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Secukinumab + ixekizumab skin IL-17A target engagement (Ayyar 2022)

Source: vignettes/articles/Ayyar_2022_il17a_target_engagement.Rmd
Ayyar_2022_il17a_target_engagement.Rmd

Model and source

Ayyar et al. (2022) combined a minimal physiologically based PK (mPBPK) model with model-based meta-analyses (MBMA) for the two marketed anti-IL-17A antibodies, secukinumab and ixekizumab. The aim was to predict free IL-17A suppression (target engagement, TE) in psoriatic skin and relate it to clinical response. The paper contributes four model files to the library:

Model Content
Ayyar_2022_secukinumab_mpbpk mPBPK + quasi-equilibrium TMDD in serum and skin for secukinumab, plus the TE-based PASI75 / PASI90 MBMA
Ayyar_2022_ixekizumab_mpbpk the same structure with the ixekizumab drug parameters
Ayyar_2022_secukinumab_mbma dose-based PASI75 / PASI90 MBMA for secukinumab (Figure 2)
Ayyar_2022_ixekizumab_mbma dose-based PASI75 / PASI90 MBMA for ixekizumab (Figure 2)
  • Citation: Ayyar VS, Lee JB, Wang W, Pryor M, Zhuang Y, Wilde T, Vermeulen A. Minimal Physiologically-Based Pharmacokinetic (mPBPK) Metamodeling of Target Engagement in Skin Informs Anti-IL17A Drug Development in Psoriasis. Front Pharmacol. 2022;13:862291. doi:10.3389/fphar.2022.862291
  • Article: https://doi.org/10.3389/fphar.2022.862291 (open access)
  • Supplementary Material: DataSheet1 (model equations) and DataSheet2 (NONMEM control stream for secukinumab), linked from the article page.
mod_sec <- rxode2::rxode2(readModelDb("Ayyar_2022_secukinumab_mpbpk"))
mod_ixe <- rxode2::rxode2(readModelDb("Ayyar_2022_ixekizumab_mpbpk"))
mbma_sec <- rxode2::rxode2(readModelDb("Ayyar_2022_secukinumab_mbma"))
mbma_ixe <- rxode2::rxode2(readModelDb("Ayyar_2022_ixekizumab_mbma"))

Population

All data came from published reports and regulatory reviews, extracted as group means (Table 1 of the paper). Secukinumab data came from the Phase 2 proof-of-concept study (FDA 2015, N = 36), the low dose-ranging study (Papp 2013, N = 125), the high dose-ranging and regimen-finding studies (Rich 2013, N = 100 and 404), the Phase 3 ERASURE, FIXTURE and FEATURE studies (N = 734, 974, 176), and the open-flow microperfusion skin biodistribution study (Dragatin 2016). Ixekizumab data came from the Phase 1 dose-escalation study (FDA 2016, N = 46), the Phase 2 dose-ranging study (Leonardi 2012, N = 141) and the Phase 3 UNCOVER-1/2/3 studies (Papp 2018). All participants were adults with moderate-to-severe plaque psoriasis. Individual demographics were not available. The mPBPK model was fitted in NONMEM 7.4 (FOCE-I) to mean data (naive pooling) with the omega matrix fixed to zero, so the models have no between-subject variability. The MBMAs were fitted by nls to placebo-adjusted week-12 response rates. The same information is in each model’s population metadata.

Model structure and units

The model follows Supplementary Eq. 1-12 and the $DES block of the secukinumab control stream:

  • An SC depot drains into plasma at ka, with bioavailability F.
  • Free drug leaves plasma by convection into three interstitial spaces: skin (is_skin), muscle (is_muscle) and lumped leaky tissues (leaky). The flows are L * (1 - sigma). Drug returns through lymph at L * (1 - sigma_L).
  • Free drug is cleared from plasma at the linear clearance CL.
  • IL-17A binds drug under quasi-equilibrium (QE) in plasma and in skin ISF, with the drug’s KD. Total IL-17A follows dRtot/dt = ksyn - kdeg * (Rtot - AR) - kint * AR, where AR is the complex. The complex is eliminated at kint, so the total-drug states lose kint * AR * V.

The paper reports KD and the IL-17A baselines in pM, and its QE equations mix them with drug concentrations. The NONMEM model must therefore have run in molar units. The models here take doses in pmol and report every concentration in pM. The paper does not state the antibody molecular weight. This vignette uses a nominal 150 kDa IgG (1 mg = 6667 pmol), the same convention as Zheng_2020_CNTO5048_mouse_mpbpk, which comes from the same group. Drug concentrations are converted back to ug/mL with the same factor. With this factor the model reproduces the paper’s printed TE values to the printed precision (below).

mw_mab <- 150000 # g/mol, nominal IgG (not stated in the paper)
mg_to_pmol <- function(mg) mg * 1e9 / mw_mab
pM_to_ugmL <- function(pM) pM * mw_mab / 1e9
# Empirical display factors for IL-17A, read from the paper's own figures.
# TOTAL IL-17A: the serum line of Figure 3E runs from about 2.8 pg/mL (baseline
# 0.015 pM) to about 100 pg/mL (plateau 0.54 pM), and the skin line of Figure
# 5B from about 50 to about 360 pg/mL (0.28 to 2.0 pM): about 185 pg/mL per pM.
# FREE IL-17A: the skin line of Figure 5A starts at about 10 pg/mL (0.28 pM)
# and the secukinumab 300 mg line sits at about 0.04 pg/mL at week 8 (0.0012
# pM): about 35 pg/mL per pM.
il17_total_pgmL_per_pM <- 185
il17_free_pgmL_per_pM <- 35

The two factors are used only to put IL-17A on the same pg/mL axes as the paper’s figures. They are not part of any model. 35 pg/mL per pM is the mass of the ~35 kDa IL-17A homodimer. 185 pg/mL per pM equals the mass of a 150 kDa antibody bound to IL-17A, as if total IL-17A (almost all of it complexed) had been converted with the complex mass.

Source trace

Equation / parameter Secukinumab Ixekizumab Source location
ODE system – – Supplementary DataSheet1 Eq. 1-8; DataSheet2 $DES
QE free drug, complex – – DataSheet1 Eq. 9-12; DataSheet2 $DES
lcl (CL) 0.154 L/day 0.379 L/day Table 2 (estimated)
lfdepot (F) 0.729 0.81 Table 2 (fixed; popPK)
lka (ka) 0.18 1/day 0.24 1/day Table 2 (fixed; popPK)
llymphflow (L) 2.9 L/day shared Table 2
llymphflow_skin / _muscle / _leaky 0.247 / 0.71 / 1.943 L/day shared Table 2
lvc (Vp) 2.6 L shared Table 2
lvleaky (V2) 4.368 L shared DataSheet2 THETA(9); Table 2 prints 4.37
lvskin / lvmuscle / lvlymph 1.81 / 6.3 / 2.6 L shared Table 2
sigma_skin 0.630 0.63 (fixed to secukinumab) Table 2, footnote b
sigma_muscle / sigma_l 0.95 / 0.2 shared Table 2
sigma_leaky 0.363 0.524 Table 2 (estimated for each drug)
lkdeg / lkdeg_skin 45.5 / 2.44 1/day shared Table 2
lkint 1.24 1/day 1.24 (fixed to secukinumab) Table 2, footnote b
lkint_skin 0.34 1/day shared Table 2 (= 2.5 x Ls / Vs)
lr0 / lr0_skin 0.015 / 0.28 pM shared Table 2
ksyn (derived) 0.683 pM/day shared Table 2; DataSheet2 $PK
lkd (KD) 129 pM 1.8 pM Table 2
TE-MBMA PASI75 e0, emax, ec50, hill 78.11, -26.94, 8.851 %, 1.094 shared Eq. 1; digitised Figure 6
TE-MBMA PASI90 e0, emax, ec50, hill 66.46, -6.839, 3.062 %, 1.262 shared Eq. 1; digitised Figure 6
Dose-MBMA PASI75 e0, emax, ed50, hill -1.245, 75.82, 16.94 mg/wk, 1.898 -7.944, 82.12, 2.605 mg/wk, 1.653 Eq. 1; digitised Figure 2
Dose-MBMA PASI90 e0, emax, ed50, hill -0.821, 66.61, 36.91 mg/wk, 1.868 -6.973, 68.56, 2.888 mg/wk, 1.474 Eq. 1; digitised Figure 2
Residual errors not reported not reported Methods (proportional, on log concentrations)

The paper prints none of the MBMA parameter values. The maintainers digitised each trend line from the figure images in the article PDF and refitted Eq. 1, Response = E0 + x^hill * (Emax - E0) / (x^hill + E50^hill), by nls to the digitised points. The refits reproduce the digitised lines to within 1 percentage point (residual SD 0.14-0.9 points). The MBMA outputs are fractions on a 0-1 scale (the paper uses percent).

Helpers 

# One-subject event table: SC doses into the depot (pmol) and an observation
# grid. Observation rows carry no compartment; with several declared
# endpoints rxode2 then returns every output on each row.
make_events <- function(dose_mg, dose_day, obs_days, cmt = "depot", id = 1L) {
  doses <- data.frame(
    id = id, time = dose_day, amt = mg_to_pmol(dose_mg), cmt = cmt,
    evid = 1L, dvid = NA_integer_
  )
  obs <- data.frame(
    id = id, time = obs_days, amt = 0, cmt = NA_character_,
    evid = 0L, dvid = 1L
  )
  out <- rbind(doses, obs)
  out[order(out$time, -out$evid), ]
}

solve_regimen <- function(mod, dose_mg, dose_day, obs_days, label) {
  ev <- make_events(dose_mg, dose_day, obs_days)
  rxode2::rxSolve(mod, ev, returnType = "data.frame") |>
    dplyr::mutate(regimen = label)
}

# Approved-regimen dosing up to (not including) the week-12 read-out
sec_days <- c(0, 7, 14, 21, 28, 56) # weeks 0, 1, 2, 3, 4, then q4w
ixe_days <- c(0, 14, 28, 42, 56, 70) # 160 mg at week 0, then 80 mg q2w
grid12 <- seq(0, 84, by = 0.25)

Baseline steady state

Without drug, total IL-17A must stay at its baseline in both serum and skin: ksyn = kdeg * R0 in each space.

nodose <- data.frame(
  id = 1L, time = c(0, 28, 84), amt = 0, cmt = NA_character_,
  evid = 0L, dvid = 1L
)
bl <- rxode2::rxSolve(mod_sec, nodose, returnType = "data.frame")
bl[, c("time", "Ctotal_target", "Ctotal_target_skin", "Cfree_target_skin")]
#>   time Ctotal_target Ctotal_target_skin Cfree_target_skin
#> 1    0         0.015               0.28              0.28
#> 2   28         0.015               0.28              0.28
#> 3   84         0.015               0.28              0.28
stopifnot(
  all(abs(bl$Ctotal_target / 0.015 - 1) < 1e-6),
  all(abs(bl$Ctotal_target_skin / 0.28 - 1) < 1e-6)
)

Target engagement at the approved regimens (Results, Figure 6)

The paper’s headline numbers are the 12-week average skin TE: 98.6% for secukinumab 300 mg at weeks 0, 1, 2, 3, 4 then q4w, and 99.9% for ixekizumab 160 mg then 80 mg q2w (Results and Discussion). In other words, free skin IL-17A averaged 1.4% and 0.1% of baseline. Figure 6 also marks secukinumab 150 mg at about 2.4% of baseline. Each model carries the running average as free_target_skin_avg_pct, defined as the cumulative free skin IL-17A divided by elapsed time. Its value at day 84 is the paper’s metric.

sims12 <- dplyr::bind_rows(
  solve_regimen(mod_sec, rep(300, 6), sec_days, grid12, "Secukinumab 300 mg"),
  solve_regimen(mod_sec, rep(150, 6), sec_days, grid12, "Secukinumab 150 mg"),
  solve_regimen(mod_ixe, c(160, rep(80, 5)), ixe_days, grid12, "Ixekizumab 160/80 mg q2w")
)
te <- sims12 |>
  dplyr::filter(time == 84) |>
  dplyr::transmute(
    regimen,
    `Average free skin IL-17A (% baseline)` = round(free_target_skin_avg_pct, 3),
    `Average TE (%)` = round(te_skin_avg_pct, 2),
    `Published` = c("98.6% TE", "about 2.4% free (Figure 6 arrow)", "99.9% TE"),
    `PASI75 (placebo-adjusted)` = round(prob_pasi75_pbo_adj, 3),
    `PASI90 (placebo-adjusted)` = round(prob_pasi90_pbo_adj, 3)
  )
knitr::kable(te)
regimen Average free skin IL-17A (% baseline) Average TE (%) Published PASI75 (placebo-adjusted) PASI90 (placebo-adjusted)
Secukinumab 300 mg 1.381 98.62 98.6% TE 0.659 0.468
Secukinumab 150 mg 2.376 97.62 about 2.4% free (Figure 6 arrow) 0.580 0.356
Ixekizumab 160/80 mg q2w 0.105 99.89 99.9% TE 0.773 0.654 

# The solve is deterministic (no random effects), so these bounds can be tight;
# they still leave room for the nominal 150 kDa molecular weight.
stopifnot(
  abs(te$`Average TE (%)`[1] - 98.6) < 0.1,
  abs(te$`Average free skin IL-17A (% baseline)`[2] - 2.4) < 0.25,
  abs(te$`Average TE (%)`[3] - 99.9) < 0.05
)

All three reproduce the paper. The TE result depends only weakly on the assumed molecular weight:

mw_sens <- lapply(c(140000, 150000, 160000), function(mw) {
  ev <- make_events(rep(300, 6), sec_days, 84)
  ev$amt[ev$evid == 1] <- 300 * 1e9 / mw
  s <- rxode2::rxSolve(mod_sec, ev, returnType = "data.frame")
  data.frame(`MW (kDa)` = mw / 1000, `Secukinumab 300 mg average TE (%)` = round(s$te_skin_avg_pct, 3), check.names = FALSE)
})
knitr::kable(do.call(rbind, mw_sens))
MW (kDa) Secukinumab 300 mg average TE (%)
140 98.691
150 98.619
160 98.547

Figure 5: skin IL-17A for the approved regimens 

sims12 |>
  dplyr::select(time, regimen, Cfree_target_skin, Ctotal_target_skin) |>
  tidyr::pivot_longer(c(Cfree_target_skin, Ctotal_target_skin), names_to = "species", values_to = "pM") |>
  dplyr::mutate(
    species = dplyr::recode(species, Cfree_target_skin = "A: skin free IL-17A", Ctotal_target_skin = "B: skin total IL-17A"),
    pgmL = pM * ifelse(grepl("free", species), il17_free_pgmL_per_pM, il17_total_pgmL_per_pM)
  ) |>
  ggplot(aes(time / 7, pgmL, colour = regimen)) +
  geom_line(linewidth = 0.8) +
  facet_wrap(~species, scales = "free_y") +
  scale_y_log10() +
  labs(
    x = "Time (weeks)", y = "IL-17A (pg/mL)", colour = NULL,
    caption = "Replicates Figure 5 of Ayyar 2022."
  ) +
  theme(legend.position = "bottom")

In Figure 5A at week 8, free skin IL-17A is about 0.08 pg/mL for secukinumab 150 mg, 0.04 pg/mL for 300 mg and 0.0035 pg/mL for ixekizumab. In Figure 5B, total skin IL-17A plateaus within 2-3 weeks, at about 360 / 50 times its baseline. These ratios do not depend on any unit conversion, so they are checked here.

wk8 <- sims12 |>
  dplyr::filter(time == 56) |>
  dplyr::select(regimen, Cfree_target_skin, Ctotal_target_skin)
wk8
#>                    regimen Cfree_target_skin Ctotal_target_skin
#> 1       Secukinumab 300 mg      0.0011797405           2.002353
#> 2       Secukinumab 150 mg      0.0023499601           1.995349
#> 3 Ixekizumab 160/80 mg q2w      0.0001029483           2.008799
free_300 <- wk8$Cfree_target_skin[wk8$regimen == "Secukinumab 300 mg"]
ratio_150 <- wk8$Cfree_target_skin[wk8$regimen == "Secukinumab 150 mg"] / free_300
ratio_ixe <- wk8$Cfree_target_skin[wk8$regimen == "Ixekizumab 160/80 mg q2w"] / free_300
c(ratio_150 = ratio_150, ratio_ixe = ratio_ixe)
#>  ratio_150  ratio_ixe 
#> 1.99192969 0.08726354
stopifnot(
  abs(log(ratio_150 / (0.08 / 0.04))) < log(1.3),
  abs(log(ratio_ixe / (0.0035 / 0.04))) < log(1.3),
  all(abs(wk8$Ctotal_target_skin / 0.28 / (360 / 50) - 1) < 0.15)
)

Figure 3: secukinumab serum PK, serum total IL-17A and skin PK

The SC panels need no body weight, so they are reproduced here: the regimen-finding study (Figure 3B), the low dose-ranging study (Figure 3C), the Phase 3 regimens (Figure 3E) and the biodistribution study (Figure 3F). The IV panels (Figure 3A, 3D) are dosed in mg/kg, and the paper does not state the body weight it used, so they are not reproduced.

g40 <- seq(0, 280, by = 1)
fig3 <- dplyr::bind_rows(
  solve_regimen(mod_sec, 150, 0, g40, "B: 150 mg SC at wk 0"),
  solve_regimen(mod_sec, rep(150, 3), c(0, 28, 56), g40, "B: 150 mg SC at wk 0,4,8"),
  solve_regimen(mod_sec, rep(150, 4), c(0, 7, 14, 28), g40, "B: 150 mg SC at wk 0,1,2,4"),
  solve_regimen(mod_sec, rep(25, 3), c(0, 28, 56), g40, "C: 25 mg at wk 0,4,8"),
  solve_regimen(mod_sec, rep(75, 3), c(0, 28, 56), g40, "C: 75 mg at wk 0,4,8"),
  solve_regimen(mod_sec, rep(150, 3), c(0, 28, 56), g40, "C: 150 mg at wk 0,4,8")
) |>
  dplyr::mutate(panel = substr(regimen, 1, 1), Cc_ugmL = pM_to_ugmL(Cc))
ggplot(fig3, aes(time / 7, Cc_ugmL, colour = regimen)) +
  geom_line() +
  facet_wrap(~panel, ncol = 2) +
  scale_y_log10(limits = c(0.01, 100)) +
  labs(
    x = "Time (weeks)", y = "Serum secukinumab (ug/mL)", colour = NULL,
    caption = "Replicates Figure 3B-C of Ayyar 2022."
  ) +
  theme(legend.position = "bottom") +
  guides(colour = guide_legend(ncol = 2))
#> Warning in scale_y_log10(limits = c(0.01, 100)): log-10 transformation
#> introduced infinite values.

sec_p3_days <- c(0, 7, 14, 21, seq(28, 364, by = 28))
g56 <- seq(0, 392, by = 1)
fig3e <- dplyr::bind_rows(
  solve_regimen(mod_sec, rep(150, length(sec_p3_days)), sec_p3_days, g56, "150 mg SC at wk 0,1,2,3,4; then q4w"),
  solve_regimen(mod_sec, rep(300, length(sec_p3_days)), sec_p3_days, g56, "300 mg SC at wk 0,1,2,3,4; then q4w")
)
fig3e |>
  dplyr::transmute(
    time, regimen,
    `Serum secukinumab (ug/mL)` = pM_to_ugmL(Cc),
    `Serum total IL-17A (pg/mL)` = Ctotal_target * il17_total_pgmL_per_pM
  ) |>
  tidyr::pivot_longer(-c(time, regimen)) |>
  ggplot(aes(time / 7, value, colour = regimen)) +
  geom_line() +
  facet_wrap(~name, scales = "free_y") +
  scale_y_log10() +
  labs(x = "Time (weeks)", y = NULL, colour = NULL, caption = "Replicates Figure 3E of Ayyar 2022.") +
  theme(legend.position = "bottom")
#> Warning in scale_y_log10(): log-10 transformation introduced infinite values.

In Figure 3E, the 300 mg troughs run at about 30-40 ug/mL and the 150 mg troughs at about 15-20 ug/mL, and serum total IL-17A rises from about 2.8 to about 100 pg/mL. The conversion-free check is the IL-17A plateau/baseline ratio (about 100 / 2.8) and the trough concentrations:

e_chk <- fig3e |>
  dplyr::filter(time == 364) |>
  dplyr::mutate(trough_ugmL = pM_to_ugmL(Cc), il17_ratio = Ctotal_target / 0.015)
e_chk[, c("regimen", "trough_ugmL", "il17_ratio")]
#>                               regimen trough_ugmL il17_ratio
#> 1 150 mg SC at wk 0,1,2,3,4; then q4w    20.69945   35.52510
#> 2 300 mg SC at wk 0,1,2,3,4; then q4w    41.40166   36.09977
stopifnot(
  all(abs(e_chk$il17_ratio / (100 / 2.8) - 1) < 0.15),
  e_chk$trough_ugmL[1] > 12, e_chk$trough_ugmL[1] < 25,
  e_chk$trough_ugmL[2] > 25, e_chk$trough_ugmL[2] < 50
)
fig3f <- solve_regimen(mod_sec, 300, 0, seq(0, 56, by = 0.25), "300 mg SC")
fig3f |>
  dplyr::transmute(time, `Serum` = pM_to_ugmL(Cc), `Skin ISF` = pM_to_ugmL(Cis_skin)) |>
  tidyr::pivot_longer(-time) |>
  ggplot(aes(time / 7, value, colour = name)) +
  geom_line() +
  scale_y_log10(limits = c(0.1, 100)) +
  labs(x = "Time (weeks)", y = "Secukinumab (ug/mL)", colour = NULL, caption = "Replicates Figure 3F of Ayyar 2022.")
#> Warning in scale_y_log10(limits = c(0.1, 100)): log-10 transformation
#> introduced infinite values.

Figure 3F shows serum secukinumab at about 20 ug/mL over weeks 1-2. Skin rises to about 8 ug/mL by week 3 and then declines in parallel with serum.

Figure 4: ixekizumab serum PK 

ixe_q2w <- c(0, seq(14, 84, by = 14))
ixe_q4w <- c(0, seq(28, 84, by = 28))
g12 <- seq(0, 112, by = 0.5)
fig4 <- dplyr::bind_rows(
  solve_regimen(mod_ixe, c(160, rep(80, length(ixe_q2w) - 1)), ixe_q2w, g12, "160 mg, then 80 mg q2w"),
  solve_regimen(mod_ixe, c(160, rep(80, length(ixe_q4w) - 1)), ixe_q4w, g12, "160 mg, then 80 mg q4w"),
  solve_regimen(mod_ixe, rep(150, 3), c(0, 14, 28), g12, "150 mg SC at wk 0,2,4")
)
ggplot(fig4, aes(time / 7, pM_to_ugmL(Cc), colour = regimen)) +
  geom_line() +
  scale_y_log10() +
  labs(x = "Time (weeks)", y = "Serum ixekizumab (ug/mL)", colour = NULL, caption = "Replicates Figure 4 of Ayyar 2022 (regimens of Table 1).") +
  theme(legend.position = "bottom")
#> Warning in scale_y_log10(): log-10 transformation introduced infinite values.

Figure 6: TE-based MBMA

The TE-MBMA trend line is shared by the two drugs. Here it is evaluated directly from the model parameters over the plotted range of free skin IL-17A, and the three approved regimens are placed on it at their simulated TE.

p <- as.list(mod_sec$theta)
te_curve <- function(x, e0, emax, ec50, h) e0 + x^h * (emax - e0) / (x^h + ec50^h)
xx <- 10^seq(-1, log10(25), length.out = 200)
curves <- dplyr::bind_rows(
  data.frame(endpoint = "PASI75", x = xx, y = te_curve(xx, p$e0_pasi75, p$emax_pasi75, exp(p$lec50_pasi75), p$hill_pasi75)),
  data.frame(endpoint = "PASI90", x = xx, y = te_curve(xx, p$e0_pasi90, p$emax_pasi90, exp(p$lec50_pasi90), p$hill_pasi90))
)
pts <- sims12 |>
  dplyr::filter(time == 84) |>
  dplyr::select(regimen, x = free_target_skin_avg_pct, PASI75 = prob_pasi75_pbo_adj, PASI90 = prob_pasi90_pbo_adj) |>
  tidyr::pivot_longer(c(PASI75, PASI90), names_to = "endpoint", values_to = "y") |>
  dplyr::mutate(y = 100 * y)
ggplot(curves, aes(x, y)) +
  geom_line(colour = "darkred", linewidth = 1) +
  geom_point(data = pts, aes(colour = regimen), size = 3) +
  facet_wrap(~endpoint) +
  scale_x_log10() +
  coord_cartesian(ylim = c(0, 100)) +
  labs(
    x = "Average predicted free IL-17A in skin (% baseline)",
    y = "Placebo-adjusted response (%)", colour = NULL,
    caption = "Replicates the trend lines of Figure 6 of Ayyar 2022."
  ) +
  theme(legend.position = "bottom")

Figure 6 shows the trend line at about 77% (PASI75) and 66% (PASI90) near 0.1% free IL-17A, and crossing 0 at about 23% (PASI75) and 18% (PASI90). The digitised parameters reproduce this:

chk6 <- c(
  p75_at_0.1 = te_curve(0.1, p$e0_pasi75, p$emax_pasi75, exp(p$lec50_pasi75), p$hill_pasi75),
  p90_at_0.1 = te_curve(0.1, p$e0_pasi90, p$emax_pasi90, exp(p$lec50_pasi90), p$hill_pasi90),
  p75_at_23 = te_curve(23, p$e0_pasi75, p$emax_pasi75, exp(p$lec50_pasi75), p$hill_pasi75),
  p90_at_18 = te_curve(18, p$e0_pasi90, p$emax_pasi90, exp(p$lec50_pasi90), p$hill_pasi90)
)
round(chk6, 1)
#> p75_at_0.1 p90_at_0.1  p75_at_23  p90_at_18 
#>       77.3       65.5        0.4        0.2
stopifnot(
  abs(chk6[["p75_at_0.1"]] - 77) < 2, abs(chk6[["p90_at_0.1"]] - 66) < 2,
  abs(chk6[["p75_at_23"]]) < 3, abs(chk6[["p90_at_18"]]) < 3
)

Figure 2: dose-based MBMA

The dose-based models are purely algebraic. They take the arm’s average weekly dose over 12 weeks as the covariate DOSE_<drug>_MGWK and need no dose events.

dose_grid <- function(drug, xmax) {
  x <- 10^seq(log10(0.5), log10(xmax), length.out = 100)
  d <- data.frame(id = seq_along(x), time = 0, evid = 0L, amt = 0, x = x)
  if (drug == "Secukinumab") d$DOSE_SECUKINUMAB_MGWK <- x else d$DOSE_IXEKIZUMAB_MGWK <- x
  d
}
fig2 <- dplyr::bind_rows(
  rxode2::rxSolve(mbma_sec, dose_grid("Secukinumab", 300), keep = "x", returnType = "data.frame") |> dplyr::mutate(drug = "Secukinumab"),
  rxode2::rxSolve(mbma_ixe, dose_grid("Ixekizumab", 55), keep = "x", returnType = "data.frame") |> dplyr::mutate(drug = "Ixekizumab")
) |>
  dplyr::select(drug, x, PASI75 = prob_pasi75_pbo_adj, PASI90 = prob_pasi90_pbo_adj) |>
  tidyr::pivot_longer(c(PASI75, PASI90), names_to = "endpoint", values_to = "y")
#> Warning: multi-subject simulation without without 'omega'
#> Warning: multi-subject simulation without without 'omega'
ggplot(fig2, aes(x, 100 * y, colour = drug)) +
  geom_line(linewidth = 1) +
  facet_wrap(~endpoint) +
  scale_x_log10() +
  coord_cartesian(ylim = c(0, 100)) +
  labs(
    x = "Dose (mg/week)", y = "Placebo-adjusted response (%)", colour = NULL,
    caption = "Replicates the trend lines of Figure 2 of Ayyar 2022."
  )

# Figure 2 trend lines read at 10 and 100 mg/week (secukinumab) and at 2 and
# 20 mg/week (ixekizumab).
f2 <- fig2 |>
  dplyr::group_by(drug, endpoint) |>
  dplyr::summarise(
    lo = approx(log(x), 100 * y, xout = log(ifelse(drug[1] == "Secukinumab", 10, 2)))$y,
    hi = approx(log(x), 100 * y, xout = log(ifelse(drug[1] == "Secukinumab", 100, 20)))$y,
    .groups = "drop"
  )
knitr::kable(f2, digits = 1)
drug endpoint lo hi
Ixekizumab PASI75 27.4 79.1
Ixekizumab PASI90 20.8 64.4
Secukinumab PASI75 19.5 73.3
Secukinumab PASI90 4.6 57.5 
# Digitised Figure 2 trend-line values at the same doses
fig2_read <- c(
  "Ixekizumab.PASI75" = 25.5, "Ixekizumab.PASI90" = 20.2,
  "Secukinumab.PASI75" = 19.8, "Secukinumab.PASI90" = 3.8
)
fig2_read_hi <- c(
  "Ixekizumab.PASI75" = 78.9, "Ixekizumab.PASI90" = 64.6,
  "Secukinumab.PASI75" = 72.7, "Secukinumab.PASI90" = 56.8
)
key <- paste(f2$drug, f2$endpoint, sep = ".")
stopifnot(
  all(abs(f2$lo - fig2_read[key]) < 3),
  all(abs(f2$hi - fig2_read_hi[key]) < 3)
)

Read off Figure 2, the secukinumab PASI75 line is at about 20% at 10 mg/week and 73% at 100 mg/week. The ixekizumab line is at about 26% at 2 mg/week and 79% at 20 mg/week.

The Methods define the dose regressor as the total dose given over the first 12 weeks divided by 12 weeks. The trial arms in Figure 2 are plotted below that definition. The Phase 3 secukinumab 150 mg and 300 mg arms sit at about 55 and 110 mg/week rather than 900 / 12 = 75 and 1800 / 12 = 150. The Phase 3 ixekizumab 80 mg q4w and q2w arms sit at about 22 and 38 mg/week rather than 320 / 12 = 27 and 560 / 12 = 47. The paper does not explain the difference. The trend lines are reproduced as plotted, so a user who computes the regressor from the stated definition will place a regimen to the right of where the paper plotted it.

PKNCA: single-dose serum PK

The paper reports no NCA. The table below gives the single-dose serum NCA of the typical-value model for each drug’s SC doses, for reference. The model clearances (0.154 and 0.379 L/day) are close to the popPK total clearances the paper cites (0.19 and 0.29 L/day).

nca_days <- seq(0, 168, by = 1)
nca_sim <- dplyr::bind_rows(
  solve_regimen(mod_sec, 150, 0, nca_days, "Secukinumab 150 mg SC"),
  solve_regimen(mod_sec, 300, 0, nca_days, "Secukinumab 300 mg SC"),
  solve_regimen(mod_ixe, 80, 0, nca_days, "Ixekizumab 80 mg SC"),
  solve_regimen(mod_ixe, 160, 0, nca_days, "Ixekizumab 160 mg SC")
) |>
  dplyr::mutate(id = as.integer(factor(regimen)), Cc = pM_to_ugmL(Cc)) |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, treatment = regimen)
dose_df <- nca_sim |>
  dplyr::distinct(id, treatment) |>
  dplyr::mutate(time = 0, dose = as.numeric(sub(".* ([0-9]+) mg SC", "\\1", treatment)))

conc_obj <- PKNCA::PKNCAconc(nca_sim, Cc ~ time | treatment + id)
dose_obj <- PKNCA::PKNCAdose(dose_df, dose ~ time | treatment + id)
intervals <- data.frame(start = 0, end = Inf, cmax = TRUE, tmax = TRUE, aucinf.obs = TRUE, half.life = TRUE)
nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))
nca_tab <- as.data.frame(nca_res$result) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "aucinf.obs", "half.life")) |>
  dplyr::select(treatment, PPTESTCD, PPORRES) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES) |>
  dplyr::rename(
    Regimen = treatment,
    `Cmax (ug/mL)` = cmax,
    `Tmax (day)` = tmax,
    `AUC0-inf (ug*day/mL)` = aucinf.obs,
    `t1/2 (day)` = half.life
  )
knitr::kable(nca_tab, digits = 2)
Regimen Cmax (ug/mL) Tmax (day) t1/2 (day) AUC0-inf (ug*day/mL)
Ixekizumab 160 mg SC 12.80 4 15.89 340.74
Ixekizumab 80 mg SC 6.40 4 15.80 170.29
Secukinumab 150 mg SC 10.20 9 40.58 708.87
Secukinumab 300 mg SC 20.41 9 40.60 1418.13

Assumptions and deviations

  • Molar units and molecular weight. The models take doses in pmol and report pM. The paper does not state the antibody molecular weight; this vignette uses a nominal 150 kDa (1 mg = 6667 pmol). With it, the printed TE values (98.6% and 99.9%) are reproduced to the printed precision, and a +/- 10 kDa change moves them by less than 0.1 percentage point.
  • IL-17A pg/mL axes. The paper’s figures report IL-17A in pg/mL but do not state the conversion from the model’s pM, and the panels are not consistent. The free IL-17A of Figure 5A corresponds to about 35 pg/mL per pM, the mass of the IL-17A homodimer. The total IL-17A of Figures 3E and 5B corresponds to about 185 pg/mL per pM, the mass of an antibody-IL-17A complex. The models reproduce the paper’s pM values (Table 2 baselines). The two factors are used here only for display, and the checks above use conversion-free ratios. A user comparing total IL-17A with assay data in pg/mL of IL-17A should note that the fitted serum total IL-17A data appear to have been converted with the complex mass.
  • Ixekizumab parameters fixed to secukinumab. Table 2 footnote b states that the ixekizumab serum complex elimination rate kint and skin reflection coefficient sigma_skin were “assumed to be the same as for secukinumab”. They are coded as fixed() in the ixekizumab model.
  • Leaky-tissue volume. The control stream fixes V2 = 4.368 L; Table 2 prints the rounded 4.37 L. The unrounded value is used.
  • Lymph equation. DataSheet1 Eq. 6 writes the lymph inflows with the vascular reflection coefficients (1 - sigma_sk), (1 - sigma_1) and (1 - sigma_2). The tissue equations (Eq. 3-5) and the control stream use the lymphatic coefficient (1 - sigma_L) for the same flows. The control stream form is used because it conserves mass between tissue and lymph.
  • IIV. The Methods state that IIV was not considered and the variance-covariance matrix was fixed to zero. The supplementary control stream nevertheless carries two estimable omegas (CL and Vp) with initial value 0.1. The models follow the Methods and have no random effects.
  • Residual error. Each fitted output had an additive error on log concentrations, which the paper calls proportional. The magnitudes are not reported, so they are lnorm() with standard deviation fixed(0).
  • QE free-drug formula. The printed 0.5 * (a + sqrt(a^2 + 4 KD C)), with a = C - KD - Rtot, loses precision at low drug concentration. The models use the algebraically identical 2 KD C / (sqrt(a^2 + 4 KD C) - a) when a < 0.
  • SC bioavailability. The control stream multiplies the absorption rate by F. The models put F on the depot dose instead (f(depot)), which gives the same plasma input.
  • MBMA parameters digitised. None of the MBMA parameters are printed. They were digitised from the Figure 2 and Figure 6 trend lines and refitted. The dose-based E0 values are slightly negative because the plotted lines start at 0.5 mg/week and do not identify E0. Below 3 mg/week the ixekizumab line is a polyline through a coarse simulation grid, so only its vertices were used. The TE-based PASI75 emax (-26.9) lies below zero because the authors’ fit reaches 0% response at about 23% free IL-17A without leveling off. Use the MBMAs only over the plotted ranges.
  • IV panels not reproduced. Figures 3A and 3D use mg/kg IV doses, and the paper gives no body weight to convert them.