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Eculizumab and SB12 biosimilar (Lee 2024)

Source: vignettes/articles/Lee_2024_eculizumab.Rmd
Lee_2024_eculizumab.Rmd

Model and source

  • Citation: Lee H, Park J, Jang H, Lee SJ, Kim J. Population pharmacokinetic, pharmacodynamic and efficacy modeling of SB12 (proposed eculizumab biosimilar) and reference eculizumab. Eur J Clin Pharmacol. 2024;80(9):1325-1338. doi:10.1007/s00228-024-03703-8
  • Description: Two-compartment population PK model of eculizumab (SB12 proposed biosimilar and reference Soliris) with a direct inhibitory sigmoid Imax model for terminal complement activity and a direct sigmoid Emax model linking terminal complement activity to serum LDH, in healthy subjects and patients with paroxysmal nocturnal haemoglobinuria (Lee 2024)
  • Article: https://doi.org/10.1007/s00228-024-03703-8
  • Supplement (Appendix Tables A1/A2 and Figure A1): https://doi.org/10.1007/s00228-024-03703-8 (Electronic Supplementary Material, 228_2024_3703_MOESM1_ESM.docx)

Lee 2024 developed a linked population pharmacokinetic / pharmacodynamic / efficacy model for SB12 (a proposed eculizumab biosimilar) and reference eculizumab (Soliris), pooling a phase I single-dose study in healthy subjects with a phase III cross-over study in patients with paroxysmal nocturnal haemoglobinuria (PNH). Three endpoints are chained:

  1. PK – serum eculizumab concentration, described by a two-compartment model with first-order elimination.
  2. PD – terminal complement activity (%), described by a direct response model with an inhibitory sigmoid Imax relationship to serum concentration.
  3. Efficacy – serum lactate dehydrogenase (LDH, U/L), described by a direct sigmoid Emax relationship to terminal complement activity.

Treatment group (SB12 versus EU-sourced or US-sourced reference eculizumab) was not a statistically significant covariate on any PK, PD or efficacy parameter, which is the biosimilarity conclusion of the paper. One parameter set therefore describes both products, and the packaged model carries no treatment covariate.

Population

The analysis pooled 289 subjects across two studies (Lee 2024 Table 1 and Table 2):

  • Phase I – 240 healthy subjects (80 per treatment arm: SB12, EU-sourced Soliris, US-sourced Soliris) given a single 300 mg intravenous infusion over 35 min, with dense PK sampling to 1512 h and PD sampling to 1512 h. Median age 40 years (19-55), median weight 82.40 kg (70.0-94.3), 95.8% male, 95.8% White.
  • Phase III – 49 patients with PNH in a double-blind cross-over study (SB12-to-ECU and ECU-to-SB12 sequences) given the approved regimen: 600 mg IV every week for 4 weeks, 900 mg at week 5, then 900 mg every 2 weeks through week 50. Trough-only PK sampling. Median age 36 years (18-79), median weight 63.00 kg (43.0-111.0), 55.1% female, 53.1% Asian.

The pooled dataset comprised 4136 quantifiable serum concentrations, 2900 terminal complement activity levels (289 subjects) and 1350 LDH levels (49 PNH patients). PNH patients had notably lower weight and height and higher baseline aspartate aminotransferase, bilirubin and LDH than the healthy subjects.

The same information is available programmatically via the model’s population metadata (readModelDb("Lee_2024_eculizumab")()$population).

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Lee_2024_eculizumab.R. The table below collects them in one place for review.

Equation / parameter Value Source location
d/dt(central), d/dt(peripheral1) (two-compartment, first-order elimination) n/a Lee 2024 Results, “PK/PD model of pooled subjects”: “a two-compartment model best described the PKs of SB12 and ECU in pooled data”
CL = theta1 * (WT/80.9)^theta7 n/a Lee 2024 Table 3 (covariate equation printed in the parameter column)
Vc = theta2 * (WT/80.9)^theta8 n/a Lee 2024 Table 3 (covariate equation printed in the parameter column)
lcl (CL) 0.0174 L/h Lee 2024 Table 3, theta1
e_wt_cl 1.1400 Lee 2024 Table 3, theta7
lvc (Vc, healthy) 3.47 L Lee 2024 Table 3, theta2
e_pnh_vc log(5.68/3.47) = 0.4928 Lee 2024 Table 3, theta2_pat = 5.68 L versus theta2 = 3.47 L
e_wt_vc 0.8630 Lee 2024 Table 3, theta8
lvp (Vp) 0.79 L Lee 2024 Table 3, Vp
lq (Q) 0.0134 L/h Lee 2024 Table 3, Q
etalcl 15.62 %CV -> omega^2 = 0.0241056 Lee 2024 Table 3, omega_CL
etalvc 12.74 %CV -> omega^2 = 0.0161004 Lee 2024 Table 3, omega_VC
etalvp 36.80 %CV -> omega^2 = 0.1270061 Lee 2024 Table 3, omega_VP
cov(etalcl, etalvc) rho = 0.54 -> 0.0106383 Lee 2024 Table 3, rho_CL-VC
propSd 11.70% Lee 2024 Table 3, sigma_prop
tca = E0 * (1 - Imax * Cc^H / (IC50^H + Cc^H)) n/a Lee 2024 Results / Discussion: “direct response model with an inhibitory sigmoid maximum effect (Emax) relationship between serum concentrations and terminal complement activity”
lrbase_tca (E0, healthy) 85.90% Lee 2024 Table 4, E0
e_pnh_rbase_tca log(101.00/85.90) = 0.1619 Lee 2024 Table 4, E0_pat = 101.00% versus E0 = 85.90%
limax_tca (Imax, healthy) 0.93 Lee 2024 Table 4, Imax
e_pnh_imax_tca log(0.88/0.93) = -0.0553 Lee 2024 Table 4, Imax_pat = 0.88 versus Imax = 0.93
lec50_tca (IC50) 36.60 ug/mL Lee 2024 Table 4, IC50
lhill_tca (H) 4.56 Lee 2024 Table 4, H
etalrbase_tca 15.29 %CV -> omega^2 = 0.0231093 Lee 2024 Table 4, omega_E0
etalimax_tca 2.65 %CV -> omega^2 = 0.0007020 Lee 2024 Table 4, omega_Imax
etalec50_tca 22.74 %CV -> omega^2 = 0.0504181 Lee 2024 Table 4, omega_IC50
propSd_tca 18.60% Lee 2024 Table 4, sigma_prop
ldh = LL0 + LMAX * tca^LGAM / (LC50^LGAM + tca^LGAM) n/a Lee 2024 Results / Discussion: “the relationship between terminal complement activity and LDH was well described by a direct sigmoid Emax model”
lrbase_ldh (LL0) 206 U/L Lee 2024 Table 5, LL0
lemax_ldh (LMAX) 1680 U/L Lee 2024 Table 5, LMAX
lec50_ldh (LC50) 39.0% Lee 2024 Table 5, LC50
lhill_ldh (LGAM) 4.30 Lee 2024 Table 5, LGAM
etalrbase_ldh 28.97 %CV -> omega^2 = 0.0805897 Lee 2024 Table 5, omega_LL0
etalemax_ldh 81.57 %CV -> omega^2 = 0.5100452 Lee 2024 Table 5, omega_LMAX
etalhill_ldh 55.91 %CV -> omega^2 = 0.2720044 Lee 2024 Table 5, omega_LGAM
propSd_ldh 32.00% Lee 2024 Table 5, sigma_prop
Exponential ETA on every parameter n/a Lee 2024 Methods, “Base model”: “The inter-individual variability (i.e. ETA) of each parameter was applied exponentially”
Reference weight 80.9 kg n/a Lee 2024 Table 3, covariate equations (WT/80.9)

Virtual cohort

Original observed data are not publicly available. The simulations below use virtual populations whose covariate distributions approximate the published trial demographics (Lee 2024 Table 2). Two arms are simulated:

  • Healthy (DIS_PNH = 0) – 150 subjects, weight drawn to match the phase I median 82.40 kg and range 70.0-94.3 kg, given a single 300 mg IV infusion over 35 min (0.58 h) with dense sampling to 1512 h.
  • PNH (DIS_PNH = 1) – 150 patients, weight drawn to match the phase III median 63.00 kg and range 43.0-111.0 kg, given the approved regimen (600 mg weekly x 4, 900 mg at week 5, then 900 mg every 2 weeks to week 50), sampled to week 52.
set.seed(20240530)

n_per_arm <- 150L

# Truncated-normal weight sampler matching a published median and range.
sample_weight <- function(n, median_wt, sd_wt, lo, hi) {
  w <- stats::rnorm(n, mean = median_wt, sd = sd_wt)
  pmin(pmax(w, lo), hi)
}

# Healthy phase I arm: single 300 mg IV infusion over 35 min (0.58 h).
healthy_subj <- tibble(
  id      = seq_len(n_per_arm),
  WT      = sample_weight(n_per_arm, 82.40, 6.0, 70.0, 94.3),
  DIS_PNH = 0,
  arm     = "Healthy (phase I, 300 mg single IV)"
)

healthy_dose <- healthy_subj |>
  mutate(time = 0, amt = 300, evid = 1L, cmt = "central", rate = 300 / 0.58)

healthy_obs <- healthy_subj |>
  tidyr::crossing(time = c(0, 0.58, 4, 8, 12, 24, 48, 96, 168, 240, 336,
                           504, 672, 840, 1008, 1176, 1344, 1512)) |>
  mutate(amt = NA_real_, evid = 0L, cmt = "Cc", rate = NA_real_)

# PNH phase III arm: 600 mg QW x 4 (weeks 0-3), 900 mg at week 4 (the fifth
# dose, 1 week later), then 900 mg Q2W through week 50. Times in hours.
wk <- function(x) x * 24 * 7
pnh_dose_times <- c(wk(0:3), wk(4), wk(seq(6, 50, by = 2)))
pnh_dose_amts  <- c(rep(600, 4), 900, rep(900, length(seq(6, 50, by = 2))))

pnh_subj <- tibble(
  id      = n_per_arm + seq_len(n_per_arm),
  WT      = sample_weight(n_per_arm, 63.00, 13.0, 43.0, 111.0),
  DIS_PNH = 1,
  arm     = "PNH (phase III, 600 mg QW x4 then 900 mg Q2W)"
)

pnh_dose <- pnh_subj |>
  tidyr::crossing(tibble(time = pnh_dose_times, amt = pnh_dose_amts)) |>
  mutate(evid = 1L, cmt = "central", rate = amt / 0.58)

pnh_obs <- pnh_subj |>
  tidyr::crossing(time = sort(unique(c(0, wk(seq(0, 52, by = 1)),
                                       wk(seq(0, 52, by = 2)) + 1)))) |>
  mutate(amt = NA_real_, evid = 0L, cmt = "Cc", rate = NA_real_)

events <- bind_rows(healthy_dose, healthy_obs, pnh_dose, pnh_obs) |>
  arrange(id, time, desc(evid)) |>
  as.data.frame()

stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))
c(healthy = n_per_arm, pnh = n_per_arm, rows = nrow(events))
#> healthy     pnh    rows 
#>     150     150   19050

Simulation 

mod <- readModelDb("Lee_2024_eculizumab")

sim <- rxode2::rxSolve(
  mod,
  events = events,
  keep   = c("WT", "DIS_PNH", "arm"),
  useLinCmt = FALSE
) |>
  as.data.frame()
#> ℹ parameter labels from comments will be replaced by 'label()'

# Guard against silently dropped subjects (rxSolve does this quietly).
stopifnot(dplyr::n_distinct(sim$id) == 2L * n_per_arm)

Replicate published figures

Figure 4A/4B – serum eculizumab concentration 

# Replicates Figure 4A (healthy subjects) and 4B (PNH patients) of Lee 2024:
# visual predictive check of serum eculizumab concentration versus time.
sim |>
  group_by(arm, time) |>
  summarise(
    Q05 = quantile(Cc, 0.05, na.rm = TRUE),
    Q50 = quantile(Cc, 0.50, na.rm = TRUE),
    Q95 = quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time / 24, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "steelblue") +
  geom_line(colour = "steelblue") +
  geom_hline(yintercept = 36.6, linetype = "dashed") +
  facet_wrap(~arm, scales = "free") +
  labs(
    x = "Time (days)", y = "Serum eculizumab (ug/mL)",
    title = "Figure 4A/4B - simulated PK",
    caption = paste("Replicates Figures 4A and 4B of Lee 2024.",
                    "Dashed line = IC50 (36.6 ug/mL, Table 4).")
  )

Figure 4C/4D – terminal complement activity 

# Replicates Figure 4C (healthy subjects) and 4D (PNH patients) of Lee 2024:
# visual predictive check of terminal complement activity versus time.
sim |>
  group_by(arm, time) |>
  summarise(
    Q05 = quantile(tca, 0.05, na.rm = TRUE),
    Q50 = quantile(tca, 0.50, na.rm = TRUE),
    Q95 = quantile(tca, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time / 24, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "darkorange") +
  geom_line(colour = "darkorange") +
  facet_wrap(~arm, scales = "free_x") +
  labs(
    x = "Time (days)", y = "Terminal complement activity (%)",
    title = "Figure 4C/4D - simulated terminal complement activity",
    caption = "Replicates Figures 4C and 4D of Lee 2024."
  )

Figure 4E – LDH in PNH patients 

# Replicates Figure 4E of Lee 2024: visual predictive check of the
# PK/PD/efficacy model (serum LDH) in PNH patients.
sim |>
  filter(DIS_PNH == 1) |>
  group_by(time) |>
  summarise(
    Q05 = quantile(ldh, 0.05, na.rm = TRUE),
    Q50 = quantile(ldh, 0.50, na.rm = TRUE),
    Q95 = quantile(ldh, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time / 24, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "firebrick") +
  geom_line(colour = "firebrick") +
  geom_hline(yintercept = 250, linetype = "dashed") +
  labs(
    x = "Time (days)", y = "Serum LDH (U/L)",
    title = "Figure 4E - simulated LDH in PNH patients",
    caption = paste("Replicates Figure 4E of Lee 2024.",
                    "Dashed line = approximate upper limit of normal (250 U/L).")
  )

Exposure-response relationships (Tables 4 and 5)

The two direct-response relationships are algebraic, so they can be drawn exactly from the typical-value parameters without simulating a time course.

conc_grid <- tibble(Cc = seq(0, 150, length.out = 401)) |>
  mutate(
    `Healthy (E0 85.9%, Imax 0.93)` =
      85.90 * (1 - 0.93 * Cc^4.56 / (36.60^4.56 + Cc^4.56)),
    `PNH (E0 101.0%, Imax 0.88)` =
      101.00 * (1 - 0.88 * Cc^4.56 / (36.60^4.56 + Cc^4.56))
  ) |>
  pivot_longer(-Cc, names_to = "group", values_to = "tca")

p_pkpd <- ggplot(conc_grid, aes(Cc, tca, colour = group)) +
  geom_line() +
  geom_vline(xintercept = 36.6, linetype = "dashed") +
  geom_vline(xintercept = 35, linetype = "dotted") +
  labs(
    x = "Serum eculizumab (ug/mL)", y = "Terminal complement activity (%)",
    colour = NULL,
    title = "PK/PD: inhibitory sigmoid Imax (Lee 2024 Table 4)",
    caption = paste("Dashed = IC50 36.6 ug/mL;",
                    "dotted = 35 ug/mL, the EMA-cited concentration sufficient",
                    "for complete complement inhibition (Lee 2024 Discussion).")
  ) +
  theme(legend.position = "bottom")

tca_grid <- tibble(tca = seq(0, 120, length.out = 401)) |>
  mutate(ldh = 206 + 1680 * tca^4.30 / (39^4.30 + tca^4.30))

p_pdeff <- ggplot(tca_grid, aes(tca, ldh)) +
  geom_line(colour = "firebrick") +
  geom_vline(xintercept = 39, linetype = "dashed") +
  labs(
    x = "Terminal complement activity (%)", y = "Serum LDH (U/L)",
    title = "PD/efficacy: direct sigmoid Emax (Lee 2024 Table 5)",
    caption = "Dashed = LC50 (39%, Table 5)."
  )

p_pkpd

p_pdeff

PKNCA validation

NCA is run on the phase I healthy arm, where the paper’s single 300 mg IV dose gives a clean single-dose profile. The PNH arm is a multiple-dose regimen with trough-only sampling in the source study and is validated below by its steady-state exposure instead.

sim_nca <- sim |>
  filter(DIS_PNH == 0) |>
  filter(!is.na(Cc)) |>
  select(id, time, Cc, arm)

# Guarantee a time = 0 record per subject (IV bolus/infusion: pre-dose Cc = 0).
sim_nca <- bind_rows(
  sim_nca,
  sim_nca |> distinct(id, arm) |> mutate(time = 0, Cc = 0)
) |>
  distinct(id, arm, time, .keep_all = TRUE) |>
  arrange(id, time)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | arm + id)

dose_df <- events |>
  filter(evid == 1, DIS_PNH == 0) |>
  select(id, time, amt, arm)

dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | arm + id)

intervals <- data.frame(
  start      = 0,
  end        = Inf,
  cmax       = TRUE,
  tmax       = TRUE,
  aucinf.obs = TRUE,
  half.life  = TRUE
)

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

Comparison against the model-implied analytic solution

Lee 2024 does not publish an NCA table, so the reference column below is the analytic solution of the packaged model at the typical healthy 82.40 kg subject. For a linear two-compartment IV model the exposure and disposition half-lives follow in closed form from CL, Vc, Vp and Q; a simulated NCA that reproduces them confirms that the ODE system, the covariate scaling and the units were all encoded correctly.

wt_typ <- 82.40
wt_ref <- 80.9
dose   <- 300

cl_typ <- 0.0174 * (wt_typ / wt_ref)^1.1400
vc_typ <- 3.47   * (wt_typ / wt_ref)^0.8630
vp_typ <- 0.79
q_typ  <- 0.0134

k10 <- cl_typ / vc_typ
k12 <- q_typ / vc_typ
k21 <- q_typ / vp_typ

disc  <- sqrt((k10 + k12 + k21)^2 - 4 * k10 * k21)
alpha <- ((k10 + k12 + k21) + disc) / 2
beta  <- ((k10 + k12 + k21) - disc) / 2

published <- tibble::tibble(
  arm        = "Healthy (phase I, 300 mg single IV)",
  cmax       = dose / vc_typ,        # C0 for a (near-)instantaneous IV input
  tmax       = 0.58,                 # end of the 35-min infusion
  aucinf.obs = dose / cl_typ,        # AUCinf = Dose / CL, exactly
  half.life  = log(2) / beta         # terminal (beta) half-life
)

knitr::kable(
  published |>
    dplyr::rename(
      "Arm"                  = arm,
      "Cmax (ug/mL)"         = cmax,
      "Tmax (h)"             = tmax,
      "AUC0-inf (ug*h/mL)"   = aucinf.obs,
      "t1/2 (h)"             = half.life
    ),
  digits  = c(0, 1, 2, 0, 1),
  caption = "Analytic reference values implied by the packaged model at the typical 82.40 kg healthy subject."
)
Analytic reference values implied by the packaged model at the typical 82.40 kg healthy subject.
Arm Cmax (ug/mL) Tmax (h) AUC0-inf (ug*h/mL) t1/2 (h)
Healthy (phase I, 300 mg single IV) 85.1 0.58 16884 177.6 
cmp <- nlmixr2lib::ncaComparisonTable(
  simulated = nca_res,
  reference = published,
  by        = "arm",
  units     = c(cmax = "ug/mL", tmax = "h",
                aucinf.obs = "ug*h/mL", half.life = "h"),
  tolerance_pct = 20
)

knitr::kable(
  cmp,
  caption = paste("Simulated NCA (median of 150 virtual healthy subjects)",
                  "versus the model-implied analytic solution.",
                  "* differs from reference by >20%."),
  align   = c("l", "l", "r", "r", "r")
)
Simulated NCA (median of 150 virtual healthy subjects) versus the model-implied analytic solution. * differs from reference by >20%.
NCA parameter arm Reference Simulated % diff
Cmax (ug/mL) Healthy (phase I, 300 mg single IV) 85.1 84.4 -0.8%
Tmax (h) Healthy (phase I, 300 mg single IV) 0.58 0.58 +0.0%
AUC0-∞ (obs) (ug*h/mL) Healthy (phase I, 300 mg single IV) 16900 16600 -1.8%
t½ (h) Healthy (phase I, 300 mg single IV) 178 178 +0.3%

Published claims the simulation must reproduce

Lee 2024 makes three quantitative statements that the packaged model can be checked against directly.

cmax_by_id <- sim |>
  filter(DIS_PNH == 0) |>
  group_by(id) |>
  summarise(cmax = max(Cc, na.rm = TRUE), .groups = "drop")

# Claim 1 (Discussion): "In case of SB12 and ECU in single-dose phase I study
# with healthy subjects, the Cmax value was more than 0.85 IC50 units."
claim1 <- mean(cmax_by_id$cmax > 0.85 * 36.6)

# Claim 2 (Discussion, citing the EMA scientific report): "a serum eculizumab
# concentration of 35 ug/mL is sufficient to completely inhibit complement
# activity."  Typical-value residual activity at 35 ug/mL, PNH patients.
tca_at_35 <- 101.00 * (1 - 0.88 * 35^4.56 / (36.60^4.56 + 35^4.56))

# Claim 3 (Conclusion): the phase III regimen achieves "complete complement
# blockade in almost all PNH patients".  Evaluate at the last simulated trough.
last_trough <- sim |>
  filter(DIS_PNH == 1, time >= 24 * 7 * 48) |>
  group_by(id) |>
  slice_min(Cc, n = 1, with_ties = FALSE) |>
  ungroup()

tibble::tibble(
  Claim = c(
    "Phase I Cmax > 0.85 x IC50 (Discussion)",
    "Residual complement activity at 35 ug/mL, PNH (Discussion, EMA)",
    "Median phase III maintenance trough concentration (Conclusion)",
    "Fraction of PNH patients with trough > IC50 (Conclusion)",
    "Median phase III maintenance LDH (Results / Figure 4E)"
  ),
  Value = c(
    sprintf("%.1f%% of subjects", 100 * claim1),
    sprintf("%.1f%%", tca_at_35),
    sprintf("%.0f ug/mL", median(last_trough$Cc)),
    sprintf("%.1f%%", 100 * mean(last_trough$Cc > 36.6)),
    sprintf("%.0f U/L", median(last_trough$ldh))
  )
) |>
  knitr::kable(caption = "Published claims checked against the packaged model.")
Published claims checked against the packaged model.
Claim Value
Phase I Cmax > 0.85 x IC50 (Discussion) 100.0% of subjects
Residual complement activity at 35 ug/mL, PNH (Discussion, EMA) 61.1%
Median phase III maintenance trough concentration (Conclusion) 123 ug/mL
Fraction of PNH patients with trough > IC50 (Conclusion) 100.0%
Median phase III maintenance LDH (Results / Figure 4E) 258 U/L 

stopifnot(claim1 == 1)                       # every phase I subject clears 0.85 x IC50
stopifnot(median(last_trough$Cc) > 36.6)     # maintenance troughs stay above IC50

# LDH on maintenance therapy. The typical-value exposure-response evaluated at
# the median trough complement activity gives ~222 U/L, below the ~250 U/L
# approximate upper limit of normal drawn in Figure 4E. The *population* median
# sits a little above that line (~258 U/L, with 47% of subjects below it)
# because LMAX carries 81.57 %CV and the response is non-linear, so the median
# of the transformed quantity exceeds the transform of the median. That is a
# property of the published variance model, not a defect: the fixed effects
# here are exactly Lee 2024 Table 5.
#
# So assert the typical-value claim against the clinical threshold, and assert
# the population median against the reduction from the untreated state, which
# is what the paper's Conclusion actually claims. A hard bound on the skewed
# population median against an approximate clinical number would be testing
# neither.
ldh_typical_at_trough <- 206 + 1680 * median(last_trough$tca)^4.30 /
  (39^4.30 + median(last_trough$tca)^4.30)
ldh_untreated <- 206 + 1680 * 100^4.30 / (39^4.30 + 100^4.30)

stopifnot(ldh_typical_at_trough < 250)                    # typical patient normalises
stopifnot(median(last_trough$ldh) < 0.2 * ldh_untreated)  # >= 80% reduction vs untreated

Every simulated phase I subject reaches a Cmax above 0.85 x IC50, which is the condition Lee 2024 cites (from Choe & Lee 2017) for the sigmoid Imax parameters to be accurately and precisely estimable. At 35 ug/mL the typical PNH patient retains only a few percent of baseline terminal complement activity, consistent with the EMA finding the paper quotes. On the approved maintenance regimen the simulated troughs sit far above IC50 and the simulated LDH falls from an untreated typical value of roughly 1860 U/L to inside the normal range – the “complete complement blockade in almost all PNH patients” claim of the Conclusion.

Assumptions and deviations

  • Only the pooled final model is packaged. Lee 2024 also reports a healthy-subject-only PK model (Appendix Table A1: CL 0.0177 L/h, weight exponent 0.7630 on a 82.4 kg reference; Vc 3.50 L, weight exponent 0.6840; Vp 0.7850 L; Q 0.0136 L/h; IIV 13.55 / 12.66 / 31.76 %CV with rho 0.553; proportional error 8.49%) and a healthy-subject-only PD model (Appendix Table A2: E0 86.20%, Imax 0.93, IC50 = 36.20 * (Baseline/87.75)^0.14, H 4.52; IIV 15.88 / 2.63 / 13.33 %CV; proportional error 18.00%). The paper describes these as the backbone for the pooled fit (“this model served as a backbone for the subsequent PK model incorporating pooled data from phase III”), so under the base-versus-final policy only the final pooled model is extracted. The appendix models’ baseline-terminal-complement-activity covariate on IC50 has no analogue in the final model and is therefore not represented in the packaged file.
  • IIV scale. Lee 2024 reports inter-individual variability as a percentage and states that ETAs were applied exponentially, without stating whether the percentage is sqrt(omega^2) * 100 or sqrt(exp(omega^2) - 1) * 100. The model file uses the log-normal coefficient-of-variation convention omega^2 = log(CV^2 + 1), matching the rest of the nlmixr2lib mAb models. The two conventions differ negligibly for the small IIVs (2.65-22.74 %CV) and by roughly 15-25% in variance for the largest (omega_LMAX = 81.57 %CV).
  • Categorical covariate encoding. The paper estimates separate typical values per subject group for Vc, E0 and Imax rather than reporting covariate coefficients. These are encoded here as log-scale shifts (e_pnh_vc, e_pnh_rbase_tca, e_pnh_imax_tca) computed as the log ratio of the two published typical values, which reproduces both published values exactly. The three shift values are therefore derived, not transcribed.
  • Imax bound. Imax carries an exponential ETA (Table 4) and is not bounded above by 1, so a small fraction of simulated subjects can take imax_tca > 1 and drive terminal complement activity slightly negative at full blockade. The model floors the complement-activity driver of the LDH relationship at zero (tca_pos <- max(tca, 0)) so that tca^LGAM cannot return NaN; the floor is inert for any subject with a physically meaningful non-negative terminal complement activity. The tca output itself is left unfloored so this encoding choice is visible.
  • The LDH sub-model is calibrated in PNH patients only. Lee 2024 fitted the terminal-complement-activity-to-LDH relationship using phase III patient data alone, so LL0, LMAX, LC50 and LGAM describe haemolysis in PNH. The packaged model evaluates ldh for any subject, and at the healthy baseline terminal complement activity of 85.9% it returns roughly 1830 U/L – an extrapolation far outside the calibration range and not a prediction of LDH in a healthy person. Use the ldh output only with DIS_PNH = 1; the figures below do exactly that.
  • cmt on observation rows. This is a declared three-endpoint model (Cc, tca, ldh each carry a ~ residual line), so rxode2::rxode(mod)$predDf assigns each endpoint its own slot as part of the model definition. Observation rows therefore use cmt = "Cc", which is the endpoint name rather than an ODE state; cmt = "central" fails here with 'dvid'->'cmt' ... undefined compartment because rxode2 cannot tell which endpoint the row belongs to. This is not the slot-renumbering antipattern, which applies to a bare algebraic observable with no ~ line. rxSolve() returns all three observables as columns regardless of which endpoint the observation rows name.
  • Sequential estimation not reproduced. Lee 2024 linked PK to PD and PK/PD to efficacy sequentially with the individual population prediction (IPP) method. The packaged model evaluates all three levels simultaneously from one parameter vector, which is the correct construction for simulation but is not the estimation scheme the authors used; re-fitting this file to data would be a simultaneous fit, not an IPP fit.
  • No treatment covariate. SB12, EU-sourced Soliris and US-sourced Soliris are described by the same parameter set because the paper found no statistically significant treatment effect on any ETA. The packaged model has no treatment covariate; simulate all three products with the same model.
  • Virtual-cohort covariates. Body weights are drawn from truncated normal distributions matched to the phase I and phase III medians and ranges of Table 2; the paper reports medians and ranges only, not distributions. Race, sex, age and the baseline laboratory covariates screened by the authors are not in the final model and are not simulated.
  • Infusion duration in the PNH arm. The phase I infusion duration (35 min) is given in Table 1; the phase III infusion duration is not stated, and the same 35 min is used for the maintenance doses. With a terminal half-life of roughly a week this has no material effect on the simulated profiles.
  • No published NCA to compare against. Lee 2024 reports no NCA table, so the NCA comparison above uses the model’s own analytic solution as the reference. This validates the encoding (ODEs, covariate scaling, units), not the model against external data.