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Vedolizumab (Hanzel 2022)

Source: vignettes/articles/Hanzel_2022_vedolizumab.Rmd
Hanzel_2022_vedolizumab.Rmd

Model and source

  • Citation: Hanzel J, Dreesen E, Vermeire S, Lowenberg M, Hoentjen F, Bossuyt P, Clasquin E, Baert FJ, D’Haens GR, Mathot R. Pharmacokinetic-Pharmacodynamic Model of Vedolizumab for Targeting Endoscopic Remission in Patients With Crohn Disease: Posthoc Analysis of the LOVE-CD Study. Inflamm Bowel Dis. 2022;28:689-699. doi:10.1093/ibd/izab143 (PMC9071095). A corrigendum (doi:10.1093/ibd/izab270; PMC9071094) corrects reference 36 only and does not change any parameter value.
  • Description: Two-compartment population PK model for vedolizumab (humanised anti-alpha4-beta7 integrin IgG1 monoclonal antibody) with parallel linear and Michaelis-Menten elimination in adults with active Crohn’s disease (LOVE-CD trial), with interindividual and inter-occasion variability (induction / maintenance) on linear clearance and time-varying effects of serum albumin, antibodies to vedolizumab and anti-TNF-naive status on linear clearance, plus the sequential first-order discrete-time Markov model for endoscopic remission (SES-CD < 4) and dropout at weeks 26 and 52 driven by the individual predicted week-22 trough concentration (Hanzel 2022). Q, Vp, Km and Vmax were held at the Rosario 2015 values.
  • Article: https://doi.org/10.1093/ibd/izab143 (open access, PMC9071095)
  • Corrigendum: https://doi.org/10.1093/ibd/izab270 (PMC9071094; corrects reference 36 only, no parameter values affected)

Hanzel et al. analysed the first 110 patients of LOVE-CD, a phase 4 open-label trial of intravenous vedolizumab in active Crohn’s disease. The paper has two sequentially fitted parts, both packaged in the single model file:

  1. A population PK model: two compartments with parallel linear and Michaelis-Menten elimination. The peripheral volume, intercompartmental clearance, Km and Vmax were held at the Rosario 2015 values (see modellib("Rosario_2015_vedolizumab")). Linear clearance carries interindividual and inter-occasion variability and time-varying covariate effects of serum albumin, antibodies to vedolizumab and anti-TNF-naive status.
  2. A first-order discrete-time Markov model for endoscopic remission (SES-CD < 4) at weeks 26 and 52. States are no remission (0), remission (1) and dropout (2). The exposure driver is the individual predicted vedolizumab concentration at week 22 (IPRED22), the trough before the week-22 infusion.

Population

The PK analysis included 108 of 110 LOVE-CD patients with at least one quantifiable vedolizumab sample (Hanzel 2022 Table 1). In total, 737 serum samples were drawn, all before an infusion. Of the 108 patients, 69% were women. Median age was 36 years (IQR 28-46), median weight 71 kg (IQR 61-82) and median baseline albumin 41 g/L (IQR 38-43). Baseline CDAI was 261 (IQR 238-312) and baseline SES-CD 12 (IQR 7-17). Previous exposure to anti-TNF agents was very common (89%). Antibodies to vedolizumab were detected in 4 patients (3.7%), on 10 samples.

Vedolizumab 300 mg was infused at weeks 0, 2 and 6 and every 8 weeks thereafter through week 52. An extra week-10 infusion was given to the 68 patients (63%) whose CDAI had not fallen by at least 70 points. Endoscopic remission was seen in 36 patients (33%) at week 26 and 40 (37%) at week 52. The trial ran at sites in the Netherlands and Belgium.

The same information is available programmatically:

str(readModelDb("Hanzel_2022_vedolizumab")()$population)
#> Warning: some etas defaulted to non-mu referenced, possible parsing error: etaiov_cl_1, etaiov_cl_2
#> as a work-around try putting the mu-referenced expression on a simple line
#> List of 13
#>  $ species         : chr "human"
#>  $ n_subjects      : int 108
#>  $ n_studies       : int 1
#>  $ age_median      : chr "36 years (IQR 28-46)"
#>  $ weight_median   : chr "71 kg (IQR 61-82)"
#>  $ sex_female_pct  : num 69
#>  $ disease_state   : chr "Active Crohn's disease (CDAI > 220 with mucosal ulceration at baseline ileocolonoscopy); baseline CDAI median 2"| __truncated__
#>  $ dose_range      : chr "Vedolizumab 300 mg IV at weeks 0, 2 and 6 and every 8 weeks thereafter through week 52; 68 patients (63%) recei"| __truncated__
#>  $ regions         : chr "Netherlands, Belgium"
#>  $ prior_tnf_pct   : num 89
#>  $ albumin_median  : chr "41 g/L (IQR 38-43)"
#>  $ ada_positive_pct: num 3.7
#>  $ notes           : chr "LOVE-CD (NCT02646683), a phase 4 prospective open-label multicentre trial. 108 of 110 enrolled patients had at "| __truncated__

Source trace

Every ini() value carries an in-file comment pointing at its source. The table below collects them in one place.

Equation / parameter Value Source location
lcl (CL_L) log(0.215) L/day Table 2, final model
lvc (V1) log(4.92) L Table 2, final model
lvp (V2) log(1.65) L, held constant Table 2 (‘FIX’, from Rosario 2015)
lq (Q) log(0.12) L/day, held constant Table 2 (‘FIX’, from Rosario 2015)
lkm (Km) log(0.964) mg/L, held constant Table 2 (‘FIX’, from Rosario 2015)
lvmax (Vm) log(0.265) mg/day, held constant Table 2 (‘FIX’, from Rosario 2015)
e_alb_cl -0.020 per g/L Table 2; Equation 1; Appendix THETA(9)
e_ada_cl 1.89 Table 2; Appendix THETA(11)
e_tnfnaive_cl 0.755 Table 2; Appendix THETA(10)
etalcl 0.262^2 = 0.068644 Table 2 IIV CL_L 26.2 CV% (footnote: CV = sqrt(variance))
etalvc 0, held constant Appendix ETA(2) on V1; variance not reported (see Assumptions)
etaiov_cl_1, etaiov_cl_2 0.152^2 = 0.023104 Table 2 IOV CL_L 15.2 CV%; Appendix $OMEGA BLOCK(1) SAME
addSd 0.469 mg/L Table 2 additive error
propSd 0.189 Table 2 proportional error; Appendix $ERROR W
emax_01 0.7, held constant Table 2 (70% FIX); Appendix THETA(1)
lec50_01 log(20.0) mg/L Table 2, final model
emax_02 1, held constant Table 2; Appendix THETA(3)
let50_02 log(515) days Table 2, final model
emax_10, lec50_10 1 (held constant), log(1.78) mg/L Table 2, final model
emax_12, lec50_12 1 (held constant), log(0.47) mg/L Table 2, final model
CL_L covariate model CL = TVCL * (1 + (ALB - 41) * th9) * th10^TNFNAIVE * th11^ADA * exp(eta1 + IOV) Equation 1; Appendix $PK
ODEs linear + Michaelis-Menten elimination from central Appendix $DES; Supplementary Figure 1
p01, p02 EMAX01 * C22 / (EC50 + C22), EMAX02 * DAYS / (ET50 + DAYS) Appendix $PRED; Supplementary Figure 1
DAYS VISIT * 7 + 0.01 Appendix $PRED
Transition fractions P(0->2) = P02 (1 - P01), P(1->2) = P12 (1 - P10) Appendix $PRED
p10, p12 EC50 / (EC50 + C22) (inhibitory, Emax = 1) Table 2 final; form inferred (see Assumptions)

Closed-form checks of the reported effects

The paper quotes four effect sizes that follow directly from the parameters.

mod <- readModelDb("Hanzel_2022_vedolizumab")
ini_df <- rxode2::rxode(mod)$iniDf
#> ℹ parameter labels from comments will be replaced by 'label()'
#> Warning: some etas defaulted to non-mu referenced, possible parsing error: etaiov_cl_1, etaiov_cl_2
#> as a work-around try putting the mu-referenced expression on a simple line
th <- setNames(ini_df$est, ini_df$name)

checks <- tibble::tribble(
  ~quantity, ~paper, ~model,
  "CL_L increase, albumin 41 -> 28 g/L (%)", 26,
  100 * ((1 + (28 - 41) * th[["e_alb_cl"]]) - 1),
  "CL_L increase with antibodies to vedolizumab (%)", 89,
  100 * (th[["e_ada_cl"]] - 1),
  "P(remission at week 26) at C22 = 20.0 mg/L (%)", 35,
  100 * th[["emax_01"]] * 20 / (exp(th[["lec50_01"]]) + 20)
)
knitr::kable(checks, digits = 2,
             caption = "Effect sizes quoted in the Hanzel 2022 Results.")
Effect sizes quoted in the Hanzel 2022 Results.
quantity paper model
CL_L increase, albumin 41 -> 28 g/L (%) 26 26
CL_L increase with antibodies to vedolizumab (%) 89 89
P(remission at week 26) at C22 = 20.0 mg/L (%) 35 35 
stopifnot(all(abs(checks$model - checks$paper) < 0.5))

The Results text also says CL_L is “+ 25% compared to no prior exposure” in patients previously exposed to biologics. The model carries the estimated multiplier 0.755 for anti-TNF-naive patients, i.e. a 24.5% lower CL_L than in exposed patients. Equivalently, exposed patients have 1 / 0.755 = 1.32-fold the CL_L of naive patients. The rounded “25%” in the text is the first of these.

Virtual cohort

The original data are not public. The simulations below use virtual patients whose covariates approximate Table 1. Albumin is drawn from a normal distribution with the cohort median of 41 g/L and an SD of 3.7 g/L, which matches the reported IQR of 38-43 g/L, truncated to 25-52 g/L. Antibodies to vedolizumab are set absent, since they were found on only 1.4% of samples. The occasion column follows the paper: OCC = 1 (induction) before the week-14 infusion and OCC = 2 (maintenance) from day 98 onward.

Each infusion is 300 mg over 30 minutes into central.

inf_rate <- 300 / (30 / 1440) # 300 mg over 30 min, in mg/day

make_cohort <- function(n, dose_weeks, obs_days, prior_tnf, regimen,
                        id_offset = 0L) {
  ids <- id_offset + seq_len(n)
  alb <- pmin(pmax(stats::rnorm(n, 41, 3.7), 25), 52)
  doses <- expand.grid(id = ids, time = dose_weeks * 7) |>
    mutate(evid = 1L, amt = 300, rate = inf_rate)
  obs <- expand.grid(id = ids, time = obs_days) |>
    mutate(evid = 0L, amt = NA_real_, rate = NA_real_)
  bind_rows(doses, obs) |>
    mutate(
      cmt = "central",
      ALB = alb[id - id_offset],
      ADA_POS = 0L,
      PRIOR_TNF = prior_tnf,
      OCC = ifelse(time < 98, 1L, 2L),
      regimen = regimen,
      tnf = ifelse(prior_tnf == 1, "Anti-TNF-experienced", "Anti-TNF-naive")
    ) |>
    arrange(id, time, desc(evid))
}

Endoscopic remission at week 26 by dosing regimen (Table 3)

Table 3 of the paper simulates three regimens in anti-TNF-naive and anti-TNF-experienced patients. The week-26 remission probability depends only on the week-22 trough. The cohorts below are therefore solved to day 154, using only the doses given before the week-22 infusion; the week-22 dose itself cannot change the pre-dose trough. Each of the six arms has 200 patients.

rxode2::rxSetSeed(2022)
set.seed(2022)
regimens <- list(
  "300 mg at weeks 0, 2, 6, 14, 22" = c(0, 2, 6, 14),
  "300 mg at weeks 0, 2, 6, 10, 14, 22" = c(0, 2, 6, 10, 14),
  "300 mg at weeks 0, 2, 6, 10, 14, 18, 22" = c(0, 2, 6, 10, 14, 18)
)
n_arm <- 200L
arms <- expand.grid(regimen = names(regimens), prior_tnf = c(0L, 1L),
                    stringsAsFactors = FALSE)
events_t3 <- bind_rows(lapply(seq_len(nrow(arms)), function(i) {
  make_cohort(
    n = n_arm,
    dose_weeks = regimens[[arms$regimen[i]]],
    obs_days = seq(0, 154, by = 7),
    prior_tnf = arms$prior_tnf[i],
    regimen = arms$regimen[i],
    id_offset = (i - 1L) * n_arm
  )
}))
stopifnot(!anyDuplicated(unique(events_t3[, c("id", "time", "evid")])))

sim_t3 <- rxode2::rxSolve(mod, events = events_t3,
                          keep = c("regimen", "tnf"),
                          returnType = "data.frame")
#> ℹ parameter labels from comments will be replaced by 'label()'
#> Warning: some etas defaulted to non-mu referenced, possible parsing error: etaiov_cl_1, etaiov_cl_2
#> as a work-around try putting the mu-referenced expression on a simple line
#> ℹ omega/sigma items treated as zero: 'etalvc'
published_t3 <- tibble::tribble(
  ~regimen, ~tnf, ~paper,
  "300 mg at weeks 0, 2, 6, 14, 22", "Anti-TNF-naive", 29.1,
  "300 mg at weeks 0, 2, 6, 10, 14, 22", "Anti-TNF-naive", 33.9,
  "300 mg at weeks 0, 2, 6, 10, 14, 18, 22", "Anti-TNF-naive", 46.5,
  "300 mg at weeks 0, 2, 6, 14, 22", "Anti-TNF-experienced", 20.2,
  "300 mg at weeks 0, 2, 6, 10, 14, 22", "Anti-TNF-experienced", 24.2,
  "300 mg at weeks 0, 2, 6, 10, 14, 18, 22", "Anti-TNF-experienced", 40.0
)

t3 <- sim_t3 |>
  filter(time == 154) |>
  group_by(regimen, tnf) |>
  summarise(
    c22 = median(Cc),
    simulated = 100 * median(prob_endorem_wk26),
    .groups = "drop"
  ) |>
  inner_join(published_t3, by = c("regimen", "tnf")) |>
  mutate(diff = simulated - paper) |>
  arrange(tnf, simulated)

t3 |>
  rename(
    "Regimen" = regimen,
    "Group" = tnf,
    "Median C22 (mg/L)" = c22,
    "Simulated median P(remission), %" = simulated,
    "Hanzel 2022 Table 3, %" = paper,
    "Difference (points)" = diff
  ) |>
  knitr::kable(digits = 1, caption = paste(
    "Replicates Table 3 of Hanzel 2022: median predicted probability of",
    "endoscopic remission at week 26."
  ))
Replicates Table 3 of Hanzel 2022: median predicted probability of endoscopic remission at week 26.
Regimen Group Median C22 (mg/L) Simulated median P(remission), % Hanzel 2022 Table 3, % Difference (points)
300 mg at weeks 0, 2, 6, 14, 22 Anti-TNF-experienced 8.4 20.6 20.2 0.4
300 mg at weeks 0, 2, 6, 10, 14, 22 Anti-TNF-experienced 10.8 24.6 24.2 0.4
300 mg at weeks 0, 2, 6, 10, 14, 18, 22 Anti-TNF-experienced 27.6 40.6 40.0 0.6
300 mg at weeks 0, 2, 6, 14, 22 Anti-TNF-naive 12.3 26.7 29.1 -2.4
300 mg at weeks 0, 2, 6, 10, 14, 22 Anti-TNF-naive 20.6 35.5 33.9 1.6
300 mg at weeks 0, 2, 6, 10, 14, 18, 22 Anti-TNF-naive 41.9 47.4 46.5 0.9

The simulated medians reproduce Table 3 to within a few percentage points in all six cells, including the gain from shortening the interval to every 4 weeks. The checks are on the cohort median, which is stable across rxode2 builds, and on the direction of the regimen and anti-TNF effects. Those contrasts are many times larger than the Monte-Carlo error of a 200-patient median.

stopifnot(
  # Structural: a mis-transcribed EC50, CL or covariate effect moves these
  # medians by far more than 6 points.
  all(abs(t3$diff) < 6),
  # Every-4-weeks dosing beats per-label dosing in both groups.
  all(t3 |> group_by(tnf) |>
        summarise(gain = max(simulated) - min(simulated)) |>
        pull(gain) > 10)
)

Exposure-response curve (Figure 2)

Figure 2 of the paper overlays observed and predicted week-26 outcomes on quantiles of the week-22 concentration. Below are the model-predicted curves from the typical parameters: remission, dropout without remission, and persistent non-remission.

c22_grid <- c(seq(0.5, 80, by = 0.5))
p01 <- th[["emax_01"]] * c22_grid / (exp(th[["lec50_01"]]) + c22_grid)
days26 <- 26 * 7 + 0.01
p02 <- th[["emax_02"]] * days26 / (exp(th[["let50_02"]]) + days26)
er_curve <- tibble(
  C22 = c22_grid,
  `Endoscopic remission` = p01,
  `Dropout` = p02 * (1 - p01),
  `No remission` = 1 - p01 - p02 * (1 - p01)
) |>
  pivot_longer(-C22, names_to = "state", values_to = "prob")

ggplot(er_curve, aes(C22, 100 * prob, linetype = state)) +
  geom_line() +
  geom_point(data = tibble(C22 = 20, prob = 0.35), aes(C22, 100 * prob),
             inherit.aes = FALSE, colour = "red") +
  labs(x = "Individual predicted vedolizumab concentration at week 22 (mg/L)",
       y = "Probability at week 26 (%)", linetype = NULL,
       caption = paste("Model counterpart of Figure 2 of Hanzel 2022; red point:",
                       "35% at 20.0 mg/L quoted in the Results."))

Week-52 Markov outcomes

The model also returns the marginal state probabilities at week 52. These come from chaining the baseline -> week 26 and week 26 -> week 52 transitions, all driven by the same week-22 concentration. The paper does not report simulated week-52 probabilities, so they cannot be checked against a published number. Two properties can be checked: the three state probabilities sum to 1 at each visit, and remission rises with exposure.

wk <- sim_t3 |>
  filter(time == 154) |>
  group_by(regimen, tnf) |>
  summarise(across(c(prob_endorem_wk26, prob_dropout_wk26,
                     prob_endorem_wk52, prob_dropout_wk52),
                   ~ 100 * median(.x)),
            .groups = "drop")
wk |>
  rename(
    "Regimen" = regimen, "Group" = tnf,
    "Remission wk 26, %" = prob_endorem_wk26,
    "Dropout wk 26, %" = prob_dropout_wk26,
    "Remission wk 52, %" = prob_endorem_wk52,
    "Dropout wk 52, %" = prob_dropout_wk52
  ) |>
  knitr::kable(digits = 1, caption = "Median simulated state probabilities.")
Median simulated state probabilities.
Regimen Group Remission wk 26, % Dropout wk 26, % Remission wk 52, % Dropout wk 52, %
300 mg at weeks 0, 2, 6, 10, 14, 18, 22 Anti-TNF-experienced 40.6 15.5 55.3 27.0
300 mg at weeks 0, 2, 6, 10, 14, 18, 22 Anti-TNF-naive 47.4 13.7 63.3 22.7
300 mg at weeks 0, 2, 6, 10, 14, 22 Anti-TNF-experienced 24.6 19.7 33.9 38.0
300 mg at weeks 0, 2, 6, 10, 14, 22 Anti-TNF-naive 35.5 16.8 48.9 30.3
300 mg at weeks 0, 2, 6, 14, 22 Anti-TNF-experienced 20.6 20.7 28.2 40.9
300 mg at weeks 0, 2, 6, 14, 22 Anti-TNF-naive 26.7 19.1 37.0 36.4 

at154 <- sim_t3 |> filter(time == 154)
stopifnot(
  max(abs(at154$prob_endorem_wk26 + at154$prob_dropout_wk26 +
            at154$prob_noendorem_wk26 - 1)) < 1e-10,
  max(abs(at154$prob_endorem_wk52 + at154$prob_dropout_wk52 +
            at154$prob_noendorem_wk52 - 1)) < 1e-10,
  cor(at154$Cc, at154$prob_endorem_wk52, method = "spearman") > 0.9
)

Trough concentrations over the LOVE-CD schedule (Figure 1)

Figure 1 of the paper is a prediction-corrected VPC of the trough samples. This section simulates the LOVE-CD schedule through week 52 with 200 patients. Infusions are at weeks 0, 2, 6, 14, 22, 30, 38 and 46, and 63% of patients get the extra week-10 infusion. Anti-TNF exposure is assigned at the trial rate of 89%. Troughs are read before each infusion and at week 52.

rxode2::rxSetSeed(1)
set.seed(1)
n_vpc <- 200L
trough_weeks <- c(2, 6, 10, 14, 22, 30, 38, 46, 52)
extra_wk10 <- stats::runif(n_vpc) < 0.63
prior <- as.integer(stats::runif(n_vpc) < 0.89)
events_vpc <- bind_rows(lapply(seq_len(n_vpc), function(i) {
  make_cohort(
    n = 1L,
    dose_weeks = c(0, 2, 6, if (extra_wk10[i]) 10, 14, 22, 30, 38, 46),
    obs_days = trough_weeks * 7 - 1e-4,
    prior_tnf = prior[i],
    regimen = "LOVE-CD",
    id_offset = i - 1L
  )
}))
sim_vpc <- rxode2::rxSolve(mod, events = events_vpc, returnType = "data.frame")
#> ℹ omega/sigma items treated as zero: 'etalvc'

vpc <- sim_vpc |>
  mutate(week = round(time / 7)) |>
  group_by(week) |>
  summarise(
    Q05 = quantile(Cc, 0.05), Q50 = median(Cc), Q95 = quantile(Cc, 0.95),
    .groups = "drop"
  )

ggplot(vpc, aes(week * 7, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25) +
  geom_line() +
  geom_point() +
  labs(x = "Time (days)", y = "Simulated trough vedolizumab (mg/L)",
       caption = paste("Counterpart of Figure 1 of Hanzel 2022 (median and",
                       "5th-95th percentiles of simulated troughs)."))

knitr::kable(vpc, digits = 1, caption = "Simulated trough percentiles by week.")
Simulated trough percentiles by week.
week Q05 Q50 Q95
2 18.8 28.0 35.4
6 12.0 26.8 44.0
10 11.3 28.1 51.5
14 4.6 22.7 46.5
22 3.1 11.1 31.9
30 2.5 8.9 28.1
38 2.4 8.3 27.0
46 2.4 8.2 26.0
52 4.7 12.6 32.7

In Figure 1 of the paper, the observed median trough is roughly 25 mg/L during induction (days 14-98) and roughly 10-12 mg/L during maintenance. These values were read approximately from the plot. The simulated medians fall in the same ranges. Figure 1 is prediction-corrected, so only this broad agreement is checked.

induction <- vpc |> filter(week %in% c(2, 6, 10)) |> pull(Q50)
maintenance <- vpc |> filter(week %in% c(22, 30, 38, 46)) |> pull(Q50)
stopifnot(
  all(induction > 15 & induction < 40),
  all(maintenance > 5 & maintenance < 20)
)

PKNCA validation

The paper reports no NCA. For completeness, PKNCA is run over the first dosing interval (days 0-14) in typical-albumin patients, split by anti-TNF exposure. The anti-TNF-naive group should show the larger AUC, because its linear clearance is 0.755-fold that of the experienced group.

rxode2::rxSetSeed(7)
set.seed(7)
nca_obs <- sort(unique(c(0, 30 / 1440, 0.25, 0.5, 1, 2, 3, 5, 7, 10, 14)))
events_nca <- bind_rows(
  make_cohort(100L, 0, nca_obs, 1L, "first dose", id_offset = 0L),
  make_cohort(100L, 0, nca_obs, 0L, "first dose", id_offset = 100L)
)
sim_nca <- rxode2::rxSolve(mod, events = events_nca, keep = "tnf",
                           returnType = "data.frame") |>
  filter(!is.na(Cc)) |>
  select(id, time, Cc, tnf)
#> ℹ omega/sigma items treated as zero: 'etalvc'

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | tnf + id)
dose_obj <- PKNCA::PKNCAdose(
  events_nca |> filter(evid == 1) |> select(id, time, amt, tnf),
  amt ~ time | tnf + id
)
intervals <- data.frame(start = 0, end = 14, cmax = TRUE, tmax = TRUE,
                        auclast = TRUE)
nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj,
                                          intervals = intervals))
nca_sum <- as.data.frame(nca_res$result) |>
  group_by(tnf, PPTESTCD) |>
  summarise(median = median(PPORRES), .groups = "drop") |>
  pivot_wider(names_from = PPTESTCD, values_from = median)
nca_sum |>
  rename("Group" = tnf, "Cmax (mg/L)" = cmax, "Tmax (day)" = tmax,
         "AUC0-14 (mg*day/L)" = auclast) |>
  knitr::kable(digits = 2, caption = "First-dose NCA (medians, PKNCA).")
First-dose NCA (medians, PKNCA).
Group AUC0-14 (mg*day/L) Cmax (mg/L) Tmax (day)
Anti-TNF-experienced 574.21 60.93 0.02
Anti-TNF-naive 614.30 60.94 0.02 
stopifnot(
  nca_sum$auclast[nca_sum$tnf == "Anti-TNF-naive"] >
    nca_sum$auclast[nca_sum$tnf == "Anti-TNF-experienced"]
)

Assumptions and deviations

  • Final-model form of the 1 -> 0 and 1 -> 2 transitions. The Appendix $PRED code models the transitions out of remission (P10, P12) as constant probabilities, THETA(5) and THETA(6), bounded to (0, 1). Table 2’s final-model column instead lists Emax10 = 1 FIX, EC50,10 = 1.78 mg/L, Emax12 = 1 FIX and EC50,12 = 0.47 mg/L, with RSEs and bootstrap medians. The maintainers implemented the Table 2 parameterisation, for three reasons:
    1. 1.78 lies outside the (0, 1) bound, so it cannot be a final value of the printed constant.
    2. With no random effects, the likelihood of the transitions out of state 1 separates from the state-0 transitions. Had P10 and P12 stayed constant, their final estimates would equal the base-model 10.0% and 2.8%. The paper reports different numbers with separate bootstrap results.
    3. Only an inhibitory form, P = EC50 / (EC50 + C22), is consistent with the estimates. At a typical week-22 trough of about 20 mg/L it gives about 8% (1 -> 0) and 2% (1 -> 2), close to the base-model constants. A stimulatory form C22 / (EC50 + C22) with Emax = 1 would give about 90%, which the data could not support.
    The driver is taken to be the same week-22 concentration (IPRED22) that drives the 0 -> 1 transition. It is the only concentration metric of the selected model that the Appendix data set carries. Only the week-52 outputs depend on this choice. The week-26 outputs, Table 3 and Figure 2 do not.
  • Table 2 bootstrap interval for EC50,12. The printed bootstrap interval for EC50,12 (0.36-3.62) is identical to the one for EC50,10. It is probably a copy error, and it does not affect the point estimate used here.
  • Interindividual variability on V1. The Appendix control stream carries ETA(2) on V1, with an initial variance of 0.03. Table 2 and the Results text report IIV only on CL_L. The slot is kept as etalvc ~ fixed(0), so simulations match the reported final model.
  • Anti-TNF covariate encoding. The source column TNFNAIVE (1 = naive) is mapped to the canonical PRIOR_TNF (1 = previously exposed) as TNFNAIVE = 1 - PRIOR_TNF. The paper’s multiplier and its reference patient (previously exposed) are unchanged.
  • Exposure driver in the model output. The probabilities p01, p10, p12 and the prob_* outputs are computed from the current Cc. They reproduce the paper only when read at day 154 (week 22), before any week-22 infusion. The 0 -> 2 dropout terms use the paper’s assessment days (VISIT * 7 + 0.01 with VISIT = 26 and 52) and do not depend on t.
  • Infusion duration. The paper does not state it; the labelled 30-minute infusion is used. It has no practical effect on trough concentrations.
  • Virtual covariates. Table 3 sampled covariates from the original data set, which is not public. Here albumin is drawn from a normal distribution that matches the reported median and IQR. Antibodies to vedolizumab are set absent: they were found on 1.4% of samples and barely move the medians.
  • Residual error. The Appendix $ERROR block W = SQRT(THETA(1)**2 + IPRED**2 * THETA(2)**2), with $SIGMA held at 1, is the nlmixr2 combined2 additive + proportional error. Table 2 labels the proportional term “%” but prints it as a fraction (0.189).