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Metformin (Chae 2012)

Source: vignettes/articles/Chae_2012_metformin.Rmd
Chae_2012_metformin.Rmd

Model and source

  • Citation: Chae JW, Baek IH, Lee BY, Cho SK, Kwon KI. Population pharmacokinetic and pharmacodynamic analysis of metformin using the signal transduction model. Br J Clin Pharmacol. 2012;74(5):815-823. doi:10.1111/j.1365-2125.2012.04260.x
  • Description: One-compartment population PK model with first-order absorption for oral metformin in healthy Korean adults, coupled to a three-transit Sun-Jusko signal-transduction PD model for the antihyperglycaemic effect (Chae 2012). Plasma drug concentration in the central compartment drives a Hill-type stimulation function DR = Emax * Cp^r / (EC50^r + Cp^r) that initiates a cascade of three secondary-messenger transit compartments (M1 -> M2 -> M3) with shared mean transit time tau. The third messenger M3 is the measured percent change in plasma glucose from baseline relative to a sugar-bolus control arm. Creatinine clearance enters CL/F as a power covariate with reference 106.5 mL/min and exponent 0.782.
  • Article: https://doi.org/10.1111/j.1365-2125.2012.04260.x

Population

Chae 2012 enrolled 42 healthy young Korean male volunteers (age 21-31 years, weight 61-78 kg, baseline fasting plasma glucose 98 +/- 7 mg/dL, creatinine clearance 90-123 mL/min) at a single centre in Daejeon. Each subject received a single 500 mg oral metformin tablet (Diabex 500 mg, Daewoong) after an overnight fast and consumed 12 g of sugar 20 min after dosing. Identical sampling without metformin was performed one week later to derive the antihyperglycaemic “percent-effect” PD endpoint as the time-matched percent change in plasma glucose relative to the no-metformin control arm. Per-subject demographics are Chae 2012 Table 1; the same information is available programmatically via rxode2::rxode2(readModelDb("Chae_2012_metformin"))$population after the model is loaded.

Source trace

Every parameter line in inst/modeldb/specificDrugs/Chae_2012_metformin.R carries an in-file source comment. The table below collects the equation and parameter origins in one place for reviewer convenience.

Equation / parameter Value Source location
1-compartment PK ODEs dX1/dt = -Ka*X1, dX2/dt = Ka*X1 - CL*X2/V2 n/a Chae 2012 p.817 (Population PK/PD analysis)
Signal-transduction PD: DR = Emax*Cp^r / (EC50^r + Cp^r), transit chain dMn/dt = (M_{n-1} - Mn)/tau, R = M3 n/a Chae 2012 p.818 (Methods) and Figure 2
Ka (lka) 0.41 1/h Chae 2012 Table 2 (RSE 2.43%)
CL/F (lcl) 52.6 L/h Chae 2012 Table 2 (RSE 4.18%); covariate equation p.819
V/F (lvc) 113 L Chae 2012 Table 2 (RSE 56.6%)
CL/F covariate (CRCL/106.5)^0.782 (e_crcl_cl) 0.782; ref 106.5 mL/min Chae 2012 p.819 final-PK structural equation
Mean transit time tau (ltau) 0.50 h Chae 2012 Table 2 (RSE 2.97%)
Emax (lemax) 19.8 (% effect) Chae 2012 Table 2 (RSE 3.17%)
EC50 (lec50) 3.68 ug/mL Chae 2012 Table 2 (RSE 3.89%)
Hill r (lhill) 0.55 Chae 2012 Table 2 (RSE 9.05%)
IIV CL 29.7% CV (etalcl ~ 0.08457) omega^2 = log(0.297^2 + 1) Chae 2012 Table 2 (IIV CV% column)
IIV V 22.1% CV (etalvc ~ 0.04769) omega^2 = log(0.221^2 + 1) Chae 2012 Table 2 (IIV CV% column)
IIV r 4.05% CV (etalhill ~ 0.001639) omega^2 = log(0.0405^2 + 1) Chae 2012 Table 2 (IIV CV% column)
PK residual (additive, ng/mL -> ug/mL) addSd = 0.023 Chae 2012 Table 2 (23 ng/mL, RSE 11.7%)
PD residual (proportional, fraction) propSd_pctEffect = 0.404 Chae 2012 Table 2 (40.4%, RSE 1.52%)

Virtual cohort

Original observed data from the 42-subject study are not publicly available. The figures below use a 200-subject virtual cohort whose covariate distributions approximate the published Korean-male healthy cohort (Chae 2012 Table 1) and whose sampling schedule matches the Chae 2012 study (0.5, 1, 1.5, 2, 2.5, 3, 4, 6, 8, 10, 12 h post-dose, augmented with t = 0 and dense early sampling for PKNCA).

set.seed(20120815)

n_sub <- 200L

pop <- tibble(
  ID   = seq_len(n_sub),
  AGE  = round(runif(n_sub, 21, 31), 1),
  WT   = round(rnorm(n_sub, mean = 68.63, sd = 8.14), 1),
  CRCL = pmax(80, pmin(130, round(rnorm(n_sub, mean = 106.5, sd = 16), 1))),
  treatment = "500 mg PO"
)

# Dose records (single 500 mg oral dose into depot at t = 0)
d_dose <- pop |>
  dplyr::mutate(TIME = 0, EVID = 1L, AMT = 500, CMT = "depot", DV = NA_real_)

# Observation records (sampling schedule covers absorption peak and terminal phase)
obs_times <- c(0, 0.25, 0.5, 1, 1.5, 2, 2.5, 3, 4, 6, 8, 10, 12)
d_obs <- pop[rep(seq_len(n_sub), each = length(obs_times)), ] |>
  dplyr::mutate(
    TIME = rep(obs_times, times = n_sub),
    EVID = 0L,
    AMT  = 0,
    CMT  = "Cc",
    DV   = NA_real_
  )

events <- dplyr::bind_rows(d_dose, d_obs) |>
  dplyr::arrange(ID, TIME, dplyr::desc(EVID))
stopifnot(!anyDuplicated(unique(events[, c("ID", "TIME", "EVID")])))

Simulation 

mod <- rxode2::rxode2(readModelDb("Chae_2012_metformin"))
#> ℹ parameter labels from comments will be replaced by 'label()'
sim <- rxode2::rxSolve(
  mod,
  events = events,
  keep   = c("treatment", "WT", "AGE", "CRCL")
) |>
  as.data.frame()

Replicate published figures

Figure 5A – metformin concentration VPC

The packaged model reproduces the plasma-metformin VPC shape reported in Chae 2012 Figure 5A: rapid first-order absorption, Cmax around 1.5-1.8 ug/mL at Tmax ~= 2 h, monotonic decline to ~0.2 ug/mL by 12 h.

sim |>
  dplyr::filter(time > 0) |>
  group_by(time) |>
  summarise(
    Q05 = quantile(Cc, 0.05, na.rm = TRUE),
    Q50 = quantile(Cc, 0.50, na.rm = TRUE),
    Q95 = quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "steelblue") +
  geom_line(linewidth = 0.8) +
  labs(
    x = "Time after metformin dosing (h)",
    y = "Metformin concentration (ug/mL)",
    title = "Figure 5A replication -- metformin VPC (500 mg single oral dose)",
    caption = "Median (line) and 5th/95th percentiles (band) from 200 virtual subjects."
  )

Figure 5B – antihyperglycaemic PD effect

Chae 2012 Figure 5B plots the absolute fasting plasma glucose vs. time, while the structural PD model itself fits the percent-effect endpoint R = M3 = pctEffect (percent change in plasma glucose relative to the no-metformin sugar-bolus control arm). The plot below shows the simulated percent-effect trajectory: a delayed onset peaking around 3-4 h post-dose with a maximum of approximately Emax = 19.8% and a return toward baseline by 12 h, matching the FPG nadir position in Chae 2012 Figure 5B (after accounting for the back-conversion of percent effect to absolute glucose).

sim |>
  dplyr::filter(time > 0) |>
  group_by(time) |>
  summarise(
    Q05 = quantile(pctEffect, 0.05, na.rm = TRUE),
    Q50 = quantile(pctEffect, 0.50, na.rm = TRUE),
    Q95 = quantile(pctEffect, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "darkorange") +
  geom_line(linewidth = 0.8) +
  labs(
    x = "Time after metformin dosing (h)",
    y = "Percent-effect on plasma glucose (M3, %)",
    title = "Figure 5B replication -- antihyperglycaemic percent-effect VPC",
    caption = "Median (line) and 5th/95th percentiles (band) from 200 virtual subjects."
  )

PKNCA validation

Chae 2012 does not tabulate NCA parameters for the metformin PK profile (the paper validates the model through bootstrap-CI parameter estimation and a visual predictive check rather than a side-by-side NCA comparison). The NCA block below reports simulated Cmax, Tmax, AUC0-inf, and t1/2 from the packaged model as an internal consistency check; the values land in the range expected for 500 mg oral metformin in healthy young adults (literature Cmax ~1.0-1.8 ug/mL at Tmax 2-3 h, t1/2 ~1.5-3 h).

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

# Guarantee a time=0 row per (id, treatment); pre-dose Cc = 0 for oral.
sim_nca <- dplyr::bind_rows(
  sim_nca,
  sim_nca |> dplyr::distinct(id, treatment) |>
    dplyr::mutate(time = 0, Cc = 0)
) |>
  dplyr::distinct(id, treatment, time, .keep_all = TRUE) |>
  dplyr::arrange(id, treatment, time)

dose_df <- events |>
  dplyr::filter(EVID == 1L) |>
  dplyr::transmute(id = ID, time = TIME, amt = AMT, treatment)

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

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

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

nca_summary <- as.data.frame(nca_res$result) |>
  dplyr::group_by(PPTESTCD) |>
  dplyr::summarise(
    median = median(PPORRES, na.rm = TRUE),
    q05    = quantile(PPORRES, 0.05, na.rm = TRUE),
    q95    = quantile(PPORRES, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  dplyr::mutate(
    Parameter = nlmixr2lib::ncaParamLabel(PPTESTCD)
  ) |>
  dplyr::select(`NCA parameter` = Parameter, Median = median,
                `P05` = q05, `P95` = q95)
#> Warning: There was 1 warning in `dplyr::mutate()`.
#> ℹ In argument: `Parameter = nlmixr2lib::ncaParamLabel(PPTESTCD)`.
#> Caused by warning:
#> ! ncaParamLabel(): unknown PKNCA code(s) returned as-is: 'adj.r.squared', 'clast.pred', 'lambda.z.time.first', 'lambda.z.time.last', 'r.squared', 'span.ratio'

knitr::kable(
  nca_summary,
  digits  = 3,
  caption = "Simulated NCA values from the 200-subject virtual cohort, single 500 mg PO dose. Chae 2012 does not report a paper-side NCA table for comparison.",
  align   = c("l", "r", "r", "r")
)
Simulated NCA values from the 200-subject virtual cohort, single 500 mg PO dose. Chae 2012 does not report a paper-side NCA table for comparison.
NCA parameter Median P05 P95
adj.r.squared 0.999 0.999 1.000
AUC0-∞ (obs) 9.597 5.891 14.772
Clast 0.117 0.035 0.353
clast.pred 0.117 0.035 0.356
Cmax 1.496 1.088 2.030
t½ 2.042 1.721 3.378
λz 0.339 0.205 0.403
λz n points 3.000 3.000 3.050
lambda.z.time.first 8.000 7.900 8.000
lambda.z.time.last 12.000 12.000 12.000
r.squared 1.000 0.999 1.000
span.ratio 1.959 1.184 2.396
Tlast 12.000 12.000 12.000
Tmax 2.500 1.500 3.000

Assumptions and deviations

  • Paper inconsistency on IIV placement. Chae 2012 p.819 writes the final PK structural equations as Ka = 0.41 * exp(eta1), CL/F = 52.6 * (CLcr/106.5)^0.782, V = 113 * exp(eta3) (eta2 omitted entirely from the equations, eta1 placed on Ka), but Chae 2012 Table 2 explicitly reports IIV CV% only for CL (29.7%), V (22.1%), and Hill r (4.05%) – Ka, tau, Emax, and EC50 IIV are marked “Not estimated” via the table’s dagger footnote. Table 2 carries paired bootstrap CIs for each estimated IIV. The packaged model follows Table 2 (IIV on CL, V, and Hill r) on the basis that the parameter-with-uncertainty table is the canonical source for IIV magnitude and the inline equations on p.819 contain an apparent typesetting error (eta1 likely intended for CL/F, with eta2 reserved for V and the Ka term carrying no eta).
  • CLcr derivation method. Chae 2012 Table 1 reports creatinine clearance in mL/min but does not state whether the values were measured directly or estimated via Cockcroft-Gault or another formula. The packaged model carries CRCL in raw mL/min (not BSA-normalised) with the reference 106.5 mL/min taken from the Chae 2012 cohort median.
  • PD endpoint scale. The structural PD output pctEffect = M3 is the percent change in plasma glucose relative to the no-metformin control arm. Chae 2012 Figure 5B shows the absolute fasting plasma glucose trajectory derived by back-conversion using each subject’s measured control-arm baseline; the packaged model emits the raw percent-effect variable so users can apply their own baseline-FPG conversion if desired.
  • No paper-side NCA table. Chae 2012 reports model validation through bootstrap-CI parameter precision and a visual predictive check (Figure 5) rather than a tabulated NCA comparison; the PKNCA section above therefore reports simulated NCA values only.
  • Virtual cohort. The 200-subject simulation uses the published cohort demographics (age 21-31 years uniform; weight 68.63 +/- 8.14 kg Gaussian; CRCL 106.5 +/- 16 mL/min Gaussian truncated to [80, 130]; all-male healthy Korean). No baseline-FPG variability is simulated because the percent-effect endpoint cancels the baseline term out of the PD ODEs.