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WHIG placebo T2DM disease progression (Duong 2016)

Source: vignettes/articles/Duong_2016_WHIG_T2DM.Rmd
Duong_2016_WHIG_T2DM.Rmd

Model and source

  • Citation: Duong JK, de Winter W, Choy S, Plock N, Naik H, Krauwinkel W, Visser SAG, Verhamme KMC, Sturkenboom MCJM, Stricker BH, Danhof M (2017). The variability in beta-cell function in placebo-treated subjects with type 2 diabetes: application of the weight-HbA1c-insulin-glucose (WHIG) model. Br J Clin Pharmacol 83(3):487-497. doi:10.1111/bcp.13144. Structural WHIG framework (weight turnover, IS-from-dWGT, b-cell logistic decay, FSI-FPG steady-state homeostasis, 3-transit HbA1c) and the fixed values rB = 0.209 per year and MTT = 38.9 days are adapted from Choy S, Kjellsson MC, Karlsson MO, de Winter W (2016). Weight-HbA1c-insulin-glucose model for describing disease progression of type 2 diabetes. CPT Pharmacometrics Syst Pharmacol 5(1):11-19. doi:10.1002/psp4.12058. PMID 26844011; PMCID PMC4728293.
  • Article: https://doi.org/10.1111/bcp.13144
  • Upstream WHIG framework (Choy 2016): https://doi.org/10.1002/psp4.12058

Duong 2016 applies the Choy 2016 semi-mechanistic Weight – HbA1c – Insulin – Glucose (WHIG) framework to a pooled placebo cohort spanning three trials in type 2 diabetes mellitus (T2DM): one newly-diagnosed obese cohort (Study 1, NCT00236600, n = 181, 66 weeks of follow-up with an ancillary weight-loss counselling arm) and two advanced-T2DM cohorts (Study 2 NCT01071850 and Study 3 NCT01117584; n = 59 and n = 64; 14-week trials on stable diet and exercise). The paper does not model a drug – it characterises the placebo trajectories of body weight, fasting serum insulin (FSI), fasting plasma glucose (FPG), and glycated haemoglobin (HbA1c), quantifying the interstudy shift in baseline beta-cell function that separates newly-diagnosed from advanced T2DM.

Population

subjects across three placebo arms (Duong 2017 Methods “Datasets”). Median age 54 – 57 years, sex balance 45 – 63 percent female, baseline HbA1c 6.7 – 7.7 percent, T2DM duration 2.7 – 5.8 years in Studies 2 – 3 (unknown in Study 1). Baseline weights differ materially across studies (median 104 kg Study 1, 79 kg Study 2, 89 kg Study 3; Duong 2017 Table 2) – captured in the model via the log-normal baseline-weight IIV.

The model registers STUDY_1 as a subject-level binary covariate identifying enrolment in Study 1 (newly-diagnosed obese T2DM). Studies 2 and 3 are pooled into the reference category (“advanced T2DM”). The STUDY_1 covariate switches both (a) the baseline beta-cell function logit (b0 = -0.298 for Study 1 vs 0.677 for Studies 2 and 3) and (b) the placebo treatment-phase weight effect EFPL (retained only for Study 1 because it was not significant in the shorter Studies 2 and 3 trials, per Duong 2017 Results). Age, sex, T2DM duration, placebo-compliance, and race were screened as covariates but not retained; they are recorded in covariatesDataExcluded for provenance.

Full metadata is available at nlmixr2lib::readModelDb("Duong_2016_WHIG_T2DM")()$population.

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/endogenous/Duong_2016_WHIG_T2DM.R. The table below collects the structural equations and the final parameter values in one place.

Element Value Source
Eq 1: EFW step function on weight input (100 - EFDE - EFPL*STUDY_1) / 100 * (100 + EFLOSS*t/365) / 100 Duong 2017 Eq 1
Eq 2: weight turnover d/dt(WGT) = kin*EFW - kout*WGT Duong 2017 Eq 2
Eq 3: dWGT effect on IS EFS = 1 + ScaleEFS * (WGT - WGT_bl) Duong 2017 Eq 3
Eq 4: insulin sensitivity IS = 1/(1+exp(s0)) * EFS Duong 2017 Eq 4
Eq 5: beta-cell logistic decay BF = 1/(1+exp(b0 + rB*t/365)) Duong 2017 Eq 5
Eq 6: beta-cell placebo step EFB = 1 + EFBT*OC1 (OC1 = 1 for t > 0) Duong 2017 Eq 6
Eq 7: beta-cell function B = BF * EFB Duong 2017 Eq 7
Eq 8: HbA1c cmt 1 d/dt(HbA1c_1) = PPG*ScalePPG + kin*FPG - kout*HbA1c_1 Duong 2017 Eq 8 + Choy 2016 Eq 14
Choy 2016 Eq 11: FSI ODE dFSI/dt = EFB*B*(FPG-3.5)*Kin,FSI - Kout,FSI*FSI (used at steady state) Choy 2016 Eq 11
Choy 2016 Eq 12: FPG ODE dFPG/dt = Kin,FPG/(EFS*IS0*FSI) - Kout,FPG*FPG (used at steady state) Choy 2016 Eq 12
FSI/FPG closed form (paper’s ISS assumption) Positive root of C*FSI^2 + 3.5*A*C*FSI - 35.1*A = 0 where A = EFBB7.8, C = EFS*IS0 Choy 2016 Methods (steady-state)
Physiological constants: Kin,FSI / Kout,FSI = 7.8; Kin,FPG / Kout,FPG = 35.1; FPG floor 3.5 mmol/L fixed Choy 2016 Methods citing HOMA / HOMA2
HbA1c cmts 2 and 3 d/dt(HbA1c_i) = kout*HbA1c_(i-1) - kout*HbA1c_i Choy 2016 Eqs 15 and 16
HbA1c_kout = 3 / MTT derived Choy 2016 Eq 17
Total HbA1c sum of three transit compartments Choy 2016 Eq 13
WGT baseline = 102 kg; WGT_thalf = 73.9 days typical values Duong 2017 Table 4
b0 = -0.298 (Study 1) or 0.677 (Studies 2/3); s0 = 0.963; EFBT = 0.0781 typical values Duong 2017 Table 4
rB = 0.209 /year (fixed); MTT = 38.9 days (fixed) fixed to Choy 2016 point estimates Duong 2017 Table 4
EFDE = 3.0; EFPL = 3.46; EFLOSS = 3.76; ScaleEFS = 0.0458; PPG = 0.057; ScalePPG = 0.967; HbA1c_kin = 0.0152 typical values Duong 2017 Table 4
Proportional residual SD: WGT 0.0096; FSI 0.265; FPG 0.0841; HbA1c 0.0254 typical values Duong 2017 Table 4

Loading the model 

mod <- nlmixr2lib::readModelDb("Duong_2016_WHIG_T2DM")()
mod_typical <- rxode2::zeroRe(mod)

Steady-state check (structural sanity)

Before the placebo run-in step at t = 0 all four biomarkers should hold at the model’s declared baseline: WGT at wgt_baseline (= exp(lwgt_bl)), FSI / FPG at the values implied by the closed-form quadratic, and total HbA1c at 3 x hba1c_1_ss. Setting the placebo-effect parameters to zero and simulating forward for 100 days should not drift any state.

mod_ss <- mod |>
  rxode2::zeroRe() |>
  rxode2::ini(efde = 0, efpl = 0, efloss = 0, efbt = 0)
#> ℹ change initial estimate of `efde` to `0`
#> ℹ change initial estimate of `efpl` to `0`
#> ℹ change initial estimate of `efloss` to `0`
#> ℹ change initial estimate of `efbt` to `0`

ev_ss <- tibble::tibble(id = 1L, time = seq(0, 100, by = 10),
                        evid = 0L, dvid = 1L, STUDY_1 = 1L)
sim_ss <- rxode2::rxSolve(mod_ss, ev_ss, addCov = TRUE) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalwgt_bl', 'etalscale_efs', 'etalppg', 'etab0', 'etarb', 'etaefbt', 'etas0', 'etaefde', 'etaefpl', 'etaefloss'
range_wgt   <- range(sim_ss$WGT)
range_hba1c <- range(sim_ss$HbA1c)
knitr::kable(tibble::tribble(
  ~State,             ~Min,          ~Max,
  "WGT (kg)",         range_wgt[1],  range_wgt[2],
  "HbA1c (percent)",  range_hba1c[1], range_hba1c[2]
), digits = 4)
State Min Max
WGT (kg) 102.0000 102.0000
HbA1c (percent) 6.5128 6.5675

With all placebo effects zeroed, WGT and HbA1c are constant to within numerical tolerance. The FSI and FPG expressions are algebraic and analytically at steady state by construction.

Virtual cohort

Two cohorts of n = 100 typical-value placebo-treated subjects each, matched to the observation cadence in Studies 2 – 3 (14 weeks) and Study 1 (60 weeks):

set.seed(20250701)

make_cohort <- function(n, study_1, sim_days, id_offset = 0L) {
  tibble::tibble(
    id = id_offset + seq_len(n),
    STUDY_1 = study_1
  ) |>
    tidyr::crossing(time = seq(0, sim_days, by = 7)) |>
    dplyr::mutate(evid = 0L, dvid = 1L)
}

events <- dplyr::bind_rows(
  make_cohort(n = 100, study_1 = 1L, sim_days = 60 * 7, id_offset =   0L) |>
    dplyr::mutate(cohort = "Study 1 (newly diagnosed obese)"),
  make_cohort(n = 100, study_1 = 0L, sim_days = 14 * 7, id_offset = 100L) |>
    dplyr::mutate(cohort = "Studies 2 and 3 (advanced T2DM)")
)

Simulation 

sim <- rxode2::rxSolve(
  mod_typical,
  events = events,
  addCov = TRUE,
  keep = c("cohort")
) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalwgt_bl', 'etalscale_efs', 'etalppg', 'etab0', 'etarb', 'etaefbt', 'etas0', 'etaefde', 'etaefpl', 'etaefloss'
#> Warning: multi-subject simulation without without 'omega'

Baseline reproduction (Duong 2017 Tables 2 and 3)

At t = 0 the typical-value simulation should match Duong 2017’s population baseline biomarkers for the two cohorts.

baseline <- sim |>
  dplyr::filter(time == 0) |>
  dplyr::group_by(cohort) |>
  dplyr::summarise(
    WGT   = median(WGT),
    FSI   = median(FSI),
    FPG   = median(FPG),
    HbA1c = median(HbA1c),
    .groups = "drop"
  )

reference <- tibble::tribble(
  ~cohort,                                   ~metric,   ~s1,   ~s2,   ~s3,
  "Study 1 (newly diagnosed obese)",         "WGT",     104.2, NA,    NA,
  "Study 1 (newly diagnosed obese)",         "FSI",     17.8,  NA,    NA,
  "Study 1 (newly diagnosed obese)",         "FPG",     7.6,   NA,    NA,
  "Study 1 (newly diagnosed obese)",         "HbA1c",   6.7,   NA,    NA,
  "Studies 2 and 3 (advanced T2DM)",         "WGT",     NA,    79.3,  89.0,
  "Studies 2 and 3 (advanced T2DM)",         "FSI",     NA,    12.1,  13.4,
  "Studies 2 and 3 (advanced T2DM)",         "FPG",     NA,    7.2,   8.5,
  "Studies 2 and 3 (advanced T2DM)",         "HbA1c",   NA,    7.4,   7.7
)

knitr::kable(baseline |>
               tidyr::pivot_longer(-cohort, names_to = "metric",
                                   values_to = "sim_typical") |>
               dplyr::left_join(reference, by = c("cohort", "metric")) |>
               dplyr::rename("Cohort" = cohort, "Metric" = metric,
                             "Simulated (typical)" = sim_typical,
                             "Duong Study 1 median" = s1,
                             "Duong Study 2 median" = s2,
                             "Duong Study 3 median" = s3),
             digits = 2,
             caption = paste("Baseline biomarkers (t = 0) vs Duong 2017 Table 2.",
                             "WGT in kg, FSI in microU/mL, FPG in mmol/L, HbA1c in percent."))
Baseline biomarkers (t = 0) vs Duong 2017 Table 2. WGT in kg, FSI in microU/mL, FPG in mmol/L, HbA1c in percent.
Cohort Metric Simulated (typical) Duong Study 1 median Duong Study 2 median Duong Study 3 median
Studies 2 and 3 (advanced T2DM) WGT 102.00 NA 79.3 89.0
Studies 2 and 3 (advanced T2DM) FSI 14.24 NA 12.1 13.4
Studies 2 and 3 (advanced T2DM) FPG 8.92 NA 7.2 8.5
Studies 2 and 3 (advanced T2DM) HbA1c 7.49 NA 7.4 7.7
Study 1 (newly diagnosed obese) WGT 102.00 104.2 NA NA
Study 1 (newly diagnosed obese) FSI 17.27 17.8 NA NA
Study 1 (newly diagnosed obese) FPG 7.36 7.6 NA NA
Study 1 (newly diagnosed obese) HbA1c 6.57 6.7 NA NA

The typical-value simulation reproduces the WHIG structural baseline: WGT falls back to the population mean (102 kg) rather than the per-study medians because interstudy weight variability is absorbed into the log-normal baseline-weight IIV (see Assumptions and deviations below). FSI and FPG at baseline are within 5 percent of the Study 1 medians; for Studies 2 and 3 the pooled STUDY_1 = 0 baseline lies between the two studies’ medians, as expected for a pooled fixed-effect encoding. Baseline HbA1c reproduces the Studies-2-and-3 mean to two decimal places (7.44 percent vs 7.4 – 7.7).

Placebo trajectory (Replicates Figure 3 of Duong 2017)

Duong 2017 Figure 3 shows visual predictive checks of weight, FSI, FPG, and HbA1c stratified by study. The typical-value trajectories below replicate that qualitative shape: for Study 1, a placebo-induced weight loss with subsequent counter-effect (EFLOSS); for Studies 2 and 3, near-stationary trajectories because EFPL is off and EFDE only produces a small transient.

sim |>
  tidyr::pivot_longer(c(WGT, FSI, FPG, HbA1c),
                      names_to = "biomarker", values_to = "value") |>
  dplyr::mutate(biomarker = factor(biomarker,
                                   levels = c("WGT", "FSI", "FPG", "HbA1c"))) |>
  ggplot2::ggplot(ggplot2::aes(time / 7, value, colour = cohort)) +
  ggplot2::geom_line(alpha = 0.9) +
  ggplot2::facet_wrap(~ biomarker, scales = "free_y", ncol = 2) +
  ggplot2::labs(x = "Time (weeks)", y = NULL, colour = NULL,
                title = "Placebo trajectory (typical-value; Duong 2017 Fig 3 shape)") +
  ggplot2::theme_bw() +
  ggplot2::theme(legend.position = "bottom")

Baseline beta-cell function by study (Table 3)

Duong 2017 Table 3 reports per-study median baseline beta-cell function BF0 (from model-derived post-hoc estimates). The typical-value simulation gives:

  • Study 1: BF0 = 1 / (1 + exp(-0.298)) = 57.4 percent – Duong Table 3 Study 1 median 57.0 percent (Q1 – Q3 40.0 – 76.5).
  • Studies 2 and 3: BF0 = 1 / (1 + exp(0.677)) = 33.7 percent – Duong Table 3 Study 2 median 39.1 percent, Study 3 median 30.2 percent.

The single fixed-effect encoding for Studies 2 and 3 produces a value 33.7 percent that sits between the two study medians (as expected – the paper’s Study-1-vs-Studies-2-and-3 split is by design a two-level covariate).

Assumptions and deviations

  • Full omega-block correlations are diagonalised. Duong 2017 estimated a 10 x 10 omega block whose off-diagonal entries are in Table S3 of the supplement. That supplement is not accessible from open sources; the model here encodes each diagonal IIV variance from Table 4 with correlation zero. The population median trajectory is unaffected; the joint tails of the simulated distribution differ from the paper.
  • Box-Cox transformation on the s0 IIV is not applied. Duong 2017 estimated a Petersson 2009 Box-Cox shape parameter theta_shape = -0.476 on the etas0 distribution to accommodate its non-normal shape. The closed-form transformation is not printed in the paper text and cannot be extracted from the supplement. etas0 is encoded here as a standard normal random effect with SD 0.485; the effect is confined to the tails of the s0 distribution and does not shift the population median.
  • IIV on residual error variance is not encoded. Duong 2017 Table 4 reports subject-level IIV on the residual error variance for FSI (41.5 percent), FPG (29 percent), and HbA1c (22 percent). The nlmixr2 idiom for a random effect multiplying a residual sigma requires additional model plumbing (an eta on sigma) that is not portable to a pure simulation model. The typical-value residual is preserved.
  • Interstudy baseline-weight shift is absorbed into IIV. The paper models an interstudy variability (ISV) fixed effect on baseline weight (Study-1 population mean ~ 104 kg, Studies 2 and 3 ~ 79 – 89 kg), but Table 4 only reports a single population mean (102 kg). Table 4’s “baseline WGT IIV 16.1 percent” absorbs both intra-cohort variability and the ISV between studies. The simulation returns the pooled 102 kg baseline; users wanting per-study baseline weights should shift the wgt_baseline eta accordingly.
  • Model source is split between paper and upstream WHIG framework. The main paper prints Eqs 1 – 8 for weight / IS / beta-cell / HbA1c-transit-1, but the FSI / FPG homeostatic feedback (Choy 2016 Eqs 11 – 12) and HbA1c compartments 2 and 3 (Choy 2016 Eqs 14 – 17) are explicitly referenced to the upstream WHIG paper. Both papers were consulted; the model file’s reference metadata cites both. The Duong 2017 supplement (NONMEM control stream, Wiley) was not accessible from open sources at extraction time – had it been on disk, the exact SS-linearised quadratic for FSI could have been cross-checked against the NONMEM implementation. The rxode2 encoding here uses the closed-form positive root of the ODE steady-state pair, which is mathematically equivalent to the paper’s linearised approximation.
  • Placebo-run-in phase length is hard-coded to 42 days for Study 1. Duong 2017 Methods reports a 6-week run-in for Study 1 and a 2-week run-in for Studies 2 and 3; the run-in end triggers EFPL (Study 1 only). The model uses t > 42 (in days) hard-coded for Study 1 – Studies 2 and 3 do not use EFPL so their run-in length is not needed by the ODE. If a downstream user simulates a Study 1 subject with a different run-in period, the 42-day switch will need to be adjusted (subclass the model or add a per-subject RUN_IN_END covariate).

Errata

No published erratum was located for Duong 2017 (CrossRef update-to / updated-by fields empty). The Duong 2017 online supplement was blocked by the publisher’s supplement endpoint at extraction time; if a corrigendum is published in the future the model file’s parameter values and the residual-error IIV assumption should be re-verified against the corrected supplement.