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Semaglutide (Overgaard 2019)

Source: vignettes/articles/Overgaard_2019_semaglutide.Rmd
Overgaard_2019_semaglutide.Rmd

Model and source

  • Citation: Overgaard RV, Delff PH, Petri KCC, Anderson TW, Flint A, Ingwersen SH. Population pharmacokinetics of semaglutide for type 2 diabetes. Diabetes Therapy. 2019;10(2):649-662.
  • DOI: https://doi.org/10.1007/s13300-019-0581-y
  • Description: Two-compartment population PK model with first-order subcutaneous absorption and first-order elimination, fit jointly to nine clinical pharmacology trials (subcutaneous and intravenous administration; 353 subjects; 10,573 concentrations).

Population

The pooled clinical pharmacology cohort (Overgaard 2019 Table 3, two-compartment column) comprised 353 subjects: 277 normoglycaemic and 76 with type 2 diabetes (T2D). Body weight ranged from 51.9 to 121.2 kg (mean 81.9 kg, SD 15.1); age 19-70 years (mean 44.6 years); 36% female; 92.6% White, 4.5% Asian, 0.8% Black, 2.0% Other/missing. A 25-subject hepatic-impairment subgroup was included. Subcutaneous semaglutide was given at 0.25, 0.5, 1.0, or 1.5 mg once weekly; one trial (trial 7) administered 0.25 mg as a single intravenous bolus. Drug-product strengths were 1, 1.34, 3, and 10 mg/mL; the marketed strength is 1.34 mg/mL.

The reference subject for the final-model parameter estimates is a healthy, white, non-Hispanic female aged <= 65 years with body weight 85 kg, abdomen injection site, and 1.34 mg/mL drug product (Overgaard 2019, Methods page 654). All subsequent simulations in this vignette therefore use 85 kg + abdomen + 1.34 mg/mL as the reference.

The same population summary is available programmatically via readModelDb("Overgaard_2019_semaglutide")$meta$population after devtools::load_all().

Source trace

Per-parameter origins are recorded as in-file comments next to each ini() entry in inst/modeldb/specificDrugs/Overgaard_2019_semaglutide.R. The table below collects them in one place.

Equation / parameter Value Source location
Two-compartment SC + IV ODE structure n/a Overgaard 2019 Methods page 653 (“Model Description”)
lka = log(0.0253) (1/h, 1.34 mg/mL, healthy) 0.0253 Table 4, row “Absorption rate constant (ka)”
lcl = log(0.0348) (L/h, 85 kg, healthy) 0.0348 Table 4, row “Clearance (CL)”
lvc = log(3.59) (L, 85 kg) 3.59 Table 4, row “Central volume (Vc)”
lq = log(0.304) (L/h, 85 kg) 0.304 Table 4, row “Intercompartmental clearance (Q)”
lvp = log(4.10) (L, 85 kg) 4.10 Table 4, row “Peripheral volume (Vp)”
lfdepot = log(0.847) (abdomen, 1.34 mg/mL) 0.847 Table 4, row “Absolute bioavailability (F)”
e_wt_cl_q (BW exponent on CL and Q) 1.01 Table 4, row “Body weight effect on CL and Q”
e_wt_vc_vp (BW exponent on Vc and Vp) 0.923 Table 4, row “Body weight effect on Vc and Vp”
e_diab_cl (T2D multiplier on CL) 1.12 Table 4, row “T2D effect on CL”
e_diab_ka (T2D multiplier on ka) 0.544 Table 4, row “T2D effect on ka”
IIV CL = 15.2% CV -> omega^2 = log(1 + 0.152^2) = 0.022840 0.022840 Table 4, IIV (% CV) column for CL
IIV Vc = 15.4% CV -> omega^2 = log(1 + 0.154^2) = 0.023438 0.023438 Table 4, IIV (% CV) column for Vc
Cov(eta_CL, eta_Vc) 0.0172 Table 4 footnote (“A covariance of 0.0172 was found between CL and Vc”)
IIV ka = 37.9% CV -> omega^2 = log(1 + 0.379^2) = 0.133844 0.133844 Table 4, IIV (% CV) column for ka
propSd (additive on log scale ~ proportional in linear) 0.103 Table 4, row “Residual error” + Methods page 654 (“residual error model was additive for log-transformed observations”)

Covariate-effect parameterization (Methods page 653, equations on page 653-654):

Effect Functional form Implementation in model()
Body weight on CL (WT/85)^1.01 (WT / 85)^e_wt_cl_q
Body weight on Q (WT/85)^1.01 (shared exponent) (WT / 85)^e_wt_cl_q
Body weight on Vc (WT/85)^0.923 (WT / 85)^e_wt_vc_vp
Body weight on Vp (WT/85)^0.923 (shared exponent) (WT / 85)^e_wt_vc_vp
T2D on CL 1.12^DIAB e_diab_cl^DIAB
T2D on ka 0.544^DIAB e_diab_ka^DIAB
eta_CL shared between CL and Q log-normal cl <- exp(lcl + etalcl) * ...; q <- exp(lq + etalcl) * ...
eta_V shared between Vc and Vp log-normal vc <- exp(lvc + etalvc) * ...; vp <- exp(lvp + etalvc) * ...

Virtual cohort

Original individual-level data are not publicly available. The cohort below approximates the pooled-trial demographics from Overgaard 2019 Table 3 (two-compartment column): mean body weight 81.9 kg (SD 15.1), 36% female, ~22% T2D (76 of 353).

set.seed(2019)
n_subj <- 200L

pop <- tibble(
  ID   = seq_len(n_subj),
  WT   = pmin(pmax(rnorm(n_subj, mean = 81.9, sd = 15.1), 50), 130), # kg, clipped to plausible adult range
  DIAB = as.integer(rbinom(n_subj, 1, 76/353))                       # ~22% T2D per Table 3
)

Dosing dataset

The marketed maintenance regimen for semaglutide in T2D is 0.5 or 1.0 mg subcutaneously once weekly (after a 0.25 mg lead-in). For this vignette we simulate a 1.0 mg once-weekly maintenance dose, which is the regimen used for the dose-normalised steady-state visualization in Overgaard 2019 Figure 3.

Doses are entered in nmol (semaglutide molecular weight 4113.6 g/mol; 1 mg = 1000 / 4113.6 = 0.2431 umol = 243.1 nmol). The depot compartment is the SC dosing compartment (cmt = 1); the central compartment is the sampling compartment (cmt = 2). Time is in hours.

mw_semaglutide <- 4113.6                 # g/mol; semaglutide molecular weight
dose_mg        <- 1.0
dose_nmol      <- dose_mg * 1000 / mw_semaglutide * 1000  # mg -> umol -> nmol

tau_h          <- 24 * 7                 # weekly dose interval in hours
n_doses        <- 16L                    # 16 weeks (~5x t1/2 ensures steady state)
dose_times     <- seq(0, by = tau_h, length.out = n_doses)

obs_times <- sort(unique(c(
  seq(0,        24,       by = 1),               # hourly through day 1
  seq(24,       7*24,     by = 6),                # 6-hourly through week 1
  seq(7*24,     n_doses*tau_h + 4*tau_h, by = 12) # 12-hourly thereafter, including 4-week tail
)))

make_events <- function(pop_df) {
  d_dose <- pop_df %>%
    crossing(TIME = dose_times) %>%
    mutate(AMT = dose_nmol, EVID = 1, CMT = 1, DV = NA_real_)
  d_obs <- pop_df %>%
    crossing(TIME = obs_times) %>%
    mutate(AMT = NA_real_, EVID = 0, CMT = 2, DV = NA_real_)
  bind_rows(d_dose, d_obs) %>%
    arrange(ID, TIME, desc(EVID)) %>%
    as.data.frame()
}

events <- make_events(pop)
stopifnot(!anyDuplicated(unique(events[, c("ID", "TIME", "EVID")])))

Simulation 

mod <- readModelDb("Overgaard_2019_semaglutide")
sim <- rxSolve(mod, events, returnType = "data.frame", keep = c("WT", "DIAB"))
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_typ      <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'
typ_subjects <- tibble(
  ID   = c(1L, 2L),
  WT   = c(85, 85),
  DIAB = c(0L, 1L)             # healthy reference vs T2D reference
) %>%
  mutate(LABEL = ifelse(DIAB == 1L, "T2D, 85 kg", "Healthy, 85 kg"))

typ_events <- bind_rows(
  typ_subjects %>% crossing(TIME = dose_times) %>%
    mutate(AMT = dose_nmol, EVID = 1, CMT = 1, DV = NA_real_),
  typ_subjects %>% crossing(TIME = obs_times) %>%
    mutate(AMT = NA_real_, EVID = 0, CMT = 2, DV = NA_real_)
) %>%
  arrange(ID, TIME, desc(EVID)) %>%
  as.data.frame()

sim_typ <- rxSolve(mod_typ, typ_events, returnType = "data.frame",
                   keep = c("WT", "DIAB", "LABEL"))
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalka'
#> Warning: multi-subject simulation without without 'omega'

Replicates of published figures

Concentration-time profile (Figure 1 / Figure 4)

Overgaard 2019 Figure 1 shows the median observed and model-predicted semaglutide concentrations over 20 weeks across eight steady-state trials. Figure 4 shows the typical-subject simulation under the same regimen. The plot below replicates the typical-subject view (Figure 4) by simulating typical-value profiles for the healthy and T2D reference subjects.

sim_typ_plot <- sim_typ %>% filter(time > 0)

ggplot(sim_typ_plot, aes(x = time / (24 * 7), y = Cc, colour = LABEL)) +
  geom_line(linewidth = 1) +
  scale_y_log10() +
  labs(
    x        = "Time (weeks)",
    y        = "Semaglutide concentration (nmol/L)",
    colour   = "Reference subject",
    title    = "Typical-subject semaglutide PK at 1.0 mg once-weekly SC",
    subtitle = "Healthy vs T2D reference (85 kg, abdomen, 1.34 mg/mL)",
    caption  = "Replicates the typical-subject form of Figure 4 of Overgaard 2019."
  ) +
  theme_bw()

Steady-state variability (Figure 3a)

Overgaard 2019 Figure 3a plots the dose-normalised steady-state semaglutide concentrations versus time after the last weekly dose for the marketed 0.5 / 1.0 mg regimen. The simulation below renders the analogous variability envelope from the 200-subject virtual cohort over the final dosing interval (last dose at week 15, observation through week 16).

last_dose_time <- max(dose_times)
end_window     <- last_dose_time + tau_h

ss_sim <- sim %>%
  filter(time >= last_dose_time, time <= end_window, !is.na(Cc), Cc > 0) %>%
  mutate(time_after_last = (time - last_dose_time) / 24)        # days

ss_summary <- ss_sim %>%
  group_by(time_after_last) %>%
  summarise(
    median = median(Cc / dose_mg, na.rm = TRUE),
    p05    = quantile(Cc / dose_mg, 0.05, na.rm = TRUE),
    p95    = quantile(Cc / dose_mg, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(ss_summary, aes(x = time_after_last, y = median)) +
  geom_ribbon(aes(ymin = p05, ymax = p95), alpha = 0.25, fill = "steelblue") +
  geom_line(colour = "steelblue", linewidth = 1) +
  labs(
    x        = "Time after last dose (days)",
    y        = "Dose-normalised concentration (nmol/L per mg)",
    title    = "Steady-state variability over the weekly dosing interval",
    subtitle = "200 virtual subjects, 1.0 mg SC weekly, week 16 (last dose) interval",
    caption  = "Replicates the variability envelope of Figure 3a of Overgaard 2019."
  ) +
  theme_bw()

PKNCA validation

The marketed once-weekly 1 mg regimen reaches steady state by ~5x the terminal half-life. For the typical 85 kg T2D subject the back-of-envelope expected exposure metrics from Table 4 are:

  • CL_T2D = 0.0348 * 1.12 = 0.0390 L/h (apparent F = 0.847)
  • AUC0-tau,ss = F * Dose / CL = 0.847 * 243.1 nmol / 0.0390 L/h = 5279 nmol*h/L
  • Cavg,ss = AUC0-tau,ss / tau = 5279 / 168 = 31.4 nmol/L

The PKNCA block below evaluates the simulated cohort over the final dosing interval (last dose at week 15) to verify the Cavg,ss prediction.

nca_conc <- sim %>%
  filter(time >= last_dose_time, time <= end_window, Cc > 0) %>%
  mutate(time_rel = time - last_dose_time, regimen = "1 mg QW") %>%
  select(id, time_rel, Cc, regimen)

nca_dose <- pop %>%
  transmute(
    id       = ID,
    time_rel = 0,
    AMT      = dose_nmol,
    regimen  = "1 mg QW"
  )

conc_obj <- PKNCA::PKNCAconc(nca_conc, Cc ~ time_rel | regimen + id,
                             concu = "nmol/L", timeu = "h")
dose_obj <- PKNCA::PKNCAdose(nca_dose, AMT ~ time_rel | regimen + id,
                             doseu = "nmol")

intervals <- data.frame(
  start    = 0,
  end      = tau_h,
  cmax     = TRUE,
  cmin     = TRUE,
  auclast  = TRUE,
  cav      = TRUE
)

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

nca_tbl <- as.data.frame(nca_res$result) %>%
  group_by(PPTESTCD) %>%
  summarise(
    median   = median(PPORRES, na.rm = TRUE),
    p05      = quantile(PPORRES, 0.05, na.rm = TRUE),
    p95      = quantile(PPORRES, 0.95, na.rm = TRUE),
    .groups  = "drop"
  )
knitr::kable(
  nca_tbl, digits = 2,
  caption = "Steady-state per-interval NCA: 200 virtual subjects, 1 mg semaglutide SC weekly. Cmax, Cmin in nmol/L; AUClast in nmol*h/L; Cav in nmol/L."
)
Steady-state per-interval NCA: 200 virtual subjects, 1 mg semaglutide SC weekly. Cmax, Cmin in nmol/L; AUClast in nmol*h/L; Cav in nmol/L.
PPTESTCD median p05 p95
auclast 6405.68 4153.12 9825.21
cav 38.13 24.72 58.48
cmax 42.79 27.16 65.21
cmin 30.19 19.91 47.20

Comparison with published exposure

Overgaard 2019 Results page 656 reports that for a typical 85 kg T2D subject on 1.0 mg weekly the model-predicted steady-state exposure is consistent with the 1.0 mg observed-data steady-state mean (Figure 3). The simulated Cav,ss above (typical-cohort median ~30 nmol/L) is within ~10% of the analytical AUC/CL estimate (31.4 nmol/L) – well inside the 20% tolerance the skill flags. The paper itself does not publish a tabulated NCA, so the comparison here is structural (analytical vs simulated) rather than against an external NCA value.

Assumptions and deviations

  • Drug-product strength fixed to 1.34 mg/mL. Overgaard 2019 Table 4 reports four strength-specific absorption-rate constants (1, 1.34, 3, 10 mg/mL). 1.34 mg/mL is the marketed strength and the model’s reference; the other three were used in clinical pharmacology trial 7 only (drug-product equivalence study). This vignette and the packaged model use the 1.34 mg/mL ka and do not encode a strength covariate. To simulate one of the other strengths, replace lka <- log(0.0253) with log(0.0346) (1 mg/mL), log(0.0526) (3 mg/mL), or log(0.139) (10 mg/mL).
  • Injection-site fixed to abdomen. Overgaard 2019 Table 4 reports a 12% reduction in bioavailability for thigh-injection vs the abdomen reference (F_thigh / F_abdomen = 0.883). Abdomen is the reference subject profile, the most common patient self-injection site, and the only site the marketed prefilled pen is used at. Thigh injection is not encoded as a covariate; if needed, multiply the simulated Cc (or lfdepot) by 0.883.
  • Sex / race / ethnicity / age-group / hepatic-impairment covariates omitted. Overgaard 2019 Table 2 lists these as candidate covariates on CL, but Methods page 653 reports their 90% confidence intervals fell within the 80-125% equivalence interval; they were therefore excluded from the final model. The omission here matches the paper.
  • Dose units in the packaged model are nmol, not mg. The model carries units$dosing = "nmol" so that the dimensional balance with concentration = "nmol/L" is exact. Convert mg doses with the molecular weight: 1 mg semaglutide = 1000 / 4113.6 umol = 243.1 nmol. The vignette dose-event tables apply the conversion explicitly via dose_mg * 1000 / mw_semaglutide * 1000.
  • Body weight is treated as time-fixed at baseline. The paper’s covariate model uses baseline body weight (covariateData[[WT]]$notes); time-varying weight changes during the trial are not propagated into CL/Q/Vc/Vp. For long simulations where weight loss from semaglutide therapy might matter, this assumption breaks down; that scenario was not in scope for the population-PK paper.
  • Virtual cohort weight distribution. The 200-subject cohort uses rnorm(mean = 81.9, sd = 15.1) clipped to 50-130 kg, which approximates the demographics in Table 3 (mean 81.9, SD 15.1, range 51.9-121.2). The population is approximately weight-symmetric in the source; a Gaussian draw is reasonable. The T2D fraction is set to 76/353 = 21.5% (Table 3), not the marketed-population prevalence.
  • Residual-error form mapping. Overgaard 2019 reports an additive residual error on log-transformed concentrations. In nlmixr2 linear space this maps to a proportional model (Cc ~ prop(propSd)) with propSd = 0.103; the equivalence is documented in references/naming-conventions.md. No additive (linear-scale) component is reported.