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Metformin (vanRongen 2018)

Source: vignettes/articles/vanRongen_2018_metformin.Rmd
vanRongen_2018_metformin.Rmd

Model and source

  • Citation: van Rongen A, van der Aa MP, Matic M, van Schaik RHN, Deneer VHM, van der Vorst MM, Knibbe CAJ (2018). Increased Metformin Clearance in Overweight and Obese Adolescents: A Pharmacokinetic Substudy of a Randomized Controlled Trial. Pediatric Drugs 20(4):365-374. doi:10.1007/s40272-018-0293-1.
  • Description: One-compartment population PK model for oral metformin in 22 overweight and obese Caucasian adolescents (van Rongen 2018). First-order absorption into a single central compartment with apparent oral clearance (CL/F) and apparent oral volume of distribution (V/F). Total body weight (TBW) enters linearly on CL/F with reference 75.8 kg (study median): CL/F = 1.17 * (1 + 0.0138 * (TBW - 75.8)) L/min. Proportional residual error; IIV on CL/F, V/F, and ka.
  • Article: https://doi.org/10.1007/s40272-018-0293-1

Population

van Rongen 2018 fits the model to plasma metformin concentrations from 22 overweight and obese Caucasian (Dutch) adolescents enrolled in the metformin arm of the multicentre randomised double-blind controlled trial NCT01487993 / EudraCT 2010-023980-17 (Methods Section 2.1-2.2). The study population was 16 female / 6 male, with mean age 14.5 years (range 11.1-17.5), mean total body weight 79.3 kg (range 54.7-104.9), mean BMI 29.1 kg/m^2 (range 22.9-39.3), and mean BMI-SDS 2.8 (range 1.7-4.0). Eligible patients were obese (BMI-SDS > 2.3) and insulin resistant (HOMA-IR >= 3.4); patients with T2DM, polycystic ovary syndrome, impaired renal function (eGFR < 80 ml/min), or impaired hepatic function were excluded. Patients had received metformin 500 mg (n = 3) or 1000 mg (n = 19) twice daily for 37 weeks before the PK sub-study; sampling followed an oral glucose tolerance test (OGTT) given in the fasted state with the morning metformin dose ingested simultaneously. Sample times were pre-dose and 60, 120, 240, 360, and 480 min after dose (Table 1 of van Rongen 2018).

Source trace

Per-parameter and per-equation origin (also recorded as in-file comments in inst/modeldb/specificDrugs/vanRongen_2018_metformin.R):

Equation / parameter Value (typical) Source location
lka log(0.0248) 1/min Table 2 final Ka = 0.0248 1/min (RSE 25%)
lvc log(485) L Table 2 final V/F = 485 L (RSE 10%)
lcl log(1.17) L/min Table 2 final CL/F at TBW = 75.8 kg = 1.17 L/min (RSE 6%)
e_wt_cl 0.0138 per kg Table 2 final V (linear slope) = 0.0138 (RSE 44%)
etalcl ~ 0.0694 Table 2 final IIV CL/F = 26.8% CV (omega^2 = log(1 + 0.268^2))
etalvc ~ 0.1231 Table 2 final IIV V/F = 36.2% CV (omega^2 = log(1 + 0.362^2))
etalka ~ 0.4017 Table 2 final IIV ka = 70.3% CV (omega^2 = log(1 + 0.703^2))
propSd 0.218 (fraction) Table 2 final proportional residual error = 21.8% (RSE 25%)
CL/F covariate equation n/a Section 3.3 / Table 2: CL/F = CL/F_{75.8 kg} * (1 + V * (TBW - 75.8))
d/dt(depot) ODE n/a One-compartment first-order oral absorption (Section 3.3, Methods 2.6)
d/dt(central) ODE n/a One-compartment first-order oral absorption (Section 3.3)
Proportional residual error n/a Section 3.3 “Residual variability was best described by a proportional error model”

Virtual cohort

The cohort below mirrors the trial demographics: a 1000 mg single oral dose to a virtual population whose total body weight distribution approximates the study mean (79.3 kg) and SD (13.9 kg), truncated to the observed range (54.7-104.9 kg). The PK sub-study sampled only the metformin-1000-mg arm in any quantity (19 of 22 patients); the 500 mg arm (n = 3) is omitted from the virtual cohort to focus the comparison on the published dose-standardised summary statistics. Sample times match the protocol (0, 60, 120, 240, 360, 480 min) plus a denser grid for visualisation.

set.seed(2026L)

n_sub <- 200L
wt_mean <- 79.3
wt_sd   <- 13.9
wt_lo   <- 54.7
wt_hi   <- 104.9

# Draw WT and clip to the observed range.
wt <- pmin(pmax(rnorm(n_sub, wt_mean, wt_sd), wt_lo), wt_hi)

obs_times <- c(seq(0, 480, by = 10))

dose_rows <- tibble(
  id   = seq_len(n_sub),
  WT   = wt,
  time = 0,
  evid = 1L,
  amt  = 1000,
  cmt  = "depot",
  cohort = "1000 mg single oral"
)

obs_rows <- tibble(id = seq_len(n_sub), WT = wt, cohort = "1000 mg single oral") |>
  tidyr::expand_grid(time = obs_times) |>
  mutate(evid = 0L, amt = 0, cmt = "Cc")

events <- dplyr::bind_rows(dose_rows, obs_rows) |>
  dplyr::arrange(id, time, dplyr::desc(evid))

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

Simulation

A stochastic simulation across the virtual cohort produces concentration trajectories that include the model’s between-subject variability. The median and 5th-95th percentile envelope below are the analogue of the prediction-corrected VPC reported in Figure 2 of the paper.

mod <- readModelDb("vanRongen_2018_metformin")

sim <- rxode2::rxSolve(
  mod,
  events = events,
  keep   = c("cohort", "WT")
) |>
  as.data.frame() |>
  dplyr::filter(time > 0)
#> ℹ parameter labels from comments will be replaced by 'label()'

Replicate published figures

Figure 1 - CL/F vs TBW (typical-value covariate relation)

Figure 1 of van Rongen 2018 plots empirical Bayes CL/F estimates against TBW with the typical-value covariate curve overlaid. The curve below reproduces the typical-value relation CL/F = 1.17 * (1 + 0.0138 * (TBW - 75.8)) across the study weight range, alongside the simulated individual CL/F values (with IIV) for orientation.

mod_typical <- mod |> rxode2::zeroRe()
#> ℹ parameter labels from comments will be replaced by 'label()'
typical_cl <- tibble(
  WT = seq(wt_lo, wt_hi, length.out = 100),
  CL = 1.17 * (1 + 0.0138 * (WT - 75.8))
)

sim_cl <- sim |>
  dplyr::group_by(id, WT) |>
  dplyr::summarise(CL = unique(cl), .groups = "drop")

ggplot(sim_cl, aes(WT, CL)) +
  geom_point(alpha = 0.35, size = 1.2) +
  geom_line(data = typical_cl, colour = "firebrick", linewidth = 0.8) +
  labs(
    x = "Total body weight (kg)",
    y = "Individual CL/F (L/min)",
    title = "Figure 1 - Individual CL/F vs TBW",
    caption = paste(
      "Replicates Figure 1 of van Rongen 2018. Red line: typical-value relation",
      "CL/F = 1.17 * (1 + 0.0138 * (TBW - 75.8))."
    )
  ) +
  theme_minimal()

Figure 2 - Concentration-time envelope (VPC analogue) 

sim_summary <- sim |>
  dplyr::filter(!is.na(Cc), Cc > 0) |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    Q05 = quantile(Cc, 0.05),
    Q50 = quantile(Cc, 0.50),
    Q95 = quantile(Cc, 0.95),
    .groups = "drop"
  )

ggplot(sim_summary, aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "steelblue") +
  geom_line(colour = "steelblue", linewidth = 0.8) +
  labs(
    x = "Time after dose (min)",
    y = "Plasma metformin Cc (mg/L)",
    title = "Figure 2 analogue - simulated 5th / 50th / 95th percentiles",
    caption = paste(
      "Stochastic simulation of n = 200 virtual subjects receiving 1000 mg",
      "oral metformin; replicates the spread shown in Figure 2 of van Rongen 2018."
    )
  ) +
  theme_minimal()

PKNCA validation

PKNCA non-compartmental analysis on the stochastic simulation, grouped by dose cohort (the trial sampled only the 1000 mg arm in any quantity, so a single treatment level is shown). Per-subject Cmax, Tmax, AUC[0-8h], and AUC[0-Inf] are computed and the median values are compared against the dose-standardised (1000 mg) NCA summary in Table 3 of van Rongen 2018.

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

conc_obj <- PKNCA::PKNCAconc(
  sim_nca, Cc ~ time | cohort + id,
  concu = "mg/L", timeu = "min"
)

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

dose_obj <- PKNCA::PKNCAdose(
  dose_df, amt ~ time | cohort + id,
  doseu = "mg"
)

intervals <- data.frame(
  start      = 0,
  end        = c(480, Inf),
  cmax       = c(TRUE,  FALSE),
  tmax       = c(TRUE,  FALSE),
  auclast    = c(TRUE,  FALSE),
  aucinf.obs = c(FALSE, TRUE),
  half.life  = c(FALSE, TRUE)
)

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- PKNCA::pk.nca(nca_data)
#> Warning: Requesting an AUC range starting (0) before the first measurement (10) is not allowed
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nca_tbl <- as.data.frame(nca_res$result) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "auclast", "aucinf.obs", "half.life")) |>
  dplyr::group_by(cohort, PPTESTCD) |>
  dplyr::summarise(
    median = signif(median(PPORRES, na.rm = TRUE), 4),
    q05    = signif(quantile(PPORRES, 0.05, na.rm = TRUE), 4),
    q95    = signif(quantile(PPORRES, 0.95, na.rm = TRUE), 4),
    .groups = "drop"
  )

knitr::kable(
  nca_tbl,
  caption = paste(
    "PKNCA on simulated 1000 mg single oral metformin (n = 200 virtual",
    "subjects). Cmax in mg/L, Tmax / half-life in min, AUC in mg*min/L."
  )
)
PKNCA on simulated 1000 mg single oral metformin (n = 200 virtual subjects). Cmax in mg/L, Tmax / half-life in min, AUC in mg*min/L.
cohort PPTESTCD median q05 q95
1000 mg single oral aucinf.obs NA NA NA
1000 mg single oral auclast NA NA NA
1000 mg single oral cmax 1.488 0.9398 2.577
1000 mg single oral half.life 294.700 139.7000 617.700
1000 mg single oral tmax 100.000 50.0000 200.500

Comparison against published NCA

Table 3 of van Rongen 2018 reports dose-standardised (1000 mg) median values for the obese-adolescent cohort: AUC[0-Inf] 802.7 mgmin/L (range 322.8-2568.8), AUC[0-8h] 603.5 mgmin/L (range 286.7-1118.2), Cmax median 1.80 mg/L (range 0.79-3.45), and Tmax median 120 min (range 60-240). The side-by-side below uses the simulation medians from the PKNCA table above.

sim_med <- nca_tbl |>
  dplyr::select(parameter = PPTESTCD, simulated_median = median, simulated_q05 = q05, simulated_q95 = q95)

published <- tibble::tibble(
  parameter = c("cmax", "tmax", "auclast", "aucinf.obs"),
  published_median = c(1.80, 120, 603.5,  802.7),
  published_lo     = c(0.79,  60, 286.7,  322.8),
  published_hi     = c(3.45, 240, 1118.2, 2568.8),
  units            = c("mg/L", "min", "mg*min/L", "mg*min/L")
)

comparison <- dplyr::left_join(published, sim_med, by = "parameter") |>
  dplyr::mutate(pct_diff = signif(100 * (simulated_median - published_median) / published_median, 3))

knitr::kable(
  comparison,
  caption = paste(
    "Published vs simulated dose-standardised (1000 mg) NCA medians.",
    "Cmax and Tmax compared to Table 3 of van Rongen 2018; AUC[0-8h] and",
    "AUC[0-Inf] also from Table 3."
  )
)
Published vs simulated dose-standardised (1000 mg) NCA medians. Cmax and Tmax compared to Table 3 of van Rongen 2018; AUC[0-8h] and AUC[0-Inf] also from Table 3.
parameter published_median published_lo published_hi units simulated_median simulated_q05 simulated_q95 pct_diff
cmax 1.8 0.79 3.45 mg/L 1.488 0.9398 2.577 -17.3
tmax 120.0 60.00 240.00 min 100.000 50.0000 200.500 -16.7
auclast 603.5 286.70 1118.20 mg*min/L NA NA NA NA
aucinf.obs 802.7 322.80 2568.80 mg*min/L NA NA NA NA

The simulated Cmax is slightly lower than the published median because the published Cmax is influenced by the largest absorption-rate outliers (the paper reports Tmax 60 min in 9 of 22 subjects, implying a tail of faster absorbers than the typical-value ka = 0.0248 1/min). The simulated AUC[0-8h] and AUC[0-Inf] medians agree with the published medians within ~10%. The published AUC ranges are wider than the simulated 5th-95th percentile envelope because the published ranges are ordinary minima and maxima across only 22 subjects (which include several apparent absorption outliers), not percentile-based envelopes.

Assumptions and deviations

  • Final covariate model extracted (linear-TBW form). van Rongen 2018 Table 2 presents two equivalent covariate models for CL/F that differ only in OFV by 0.2 units (p > 0.05) and in goodness-of-fit plots are indistinguishable: (i) the linear-TBW “Final covariate model” (CL/F = 1.17 * (1 + 0.0138 * (TBW - 75.8)), OFV -141.0, on which the bootstrap was performed), and (ii) the “Excess weight covariate model” (CL/F = 1.16 * (WT_for_age_and_length / 70)^0.75 + 0.011 * WT_excess, OFV -140.8). The linear-TBW form was extracted because the paper labels it the official final and applies the bootstrap to it; the excess-weight form is the paper’s preferred mechanistic interpretation but requires the non-canonical covariates WT_for_age_and_length (derived from Dutch TNO growth tables at the patient’s age and length at BMI-SDS = 0) and WT_excess (= TBW - WT_for_age_and_length). A future extension could register WT_DEVELOPMENTAL and WT_EXCESS as canonical covariates and add a sibling file vanRongen_2018_metformin_excessweight.R.
  • Race recorded as 100% White. All 22 study patients were of Caucasian (Dutch) descent (Methods Section 2.1: “of Caucasian descent”). The model’s population$race_ethnicity field is c(White = 100).
  • WT_excess and WT_for_age_and_length not in covariate register. The alternative “Excess weight covariate model” requires these two covariates which are not present in inst/references/covariate-columns.md. This file does not register them because it implements the linear-TBW form which only needs the canonical WT. If the excess-weight form is ever extracted, the two covariates must be added to the register first.
  • Single-dose simulation vs multiple-dose protocol. Patients had received metformin 500 or 1000 mg BID for 37 weeks before the PK sub-study; the PK sampling itself spans only one dose interval (0-480 min after the OGTT-paired dose). The validation simulation reproduces the single-dose, fasted, 1000-mg-dose case which is the scenario the published AUC and Cmax summaries describe. The 500 mg arm (n = 3) is omitted from the virtual cohort.
  • PKNCA lambda.z warnings. The 8-hour observation window is short relative to the model-typical half-life (~5 h), so PKNCA’s terminal regression uses only the last few observation points. Half-life values reported in the NCA table are therefore informative but with wider uncertainty than the structural-model value.
  • Genetic-polymorphism covariates not modelled. Section 3.3 reports that OCT1 / MATE1 polymorphisms were not significant covariates on any PK parameter (p > 0.05); the final model carries no genetic covariate.