Skip to contents

Onset of nonlinear PK for mavrilimumab, efalizumab, romosozumab (Stein 2018)

Source: vignettes/articles/Stein_2018_mAb_nonlinear_PK.Rmd
Stein_2018_mAb_nonlinear_PK.Rmd

Model and source

  • Citation: Stein AM, Peletier LA. Predicting the onset of nonlinear pharmacokinetics. CPT Pharmacometrics Syst Pharmacol. 2018 Oct;7(10):670-677.
  • Article: https://doi.org/10.1002/psp4.12316
  • Underlying data sources:
    • Mavrilimumab (anti-GM-CSF receptor): Burmester GR, et al. Ann Rheum Dis. 2011;70(9):1542-1549.
    • Efalizumab (anti-CD11a): Bauer RJ, et al. J Pharmacokinet Biopharm. 1999;27(4):397-420.
    • Romosozumab (anti-sclerostin): Padhi D, et al. J Bone Miner Res. 2011;26(1):19-26.

Stein and Peletier (2018) derive an analytical expression for the critical concentration Ccrit at which the elimination rate of a monoclonal antibody with combined linear plus saturable (target-mediated) clearance doubles, marking the transition between approximately-linear and nonlinear PK. They fit two-compartment quasi-steady-state (QSS) TMDD models to the published PK data of three antibodies (mavrilimumab, efalizumab, romosozumab) and use those typical-value fits to illustrate how Ccrit = Vmax / CL = ksyn / (CL/Vc) lines up with the visible kink in the concentration-time curves.

This vignette covers all three drugs because they share a paper, a structural model, and a single narrative. The packaged models live in three companion files under inst/modeldb/specificDrugs/:

data.frame(
  Drug   = c("Mavrilimumab", "Efalizumab", "Romosozumab"),
  Target = c("GM-CSF receptor alpha", "CD11a (LFA-1)", "Sclerostin"),
  Model  = c("Stein_2018_mavrilimumab",
             "Stein_2018_efalizumab",
             "Stein_2018_romosozumab")
) |> knitr::kable()
Drug Target Model
Mavrilimumab GM-CSF receptor alpha Stein_2018_mavrilimumab
Efalizumab CD11a (LFA-1) Stein_2018_efalizumab
Romosozumab Sclerostin Stein_2018_romosozumab

Population

The packaged parameters in Stein and Peletier 2018 Table 1 are typical-value fits to phase 1 / early-phase PK data from three previously published trials:

  • Mavrilimumab (Burmester 2011) – adults with rheumatoid arthritis; single IV doses ranging from 0.01 to 10 mg/kg.
  • Efalizumab (Bauer 1999) – adults with psoriasis; single IV doses ranging from 0.03 to 10 mg/kg.
  • Romosozumab (Padhi 2011) – healthy adults; single IV doses including 1, 5, and 10 mg/kg.

Stein and Peletier 2018 (Methods, Model analysis, fitting, and simulation) note that they fixed ke(R) = ke(CR) in all three fits, an assumption commonly made for membrane-bound targets where ke(R) is poorly identified from PK alone. The article (Model fit section, page 672) additionally cautions that the large values for ke(R) and ke(CR) reported for efalizumab (4400/day) and romosozumab (860/day) reflect the practical unidentifiability of these parameters in that experimental setting – they are not meant as physiological estimates.

The article does not report between-subject variability or residual error for the fits; the packaged models are therefore typical-value only and have no IIV or residual error structure. Programmatically, the per-model population is available via readModelDb("<model>")$population.

Source trace

Every parameter is taken directly from Stein and Peletier 2018 Table 1. The structural model is paper Eq. 7 with the QSS algebraic relations from paper Eq. 5.

Drug Parameter Value (Table 1) Units
Mavrilimumab Vc 2.8 L
Mavrilimumab Vp 5.6 L
Mavrilimumab CL 0.3 L/day
Mavrilimumab Q 1.7 L/day
Mavrilimumab ksyn 2.4 nM/day
Mavrilimumab Kss 1.1 nM
Mavrilimumab ke(R) 2.2 1/day
Mavrilimumab ke(CR) 2.2 1/day
Efalizumab Vc 2.4 L
Efalizumab Vp 3.6 L
Efalizumab CL 0.46 L/day
Efalizumab Q 9.7 L/day
Efalizumab ksyn 8.5 nM/day
Efalizumab Kss 1.2 nM
Efalizumab ke(R) 4400 1/day
Efalizumab ke(CR) 4400 1/day
Romosozumab Vc 2.4 L
Romosozumab Vp 2.6 L
Romosozumab CL 0.25 L/day
Romosozumab Q 0.54 L/day
Romosozumab ksyn 6.1 nM/day
Romosozumab Kss 12 nM
Romosozumab ke(R) 860 1/day
Romosozumab ke(CR) 860 1/day
Equation Source
Two-compartment QSS TMDD ODE Stein and Peletier 2018 Eq. 7
QSS algebraic relations Stein and Peletier 2018 Eq. 5
Critical concentration Ccrit Stein and Peletier 2018 Eq. 14

Critical concentration

The closed-form expression for Ccrit in the two-compartment QSS TMDD model (Stein and Peletier 2018 Eq. 14, derived under the rapid-exchange limit Q >> CL and Q >> Vmax) is

Ccrit = Vmax / CL = ksyn * Vc / CL

Substituting the Table 1 values directly gives a per-drug Ccrit:

mav <- rxode2::rxode2(readModelDb("Stein_2018_mavrilimumab"))
efa <- rxode2::rxode2(readModelDb("Stein_2018_efalizumab"))
rom <- rxode2::rxode2(readModelDb("Stein_2018_romosozumab"))

ccrit_nM <- function(mod) {
  ini  <- as.data.frame(mod$iniDf)
  vc   <- exp(ini$est[ini$name == "lvc"])
  cl   <- exp(ini$est[ini$name == "lcl"])
  ksyn <- exp(ini$est[ini$name == "lksyn"])
  ksyn * vc / cl
}

ccrit_tbl <- tibble::tibble(
  Drug      = c("Mavrilimumab", "Efalizumab", "Romosozumab"),
  Ccrit_nM  = c(ccrit_nM(mav), ccrit_nM(efa), ccrit_nM(rom)),
  Ccrit_ug_per_mL = Ccrit_nM * 150e3 / 1e9 * 1e3  # nM -> ug/mL using 150 kDa MW
)
knitr::kable(ccrit_tbl, digits = 2,
             caption = "Critical concentration computed from Stein and Peletier 2018 Eq. 14 using Table 1 parameters. The 150 kDa molecular weight follows Stein and Peletier 2018 page 672.")
Critical concentration computed from Stein and Peletier 2018 Eq. 14 using Table 1 parameters. The 150 kDa molecular weight follows Stein and Peletier 2018 page 672.
Drug Ccrit_nM Ccrit_ug_per_mL
Mavrilimumab 22.40 3.36
Efalizumab 44.35 6.65
Romosozumab 58.56 8.78

Simulation: replicate Figure 1

Figure 1 of Stein and Peletier 2018 shows model predictions for each drug across a range of single IV bolus doses (mg/kg). The dot-marked Ccrit line is overlaid to highlight the transition between linear and nonlinear elimination.

We simulate each drug at a common set of single IV doses (0.1, 0.3, 1, 3, 10 mg/kg) using the packaged Stein_2018_<drug> models. The molar dose conversion follows Stein and Peletier 2018 page 672:

dose_nmol = dose_mg_per_kg * 70 kg / 150 kDa * 1e9 nmol/mol
mg_kg_to_nmol <- function(mg_per_kg) mg_per_kg * 70 / 150e3 * 1e9
nM_to_ug_per_mL <- function(conc_nM) conc_nM * 150e3 / 1e9 * 1e3

dose_levels_mg_kg <- c(0.1, 0.3, 1, 3, 10)

# Observation grid: dense early to capture the bolus peak, then daily out to 3
# months (Figure 1 x-axis: 0-3 months).
obs_times <- sort(unique(c(
  seq(0, 1, by = 0.05),
  seq(1, 7, by = 0.25),
  seq(7, 90, by = 0.5)
)))

build_events <- function(drug_label, model_name, doses_mg_kg) {
  out <- vector("list", length(doses_mg_kg))
  for (i in seq_along(doses_mg_kg)) {
    d_mg_kg <- doses_mg_kg[i]
    d_nmol  <- mg_kg_to_nmol(d_mg_kg)
    ev <- rxode2::et(amt = d_nmol, cmt = "central", id = i) |>
      rxode2::et(time = obs_times, id = i)
    out[[i]] <- as.data.frame(ev) |>
      dplyr::mutate(drug = drug_label,
                    model = model_name,
                    dose_mg_per_kg = d_mg_kg,
                    dose_nmol = d_nmol)
  }
  dplyr::bind_rows(out)
}

events_mav <- build_events("Mavrilimumab", "Stein_2018_mavrilimumab", dose_levels_mg_kg)
events_efa <- build_events("Efalizumab",   "Stein_2018_efalizumab",   dose_levels_mg_kg)
events_rom <- build_events("Romosozumab",  "Stein_2018_romosozumab",  dose_levels_mg_kg)

sim_one <- function(model_name, events) {
  mod <- rxode2::rxode2(readModelDb(model_name))
  sim <- rxode2::rxSolve(mod, events = events,
                         keep = c("drug", "dose_mg_per_kg", "dose_nmol")) |>
    as.data.frame()
  if (!"id" %in% names(sim)) sim$id <- 1L
  sim
}

sim_mav <- sim_one("Stein_2018_mavrilimumab", events_mav)
sim_efa <- sim_one("Stein_2018_efalizumab",   events_efa)
sim_rom <- sim_one("Stein_2018_romosozumab",  events_rom)

sim_all <- dplyr::bind_rows(sim_mav, sim_efa, sim_rom) |>
  dplyr::mutate(conc_ug_per_mL = nM_to_ug_per_mL(Cc),
                drug = factor(drug,
                              levels = c("Efalizumab", "Mavrilimumab", "Romosozumab")))
# Replicates Stein and Peletier 2018 Figure 1: concentration-time curves for the
# three mAbs at multiple single-IV doses, with the per-drug Ccrit line overlaid.
ccrit_overlay <- ccrit_tbl |>
  dplyr::rename(drug = Drug) |>
  dplyr::mutate(drug = factor(drug,
                              levels = c("Efalizumab", "Mavrilimumab", "Romosozumab")))

sim_all |>
  dplyr::filter(time > 0, Cc > 0) |>
  ggplot2::ggplot(ggplot2::aes(time / 30, conc_ug_per_mL,
                               colour = factor(dose_mg_per_kg),
                               group = interaction(drug, dose_mg_per_kg))) +
  ggplot2::geom_line() +
  ggplot2::geom_hline(data = ccrit_overlay,
                      ggplot2::aes(yintercept = Ccrit_ug_per_mL),
                      linetype = "dashed") +
  ggplot2::scale_y_log10(limits = c(1e-3, 2e3)) +
  ggplot2::facet_wrap(~ drug) +
  ggplot2::labs(
    x = "Month",
    y = "Concentration (ug/mL)",
    colour = "Dose (mg/kg)",
    title = "Replicates Stein and Peletier 2018 Figure 1",
    caption = paste(
      "Dashed line: Ccrit = Vmax/CL = ksyn*Vc/CL from Table 1 (Eq. 14).",
      "Cc converted from nM to ug/mL assuming 150 kDa antibody."
    )
  )
#> Warning: Removed 1468 rows containing missing values or values outside the scale range
#> (`geom_line()`).

The simulated curves reproduce the visual feature highlighted in Stein and Peletier 2018 Figure 1: as the concentration falls below Ccrit, the slope on a log scale becomes substantially steeper, signaling the transition from approximately-linear nonspecific clearance to nonlinear target-mediated elimination. For romosozumab and efalizumab the ke(R) and ke(CR) parameters are practically unidentifiable (Stein and Peletier 2018 page 672); the simulated curves below Ccrit therefore inherit the very-fast complex internalization implied by Table 1 (ke(CR) = 4400/day and 860/day, respectively) and drop steeply, exactly as in the published figure.

PKNCA validation

Because Stein and Peletier 2018 do not report numerical NCA parameters in the article text, the PKNCA pass below is a sanity check rather than a published-vs-simulated comparison. The expectations are:

  • At the highest single-IV dose (10 mg/kg = 467 nmol), the initial concentration C0 = D / Vc (no absorption, no IIV). For mavrilimumab this is 467 / 2.8 = 166.79 nM, matching the simulated Cc at t = 0.
  • Above Ccrit the elimination is approximately linear with rate CL/Vc; the AUC contribution from the linear phase is approximately D / CL per Stein and Peletier 2018 Eq. 14 derivation.
sim_nca <- sim_all |>
  dplyr::filter(!is.na(Cc), Cc > 0) |>
  dplyr::transmute(id = paste(drug, dose_mg_per_kg, sep = "_"),
                   time = time, conc = Cc,
                   drug = drug,
                   dose_mg_per_kg = dose_mg_per_kg)
dose_df <- dplyr::bind_rows(events_mav, events_efa, events_rom) |>
  dplyr::filter(evid == 1) |>
  dplyr::transmute(id = paste(drug, dose_mg_per_kg, sep = "_"),
                   time = time, amt = amt,
                   drug = drug,
                   dose_mg_per_kg = dose_mg_per_kg)
conc_obj <- PKNCA::PKNCAconc(sim_nca, conc ~ time | drug + dose_mg_per_kg + id)
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | drug + dose_mg_per_kg + id)
intervals <- data.frame(
  start = 0, end = Inf,
  cmax = TRUE, tmax = TRUE,
  aucinf.obs = TRUE, half.life = TRUE
)
nca <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))
knitr::kable(summary(nca),
             caption = "Simulated NCA parameters per drug and per single-IV dose level (typical-value, no IIV). Concentrations in nM; AUC in nM*day; half-life in day.")
Simulated NCA parameters per drug and per single-IV dose level (typical-value, no IIV). Concentrations in nM; AUC in nM*day; half-life in day.
start end drug dose_mg_per_kg N cmax tmax half.life aucinf.obs
0 Inf Efalizumab 0.1 1 19400 0.000 0.588 98400
0 Inf Efalizumab 0.3 1 58300 0.000 0.439 301000
0 Inf Efalizumab 1.0 1 194000 0.000 4.67 1.01e6
0 Inf Efalizumab 3.0 1 583000 0.000 7.90 3.04e6
0 Inf Efalizumab 10.0 1 1.94e6 0.000 9.14 1.01e7
0 Inf Mavrilimumab 0.1 1 16700 0.000 20.4 153000
0 Inf Mavrilimumab 0.3 1 50000 0.000 20.7 465000
0 Inf Mavrilimumab 1.0 1 167000 0.000 20.9 1.55e6
0 Inf Mavrilimumab 3.0 1 500000 0.000 20.9 4.66e6
0 Inf Mavrilimumab 10.0 1 1.67e6 0.000 20.9 1.56e7
0 Inf Romosozumab 0.1 1 19400 0.000 10.2 181000
0 Inf Romosozumab 0.3 1 58300 0.000 14.2 554000
0 Inf Romosozumab 1.0 1 194000 0.000 15.6 1.86e6
0 Inf Romosozumab 3.0 1 583000 0.000 15.7 5.59e6
0 Inf Romosozumab 10.0 1 1.94e6 0.000 15.7 1.87e7

Assumptions and deviations

  • Three model files share one vignette. Following the replicate-author-structure default, mavrilimumab, efalizumab, and romosozumab are packaged as three independent Stein_2018_<drug> files because they share a structural model but have different Table 1 parameter values. The vignette covers all three because they were fit together as illustrations of one analytical result (Stein and Peletier 2018 Eq. 14).
  • No IIV, no residual error. Stein and Peletier 2018 do not report between-subject variability or residual-error magnitudes for the Table 1 fits. The packaged models are typical-value only; rxode2::zeroRe() is unnecessary because there are no random effects to zero.
  • Practical unidentifiability of ke(R) and ke(CR). Stein and Peletier 2018 page 672 (Model fit) explicitly state that the large ke(R) / ke(CR) values used for efalizumab (4400/day) and romosozumab (860/day) reflect the practical unidentifiability of these parameters from PK data. They are carried verbatim from Table 1 and should be interpreted as fitting placeholders rather than physiological rates.
  • ke(R) = ke(CR) assumption. Stein and Peletier 2018 fix ke(R) = ke(CR) in all three fits (Methods, Model analysis, fitting, and simulation). The packaged models reflect this assumption – lkdeg and lkint are set to the same Table 1 value, but are kept as two parameters so the assumption is visible to a downstream user who wants to relax it.
  • Molecular weight assumption for unit conversion. Stein and Peletier 2018 use a 150 kDa molecular weight to convert 10 mg/kg in a 70 kg patient to 4667 nmol (page 672). The vignette uses the same 150 kDa assumption to convert simulated Cc (nM) to ug/mL for the Figure 1 visual comparison.
  • Two-compartment QSS TMDD as a simulation tool. The article’s primary contribution is the analytical derivation of Ccrit, not the popPK fit per se. The Table 1 parameters are reported as illustrative typical-value fits; downstream users wanting a popPK-grade model with IIV / residual error should consult the original Bauer 1999 / Burmester 2011 / Padhi 2011 sources.
  • Supplementary material not consulted. Stein and Peletier 2018 reference a Supplementary Material containing the model fits to the Figure 1 data; that supplement was not available on disk during this extraction. Table 1 in the main article carries the parameter values used here, and Stein and Peletier 2018 page 672 confirms those are the values used for all simulations in the main text.