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Inter-antibody variability in mAb PK (Kumar 2024)

Source: vignettes/articles/Kumar_2024_mAb_popPBPK.Rmd
Kumar_2024_mAb_popPBPK.Rmd

Model and source

Kumar 2024 takes the Shah & Betts 2012 platform PBPK model (packaged separately as Shah_2012_mAb_PBPK) and turns it into a population model in which the “individuals” are antibodies rather than patients. Mean clinical plasma profiles were digitised for 143 mAbs; 55 showed linear PK; 44 with IV data built the IV model and the 16 that also had SC data built the SC model. Two of the four Shah & Betts system parameters (CLup, kdeg) were re-estimated with inter-antibody random effects, and two new absorption parameters (kSC, S_LU) were introduced for the SC route.

Two model files come out of this one paper, matching the two models the authors built:

File Route States Estimated parameters
Kumar_2024_mAb_popPBPK_iv IV 93 CLup, kdeg (+ 2 etas)
Kumar_2024_mAb_popPBPK_sc SC 99 CLup, kdeg, kSC, S_LU (+ 4 etas) 
mod_iv <- rxode2(readModelDb("Kumar_2024_mAb_popPBPK_iv"))
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_sc <- rxode2(readModelDb("Kumar_2024_mAb_popPBPK_sc"))
#> ℹ parameter labels from comments will be replaced by 'label()'

c(iv_states = length(mod_iv$state), sc_states = length(mod_sc$state))
#> iv_states sc_states 
#>        93        99

Population

The 44 IV antibodies and 16 SC antibodies are drawn overwhelmingly from published phase 1 single-dose and dose-escalation studies, in healthy volunteers and in patients (oncology, rheumatoid arthritis, asthma, psoriasis, infectious disease). Kumar 2024 Supplementary Table S3 lists per-antibody approval status, phase, country, and year.

Two deliberate exclusions define the model’s scope:

  • Antibodies with dose-dependent (target-mediated) disposition were removed by dose-normalising the digitised profiles, so the model describes linear mAb disposition only and carries no TMDD terms.
  • Panobacumab (an IgM) and suvratoxumab (Fc-modified) were dropped despite showing linear PK, because their FcRn affinity differs substantially from the rest of the IgG set.

The underlying human physiology is the Shah & Betts 2012 Table 4 71 kg male reference; there are no subject-level covariates in the model at all.

str(mod_iv$meta$population)
#> List of 11
#>  $ n_subjects    : num 44
#>  $ n_studies     : num 44
#>  $ age_range     : chr "adults (predominantly healthy-volunteer phase 1 cohorts)"
#>  $ weight_range  : chr "71 kg reference adult male (Shah and Betts 2012 Table 4 physiology)"
#>  $ sex_female_pct: num NA
#>  $ species       : chr "human"
#>  $ disease_state : chr "Mixed. Predominantly phase 1 single-dose studies in healthy volunteers, with some patient cohorts (oncology, rh"| __truncated__
#>  $ dose_range    : chr "multiple IV dose levels per antibody (digitized phase 1 dose-escalation profiles)"
#>  $ regions       : chr "multi-national (USA, Canada, Europe, Japan, Korea, China and others; Supplementary Table S3)"
#>  $ notes         : chr "The 'individuals' of this population model are ANTIBODIES, not human subjects. Mean (digitized) plasma concentr"| __truncated__
#>  $ scope_note    : chr "Simulating with the etas active generates a range of plausible ANTIBODY-level typical profiles (the paper's Mon"| __truncated__

Source trace

Every value in both ini() blocks, and each structural choice, traced to its source location.

Quantity Value Source
CLup (pinocytosis per unit endosomal volume) 0.32 L/h/L (RSE 5.6%) Kumar 2024 Suppl. Table S2; abstract
omega_CLup 73% (RSE 3.2%) Kumar 2024 Suppl. Table S2
kdeg (endosomal degradation, FcRn-unbound) 26.1 1/h (RSE 11%) Kumar 2024 Suppl. Table S2; abstract
omega_kdeg 46% (RSE 17.6%) Kumar 2024 Suppl. Table S2
kSC (local degradation at injection site) 0.0015 1/h (RSE 66%) Kumar 2024 Suppl. Table S2; abstract
omega_kSC 193% (RSE 24.7%) Kumar 2024 Suppl. Table S2
S_LU (lymphatic-uptake scaling) 0.54 (RSE 14%) Kumar 2024 Suppl. Table S2; abstract
omega_S_LU 49% (RSE 21.1%) Kumar 2024 Suppl. Table S2
Central plasma volume 1412 mL Kumar 2024 Section 2.5
Central blood-cell volume 1155 mL Kumar 2024 Section 2.5
Skin sub-volumes and flows (SC model) Table 1 col. “Skin (SC)” Kumar 2024 Table 1
Subcutaneous sub-volumes and flows Table 1 col. “SC (SC)” Kumar 2024 Table 1
All other organ volumes and flows (human, 71 kg) Table 4 Shah & Betts 2012 Table 4
Vascular reflection coefficients sigma_V 0.95 / 0.90 / 0.85, brain 0.99 Shah & Betts 2012 “Model parameters”
Lymphatic reflection coefficient sigma_IS 0.2 (all tissues) Shah & Betts 2012 “Model parameters”
Lymph flow plasma flow / 500 Shah & Betts 2012 “Model parameters”
Endosomal FcRn concentration 4.98e-5 M Shah & Betts 2012 Table 6
FR (recycled fraction) 0.715 Shah & Betts 2012 “Model parameters”
kon / koff (human) 5.59e8 1/M/h, 23.9 1/h Shah & Betts 2012 “Model parameters”
Lymph-node return flow 3.670 L/h Shah & Betts 2012 Table 4 row “Ly. Node”
Blood / blood-cell / lymph-node ODEs, tissue ODEs Eqs 1-3, 4-10, liver 11-12 Shah & Betts 2012
SC interstitium ODE (adds S_LU, kSC) Section 2.4 Kumar 2024

Note the provenance split: Kumar 2024 states in Section 2.5 that “all model parameters apart from pinocytosis rate (CLup) and non-specific degradation rate of unbound antibody (kdeg) are the same as those used for humans by Shah and Betts”, so the physiology rows above are read from the upstream paper, which is on disk.

Units in the ODE system

Identical to the Shah_2012_mAb_PBPK sibling.

Quantity Units
State amounts (vp_*, bc_*, eu_*, eb_*, is_*) nmol
State amounts (free FcRn, fr_*) nmol (1:1 with mAb)
Concentrations (state / volume) nmol/L (= nM)
Time h
Volumes L
Flows L/h
kon 1/(M h), converted internally to 1/(nM h)
koff, kdeg, kSC 1/h
CLup L/h per L of endosomal volume
S_LU unitless

Doses are supplied in nmol: dose_nmol = dose_mg / MW_g_per_mol * 1e6. IV doses go to plasma; SC doses go to is_subcutaneous (the interstitial space of the injection-site compartment), not to a depot.

mw_igg <- 145000  # g/mol, typical IgG1
mg_to_nmol <- function(mg) mg / mw_igg * 1e6

Three source reconciliations

The paper inherits its structure by reference rather than printing it, so three points had to be resolved against the upstream sources. Each is recorded here because each changes the numbers.

1. CLup scales by each tissue’s endosomal volume

Shah & Betts define CLup as the “rate of pinocytosis per unit endosomal space” but do not print how the tissue-level uptake clearance is formed. The Shah & Betts ADAPT deposit sets every tissue’s uptake clearance to CLup * V_endosomal,TOTAL; Chang 2019 (same laboratory, explicitly “augmenting our previously published platform PBPK model”) prints CLup_i = CLup * V_ES_i, i.e. scaling by that tissue’s endosomal volume.

The per-tissue reading is used here, for three reasons:

  1. It is the only reading consistent with the parameter’s own stated definition (“per unit endosomal space”).
  2. The total-volume reading is physically absurd for small tissues: thymus (endosomal volume 0.0321 mL) would get an endosomal turnover rate of ~3300 1/h against kdeg = 26.1 1/h.
  3. It reproduces a correct mAb clearance. The check below is the decisive one.

2. The brain compartment is in the model

Kumar 2024 Section 2.2 enumerates “fifteen tissues” and omits the brain. That enumeration is wrong: Figure 1 of the same paper draws a Brain box, and the paper’s own arithmetic requires it. Section 2.5 derives the central plasma volume as total blood plasma minus the vascular volume held in the tissues. Using Shah & Betts Table 4:

tissue_plasma <- c(heart = 13.1, lung = 55.0, muscle = 662, skin = 127,
                   adipose = 148, bone = 224, kidney = 18.2, liver = 183,
                   small_intestine = 6.15, large_intestine = 8.74,
                   pancreas = 5.70, thymus = 0.353, spleen = 26.8,
                   other = 204)
brain_plasma <- 31.9
total_plasma <- 3126

data.frame(
  scenario = c("without brain", "with brain", "Kumar 2024 Section 2.5"),
  central_plasma_mL = round(c(total_plasma - sum(tissue_plasma),
                              total_plasma - sum(tissue_plasma) - brain_plasma,
                              1412), 1)
)
#>                 scenario central_plasma_mL
#> 1          without brain            1444.0
#> 2             with brain            1412.1
#> 3 Kumar 2024 Section 2.5            1412.0

Subtracting the brain’s vascular plasma reproduces the paper’s stated 1412 mL exactly; omitting it gives 1444 mL. The same holds for blood cells (2558 - 1374.7 - 26.1 = 1157 mL vs the paper’s 1155 mL). The brain is therefore retained.

3. S_LU is applied to both ends of the SC-to-lymph flux

Kumar’s SC interstitium equation multiplies the convective efflux by S_LU, but the lymph-node equation printed just above it sums the unscaled (1 - sigma_IS) * L_SC * C_SC term. Taken literally that creates mass: with S_LU = 0.54 the lymph node would gain almost twice what the SC compartment loses. S_LU is applied to both terms here.

The paper’s own sensitivity analysis confirms this. Under the literal reading, raising S_LU would drain the SC site faster while the lymph node gains only in proportion to the (now lower) SC concentration, so exposure would fall. Figure 7 shows exposure rising with S_LU, which only the mass-balanced form produces.

Mass-balance check

With kSC and kdeg switched off, both models are closed systems and total mAb mass must be conserved. This catches structural ODE errors that numerical agreement alone would hide.

organs <- c("heart", "lung", "muscle", "skin", "adipose", "bone", "brain",
            "kidney", "liver", "small_intestine", "large_intestine",
            "pancreas", "thymus", "spleen", "other")
organs_sc <- c(organs, "subcutaneous")

mab_cols <- function(org) {
  c("plasma", "bcc", "lnode",
    paste0("vp_", org), paste0("bc_", org),
    paste0("eu_", org), paste0("eb_", org), paste0("is_", org))
}

dose_nmol <- mg_to_nmol(5 * 71)   # 5 mg/kg in a 71 kg adult

closed_iv <- mod_iv |> ini(lkdeg = log(1e-12))
#> ℹ change initial estimate of `lkdeg` to `-27.6310211159285`
ev_iv <- et(amt = dose_nmol, cmt = "plasma", time = 0) |> et(seq(0, 336, by = 8))
sim_closed_iv <- as.data.frame(rxSolve(zeroRe(closed_iv), ev_iv))
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg'
tot_iv <- rowSums(sim_closed_iv[, mab_cols(organs), drop = FALSE])

closed_sc <- mod_sc |> ini(lkdeg = log(1e-12), lksc = log(1e-12))
#> ℹ change initial estimate of `lkdeg` to `-27.6310211159285`
#> ℹ change initial estimate of `lksc` to `-27.6310211159285`
ev_sc <- et(amt = dose_nmol, cmt = "is_subcutaneous", time = 0) |>
  et(seq(0, 336, by = 8))
sim_closed_sc <- as.data.frame(rxSolve(zeroRe(closed_sc), ev_sc))
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg', 'etalksc', 'etalslu'
tot_sc <- rowSums(sim_closed_sc[, mab_cols(organs_sc), drop = FALSE])

data.frame(
  model = c("IV (93 states)", "SC (99 states)"),
  dose_nmol = round(dose_nmol, 1),
  total_at_t0 = round(c(tot_iv[1], tot_sc[1]), 1),
  total_at_14d = round(c(tail(tot_iv, 1), tail(tot_sc, 1)), 1),
  max_rel_error = signif(c(max(abs(tot_iv / dose_nmol - 1)),
                           max(abs(tot_sc / dose_nmol - 1))), 3)
)
#>            model dose_nmol total_at_t0 total_at_14d max_rel_error
#> 1 IV (93 states)    2448.3      2448.3       2448.3      2.60e-11
#> 2 SC (99 states)    2448.3      2448.3       2448.3      1.67e-10

Free plus bound FcRn is also conserved per tissue, since the FcRn ODE is the exact negative of the bound-complex formation and release terms.

fcrn_M <- 4.98e-5
v_e <- c(heart = 0.00171, muscle = 0.150, liver = 0.0107)
chk <- sapply(names(v_e), function(o) {
  tot <- sim_closed_iv[[paste0("fr_", o)]] + sim_closed_iv[[paste0("eb_", o)]]
  max(abs(tot / (fcrn_M * 1e9 * v_e[[o]]) - 1))
})
signif(chk, 3)
#>    heart   muscle    liver 
#> 3.89e-15 2.55e-15 3.22e-15

Typical-value profiles

A 5 mg/kg dose in the 71 kg reference adult, by each route. The etas are zeroed so these are the population-typical antibody.

tgrid <- c(seq(0, 48, by = 2), seq(60, 24 * 84, by = 12))

sim_iv <- as.data.frame(rxSolve(
  zeroRe(mod_iv),
  et(amt = dose_nmol, cmt = "plasma", time = 0) |> et(tgrid)
))
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg'
sim_sc <- as.data.frame(rxSolve(
  zeroRe(mod_sc),
  et(amt = dose_nmol, cmt = "is_subcutaneous", time = 0) |> et(tgrid)
))
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg', 'etalksc', 'etalslu'

typical <- bind_rows(
  sim_iv |> transmute(time, Cc, route = "IV"),
  sim_sc |> transmute(time, Cc, route = "SC")
)
typical |>
  filter(time > 0) |>
  ggplot(aes(time / 24, Cc, colour = route)) +
  geom_line(linewidth = 0.8) +
  scale_y_log10() +
  labs(x = "Time (days)", y = "Plasma mAb (nM)", colour = "Route",
       title = "Typical-value linear-mAb PK, 5 mg/kg")
Typical-value plasma profiles for a linear mAb after 5 mg/kg IV and SC, from the Kumar 2024 popPBPK model.

Typical-value plasma profiles for a linear mAb after 5 mg/kg IV and SC, from the Kumar 2024 popPBPK model.

The SC curve shows the slow, flat absorption characteristic of lymphatic uptake, peaking around a week.

PKNCA validation 

# `route` and `dose` are reserved PKNCA column names and must not be used as
# grouping variables; the arm label is carried as `treatment` and the dose
# amount as `amt`. `route` is kept for its intended PKNCA purpose.
nca_conc <- typical |>
  filter(!is.na(Cc)) |>
  mutate(treatment = route, id = ifelse(route == "IV", 1L, 2L)) |>
  select(id, treatment, time, Cc)

# Both arms must carry a time-zero record, or PKNCA warns that the AUC
# interval starts before the first measurement. The simulation grid begins
# at 0, so this is an assertion rather than a repair.
stopifnot(all(table(nca_conc$treatment[nca_conc$time == 0]) == 1))

nca_dose <- data.frame(
  id = c(1L, 2L),
  treatment = c("IV", "SC"),
  time = 0,
  amt = dose_nmol,
  route = c("intravascular", "extravascular")
)

o_conc <- PKNCAconc(nca_conc, Cc ~ time | treatment + id,
                    concu = "nmol/L", timeu = "h")
o_dose <- PKNCAdose(nca_dose, amt ~ time | treatment + id,
                    route = "route", doseu = "nmol")

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

res <- pk.nca(PKNCAdata(o_conc, o_dose, intervals = intervals))
nca_tab <- as.data.frame(res)
nca_wide <- nca_tab |>
  filter(PPTESTCD %in% c("cmax", "tmax", "auclast", "aucinf.obs",
                         "half.life", "cl.obs")) |>
  select(treatment, PPTESTCD, PPORRES) |>
  pivot_wider(names_from = PPTESTCD, values_from = PPORRES)

nca_disp <- nca_wide |>
  transmute(
    treatment,
    `Cmax (nM)`        = signif(cmax, 4),
    `Tmax (days)`      = signif(tmax / 24, 3),
    `AUClast (nM*h)`   = signif(auclast, 4),
    `AUCinf (nM*h)`    = signif(aucinf.obs, 4),
    `t1/2 (days)`      = signif(half.life / 24, 3),
    `CL/F (L/day)`     = signif(cl.obs * 24, 3)
  ) |>
  dplyr::rename(Route = treatment)

knitr::kable(nca_disp,
             caption = "PKNCA parameters from the typical-value profiles.")
PKNCA parameters from the typical-value profiles.
Route Cmax (nM) Tmax (days) AUClast (nM*h) AUCinf (nM*h) t1/2 (days) CL/F (L/day)
IV 1734.0 0.0 267400 280700 19.3 0.209
SC 221.6 5.5 173700 183500 19.5 0.320

Comparison against published values

Kumar 2024 does not tabulate NCA parameters, so the simulated values are checked against the quantities the paper and the mAb literature do pin down.

f_sc <- nca_wide$aucinf.obs[nca_wide$treatment == "SC"] /
  nca_wide$aucinf.obs[nca_wide$treatment == "IV"]
thalf <- nca_wide$half.life[nca_wide$treatment == "IV"] / 24
cl_iv <- nca_wide$cl.obs[nca_wide$treatment == "IV"] * 24
tmax_sc <- nca_wide$tmax[nca_wide$treatment == "SC"] / 24

knitr::kable(
  data.frame(
    Quantity = c("IV terminal half-life (days)",
                 "IV clearance (L/day)",
                 "SC bioavailability (AUC ratio)",
                 "SC Tmax (days)"),
    Simulated = signif(c(thalf, cl_iv, f_sc, tmax_sc), 3),
    Expected = c("~2-3 weeks for a linear IgG1",
                 "~0.2-0.3 for a linear IgG1",
                 "~0.5-0.8 typical mAb SC",
                 "~4-8 typical mAb SC")
  ),
  caption = "Simulated summary PK against canonical linear-mAb behaviour."
)
Simulated summary PK against canonical linear-mAb behaviour.
Quantity Simulated Expected
IV terminal half-life (days) 19.300 ~2-3 weeks for a linear IgG1
IV clearance (L/day) 0.209 ~0.2-0.3 for a linear IgG1
SC bioavailability (AUC ratio) 0.654 ~0.5-0.8 typical mAb SC
SC Tmax (days) 5.500 ~4-8 typical mAb SC

The clearance is the sharpest of these checks. It is an emergent quantity: nothing in the model is a clearance term, so it falls out of CLup, kdeg, FR, the FcRn binding constants and the whole-body physiology together. Recovering ~0.2 L/day is what confirms reconciliation 1 above; the total-endosomal-volume reading of CLup would inflate whole-body pinocytosis roughly 15-fold and put clearance far outside the mAb range.

Replicating Figure 7 (local sensitivity analysis)

Figure 7 reports the percent change in SC AUC when kSC and S_LU are each altered by +/-50%, using % change = (AUC_{+/-50%} - AUC_orig) / AUC_orig with all other parameters at their population values (Kumar 2024 Section 2.6). This is the paper’s only quantitative, digit-level result for the SC layer, so it is the primary validation target.

auc_sc_for <- function(ksc_mult = 1, slu_mult = 1) {
  p <- c(lksc = log(0.0015 * ksc_mult), lslu = log(0.54 * slu_mult))
  s <- as.data.frame(rxSolve(
    zeroRe(mod_sc),
    et(amt = dose_nmol, cmt = "is_subcutaneous", time = 0) |> et(tgrid),
    params = p
  ))
  sum(diff(s$time) * (head(s$Cc, -1) + tail(s$Cc, -1)) / 2)
}

auc_base <- auc_sc_for()
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg', 'etalksc', 'etalslu'
pct <- function(x) (x / auc_base - 1) * 100

fig7 <- data.frame(
  Scenario = c("kSC -50%", "kSC +50%", "S_LU -50%", "S_LU +50%"),
  Simulated = round(c(pct(auc_sc_for(ksc_mult = 0.5)),
                      pct(auc_sc_for(ksc_mult = 1.5)),
                      pct(auc_sc_for(slu_mult = 0.5)),
                      pct(auc_sc_for(slu_mult = 1.5))), 1),
  Published = c(5.4, -4.9, -15.8, 8.6)
)
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg', 'etalksc', 'etalslu'
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg', 'etalksc', 'etalslu'
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg', 'etalksc', 'etalslu'
#> Warning: No sigma parameters in the model
#> ℹ omega/sigma items treated as zero: 'etalclup', 'etalkdeg', 'etalksc', 'etalslu'
fig7$Difference <- round(fig7$Simulated - fig7$Published, 1)

knitr::kable(
  fig7 |> dplyr::rename(`% change in AUC (simulated)` = Simulated,
                        `% change in AUC (Kumar Fig 7)` = Published,
                        `Difference (pp)` = Difference),
  caption = "Replicates Kumar 2024 Figure 7."
)
Replicates Kumar 2024 Figure 7.
Scenario % change in AUC (simulated) % change in AUC (Kumar Fig 7) Difference (pp)
kSC -50% 5.4 5.4 0.0
kSC +50% -4.9 -4.9 0.0
S_LU -50% -23.7 -15.8 -7.9
S_LU +50% 12.6 8.6 4.0 
fig7 |>
  tidyr::pivot_longer(c(Simulated, Published),
                      names_to = "source", values_to = "pct") |>
  ggplot(aes(Scenario, pct, fill = source)) +
  geom_col(position = "dodge") +
  geom_hline(yintercept = 0) +
  labs(x = NULL, y = "% change in AUC", fill = NULL,
       title = "Local sensitivity analysis of the SC absorption parameters")
Replicates Kumar 2024 Figure 7: local sensitivity of SC AUC to kSC and S_LU.

Replicates Kumar 2024 Figure 7: local sensitivity of SC AUC to kSC and S_LU.

The kSC arm reproduces exactly (+5.4% and -4.9%, matching the published values to the printed precision). That agreement validates the kSC value, the SC interstitial volume, the AUC window, and the percent-change definition all at once.

The S_LU arm has the correct sign, the correct rank ordering (S_LU is the more sensitive parameter, as the paper concludes) and the correct asymmetry, but is about 1.5x too large. Inverting the published percentages through the competition between lymphatic escape and local loss implies the authors’ SC bioavailability was roughly 0.76-0.81, where this implementation gives:

signif(f_sc, 3)
#> [1] 0.654

Both lie inside the observed mAb range, and no parameter has been adjusted to close the gap. See Errata for the candidate explanations.

Monte Carlo prediction window (Figure 8)

Kumar 2024 simulated 1000 draws with inter-antibody variability on all four parameters to produce an a priori prediction window for SC mAb PK, and validated it against nine held-out antibodies. The cohort here is capped at 200 draws per the library’s simulation guidance; the window is essentially unchanged.

Because the random effects are inter-antibody, each draw is a different hypothetical antibody, not a different patient.

n_ab <- 200
set.seed(20240709)

ev_mc <- et(amt = mg_to_nmol(70), cmt = "is_subcutaneous", time = 0) |>
  et(seq(0, 24 * 84, by = 12)) |>
  et(id = seq_len(n_ab))

mc <- as.data.frame(rxSolve(mod_sc, ev_mc,
                            omega = mod_sc$omega,
                            returnType = "data.frame"))

stopifnot(length(unique(mc$id)) == n_ab)

window <- mc |>
  filter(time > 0) |>
  group_by(time) |>
  summarise(
    lo  = quantile(Cc, 0.05),
    med = median(Cc),
    hi  = quantile(Cc, 0.95),
    .groups = "drop"
  )
ggplot(window, aes(time / 24)) +
  geom_ribbon(aes(ymin = lo, ymax = hi), alpha = 0.25) +
  geom_line(aes(y = med), linewidth = 0.8) +
  scale_y_log10() +
  labs(x = "Time (days)", y = "Plasma mAb (nM)",
       title = "A priori prediction window, 70 mg SC",
       caption = "Replicates Kumar 2024 Figure 8 panel A geometry.")
Replicates the prediction window of Kumar 2024 Figure 8 (panel A geometry: 70 mg SC). Band is the 5th-95th percentile across 200 simulated antibodies; line is the median.

Replicates the prediction window of Kumar 2024 Figure 8 (panel A geometry: 70 mg SC). Band is the 5th-95th percentile across 200 simulated antibodies; line is the median. 

w <- window |>
  filter(time %in% c(24 * 7, 24 * 28, 24 * 56)) |>
  transmute(`Day` = time / 24,
            `5th pct (nM)` = signif(lo, 3),
            `Median (nM)` = signif(med, 3),
            `95th pct (nM)` = signif(hi, 3),
            `95th/5th` = signif(hi / lo, 3))
knitr::kable(w, caption = "Width of the simulated inter-antibody prediction window.")
Width of the simulated inter-antibody prediction window.
Day 5th pct (nM) Median (nM) 95th pct (nM) 95th/5th
7 5.67 42.70 70.6 12.4
28 2.56 21.10 40.0 15.6
56 0.45 6.85 23.2 51.5

The observed profiles of the nine validation antibodies are not reproduced here: Kumar 2024 overlays digitised data that are not tabulated in the paper or its supplement, so only the window itself can be replicated.

Assumptions and deviations

  • CLup scales by per-tissue endosomal volume. Resolved in favour of the Chang 2019 printed form over the Shah & Betts ADAPT deposit’s total-endosomal-volume form; see reconciliation 1. Validated by the emergent clearance (~0.2 L/day).
  • Brain compartment retained. Kumar 2024 Section 2.2’s tissue list omits the brain, but Figure 1 draws it and the Section 2.5 central-volume arithmetic requires it (1412 mL is reproduced exactly only with the brain subtracted). Treated as a prose omission in the paper.
  • S_LU applied to the lymph-node inflow as well as the SC efflux. The literal reading of the two printed equations is not mass-conserving and produces the wrong sign in the Figure 7 sensitivity analysis.
  • S_LU sensitivity is ~1.5x the published magnitude. The kSC arm of Figure 7 matches exactly, which isolates the discrepancy to how mAb escapes the SC site. The most likely contributors are the SC sub-compartment volumes in Table 1, which are printed to one or two significant figures (the endosomal volume is given as “0.03” mL, where preserving the skin sub-compartment ratios the paper describes gives 0.034 mL), and the SC lymph flow, which the paper says was “scaled down” but never tabulates - it is taken here as plasma flow / 500, the Shah & Betts rule. No parameter was adjusted to close the gap.
  • omega values read as log-scale standard deviations. The paper reports omega_CLup = 73%, omega_kdeg = 46%, omega_kSC = 193%, omega_S_LU = 49% without stating whether these are SDs of the random effect or CV%. They are encoded as SDs (the Monolix native output and the usual reading of omega), so ini() carries the squares. The empirical spread of the individual estimates in Kumar 2024 Figure 6 is consistent with this reading; a CV% reading would give omega_CLup = 0.65 rather than 0.73, a modest narrowing of the prediction window.
  • Random effects are inter-ANTIBODY, not between-subject. Each antibody was treated as an individual during fitting. Simulating with the etas active generates a range of plausible antibody-level typical profiles, which is exactly the paper’s intended use (the Figure 8 prediction window). It does not describe patient-to-patient variability for a single antibody, and these models should not be used to size a study.
  • No residual error model. Kumar 2024 specified a combined error model in Monolix but does not report its coefficients anywhere in the paper or supplement, so no propSd / addSd is included. Both files are typical-value plus inter-antibody simulators.
  • Lung plasma flow derived for mass-balance closure. As in the Shah_2012_mAb_PBPK sibling, the Table 4 lung flows are ~1.8% higher than the sum of the tissue inflows they must equal; using them literally leaks mass at the lung-arterial junction. q_lu is derived as sum(q_X) / (1 - 1/500) and bcq_lu as sum(bcq_X).
  • C_LNLF not carried. Kumar 2024 inherits Shah & Betts’ C_LNLF = 9.1, but the lymph-node return flow is tabulated directly (3.670 L/h, Table 4) and that value is used. C_LNLF does not enter the ODEs and is omitted from ini().
  • No antigen binding, no TMDD. By construction: antibodies with nonlinear PK were excluded from the dataset.
  • compartmentData metadata absent. Both files omit the per-state compartmentData block, matching the Shah_2012_mAb_PBPK sibling. checkModelConventions() raises the same warning for that registered model; it is tracked in nlmixr2lib issue #482 for the PBPK family as a whole rather than being specific to this extraction.
  • Approval status is documented, not modelled. Kumar 2024 compares the individual parameter distributions of FDA-approved versus clinically tested mAbs (Figure 6) and reports the medians, but concludes there is no clear distinction and never enters approval status into the model. It is recorded in covariatesDataExcluded for provenance.