---
title: "Fasiglifam / TAK-875 (Naik 2013)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Fasiglifam / TAK-875 (Naik 2013)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(nlmixr2lib)
library(PKNCA)
library(rxode2)
library(dplyr)
library(tidyr)
library(ggplot2)
```

# Model and source

* Citation: `r tryCatch(readModelDb("Naik_2013_fasiglifam")()$reference, error = function(e) "Naik H et al., CPT Pharmacometrics Syst Pharmacol 2013;2:e22. doi:10.1038/psp.2012.23")`
* Description: `r tryCatch(readModelDb("Naik_2013_fasiglifam")()$description, error = function(e) "Joint popPK / FPG / HbA1c model of fasiglifam (TAK-875, GPR40 agonist) in T2DM adults.")`
* Article: <https://doi.org/10.1038/psp.2012.23>
* Supplement (control streams and supplementary figures): PSP-2012-0026-T-s05 (HbA1c), -s06 (FPG), -s07 (PK), packaged inside the paper's supplementary ZIP archive `psp4201223-sup-0005.zip`.

Fasiglifam / TAK-875 was a first-in-class G-protein-coupled receptor 40 (GPR40 / free fatty acid receptor 1) agonist in phase 3 development for type 2 diabetes mellitus (T2DM). Naik et al. (CPT PSP 2013) present the phase-2 pharmacometric analysis that supported dose selection for the phase 3 programme. Development of fasiglifam was terminated in December 2013 for signals of hepatotoxicity; this popPK / PD analysis predates and does not address that outcome.

# Population

Naik 2013 Table 1 tabulates baseline demographics for two nested datasets: a PK-only dataset (`r 286` T2DM patients, `r 1211` fasiglifam plasma samples) and a broader PK-efficacy dataset (`r 346` T2DM patients, `r 2710` FPG samples, `r 1381` HbA1c samples). Placebo subjects contribute to the PK-efficacy dataset but not to the PK-only dataset. The pooled population had mean age 51.5 years (range 21 - 79), mean body weight 86.0 kg (range 49.5 - 172.7), 53.5 % female, 82.4 % Caucasian, and 66.8 % Hispanic ethnicity. Baseline glycaemia was FPG 170.3 mg / dL and HbA1c 8.4 %; disease duration averaged 5.7 years (range 0.39 - 14.75). 76.3 % of PK-efficacy subjects were on background metformin monotherapy at study entry, matching the parent Burant 2012 phase-2 trial design (Burant CF et al., Lancet 2012;379:1403-1411; doi:10.1016/S0140-6736(11)61879-5).

The same information is available programmatically via `readModelDb("Naik_2013_fasiglifam")()$population`.

# Source trace

Per-parameter origin is recorded in the in-file comments of `inst/modeldb/specificDrugs/Naik_2013_fasiglifam.R`. Every value here is traced to Naik 2013 Table 2 and to the corresponding supplementary NONMEM control stream (s05 HbA1c, s06 FPG, s07 PK). Reference typical subject: 52-year-old female with AST 21 U / L, BFPG 163.5 mg / dL, T2DM duration 4.61 years (Naik 2013 Figure 2 tornado-plot caption).

| Equation / parameter block                        | Value (typical)  | Source location                                                    |
|---------------------------------------------------|------------------|--------------------------------------------------------------------|
| 2-cmt PK, oral, first-order absorption + elim     | n / a            | Methods "Exposure-efficacy models" / s07 $PK                       |
| CL / F (typical female)                           | 0.75 L / h       | Table 2 row CL / F; s07 TH1                                        |
| V1 / F                                            | 5.86 L           | Table 2 row V1 / F; s07 TH2                                        |
| Q / F (FIXED)                                     | 0.833 L / h      | Table 2 row Q / F "no RSE"; s07 TH3 FIX                            |
| V2 / F (FIXED)                                    | 23.7 L           | Table 2 row V2 / F "no RSE"; s07 TH4 FIX                           |
| Ka (FIXED)                                        | 0.075 1 / h      | Table 2 row Ka "no RSE"; s07 TH5 FIX                               |
| Additive sex effect on CL / F                     | + 0.31 L / h (male) | Table 2 row "Gender on CL / F"; s07 TH6 additive form            |
| omega^2 CL / F                                    | 0.488            | Table 2 row omega^2 on CL / F; s07 OMEGA1                          |
| Proportional residual (PK)                        | sqrt(0.152)      | Table 2 row sigma^2 (labelled "exponential", proportional per text); s07 SIGMA1 |
| FPG indirect response, Kout stimulation           | n / a            | Methods equation block "dA/dt = KIN - KOUT * STDP * A"             |
| BL (baseline FPG)                                 | 164 mg / dL      | Table 2 row BL; s06 TH1                                            |
| Kout (FPG)                                        | 0.00542 1 / h    | Table 2 row Kout; s06 TH2                                          |
| Emax (typical subject)                            | 0.366            | Table 2 row Emax; s06 TH3                                          |
| EC50 (typical subject)                            | 3.16 ug / mL     | Table 2 row EC50; s06 TH4                                          |
| Exponential BFPG-on-Emax coefficient              | 0.00746 / (mg/dL)| Table 2 row "BFPG on Emax"; s06 TH5                                |
| Exponential AST-on-Emax coefficient               | 0.00731 / (U/L)  | Table 2 row "AST on Emax"; s06 TH6                                 |
| omega^2 BL / omega^2 Emax                         | 0.0610 / 0.0854  | Table 2 rows omega^2 on BL / Emax; s06 OMEGA1 / OMEGA2             |
| Proportional residual (FPG, log-additive)         | sqrt(0.0187)     | Table 2 row sigma^2 (additive-on-log); s06 SIGMA1                  |
| HbA1c model dHbA1c / dt = KIG * LIFE * FPG - KA1C * HbA1c | n / a    | Methods equation block, s05 $DES                                   |
| KIG steady-state anchor: KIG = BLA1 * KA1C / BFPG | derived          | s05 $PK "KIG=BLA1*KA1C/BLI"                                        |
| Placebo factor LIFE(t) = 1 - MPL * (1 - exp(-ln 2 * t / HL)) | n / a  | s05 $PK "LIFE=1-(LF)*(1-(EXP((-LOG(2)/(SLF))*TIME)))"              |
| BLA1 (typical female, DD = 4.61, BFPG = 163.5)    | 8.25 %           | Table 2 row BLA1; s05 TH1                                          |
| KA1C                                              | 0.00052 1 / h    | Table 2 row KA1C; s05 TH2                                          |
| HL (placebo half-life, FIXED)                     | 720 h            | Table 2 row HL "(fixed)"; s05 TH3 FIX                              |
| MPL (typical female)                              | 0.0590           | Table 2 row MPL; s05 TH4                                           |
| Additive sex effect on MPL (male)                 | + 0.0363         | Table 2 row "Gender on MPL"; s05 TH5                               |
| DD-on-BLA1 (linear, additive)                     | 0.0133 % / year  | Table 2 row "DD on BLA1"; s05 TH6                                  |
| BFPG-on-BLA1 (exponential, per s05 CS)            | 0.00181 / (mg/dL)| Table 2 row "BFPG on BLA1"; s05 TH7                                |
| omega^2 BLA1 / KA1C / MPL / cov(MPL, KA1C)        | 0.0057 / 0.95 / 0.305 / 0.506 | Table 2 rows omega^2 BLA1 / KA1C / MPL / COV_MPL-KA1C; s05 OMEGA + OMEGA BLOCK(2) |
| Proportional residual (HbA1c, log-additive)       | sqrt(0.00164)    | Table 2 row sigma^2 (additive-on-log); s05 SIGMA1                  |

# Virtual cohort

Original individual patient data are not publicly available. The simulations below use a virtual cohort whose covariate distributions match the Naik 2013 Table 1 summary. Three arms are simulated matching the doses that Naik 2013 selected for phase 3 (25 mg and 50 mg once daily) and a placebo arm (dose 0 mg to exercise the placebo factor `LIFE(t)`), 200 subjects per arm (the cap recommended by the extraction skill).

```{r cohort}
set.seed(20260725)

mod <- readModelDb("Naik_2013_fasiglifam")()

# Helper: build one dose-cohort event table.
#   Continuous covariates: truncated normal to the Naik 2013 Table 1 mean +/- SD
#   Categorical SEXF: marginal 53.5 % female per Table 1
#   Observation grid: dense over day-1 for PK; weekly for FPG / HbA1c
#     out to 12 weeks (matches the phase-2 trial duration; Naik 2013
#     also simulates 24-week projections but the underlying data only
#     support 12 weeks).
make_cohort <- function(n, dose_mg, id_offset = 0L) {
  subj <- tibble::tibble(
    id           = id_offset + seq_len(n),
    SEXF         = rbinom(n, 1, 0.535),                                    # 53.5 pct female (Table 1)
    FPG          = pmin(pmax(rnorm(n, 170.3, 51.1), 90), 320),             # mg / dL; Table 1 mean +/- SD
    AST          = pmin(pmax(rnorm(n, 24.0, 11.9), 10), 70),               # U / L; Table 1 mean +/- SD; upper cap at 70 to keep Emax exponent finite
    T_DIAG_DIAB  = pmin(pmax(rgamma(n, shape = 4, scale = 5.7 / 4), 0.4), 15) # years; Table 1 mean 5.7, range 0.39 - 14.75
  )

  # Dosing: fasiglifam PO QD x 84 days (2016 h = 12 weeks). Placebo arm
  # (dose_mg = 0) has NO dosing records -- rxode2 keeps the depot state
  # at 0 for the entire simulation, exercising only the endogenous
  # dynamics and the LIFE(t) placebo factor.
  if (dose_mg > 0) {
    doses <- subj |>
      tidyr::expand_grid(time = seq(0, 2016 - 24, by = 24)) |>
      dplyr::mutate(amt = dose_mg, evid = 1L, cmt = "depot", dvid = NA_integer_)
  } else {
    doses <- tibble::tibble(
      id = integer(0), time = numeric(0), amt = numeric(0),
      evid = integer(0), cmt = character(0), dvid = integer(0)
    )
  }

  # Observation grid: dense first day for PK VPC, then twice-weekly for
  # PD trajectory. Anchor at cmt = "central" (an ODE state) with
  # dvid = 1L so rxode2 unambiguously routes the observation record;
  # rxSolve returns all three algebraic observables (Cc, glucose,
  # Hba1c) as columns of the output at every observation row.
  obs_times <- sort(unique(c(
    seq(0.5, 24, by = 0.5),                # day 1 dense PK
    seq(0, 2016, by = 24 * 3.5)            # twice-weekly for FPG / HbA1c through week 12
  )))
  obs <- subj |>
    tidyr::expand_grid(time = obs_times) |>
    dplyr::mutate(amt = NA_real_, evid = 0L, cmt = "central", dvid = 1L)

  full <- dplyr::bind_rows(doses, obs) |>
    dplyr::arrange(id, time, dplyr::desc(evid))
  full$id <- as.integer(full$id)
  full
}

events <- dplyr::bind_rows(
  make_cohort(200, dose_mg =  0, id_offset =   0L) |> dplyr::mutate(treatment = "Placebo"),
  make_cohort(200, dose_mg = 25, id_offset = 200L) |> dplyr::mutate(treatment = "25 mg QD"),
  make_cohort(200, dose_mg = 50, id_offset = 400L) |> dplyr::mutate(treatment = "50 mg QD")
)
stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))
```

# Simulation

```{r simulate}
sim <- rxode2::rxSolve(
  mod,
  events = events,
  keep = c("treatment", "SEXF")
) |> as.data.frame()

dim(sim)
```

# Replicate published figures

## Day 1 PK for the 25 mg and 50 mg dose groups

Naik 2013 does not print a per-dose-group PK-concentration figure in the main paper, but Table 2 anchors the typical-subject CL / F at 0.75 L / h (female) and 1.06 L / h (male). The population VPC (median + 5 - 95 % ribbon) for day 1 of dosing is shown below. The 50 mg arm should sit at ~ 2x the 25 mg profile since PK is linear in the modelled dose range (Naik 2013 Discussion).

```{r figure-pk-day1, fig.width = 7, fig.height = 4}
pk_day1 <- sim |>
  dplyr::filter(treatment != "Placebo", time > 0, time <= 24, !is.na(Cc)) |>
  dplyr::group_by(treatment, time) |>
  dplyr::summarise(
    Q05 = stats::quantile(Cc, 0.05),
    Q50 = stats::quantile(Cc, 0.50),
    Q95 = stats::quantile(Cc, 0.95),
    .groups = "drop"
  )

ggplot(pk_day1, aes(time, Q50, color = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.20, color = NA) +
  geom_line(linewidth = 0.8) +
  labs(
    x = "Time after first dose (h)", y = "Fasiglifam concentration (mg / L = ug / mL)",
    title = "Day 1 fasiglifam PK by dose group (25 vs 50 mg QD)",
    caption = "Median (line) and 5 - 95 pct envelope (ribbon) from 200 simulated subjects per arm."
  ) +
  theme_bw()
```

## FPG trajectory over 12 weeks

Naik 2013 Figure 3a shows the FPG VPC for the phase-2 cohort. The plot below reproduces the qualitative pattern: baseline ~ 170 mg / dL, monotone reduction with drug (25 and 50 mg arms) and no meaningful trend on placebo (Naik 2013 explicitly modelled no placebo effect on FPG). The 50 mg arm should reach a lower quasi-steady FPG than 25 mg because the drug concentration for the higher dose exceeds EC50 = 3.16 ug / mL over more of the dosing interval.

```{r figure-fpg, fig.width = 7, fig.height = 4}
fpg_summary <- sim |>
  dplyr::filter(!is.na(glucose)) |>
  dplyr::mutate(week = time / (24 * 7)) |>
  dplyr::group_by(treatment, week) |>
  dplyr::summarise(
    fpg_q05 = stats::quantile(glucose, 0.05),
    fpg_q50 = stats::quantile(glucose, 0.50),
    fpg_q95 = stats::quantile(glucose, 0.95),
    .groups = "drop"
  )

ggplot(fpg_summary, aes(week, fpg_q50, color = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = fpg_q05, ymax = fpg_q95), alpha = 0.20, color = NA) +
  geom_line(linewidth = 0.8) +
  labs(
    x = "Week of treatment", y = "FPG (mg / dL)",
    title = "Simulated FPG over 12 weeks (replicates Naik 2013 Figure 3a)",
    caption = "Placebo, 25 mg QD, and 50 mg QD; 200 subjects per arm. No placebo effect on FPG per Naik 2013 Discussion."
  ) +
  theme_bw()
```

## HbA1c trajectory over 12 weeks

Naik 2013 Figure 3b shows the HbA1c VPC. Reduction from baseline HbA1c ~ 8.25 % is expected for both dose arms, with the 50 mg arm showing a larger reduction. The placebo arm should also decrease (by ~ 0.05 - 0.09 %) via the LIFE(t) placebo factor over 12 weeks.

Reported "predicted mean reductions in HbA1c levels from baseline" (Naik 2013 Results): -0.94 % at month 3 for the 25 mg arm (observed -0.84 %) and -1.16 % at month 3 for the 50 mg arm (observed -1.05 %).

```{r figure-hba1c, fig.width = 7, fig.height = 4}
hba1c_summary <- sim |>
  dplyr::filter(!is.na(Hba1c)) |>
  dplyr::mutate(week = time / (24 * 7)) |>
  dplyr::group_by(treatment, week) |>
  dplyr::summarise(
    hba_q05 = stats::quantile(Hba1c, 0.05),
    hba_q50 = stats::quantile(Hba1c, 0.50),
    hba_q95 = stats::quantile(Hba1c, 0.95),
    .groups = "drop"
  )

ggplot(hba1c_summary, aes(week, hba_q50, color = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = hba_q05, ymax = hba_q95), alpha = 0.20, color = NA) +
  geom_line(linewidth = 0.8) +
  labs(
    x = "Week of treatment", y = "HbA1c (%)",
    title = "Simulated HbA1c over 12 weeks (replicates Naik 2013 Figure 3b)",
    caption = "Placebo, 25 mg QD, and 50 mg QD; 200 subjects per arm."
  ) +
  theme_bw()
```

# Comparison against Naik 2013 published month-3 HbA1c reductions

Naik 2013 Results explicitly report (Naik 2013 Results paragraph "Simulations outcome"):

* 25 mg QD: predicted -0.94 % at month 3 (observed -0.84 %), predicted -1.24 % at month 6.
* 50 mg QD: predicted -1.16 % at month 3 (observed -1.05 %), predicted -1.51 % at month 6.

We compute the simulated median change-from-baseline (dHbA1c = HbA1c_week12 - HbA1c_baseline) per treatment arm and compare.

```{r hba1c-compare}
baseline <- sim |>
  dplyr::filter(!is.na(Hba1c), time == 0) |>
  dplyr::select(id, treatment, hba1c_baseline = Hba1c)

week12 <- sim |>
  dplyr::filter(!is.na(Hba1c), time == 24 * 7 * 12) |>
  dplyr::select(id, treatment, hba1c_week12 = Hba1c)

cfb <- dplyr::inner_join(baseline, week12, by = c("id", "treatment")) |>
  dplyr::mutate(delta_hba1c = hba1c_week12 - hba1c_baseline)

simulated <- cfb |>
  dplyr::group_by(treatment) |>
  dplyr::summarise(
    n                  = dplyr::n(),
    delta_hba1c_median = stats::median(delta_hba1c),
    delta_hba1c_q025   = stats::quantile(delta_hba1c, 0.025),
    delta_hba1c_q975   = stats::quantile(delta_hba1c, 0.975),
    .groups = "drop"
  )

published <- tibble::tibble(
  treatment          = c("Placebo", "25 mg QD", "50 mg QD"),
  published_model    = c(NA_real_, -0.94, -1.16),
  published_observed = c(NA_real_, -0.84, -1.05)
)

cmp <- dplyr::full_join(simulated, published, by = "treatment") |>
  dplyr::rename(
    "Treatment"                              = treatment,
    "N (simulated)"                          = n,
    "Simulated median dHbA1c (%)"            = delta_hba1c_median,
    "Simulated 2.5 pct (%)"                  = delta_hba1c_q025,
    "Simulated 97.5 pct (%)"                 = delta_hba1c_q975,
    "Naik 2013 predicted month-3 dHbA1c (%)" = published_model,
    "Naik 2013 observed month-3 dHbA1c (%)"  = published_observed
  )

knitr::kable(
  cmp,
  caption = paste(
    "Simulated median change-from-baseline HbA1c at week 12 vs Naik 2013",
    "predicted / observed month-3 dHbA1c values (Naik 2013 Results",
    "'Simulations outcome' paragraph). Placebo has no published dHbA1c",
    "value in that paragraph (the paper focused on the drug arms)."
  ),
  digits = 3
)
```

# PKNCA validation (steady-state Cmax, Tmax, AUC0-24, half-life on week-12 dose)

Naik 2013 does not tabulate NCA parameters directly for the phase-2 cohort (the primary published NCA is from the phase-1 Naik 2012 and Leifke 2012 dose-ranging studies). The block below computes NCA on the week-12 dose interval (2 weeks-worth of accumulation past the operative-half-life plateau) for each active-drug arm and compares to the reference-typical AUC that the model implies analytically: AUC_ss ~ dose / CL_typ. For the typical female subject with CL / F = 0.75 L / h, the analytic AUC_ss for 25 mg QD is 25 / 0.75 = 33.3 mg * h / L (33.3 ug * h / mL) and for 50 mg QD is 66.7 mg * h / L (66.7 ug * h / mL).

```{r pknca-week12}
sim_nca <- sim |>
  dplyr::filter(treatment != "Placebo", !is.na(Cc),
                time >= 24 * 7 * 11, time <= 24 * 7 * 12) |>
  dplyr::mutate(rel_time = time - 24 * 7 * 11) |>
  dplyr::select(id, rel_time, Cc, treatment)

# Guarantee a rel_time = 0 row per (id, treatment) so PKNCA anchors AUC at 0
sim_nca <- dplyr::bind_rows(
  sim_nca,
  sim_nca |> dplyr::distinct(id, treatment) |>
    dplyr::mutate(rel_time = 0, Cc = 0)
) |>
  dplyr::distinct(id, treatment, rel_time, .keep_all = TRUE) |>
  dplyr::arrange(id, treatment, rel_time)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ rel_time | treatment + id)

# One "steady-state" dose per subject: the dose at the start of the week-11 interval.
# The rel_time frame is centred at 0 = beginning of the day-77 dose.
dose_df <- events |>
  dplyr::filter(evid == 1L, treatment != "Placebo", time == 24 * 7 * 11) |>
  dplyr::mutate(rel_time = 0) |>
  dplyr::select(id, rel_time, amt, treatment)

dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ rel_time | treatment + id)

intervals <- data.frame(
  start     = 0,
  end       = 24,
  cmax      = TRUE,
  tmax      = TRUE,
  auclast   = TRUE,
  half.life = TRUE
)

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

# Reference: model-analytic AUC0-24 at steady state for a typical female
# with CL / F = 0.75 L / h; approximate Cmax at day-77 dose from the
# typical PK simulation (checked at extraction time to be ~ 5 mg / L
# for 25 mg and ~ 10 mg / L for 50 mg -- the model is linear in dose).
# tmax is anchored by the FIXED Ka = 0.075 / h so Tmax is late in the
# dosing interval (roughly 8 - 12 h).
published <- tibble::tribble(
  ~treatment,  ~cmax,  ~tmax, ~auclast, ~half.life,
  "25 mg QD",   5.0,    10,    33.3,     45,
  "50 mg QD",  10.0,    10,    66.7,     45
)

cmp_nca <- nlmixr2lib::ncaComparisonTable(
  simulated = nca_res,
  reference = published,
  by        = "treatment",
  units     = c(cmax = "mg / L", auclast = "mg * h / L", tmax = "h", half.life = "h"),
  tolerance_pct = 25
)

knitr::kable(
  cmp_nca,
  caption = paste(
    "Week-12 (day-77 dose) NCA vs model-analytic reference AUC_ss = dose / CL",
    "at CL / F = 0.75 L / h and reference Cmax / Tmax / t1/2 from the typical",
    "PK simulation (Naik 2013 does not tabulate per-dose NCA values in the",
    "phase-2 cohort). Starred rows differ from the reference by > 25 pct;",
    "moderate discrepancies are expected because the reference values",
    "assume the typical female subject whereas the simulated cohort includes",
    "46.5 pct males with 41 pct higher CL / F."
  ),
  align = c("l", "l", "r", "r", "r", "r")
)
```

# Assumptions and deviations

- **AST-on-Emax encoded as exponential per s06 control stream, not "linear" as the paper prose describes.** Naik 2013 Results paragraph "Exposure-efficacy response analysis" states that Emax "increased linearly with increasing aspartate aminotransferase (AST) levels", but the supplement control stream `PSP-2012-0026-T-s06.doc` encodes it as `TVEMAX = THETA(3) * EXP(THETA(5) * (FPG - 163.5) + THETA(6) * (SGOT - 21))`. Both BFPG and AST effects sit inside the same `EXP()`, so both are exponential-additive on the log-Emax scale. The control stream is the definitive source; the packaged model uses the exponential form. The impact of the discrepancy is small (0.00731 * (AST - 21) is close to zero for realistic AST values), but the encoding matters for AST outliers.
- **AST outlier caveat.** Naik 2013 Discussion notes that removing seven patients with AST >= 70 U / L rendered the AST-on-Emax effect statistically non-significant (P >= 0.005). The coefficient is retained here as reported for the full cohort. The virtual cohort caps AST at 70 U / L to keep the exponent numerically well-behaved.
- **BFPG-on-BLA1 encoded as exponential per s05 control stream, not additive as the units in Table 2 would suggest.** Naik 2013 Table 2 lists "BFPG on BLA1" with units of "% / (mg / dL)", which reads as an additive slope. The supplement control stream `PSP-2012-0026-T-s05.doc` encodes it as `TVBLA1 = (THETA(1) + THETA(6) * (DD - 4.61)) * EXP(THETA(7) * (BLI - 163.5))`, i.e., a linear-in-DD, exponential-in-BFPG form. The packaged model uses the exponential form per the control stream.
- **Additive sex effects on CL / F and MPL.** The paper reports "Gender on CL / F = 0.31 L / h" and "Gender on MPL = 0.0363" as additive shifts on the linear scale. The packaged model encodes them as `cl_typ = 0.75 + 0.31 * (1 - SEXF)` and `mpl_typ = 0.0590 + 0.0363 * (1 - SEXF)`, then applies the log-normal IIV multiplicatively to the sum: `cl_i = cl_typ * exp(etalcl)`. This matches the NONMEM specification `TVCL = TH1 + TH6 * SEX; CL = TVCL * EXP(ETA)`.
- **Q / F, V2 / F, Ka FIXED to the Leifke 2012 phase-1 multiple-rising-dose (MRD) values.** Naik 2013 Discussion explicitly states these three parameters "were fixed to values estimated in an earlier population PK analysis performed using frequently collected PK samples in a multiple rising dose study in patients with T2DM" (Leifke 2012 Clin Pharmacol Ther 92:29-39; doi:10.1038/clpt.2012.43). The phase-2 sparse-sampling design could not identify them. The Leifke 2012 paper is not on disk; the fixed values reproduced here are those printed in Naik 2013 Table 2.
- **Placebo half-life HL FIXED at 720 h.** Naik 2013 Methods: "the half-life of MPL was fixed to the value (720 h) observed based upon the graphical analysis of data" (short-trial-duration rationale). This is encoded as `hl_placebo <- fixed(720)`.
- **Virtual cohort.** Continuous covariates (FPG, AST, T_DIAG_DIAB) are generated from Table-1 mean +/- SD summaries via truncated normal / gamma distributions; the joint distribution of correlated covariates is not preserved. SEXF is Bernoulli(0.535) per Table 1. The published cohort size (346 PK-efficacy patients) is not reproduced; 200 subjects per arm was chosen because it is the cap recommended by the extraction skill and is ample for a VPC.
- **Simulation horizon.** The simulations run 12 weeks (2016 h) to match the phase-2 trial duration; Naik 2013 also projected 24-week trajectories but explicitly noted that "the projection beyond 12 weeks may be slightly over or under estimated" because HbA1c did not reach a plateau by month 3 in the trial. The packaged vignette stops at 12 weeks.
- **Model-based meta-analysis (MBMA) NOT extracted.** Naik 2013 also reports a model-based meta-analysis that fits an Emax dose-response model to HbA1c change from 74 external trials of DPP-4 inhibitors, glimepiride, and TAK-875 (Naik 2013 Methods "Model-based meta-analysis"). The paper does not tabulate the numeric Emax, ED50, or between-trial-variance parameter estimates for this analysis (only the qualitative results "50 mg of TAK-875 is expected to produce ~ 85 % of maximum effect on HbA1c" and a Figure 4b dose-response curve are reported). Without printed parameter values the MBMA cannot be reproduced as a modeldb entry; the primary popPK / FPG / HbA1c joint model is the extractable content of the paper.
- **Post-2013 fasiglifam withdrawal note.** Fasiglifam / TAK-875 phase-3 development was terminated in December 2013 for signals of hepatotoxicity, an outcome that post-dates this popPK / PD analysis. The packaged model reproduces the phase-2 dose-selection analysis as published; it does not encode any liver-toxicity signal.
