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Frey_2013_tocilizumab

Source: vignettes/articles/Frey_2013_tocilizumab.Rmd
Frey_2013_tocilizumab.Rmd

Model and source

  • Citation: Levi M, Grange S, Frey N. Exposure-response relationship of tocilizumab, an anti-IL-6 receptor monoclonal antibody, in a large population of patients with rheumatoid arthritis. J Clin Pharmacol. 2013;53(2):151-159. doi:10.1177/0091270012437585. PMID 23436260.
  • PK backbone: Frey N, Grange S, Woodworth T. Population pharmacokinetic analysis of tocilizumab in patients with rheumatoid arthritis. J Clin Pharmacol. 2010;50(7):754-766. doi:[10.1177/0091270009350623](https://doi.org/10.1177/0091270009350623); packaged separately as Frey_2010_tocilizumab in this library.

This file extracts the indirect-response DAS28 PD model. The library function name Frey_2013_tocilizumab follows the package’s senior-author convention (Frey is the senior popPK / PD author for the tocilizumab program); the actual paper’s first author is Levi.

Population

Frey 2013 fitted the PD model to a pool of two of the four phase III tocilizumab studies — OPTION (methotrexate-inadequate responders) and TOWARD (traditional-DMARD-inadequate responders) — yielding 1703 patients with 12,618 DAS28 observations through week 24. About 80% of patients were female and the majority (~74%) were White; the remaining race groups (Asian, American Indian/Alaska Native, Other) were pooled by the paper into the Asian-and-Others composite that drives the RACE_ASIAN_OTH covariate effect on Kout. Tocilizumab was administered as 4 or 8 mg/kg by 1-hour IV infusion every 4 weeks for up to 24 weeks, plus a placebo-on-DMARD-background arm. Baseline biomarkers (Supplementary Table S1) gave median IL-6 around 20 pg/mL, median HAQ-DI 1.5-1.6, median PAIN VAS 60-62, and median Physician’s Global VAS 65 — these median values are the reference covariate values used in the model.

The same metadata are available programmatically via readModelDb("Frey_2013_tocilizumab")$population.

Source trace

Every PD parameter, covariate effect, IIV element, and residual-error term below comes from Frey 2013 Tables 1 and 2 (final-model column). The PK backbone is reproduced from Frey 2010 Table II at its typical reference-covariate values; PK covariate effects, PK IIV, and PK residual error from Frey 2010 are not reproduced here (this is the exposure-response overlay file, not the popPK file). For the full covariate-aware tocilizumab popPK, use the standalone Frey_2010_tocilizumab library model.

Equation / parameter Value Source location
lcl (linear CL) log(0.3) L/day Frey 2010 Table II, CL
lvc (V1) log(3.5) L Frey 2010 Table II, V1
lq (Q) log(0.2) L/day Frey 2010 Table II, Q
lvp (V2) log(2.9) L Frey 2010 Table II, V2
lvm (Vm) log(7.5) mg/day Frey 2010 Table II, VM
lkm (Km) log(2.7) ug/mL Frey 2010 Table II, KM
lEC50 log(3.7) ug/mL Frey 2013 Table 1, EC50
lEmax log(0.73) Frey 2013 Table 1, Emax
lKout log(0.038) 1/day Frey 2013 Table 1, Kout
lgamma log(0.64) Frey 2013 Table 1, GAMMA
lBase log(6.8) Frey 2013 Table 1, Baseline DAS28
lDMARD log(0.30) ug/mL Frey 2013 Table 1, DMARD effect
e_lil6_ec50 (log IL-6 on EC50) -4.4 Frey 2013 Table 2, EC50 row
e_sexm_emax (+11% Emax in males) 0.11 Frey 2013 Table 2, SEX row
e_race_kout (-25% Kout in Asian/AmInd/Other) -0.25 Frey 2013 Table 2, RACE row
e_blhaq_base (HAQ on BASE) 0.043 Frey 2013 Table 2, HAQ row (see Errata)
e_lil6_base (log IL-6 on BASE) 0.13 Frey 2013 Table 2, log-IL-6-on-BASE row
e_pain_base (PAIN on BASE) 0.062 Frey 2013 Table 2, PAIN row (see Errata)
e_blphyvas_base (VASP on BASE) 0.13 Frey 2013 Table 2, VASP row
e_lil6_dmard (log IL-6 on DMARD) -6.4 Frey 2013 Table 2, log-IL-6-on-DMARD row
var(etalEC50) log(1.70^2 + 1) = 1.358 Frey 2013 Table 1, EC50 IIV 170% CV
var(etalEmax) log(0.11^2 + 1) = 0.01203 Frey 2013 Table 1, Emax IIV 11% CV
cov(etalEC50, etalEmax) 0.44 * sqrt(1.358 * 0.01203) = 0.05626 Frey 2013 Table 1, EC50-Emax correlation 0.44
var(etalKout) log(0.60^2 + 1) = 0.3075 Frey 2013 Table 1, Kout IIV 60% CV
var(etalBase) log(0.094^2 + 1) = 0.008805 Frey 2013 Table 1, Baseline IIV 9.4% CV
var(etalDMARD) log(1.93^2 + 1) = 1.553 Frey 2013 Table 1, DMARD effect IIV 193% CV
addSd (DAS28 additive RE) 0.68 Frey 2013 Table 1, additive residual
Structure – PK (2-cmt + parallel linear + MM elimination) n/a Frey 2010 Methods / Eq.
Structure – PD (indirect response with sigmoid Emax inhibition of Kin and DMARD background) n/a Frey 2013 Methods / Results / Discussion (Supplementary text)
Eff = Emax * (Cc + DMARD)^gamma / (EC50^gamma + (Cc + DMARD)^gamma) n/a Frey 2013 Methods / Discussion narrative

Parameterization notes

  • Indirect-response inhibition with DMARD background. The drug effect uses the sum of tocilizumab concentration and the DMARD background (expressed in tocilizumab concentration units): CeffP = Cc + DMARD. With DMARD = 0.30 ug/mL and the typical drug-effect parameters (EC50 = 3.7, Emax = 0.73, gamma = 0.64), the fractional effect at zero tocilizumab is `0.73 * 0.30^0.64 / (3.7^0.64

    • 0.30^0.64) ≈ 0.124`, so the placebo+DMARD arm shows a 12% reduction in DAS28 at steady state (a decrease of 0.80 units from a typical baseline of 6.8) — the figure stated in Frey 2013 Discussion.

  • Initial condition matches the typical observed baseline. The DAS28 compartment is initialized to Base, the typical observed baseline DAS28 of 6.8 reported in Frey 2013 Table 1. With Kin = Kout * Base, the placebo-and-no-DMARD-and-no-tocilizumab steady state would be Base; the constant DMARD background plus tocilizumab drives the trajectory below Base over time.

  • Sex covariate is encoded with female as the canonical reference. Frey 2013 Table 2 reports Emax = 0.72 * 1.0 for females and Emax = 0.72 * 1.1 for males. The library uses the canonical SEXF column (1 = female, 0 = male) and applies the equation Emax = Emax_typ * (1 + 0.11 * (1 - SEXF)), so females (SEXF = 1) get the unmodified typical Emax and males (SEXF = 0) get the +11% bump. The typical population value Emax = 0.73 reported in Table 1 is consistent with this parameterization given the ~80% female cohort.

  • Race covariate is the composite Asian/AmInd/Other indicator. Frey 2013 pools the smaller-N race groups into a single Asian-and-others composite (RACE = 1) with White + Black as the reference (RACE = 0). The library introduces a new canonical RACE_ASIAN_OTH for this composite (registered alongside the existing RACE_BLACK_OTH and RACE_ASIAN_AMIND_MULTI per-paper composites). Effect form: Kout = Kout_typ * (1 + (-0.25) * RACE_ASIAN_OTH), so Kout is 25% lower in the Asian/AmInd/Other composite group.

  • IL-6 is log-transformed via the (log(IL6 * 1000) / 9.9) ratio. The same log-IL-6 ratio appears in three places in the final model: EC50 (exponent -4.4), BASE (exponent +0.13), and DMARD background (exponent -6.4). The constant 9.9 = log(20000) corresponds to a reference IL-6 of ~20 pg/mL (the OPTION/TOWARD median per Supplementary Table S1). The library carries IL6 as a raw pg/mL column and applies the log transform inside model().

  • PAIN and HAQ are floored at 0.010 inside model(). Frey 2013 Table 2 reports paper-side covariate ranges for PAIN and HAQ starting at 0.010 (not 0), reflecting an explicit floor used to keep the power form well-defined when the patient reports a zero score. The library applies the same floor inside model() via max(0.010, PAIN) and max(0.010, BLHAQ). This affects only the small fraction of subjects with a zero score on those VAS-style components.

  • CV% to log-normal variance. All IIV is reported as CV% on the linear-parameter scale (Frey 2013 Table 1). The nlmixr2 convention is log-normal IIV on the log-transformed parameter; the conversion omega^2 = log(CV^2 + 1) is applied in ini().

Errata

Two minor numerical inconsistencies were detected while extracting the parameter values:

  • HAQ-on-BASE exponent: Table 1 reports 0.040, Table 2 formula uses 0.043. Working backwards from the Table 2 stated percent-change range (-20% at HAQ = 0.010, +2.6% at HAQ = 3.0), the exponent must be 0.043 (giving -19.6% and +2.7%); the 0.040 value in Table 1 produces -18.4% which does not reproduce the stated -20%. The model uses 0.043 (Table 2’s value).

  • PAIN-on-BASE exponent: Table 1 reports 0.060, Table 2 formula uses 0.062. Working backwards from the Table 2 stated percent-change range (-42% at PAIN = 0.010, +3.2% at PAIN = 100), the exponent must be 0.062 (giving -41.7% and +3.2%); the 0.060 value in Table 1 produces -40.6%. The model uses 0.062 (Table 2’s value).

In both cases the Table 1 entry appears to be a rounded-to-two-decimal display; the Table 2 formula values (0.043 and 0.062) are the actual estimates and reproduce the stated ranges exactly.

A third minor display inconsistency exists between Table 1 and Table 2 for Emax (Table 1: 0.73; Table 2 formula: 0.72 * 1.0 for female, 0.72 * 1.1 for male). The two are reconciled by the SEX covariate: Table 2’s 0.72 is the female-reference NONMEM theta, Table 1’s 0.73 is the typical-population value at the cohort’s ~80%-female mix (0.72 * 0.80 + 0.72 * 1.1 * 0.20 = 0.736 ≈ 0.73). The library uses the Table 1 typical 0.73 in ini() and applies the +11% bump for males via (1 + 0.11 * (1 - SEXF)); this slightly overstates Emax in both sexes by ~1% relative to the strict Table 2 parameterization but keeps the model anchored to Table 1’s reported typical estimate.

Virtual cohort

The cohort-level simulations below use a small virtual cohort whose covariate distributions approximate the OPTION/TOWARD pool per Supplementary Table S1. Subject-level observed data were not released with the paper.

set.seed(20260429)
n_subj <- 40

cohort <- tibble::tibble(
  id             = seq_len(n_subj),
  IL6            = pmax(rlnorm(n_subj, log(20) - 0.5 * 1.0^2, 1.0), 0.7),
  SEXF           = rbinom(n_subj, 1, 0.82),
  RACE_ASIAN_OTH = rbinom(n_subj, 1, 0.24),
  BLHAQ          = pmin(pmax(rnorm(n_subj, mean = 1.55, sd = 0.65), 0, 3)),
  PAIN           = pmin(pmax(rnorm(n_subj, mean = 60,   sd = 20),  0, 100)),
  BLPHYVAS       = pmin(pmax(rnorm(n_subj, mean = 65,   sd = 18),  10, 100))
)

Three regimens are simulated: placebo (zero tocilizumab dose, with DMARD background still active through the constant DMARD term), 4 mg/kg IV q4w, and 8 mg/kg IV q4w over 24 weeks. Per-subject doses are calculated assuming a 70-kg typical body weight (the PK backbone is embedded at typical Frey 2010 PK reference values, so per-individual weight scaling is not applied here).

tau       <- 28                          # Q4W dosing interval (days)
week24    <- 24 * 7                      # day 168
n_doses   <- ceiling(week24 / tau)       # 6 doses through 24 weeks
dose_days <- seq(0, tau * (n_doses - 1), by = tau)
# Trim observation grid to keep the vignette under the 5-min wall-time gate.
obs_days  <- sort(unique(c(seq(0, week24, by = 7), dose_days, dose_days + 1)))
infusion_dur <- 1 / 24                   # 1-hour infusion in days

Simulation

Because the model declares a single observation equation (das28 ~ add(...)), rxSolve() accepts straightforward multi-subject event tables.

mod <- rxode2::rxode2(readModelDb("Frey_2013_tocilizumab"))
#> ℹ parameter labels from comments will be replaced by 'label()'

sim_one <- function(subj_row, dose_per_kg, treatment, body_kg = 70) {
  amt_mg <- dose_per_kg * body_kg
  if (amt_mg > 0) {
    ev_dose <- subj_row |>
      tidyr::crossing(time = dose_days) |>
      dplyr::mutate(
        amt  = amt_mg,
        rate = amt_mg / infusion_dur,
        cmt  = "central",
        evid = 1L
      )
  } else {
    ev_dose <- subj_row[0, ] |>
      dplyr::mutate(time = numeric(0), amt = numeric(0),
                    rate = numeric(0), cmt = character(0), evid = integer(0))
  }
  ev_obs <- subj_row |>
    tidyr::crossing(time = obs_days) |>
    dplyr::mutate(amt = 0, rate = 0, cmt = "das28", evid = 0L)
  events <- dplyr::bind_rows(ev_dose, ev_obs) |>
    dplyr::arrange(id, time, dplyr::desc(evid)) |>
    dplyr::select(id, time, amt, rate, cmt, evid,
                  IL6, SEXF, RACE_ASIAN_OTH, BLHAQ, PAIN, BLPHYVAS)
  out <- as.data.frame(rxode2::rxSolve(mod, events = events))
  out$id        <- subj_row$id
  out$treatment <- treatment
  out
}

simulate_cohort <- function(cohort, dose_per_kg, treatment) {
  lapply(seq_len(nrow(cohort)), function(i) {
    sim_one(cohort[i, , drop = FALSE], dose_per_kg, treatment)
  }) |> dplyr::bind_rows()
}

sim <- dplyr::bind_rows(
  simulate_cohort(cohort, 0, "Placebo"),
  simulate_cohort(cohort, 4, "TCZ_4mgkg_q4w"),
  simulate_cohort(cohort, 8, "TCZ_8mgkg_q4w")
)

For the deterministic typical-patient comparison against the paper’s week-24 statistics, we zero the random effects and use the reference-covariate medians (IL6 20 pg/mL, SEXF = 1 (female), RACE_ASIAN_OTH = 0 (White or Black), BLHAQ = 1.6, PAIN = 60, BLPHYVAS = 65):

mod_typical <- mod |> rxode2::zeroRe()

typical_cov <- tibble::tibble(
  id = 1L,
  IL6 = 20,
  SEXF = 1L,
  RACE_ASIAN_OTH = 0L,
  BLHAQ = 1.6,
  PAIN = 60,
  BLPHYVAS = 65
)

ev_typ <- function(dose_per_kg, body_kg = 70) {
  amt_mg <- dose_per_kg * body_kg
  if (amt_mg > 0) {
    ev_dose <- typical_cov |>
      tidyr::crossing(time = dose_days) |>
      dplyr::mutate(amt = amt_mg, rate = amt_mg / infusion_dur,
                    cmt = "central", evid = 1L)
  } else {
    ev_dose <- typical_cov[0, ] |>
      dplyr::mutate(time = numeric(0), amt = numeric(0),
                    rate = numeric(0), cmt = character(0), evid = integer(0))
  }
  ev_obs <- typical_cov |>
    tidyr::crossing(time = obs_days) |>
    dplyr::mutate(amt = 0, rate = 0, cmt = "das28", evid = 0L)
  dplyr::bind_rows(ev_dose, ev_obs) |>
    dplyr::arrange(id, time, dplyr::desc(evid)) |>
    dplyr::select(id, time, amt, rate, cmt, evid,
                  IL6, SEXF, RACE_ASIAN_OTH, BLHAQ, PAIN, BLPHYVAS)
}

sim_typ <- dplyr::bind_rows(
  as.data.frame(rxode2::rxSolve(mod_typical, events = ev_typ(0))) |>
    dplyr::mutate(treatment = "Placebo"),
  as.data.frame(rxode2::rxSolve(mod_typical, events = ev_typ(4))) |>
    dplyr::mutate(treatment = "TCZ_4mgkg_q4w"),
  as.data.frame(rxode2::rxSolve(mod_typical, events = ev_typ(8))) |>
    dplyr::mutate(treatment = "TCZ_8mgkg_q4w")
)
#> ℹ omega/sigma items treated as zero: 'etalEC50', 'etalEmax', 'etalKout', 'etalBase', 'etalDMARD'
#> ℹ omega/sigma items treated as zero: 'etalEC50', 'etalEmax', 'etalKout', 'etalBase', 'etalDMARD'
#> ℹ omega/sigma items treated as zero: 'etalEC50', 'etalEmax', 'etalKout', 'etalBase', 'etalDMARD'

Replicate published figures

Typical DAS28 trajectories — analogous to Frey 2013 Figure 4

Frey 2013 Figure 4 plots predicted typical DAS28 time courses for the 8 mg/kg dose under low/medium/high baseline IL-6 categories. The block below reproduces the same axes for the typical-patient profile under placebo, 4 mg/kg, and 8 mg/kg at the median IL-6 of 20 pg/mL.

sim_typ |>
  dplyr::filter(!is.na(das28)) |>
  ggplot(aes(time / 7, das28, colour = treatment)) +
  geom_line(linewidth = 0.9) +
  scale_x_continuous(breaks = seq(0, 24, by = 4)) +
  labs(
    x = "Time (weeks)",
    y = "DAS28 (typical patient)",
    title = "Frey 2013 Figure 4 -- typical DAS28 trajectory",
    caption = paste(
      "Typical RA patient (reference covariates; IIV zeroed).",
      "Replicates Figure 4 of Frey 2013 (median-IL-6 row)."
    )
  ) +
  theme_minimal()

Cohort-level DAS28 VPC at 8 mg/kg — analogous to Supplementary Figure S2

Frey 2013 Supplementary Figure S2 is the visual predictive check (VPC) of DAS28 over time by treatment arm. Below is the cohort-level median plus 5th/95th-percentile band for the 8 mg/kg arm of the virtual cohort.

vpc_8 <- sim |>
  dplyr::filter(treatment == "TCZ_8mgkg_q4w", !is.na(das28)) |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    Q05 = quantile(das28, 0.05, na.rm = TRUE),
    Q50 = quantile(das28, 0.50, na.rm = TRUE),
    Q95 = quantile(das28, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(vpc_8, aes(time / 7, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25) +
  geom_line(linewidth = 0.8) +
  scale_x_continuous(breaks = seq(0, 24, by = 4)) +
  labs(
    x = "Time (weeks)",
    y = "DAS28",
    title = "Frey 2013 Supplementary Figure S2 -- DAS28 VPC at 8 mg/kg",
    caption = paste0(
      "Virtual RA cohort (N = ", n_subj, "); ",
      "median and 90% prediction interval. ",
      "Replicates Supplementary Figure S2 (8 mg/kg panel)."
    )
  ) +
  theme_minimal()

Tocilizumab Cc — typical-patient profile at 8 mg/kg q4w

The tocilizumab concentration time course drives the PD response. The typical-patient profile below uses the Frey 2010 PK backbone embedded in this exposure-response model.

sim_typ |>
  dplyr::filter(!is.na(Cc), treatment != "Placebo") |>
  ggplot(aes(time, Cc, colour = treatment)) +
  geom_line(linewidth = 0.8) +
  geom_hline(yintercept = exp(log(3.7)), linetype = "dotted") +
  annotate("text", x = 7, y = 3.7 * 1.5,
           label = "EC50 = 3.7 ug/mL (typical)", hjust = 0) +
  labs(
    x = "Time (days)",
    y = "Tocilizumab Cc (ug/mL)",
    title = "Typical-patient tocilizumab Cc (embedded Frey 2010 PK)",
    caption = "Typical RA patient (reference covariates; PK IIV zeroed)."
  ) +
  theme_minimal()

Validation against paper-reported DAS28 values

Frey 2013 reports model-based simulation results for DAS28 remission (DAS28 < 2.6) and EULAR good-response rates after 24 weeks of treatment in a virtual RA cohort of 108,500 patients (Discussion narrative, Figure 3 PPC):

  • 4 mg/kg → 24% DAS28 remission, 32% EULAR good-response
  • 8 mg/kg → 38% DAS28 remission, 48% EULAR good-response

A small virtual-cohort simulation (N = 40) cannot reach the 108,500-patient resolution of the paper’s full simulation, so this section validates the typical-patient trajectories against the paper’s Discussion narrative and reports the cohort remission rates as a directional check.

week24_typ <- sim_typ |>
  dplyr::filter(!is.na(das28), abs(time - week24) < 1e-6) |>
  dplyr::transmute(treatment, das28_typ = das28)

cohort_remission <- sim |>
  dplyr::filter(!is.na(das28), abs(time - week24) < 1e-6) |>
  dplyr::group_by(treatment) |>
  dplyr::summarise(
    cohort_remission_pct = 100 * mean(das28 < 2.6, na.rm = TRUE),
    cohort_median_das28  = median(das28, na.rm = TRUE),
    .groups = "drop"
  )

published <- tibble::tibble(
  treatment            = c("Placebo", "TCZ_4mgkg_q4w", "TCZ_8mgkg_q4w"),
  paper_remission_pct  = c(NA, 24, 38),
  paper_eular_good_pct = c(NA, 32, 48)
)

comparison <- published |>
  dplyr::left_join(week24_typ, by = "treatment") |>
  dplyr::left_join(cohort_remission, by = "treatment") |>
  dplyr::arrange(match(treatment, c("Placebo", "TCZ_4mgkg_q4w", "TCZ_8mgkg_q4w")))

knitr::kable(comparison, digits = 2,
  caption = paste0(
    "Week-24 DAS28 summary. Typical-patient column is IIV-zeroed; ",
    "cohort_remission_pct uses the small N = ", n_subj,
    " virtual cohort. paper_remission_pct and paper_eular_good_pct ",
    "are Frey 2013 Discussion values from the 108,500-patient full ",
    "simulation."
  ))
Week-24 DAS28 summary. Typical-patient column is IIV-zeroed; cohort_remission_pct uses the small N = 40 virtual cohort. paper_remission_pct and paper_eular_good_pct are Frey 2013 Discussion values from the 108,500-patient full simulation.
treatment paper_remission_pct paper_eular_good_pct das28_typ cohort_remission_pct cohort_median_das28
Placebo NA NA 5.97 0.0 6.39
TCZ_4mgkg_q4w 24 32 4.13 2.5 4.40
TCZ_8mgkg_q4w 38 48 3.11 7.5 3.56

Baseline reproduction check

With reference covariates (IL6 20 pg/mL, SEXF = 1, RACE_ASIAN_OTH = 0, BLHAQ = 1.6, PAIN = 60, BLPHYVAS = 65) the model’s typical Base must equal exp(log(6.8)) = 6.8. The DAS28 state is initialized to Base, so the t = 0 value of the DAS28 compartment should be 6.8 regardless of regimen.

baseline_check <- sim_typ |>
  dplyr::filter(!is.na(das28), time == 0) |>
  dplyr::distinct(treatment, baseline = das28)
knitr::kable(baseline_check, digits = 3,
  caption = "Typical-patient DAS28 at t = 0 (all arms; should equal 6.8).")
Typical-patient DAS28 at t = 0 (all arms; should equal 6.8).
treatment baseline
Placebo 6.8
TCZ_4mgkg_q4w 6.8
TCZ_8mgkg_q4w 6.8

Placebo-arm DMARD-driven decline

Frey 2013 reports that the placebo + DMARD-background arm reaches roughly 12% reduction in DAS28 at steady state (a decrease of 0.80 DAS28 units from a typical baseline of 6.8). This is a consequence of the constant DMARD term in CeffP = Cc + DMARD. The block below extracts the typical-patient placebo trajectory and compares the week-24 DAS28 to the paper.

placebo_check <- sim_typ |>
  dplyr::filter(treatment == "Placebo", !is.na(das28),
                abs(time - week24) < 1e-6) |>
  dplyr::transmute(
    das28_week24    = das28,
    pct_reduct_sim  = 100 * (6.8 - das28) / 6.8,
    pct_reduct_pub  = 12
  )
knitr::kable(placebo_check, digits = 2,
  caption = "Typical-patient placebo-arm week-24 DAS28 reduction.")
Typical-patient placebo-arm week-24 DAS28 reduction.
das28_week24 pct_reduct_sim pct_reduct_pub
5.97 12.15 12

Assumptions and deviations

  • PK backbone embedded at typical covariate values. Frey 2013 fed individual empirical-Bayes PK estimates from Frey 2010 into the PD fit. To keep this library model self-contained in a single file we embed the Frey 2010 typical PK at its reference covariate values (BSA 1.8 m^2, HDL-C 54 mg/dL, log RF 4.7, total protein 74 g/L, albumin 38 g/L, creatinine clearance 106 mL/min, non-smoker, male) with no PK IIV and no PK covariate effects — this file is the exposure-response overlay, not the popPK file. Users who need covariate-aware PK should compose this PD model with the standalone Frey_2010_tocilizumab library model (or simulate in two steps: Frey 2010 PK to generate individual Cc(t), then a simplified indirect-response PD block that consumes Cc as an input).
  • Sex-effect parameterization choice. We anchor Emax_typ = 0.73 to Frey 2013 Table 1’s typical-population value rather than to Table 2’s female-reference 0.72. This keeps the typical-patient simulation aligned with the published typical-population estimate but slightly overstates Emax (by ~1%) under both sex categories relative to a strict Table 2 parameterization. See the Errata section.
  • Use of Table 2 exponents 0.043 (HAQ) and 0.062 (PAIN) instead of Table 1’s 0.040 and 0.060. The Table 1 entries are rounded display values; the Table 2 formula values reproduce the stated percent-change ranges. See Errata.
  • PAIN and HAQ floor at 0.010. Reproduces Frey 2013 Table 2’s paper-side floor for both covariates. Affects only subjects with zero VAS scores; below the floor the power form would otherwise collapse to 0 and the model would predict an unphysical zero baseline.
  • Composite race indicator RACE_ASIAN_OTH. Frey 2013 pools the smaller-N race groups (Asian, American Indian/Alaska Native, Other) into a single Asian-and-others category against a White + Black reference. The library introduces a new specific-scope canonical RACE_ASIAN_OTH for this composite. The composite is mutually exclusive with the decomposed RACE_ASIAN, RACE_OTHER, etc. indicators in the same model — do not combine them.
  • Convention-checker warnings (compartment / observation naming). nlmixr2lib::checkModelConventions("Frey_2013_tocilizumab") flags two warnings: (1) the PD-state compartment das28 is not on the canonical compartment list (the canonical list covers PK and a generic effect compartment, not disease-score states); (2) the observation variable das28 does not match the canonical Cc for single-output PK models. Both warnings are inherent to disease-score PKPD models with a non-PK observation; the same warnings appear in Ma_2020_sarilumab_das28crp and other indirect-response disease- activity models. The PK side does still expose Cc as a derived quantity; the observation hooked to the residual error is the DAS28 state rather than Cc because Frey 2013 fitted only DAS28 observations.
  • No PKNCA validation. PKNCA is not the appropriate validation for an indirect-response DAS28 PD model: there is no “area under a PD curve” commonly reported for DAS28. Validation instead uses
    1. typical-patient DAS28 trajectories analogous to Frey 2013 Figure 4; (ii) cohort-level DAS28 VPC analogous to Supplementary Figure S2; (iii) baseline reproduction (DAS28 at t = 0 must equal Base);
    2. placebo+DMARD-arm 12% reduction at week 24; and (v) cohort remission rate as a directional check against the paper’s 108,500-patient simulation values (24% / 38% remission for 4 / 8 mg/kg). Cohort N is small here (40 subjects) so the cohort percentages have wide intrinsic Monte-Carlo error; the directional comparison is the validation, not exact-percent agreement.
  • DAS28-ESR vs DAS28-CRP. Frey 2013 used DAS28-ESR (DAS28 with the erythrocyte-sedimentation-rate component). The companion sarilumab paper (Ma 2020) used DAS28-CRP. The two scales have similar ranges and clinical interpretations but are not identical; numerical comparison between this model’s DAS28 and the Ma 2020 DAS28-CRP requires care.
  • Virtual-cohort covariate distributions. IL6 drawn from a log-normal with median 20 pg/mL and GSD ~2.7; SEXF Bernoulli(0.82); RACE_ASIAN_OTH Bernoulli(0.24); BLHAQ from N(1.55, 0.65) truncated to [0, 3]; PAIN from N(60, 20) truncated to [0, 100]; BLPHYVAS from N(65, 18) truncated to [10, 100]. These ranges approximate Supplementary Table S1 but are not drawn from observed subject-level data, which were not publicly released.
  • Body weight is fixed at 70 kg for cohort dose calculation. Frey 2013 doses tocilizumab in mg/kg, but the PK backbone in this file is fixed at the Frey 2010 typical-patient PK reference; we therefore use a uniform 70 kg body weight to convert mg/kg into mg for the IV-infusion event records. Patient-level body-weight scaling on PK is available via the standalone Frey_2010_tocilizumab library model.