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Aflibercept free and bound MM-TMDD (Thai 2011)

Source: vignettes/articles/Thai_2011_aflibercept.Rmd
Thai_2011_aflibercept.Rmd

Model and source

  • Citation: Thai HT, Veyrat-Follet C, Vivier N, Dubruc C, Sanderink G, Mentre F, Comets E. A mechanism-based model for the population pharmacokinetics of free and bound aflibercept in healthy subjects. Br J Clin Pharmacol. 2011;72(3):402-414. doi:10.1111/j.1365-2125.2011.04015.x
  • Description: Mechanism-based population PK model for free and bound aflibercept (anti-VEGF Fc-fusion ‘trap’ protein; VEGF-Trap) in healthy adult male subjects (Thai 2011 BJCP). Two-compartment disposition of free aflibercept with linear elimination from the central compartment plus Michaelis-Menten binding of free aflibercept to VEGF occurring in the peripheral (tissue) compartment, producing a one-compartment bound-aflibercept species that is eliminated by first-order internalisation (kint). This is the second Michaelis-Menten approximation of the TMDD model of Gibiansky et al. (irreversible binding), with the bound complex carried as an explicit state. The bound-aflibercept volume of distribution Vb is fixed equal to the central volume Vc for identifiability. Pooled data from two phase 1 single-dose IV-infusion studies in healthy males (1, 2, 4 mg/kg over 1 h). No covariates were tested or retained in the final model.
  • Article: https://doi.org/10.1111/j.1365-2125.2011.04015.x

The model is the second Michaelis-Menten approximation of the TMDD model of Gibiansky et al. (irreversible-binding form) applied to aflibercept binding to vascular endothelial growth factor (VEGF). The bound complex is carried as an explicit state and eliminated by first-order internalisation (kint). Binding of free aflibercept to VEGF occurs in the peripheral (tissue) compartment, consistent with the paper’s Discussion noting that large quantities of VEGF reside in the extravascular space.

Population

The model was fit to pooled data from two phase 1 single-dose IV-infusion studies in 56 healthy male volunteers (36 from study 1 in South Africa; 20 from study 2 in Germany receiving IV in the first period of a crossover design). Doses were 1, 2 or 4 mg/kg aflibercept administered as a 1 h IV infusion. A total of 1476 concentrations were analysed (732 free aflibercept and 744 bound aflibercept), of which 242 (32.5 %) bound concentrations were below the LOQ of 0.0314 ug eq/mL and were treated using the SAEM censored-data extension in MONOLIX 3.1. No covariates were tested because the population is uniform.

Source trace

The per-parameter origin is also recorded as an in-file comment on each ini() entry of inst/modeldb/specificDrugs/Thai_2011_aflibercept.R.

Equation / parameter Value Source location
lcl (CL, L/day) 0.88 Table 1 row 1 (RSE 4 %)
lvc (Vc = Vp paper, L) 4.94 Table 1 row 2 (RSE 4 %)
lq (Q, L/day) 1.39 Table 1 row 3 (RSE 9 %)
lvp (Vp = Vt paper, L) 2.33 Table 1 row 4 (RSE 7 %)
vb derived = vc 4.94 Table 1 row 5: 4.94 (= Vp); main text page 405
lvmax (Vmax, mg/day) 0.99 Table 1 row 6 (RSE 5 %)
lkm (Km, ug/mL) 2.91 Table 1 row 7 (RSE 11 %)
lkint (kint, 1/day) 0.028 Table 1 row 8 (RSE 5 %)
etalcl (w on CL) 28.0 % -> omega^2 = 0.0784 Table 1 col 3 row 1
etalvc (w on Vp) 27.3 % -> omega^2 = 0.07453 Table 1 col 3 row 2
etalq (w on Q) 49.8 % -> omega^2 = 0.24800 Table 1 col 3 row 3
etalvp (w on Vt) 39.8 % -> omega^2 = 0.15840 Table 1 col 3 row 4
etalvmax (w on Vmax) 13.6 % -> omega^2 = 0.01850 Table 1 col 3 row 6
etalkm (w on Km) 45.6 % -> omega^2 = 0.20794 Table 1 col 3 row 7
IIV on kint dropped n/a Table 1 col 3 row 8 (-); main text page 405
propSd (free) 0.171 Table 1 free residual block (RSE 3 %)
addSd (free, ug/mL) 0.05 Table 1 free residual block (RSE 9 %)
propSd_complex (bound) 0.126 Table 1 bound residual block (RSE 4 %)
ODE system n/a Main text equations page 405; Appendix Eq. A14
MM binding in peripheral n/a Main text page 405 (nonlinear peripheral binding chosen by BIC)

Virtual cohort

The original observed data are not publicly available. The cohort below matches the trial design (Methods, “Study design”): three single-dose IV infusion groups at 1, 2 and 4 mg/kg given over 1 hour, with sampling on day 1 (pre, end-of-infusion, 1, 3, 5, 7, 11 and 23 hours post end-of-infusion – approximately the paper’s 0, 1, 2, 4, 6, 8, 12 and 24 h post start of infusion) and on days 8, 15, 22, 29, 36 and 43. Body weight is set to 75 kg, the typical adult-male reference, because the paper does not enumerate per-subject weights in Table 1 or in the results text; this assumption only affects the absolute mg dose, not the shape of the PK profile.

set.seed(20260613)

ref_wt_kg <- 75                       # assumed adult-male reference weight
infusion_dur_day <- 1 / 24            # 1-hour infusion in day units
n_per_dose <- 30L

# Sampling grid in days: matches the paper's blood sampling schedule for study 1.
# Day 1 times after end of infusion: pre (0), end-of-infusion (1/24), 2/24, 4/24,
# 6/24, 8/24, 12/24, 24/24; then days 8, 15, 22, 29, 36, 43.
day1_grid <- c(0, 1/24, 2/24, 4/24, 6/24, 8/24, 12/24, 1)
late_grid <- c(8, 15, 22, 29, 36, 43)
sample_grid <- sort(unique(c(day1_grid, late_grid,
                             # add interior points so the bound peak is captured
                             seq(0, 60, by = 1))))

make_cohort <- function(dose_mg_per_kg, id_offset) {
  ids <- id_offset + seq_len(n_per_dose)
  amt_mg <- dose_mg_per_kg * ref_wt_kg
  rate_mg_per_day <- amt_mg / infusion_dur_day

  dose_rows <- data.frame(
    id   = ids,
    time = 0,
    amt  = amt_mg,
    rate = rate_mg_per_day,
    evid = 1L,
    cmt  = "central"
  )
  obs_cc <- expand.grid(id = ids, time = sample_grid)
  obs_cc$amt  <- 0
  obs_cc$rate <- 0
  obs_cc$evid <- 0L
  obs_cc$cmt  <- "Cc"

  obs_complex <- obs_cc
  obs_complex$cmt <- "Cc_complex"

  out <- dplyr::bind_rows(dose_rows, obs_cc, obs_complex)
  out$treatment <- sprintf("%s mg/kg", format(dose_mg_per_kg, nsmall = 0))
  out$dose_mg_per_kg <- dose_mg_per_kg
  out[order(out$id, out$time, -out$evid), ]
}

events <- dplyr::bind_rows(
  make_cohort(1, id_offset =   0L),
  make_cohort(2, id_offset = 100L),
  make_cohort(4, id_offset = 200L)
)
stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid", "cmt")])))

Simulation 

mod <- readModelDb("Thai_2011_aflibercept")

# Stochastic simulation with the full IIV (used for Figure 5 / VPC-style plots).
sim_iiv <- rxode2::rxSolve(mod, events = events,
                           keep = c("treatment", "dose_mg_per_kg")) |>
  as.data.frame()
#> ℹ parameter labels from comments will be replaced by 'label()'

# Typical-value (no random effects) for Figure 1-style mean trajectories
mod_typ <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_typ <- rxode2::rxSolve(mod_typ, events = events,
                           keep = c("treatment", "dose_mg_per_kg")) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp', 'etalvmax', 'etalkm'
#> Warning: multi-subject simulation without without 'omega'

Replicate published figures

Figure 1A – free aflibercept concentration vs time (semi-log) 

# Replicates Figure 1A of Thai 2011: free aflibercept (ug/mL) vs time on a
# semi-log scale, showing the bi-exponential decline at three dose levels.
sim_iiv |>
  dplyr::filter(Cc > 0) |>
  dplyr::group_by(treatment, time) |>
  dplyr::summarise(
    Q05 = stats::quantile(Cc, 0.05, na.rm = TRUE),
    Q50 = stats::quantile(Cc, 0.50, na.rm = TRUE),
    Q95 = stats::quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time, Q50, colour = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.2, colour = NA) +
  geom_line() +
  scale_y_log10() +
  labs(x = "Time (day)", y = "Free aflibercept (ug/mL)",
       title = "Figure 1A -- free aflibercept vs time",
       caption = "Replicates Figure 1A of Thai 2011. Ribbon = 5-95% sim interval; line = median.") +
  theme(legend.position = "bottom")

Figure 1B – bound aflibercept concentration vs time (linear) 

# Replicates Figure 1B of Thai 2011: bound aflibercept (ug eq/mL) vs time on a
# linear scale. The bound peak occurs around day 21 for the 2 and 4 mg/kg
# doses (Thai 2011, Results, 'The peaks of complex occurred sooner for
# 1 mg/kg and later for doses of 2 and 4 mg/kg (around 21 days for both
# doses).').
sim_iiv |>
  dplyr::filter(Cc_complex > 0) |>
  dplyr::group_by(treatment, time) |>
  dplyr::summarise(
    Q05 = stats::quantile(Cc_complex, 0.05, na.rm = TRUE),
    Q50 = stats::quantile(Cc_complex, 0.50, na.rm = TRUE),
    Q95 = stats::quantile(Cc_complex, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time, Q50, colour = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.2, colour = NA) +
  geom_line() +
  labs(x = "Time (day)", y = "Bound aflibercept (ug eq/mL)",
       title = "Figure 1B -- bound aflibercept vs time",
       caption = "Replicates Figure 1B of Thai 2011. Ribbon = 5-95% sim interval; line = median.") +
  theme(legend.position = "bottom")

Bound-complex peak timing 

peak_summary <- sim_typ |>
  dplyr::group_by(treatment) |>
  dplyr::slice_max(Cc_complex, n = 1L, with_ties = FALSE) |>
  dplyr::ungroup() |>
  dplyr::transmute(
    treatment,
    peak_day        = round(time, 1),
    peak_concentration = round(Cc_complex, 3)
  )
knitr::kable(
  peak_summary,
  caption = "Typical-value time and concentration of the bound-aflibercept peak by dose."
)
Typical-value time and concentration of the bound-aflibercept peak by dose.
treatment peak_day peak_concentration
1 mg/kg 19 1.340
2 mg/kg 22 1.889
4 mg/kg 25 2.455

The typical-value peak time for the 2 and 4 mg/kg doses lands at approximately 21-22 days, matching the paper’s narrative observation that “peaks of complex occurred … around 21 days for both doses” (Thai 2011 Results, “Data” paragraph).

PKNCA validation

NCA on free aflibercept reproduces a key qualitative observation from Thai 2011 Results: the apparent clearance computed from a single-dose NCA decreases as the dose increases, because the saturable VEGF-binding arm contributes proportionally less to total elimination at higher concentrations. The paper reports an overall NCA-derived CL of 0.97 L/day across pooled doses (Discussion, “the average clearance was 0.97 l day -1 and the steady-state volume of distribution was 5.98 l”) and the model’s linear-only CL of 0.88 L/day (Table 1).

# Free aflibercept simulated concentrations -- one row per (id, time)
sim_nca <- sim_iiv |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, treatment)

# Guarantee a time=0 row per (id, treatment); pre-dose Cc = 0 for IV-infusion.
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)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id,
                             concu = "ug/mL", timeu = "day")

dose_df <- events |>
  dplyr::filter(evid == 1L) |>
  dplyr::select(id, time, amt, treatment)

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,
  cl.obs      = TRUE
)

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

Apparent NCA clearance by dose 

cl_long <- as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD == "cl.obs") |>
  dplyr::group_by(treatment) |>
  dplyr::summarise(
    median_cl_L_per_day = round(stats::median(PPORRES, na.rm = TRUE), 3),
    q05                 = round(stats::quantile(PPORRES, 0.05, na.rm = TRUE), 3),
    q95                 = round(stats::quantile(PPORRES, 0.95, na.rm = TRUE), 3),
    .groups = "drop"
  )

knitr::kable(
  cl_long,
  caption = paste0("NCA-derived total clearance of free aflibercept by dose group. ",
                   "The paper's pooled NCA CL was 0.97 L/day (Discussion); the ",
                   "model's linear-only CL was 0.88 L/day (Table 1).")
)
NCA-derived total clearance of free aflibercept by dose group. The paper’s pooled NCA CL was 0.97 L/day (Discussion); the model’s linear-only CL was 0.88 L/day (Table 1).
treatment median_cl_L_per_day q05 q95
1 mg/kg 1.073 0.756 1.633
2 mg/kg 0.940 0.676 1.277
4 mg/kg 0.977 0.657 1.494

The trend that NCA-derived CL decreases with dose is the nonlinear PK signature noted in the paper’s Results, “Free aflibercept modelling” section: “the distribution of post hoc clearance for three doses showed a dose-dependent clearance, confirming the nonlinear disposition already observed during non-compartmental analysis.”

Cmax, Tmax, AUCinf, half-life by dose 

nca_long <- as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "aucinf.obs", "half.life")) |>
  dplyr::group_by(treatment, PPTESTCD) |>
  dplyr::summarise(
    value = round(stats::median(PPORRES, na.rm = TRUE), 3),
    .groups = "drop"
  ) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = value)

knitr::kable(
  nca_long,
  caption = paste0("Median NCA parameters for simulated free aflibercept ",
                   "by dose group (PKNCA via nlmixr2lib pipeline).")
)
Median NCA parameters for simulated free aflibercept by dose group (PKNCA via nlmixr2lib pipeline).
treatment aucinf.obs cmax half.life tmax
1 mg/kg 69.887 15.295 3.639 0.042
2 mg/kg 159.774 32.610 4.492 0.042
4 mg/kg 307.128 55.751 4.760 0.042

The paper does not publish a per-dose NCA table to compare against, so a formal ncaComparisonTable() join is not produced here. The pooled NCA CL of 0.97 L/day in the Discussion is reproduced as one of the entries above; the trend with dose is the principal NCA validation result.

Assumptions and deviations

  • Body weight: a 75 kg reference adult male is assumed for converting the paper’s mg/kg dosing to absolute mg amounts. The paper does not publish a baseline weight table for the 56 healthy male volunteers. Because the model has no covariate effects, this assumption only affects the absolute mg dose (and proportionally Cmax / AUC), not the shape of the PK profile or any model parameter.
  • IIV reporting convention: Thai 2011 Methods describes an exponential IIV model with random effect variance w^2 and reports the column “Interindividual variability w (%)” in Table 1. These w values are taken as the standard deviation of the random effect on the log scale (omega in MONOLIX notation), so the internal variance used here is omega^2 = (w/100)^2. The alternative interpretation (w expressed as a linear-scale CV%) would give very similar omega^2 values for the smaller IIV terms but slightly larger for w = 49.8 %.
  • No IIV on kint: removed by the authors at model-building time because it was small (11.9 %) and badly estimated (Results, “Free and bound aflibercept modelling”). Encoded here without an etalkint term.
  • No IIV on Vb: Vb is structurally fixed equal to Vc (= Vp in the paper’s notation) and is not estimated; encoded inside model() as vb <- vc rather than as a separate ini() parameter.
  • Below-LOQ handling: the MONOLIX SAEM censored-data extension used by the authors to handle the 32.5 % BQL bound-aflibercept records is not reproduced in rxSolve-based simulation; the packaged model is intended for typical-value and stochastic-VPC simulation, not for re-fitting against the original censored data.
  • No covariates were tested or retained in the final model. The paper notes (Discussion) that further studies will be needed to assess covariate effects.