Skip to contents

Flip-flop and local identifiability (Kuan 2023)

Source: vignettes/articles/Kuan_2023_flipflop.Rmd
Kuan_2023_flipflop.Rmd

Model and source

  • Citation: Kuan IHS, Wright DFB, Duffull SB. The influence of flip-flop in population pharmacokinetic analyses. CPT Pharmacometrics Syst Pharmacol. 2023;12(3):285-287. doi:10.1002/psp4.12909. PMID 36647235. PMCID PMC10014047. Parameter values transcribed from Supplement Appendix S1 Table S1 (Permutation 1 column); the permutation algebra is main-text Table 1.
  • Article: https://doi.org/10.1002/psp4.12909
  • Supplement (Appendices S1-S3): https://doi.org/10.1002/psp4.12909, Supporting Information

This paper contributes two model files, one per permutation of the same one-compartment model:

  • Kuan_2023_oral1cmt_permutation1 – the ka > k branch.
  • Kuan_2023_oral1cmt_permutation2 – the ka < k branch.
m1 <- rxode2::rxode(readModelDb("Kuan_2023_oral1cmt_permutation1"))
m2 <- rxode2::rxode(readModelDb("Kuan_2023_oral1cmt_permutation2"))

tibble::tibble(
  Model = c("Kuan_2023_oral1cmt_permutation1", "Kuan_2023_oral1cmt_permutation2"),
  ka    = c(exp(m1$theta[["lka"]]), exp(m2$theta[["lka"]])),
  V     = c(exp(m1$theta[["lvc"]]), exp(m2$theta[["lvc"]])),
  CL    = c(exp(m1$theta[["lcl"]]), exp(m2$theta[["lcl"]]))
) |>
  mutate(k = CL / V) |>
  relocate(k, .after = ka) |>
  knitr::kable(
    digits  = 3,
    caption = "Appendix S1 Table S1. Both permutations of the one-compartment model."
  )
Appendix S1 Table S1. Both permutations of the one-compartment model.
Model ka k V CL
Kuan_2023_oral1cmt_permutation1 0.5 0.1 10 1
Kuan_2023_oral1cmt_permutation2 0.1 0.5 2 1

What the paper is about

“Flip-flop” describes a model whose absorption and elimination rate constants appear to be switched. Kuan 2023 argues that this is not really a switch but a permutation of the rank order of the parameter values, and therefore an issue of local identifiability: a finite set of parameter vectors – not a single one – reproduces the same input-output relationship. For a mammillary model with n compartments and a depot, there are n + 1 such permutations.

The one-compartment case has two, and Appendix S1 Table S1 gives both numerically. The whole point of this vignette is that those two parameter sets must produce the same concentration-time curve.

Population

There is no population. Both model files encode an author-chosen mathematical demonstration – no drug, no patients, no fitted estimates – so every value is wrapped in fixed(). This follows the precedent of Beal_2001_iv1cmt_bql, the other methodology-reference toy model in the library.

str(m1$population)
#> List of 7
#>  $ species      : chr "None (methodology paper; a single deterministic simulated profile of a hypothetical drug, not a fit of any real molecule)."
#>  $ n_subjects   : int 1
#>  $ disease_state: chr "N/A (algebraic demonstration of local identifiability)."
#>  $ dose_range   : chr "Single unit dose, D = 1, with F = 1 (Appendix S1 Table S1)."
#>  $ regions      : chr "N/A"
#>  $ scope_note   : chr "Filed under inst/modeldb/pharmacokinetics/ rather than specificDrugs/ because there is no drug, following the p"| __truncated__
#>  $ notes        : chr "Appendix S1: 'Simulations using all the possible permutations for a one-compartment model was performed to illu"| __truncated__

The paper’s motivating analysis (main text and Appendix S3) is a separate metformin population fit: 55 participants contributing 426 plasma concentrations pooled from Kuan 2021, Dissanayake 2017 and Pentikainen 1979, with creatinine clearance (Cockcroft-Gault, computed on ideal body weight) spanning 9.5 to 167.0 mL/min. That model is not encodable and is not included here – see Errata below.

Source trace

Equation / parameter Value Source location
Structural model 1-compartment, first-order in and out Main text, p. 286 (“A one-compartment with first-order input and output model was fit to the data”); Appendix S1
lka (permutation 1) log(0.5) Appendix S1 Table S1, Permutation 1: ka = 0.5
lvc (permutation 1) log(10) Appendix S1 Table S1, Permutation 1: V = 10
lcl (permutation 1) log(1) Derived: CL = V * k = 10 * 0.1; Table S1 gives k = 0.1. Main text p. 286: “CL remains unchanged in both permutations and is therefore invariant to flip-flop”
lfdepot (permutation 1) log(1) Appendix S1 Table S1, Permutation 1: F = 1
lka (permutation 2) log(0.1) Appendix S1 Table S1, Permutation 2: ka = 0.1
lvc (permutation 2) log(2) Appendix S1 Table S1, Permutation 2: V = 2
lcl (permutation 2) log(1) Derived: CL = V * k = 2 * 0.5; Table S1 gives k = 0.5
lfdepot (permutation 2) log(1) Appendix S1 Table S1, Permutation 2: F = 1
Dose D = 1 at time = 0 Appendix S1 Table S1
Observation grid t = 0:100 Appendix S1 Table S1
1-cmt permutation algebra see below Main text Table 1
2-cmt permutation algebra see below Main text Table 1; Appendix S2

Virtual cohort

The source simulation is deterministic: one profile per permutation, no between-subject variability and no residual error (none is reported, so none is encoded). The “cohort” is therefore two single profiles, far below the 200-per-arm cap.

# Appendix S1 Table S1 specifies t = 0:100. A step of 0.1 rather than 1 is used
# so that the trapezoidal NCA below resolves Tmax (analytically 4.02); the
# plotted curve is unaffected. See "Assumptions and deviations".
tgrid <- seq(0, 100, by = 0.1)

events <- rxode2::et(amt = 1, cmt = "depot", time = 0) |>
  rxode2::et(time = tgrid, cmt = "central")

Note that observation rows use cmt = "central" – the ODE state – not the algebraic observable Cc; rxode2 returns Cc as a column regardless.

Simulation 

# Neither model declares an eta, so `omega = NA` must NOT be passed (it would
# fail in rep(0, dim(NA)[1])), and zeroRe() is unnecessary.
solve_perm <- function(mod, label) {
  rxode2::rxSolve(mod, events = events) |>
    as.data.frame() |>
    # rxSolve omits `id` for a single-subject event table; PKNCA needs it.
    mutate(id = 1L, permutation = label)
}

sim <- bind_rows(
  solve_perm(m1, "Permutation 1"),
  solve_perm(m2, "Permutation 2")
)

stopifnot(
  nrow(sim) == 2 * length(tgrid),
  !anyNA(sim$Cc),
  # PKNCA takes log() of the tail; a solve that dipped negative would give NaN.
  all(sim$Cc >= 0)
)

Replicate Figure S1 

# Replicates Figure S1 of Kuan 2023: "Simulations of the permutations that
# provide the same input-output relationship for a one compartment
# pharmacokinetic model." The curves are plotted with different linetypes and
# widths so that the (exact) overlap is visible rather than one hiding the other.
ggplot(sim, aes(time, Cc, colour = permutation, linetype = permutation)) +
  geom_line(aes(linewidth = permutation)) +
  scale_linewidth_manual(values = c("Permutation 1" = 1.6, "Permutation 2" = 0.6)) +
  scale_linetype_manual(values = c("Permutation 1" = "solid", "Permutation 2" = "22")) +
  coord_cartesian(xlim = c(0, 60)) +
  labs(
    x = "Time (time_unit)", y = "Cc (dose_unit / vol_unit)",
    title = "Figure S1 - the two permutations superimpose",
    caption = "Replicates Figure S1 of Kuan 2023."
  ) +
  theme(legend.position = "bottom")

Validation

Every check below is deterministic – there is no random effect anywhere in either model, so both sides of each comparison are exact solves of the same system and the only difference is numerical. Tight tolerances are therefore correct here, and are deliberately not the loose cohort-quantile bounds used in vignettes that simulate a random population.

Gate 1: the permutations superimpose (the paper’s central claim) 

w <- sim |>
  select(time, permutation, Cc) |>
  pivot_wider(names_from = permutation, values_from = Cc)

abs_diff <- max(abs(w$`Permutation 1` - w$`Permutation 2`))
big      <- w$`Permutation 1` > 1e-6
rel_diff <- max(abs((w$`Permutation 1` - w$`Permutation 2`) / w$`Permutation 1`)[big])

c(max_abs_difference = abs_diff, max_relative_difference = rel_diff)
#>      max_abs_difference max_relative_difference 
#>            4.857226e-17            3.038369e-15

# Achieved 4.9e-17 (absolute) and 3.0e-15 (relative): machine precision.
# Bound set many orders of magnitude above that but still far below any real
# structural error -- a single mis-transcribed parameter moves these to O(1).
stopifnot(abs_diff < 1e-10, rel_diff < 1e-8)

Gate 2: each permutation matches the analytic solution 

# C(t) = F * D * ka / (V * (ka - k)) * (exp(-k t) - exp(-ka t))
Canalytic <- function(t, F, D, ka, k, V) {
  (F * D * ka) / (V * (ka - k)) * (exp(-k * t) - exp(-ka * t))
}

closed_form <- sim |>
  group_by(permutation) |>
  mutate(
    Cref = Canalytic(
      time, F = 1, D = 1,
      ka = first(ka), k = first(kel), V = first(vc)
    )
  ) |>
  summarise(
    max_abs_error = max(abs(Cc - Cref)),
    max_rel_error = max((abs(Cc - Cref) / Cref)[Cref > 1e-6]),
    .groups = "drop"
  )

knitr::kable(closed_form, digits = 20,
             caption = "Numerical solve vs. the closed-form solution.")
Numerical solve vs. the closed-form solution.
permutation max_abs_error max_rel_error
Permutation 1 9.021e-17 1.691232e-14
Permutation 2 6.939e-17 1.479106e-14 

# Achieved ~2e-16 absolute / ~5e-14 relative for both permutations.
stopifnot(closed_form$max_abs_error < 1e-10, closed_form$max_rel_error < 1e-8)

Gate 3: main-text Table 1, one-compartment permutation algebra

Table 1 states the permutation rules in two parameterizations. Applying them to permutation 1 must reproduce permutation 2 exactly.

ka <- 0.5; V <- 10; k <- 0.1        # Table S1, Permutation 1
CL <- V * k

t1 <- tibble::tibble(
  Parameterization = c("(k, V, ka)", "(k, V, ka)", "(k, V, ka)",
                       "(CL, V, ka)", "(CL, V, ka)", "(CL, V, ka)"),
  Rule = c("k' = ka", "V' = (V * k) / ka", "ka' = k",
           "CL' = CL", "V' = CL / ka", "ka' = CL / V"),
  Derived  = c(ka, (V * k) / ka, k, CL, CL / ka, CL / V),
  Expected = c(0.5, 2, 0.1, 1, 2, 0.1)   # Table S1 Permutation 2 (CL = V * k = 2 * 0.5)
)

knitr::kable(t1, digits = 6,
             caption = "Table 1 permutation rules applied to Permutation 1.")
Table 1 permutation rules applied to Permutation 1.
Parameterization Rule Derived Expected
(k, V, ka) k’ = ka 0.5 0.5
(k, V, ka) V’ = (V * k) / ka 2.0 2.0
(k, V, ka) ka’ = k 0.1 0.1
(CL, V, ka) CL’ = CL 1.0 1.0
(CL, V, ka) V’ = CL / ka 2.0 2.0
(CL, V, ka) ka’ = CL / V 0.1 0.1 

stopifnot(max(abs(t1$Derived - t1$Expected)) < 1e-12)

Note the asymmetry the paper highlights: under (k, V, ka) the permutation is complete – the two rate constants simply exchange – whereas under (CL, V, ka) clearance is unchanged, and it is V and ka that are functions of the other parameters. Clearance is the flip-flop invariant.

Gate 4: Table 1 / Appendix S2, two-compartment permutation algebra

The two-compartment model has n + 1 = 3 permutations. The paper gives these symbolically only (Table 1 and the substitutions of Appendix S2) with no numeric values, so no model file is generated for this case. The identity can still be checked: the values below are chosen here purely to evaluate the paper’s algebra and are not from the paper.

C2 <- function(t, F, D, ka, alpha, beta, k21, Vc) {
  (F * D * ka / Vc) * (
    (k21 - alpha) * exp(-alpha * t) / ((ka - alpha) * (beta - alpha)) +
    (k21 - beta)  * exp(-beta  * t) / ((ka - beta)  * (alpha - beta)) +
    (k21 - ka)    * exp(-ka    * t) / ((alpha - ka) * (beta - ka))
  )
}

tt <- seq(0, 60, by = 0.05)
ka2 <- 0.3; alpha <- 1.0; beta <- 0.1; k21 <- 0.2; Vc <- 5
# Sanity: a valid 2-cmt disposition requires beta < k21 < alpha and k10, k12 > 0.
stopifnot(beta < k21, k21 < alpha,
          alpha * beta / k21 > 0,
          alpha + beta - k21 - alpha * beta / k21 > 0)

p1 <- C2(tt, 1, 1, ka2,  alpha, beta,  k21, Vc)
# Permutation 2: alpha' = ka, beta' = beta, ka' = alpha, Vc' = Vc * alpha / ka
p2 <- C2(tt, 1, 1, alpha, ka2,  beta,  k21, Vc * alpha / ka2)
# Permutation 3: alpha' = alpha, beta' = ka, ka' = beta, Vc' = Vc * beta / ka
p3 <- C2(tt, 1, 1, beta,  alpha, ka2,   k21, Vc * beta / ka2)

relmax <- function(x, y) max((abs(x - y) / abs(y))[abs(y) > 1e-9])
two_cmt <- c(perm2_vs_perm1 = relmax(p2, p1), perm3_vs_perm1 = relmax(p3, p1))
two_cmt
#> perm2_vs_perm1 perm3_vs_perm1 
#>   1.227647e-15   2.308361e-16

# Achieved 1.2e-15 and 2.3e-16.
stopifnot(all(two_cmt < 1e-8))

PKNCA validation

The paper makes a specific, testable claim about noncompartmental analysis (main text, p. 286):

“noncompartmental analyses are unaffected by a model being in either a ‘flip’ or a ‘flop’ state”

and

“the standard exposure relationship AUC = Dose/CL remains true irrespective of whether the system is in a state of ‘flip’ or ‘flop’”.

NCA is therefore the right validation tool here, and the gate is that every NCA parameter must be identical across the two permutations.

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

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

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | permutation + id)

dose_df <- sim_nca |>
  dplyr::distinct(id, permutation) |>
  dplyr::mutate(time = 0, amt = 1)
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | permutation + id)

# Two intervals. The terminal-slope interval starts at t = 40 because a
# lambda.z window opening just after Tmax still contains the fast exponential
# and biases the estimate (0.0997 rather than 0.1000, so half-life reads 6.95
# instead of 6.93). By t = 40 the fast term has decayed to ~1e-6 of the slow
# one. See failure pattern 11: fit the terminal slope well clear of the
# distribution phase.
intervals <- data.frame(
  start      = c(0, 40),
  end        = c(Inf, Inf),
  cmax       = c(TRUE,  FALSE),
  tmax       = c(TRUE,  FALSE),
  aucinf.obs = c(TRUE,  FALSE),
  cl.obs     = c(TRUE,  FALSE),
  half.life  = c(FALSE, TRUE),
  lambda.z   = c(FALSE, TRUE)
)

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

nca_tidy <- as.data.frame(nca_res) |>
  # lambda.z / half.life are computed on BOTH intervals (aucinf needs lambda.z);
  # keep the terminal-window value for those and the full-window value otherwise.
  dplyr::filter(
    (PPTESTCD %in% c("half.life", "lambda.z") & start == 40) |
      (PPTESTCD %in% c("cmax", "tmax", "aucinf.obs", "cl.obs") & start == 0)
  ) |>
  dplyr::select(permutation, PPTESTCD, PPORRES)

stopifnot(nrow(nca_tidy) == 12L)   # 6 parameters x 2 permutations; guard pattern 10

NCA is invariant to the permutation 

nca_wide <- nca_tidy |>
  pivot_wider(names_from = permutation, values_from = PPORRES)

nca_wide |>
  mutate(`Absolute difference` = abs(`Permutation 1` - `Permutation 2`)) |>
  dplyr::rename("NCA parameter" = PPTESTCD) |>
  knitr::kable(digits = 10,
               caption = "Every NCA parameter is identical across permutations.")
Every NCA parameter is identical across permutations.
NCA parameter Permutation 1 Permutation 2 Absolute difference
cmax 0.0668731 0.0668731 0
tmax 4.0000000 4.0000000 0
aucinf.obs 0.9999535 0.9999535 0
cl.obs 1.0000465 1.0000465 0
lambda.z 0.1000000 0.1000000 0
half.life 6.9314718 6.9314718 0 

stopifnot(
  nrow(nca_wide) == 6L,
  max(abs(nca_wide$`Permutation 1` - nca_wide$`Permutation 2`)) < 1e-10
)

Simulated NCA against the analytic reference

The paper reports no NCA table, so the reference column below is the analytic value implied by the paper’s own Table S1 parameters, not a transcribed published number.

val <- function(p) nca_wide$`Permutation 1`[nca_wide$PPTESTCD == p]

cmp <- tibble::tibble(
  `NCA parameter` = c("Cmax", "Tmax", "AUC0-inf (obs)", "CL/F (obs)", "t-half", "lambda-z"),
  Simulated = c(val("cmax"), val("tmax"), val("aucinf.obs"),
                val("cl.obs"), val("half.life"), val("lambda.z")),
  `Analytic reference` = c(
    Canalytic(log(0.5 / 0.1) / (0.5 - 0.1), 1, 1, 0.5, 0.1, 10),  # Cmax
    log(0.5 / 0.1) / (0.5 - 0.1),                                  # Tmax
    1,                                                             # AUC = Dose/CL = 1/1
    1,                                                             # CL = V * k = 1
    log(2) / 0.1,                                                  # slower rate constant
    0.1                                                            # slower rate constant
  ),
  Source = c(
    "Closed form at Tmax", "ln(ka/k)/(ka-k)",
    "Dose/CL, main text p. 286", "Table S1: CL = V * k",
    "ln(2)/0.1", "the SLOWER of ka and k"
  )
) |>
  mutate(`Percent difference` = 100 * (Simulated - `Analytic reference`) /
           `Analytic reference`)

knitr::kable(cmp, digits = 6,
             caption = "Simulated NCA vs. the analytic values implied by Table S1.")
Simulated NCA vs. the analytic values implied by Table S1.
NCA parameter Simulated Analytic reference Source Percent difference
Cmax 0.066873 0.066874 Closed form at Tmax -0.001398
Tmax 4.000000 4.023595 ln(ka/k)/(ka-k) -0.586410
AUC0-inf (obs) 0.999953 1.000000 Dose/CL, main text p. 286 -0.004652
CL/F (obs) 1.000047 1.000000 Table S1: CL = V * k 0.004652
t-half 6.931472 6.931472 ln(2)/0.1 0.000000
lambda-z 0.100000 0.100000 the SLOWER of ka and k 0.000000 

# Tmax is exempt from the percentage gate: the observation grid has a 0.1 step
# and the true Tmax is 4.0236, so the grid maximum necessarily lands at 4.0.
# That is a discretisation artefact of the sampling schedule, not a model
# error, and the honest assertion is that it agrees to within one grid step.
tight <- cmp |> dplyr::filter(`NCA parameter` != "Tmax")

# Achieved: 0.0047 percent (AUC and CL/F), 0.0014 percent (Cmax), 4e-7 percent
# (half-life, lambda.z). These are trapezoidal-discretisation errors on a fixed
# grid, not solver noise, so they are reproducible -- but the bound is set an
# order of magnitude above the worst of them so that a future PKNCA AUC-method
# change does not turn this red. It still goes red loudly for what it is meant
# to catch: a mis-transcribed CL, V, ka or dose moves these by tens of percent.
stopifnot(max(abs(tight$`Percent difference`)) < 0.05)

tmax_sim <- cmp$Simulated[cmp$`NCA parameter` == "Tmax"]
tmax_ref <- cmp$`Analytic reference`[cmp$`NCA parameter` == "Tmax"]
stopifnot(abs(tmax_sim - tmax_ref) <= 0.1 + 1e-12)   # within one observation-grid step

Both permutations return AUC0-inf = 1 and CL/F = 1, confirming AUC = Dose/CL in each state, and both return the same terminal half-life – because the terminal slope always recovers the slower of the two rate constants, which is k = 0.1 under permutation 1 and ka = 0.1 under permutation 2. That is precisely the flip-flop ambiguity, and it is invisible to NCA.

What NCA does not recover

Clearance is invariant, but volume is not: Table 1 gives V' = V * k / ka, so the two permutations have genuinely different volumes (10 and 2). The usual NCA volume Vz = CL / lambda_z returns a single number, which can only be right for one of them.

Vz <- val("cl.obs") / val("lambda.z")

tibble::tibble(
  Quantity = c("Vz from NCA (both permutations)",
               "True V, permutation 1", "True V, permutation 2"),
  Value = c(Vz, exp(m1$theta[["lvc"]]), exp(m2$theta[["lvc"]]))
) |>
  knitr::kable(digits = 4,
               caption = "Volume is permuted by flip-flop; NCA cannot resolve it.")
Volume is permuted by flip-flop; NCA cannot resolve it.
Quantity Value
Vz from NCA (both permutations) 10.0005
True V, permutation 1 10.0000
True V, permutation 2 2.0000 

# Vz matches permutation 1's volume and NOT permutation 2's. Without
# intravenous data there is no way to tell which state the system is in.
stopifnot(
  abs(Vz - exp(m1$theta[["lvc"]])) / exp(m1$theta[["lvc"]]) < 0.01,
  abs(Vz - exp(m2$theta[["lvc"]])) / exp(m2$theta[["lvc"]]) > 1
)

Assumptions and deviations

  • Observation grid. Appendix S1 Table S1 specifies t = 0:100. A step of 0.1 rather than 1 is used so trapezoidal NCA resolves Tmax = 4.02; on a unit grid the AUC is understated (failure pattern 11). The plotted profile is unchanged, and the closed-form gates hold on either grid.
  • Clearance is derived, not printed. Table S1 reports k and V; the model files store the canonical (CL, V, ka) parameterization with CL = V * k = 1. Both permutations therefore carry lcl = log(1), which is the paper’s own invariance claim made explicit in the ini() block. The paper’s k is recovered in model() as kel = cl / vc.
  • No IIV and no residual error are encoded, because the Appendix S1 demonstration is deterministic and reports none. The metformin model (Appendix S3) is described as using exponential between-subject variability and a combined residual error model, but no variance or error value is given for either.
  • Terminal-slope window. lambda.z is fit from t = 40 rather than from the default post-Tmax window, which would still contain the fast exponential and bias half-life by about 0.3 percent.
  • The two-compartment check uses non-paper values. Gate 4 evaluates the paper’s Table 1 / Appendix S2 algebra at ka = 0.3, alpha = 1.0, beta = 0.1, k21 = 0.2, Vc = 5. These were chosen for this vignette solely to evaluate an identity the paper states symbolically; they appear in no model file and are not attributable to the authors.
  • Compartment metadata is unverified (verified = FALSE). Appendix S1 names no molecule and no biological matrix, so analyte and specimen cannot be confirmed against the source.

Errata and scope

  • The metformin population model is not extractable. The main text and Appendix S3 describe it in full – data sources, BQL handling (M6 per Beal 2001), NONMEM 7.3 with FOCE-I, a one-compartment structure, exponential BSV, combined residual error, and a creatinine-clearance covariate screen on ka and on CL – but report no parameter estimate of any kind. There is no fixed-effect value, no OMEGA, no SIGMA and no covariate coefficient anywhere in the paper or its supplement; only the direction of the objective-function change is stated. Nothing can be placed in an ini() block, so no metformin model file is generated. The covariate screen is preserved as covariatesDataExcluded$CRCL in both model files.
  • Related metformin models already in the library. The paper’s own data sources are separately published. Chae_2012_metformin, Choi_2018_metformin, vanRongen_2018_metformin and Yoon_2013_metformin are already packaged; the upstream source for this paper’s largest cohort, Kuan 2021 (PLOS One, doi:10.1371/journal.pone.0246247), is not, and is the natural candidate for a future extraction.
  • No erratum. No correction or corrigendum was found for doi:10.1002/psp4.12909.
  • Article type. This is a three-page Perspective. It is extracted because Appendix S1 Table S1 fully specifies a numeric model, matching the precedent set by Beal_2001_iv1cmt_bql (a methodology paper’s simulation-only toy model, filed under pharmacokinetics/ by operator decision).