. The paper discards their renal elimination and treats both as fully converted onward, so their transfer rate constants are their total elimination rate constants and take the canonical kel / kel_6mam names.

Population

The model was estimated on 385 plasma concentrations of diamorphine, 6-MAM and morphine from two NIDA studies (Cone et al. and Skopp et al.; paper Table 1) that shared dosing regimen, study medication, GC-MS assay and sampling times. Ten adult male regular heroin users, opioid-free after at least three consecutive days of negative tests, each received single doses in a double-blind double-dummy crossover with a one-week washout: 6 mg intramuscular (the reference route), and 6 mg and 12 mg intranasal. Weight ranged 60.4-81.4 kg and age 23-41 years; no females were included (paper Table S2 and Discussion).

Between-occasion variability could not be estimated, so each of the 28 treatment sessions was treated as one “modeled subject” and all random effects are between-subject variability on that basis.

Two further studies contributed no parameter estimates and were used only for external verification: Girardin et al. (8 adults, intramuscular 181, 366 and 548 umol) and Kidd et al. (12 children aged 4-13 years, intranasal 0.1 mg/kg). Those are the four groups reproduced below.

The same information is available programmatically via readModelDb("Cai_2025_diamorphine")()$population.

Source trace

Per-parameter origin is recorded next to each ini() entry in inst/modeldb/specificDrugs/Cai_2025_diamorphine.R. Collected here for review. “S1” is Supplementary Material S1 (the Mlxtran listing of the final model) and “S2” is Supplementary Material S2 (the covariate equations).

Equation / parameter Value Source location
lka 3.04 /h Table 2, Ka (RSE 12.9%)
logitfdepot logit(0.519) Table 2, F% (RSE 13.6%)
lvc 8.21 L / 70 kg Table 2, V1 (RSE 28.9%); V2 = V1
lkel 103 /h Table 2, K12 (RSE 23.5%)
lkel_6mam 106 /h Table 2, K23 (RSE 23.3%)
lvc_morphine 32.5 L / 70 kg Table 2, V3 (RSE 13.5%)
lcl_morphine 132 L/h / 70 kg Table 2, CL (RSE 19.0%)
lk12_morphine 24.2 /h Table 2, K3p (RSE 14.1%)
lk21_morphine 2.69 /h Table 2, Kp3 (RSE 14.3%)
e_wt_cl_q 0.75 (fixed) S2 Eq. 1
e_wt_vc_vp 1 (fixed) S2 Eq. 3
rate-constant allometric exponent -0.25 S2 Eq. 2; equals Eq. 1 minus Eq. 3
ltm50_cl 58.1 weeks (fixed) S2 Eq. 5, from Holford 2012
e_age_cl_hill 3.58 (fixed) S2 Eq. 5, from Holford 2012
PMA = 40 + AGE * 52 n/a S2 Eq. 4; also the S1 [COVARIATE] block
maturation 1 / (1 + (TM50/PMA)^Hill) n/a S2 Eq. 5; S1 Maturation = log(1/(1+(58.1/PMA)^3.58))
etalka + etalvc block 0.609, 0.478, r = 0.854 Table 2, BSV column and corr_V1_Ka
etalogitfdepot 0.568 Table 2, BSV F%
etalkel, etalkel_6mam 0.400, 0.295 Table 2, BSV K12, K23
etalvc_morphine, etalcl_morphine 0.279, 0.297 Table 2, BSV V3, CL
etalk12_morphine, etalk21_morphine 0.125, 0.385 Table 2, BSV K3p, Kp3
propSd, propSd_6mam, propSd_morphine 0.430, 0.215, 0.236 Table 2, RUV1-RUV3
two depots, common ka, p = 1 / p = F n/a S1 [LONGITUDINAL] PK: block
d/dt(A1..A4) chain, k30 = CL/V3 n/a S1 EQUATION: block; paper Figure 1b
C2 = A2/V1 (6-MAM uses the diamorphine volume) n/a S1 EQUATION: block

The Table 2 “BSV” entries are Monolix omega standard deviations on the transformed scale, not variances: S1 declares every parameter with sd = omega_*, and Table S1 records that the Monolix default OMEGA initial value of 1 was used. Each ini() variance below is therefore the squared table entry.

mod <- readModelDb("Cai_2025_diamorphine")

# Molecular weight of diamorphine free base, used to convert mg doses to the
# model's nmol dosing unit (paper Methods, "Dataset preparation": 369.4,
# 327.4 and 285.34 g/mol for diamorphine, 6-MAM and morphine).
MW_DIAM <- 369.4
mg_to_nmol <- function(mg) mg / MW_DIAM * 1e6

Structural checks against the paper’s own prose

Three quantities are quoted in the text as consequences of the fitted rate constants. They are deterministic functions of Table 2 and are checked exactly.

tab2 <- c(ka = 3.04, kel = 103, kel_6mam = 106)

# "an approximate half-life of absorption of 13.6 min" (Results)
t_half_abs <- log(2) / tab2[["ka"]] * 60
# "The estimated metabolic conversion half-life for deacetylation was 0.4 min"
t_half_deacet <- log(2) / tab2[["kel"]] * 60
t_half_deacet2 <- log(2) / tab2[["kel_6mam"]] * 60

prose <- tibble::tibble(
  Quantity = c("Absorption half-life (min)",
               "Diamorphine deacetylation half-life (min)",
               "6-MAM deacetylation half-life (min)"),
  Paper = c(13.6, 0.4, 0.4),
  Model = round(c(t_half_abs, t_half_deacet, t_half_deacet2), 3)
)
knitr::kable(prose, caption = "Half-lives quoted in the text versus log(2)/rate from Table 2.")
Half-lives quoted in the text versus log(2)/rate from Table 2.
Quantity Paper Model
Absorption half-life (min) 13.6 13.681
Diamorphine deacetylation half-life (min) 0.4 0.404
6-MAM deacetylation half-life (min) 0.4 0.392 

# Deterministic: these are arithmetic on the published table, so a tight bound
# is correct here (no simulated cohort is involved).
stopifnot(
  abs(t_half_abs - 13.6) < 0.1,
  abs(t_half_deacet - 0.4) < 0.01,
  abs(t_half_deacet2 - 0.4) < 0.01
)

Exact mass-balance validation

Because diamorphine and 6-MAM have no elimination pathway other than conversion, and morphine’s only sink is cl_morphine, the model obeys three exact amount balances at any time T – no extrapolation to infinity and no steady state required:

kel      * vc          * AUC_diam[0,T]  = D*F - depot(T) - central(T)
kel_6mam * vc          * AUC_6mam[0,T]  = D*F - depot(T) - central(T) - central_6mam(T)
cl_morphine            * AUC_mor[0,T]   = D*F - depot(T) - central(T) - central_6mam(T)
                                          - central_morphine(T) - peripheral1_morphine(T)

These are the strongest available gate: they test the ODE topology, both volumes, every rate constant, the clearance and the relative bioavailability simultaneously. The right-hand sides are built from the published Table 2 numbers written literally below, so a mis-transcribed value in the model file makes the check go red rather than cancelling out.

# Typical-value subject at the allometric reference: 70 kg, 30 years old, so
# every allometric term is exactly 1 and the maturation multiplier is ~1.
WT_REF  <- 70
AGE_REF <- 30

dose_mg   <- 6
dose_nmol <- mg_to_nmol(dose_mg)

# Fine early grid (the diamorphine and 6-MAM peaks are minutes wide), then a
# coarser tail. 8 h is ~30 terminal half-lives, long enough for a near-complete
# balance without decaying into solver noise.
tgrid <- sort(unique(c(seq(0, 0.5, by = 0.002), seq(0.5, 8, by = 0.01))))

# The model has three endpoints (Cc, Cc_6mam, Cc_morphine), so rxode2 builds a
# dvid -> cmt map and requires observation rows to carry a `dvid`. `dvid = 1L`
# anchors the timing on the first endpoint; every observable is still returned
# as its own column, so one set of observation rows yields all three analytes.
ev_typ <- bind_rows(
  tibble(id = 1L, time = 0, amt = dose_nmol, evid = 1L, cmt = "depot",
         dvid = NA_integer_, route = "IM 6 mg"),
  tibble(id = 1L, time = tgrid, amt = NA_real_, evid = 0L, cmt = "central",
         dvid = 1L, route = "IM 6 mg"),
  tibble(id = 2L, time = 0, amt = dose_nmol, evid = 1L, cmt = "depot2",
         dvid = NA_integer_, route = "IN 6 mg"),
  tibble(id = 2L, time = tgrid, amt = NA_real_, evid = 0L, cmt = "central",
         dvid = 1L, route = "IN 6 mg")
) |>
  mutate(WT = WT_REF, AGE = AGE_REF) |>
  arrange(id, time, evid)

sim_typ <- rxode2::rxSolve(
  rxode2::zeroRe(mod), events = ev_typ,
  keep = c("route", "WT", "AGE"),
  useLinCmt = FALSE  # ODE -> linCmt auto-conversion breaks multi-output models
) |>
  as.data.frame() |>
  filter(!is.na(Cc))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalka', 'etalvc', 'etalogitfdepot', 'etalkel', 'etalkel_6mam', 'etalvc_morphine', 'etalcl_morphine', 'etalk12_morphine', 'etalk21_morphine'
#> Warning: multi-subject simulation without without 'omega'

# Published values, written literally so a model-file typo cannot hide.
P <- list(ka = 3.04, F = 0.519, vc = 8.21, kel = 103, kel_6mam = 106,
          vc_morphine = 32.5, cl_morphine = 132, tm50 = 58.1, hill = 3.58)
pma_ref <- 40 + AGE_REF * 52
mat_ref <- 1 / (1 + (P$tm50 / pma_ref)^P$hill)

balance <- sim_typ |>
  group_by(route) |>
  filter(time == max(time)) |>
  ungroup() |>
  mutate(
    Fin        = ifelse(route == "IM 6 mg", 1, P$F),
    input      = dose_nmol * Fin,
    outstanding_diam = depot + depot2 + central,
    outstanding_6mam = outstanding_diam + central_6mam,
    outstanding_mor  = outstanding_6mam + central_morphine + peripheral1_morphine
  )

# Trapezoidal AUC of each observable over the same window, from the simulation.
auc_of <- function(df, col) {
  d <- df[order(df$time), ]
  sum(diff(d$time) * (head(d[[col]], -1) + tail(d[[col]], -1)) / 2)
}
aucs <- sim_typ |>
  group_by(route) |>
  summarise(
    auc_diam = auc_of(pick(everything()), "Cc"),
    auc_6mam = auc_of(pick(everything()), "Cc_6mam"),
    auc_mor  = auc_of(pick(everything()), "Cc_morphine"),
    .groups = "drop"
  )

mb <- balance |>
  select(route, input, outstanding_diam, outstanding_6mam, outstanding_mor) |>
  left_join(aucs, by = "route") |>
  mutate(
    elim_diam_pred = P$kel * P$vc * auc_diam,
    elim_diam_obs  = input - outstanding_diam,
    elim_6mam_pred = P$kel_6mam * P$vc * auc_6mam,
    elim_6mam_obs  = input - outstanding_6mam,
    elim_mor_pred  = P$cl_morphine * mat_ref * auc_mor,
    elim_mor_obs   = input - outstanding_mor,
    pct_diam = 100 * (elim_diam_pred - elim_diam_obs) / elim_diam_obs,
    pct_6mam = 100 * (elim_6mam_pred - elim_6mam_obs) / elim_6mam_obs,
    pct_mor  = 100 * (elim_mor_pred  - elim_mor_obs)  / elim_mor_obs
  )

mb |>
  select(route, pct_diam, pct_6mam, pct_mor) |>
  rename("Route" = route,
         "Diamorphine (% diff)" = pct_diam,
         "6-MAM (% diff)"       = pct_6mam,
         "Morphine (% diff)"    = pct_mor) |>
  knitr::kable(digits = 3,
               caption = "Amount-balance identity: rate constant x volume x AUC versus dose in minus amount remaining. Deviations are trapezoidal-integration error only.")
Amount-balance identity: rate constant x volume x AUC versus dose in minus amount remaining. Deviations are trapezoidal-integration error only.
Route Diamorphine (% diff) 6-MAM (% diff) Morphine (% diff)
IM 6 mg -0.009 0.002 0
IN 6 mg -0.009 0.002 0 

# Deterministic identity on a typical-value solve, so the only error source is
# the trapezoidal rule on the observation grid and the bound is tightened to
# the accuracy actually achieved (realised: 0.009% / 0.002% / 0.000%). A
# mis-transcribed rate constant, volume, clearance or F moves these by tens of
# percent.
stopifnot(
  max(abs(mb$pct_diam)) < 0.05,
  max(abs(mb$pct_6mam)) < 0.05,
  max(abs(mb$pct_mor))  < 0.05
)

Relative bioavailability

The intranasal-to-intramuscular exposure ratio must reproduce the estimated F% exactly, for every analyte, because the two routes differ only by the amount entering the identical downstream chain.

frel <- aucs |>
  tidyr::pivot_longer(-route, names_to = "analyte", values_to = "auc") |>
  tidyr::pivot_wider(names_from = route, values_from = auc) |>
  mutate(ratio = `IN 6 mg` / `IM 6 mg`)

frel |>
  rename("Analyte" = analyte, "AUC IM" = `IM 6 mg`,
         "AUC IN" = `IN 6 mg`, "IN / IM" = ratio) |>
  knitr::kable(digits = 4,
               caption = "Intranasal / intramuscular AUC ratio; the paper's F% is 0.519 (Table 2).")
Intranasal / intramuscular AUC ratio; the paper’s F% is 0.519 (Table 2).
Analyte AUC IM AUC IN IN / IM
auc_diam 19.2060 9.9679 0.519
auc_6mam 18.6643 9.6868 0.519
auc_mor 115.9736 60.1903 0.519 

stopifnot(max(abs(frel$ratio - P$F)) < 1e-6)

External verification (paper Table 3)

The paper’s Table 3 compares observed and predicted morphine PK in four external groups: three adult intramuscular dose levels from Girardin et al. and one paediatric intranasal group from Kidd et al. Virtual subjects are drawn over the same weight and age ranges the paper used.

Note on units: Table 3 prints the observed Cmax column in umol/L and the predicted column header as ng/mL, but the predicted/observed ratios in the same row only reconcile if both are umol/L – the “ng/mL” header is a typo. Values here are converted to nmol/L (1 umol/L = 1000 nmol/L).

# set.seed() seeds R's RNG, not rxode2's; rxode2 partitions its streams per
# solver thread, so CI draws a different cohort than this machine does. Every
# assertion below is written to hold for any cohort the model can produce.
set.seed(20250907)

N_ARM <- 100

# Weights are drawn from a log-normal truncated to the study's reported range
# and centred on its reported median, rather than uniformly over the range: the
# paper reports weight as "median [range]" (Table 1), and a uniform draw over
# 19-59 kg would put the cohort median at 39 kg where Kidd et al. report 30.1 kg,
# systematically over-sizing the children and depressing every concentration.
# The spread is set so the reported range spans about four SDs on the log scale.
rtrunc_lnorm <- function(n, med, lo, hi) {
  sdlog <- (log(hi) - log(lo)) / 4
  qlnorm(runif(n, plnorm(lo, log(med), sdlog), plnorm(hi, log(med), sdlog)),
         log(med), sdlog)
}

make_arm <- function(n, wt_med, wt_lo, wt_hi, age_lo, age_hi, dose_nmol_fun,
                     cmt_dose, label, tmax_obs, dt, id_offset) {
  subj <- tibble(
    id  = id_offset + seq_len(n),
    WT  = rtrunc_lnorm(n, wt_med, wt_lo, wt_hi),
    AGE = runif(n, age_lo, age_hi),
    treatment = label
  ) |>
    mutate(dose_nmol = dose_nmol_fun(WT))

  doses <- subj |>
    transmute(id, WT, AGE, treatment, time = 0,
              amt = dose_nmol, evid = 1L, cmt = cmt_dose, dvid = NA_integer_)
  obs <- subj |>
    select(id, WT, AGE, treatment) |>
    tidyr::crossing(time = seq(0, tmax_obs, by = dt)) |>
    mutate(amt = NA_real_, evid = 0L, cmt = "central", dvid = 1L)

  bind_rows(doses, obs) |> arrange(id, time, evid)
}

# Girardin adults: 43-85 kg, 24-39 years, IM 181 / 366 / 548 umol.
adult_arms <- list(
  list(lab = "1: 181 umol IM", d = 181e3, off =    0L),
  list(lab = "2: 366 umol IM", d = 366e3, off =  100L),
  list(lab = "3: 548 umol IM", d = 548e3, off =  200L)
)
events_adult <- bind_rows(lapply(adult_arms, function(a) {
  make_arm(N_ARM, wt_med = 67, wt_lo = 43, wt_hi = 85,
           age_lo = 24, age_hi = 39,
           dose_nmol_fun = function(wt) rep(a$d, length(wt)),
           cmt_dose = "depot", label = a$lab,
           tmax_obs = 6, dt = 0.02, id_offset = a$off)
}))

# Kidd children: median 30.1 kg [19-59], 4-13 years, IN 0.1 mg/kg, within 1 h.
events_child <- make_arm(
  N_ARM, wt_med = 30.1, wt_lo = 19, wt_hi = 59,
  age_lo = 4, age_hi = 13,
  dose_nmol_fun = function(wt) mg_to_nmol(0.1 * wt),
  cmt_dose = "depot2", label = "4: 0.1 mg/kg IN",
  tmax_obs = 1, dt = 0.005, id_offset = 300L
)

events_ext <- bind_rows(events_adult, events_child)
stopifnot(!anyDuplicated(unique(events_ext[, c("id", "time", "evid")])))
sim_ext <- rxode2::rxSolve(
  mod, events = events_ext,
  keep = c("treatment", "WT", "AGE"),
  useLinCmt = FALSE
) |>
  as.data.frame()
#> ℹ parameter labels from comments will be replaced by 'label()'

# Solver noise in the far tail can push a decayed concentration slightly
# negative, which would make PKNCA's log-linear steps produce NaN.
stopifnot(all(sim_ext$Cc_morphine > -1e-8, na.rm = TRUE))
# Replicates Figure 3b of Cai 2025: simulated morphine concentration-time
# profile in children after a single 0.1 mg/kg intranasal dose, against the
# observed group mean of Kidd et al.
kidd_obs <- tibble(time = 1, Cc_morphine = 36.1)  # Table 3 group 4 Cmax, umol/L -> nmol/L

sim_ext |>
  filter(treatment == "4: 0.1 mg/kg IN", !is.na(Cc_morphine)) |>
  group_by(time) |>
  summarise(Q05 = quantile(Cc_morphine, 0.05),
            Q50 = quantile(Cc_morphine, 0.50),
            Q95 = quantile(Cc_morphine, 0.95),
            .groups = "drop") |>
  ggplot(aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25) +
  geom_line(colour = "red") +
  geom_hline(yintercept = c(13.7, 82.7), linetype = "dashed") +
  labs(x = "Time (h)", y = "Morphine (nmol/L)",
       title = "Figure 3b - morphine after 0.1 mg/kg intranasal diamorphine",
       caption = paste("Replicates Figure 3b of Cai 2025. Dashed lines are the",
                       "reported Cmax range 13.7-82.7 nmol/L (Table S3)."))

nca_frame <- sim_ext |>
  filter(!is.na(Cc_morphine)) |>
  transmute(id, time, treatment, Cc = Cc_morphine)

# Guarantee a time = 0 row per subject; pre-dose morphine is 0 for both
# extravascular routes. Filter is !is.na() only -- adding time > 0 or Cc > 0
# would drop this anchor row.
nca_frame <- bind_rows(
  nca_frame,
  nca_frame |> distinct(id, treatment) |> mutate(time = 0, Cc = 0)
) |>
  distinct(id, treatment, time, .keep_all = TRUE) |>
  arrange(id, treatment, time)

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

dose_df <- events_ext |>
  filter(evid == 1) |>
  select(id, time, amt, treatment)
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id)

# One interval per group, over that group's own observation window. Carrying
# the grouping column in the interval frame keeps each request scoped to its
# group; a single ungrouped 0-6 h interval would ask for six hours of data the
# paediatric arm does not have. AUC is requested only for the paediatric group,
# the one whose window (0-1 h) the paper actually reports.
intervals <- data.frame(
  treatment = c("1: 181 umol IM", "2: 366 umol IM",
                "3: 548 umol IM", "4: 0.1 mg/kg IN"),
  start     = 0,
  end       = c(6, 6, 6, 1),
  cmax      = TRUE,
  tmax      = TRUE,
  auclast   = c(FALSE, FALSE, FALSE, TRUE),
  stringsAsFactors = FALSE
)

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

Comparison against published NCA 

# Table 3 observed morphine Cmax, umol/L converted to nmol/L.
published <- tibble::tribble(
  ~treatment,          ~cmax,
  "1: 181 umol IM",    1100,
  "2: 366 umol IM",    1700,
  "3: 548 umol IM",    1700,
  "4: 0.1 mg/kg IN",   36.1
)

cmp <- nlmixr2lib::ncaComparisonTable(
  simulated     = nca_res,
  reference     = published,
  by            = "treatment",
  units         = c(cmax = "nmol/L"),
  tolerance_pct = 20
)

knitr::kable(
  cmp,
  caption = paste("Simulated versus published morphine Cmax (Cai 2025 Table 3).",
                  "* differs from the reference by more than 20%."),
  align = c("l", "l", "r", "r", "r")
)
Simulated versus published morphine Cmax (Cai 2025 Table 3). * differs from the reference by more than 20%.
NCA parameter treatment Reference Simulated % diff
Cmax (nmol/L) 1: 181 umol IM 1100 637 -42.1%*
Cmax (nmol/L) 2: 366 umol IM 1700 1250 -26.4%*
Cmax (nmol/L) 3: 548 umol IM 1700 1960 +15.5%
Cmax (nmol/L) 4: 0.1 mg/kg IN 36.1 26.6 -26.3%*

The paper’s own predicted/observed ratios for these four groups are 0.69, 0.91, 1.36 and 1.21 (Table 3), i.e. its own simulation misses the observed group means by up to 36% while still falling inside the twofold band it adopted as the acceptance criterion. A reproduction of that model therefore cannot be held to a tighter standard than the paper achieves against the same data, so the gate below asserts the twofold criterion the paper actually used, applied to the group medians.

sim_cmax <- as.data.frame(nca_res) |>
  filter(PPTESTCD == "cmax") |>
  group_by(treatment) |>
  summarise(cmax_sim = median(PPORRES), .groups = "drop") |>
  left_join(published, by = "treatment") |>
  mutate(fold = cmax_sim / cmax)

# Guard: the filter must have found every group, or the gate below would pass
# vacuously on an empty frame (all(logical(0)) is TRUE).
stopifnot(nrow(sim_cmax) == nrow(published), !anyNA(sim_cmax$fold))

paper_fold <- c(0.69, 0.91, 1.36, 1.21)  # Table 3, predicted/observed column

sim_cmax |>
  mutate(paper = paper_fold) |>
  rename("Group" = treatment, "Simulated Cmax (nmol/L)" = cmax_sim,
         "Observed Cmax (nmol/L)" = cmax,
         "This reproduction: pred/obs" = fold,
         "Paper Table 3: pred/obs" = paper) |>
  knitr::kable(digits = c(0, 1, 1, 2, 2),
               caption = "Median simulated morphine Cmax against Table 3 observed means, alongside the paper's own predicted/observed ratios.")
Median simulated morphine Cmax against Table 3 observed means, alongside the paper’s own predicted/observed ratios.
Group Simulated Cmax (nmol/L) Observed Cmax (nmol/L) This reproduction: pred/obs Paper Table 3: pred/obs
1: 181 umol IM 637.1 1100.0 0.58 0.69
2: 366 umol IM 1250.5 1700.0 0.74 0.91
3: 548 umol IM 1963.8 1700.0 1.16 1.36
4: 0.1 mg/kg IN 26.6 36.1 0.74 1.21 

# Weighted average fold error, the paper's own summary statistic (its value for
# Cmax is 1.06). Computed as a geometric mean so that over- and
# under-prediction are treated symmetrically.
afe <- exp(mean(log(sim_cmax$fold)))

# Two gates. The per-group bound is deliberately wider than the paper's 0.5-2
# criterion because these are cohort medians: the eta draw varies with the
# solver thread count, so a bound sitting just outside one observed run would
# flicker in CI. It is still far tighter than any transcription error would
# survive -- a wrong molecular weight, salt form, volume or dose unit moves
# Cmax by two-fold or more. Realised folds were 0.58 / 0.74 / 1.16 / 0.74 with
# AFE 0.78 (they were 0.61 / 0.75 / 1.18 / 0.74 with the earlier uniform weight
# draw, so the bound is not sensitive to that choice). The lowest, 0.58, sits
# well clear of the 0.4 bound.
stopifnot(
  all(sim_cmax$fold > 0.4 & sim_cmax$fold < 2.5),
  afe > 0.5, afe < 2
)

Paediatric AUC endpoint

For the paediatric group the paper defines an explicit window – morphine AUC over the first hour – and reports the observed range as 10.13-58.36 nmol*h/L (Table S3, endpoint 1). The adult groups’ AUC0-t windows are not reported, so only this group can be compared on exposure.

auc_child <- as.data.frame(nca_res) |>
  filter(PPTESTCD == "auclast", treatment == "4: 0.1 mg/kg IN") |>
  pull(PPORRES)

stopifnot(length(auc_child) == N_ARM)  # guard: the filter must have found rows

pct_in_window <- 100 * mean(auc_child >= 10.13 & auc_child <= 58.36)

tibble(
  Quantity = c("Median simulated AUC0-1h (nmol*h/L)",
               "Observed AUC0-1h from Kidd et al. (nmol*h/L)",
               "Percent of virtual children inside the reported window",
               "Percent reported by the paper (Table S3, endpoint 1)"),
  Value = c(round(median(auc_child), 1), 29.9, round(pct_in_window, 1), 82.6)
) |>
  knitr::kable(caption = "Morphine AUC over the first hour after 0.1 mg/kg intranasal diamorphine.")
Morphine AUC over the first hour after 0.1 mg/kg intranasal diamorphine.
Quantity Value
Median simulated AUC0-1h (nmol*h/L) 17.6
Observed AUC0-1h from Kidd et al. (nmol*h/L) 29.9
Percent of virtual children inside the reported window 95.0
Percent reported by the paper (Table S3, endpoint 1) 82.6 

# Cohort-derived, so bounds are robust rather than tight. The paper reports
# 82.6% of its virtual children inside the window; a reproduction that put
# fewer than half of them there would indicate a structural error.
#
# The median simulated AUC0-1h (about 18 nmol*h/L) sits below the observed
# 29.9, a fold error of roughly 0.6 against the paper's own 0.87 for this
# group. Both under-predict, and both stay inside the paper's twofold
# acceptance criterion; the paper attributes its own uniform AUC
# under-prediction to full extrapolation from adults, a single paediatric
# dataset, and three different bioanalytical methods across the studies
# (Discussion). Part of the residual gap here is the median-versus-mean
# difference noted in Assumptions.
stopifnot(
  median(auc_child) > 10.13, median(auc_child) < 58.36,
  pct_in_window > 50
)

Paediatric dosing simulation (paper Figure 4)

The paper’s conclusion is that a 0.3 mg/kg intranasal loading dose followed by 0.1 mg/kg hourly holds median morphine concentrations inside the 35-70 nmol/L steady-state target across four FDA paediatric age bands. Representative weights are the Table S3 medians.

set.seed(20250908)

peds <- tibble::tribble(
  ~band,         ~wt_lo, ~wt_hi, ~age_lo,  ~age_hi,
  "Neonates",      3.07,   3.86,  0.0,     0.083,
  "Infants",       7.83,  12.30,  0.083,   2.0,
  "Children",     17.70,  28.60,  2.0,    12.0,
  "Adolescents",  39.70,  49.90, 12.0,    16.0
)

make_peds <- function(row, id_offset) {
  subj <- tibble(
    id  = id_offset + seq_len(N_ARM),
    WT  = runif(N_ARM, row$wt_lo, row$wt_hi),
    AGE = runif(N_ARM, row$age_lo, row$age_hi),
    band = row$band
  )
  dose_times <- 0:5
  doses <- subj |>
    tidyr::crossing(time = dose_times) |>
    mutate(mgkg = ifelse(time == 0, 0.3, 0.1),
           amt = mg_to_nmol(mgkg * WT), evid = 1L, cmt = "depot2",
           dvid = NA_integer_) |>
    select(id, WT, AGE, band, time, amt, evid, cmt, dvid)
  obs <- subj |>
    tidyr::crossing(time = seq(0, 6, by = 0.02)) |>
    mutate(amt = NA_real_, evid = 0L, cmt = "central", dvid = 1L)
  bind_rows(doses, obs) |> arrange(id, time, evid)
}

events_peds <- bind_rows(lapply(seq_len(nrow(peds)), function(i) {
  make_peds(peds[i, ], id_offset = (i - 1L) * 1000L)
}))
stopifnot(!anyDuplicated(unique(events_peds[, c("id", "time", "evid")])))

sim_peds <- rxode2::rxSolve(
  mod, events = events_peds,
  keep = c("band", "WT", "AGE"),
  useLinCmt = FALSE
) |>
  as.data.frame() |>
  mutate(band = factor(band, levels = peds$band))
# Replicates Figure 4 of Cai 2025: median morphine concentration-time profiles
# for four paediatric age groups on 0.3 mg/kg loading + 0.1 mg/kg hourly
# intranasal diamorphine, against the 35-70 nmol/L steady-state target.
sim_peds |>
  filter(!is.na(Cc_morphine)) |>
  group_by(band, time) |>
  summarise(Q05 = quantile(Cc_morphine, 0.05),
            Q50 = quantile(Cc_morphine, 0.50),
            Q95 = quantile(Cc_morphine, 0.95),
            .groups = "drop") |>
  ggplot(aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.2) +
  geom_line() +
  geom_hline(yintercept = c(35, 70), colour = "red") +
  facet_wrap(~band) +
  labs(x = "Time (h)", y = "Morphine (nmol/L)",
       title = "Figure 4 - paediatric intranasal dosing simulation",
       caption = paste("Replicates Figure 4 of Cai 2025. Red lines are the",
                       "35-70 nmol/L steady-state target."))

# The paper's claim concerns the MEDIAN profile at steady state. Use the last
# dosing interval (5-6 h) and summarise the median profile over it.
ss <- sim_peds |>
  filter(!is.na(Cc_morphine), time >= 5, time <= 6) |>
  group_by(band, time) |>
  summarise(median_c = median(Cc_morphine), .groups = "drop") |>
  group_by(band) |>
  summarise(median_min = min(median_c),
            median_max = max(median_c),
            median_avg = mean(median_c), .groups = "drop")

ss <- ss |>
  mutate(in_target = median_avg >= 35 & median_avg <= 70)

ss |>
  rename("Age band" = band,
         "Median trough (nmol/L)" = median_min,
         "Median peak (nmol/L)"   = median_max,
         "Median average (nmol/L)" = median_avg,
         "Inside 35-70 nmol/L" = in_target) |>
  knitr::kable(digits = 1,
               caption = "Median morphine concentration over the last dosing interval (5-6 h), by age band.")
Median morphine concentration over the last dosing interval (5-6 h), by age band.
Age band Median trough (nmol/L) Median peak (nmol/L) Median average (nmol/L) Inside 35-70 nmol/L
Neonates 93.6 121.0 107.8 FALSE
Infants 42.3 70.6 55.1 TRUE
Children 41.7 69.1 53.0 TRUE
Adolescents 46.0 74.9 58.5 TRUE

The infant, child and adolescent bands reproduce the paper’s claim: their median profiles over the last dosing interval sit inside the 35-70 nmol/L target. That is what the weight-proportional dose plus allometric clearance is designed to deliver, and it is the paper’s central result.

Documented deviation - the neonatal band. This reproduction puts the neonatal median profile at roughly twice the top of the target band, not inside it as Figure 4 of the paper shows. The mechanism is visible in the model: at a postmenstrual age of about 42 weeks the maturation multiplier 1 / (1 + (58.1/PMA)^3.58) is near 0.24, so a neonate clears morphine at about a quarter of the weight-adjusted mature rate, while the dose stays strictly proportional to weight. Weight-proportional dosing with less-than-proportional clearance necessarily over-exposes the youngest band. The gate below therefore asserts the paper’s target only for the three bands that reproduce it, and records the neonatal result rather than widening the band until it passes.

This deviation does not contradict the paper’s own conclusions so much as sharpen them: the Discussion states that “caution is still advised when considering neonates and infants” and that “the intranasal dosage prediction for neonates and infants remains speculative”, and the paper notes no clinical data exist to validate the model in these bands.

# Cohort-derived quantities, so the bounds are magnitude bounds rather than
# tight ones, and none is taken from a single observed run.
target <- ss |> filter(band != "Neonates")
stopifnot(
  # The paper's claim, asserted where it reproduces.
  all(target$in_target),
  # The neonatal band is over-exposed rather than merely noisy: assert the
  # direction of the documented deviation so that a future change which
  # silently "fixes" it is noticed rather than passing unremarked.
  ss$median_avg[ss$band == "Neonates"] > 70,
  # Nothing anywhere should be at a physically implausible level; a wrong dose
  # unit or molecular weight moves these by orders of magnitude.
  all(ss$median_avg > 10), all(ss$median_avg < 500)
)

Assumptions and deviations

  • Dose unit and salt form. Doses are converted from milligrams to nmol with the diamorphine free-base molecular weight of 369.4 g/mol, the value the paper gives in Methods for the concentration conversion. The study medication is diamorphine hydrochloride, and the paper does not state whether its milligram doses refer to salt or base; a salt-based reading would scale every simulated concentration here by about 0.92. Table 3’s dose column is already molar (181, 366, 548 umol) so the three adult groups are unaffected by this choice; only the 0.1 mg/kg paediatric arms depend on it.
  • Table 3 unit typo. Table 3 labels the observed morphine Cmax column umol/L and the predicted column ng/mL. Only the umol/L reading reconciles with the predicted/observed ratios printed in the same rows, so both are treated as umol/L. The same applies to the AUC0-t columns, labelled ngmin/mL; the group 4 value of 1.794 corresponds to 29.9 nmolh/L, which sits inside the paper’s own 10.13-58.36 nmol*h/L AUC0-1h window and confirms the molar reading.
  • Covariate distributions. Age is drawn uniformly over the range the paper states for each verification cohort; weight is drawn from a log-normal truncated to the stated range and centred on the stated median (adults 67 kg [43-85]; children 30.1 kg [19-59]; Table 1). A uniform draw over the weight range was tried first and is wrong for this purpose: it puts the paediatric cohort median at 39 kg where Kidd et al. report 30.1 kg, which over-sizes the children and depresses every simulated concentration. The paper generated its virtual subjects in the Simcyp paediatric module, which correlates weight with age; independent draws here produce a wider covariate spread, widening the simulated percentiles but not the medians.
  • Medians versus means. Table 3 reports the paper’s predictions as means across simulated trials, whereas the comparison here pools by median (the convention of nlmixr2lib::ncaComparisonTable()). For the log-normally distributed exposures this model produces, the mean sits above the median by roughly exp(sigma^2/2), of order 10%, so this reproduction is expected to read slightly lower than the paper’s own predicted column even where the underlying model agrees exactly.
  • Rate-constant allometric exponent. Supplementary Material S2 Eq. 2 states the exponent as -0.25. The model file derives it as e_wt_cl_q - e_wt_vc_vp (0.75 - 1), which is numerically identical and keeps the three theory-based exponents mutually consistent rather than declaring the same constant a third time.
  • Maturation form. S2 Eq. 5 prints the multiplier as 1 / (1 + (PMA/TM50)^-Hill) while the S1 Mlxtran code writes 1 / (1 + (TM50/PMA)^Hill). These are the same function; the Mlxtran form is used. TM50 and Hill are fixed literature values from Holford 2012 and carry no uncertainty in the paper.
  • fixed() encoding. The two allometric exponents and the two maturation constants are wrapped in fixed(); everything else in Table 2 is an estimated value with a reported RSE.
  • No between-occasion variability. The paper could not estimate BOV and instead treated each of the 28 treatment sessions as an independent subject. The packaged model therefore carries only between-subject variability, which will overstate within-subject reproducibility if used to simulate a crossover.
  • Females. The development data are all male; the paper flags in its Discussion that the absence of a sex effect is an untested assumption when extrapolating to a paediatric population that was 67% female.
  • Neonates and infants. The paper states plainly that no clinical data validate the model in these two bands and that its dose predictions there are speculative. The Figure 4 reproduction above inherits that limitation, and additionally does not reproduce the paper’s claim for the neonatal band: the median neonatal profile here sits at roughly twice the top of the 35-70 nmol/L target rather than inside it, because the maturation function cuts morphine clearance to about a quarter of the mature weight-adjusted value at 42 weeks postmenstrual age while the dose remains proportional to weight. This is recorded as a deviation and excluded from the gate rather than absorbed by widening the target band. The infant, child and adolescent bands do reproduce the paper’s claim.
  • New canonical name. The 6-MAM metabolite suffix 6mam is introduced by this extraction and registered in inst/references/compartment-names.md alongside the already-registered morphine suffix.