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Soybean oil-based lipid emulsion (Fukushima 2022)

Source: vignettes/articles/Fukushima_2022_soybeanOilLipidEmulsion.Rmd
Fukushima_2022_soybeanOilLipidEmulsion.Rmd

Model and source

  • Citation: Fukushima K, Omura K, Goshi S, Okada A, Tanaka M, Tsujimoto T, Iriyama K, Sugioka N. Individualization of the infusion rate of a soybean oil-based intravenous lipid emulsion for inpatients, based on baseline triglyceride concentrations: A population pharmacokinetic approach. JPEN J Parenter Enteral Nutr. 2022;46:104-113. doi:10.1002/jpen.2111.
  • Description: One-compartment plasma triglyceride (TG) kinetic model for a soybean oil-based intravenous lipid emulsion (Intralipos 20%) in adult Japanese inpatients receiving parenteral nutrition: zero-order endogenous TG production (EndVLDL) plus the zero-order emulsion infusion into a plasma volume fixed at 0.422 dL/kg, with first-order elimination whose rate constant falls with baseline TG and body weight (Fukushima 2022).
  • Article (open access): https://doi.org/10.1002/jpen.2111 (PMC9290941); the Supplementary Materials document is distributed with the article on Europe PMC.

Fukushima et al. describe plasma triglyceride (TG) during and after an infusion of a 20% soybean oil-based intravenous lipid emulsion (SO-ILE, Intralipos 20%) with a single plasma compartment (Eq 1):

dAdt=EndVLDL−A⋅Kel+InfRate,C=A/V\frac{dA}{dt} = \mathrm{EndVLDL} - A \cdot K_{el} + \mathrm{InfRate}, \qquad C = A / V

where EndVLDL is the zero-order endogenous TG (VLDL) production rate, Kel the first-order elimination rate constant, InfRate the emulsion infusion rate and V the plasma volume, fixed at 0.422 dL/kg. In the model file EndVLDL is kin, Kel is kel and V is vc; the infusion is an ordinary zero-order infusion (amt and rate) into central, with amt in mg of TG (1 g of fat = 1000 mg of TG). The paper uses the model to derive an individual maximum acceptable infusion rate (Eq 7) and a nomogram (Figure 2).

Population

Eighty-three adult Japanese inpatients (44 male / 39 female) receiving peripheral (n = 75) or central venous (n = 8) parenteral nutrition at two hospitals in Japan (October 2016 - March 2018) were analysed (Table 1). The median age was 74 years (range 20-95) and median body weight 55.8 kg (31.2-86.1). Most had acute inflammatory gastrointestinal disease (colon diverticulitis, acute cholecystitis, peptic ulcer, ischemic colitis, colorectal cancer, …) with low serum albumin (median 3.52 g/dL) and raised CRP (median 1.56 mg/dL). Baseline TG had a median of 94 mg/dL (30-291); patients with baseline TG above 300 mg/dL were excluded. Each patient received one SO-ILE infusion of 20-120 g fat (median 50 g) at 0.053-0.363 g/kg/h (median 0.217) for 0.75-19.8 h (median 5.2 h). Two TG samples were drawn during the infusion (the first within 0-1, 1-2 or 2-3 h by randomised group, the second at the end of infusion), and 72 next-morning routine samples were added, giving 238 concentrations.

The same information is available programmatically via readModelDb("Fukushima_2022_soybeanOilLipidEmulsion")()$population.

Source trace

Equation / parameter Value Source location
d/dt(central) <- kin - kel * central (+ infusion) n/a Eq 1; Supplementary Scheme 2, Model 3
Cc <- central / vc n/a Methods, ‘C = A / V’
central(0) <- TRIG * vc n/a Not printed; established from Supplementary Figure S7 (see below)
vc = 0.422 * WT (lvc, fixed) 0.422 dL/kg Table 4 final model structure; Methods (fixed from healthy-subject literature)
lkin (EndVLDL, theta1) 5709 mg/h Table 4
lkel (Kel, theta2) 2.51 1/h Table 4
e_trig_kel (theta3) -0.833 Table 4; reference TG 94 mg/dL (Table 4 note)
e_wt_kel (theta4) -1.27 Table 4; reference WT 55.8 kg (Table 4 note)
etalkel 0.433^2 = 0.1875 Table 4 ‘omega Kel 43.3%’, Eq 2 (exponential IIV)
propSd 0.318 Table 4 ‘delta 31.8%’, Eq 3 ‘CObs = C x (1 + eps)’
No IIV on EndVLDL or V n/a Table 3 model 9; Supplementary ‘Model Selection’
Eq 6, Cmax = (EndVLDL + InfRate) / (V x Kel) n/a Methods, Eq 6
Eq 7, maximum acceptable infusion rate n/a Results, Eq 7 (target mean Cmax 211 mg/dL = 400 / 1.894)

Typical-patient profile and the initial condition (Supplementary Figure S7)

The paper does not print the initial condition of Eq 1. Two choices are natural: start at the measured baseline, A(0) = TGbase x V, or start at the model’s own steady state, A(0) = EndVLDL / Kel. For the typical patient they coincide to within 3% (94 vs 96.6 mg/dL), but they spread differently across patients: the steady-state start inherits the full between-patient variability of Kel, the baseline start does not.

Supplementary Figure S7 shows the median and 90th percentile of 1,000 simulated profiles for a typical patient (TGbase 94 mg/dL, 55.8 kg) given 0.121 g/kg/h for 6 h, with the residual error included. At the first plotted time (about 0.1 h), the 90th-percentile curve for V = 0.422 x BW starts at about 165 mg/dL (all five plasma-volume variants start between about 155 and 185 mg/dL). The percentiles below are computed without Monte Carlo noise. The typical-value model is solved on a deterministic grid of 200 quantiles of etalkel, and the proportional residual error is then integrated out analytically.

mod <- readModelDb("Fukushima_2022_soybeanOilLipidEmulsion")
mod_ui <- rxode2::rxode2(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_typ <- rxode2::zeroRe(mod_ui)

ini_df <- mod_ui$iniDf
omega_kel <- sqrt(ini_df$est[ini_df$name == "etalkel"])
prop_sd <- ini_df$est[ini_df$name == "propSd"]
lkel_typ <- ini_df$est[ini_df$name == "lkel"]

# Deterministic quadrature grid over the between-patient variability of kel
eta_grid <- omega_kel * qnorm(ppoints(200))

wt_typ <- 55.8
tg_typ <- 94
rate_s7 <- 0.121 * wt_typ * 1000 # mg TG / h
t_obs <- c(0, 0.1, 0.25, 0.5, 1, 2, 3, 4, 5, 6, 6.25, 6.5, 7, 7.5, 8, 9)

make_grid_events <- function(rate_mg_h, dur_h, times, tg, wt, n) {
  dose <- data.frame(
    id = seq_len(n), time = 0, evid = 1L, amt = rate_mg_h * dur_h,
    rate = rate_mg_h, cmt = "central"
  )
  obs <- expand.grid(id = seq_len(n), time = times) |>
    dplyr::mutate(evid = 0L, amt = 0, rate = 0, cmt = "central")
  dplyr::bind_rows(dose, obs) |>
    dplyr::mutate(TRIG = tg, WT = wt) |>
    dplyr::arrange(id, time, dplyr::desc(evid))
}

ev_grid <- make_grid_events(rate_s7, 6, t_obs, tg_typ, wt_typ, length(eta_grid))
par_grid <- data.frame(lkel = lkel_typ + eta_grid)

# The grid solves are deliberately multi-subject with zeroed omega (each
# subject is one quadrature node); muffle only that expected warning.
solve_quadrature <- function(model, events, params) {
  withCallingHandlers(
    rxode2::rxSolve(
      model,
      events = events, params = params,
      rtol = 1e-10, atol = 1e-12, returnType = "data.frame"
    ),
    warning = function(w) {
      if (grepl("without 'omega'", conditionMessage(w), fixed = TRUE)) {
        invokeRestart("muffleWarning")
      }
    }
  )
}
solve_grid <- function(model) solve_quadrature(model, ev_grid, par_grid)

# Quantile of C x (1 + eps), eps ~ N(0, propSd^2), mixed over the eta grid
mix_quantile <- function(conc, p, sd = prop_sd) {
  stats::uniroot(
    function(x) mean(stats::pnorm((x / conc - 1) / sd)) - p,
    interval = c(0, 10 * max(conc))
  )$root
}

summarise_grid <- function(sim) {
  sim |>
    dplyr::group_by(time) |>
    dplyr::summarise(
      median = mix_quantile(Cc, 0.5),
      p90 = mix_quantile(Cc, 0.9),
      .groups = "drop"
    )
}

# Shipped model: central(0) <- TRIG * vc
s7_base <- summarise_grid(solve_grid(mod_typ)) |>
  dplyr::mutate(start = "A(0) = TGbase x V (shipped)")
#> ℹ omega/sigma items treated as zero: 'etalkel'

# Alternative reading: start at the model steady state
mod_ss <- mod_typ |> rxode2::model(central(0) <- kin / kel)
s7_ss <- summarise_grid(solve_grid(mod_ss)) |>
  dplyr::mutate(start = "A(0) = EndVLDL / Kel (alternative)")
#> ℹ omega/sigma items treated as zero: 'etalkel'

s7 <- dplyr::bind_rows(s7_base, s7_ss)
s7 |>
  dplyr::filter(time %in% c(0.1, 1, 6, 9)) |>
  dplyr::arrange(start, time) |>
  dplyr::rename(
    "Initial condition" = start,
    "Time (h)" = time,
    "Median (mg/dL)" = median,
    "90th percentile (mg/dL)" = p90
  ) |>
  knitr::kable(digits = 1)
Time (h) Median (mg/dL) 90th percentile (mg/dL) Initial condition
0.1 119.1 216.3 A(0) = EndVLDL / Kel (alternative)
1.0 194.2 355.8 A(0) = EndVLDL / Kel (alternative)
6.0 203.9 400.5 A(0) = EndVLDL / Kel (alternative)
9.0 93.6 186.1 A(0) = EndVLDL / Kel (alternative)
0.1 117.4 169.1 A(0) = TGbase x V (shipped)
1.0 192.8 340.3 A(0) = TGbase x V (shipped)
6.0 203.9 400.5 A(0) = TGbase x V (shipped)
9.0 93.6 186.1 A(0) = TGbase x V (shipped) 
s7 |>
  tidyr::pivot_longer(c(median, p90), names_to = "stat", values_to = "Cc") |>
  ggplot(aes(time, Cc, colour = start, linetype = stat)) +
  geom_line() +
  annotate("point", x = 0.1, y = 165, shape = 4, size = 3) +
  labs(
    x = "Time after start of infusion (h)", y = "TG concentration (mg/dL)",
    colour = NULL, linetype = NULL,
    title = "Typical patient, 0.121 g/kg/h for 6 h",
    caption = paste(
      "Replicates Supplementary Figure S7 of Fukushima 2022 (V = 0.422 x BW).",
      "Cross: 90th percentile at 0.1 h read from the figure."
    )
  ) +
  theme(legend.position = "bottom", legend.direction = "vertical")

p90_early <- function(x) x$p90[x$time == 0.1]
med_at <- function(x, t) x$median[x$time == t]
stopifnot(
  # Early 90th percentile: the shipped start matches the figure (~165 mg/dL);
  # the steady-state start (about 217 mg/dL) cannot.
  abs(p90_early(s7_base) / 165 - 1) < 0.10,
  abs(p90_early(s7_ss) / 165 - 1) > 0.20,
  # Plateau 90th percentile ~400 mg/dL, the design target of the dosing rule.
  abs(s7_base$p90[s7_base$time == 6] / 400 - 1) < 0.02,
  # Medians read from Figure S7: ~210 mg/dL plateau, ~97 mg/dL at 9 h. The
  # exact medians here are 204 and 94 (proportional error on a log-normal
  # kel pulls the median below the typical value); the figure's 1,000-subject
  # Monte Carlo median carries an SE of ~2% (~4.6 mg/dL at the plateau).
  abs(med_at(s7_base, 6) / 210 - 1) < 0.06,
  abs(med_at(s7_base, 9) / 97 - 1) < 0.06
)

The baseline start reproduces the early spread in Figure S7, and the steady-state start overshoots it by about a third. The model therefore starts each patient at the measured baseline TG. After the infusion stops, TG returns to the model steady state EndVLDL / (Kel x V), not to TGbase. For the typical patient that is 96.6 mg/dL, consistent with the ~97 mg/dL tail of Figure S7. The plateau of the typical-value profile is Eq 6, 210.8 mg/dL (checked to 0.2% in the PKNCA section below). The median in the table is lower (204 mg/dL) because a proportional residual error applied to a log-normal Kel pulls the median below the typical value. Figure S7’s plateau median (~210 mg/dL) comes from 1,000 simulated patients and lies within its Monte Carlo noise.

Variability propagation: the 1.894 ratio and the omega scale

The paper reports that, by 10,000-patient Monte Carlo simulation, the 90th percentile of Cmax is 1.894 +/- 0.014 times the population-mean Cmax of Eq 6, so the target mean Cmax is 400 / 1.894 = 211 mg/dL. This ratio depends only on the between-patient and residual variability. Table 4 gives “omega Kel = 43.3%”, which could be the SD of eta (variance 0.1875) or a CV% (variance log(1 + 0.433^2) = 0.172). The ratio is computed on the same quadrature grid at the end of a 6-h infusion, when TG is at steady state for every patient (the slowest grid patient has a half-life of about 1 h):

ratio_for_omega <- function(omega) {
  ev <- make_grid_events(rate_s7, 6, 6, tg_typ, wt_typ, 200)
  sim <- solve_quadrature(
    mod_typ, ev,
    data.frame(lkel = lkel_typ + omega * qnorm(ppoints(200)))
  )
  typical <- (exp(ini_df$est[ini_df$name == "lkin"]) + rate_s7) /
    (exp(lkel_typ) * 0.422 * wt_typ) # Eq 6
  mix_quantile(sim$Cc, 0.9) / typical
}
ratio_sd <- ratio_for_omega(omega_kel)
#> ℹ omega/sigma items treated as zero: 'etalkel'
ratio_cv <- ratio_for_omega(sqrt(log(1 + 0.433^2)))
#> ℹ omega/sigma items treated as zero: 'etalkel'
data.frame(
  reading = c("omega = 0.433 is the SD of eta (shipped)", "43.3% is a CV"),
  ratio = c(ratio_sd, ratio_cv),
  published = 1.894
) |>
  dplyr::rename(
    "Reading of 'omega Kel 43.3%'" = reading,
    "p90 / mean Cmax" = ratio,
    "Published" = published
  ) |>
  knitr::kable(digits = 3)
Reading of ‘omega Kel 43.3%’ p90 / mean Cmax Published
omega = 0.433 is the SD of eta (shipped) 1.900 1.894
43.3% is a CV 1.863 1.894 

stopifnot(
  # The SD reading lands within the paper's +/-0.014 Monte Carlo spread;
  # the CV reading is two spreads away.
  abs(ratio_sd - 1.894) < 0.014,
  abs(ratio_cv - 1.894) > 0.025
)

Maximum acceptable infusion rate (Eq 7 and Supplementary Table S6)

Eq 7 rearranges Eq 6 for the infusion rate that gives a mean Cmax of 211 mg/dL. Here it is evaluated with the packaged model’s individual kel and vc, read from a typical-value solve:

mod_params <- function(tg, wt) {
  ev <- data.frame(
    id = seq_along(tg), time = 0, evid = 0L, amt = 0, cmt = "central",
    TRIG = tg, WT = wt
  )
  solve_quadrature(mod_typ, ev, NULL) |>
    dplyr::select(kin, kel, vc)
}
# Eq 7, expressed per kg (g/kg/h)
max_rate <- function(tg, wt, target = 211) {
  p <- mod_params(tg, wt)
  (target * p$kel * p$vc - p$kin) / 1000 / wt
}
rate_typ <- max_rate(tg_typ, wt_typ)
#> ℹ omega/sigma items treated as zero: 'etalkel'
rate_typ
#> [1] 0.1211836

Supplementary Table S6 refits the model with V fixed at 0.3-0.6 dL/kg and reports the typical-patient Eq 7 rate for each fit. The 0.422 dL/kg column is the packaged model. The other columns check that Eq 7 has been read correctly (same formula, Table S6 estimates):

s6 <- tibble::tribble(
  ~v_per_kg, ~theta1, ~theta2, ~theta3, ~theta4, ~published,
  0.3,       6005,    3.70,    -0.826,  -1.262,  0.127,
  0.4,       5770,    2.67,    -0.833,  -1.270,  0.122,
  0.5,       5540,    2.06,    -0.835,  -1.273,  0.118,
  0.6,       5336,    1.65,    -0.840,  -1.286,  0.113
) |>
  dplyr::mutate(
    eq7 = (211 * theta2 * (tg_typ / 94)^theta3 * (wt_typ / 55.8)^theta4 *
      v_per_kg * wt_typ - theta1) / 1000 / wt_typ
  ) |>
  dplyr::bind_rows(
    tibble::tibble(v_per_kg = 0.422, published = 0.121, eq7 = rate_typ)
  ) |>
  dplyr::arrange(v_per_kg)

s6 |>
  dplyr::select(v_per_kg, eq7, published) |>
  dplyr::rename(
    "V (dL/kg)" = v_per_kg,
    "Eq 7 rate (g/kg/h)" = eq7,
    "Table S6 (g/kg/h)" = published
  ) |>
  knitr::kable(digits = 4)
V (dL/kg) Eq 7 rate (g/kg/h) Table S6 (g/kg/h)
0.300 0.1266 0.127
0.400 0.1219 0.122
0.422 0.1212 0.121
0.500 0.1180 0.118
0.600 0.1133 0.113 

# Table S6 prints three decimals, so a correct reading is within 0.0005.
stopifnot(all(abs(s6$eq7 - s6$published) < 0.0006))

Nomogram (Figure 2)

Figure 2 draws the Eq 7 rate as contours over baseline TG and body weight. The contour heights below were digitised by the maintainers from the published figure at three body weights:

digitised <- tibble::tribble(
  ~wt,  ~rate, ~tg,
  30.6, 0.300,  97.4, 30.6, 0.200, 128.5, 30.6, 0.150, 151.8,
  30.6, 0.125, 166.5, 30.6, 0.100, 184.5, 30.6, 0.075, 205.9,
  30.6, 0.050, 232.6, 30.6, 0.025, 266.0, 30.6, 0.010, 290.7,
  40.1, 0.300,  72.1, 40.1, 0.200,  98.1, 40.1, 0.150, 118.8,
  40.1, 0.125, 132.5, 40.1, 0.100, 149.1, 40.1, 0.075, 169.8,
  40.1, 0.050, 197.2, 40.1, 0.025, 232.9, 40.1, 0.010, 261.0,
  89.4, 0.200,  39.0, 89.4, 0.150,  50.4, 89.4, 0.125,  58.7,
  89.4, 0.100,  69.7, 89.4, 0.075,  85.4, 89.4, 0.050, 108.4,
  89.4, 0.025, 146.1, 89.4, 0.010, 182.9
)

# Baseline TG at which the packaged model's Eq 7 rate equals each contour
tg_on_contour <- function(rate, wt) {
  stats::uniroot(
    function(tg) max_rate(tg, wt) - rate,
    interval = c(1, 3000), tol = 1e-6
  )$root
}
digitised$tg_model <- mapply(tg_on_contour, digitised$rate, digitised$wt)
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digitised$tg_ratio <- digitised$tg / digitised$tg_model

grid <- expand.grid(wt = seq(30, 90, by = 1), tg = seq(30, 300, by = 2))
grid$rate <- max_rate(grid$tg, grid$wt)
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ggplot(grid, aes(wt, tg)) +
  geom_contour(
    aes(z = rate),
    breaks = c(0.01, 0.025, 0.05, 0.075, 0.1, 0.125, 0.15, 0.2, 0.3),
    colour = "black"
  ) +
  geom_point(data = digitised, colour = "red", shape = 4) +
  annotate("point", x = wt_typ, y = tg_typ, shape = 16, size = 2.5, colour = "blue") +
  coord_cartesian(xlim = c(30, 90), ylim = c(30, 300)) +
  labs(
    x = "Body weight (kg)", y = "Baseline TG (mg/dL)",
    title = "Eq 7 contours, 0.01-0.3 g/kg/h",
    caption = paste(
      "Replicates Figure 2 of Fukushima 2022. Lines: packaged model;",
      "red crosses: digitised contours; blue dot: typical patient."
    )
  )


knitr::kable(
  digitised |>
    dplyr::group_by(wt) |>
    dplyr::summarise(
      min_ratio = min(tg_ratio), median_ratio = median(tg_ratio),
      max_ratio = max(tg_ratio), .groups = "drop"
    ) |>
    dplyr::rename(
      "Body weight (kg)" = wt,
      "Min ratio" = min_ratio,
      "Median ratio" = median_ratio,
      "Max ratio" = max_ratio
    ),
  digits = 3,
  caption = "Digitised / model baseline TG on each contour, by body weight."
)
Digitised / model baseline TG on each contour, by body weight.
Body weight (kg) Min ratio Median ratio Max ratio
30.6 1.055 1.058 1.061
40.1 1.052 1.056 1.059
89.4 1.039 1.050 1.057 

stopifnot(
  # Shape: one constant ratio across all 26 digitised contour points means
  # the TG exponent (theta3) and body-weight exponent (theta4) carry through
  # unchanged.
  stats::sd(log(digitised$tg_ratio)) < 0.02,
  # Offset: the published contours sit a uniform ~5% higher in baseline TG
  # (see Assumptions and deviations).
  abs(median(digitised$tg_ratio) - 1.055) < 0.02
)

The digitised contours lie on the model’s contours up to one uniform factor of about 1.055 in baseline TG. A single constant factor across rates and weights is what a slightly different target Cmax or Kel in the drawing would produce: (1.055)^0.833 corresponds to about 220 instead of 211 mg/dL. A misread exponent or a different EndVLDL would distort the contour shape instead. The paper’s own tabulated typical-patient rate (0.121 g/kg/h, Table S6) agrees with Eq 7 and the packaged model, not with the drawn contours, and the shift is recorded below as a deviation of the published figure. Because of it, the typical patient (blue dot) falls just above the 0.125 contour here (0.121 g/kg/h). The paper’s caption places it in the “0.125 g/kg/h” area.

Virtual cohort: observed profiles (Figure 1) and individual rates (Figure 3)

Individual data are not available. The virtual cohort draws each covariate and dosing variable independently from a piecewise-linear interpolation of the Table 1 quantiles (minimum, quartiles, maximum).

set.seed(2111)
n_sub <- 200L
r_table1 <- function(n, q) {
  stats::approx(c(0, 0.25, 0.5, 0.75, 1), q, xout = stats::runif(n))$y
}
cohort <- tibble::tibble(
  id = seq_len(n_sub),
  WT = r_table1(n_sub, c(31.2, 46.2, 55.8, 65, 86.1)),
  TRIG = r_table1(n_sub, c(30, 68, 94, 119, 291)),
  rate_gkgh = r_table1(n_sub, c(0.053, 0.105, 0.217, 0.278, 0.363)),
  dur = r_table1(n_sub, c(0.75, 3.0, 5.2, 7.2, 19.8))
) |>
  dplyr::mutate(rate = rate_gkgh * WT * 1000, amt = rate * dur)

ev_cohort <- dplyr::bind_rows(
  cohort |>
    dplyr::transmute(id, time = 0, evid = 1L, amt, rate, cmt = "central", TRIG, WT),
  cohort |>
    dplyr::select(id, TRIG, WT, dur) |>
    tidyr::crossing(time = seq(0, 24, by = 0.25)) |>
    dplyr::bind_rows(cohort |> dplyr::transmute(id, TRIG, WT, dur, time = dur)) |>
    dplyr::distinct(id, time, .keep_all = TRUE) |>
    dplyr::transmute(id, time, evid = 0L, amt = 0, rate = 0, cmt = "central", TRIG, WT)
) |>
  dplyr::arrange(id, time, dplyr::desc(evid))

rxode2::rxSetSeed(2111)
sim_cohort <- rxode2::rxSolve(mod, events = ev_cohort, returnType = "data.frame") |>
  dplyr::left_join(cohort |> dplyr::select(id, dur), by = "id") |>
  dplyr::mutate(phase = ifelse(time <= dur, "during infusion", "after infusion"))
#> ℹ parameter labels from comments will be replaced by 'label()'
ggplot(sim_cohort, aes(time, Cc, group = interaction(id, phase))) +
  geom_line(aes(linetype = phase), alpha = 0.3) +
  scale_linetype_manual(values = c("during infusion" = "solid", "after infusion" = "dashed")) +
  labs(
    x = "Time (h)", y = "Triglyceride concentration (mg/dL)", linetype = NULL,
    title = "Simulated TG profiles, Table 1-like virtual cohort",
    caption = "Compare with Figure 1 of Fukushima 2022 (observed data). Individual predictions without residual error."
  ) +
  theme(legend.position = "bottom")

As in Figure 1, most simulated profiles stay below about 500 mg/dL and TG falls back near baseline within a few hours of the end of the infusion. The upper tail is thinner than observed. When the maintainers ran this, the simulated peaks had a median of about 250 mg/dL and only about 1-2% of patients exceeded 1,000 mg/dL, reaching about 1,500 mg/dL. Figure 1 shows roughly six of 83 patients above 1,000 mg/dL, reaching about 2,450 mg/dL. This matches the authors’ own finding that the first-order model underpredicts concentrations above about 1,000 mg/dL, where TG elimination appears saturated (Supplementary Model Selection and Figure S2).

Figure 3 histograms the Eq 7 rate for the 83 study patients. The same calculation on the virtual cohort is compared below. The virtual cohort draws baseline TG and body weight independently, which the real cohort need not do, so this comparison is descriptive and has no gate:

bins <- c(-Inf, 0.01, 0.025, 0.05, 0.075, 0.1, 0.125, 0.15, 0.2, 0.3, Inf)
bin_labels <- c(
  "<0.01", "0.01-0.025", "0.026-0.05", "0.06-0.075", "0.076-0.1",
  "0.11-0.125", "0.126-0.15", "0.16-0.2", "0.21-0.3", ">=0.3"
)
published_fig3 <- c(3, 2, 3, 10, 11, 7, 10, 7, 15, 15) # Figure 3, n = 83
cohort$max_rate <- max_rate(cohort$TRIG, cohort$WT)
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virtual_counts <- as.vector(table(cut(cohort$max_rate, bins, right = FALSE)))
tibble::tibble(
  bin = bin_labels,
  published_pct = 100 * published_fig3 / sum(published_fig3),
  virtual_pct = 100 * virtual_counts / n_sub
) |>
  dplyr::rename(
    "Max acceptable rate (g/kg/h)" = bin,
    "Figure 3, % of 83" = published_pct,
    "Virtual cohort, %" = virtual_pct
  ) |>
  knitr::kable(digits = 1)
Max acceptable rate (g/kg/h) Figure 3, % of 83 Virtual cohort, %
<0.01 3.6 13.5
0.01-0.025 2.4 5.5
0.026-0.05 3.6 2.5
0.06-0.075 12.0 10.5
0.076-0.1 13.3 9.5
0.11-0.125 8.4 11.0
0.126-0.15 12.0 5.5
0.16-0.2 8.4 11.5
0.21-0.3 18.1 14.0
>=0.3 18.1 16.5

The virtual cohort over-fills the lowest bins. The interpolated Table 1 quantiles put a quarter of the virtual patients uniformly between 119 and 291 mg/dL baseline TG, which over-represents very high baselines (where Eq 7 falls below 0.01 g/kg/h) compared with the right-skewed real distribution. Above about 0.05 g/kg/h, the virtual and published distributions are broadly similar.

PKNCA validation

The paper reports no NCA. Its one published concentration is the population-mean Cmax of 211 mg/dL at the typical patient’s maximum acceptable rate. Eq 6 gives the population-mean Cmax at any other infusion rate. Typical-value profiles are simulated for five 6-hour infusion rates, and PKNCA Cmax is compared with Eq 6 (and with 211 mg/dL for the 0.121 g/kg/h arm). An infusion of 6 h is about 20 half-lives of the typical kel, so each Cmax is the Eq 6 plateau.

arms <- tibble::tibble(rate_gkgh = c(0.05, 0.1, 0.121, 0.2, 0.3)) |>
  dplyr::mutate(
    id = dplyr::row_number(),
    treatment = sprintf("%.3f g/kg/h", rate_gkgh),
    rate = rate_gkgh * wt_typ * 1000,
    amt = rate * 6
  )
ev_nca <- dplyr::bind_rows(
  arms |> dplyr::transmute(id, treatment, time = 0, evid = 1L, amt, rate, cmt = "central"),
  arms |>
    dplyr::select(id, treatment) |>
    tidyr::crossing(time = seq(0, 24, by = 0.25)) |>
    dplyr::mutate(evid = 0L, amt = 0, rate = 0, cmt = "central")
) |>
  dplyr::mutate(TRIG = tg_typ, WT = wt_typ) |>
  dplyr::arrange(id, time, dplyr::desc(evid))

sim_nca <- rxode2::rxSolve(
  mod_typ,
  events = ev_nca, keep = "treatment",
  rtol = 1e-10, atol = 1e-12, returnType = "data.frame"
) |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, treatment)
#> ℹ omega/sigma items treated as zero: 'etalkel'
#> Warning: multi-subject simulation without without 'omega'

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id)
dose_obj <- PKNCA::PKNCAdose(
  arms |> dplyr::mutate(time = 0, dur = 6) |>
    dplyr::select(id, treatment, time, amt, dur),
  amt ~ time | treatment + id,
  route = "intravascular", duration = "dur"
)
intervals <- data.frame(start = 0, end = 24, cmax = TRUE, tmax = TRUE)
nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))

typ <- mod_params(tg_typ, wt_typ)
#> ℹ omega/sigma items treated as zero: 'etalkel'
reference <- arms |>
  dplyr::transmute(
    treatment,
    cmax = ifelse(rate_gkgh == 0.121, 211, (typ$kin + rate) / (typ$kel * typ$vc))
  )

cmp <- nlmixr2lib::ncaComparisonTable(
  simulated = nca_res,
  reference = reference,
  by = "treatment",
  params = "cmax",
  units = c(cmax = "mg/dL"),
  tolerance_pct = 20
)
knitr::kable(
  cmp,
  caption = paste(
    "Simulated vs. reference Cmax for a typical patient (94 mg/dL, 55.8 kg).",
    "Reference: 211 mg/dL published for 0.121 g/kg/h; Eq 6 otherwise.",
    "* differs from reference by >20%."
  )
)
Simulated vs. reference Cmax for a typical patient (94 mg/dL, 55.8 kg). Reference: 211 mg/dL published for 0.121 g/kg/h; Eq 6 otherwise. * differs from reference by >20%.
NCA parameter treatment Reference Simulated % diff
Cmax (mg/dL) 0.050 g/kg/h 144 144 -0.0%
Cmax (mg/dL) 0.100 g/kg/h 191 191 -0.0%
Cmax (mg/dL) 0.121 g/kg/h 211 211 -0.1%
Cmax (mg/dL) 0.200 g/kg/h 285 285 -0.0%
Cmax (mg/dL) 0.300 g/kg/h 380 380 -0.0% 

cmax_sim <- as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD == "cmax") |>
  dplyr::left_join(reference, by = "treatment")
stopifnot(
  # Same parameters on both sides (Eq 6 plateau vs the ODE solve), so a tight
  # bound is correct; 211 is the paper's rounding of 210.8.
  all(abs(cmax_sim$PPORRES / cmax_sim$cmax - 1) < 0.002)
)

Assumptions and deviations

  • Initial condition. Eq 1 is printed without an initial condition. The maintainers set central(0) <- TRIG * vc, the measured baseline TG times the fixed plasma volume. Supplementary Figure S7 selects this choice: its early 90th percentile (~165 mg/dL at 0.1 h) is matched by the baseline start (~169 mg/dL) and not by a steady-state start EndVLDL / Kel (~217 mg/dL). After the infusion, TG relaxes to the model’s own steady state EndVLDL / (Kel x V), which differs from TRIG patient by patient. TRIG must therefore be the pre-infusion baseline, held constant.
  • Omega scale. Table 4 prints “omega Kel = 43.3%”. The maintainers read it as the SD of the exponential random effect (variance 0.433^2 = 0.1875), because it reproduces the paper’s variability-propagation ratio (1.900 vs published 1.894 +/- 0.014). The CV reading gives 1.864.
  • Nomogram offset. The published Figure 2 contours sit a uniform ~5.5% higher in baseline TG than Eq 7 evaluated with the Table 4 estimates (contour shape agrees to within a 2% log-SD over 26 digitised points). Eq 7 with the Table 4 estimates reproduces the paper’s own typical-patient rate (0.121 g/kg/h, Supplementary Table S6) and all four Table S6 sensitivity fits, so the packaged model follows Table 4 and Eq 7. The figure offset is recorded here as a deviation of the published drawing.
  • Units. amt and rate are in mg of TG (1 g of emulsion fat = 1000 mg TG), the plasma volume is in dL and concentrations are in mg/dL, matching the paper’s EndVLDL (mg/h) and V (dL).
  • Virtual cohort. Body weight, baseline TG, infusion rate and infusion duration were drawn independently from piecewise-linear interpolations of the Table 1 quantiles. Correlations among them are not reported.
  • Covariate screen. Apolipoprotein C-III, apolipoprotein C-II and serum albumin were significant on Kel in the univariate screen (Table 2), but none was retained once baseline TG was in the model (Table 3). Apolipoprotein C-III and albumin are listed in the model’s covariatesDataExcluded metadata. Apolipoprotein C-II, which has no registered covariate column, is documented here only.
  • Scope. The model assumes first-order TG elimination. The authors note that elimination appears saturated above about 1000 mg/dL (Supplementary Model Selection, Figure S2), so predictions in that range are underestimates by design.
  • No erratum or correction notice for this article was found on Europe PMC as of 2026-10-02.