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Gadoxetate rat PBPK (Scotcher 2021)

Source: vignettes/articles/Scotcher_2021_gadoxetate_rat_pbpk.Rmd
Scotcher_2021_gadoxetate_rat_pbpk.Rmd

Model and source

  • Citation: Scotcher D, Melillo N, Tadimalla S, Darwich AS, Ziemian S, Ogungbenro K, Schutz G, Sourbron S, Galetin A. Physiologically Based Pharmacokinetic Modeling of Transporter-Mediated Hepatic Disposition of Imaging Biomarker Gadoxetate in Rats. Mol Pharm. 2021;18(8):2997-3009. doi:10.1021/acs.molpharmaceut.1c00206 (PMC8397403). Model equations (S1-S3) and physiological parameters (Table S2) are in the paper’s Supporting Information (mp1c00206_si_001.pdf).
  • Description: Preclinical (rat). Reduced PBPK model (seven compartments, permeability-limited liver) for the hepatobiliary MRI contrast agent gadoxetate in the 250 g male Wistar-Han rat after a 25 umol/kg intravenous dose, given alone or 1 h after a single 10 mg/kg intravenous rifampicin dose. Blood, spleen and splanchnic extracellular spaces are perfusion-limited; the rest of the body is split into a vascular and an interstitial space linked by a permeability-surface product; the liver has an extracellular space and a hepatocyte space linked by saturable OATP-mediated active uptake (linearised, CLactive) and bidirectional passive diffusion, with biliary (Mrp2) excretion from the hepatocyte and renal clearance from blood. Active uptake, biliary clearance and PS were fitted (naive pooled) to dynamic contrast-enhanced MRI Delta R1 profiles of blood, spleen and liver at 4.7 T and 7 T; the defaults are the simultaneous control plus rifampicin fit (Table 4), in which rifampicin inhibits active uptake by 96 percent. Delta R1 outputs at both field strengths are derived with the ex vivo relaxivities of Table 1. Deterministic: no between-animal variability or residual error was estimated.
  • Article: https://doi.org/10.1021/acs.molpharmaceut.1c00206 (open access, PMC8397403)
  • Supporting Information (model equations S1-S3, Table S2 physiology): https://pubs.acs.org/doi/10.1021/acs.molpharmaceut.1c00206?goto=supporting-info

Gadoxetate is a hepatobiliary MRI contrast agent taken up into hepatocytes by OATP (rat Oatp1a1) and excreted into bile by MRP2 (rat Mrp2). Scotcher et al. built a reduced PBPK model of gadoxetate in the rat and used dynamic contrast-enhanced MRI (DCE-MRI) images of blood, spleen and liver to refine the in vitro-in vivo extrapolation (IVIVE) of its hepatic transporter kinetics, and then to quantify the interaction with a single intravenous dose of the OATP inhibitor rifampicin.

The paper reports one model structure with several parameter sets. The packaged model uses the final simultaneous fit of the control and rifampicin phases (Table 4); the bottom-up IVIVE values and the control-only top-down fit (both Table 3) are reproduced below by overriding three ini() values.

Population

The DCE-MRI data come from male Wistar-Han rats in a multicentre preclinical study, scanned at two magnetic field strengths (4.7 T and 7 T) after a 25 umol/kg intravenous gadoxetate dose, given either alone (control: 43 profiles at 4.7 T from 33 animals and 52 profiles at 7 T from 43 animals, some animals scanned twice) or 1 h after 10 mg/kg intravenous rifampicin (7 profiles at 4.7 T and 6 at 7 T). The PBPK model describes an average 250 g rat. Separately, in vitro hepatocyte uptake kinetics were measured in plated hepatocytes from four male Sprague-Dawley rats (250-300 g, Table 2) and supply the passive diffusion clearance and the intracellular unbound fraction. All fits were naive pooled, so the model carries no between-animal variability.

The same information is available programmatically via readModelDb("Scotcher_2021_gadoxetate_rat_pbpk")()$population.

Model structure

Seven gadoxetate compartments (Figure 2; Supporting Information equation system S1), all written in amounts:

  • blood – systemic blood, receiving the intravenous dose and cleared renally (CLr).
  • vp_remainder / is_remainder – the rest of the body (lungs, brain, heart, kidneys, bone, muscle, skin, fat): a vascular space perfused at Qrob and an interstitial space exchanging with it through the permeability-surface product PS (eq 5).
  • is_spleen, is_splanchnic – perfusion-limited extracellular spaces (organ blood plus interstitium) of the spleen and of the stomach + gut + pancreas, draining into the liver through the portal vein.
  • is_liver / int_liver – liver extracellular space and hepatocytes (eq 6). Uptake is by linearised active transport CLactive on the total extracellular concentration plus bidirectional passive diffusion CLpassive (efflux on the unbound intracellular concentration, fu,liv,cell); biliary excretion CLbiliary acts on the unbound intracellular concentration.

Two cumulative records (urine, a_bile) are added so the excreted fractions can be read off a solve; they do not feed back.

The extracellular-to-blood partition coefficients follow equation S3, which assumes that gadoxetate does not enter red cells and that plasma and interstitial fluid equilibrate: K = (V_int + V_b (1 - Hct)) / ((V_int + V_b) (1 - Hct)).

The observed quantity is Delta R1 (1/s), linear in concentration through the ex vivo relaxivities of Table 1 (eq 7): blood and spleen use the blood relaxivity; the liver signal is the volume-weighted sum of the extracellular (blood relaxivity) and hepatocyte (hepatocyte relaxivity) contributions. Relaxivities depend on field strength, so the model returns Delta R1 at both 4.7 T and 7 T.

Source trace

Equation / parameter Value Source location
lps_act_inf (CLactive, control) log(2.38) L/h Table 4, control column
lps_act_inf_inh (CLactive, rifampicin) log(0.095) L/h Table 4, rifampicin column
lcl_bile (CLbiliary, control) log(0.07) L/h Table 4, control column
lcl_bile_inh (CLbiliary, rifampicin) log(0.08) L/h Table 4, rifampicin column
lps_remainder (PS, both phases) log(0.71) L/h Table 4
lps_dif (CLpassive) log(0.014) L/h, held constant Table 3 footnote d; Table 2 mean 0.193 uL/min/10^6 cells scaled by eq 3
fu_liver_cell 0.648, held constant Table 2 mean; Table 3
lcl_renal (CLr) log(0.17) L/h, held constant Table 3 footnote e (36.7 mL/min/kg x fe 0.305 x 0.25 kg)
qc (cardiac output) 6.62 L/h Table S2, blood / lungs blood flow
hct 0.4183 Supporting Information Section 5
Organ masses, densities, vascular and interstitial fractions, blood flows constants in model() Table S2
Lumping of volumes, flows and partition coefficients n/a Equation S2
Partition coefficients kp_spleen, kp_splanchnic, kp_liver derived Equation S3 (derivation in Supporting Information Section 3)
d/dt(...) for all seven compartments n/a Equation system S1; eqs 5 and 6
Relaxivities (blood 6.4 / 6.2, hepatocytes 7.6 / 6 per s per mM at 4.7 / 7 T) constants in model() Table 1 and its footnote
Delta R1 of blood, spleen, liver n/a Equation 7
Rifampicin phase indicator CONMED_RIFAMPICIN_SD 0 / 1 Table 4 (separate control and rifampicin estimates)

Simulation

A single typical rat (the model has no random effects) receives 25 umol/kg x 0.25 kg = 6.25 umol as an intravenous bolus into blood. The figures in the paper place the injection about 4.6 minutes into the image acquisition (after a baseline); the plots below use time since injection.

mod <- readModelDb("Scotcher_2021_gadoxetate_rat_pbpk")

dose_umol <- 25 * 0.25 # 25 umol/kg x 0.25 kg reference rat

make_events <- function(rif, times) {
  ev <- rxode2::et(amt = dose_umol, cmt = "blood") |>
    rxode2::et(times) |>
    as.data.frame()
  ev$CONMED_RIFAMPICIN_SD <- rif
  ev
}

# 1-s grid over the 30-min imaging window (hours)
t_img <- seq(0, 0.5, by = 1 / 3600)

# Parameter sets. Table 4 is the packaged default; Table 3 gives the
# control-only top-down fit and the bottom-up IVIVE (Monte Carlo mean) values.
mod_topdown <- mod |>
  rxode2::ini(lps_act_inf = log(2.17), lcl_bile = log(0.07), lps_remainder = log(0.62))
#> ℹ change initial estimate of `lps_act_inf` to `0.774727167552368`
#> ℹ change initial estimate of `lcl_bile` to `-2.65926003693278`
#> ℹ change initial estimate of `lps_remainder` to `-0.478035800943`
mod_bottomup <- mod |>
  rxode2::ini(lps_act_inf = log(0.23), lcl_bile = log(0.014), lps_remainder = log(0.014))
#> ℹ change initial estimate of `lps_act_inf` to `-1.46967597005894`
#> ℹ change initial estimate of `lcl_bile` to `-4.26869794936688`
#> ℹ change initial estimate of `lps_remainder` to `-4.26869794936688`

solve_img <- function(m, rif, label) {
  rxode2::rxSolve(m, make_events(rif, t_img), rtol = 1e-10, atol = 1e-12, returnType = "data.frame") |>
    dplyr::mutate(scenario = label)
}

sim <- dplyr::bind_rows(
  solve_img(mod_bottomup, 0, "Bottom-up IVIVE (Table 3), control"),
  solve_img(mod_topdown, 0, "Top-down fit (Table 3), control"),
  solve_img(mod, 0, "Simultaneous fit (Table 4), control"),
  solve_img(mod, 1, "Simultaneous fit (Table 4), rifampicin")
)

dr1_long <- sim |>
  dplyr::select(scenario, time, dplyr::starts_with("dR1_")) |>
  tidyr::pivot_longer(dplyr::starts_with("dR1_"), names_to = "output", values_to = "dR1") |>
  dplyr::mutate(
    tissue = dplyr::case_when(
      grepl("blood", output) ~ "Blood",
      grepl("spleen", output) ~ "Spleen",
      TRUE ~ "Liver"
    ),
    tissue = factor(tissue, levels = c("Blood", "Spleen", "Liver")),
    field = ifelse(grepl("4p7t", output), "4.7 T", "7 T"),
    tmin = time * 60
  )

plot_dr1 <- function(which, title, caption) {
  dr1_long |>
    dplyr::filter(scenario == which) |>
    ggplot(aes(tmin, dR1)) +
    geom_line(colour = "steelblue", linewidth = 0.9) +
    facet_grid(field ~ tissue, scales = "free_y") +
    labs(
      x = "Time since gadoxetate injection (min)",
      y = expression(Delta * R[1] ~ (s^-1)),
      title = title,
      caption = caption
    ) +
    theme_bw()
}

Replicate published figures 

plot_dr1(
  "Bottom-up IVIVE (Table 3), control",
  "Bottom-up PBPK prediction",
  "Replicates the median lines of Figure 4 of Scotcher 2021 (deterministic, at the Table 3 bottom-up values)."
)

The bottom-up prediction captures the rapid disappearance from blood and spleen, but the liver signal keeps rising over the 30-minute window because the in vitro active uptake (0.23 L/h) and the assumed biliary clearance (0.014 L/h, set equal to CLpassive) are far too low – the misfit the paper reports in Figure 4. The paper’s curve is the median of a 10,000-sample Monte Carlo over the uncertain in vitro parameters, so the deterministic curve at the Monte Carlo mean values is a close but not identical comparison.

plot_dr1(
  "Top-down fit (Table 3), control",
  "Top-down refinement using the liver-imaging data",
  "Replicates Figure 5 of Scotcher 2021 (Table 3 top-down estimates)."
)

plot_dr1(
  "Simultaneous fit (Table 4), rifampicin",
  "Gadoxetate given 1 h after rifampicin 10 mg/kg IV",
  "Replicates Figure 6 of Scotcher 2021 (Table 4, rifampicin-phase estimates)."
)

plot_dr1(
  "Simultaneous fit (Table 4), control",
  "Control phase of the simultaneous fit",
  "Replicates Figure S9 of Scotcher 2021 (Table 4, control-phase estimates)."
)

Quantitative check against Figures 5 and 6

The maintainers read the liver Delta R1 of the model curves in Figures 5 and 6 off the published figure panels (peak value, and the value at the end of the panel, 30 min on the figure axis = about 25.4 min after injection). The liver signal is smooth, so its reading is precise to about 0.05 1/s. A mis-transcribed clearance, volume or relaxivity moves these values by far more than the 15% tolerance.

at_min <- function(d, tm) d$value[which.min(abs(d$tmin - tm))]

digitised <- tibble::tribble(
  ~figure, ~scenario, ~output, ~quantity, ~published,
  "Figure 5", "Top-down fit (Table 3), control", "dR1_liver_4p7t", "peak", 2.65,
  "Figure 5", "Top-down fit (Table 3), control", "dR1_liver_7t", "peak", 2.10,
  "Figure 5", "Top-down fit (Table 3), control", "dR1_liver_4p7t", "25.4 min", 0.33,
  "Figure 5", "Top-down fit (Table 3), control", "dR1_liver_7t", "25.4 min", 0.25,
  "Figure 6", "Simultaneous fit (Table 4), rifampicin", "dR1_liver_4p7t", "peak", 0.87,
  "Figure 6", "Simultaneous fit (Table 4), rifampicin", "dR1_liver_7t", "peak", 0.71,
  "Figure 6", "Simultaneous fit (Table 4), rifampicin", "dR1_liver_4p7t", "25.4 min", 0.28,
  "Figure 6", "Simultaneous fit (Table 4), rifampicin", "dR1_liver_7t", "25.4 min", 0.22
)

sim_value <- function(scn, out, qty) {
  d <- dr1_long |>
    dplyr::filter(scenario == scn, output == out) |>
    dplyr::rename(value = dR1)
  if (nrow(d) == 0) stop("no simulated rows for ", scn, " / ", out)
  if (qty == "peak") max(d$value) else at_min(d, 25.4)
}

digitised$simulated <- mapply(sim_value, digitised$scenario, digitised$output, digitised$quantity)
digitised$pct_diff <- 100 * (digitised$simulated / digitised$published - 1)

digitised |>
  dplyr::mutate(simulated = signif(simulated, 3), pct_diff = round(pct_diff, 1)) |>
  dplyr::select(-scenario) |>
  dplyr::rename(
    "Figure" = figure,
    "Output" = output,
    "Quantity" = quantity,
    "Published (digitised, 1/s)" = published,
    "Simulated (1/s)" = simulated,
    "Difference (%)" = pct_diff
  ) |>
  knitr::kable(caption = "Liver Delta R1: simulated vs. read from the published model curves.")
Liver Delta R1: simulated vs. read from the published model curves.
Figure Output Quantity Published (digitised, 1/s) Simulated (1/s) Difference (%)
Figure 5 dR1_liver_4p7t peak 2.65 2.580 -2.7
Figure 5 dR1_liver_7t peak 2.10 2.040 -2.9
Figure 5 dR1_liver_4p7t 25.4 min 0.33 0.348 5.4
Figure 5 dR1_liver_7t 25.4 min 0.25 0.275 9.9
Figure 6 dR1_liver_4p7t peak 0.87 0.819 -5.8
Figure 6 dR1_liver_7t peak 0.71 0.672 -5.3
Figure 6 dR1_liver_4p7t 25.4 min 0.28 0.295 5.2
Figure 6 dR1_liver_7t 25.4 min 0.22 0.238 8.0 

stopifnot(nrow(digitised) == 8, all(abs(digitised$pct_diff) < 15))

All eight readings agree within about 10%. The blood and spleen peaks are not compared: gadoxetate is injected as a bolus into a 16 mL blood pool, so the simulated blood Delta R1 falls from 2.5 1/s to under 0.6 1/s within the first minute, and the peak height of a plotted curve depends entirely on the time resolution at which it was drawn (the paper computes residuals against the mean over each 57-s acquisition frame, Supporting Information Section 4). Later blood values agree with Figure 6 (about 0.2 1/s at 15 min after injection on the published curve; simulated 0.17 1/s).

Excretion and rifampicin inhibition

The paper states that, with the top-down estimates, the simulated percentages of the dose excreted in urine and bile were 17% and 83%.

t_long <- c(seq(0, 1, by = 1 / 600), seq(1.1, 48, by = 0.1))
exc <- dplyr::bind_rows(
  rxode2::rxSolve(mod_topdown, make_events(0, t_long), rtol = 1e-10, atol = 1e-12, returnType = "data.frame") |>
    dplyr::mutate(scenario = "Top-down fit (Table 3), control"),
  rxode2::rxSolve(mod, make_events(0, t_long), rtol = 1e-10, atol = 1e-12, returnType = "data.frame") |>
    dplyr::mutate(scenario = "Simultaneous fit (Table 4), control"),
  rxode2::rxSolve(mod, make_events(1, t_long), rtol = 1e-10, atol = 1e-12, returnType = "data.frame") |>
    dplyr::mutate(scenario = "Simultaneous fit (Table 4), rifampicin")
) |>
  dplyr::group_by(scenario) |>
  dplyr::slice_max(time, n = 1) |>
  dplyr::ungroup() |>
  dplyr::mutate(
    urine_pct = 100 * urine / dose_umol,
    bile_pct = 100 * a_bile / dose_umol
  )

exc |>
  dplyr::select(scenario, urine_pct, bile_pct) |>
  dplyr::mutate(dplyr::across(c(urine_pct, bile_pct), ~ round(.x, 1))) |>
  dplyr::rename(
    "Scenario" = scenario,
    "Urine (% of dose, 48 h)" = urine_pct,
    "Bile (% of dose, 48 h)" = bile_pct
  ) |>
  knitr::kable()
Scenario Urine (% of dose, 48 h) Bile (% of dose, 48 h)
Simultaneous fit (Table 4), control 17.2 82.8
Simultaneous fit (Table 4), rifampicin 59.0 41.0
Top-down fit (Table 3), control 17.6 82.4 

td <- exc[exc$scenario == "Top-down fit (Table 3), control", ]
stopifnot(
  nrow(td) == 1,
  # Complete recovery: every route of loss is urine or bile.
  abs(td$urine_pct + td$bile_pct - 100) < 0.1,
  # Paper: 17% urine / 83% bile.
  abs(td$urine_pct - 17) < 1.5
)

inhibition_pct <- 100 * (1 - 0.095 / 2.38)
inhibition_pct
#> [1] 96.0084
stopifnot(round(inhibition_pct) == 96) # paper: 96% inhibition of CLactive

The simulated split is 17.6% / 82.4% (the paper rounds to 17 / 83). Rifampicin shifts elimination towards the kidney (about 59% of the dose in urine) because hepatic uptake falls to 4% of control.

Mass balance

With renal and biliary clearance switched off, the total amount over all states must stay equal to the dose.

mod_closed <- mod |> rxode2::ini(lcl_renal = log(1e-12), lcl_bile = log(1e-12), lcl_bile_inh = log(1e-12))
#> ℹ change initial estimate of `lcl_renal` to `-27.6310211159285`
#> ℹ change initial estimate of `lcl_bile` to `-27.6310211159285`
#> ℹ change initial estimate of `lcl_bile_inh` to `-27.6310211159285`
closed <- rxode2::rxSolve(mod_closed, make_events(0, t_long), rtol = 1e-10, atol = 1e-12, returnType = "data.frame")
states <- c(
  "blood", "vp_remainder", "is_remainder", "is_spleen", "is_splanchnic",
  "is_liver", "int_liver", "urine", "a_bile"
)
total <- rowSums(closed[, states])
max_dev <- max(abs(total / dose_umol - 1))
max_dev
#> [1] 3.974598e-14
stopifnot(max_dev < 1e-6)

PKNCA validation

The paper reports no non-compartmental results, so PKNCA is used to summarise the simulated blood exposure of the control and rifampicin phases (Table 4 parameters) and to cross-check the urinary recovery: CLr x AUCinf / Dose must reproduce the urine fraction read directly from the cumulative urine state.

t_nca <- c(seq(0, 0.5, by = 1 / 3600), seq(0.51, 3, by = 0.01))
sim_nca_raw <- dplyr::bind_rows(
  rxode2::rxSolve(mod, make_events(0, t_nca), rtol = 1e-10, atol = 1e-14, returnType = "data.frame") |>
    dplyr::mutate(treatment = "control"),
  rxode2::rxSolve(mod, make_events(1, t_nca), rtol = 1e-10, atol = 1e-14, returnType = "data.frame") |>
    dplyr::mutate(treatment = "rifampicin")
) |>
  dplyr::mutate(id = 1L)

# Solver undershoot must be noise; then floor it.
stopifnot(all(sim_nca_raw$Cc >= -1e-6 * max(sim_nca_raw$Cc)))
sim_nca <- sim_nca_raw |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::mutate(Cc = pmax(Cc, 0)) |>
  dplyr::select(id, time, Cc, treatment)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id)
dose_df <- data.frame(id = 1L, time = 0, amt = dose_umol, treatment = c("control", "rifampicin"))
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id, route = "intravascular")

intervals <- data.frame(
  start = 0,
  end = Inf,
  cmax = TRUE,
  aucinf.obs = TRUE,
  cl.obs = TRUE,
  half.life = TRUE
)
nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))

nca_tab <- as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "aucinf.obs", "cl.obs", "half.life")) |>
  dplyr::select(treatment, PPTESTCD, PPORRES) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES)

nca_tab |>
  dplyr::mutate(
    fe_urine_pct = 100 * 0.17 * aucinf.obs / dose_umol,
    half.life = half.life * 60,
    dplyr::across(where(is.numeric), ~ signif(.x, 3))
  ) |>
  dplyr::select(treatment, cmax, aucinf.obs, cl.obs, half.life, fe_urine_pct) |>
  dplyr::rename(
    "Phase" = treatment,
    "Cmax (umol/L)" = cmax,
    "AUC0-inf (umol*h/L)" = aucinf.obs,
    "CL blood (L/h)" = cl.obs,
    "Terminal t1/2 (min)" = half.life,
    "CLr x AUC / Dose (%)" = fe_urine_pct
  ) |>
  knitr::kable(caption = "PKNCA summary of simulated blood gadoxetate (Table 4 parameters).")
PKNCA summary of simulated blood gadoxetate (Table 4 parameters).
Phase Cmax (umol/L) AUC0-inf (umol*h/L) CL blood (L/h) Terminal t1/2 (min) CLr x AUC / Dose (%)
control 396 6.32 0.990 5.72 17.2
rifampicin 396 21.70 0.288 9.12 59.0 

fe_nca <- 100 * 0.17 * nca_tab$aucinf.obs / dose_umol
fe_state <- exc$urine_pct[match(
  c("Simultaneous fit (Table 4), control", "Simultaneous fit (Table 4), rifampicin"),
  exc$scenario
)]
stopifnot(length(fe_nca) == 2, all(abs(fe_nca / fe_state - 1) < 0.02))

Blood clearance of the control phase is about 1 L/h (66 mL/min/kg), almost twice the literature total blood clearance of 36.7 mL/min/kg that the paper used, with the literature urinary fraction of 0.305, to derive CLr. The fitted model therefore implies a urinary recovery of about 17%, not 30.5%; this is a property of the published parameter set (the paper notes that CLr and CLactive could not be estimated together without urinary or biliary data), not of the implementation.

Assumptions and deviations

  • Rest-of-body blood flow. Table S2 lists a cardiac output of 6.62 L/h (the blood / lungs row), and equation S2 lumps flows as sums, but the systemic organs of the rest of the body sum to 4.89 L/h while the hepatic inflow sums to 1.153 L/h – 0.58 L/h short of cardiac output (lungs are in series and cannot be summed). The maintainers set Qrob = QCO - Qh (5.47 L/h) so that blood leaving and returning to the blood pool balance. Taking the organ sum 4.89 L/h with cardiac output 4.89 + 1.153 = 6.04 L/h instead changes every Delta R1 output by less than 1.3% over the 30-min imaging window. Taking the organ sum with cardiac output 6.62 L/h would destroy 0.58 L/h x c_blood of drug and leave only about 63% of the dose recovered in urine plus bile, contradicting the paper’s statement that 17% and 83% of the dose are excreted.
  • Hepatic return term in S1. The printed blood equation of system S1 returns Q_liv a_liv,int / (V_liv,int K_liv,int-b) from the liver; the subscripts are read as the hepatic venous outflow of the liver extracellular space, Qh a_liv,extr / (V_liv,extr K_liv,extr-b), which is the outflow term of the liver extracellular equation in the same system and eq 6.
  • Blood volume. The blood compartment volume is the Table S2 total blood volume (15.77 mL); organ blood volumes are also counted inside the organ compartments, as in the paper.
  • Splanchnic partition coefficient. Equation S2 averages organ partition coefficients weighted by volume. The maintainers used the organs’ extracellular volumes as the weights, which makes the lumped coefficient identical to equation S3 applied to the summed interstitial and blood volumes.
  • Relaxivities in model(). The ex vivo relaxivities of Table 1 are constants of the measurement, not model parameters, so the model returns Delta R1 at both field strengths instead of switching on a field-strength covariate.
  • Rifampicin phase. The paper estimated the rifampicin-phase active uptake and biliary clearance as separate values, not as a function of rifampicin concentration. CONMED_RIFAMPICIN_SD = 1 therefore represents only the studied design (single 10 mg/kg intravenous rifampicin 1 h before gadoxetate).
  • Parameter sets not packaged. The blood-only fits (Tables S3 and S5), the fit that also estimated CLr (Table S4, in which CLactive was unidentifiable) and the five-compartment liver variant (Table S6) are sensitivity analyses the authors did not adopt, and are not included.
  • No variability. The fits were naive pooled with unweighted least squares on Delta R1; no between-animal variability or residual-error magnitude was reported, so the model is deterministic.
  • Correction search. No erratum or correction notice for this article was linked to this article in Europe PMC as of 2026-09-28.