Cetuximab (Dirks 2008) • nlmixr2lib

library(nlmixr2lib)
library(rxode2)
#> rxode2 5.1.6 using 2 threads (see ?getRxThreads)
#>   no cache: create with `rxCreateCache()`
library(PKNCA)
#> 
#> Attaching package: 'PKNCA'
#> The following object is masked from 'package:stats':
#> 
#>     filter
library(dplyr)
#> 
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union
library(tidyr)
library(ggplot2)

Cetuximab population PK replication (Dirks 2008)

Dirks et al. (2008) characterised cetuximab pharmacokinetics in 143 adults with recurrent or metastatic squamous cell carcinoma of the head and neck (SCCHN) pooled across two phase I/II studies. Cetuximab is a chimeric IgG1 directed against the epidermal growth factor receptor (EGFR). Serum concentrations were best described by a two-compartment model with Michaelis-Menten (target-mediated) elimination from the central compartment; a parallel first-order clearance pathway was tested but produced no OFV improvement and was dropped in favour of the more parsimonious purely nonlinear model. Ideal body weight (IBW) and white blood cell count (WBC) enter as additive linear-deviation covariates on Vmax; total body weight (WT) enters on V1.

This vignette reproduces the population profile under the approved regimen (400 mg/m^2 loading over 2 h, followed by weekly 250 mg/m^2 maintenance over 1 h), documents the parameter provenance in a source-trace table, and validates the simulated NCA against the published peak and trough summaries.

Population studied

Dirks 2008 Table 3-1 (thesis Chapter 3, p. 51):

Field	Value
N subjects	143 (Study A n = 47; Study B n = 96)
Age	23-77 years (median 56)
Weight	34-113 kg (median 60)
Sex	120 M / 23 F (16.1 percent female)
Race / ethnicity	131 Caucasian / 12 unknown
Disease state	Recurrent and/or metastatic SCCHN
EGFR-expressing	138 / 143 tumours (immunohistochemistry)
Baseline IBW	43.3-81.3 kg (median 64.2)
Baseline WBC	1.7-41.8 x 10⁹ /L (median 6.8)
Dose range	400 mg/m^2 loading over 2 h, then 250 mg/m^2 weekly over 1 h
Duration of therapy	1-54 weeks (median 6)
Concomitant 5-FU	47 / 143
Concomitant platinum	89 / 143

Human anti-chimeric antibodies were not detected in either study, so an immunogenicity effect on clearance is not modelled.

Source trace

Every numeric value in the model file inst/modeldb/specificDrugs/Dirks_2008_cetuximab.R comes from the following locations in Dirks NL, Nolting A, Kovar A, Meibohm B, Journal of Clinical Pharmacology 2008;48(3):267-278 (doi:10.1177/0091270007313393) – which is reproduced verbatim as Chapter 3 (pp. 45-64) of the Dirks 2010 UTHSC PhD dissertation (doi:10.21007/etd.cghs.2010.0072) – with the NONMEM control stream in Appendix A (pp. 141-149).

Quantity	Source location	Value used
Two-compartment MM PK structure	Chapter 3 Sec. Structural Model, p. 50	two-cmt with Michaelis-Menten elimination
Parallel linear pathway	Chapter 3 Sec. Structural Model, p. 50	tested; dropped (no OFV improvement)
V_max typical (reference covariates)	Table 3-2 (p. 53), theta1	4.38 mg h^-1
K_m	Table 3-2 (p. 53), theta4	74 ug mL^-1
V₁ typical (reference WT)	Table 3-2 (p. 53), theta5	2.83 L
V₂	Table 3-2 (p. 53), theta7	2.43 L
Q	Table 3-2 (p. 53), theta8	0.103 L h^-1
IBW covariate form	Table 3-2, Chapter 3 Covariate Model	V_max = TVV_max * (1 + 0.0108 * (IBW - 64))
WBC covariate form	Table 3-2, Chapter 3 Covariate Model	V_max = TVV_max * (1 + 0.0216 * (WBC - 6.8))
WT (WGT) covariate form	Table 3-2, Chapter 3 Covariate Model	V₁ = TVV₁ * (1 + 0.0083 * (WT - 60))
Reference IBW	Table 3-2 formula centring	64 kg
Reference WBC	Table 3-2 formula centring	6.8 x 10⁹ /L
Reference WT	Table 3-2 formula centring	60 kg
Coefficient theta2 (IBW on V_max)	Table 3-2 (p. 53)	0.0108 per kg
Coefficient theta3 (WBC on V_max)	Table 3-2 (p. 53)	0.0216 per 10⁹ /L
Coefficient theta6 (WT on V₁)	Table 3-2 (p. 53)	0.0083 per kg
IIV on V_max	Table 3-2	15.4 percent CV (90 percent CI 12.0-19.1)
IIV on V₁	Table 3-2	18.6 percent CV (90 percent CI 12.5-22.2)
IIV on V₂	Table 3-2	56.4 percent CV (90 percent CI 18.0-72.8)
IIV on Q	Table 3-2	97.2 percent CV (90 percent CI 40.2-133)
IIV on K_m	Table 3-2 / p. 50 Structural Model	Not estimated (Km treated as fixed across subjects)
Omega structure	Appendix A NM-TRAN (p. 149)	diagonal (no covariance)
Residual error (Study A)	Table 3-2	14.6 percent CV (90 percent CI 12.3-16.8)
Residual error (Study B)	Table 3-2 / p. 61 Results	21.2 percent CV
Peak concentration (Figure 3-2, week 4)	Figure 3-2 (p. 56)	~200 ug/mL typical peak
Trough concentration (Figure 3-3)	Figures 3-3 / 3-4 (pp. 58-59)	~50-60 ug/mL typical trough
Approved dose	Chapter 3 Discussion, p. 61	400 mg/m^2 loading + 250 mg/m^2 weekly maintenance

Virtual cohort

N = 200 virtual SCCHN patients. WT, IBW, and WBC are drawn from log-normal distributions anchored on the medians reported in Table 3-1 (60 kg, 64.2 kg, 6.8 x 10⁹ /L respectively), with per-covariate SDs chosen so that the simulated ranges cover the reported Table 3-1 extremes. BSA is derived from WT and a random height using DuBois’ formula so the BSA-based dose calculation stays consistent with the source.

set.seed(2008)
n_subj <- 200

wt_med  <- 60
ibw_med <- 64.2
wbc_med <- 6.8

wt  <- pmax(34,  pmin(exp(log(wt_med)  + rnorm(n_subj, 0, 0.25)), 113))
ibw <- pmax(43.3, pmin(exp(log(ibw_med) + rnorm(n_subj, 0, 0.15)), 81.3))
wbc <- pmax(1.7, pmin(exp(log(wbc_med) + rnorm(n_subj, 0, 0.50)), 41.8))

ht_cm <- pmax(150, pmin(160 + (wt - wt_med) * 0.6 + rnorm(n_subj, 0, 6), 200))
bsa   <- 0.007184 * ht_cm^0.725 * wt^0.425

pop <- tibble(
  ID  = seq_len(n_subj),
  WT  = wt,
  IBW = ibw,
  WBC = wbc,
  BSA = bsa
)

summary(pop[, c("WT", "IBW", "WBC", "BSA")])
#>        WT              IBW             WBC              BSA       
#>  Min.   : 34.00   Min.   :43.30   Min.   : 2.051   Min.   :1.216  
#>  1st Qu.: 50.02   1st Qu.:57.82   1st Qu.: 5.048   1st Qu.:1.466  
#>  Median : 59.88   Median :63.86   Median : 7.248   Median :1.615  
#>  Mean   : 60.49   Mean   :64.21   Mean   : 7.950   Mean   :1.634  
#>  3rd Qu.: 68.96   3rd Qu.:70.48   3rd Qu.: 9.627   3rd Qu.:1.777  
#>  Max.   :113.00   Max.   :81.30   Max.   :31.471   Max.   :2.462

Dataset construction

Simulate an 8-week course of the approved cetuximab regimen: a 400 mg/m^2 loading dose infused over 2 h at week 0, then 250 mg/m^2 weekly maintenance doses infused over 1 h at weeks 1-7. Observations are placed at end-of-infusion, at 6 h and 24 h post-dose, at every pre-dose trough, and on a dense early grid over the first 72 h.

week_h <- 7 * 24
maint_times <- (seq_len(7)) * week_h
tmax_h <- max(maint_times) + week_h

d_load <- pop |>
  mutate(
    TIME = 0,
    AMT  = 400 * BSA,
    EVID = 1,
    CMT  = "central",
    RATE = (400 * BSA) / 2,
    DV   = NA_real_
  )

d_maint <- pop |>
  tidyr::crossing(TIME = maint_times) |>
  mutate(
    AMT  = 250 * BSA,
    EVID = 1,
    CMT  = "central",
    RATE = 250 * BSA,
    DV   = NA_real_
  )

obs_only <- sort(unique(c(
  seq(0, 72, by = 1),
  c(0, maint_times) + 2,
  c(0, maint_times) + 6,
  c(0, maint_times) + 24,
  seq(0, tmax_h, by = 6)
)))
obs_only <- setdiff(obs_only, c(0, maint_times))

d_obs <- pop |>
  tidyr::crossing(TIME = obs_only) |>
  mutate(
    AMT  = 0,
    EVID = 0,
    CMT  = "Cc",
    RATE = 0,
    DV   = NA_real_
  )

d_sim <- bind_rows(d_load, d_maint, d_obs) |>
  arrange(ID, TIME, desc(EVID)) |>
  select(ID, TIME, AMT, EVID, CMT, RATE, DV, WT, IBW, WBC)

stopifnot(sum(d_sim$EVID == 1) == n_subj * 8)

Simulation

Two passes are run: a stochastic simulation with the full IIV and residual error (for VPC-style plots and NCA) and a typical-value simulation with rxode2::zeroRe() for the reference profile.

mod <- readModelDb("Dirks_2008_cetuximab")

set.seed(20080315)
sim_full <- rxode2::rxSolve(mod, events = d_sim) |>
  as.data.frame() |>
  mutate(time_day = time / 24)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalkm'

mod_typ <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_typ <- rxode2::rxSolve(mod_typ, events = d_sim) |>
  as.data.frame() |>
  mutate(time_day = time / 24)
#> ℹ omega/sigma items treated as zero: 'etalvmax', 'etalkm', 'etalvc', 'etalvp', 'etalq'
#> Warning: multi-subject simulation without without 'omega'

Figure-equivalent 1 – Early concentration-time profile

Dirks 2008 Figure 3-2 (p. 56) shows six representative patients over 28 days after initiation of therapy. The panel below shows the first 28-day profile for the virtual cohort overlaid with the typical-value trajectory of a subject with the median covariates (WT 60 kg, IBW 64.2 kg, WBC 6.8 x 10⁹ /L).

first_month <- sim_full |> filter(time <= 28 * 24)
first_typ   <- sim_typ  |> filter(time <= 28 * 24, id == 1)

ggplot() +
  geom_line(
    data = first_month, aes(time_day, Cc, group = id),
    colour = "grey70", alpha = 0.25, linewidth = 0.25
  ) +
  geom_line(
    data = first_typ, aes(time_day, Cc), colour = "firebrick", linewidth = 1
  ) +
  labs(
    x     = "Time after first dose (days)",
    y     = "Cetuximab Cc (ug/mL)",
    title = "Cetuximab first 28 days -- virtual cohort vs typical patient",
    caption = "Dirks 2008 Figure 3-2-equivalent (400 mg/m^2 loading + 250 mg/m^2 weekly maintenance)"
  ) +
  theme_bw()

Figure-equivalent 2 – Week 4 peak and trough summaries

Dirks 2008 Figure 3-3 (p. 58) summarises simulated peak and trough concentrations at week 4 across 500 bootstrap replicates and compares them to observed peaks / troughs. The panel below plots the distribution of the simulated peak (end of the week-4 infusion) and trough (just before the week-4 dose) for the virtual cohort.

w4_dose_time <- 3 * week_h
w4_peak_target   <- w4_dose_time + 1     # end of infusion
w4_trough_target <- w4_dose_time         # right at dose (pre-infusion trough)

pick_nearest <- function(df, target) {
  df |>
    group_by(id) |>
    slice_min(abs(time - target), n = 1, with_ties = FALSE) |>
    ungroup()
}

pt <- bind_rows(
  pick_nearest(sim_full, w4_peak_target)   |> mutate(label = "Peak (end of week-4 infusion)"),
  pick_nearest(sim_full, w4_trough_target) |> mutate(label = "Trough (just before week-4 dose)")
)

ggplot(pt, aes(x = Cc)) +
  geom_histogram(bins = 30, fill = "steelblue", colour = "grey30") +
  facet_wrap(~ label, scales = "free") +
  labs(
    x     = "Cetuximab Cc (ug/mL)",
    y     = "Number of virtual subjects",
    title = "Week 4 peak and trough distributions",
    caption = "Dirks 2008 Figure 3-3-equivalent"
  ) +
  theme_bw()

PKNCA validation over the week-4 dosing interval

Run NCA on the week-4 dosing interval (168 h after the start of the week-3 maintenance infusion) and compare the simulated typical patient’s peak and trough to the published week-4 peak (~200 ug/mL) and trough (~50-60 ug/mL) reported in Figures 3-2 through 3-4.

The PKNCA formula includes a grouping variable (treatment) so each regimen’s NCA results are rolled up independently, per the library’s PKNCA-recipe convention.

tau_h <- week_h
start_ss <- 3 * week_h
end_ss   <- start_ss + tau_h

sim_nca <- sim_full |>
  filter(!is.na(Cc), time >= start_ss - 1, time <= end_ss + 1) |>
  transmute(
    id        = id,
    time      = time,
    Cc        = Cc,
    treatment = "250 mg/m^2 weekly maintenance"
  )

dose_nca <- d_sim |>
  filter(EVID == 1) |>
  transmute(
    id        = ID,
    time      = TIME,
    amt       = AMT,
    treatment = "250 mg/m^2 weekly maintenance"
  )

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id)
dose_obj <- PKNCA::PKNCAdose(dose_nca, amt ~ time | treatment + id)

intervals <- data.frame(
  start   = start_ss,
  end     = end_ss,
  cmax    = TRUE,
  tmax    = TRUE,
  cmin    = TRUE,
  auclast = TRUE,
  cav     = TRUE
)

res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))
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#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
nca_summary <- as.data.frame(res$result) |>
  filter(PPTESTCD %in% c("cmax", "tmax", "cmin", "auclast", "cav")) |>
  group_by(PPTESTCD) |>
  summarise(
    median = stats::median(PPORRES, na.rm = TRUE),
    p05    = stats::quantile(PPORRES, 0.05, na.rm = TRUE),
    p95    = stats::quantile(PPORRES, 0.95, na.rm = TRUE),
    .groups = "drop"
  )
nca_summary
#> # A tibble: 5 × 4
#>   PPTESTCD median   p05   p95
#>   <chr>     <dbl> <dbl> <dbl>
#> 1 auclast    NA    NA     NA 
#> 2 cav        NA    NA     NA 
#> 3 cmax      183.  130.   245.
#> 4 cmin       52.8  16.5  105.
#> 5 tmax        2     2      2

Comparison against the published exposure summaries

sim_cmax <- nca_summary |> filter(PPTESTCD == "cmax") |> pull(median)
sim_cmin <- nca_summary |> filter(PPTESTCD == "cmin") |> pull(median)

comparison <- tibble::tribble(
  ~metric,                     ~published, ~simulated, ~units,
  "Week-4 peak (typical)",     200,        sim_cmax,   "ug/mL",
  "Week-4 trough (typical)",   55,         sim_cmin,   "ug/mL"
)
comparison
#> # A tibble: 2 × 4
#>   metric                  published simulated units
#>   <chr>                       <dbl>     <dbl> <chr>
#> 1 Week-4 peak (typical)         200     183.  ug/mL
#> 2 Week-4 trough (typical)        55      52.8 ug/mL

Interpretation: Dirks 2008 Figures 3-2 to 3-4 present cetuximab peak concentrations near 200 ug/mL and trough concentrations in the 50-60 ug/mL range at week 4 of therapy for a typical SCCHN patient on the approved 400 mg/m^2 loading + 250 mg/m^2 weekly maintenance regimen. The simulated medians above land in that range, confirming that the packaged model reproduces the published typical exposure at the approved dose.

Michaelis-Menten dose dependence

A hallmark of cetuximab’s target-mediated elimination is a decrease in apparent clearance with increasing dose (Dirks 2008 Discussion, p. 60-61, citing prior phase I dose-escalation data). The panel below repeats the week-4 typical-patient NCA over a range of maintenance dose levels (50, 100, 200, 400, 600 mg) and plots apparent CL (dose / AUC_0-tau) against dose. Consistent with a saturable Michaelis-Menten pathway, apparent CL declines as the maintenance dose increases past the K_m.

one_dose <- function(dose_mg) {
  d_dose_typ <- tibble(
    ID = 1L,
    TIME = c(0, week_h),
    AMT  = c(dose_mg, dose_mg),
    EVID = 1L,
    CMT  = "central",
    RATE = c(dose_mg, dose_mg),
    DV   = NA_real_,
    WT   = wt_med,
    IBW  = ibw_med,
    WBC  = wbc_med
  )
  d_obs_typ <- tibble(
    ID = 1L,
    TIME = sort(unique(c(seq(0, 2 * week_h, by = 1), c(0, week_h) + 1))),
    AMT = 0, EVID = 0L, CMT = "Cc", RATE = 0, DV = NA_real_,
    WT = wt_med, IBW = ibw_med, WBC = wbc_med
  )
  d <- bind_rows(d_dose_typ, d_obs_typ) |> arrange(TIME, desc(EVID))
  sim <- rxode2::rxSolve(mod_typ, events = d) |> as.data.frame()
  interval <- sim |> filter(time >= week_h, time <= 2 * week_h)
  auc <- pracma::trapz(interval$time, interval$Cc)
  tibble(dose_mg = dose_mg, AUC = auc, CL_apparent = dose_mg / auc)
}

if (requireNamespace("pracma", quietly = TRUE)) {
  dose_levels <- c(50, 100, 200, 400, 600)
  nonlin <- purrr::map_dfr(dose_levels, one_dose)

  ggplot(nonlin, aes(dose_mg, CL_apparent)) +
    geom_line(colour = "firebrick", linewidth = 1) +
    geom_point(size = 2) +
    scale_x_log10() +
    labs(
      x     = "Weekly maintenance dose (mg)",
      y     = "Apparent CL over week-2 interval (L/hour)",
      title = "Dose-dependent apparent CL (Michaelis-Menten saturation)",
      caption = "Dirks 2008 Discussion Figure equivalent"
    ) +
    theme_bw()
} else {
  message("Package 'pracma' not installed; skipping the non-linearity panel.")
}
#> Package 'pracma' not installed; skipping the non-linearity panel.

Assumptions and deviations

Single residual error. The source paper fitted separate residual errors for Study A (14.6 percent CV) and Study B (21.2 percent CV) as a stability aid, not a biological covariate; the packaged model uses a single proportional residual of 14.6 percent CV (the Study A value, chosen because Study A contributed the denser sampling: median 13 observations per patient versus 4 per patient in Study B).
Km BSV not estimated. Dirks 2008 held K_m fixed across subjects because between-subject variability in both V_max and K_m could not be simultaneously identified (adding etakm cost only 0.3 OFV); etalkm is wrapped in fixed(0) so K_m varies only with the typical value.
Diagonal Omega. Correlations between V_max and the volume parameters were tested but produced no OFV improvement (Chapter 3 Covariate Model, p. 52), so the packaged Omega is diagonal. The 0.777 / 0.875 correlations that appear in Chapter 1 Tables 1-3 / 1-4 of the thesis are from a literature-survey review, not from this cetuximab model.
Inter-occasion variability. Not modelled by Dirks 2008; not simulated here.
BSA computation. The virtual cohort computes BSA from a simulated height and WT via DuBois; the source dataset used the recorded BSA from each patient. This affects only the per-subject dose amount (mg per BSA) and not the model’s PK parameters.
Gender and age. Tested but not retained as covariates on any PK parameter; not simulated.
Concomitant chemotherapy (5-FU, platinum). Tested but not retained; not simulated.
Immunogenicity. Human anti-chimeric antibodies were not detected in either study; not modelled.
Parallel first-order clearance. Considered and rejected (no OFV improvement); the packaged model has purely Michaelis-Menten elimination.

Reference

Dirks NL, Nolting A, Kovar A, Meibohm B. Population pharmacokinetics of cetuximab in patients with squamous cell carcinoma of the head and neck. J Clin Pharmacol. 2008;48(3):267-278. doi:10.1177/0091270007313393. See also Chapter 3 (pp. 45-64) and Appendix A (NONMEM control stream, pp. 141-149) of Dirks NL, Population Pharmacokinetics of Therapeutic Monoclonal Antibodies: Examples and Estimation Method Performance Differences, PhD dissertation, University of Tennessee Health Science Center, May 2010, ETD paper 63, doi:10.21007/etd.cghs.2010.0072.