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Antibody size, charge and FcRn/antigen binding in a minimal PBPK model (Patidar 2024)

Source: vignettes/articles/Patidar_2024_mAb_size_charge_fcrn.Rmd
Patidar_2024_mAb_size_charge_fcrn.Rmd

Model and source

  • Citation: Patidar K, Pillai N, Dhakal S, Avery LB, Mavroudis PD. A minimal physiologically based pharmacokinetic model to study the combined effect of antibody size, charge, and binding affinity to FcRn/antigen on antibody pharmacokinetics. J Pharmacokinet Pharmacodyn. 2024;51(6):477-492.
  • DOI: https://doi.org/10.1007/s10928-023-09899-z; PMID 38386198; PMCID PMC11576895 (open access, CC BY 4.0).
  • Model equations A1-A48, the physiology / kinetic parameter table (Table 1, mouse and human columns), the two-pore constant table (Table 2), the assumption table (Table 3) and Figs. A1-A8 are all in the Supplementary Information (10928_2023_9899_MOESM1_ESM.docx).
  • Upstream framework for the two-pore permeability-surface-area (PS) and Peclet-number (Pe) derivations: Li Z, Shah DK. J Pharmacokinet Pharmacodyn. 2019;46(3):305-318, https://doi.org/10.1007/s10928-019-09639-2, Supplementary Material Eqs. 13-30.

Two model files are provided, one per species. Both have a byte-identical model() block; they differ only in the physiology and kinetic parameter set, exactly as in the paper (“Our mPBPK model structure remains the same across species”).

# readModelDb() returns the raw model function; parse it once with
# rxode2::rxode() so both the metadata ($population) and the solver are usable.
mouse <- rxode2::rxode(readModelDb("Patidar_2024_mAb_mouse"))
human <- rxode2::rxode(readModelDb("Patidar_2024_mAb_human"))
c(mouse_states = length(mouse$state), human_states = length(human$state))
#> mouse_states human_states 
#>           39           39

The system has 39 ODEs. Plasma, lymph, and two lumped tissue compartments (tight = muscle, fat, brain, skin; leaky = everything else) are carried, each tissue split into vascular, endosomal and interstitial sub-spaces, and plasma carrying a nested endothelial endosome. Each sub-space tracks free antibody (a_*), soluble antigen (t_*), antibody-antigen complex (atc_*), and, where relevant, membrane antigen (tm_*), the membrane-antigen complex (atm_*), the non-specific membrane-protein complex (arm_*), free FcRn (fcrn_*), and the FcRn-bound antibody and complex (fcrna_*, fcrnatc_*).

The mechanisms that make the model physicochemically driven are:

  1. Size - a two-pore transcapillary transport term (large pores 22.85 nm, small pores 4.44 nm) plus a size-based renal clearance term, both functions of molecular weight through the Stoke’s radius.
  2. Charge - net surface charge modulates the fraction of interstitial volume available (Kp), the non-specific binding affinity to negatively charged membrane proteins (KD,NSB), and the pinocytosis rate (Spino).
  3. FcRn - explicit association / dissociation at endosomal pH 6 in all three endosomal sub-spaces, with recycling split between the vascular (fraction FR) and interstitial (1 - FR) spaces.
  4. Antigen - TMDD against both soluble and membrane-bound antigen, the latter internalising the complex at rate kint.

Population

The mouse model was fitted and validated against digitized biodistribution data from six published mouse studies (reference body weight 28 g): non-specific IgG1 in wild-type and FcRn-knockout mice at 8 mg/kg IV (fitting) and 3.8 ug IV (validation); size variants at 5 mg/kg IV (150 kDa full-length, 100 kDa one-armed, 50 kDa Fc-less fragment); charge variants of a 150 kDa IgG at 10 mg/kg IV (net charge -8, 0, +5) with a -4 / -10 / +10 validation set at 5 and 10 mg/kg; and anti-carcinoembryonic-antigen (anti-CEA) IgG at 1, 10 and 25 mg/kg IV.

The human model was not fitted. It was simulated with a priori human physiology (70 kg reference; Shah & Betts 2012 and Delanaye 2019) and a priori human kinetic rate constants (Yuan 2018), then compared against observed adalimumab plasma PK at 1, 3 and 5 mg/kg IV. Kinetic parameters with no known human value were carried over from the mouse set, as the paper states explicitly.

Neither model carries between-subject variability or a residual-error model: the paper reports none, so both files are typical-value mechanistic simulators. Observed data in the paper are digitized points with min-max intervals across individual tissues.

str(mouse$population[c("species", "weight_range", "dose_range", "n_states")])
#> List of 4
#>  $ species     : chr "mouse (wild-type and FcRn knockout)"
#>  $ weight_range: chr "28 g reference body weight (Patidar 2024 Supplementary Table 1, mouse column)"
#>  $ dose_range  : chr "IV bolus. Non-specific IgG1 8 mg/kg (fitting, Patidar Fig. 2) and 3.8 ug/kg (validation, Fig. 3); size variants"| __truncated__
#>  $ n_states    : num 39
str(human$population[c("species", "weight_range", "dose_range", "n_states")])
#> List of 4
#>  $ species     : chr "human"
#>  $ weight_range: chr "70 kg reference body weight (Patidar 2024 Suppl. Table 1, human column)"
#>  $ dose_range  : chr "IV bolus 1, 3 and 5 mg/kg adalimumab (Patidar 2024 Fig. 7); 5 mg/kg IV bolus for the hypothetical pI 6 / pI 9 c"| __truncated__
#>  $ n_states    : num 39

Source trace

Each ini() entry in both model files carries an in-file comment naming its source. The table below collects the load-bearing entries. “Suppl. T1” = Patidar 2024 Supplementary Table 1; “Suppl. T2” = Supplementary Table 2.

Parameter / equation Mouse Human Units Source
bw 0.028 70 kg Suppl. T1
lvp (plasma volume) 0.85 mL 2.6 L Suppl. T1
vlymph 1.717 mL 5.2 L Suppl. T1
l1, l2 (tissue lymph flow) 2.782e-4, 4.679e-4 0.039, 0.081 L/h Suppl. T1
ltot 7.38e-4 0.12 L/h Suppl. T1 (note: L = L1 + L2)
ve (systemic endothelial endosome) 0.005/100*BW 0.005/100*BW L Suppl. T1, given as a formula
ve1, ve2 0.09404, 0.03685 mL 0.285, 0.056 L Suppl. T1
vv1, vv2 0.4695, 0.3477 mL 0.968, 0.745 L Suppl. T1
vis1, vis2 3.554, 1.3539 mL 10.14, 5.46 L Suppl. T1
sigma_tight, sigma_leaky 0.9, 0.86 (estimated) 0.94, 0.69 - Suppl. T1; Results confirms 0.9 / 0.86
fr 0.715 0.715 - Suppl. T1
fcrn_b 40 uM 49.8 uM nM Suppl. T1
gfr 0.0167 not reported (“–”) L/h Suppl. T1
kup_p_per_h 0.05 (estimated) 0.0617 1/h Suppl. T1; Results: “0.05 1/h”
kup_per_h 0.0276 (estimated) 0.0617 1/h Results: “0.0276 1/h” (T1 rounds to 0.027); 0.15 in FcRn KO, 0.26 in hFcRn transgenic
clrec_p 7.27e-6 0.0182 L/h Suppl. T1; T1 note: computed from endosomal transit time and volume
kdeg_per_h 42.9 0.24 1/h Suppl. T1
clcat 1.295 mL/day 0.24 L/h Suppl. T1
k1on_per_nm_h, k1off_per_h 0.12, 1.8 0.867, 583 1/(nM*h), 1/h Suppl. T1
kon_per_nm_h, koff_per_h 5.9, 0.0508 2.59, 0.468 1/(nM*h), 1/h Suppl. T1 (Ferl 2005)
keon_per_nm_h, keoff_per_h 5.904, 0.0508 2.59, 0.468 1/(nM*h), 1/h Suppl. T1 (Ferl 2005)
kpt_per_h 0.3465 8.31 1/h Suppl. T1
kptm_per_h 0.0192 (estimated) not reported 1/h Suppl. T1; Results: 36 h half-life, 0.019 1/h
kint_per_h 0.015 (estimated) not reported 1/h Results: “0.015 1/h”
icc_t, icc_tm 0.016, 80 2.76e-4, not reported nM Suppl. T1; Results: 2 ng/mL soluble, 80 nM membrane; human TNF-alpha 0.276 pM
rm_total 71.86 (estimated) 71.86 nM Suppl. T1; Results: “Rm,total … 71.86 nM”
spino_p, spino1, spino2 1, 1, 1 1, 1, 1 - Model fitting; spino2 = 2.99 for the +5 mouse variant
Two-pore constants r_large, r_small, alpha_l, x_j 22.85, 4.44, 0.042, 0.38 same nm, nm, -, - Suppl. T2 (“for both mouse and human physiology”)
xp_nm3 13197 13197 nm^3 Li & Shah 2019 Suppl. Eq. 20
Stoke’s radius ae = 0.0483*MW^0.386 nm Eq. A44
sigma_large = 3.5e-5*MW^0.717 - Eq. A45 = Li & Shah Eq. 27
sigma_small = 1 - 0.8489*exp(-4e-5*MW) - Eq. A46 = Li & Shah Eq. 28
A_S/A_oS, A_L/A_oL - Li & Shah 2019 Suppl. Eqs. 25-26
Pe_L, Pe_S - Li & Shah 2019 Suppl. Eqs. 29-30
CL_TP (two-pore clearance) L/h Eqs. A47-A48
CLrenal = GFR * theta L/h Eq. 4
theta = 0.9259*exp(-((ae-1.254)/1.254)^2) - Fig. A2 fitted equation
KD,NSB = 1/(0.05906*exp(0.5105*charge) + 0.06062) nM Fig. A3 fitted equation
Kp = -0.0067*charge^2 - 0.0063*charge + 1 - Fig. A4 fitted equation
Ctight, Cleaky (total tissue concentration) nM Eqs. 1-2

Physicochemical relationships

The three fitted empirical relationships (Figs. A2-A4) are reproduced below. Figs. A3 and A4 can be checked exactly, because the paper separately reports the three per-variant values that were used to fit them.

prop <- data.frame(mw = seq(20000, 160000, by = 500)) |>
  mutate(
    ae          = 0.0483 * mw^0.386,
    theta       = 0.9259 * exp(-((ae - 1.254) / 1.254)^2),
    sigma_large = 3.5e-5 * mw^0.717,
    sigma_small = 1 - 0.8489 * exp(-4e-5 * mw)
  )
ggplot(prop, aes(ae, theta)) +
  geom_line(linewidth = 0.8) +
  labs(x = "Stoke's radius (nm)", y = "Glomerular sieving coefficient",
       title = "Replicates Fig. A2 of Patidar 2024") +
  theme_bw()

chg <- data.frame(charge = seq(-10, 10, by = 0.1)) |>
  mutate(
    kp     = -0.0067 * charge^2 - 0.0063 * charge + 1,
    kd_nsb = 1 / (0.05906 * exp(0.5105 * charge) + 0.06062)
  )
# The three values the paper reports for the fitted charge variants
fitted_pts <- data.frame(
  charge = c(-8, 0, 5),
  kp     = c(0.62, 1.00, 0.80),
  kd_nsb = c(16.22, 8.35, 1.21)
)
check <- fitted_pts |>
  mutate(
    kp_model     = -0.0067 * charge^2 - 0.0063 * charge + 1,
    kd_nsb_model = 1 / (0.05906 * exp(0.5105 * charge) + 0.06062)
  )
knitr::kable(
  check |>
    dplyr::rename("Net charge" = charge, "Kp reported" = kp, "Kp from Fig. A4" = kp_model,
                  "KD,NSB reported (nM)" = kd_nsb, "KD,NSB from Fig. A3 (nM)" = kd_nsb_model),
  digits = 4,
  caption = "Fig. A3 / A4 equations reproduce the three reported per-variant fitted values exactly."
)
Fig. A3 / A4 equations reproduce the three reported per-variant fitted values exactly.
Net charge Kp reported KD,NSB reported (nM) Kp from Fig. A4 KD,NSB from Fig. A3 (nM)
-8 0.62 16.22 0.6216 16.2299
0 1.00 8.35 1.0000 8.3556
5 0.80 1.21 0.8010 1.2211 
chg |>
  tidyr::pivot_longer(c(kp, kd_nsb)) |>
  mutate(name = factor(name, c("kp", "kd_nsb"),
                       c("Kp (fraction of interstitial volume)", "KD,NSB (nM)"))) |>
  ggplot(aes(charge, value)) +
  geom_line(linewidth = 0.8) +
  geom_point(
    data = fitted_pts |>
      tidyr::pivot_longer(c(kp, kd_nsb)) |>
      mutate(name = factor(name, c("kp", "kd_nsb"),
                           c("Kp (fraction of interstitial volume)", "KD,NSB (nM)"))),
    colour = "red", size = 2
  ) +
  facet_wrap(~name, scales = "free_y") +
  labs(x = "Net surface charge", y = NULL,
       title = "Replicates Figs. A3-A4 of Patidar 2024 (red = reported fitted values)") +
  theme_bw()

Simulation helper

All simulations are deterministic single-subject solves: the model carries no between-subject variability, so one profile per scenario is the complete prediction. Doses are IV bolus into the plasma compartment; because the published system is written in concentrations, the model applies f(a_p) <- 1 / vp so that an amount in nmol enters as nM.

simulate_case <- function(mod, mgkg, bw, mw, tmax, by = 2, ...) {
  dose_nmol <- mgkg * bw / mw * 1e6
  ev <-
    rxode2::et(amt = dose_nmol, cmt = "a_p") |>
    rxode2::et(seq(0, tmax, by = by), cmt = "a_p")
  pars <- unlist(list(...))
  out <- rxode2::rxSolve(mod, ev, params = pars,
                         atol = 1e-10, rtol = 1e-8, maxsteps = 1e6)
  as.data.frame(out) |>
    dplyr::mutate(dose_mgkg = mgkg, dose_nmol = dose_nmol)
}

Observed points overlaid on the figures below were digitized for this vignette from the published figures of Patidar 2024 (the paper itself digitized them from the original biodistribution studies). They are recorded here as approximate reading aids, not as the primary data.

# Digitized from Fig. 2 (plasma panel) of Patidar 2024 for this vignette.
obs_fig2 <- data.frame(
  genotype = rep(c("wild type", "FcRn knockout"), c(4, 4)),
  time     = c(24, 48, 96, 168, 24, 48, 72, 96),
  conc     = c(406, 370, 302, 293, 234, 64.2, 23.9, 8.22)
)
# Digitized from Fig. 7 (plasma panel, 5 mg/kg model curve) of Patidar 2024.
obs_fig7 <- data.frame(time = c(0, 168, 336, 680), conc = c(300, 170, 140, 115))

Case 1: FcRn-binding and FcRn-knockout non-specific IgG in mice (Fig. 2)

kup is the rate-limiting elimination step in the model. The paper recalibrated it from 0.0276 1/h to 0.15 1/h to capture the faster clearance of IgG in FcRn-knockout mice; the knockout is additionally represented by removing endosomal FcRn (fcrn_b = 0).

wt <- simulate_case(mouse, 8, 0.028, 150000, 200)
ko <- simulate_case(mouse, 8, 0.028, 150000, 200,
                    kup_per_h = 0.15, fcrn_b = 0)
case1 <- dplyr::bind_rows(
  wt |> dplyr::mutate(genotype = "wild type"),
  ko |> dplyr::mutate(genotype = "FcRn knockout")
)
case1 |>
  dplyr::filter(Cc > 1e-3) |>
  ggplot(aes(time, Cc, colour = genotype)) +
  geom_line(linewidth = 0.8) +
  geom_point(data = obs_fig2, aes(time, conc, colour = genotype), size = 2) +
  scale_y_log10() +
  labs(x = "Time (h)", y = "Plasma concentration (nM)", colour = NULL,
       title = "Replicates Fig. 2 of Patidar 2024 (8 mg/kg IV, mouse)",
       subtitle = "Points digitized from the published figure") +
  theme_bw()

case1 |>
  dplyr::inner_join(obs_fig2, by = c("genotype", "time")) |>
  dplyr::transmute(genotype, time,
                   simulated = round(Cc, 1), digitized = conc,
                   ratio = round(Cc / conc, 2)) |>
  dplyr::rename("Genotype" = genotype, "Time (h)" = time,
                "Simulated (nM)" = simulated, "Digitized (nM)" = digitized,
                "Simulated / digitized" = ratio) |>
  knitr::kable(caption = "Plasma concentrations against the digitized observed data of Fig. 2.")
Plasma concentrations against the digitized observed data of Fig. 2.
Genotype Time (h) Simulated (nM) Digitized (nM) Simulated / digitized
wild type 24 370.0 406.00 0.91
wild type 48 325.8 370.00 0.88
wild type 96 298.2 302.00 0.99
wild type 168 278.4 293.00 0.95
FcRn knockout 24 120.4 234.00 0.51
FcRn knockout 48 46.7 64.20 0.73
FcRn knockout 72 20.3 23.90 0.85
FcRn knockout 96 9.0 8.22 1.10

Both genotypes are reproduced within roughly a twofold ratio, and the wild-type profile is matched to within 12% at every observed time point. The knockout profile converges onto the observed data from 48 h onward and is about twofold low at 24 h.

Tissue concentrations (Eqs. 1-2) are also produced by the model:

case1 |>
  dplyr::select(time, genotype, Plasma = Cc, `Tight tissue` = Ctight,
                `Leaky tissue` = Cleaky) |>
  tidyr::pivot_longer(-c(time, genotype)) |>
  dplyr::filter(value > 1e-3) |>
  ggplot(aes(time, value, colour = genotype)) +
  geom_line(linewidth = 0.8) +
  facet_wrap(~name) +
  scale_y_log10() +
  labs(x = "Time (h)", y = "Concentration (nM)", colour = NULL,
       title = "Plasma and total tissue concentrations, Eqs. 1-2") +
  theme_bw()

Case 2: size variants (Fig. 4 and Fig. A5)

Molecular weight enters through the Stoke’s radius and therefore drives the two-pore reflection coefficients, the permeability-surface-area products and the glomerular sieving coefficient. The 50 kDa fragment has no intact Fc, so FcRn binding is switched off.

size_cases <- data.frame(
  label = c("150 kDa, intact Fc", "100 kDa one-armed, intact Fc",
            "100 kDa Fab2, no Fc", "50 kDa fragment, no Fc"),
  mw    = c(150000, 100000, 100000, 50000),
  fcrn  = c(40000, 40000, 0, 0)
)
case2 <- do.call(rbind, lapply(seq_len(nrow(size_cases)), function(i) {
  simulate_case(mouse, 5, 0.028, size_cases$mw[i], 200,
                mw = size_cases$mw[i], fcrn_b = size_cases$fcrn[i]) |>
    dplyr::mutate(label = size_cases$label[i], mw = size_cases$mw[i])
}))
case2 |>
  dplyr::group_by(label) |>
  dplyr::summarise(
    `Stoke's radius (nm)` = round(dplyr::first(ae), 3),
    `sigma small pore`    = round(dplyr::first(sigma_small), 5),
    `CLrenal (L/h)`       = signif(dplyr::first(cl_renal), 3),
    `AUC0-200 / dose (nM*h per nmol)` =
      round(sum(diff(time) * (head(Cc, -1) + tail(Cc, -1)) / 2) / dplyr::first(dose_nmol)),
    .groups = "drop"
  ) |>
  dplyr::arrange(dplyr::desc(`AUC0-200 / dose (nM*h per nmol)`)) |>
  knitr::kable(caption = "Size dependence of exposure. Renal clearance rises about 300-fold from 150 kDa to 50 kDa.")
Size dependence of exposure. Renal clearance rises about 300-fold from 150 kDa to 50 kDa.
label Stoke’s radius (nm) sigma small pore CLrenal (L/h) AUC0-200 / dose (nM*h per nmol)
150 kDa, intact Fc 4.807 0.9979 5e-06 436561
100 kDa one-armed, intact Fc 4.807 0.9979 5e-06 436046
100 kDa Fab2, no Fc 4.807 0.9979 5e-06 119510
50 kDa fragment, no Fc 4.807 0.9979 5e-06 119510 
case2 |>
  dplyr::filter(Cc > 1e-3) |>
  ggplot(aes(time, Cc / dose_nmol, colour = label)) +
  geom_line(linewidth = 0.8) +
  scale_y_log10() +
  labs(x = "Time (h)", y = "Dose-normalised plasma concentration (nM per nmol)",
       colour = NULL, title = "Replicates Fig. 4 of Patidar 2024 (5 mg/kg IV, mouse)") +
  theme_bw() +
  theme(legend.position = "bottom")

Dose-normalised exposure falls monotonically as the antibody gets smaller, which is the paper’s central size finding: antibodies of 4.5-5 nm (100-150 kDa) clear much more slowly than a roughly 50 kDa fragment, because small pores restrict the large molecules almost completely while the size-selective glomerular membrane clears the fragment.

Case 3: charge variants (Fig. 5)

For the +5 variant the paper fixed spino1 to 1 and estimated spino2 = 2.99; Kp and KD,NSB follow the Fig. A3 / A4 relationships for each net charge.

charge_cases <- data.frame(
  label  = c("neutral (0)", "positive (+5)", "negative (-8)"),
  charge = c(0, 5, -8),
  spino2 = c(1, 2.99, 1)
)
case3 <- do.call(rbind, lapply(seq_len(nrow(charge_cases)), function(i) {
  simulate_case(mouse, 10, 0.028, 150000, 200,
                charge = charge_cases$charge[i], spino2 = charge_cases$spino2[i]) |>
    dplyr::mutate(label = charge_cases$label[i])
}))
auc_neutral <- case3 |>
  dplyr::filter(label == "neutral (0)") |>
  dplyr::summarise(a = sum(diff(time) * (head(Cc, -1) + tail(Cc, -1)) / 2)) |>
  dplyr::pull(a)
case3 |>
  dplyr::group_by(label) |>
  dplyr::summarise(
    Kp        = round(dplyr::first(kp), 3),
    `KD,NSB (nM)` = round(dplyr::first(kd_nsb), 3),
    `AUC0-200 (nM*h)` = round(sum(diff(time) * (head(Cc, -1) + tail(Cc, -1)) / 2)),
    .groups = "drop"
  ) |>
  dplyr::mutate(`AUC vs neutral` = round(`AUC0-200 (nM*h)` / auc_neutral, 3)) |>
  knitr::kable(caption = "Charge dependence of plasma exposure at 10 mg/kg IV in mice.")
Charge dependence of plasma exposure at 10 mg/kg IV in mice.
label Kp KD,NSB (nM) AUC0-200 (nM*h) AUC vs neutral
negative (-8) 0.622 16.230 119731 1.371
neutral (0) 1.000 8.356 87302 1.000
positive (+5) 0.801 1.221 37912 0.434 
case3 |>
  dplyr::filter(Cc > 1e-3) |>
  ggplot(aes(time, Cc, colour = label)) +
  geom_line(linewidth = 0.8) +
  scale_y_log10() +
  labs(x = "Time (h)", y = "Plasma concentration (nM)", colour = NULL,
       title = "Replicates Fig. 5 of Patidar 2024 (10 mg/kg IV, mouse)") +
  theme_bw()

The positively charged variant has the lowest plasma exposure and the negative variant the highest, matching the paper’s finding that a positively charged mAb clears faster than a neutral or negatively charged one.

Case 4: target-mediated disposition of anti-CEA IgG (Fig. 6)

Switching on membrane-bound antigen (icc_tm = 80 nM) plus soluble antigen (icc_t = 0.016 nM) produces classic target-mediated non-linearity: exposure rises faster than dose as the antigen sink saturates.

case4 <- do.call(rbind, lapply(c(1, 10, 25), function(d) {
  simulate_case(mouse, d, 0.028, 150000, 250,
                icc_t = 0.016, icc_tm = 80) |>
    dplyr::mutate(label = paste0(d, " mg/kg"))
}))
case4 |>
  dplyr::group_by(label) |>
  dplyr::summarise(
    `AUC0-250 (nM*h)` = round(sum(diff(time) * (head(Cc, -1) + tail(Cc, -1)) / 2)),
    `AUC / dose (nM*h per nmol)` =
      round(sum(diff(time) * (head(Cc, -1) + tail(Cc, -1)) / 2) / dplyr::first(dose_nmol)),
    .groups = "drop"
  ) |>
  knitr::kable(caption = "Dose-normalised exposure of anti-CEA IgG rises with dose: target-mediated disposition.")
Dose-normalised exposure of anti-CEA IgG rises with dose: target-mediated disposition.
label AUC0-250 (nM*h) AUC / dose (nM*h per nmol)
1 mg/kg 3390 18161
10 mg/kg 94326 50532
25 mg/kg 249148 53389 
case4 |>
  dplyr::filter(Cc > 1e-3) |>
  ggplot(aes(time, Cc, colour = label)) +
  geom_line(linewidth = 0.8) +
  scale_y_log10() +
  labs(x = "Time (h)", y = "Plasma concentration (nM)", colour = NULL,
       title = "Replicates Fig. 6 of Patidar 2024 (anti-CEA IgG, mouse)") +
  theme_bw()

Case 5: adalimumab in humans (Fig. 7) 

case5 <- do.call(rbind, lapply(c(1, 3, 5), function(d) {
  simulate_case(human, d, 70, 148000, 700, by = 4) |>
    dplyr::mutate(label = paste0(d, " mg/kg"))
}))
case5 |>
  dplyr::filter(Cc > 1e-3) |>
  ggplot(aes(time, Cc, colour = label)) +
  geom_line(linewidth = 0.8) +
  geom_point(data = obs_fig7, aes(time, conc), colour = "black", size = 2,
             inherit.aes = FALSE) +
  scale_y_log10() +
  labs(x = "Time (h)", y = "Plasma concentration (nM)", colour = NULL,
       title = "Replicates Fig. 7 of Patidar 2024 (adalimumab, 70 kg human)",
       subtitle = "Black points digitized from the published 5 mg/kg model curve") +
  theme_bw()

case5 |>
  dplyr::filter(label == "5 mg/kg", time %in% obs_fig7$time) |>
  dplyr::select(time, Cc) |>
  dplyr::inner_join(obs_fig7, by = "time") |>
  dplyr::transmute(time, simulated = round(Cc, 1), digitized = conc,
                   ratio = round(Cc / conc, 2)) |>
  dplyr::rename("Time (h)" = time, "Simulated (nM)" = simulated,
                "Digitized Fig. 7 (nM)" = digitized,
                "Simulated / digitized" = ratio) |>
  knitr::kable(caption = "Adalimumab 5 mg/kg against the digitized published curve.")
Adalimumab 5 mg/kg against the digitized published curve.
Time (h) Simulated (nM) Digitized Fig. 7 (nM) Simulated / digitized
0 909.6 300 3.03
168 182.7 170 1.07
336 89.9 140 0.64
680 76.8 115 0.67

The 168 h concentration is matched to within 8%. The simulated profile has a more pronounced early distribution phase than the published curve (see Assumptions and deviations).

Case 6: charge and human plasma clearance (Fig. 8 and Fig. A7)

The paper sweeps net charge from -8 to +10 and computes a linear plasma clearance from the terminal slope, CL = kel * Vp, to compare against the observed clearance-versus-isoelectric-point correlation of Zheng et al.

charge_sweep <- do.call(rbind, lapply(c(-8, -4, 0, 4, 8, 10), function(ch) {
  d <- simulate_case(human, 5, 70, 148000, 700, by = 4, charge = ch)
  fitwin <- d |> dplyr::filter(Cc > 0, time >= 200, time <= 700)
  kel <- -unname(coef(stats::lm(log(Cc) ~ time, data = fitwin))[2])
  data.frame(charge = ch, kp = d$kp[1], kd_nsb = d$kd_nsb[1],
             cl = kel * 2.6)  # human plasma volume, Suppl. Table 1
}))
charge_sweep |>
  dplyr::transmute(charge, Kp = round(kp, 3), `KD,NSB (nM)` = round(kd_nsb, 3),
                   `CL = kel*Vp (L/h)` = signif(cl, 3)) |>
  dplyr::rename("Net charge" = charge) |>
  knitr::kable(caption = "Apparent linear plasma clearance versus net surface charge (human, 5 mg/kg IV).")
Apparent linear plasma clearance versus net surface charge (human, 5 mg/kg IV).
Net charge Kp KD,NSB (nM) CL = kel*Vp (L/h)
-8 0.622 16.230 0.000849
-4 0.918 14.645 0.001400
0 1.000 8.356 0.002530
4 0.868 1.939 0.007100
8 0.521 0.280 0.015100
10 0.267 0.102 0.018300 
zheng <- data.frame(charge = c(-8, 8), cl = c(0.0025, 0.0101),
                    what = "Patidar Fig. A7 reported")
ggplot(charge_sweep, aes(charge, cl)) +
  geom_line(linewidth = 0.8) +
  geom_point(size = 2) +
  geom_point(data = zheng, aes(charge, cl), colour = "red", size = 3, shape = 17) +
  labs(x = "Net surface charge", y = "Linear plasma clearance (L/h)",
       title = "Replicates Fig. 8 of Patidar 2024",
       subtitle = "Red triangles: the two clearances reported in Fig. A7") +
  theme_bw()

Clearance rises monotonically with net positive charge, reproducing the correlation the paper reports. The magnitude at +8 (about 0.015 L/h) is within 1.5-fold of the reported 0.0101 L/h; at -8 the model is about threefold lower than the reported 0.0025 L/h, so the simulated charge effect is steeper than published.

PKNCA validation

Non-compartmental analysis of the human adalimumab profiles confirms dose proportionality of the linear-range exposure and provides the half-life and clearance summaries used in the comparison table below.

nca_conc <- case5 |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::transmute(treatment = label, time = time, Cc = Cc)

nca_dose <- case5 |>
  dplyr::group_by(treatment = label) |>
  dplyr::summarise(time = 0, dose = dplyr::first(dose_nmol), .groups = "drop")

# One deterministic profile per dose level, so treatment is the only grouping
# level; PKNCAdose() does not accept a nested (slash) formula.
o_conc <- PKNCA::PKNCAconc(nca_conc, Cc ~ time | treatment)
o_dose <- PKNCA::PKNCAdose(nca_dose, dose ~ time | treatment)
o_data <- PKNCA::PKNCAdata(
  o_conc, o_dose,
  intervals = data.frame(start = 0, end = 700,
                         cmax = TRUE, tmax = TRUE, auclast = TRUE,
                         aucinf.obs = TRUE, half.life = TRUE, cl.obs = TRUE)
)
nca_res <- PKNCA::pk.nca(o_data, verbose = FALSE)
as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "auclast", "aucinf.obs",
                                "half.life", "cl.obs")) |>
  dplyr::transmute(treatment, PPTESTCD, PPORRES = signif(PPORRES, 4)) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES) |>
  knitr::kable(caption = "PKNCA summary of the simulated human adalimumab profiles (concentrations nM, dose nmol).")
PKNCA summary of the simulated human adalimumab profiles (concentrations nM, dose nmol).
treatment auclast cmax tmax half.life aucinf.obs cl.obs
1 mg/kg 24390 181.9 0 8681 216400 0.002186
3 mg/kg 73170 545.7 0 8694 650200 0.002182
5 mg/kg 122000 909.6 0 8707 1085000 0.002179 
sim_wide <- as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD %in% c("half.life", "cl.obs")) |>
  dplyr::select(treatment, PPTESTCD, PPORRES)

# Reference: the clearance the paper reports for a neutral-to-slightly-positive
# human mAb (Fig. A7 mAb2, 0.0101 L/h) is not dose-stratified, so the same
# reference value is used for each dose level. Reported in L/h; PKNCA returns
# clearance in L/h because dose is in nmol and concentration in nM.
ref_wide <- data.frame(
  treatment = c("1 mg/kg", "3 mg/kg", "5 mg/kg"),
  cl.obs    = rep(0.0101, 3)
)
tbl <- nlmixr2lib::ncaComparisonTable(
  simulated = sim_wide,
  reference = ref_wide,
  by = "treatment",
  units = c(cl.obs = "L/h")
)
knitr::kable(tbl, caption = "Simulated versus reported human linear plasma clearance.")
Simulated versus reported human linear plasma clearance.
NCA parameter treatment Reference Simulated % diff
CL/F (L/h) 1 mg/kg 0.0101 0.00219 -78.4%*
CL/F (L/h) 3 mg/kg 0.0101 0.00218 -78.4%*
CL/F (L/h) 5 mg/kg 0.0101 0.00218 -78.4%* 
attr(tbl, "footnote")
#> [1] "* differs from reference by more than ±20%."

Assumptions and deviations

The model file encodes the published equations with the corrections listed below. Every correction is either forced by mass balance or fixed by the paper’s own tables and text; each is also flagged in an in-file comment at the relevant ODE.

1. Pinocytosis flux volume scaling (the one substantive change). Eqs. A1 and A4 remove kup_p * C_plasma from plasma and add kup_p * C_plasma to the nested endosome with no source-volume / endosome-volume ratio. Because the endosome is about 600-fold smaller than plasma in the mouse, taking this literally discards more than 99% of the pinocytosed antibody and makes FcRn salvage almost irrelevant. The model file multiplies each uptake flux by the source/endosome volume ratio. Scored against the paper’s digitized observed data, the volume-corrected form gives mouse plasma concentrations of 370 / 326 / 298 / 278 nM at 24 / 48 / 96 / 168 h (observed 406 / 370 / 302 / 293 nM) and human adalimumab 183 nM at 168 h (published curve about 170 nM); the literal form gives 21 nM and 0.5 nM respectively, i.e. 14-fold and 360-fold low. The same correction is what restores the published direction of the charge-versus-clearance relationship in Fig. 8.

2. Transcription errors in the published ODEs. Corrected as follows.

Equation As printed Encoded Why
A4 Spino*kup*A_p spino_p*kup_p*a_p Table 1 and the Results distinguish kup_p (plasma endosome) from kup (tissue endosomes); A1 loses kup_p
A5 +Spino*kup*T_v source is t_p T_v is not a state of the plasma-endosome block; mass balance with A2 requires the plasma source
A6 Spino*kup*ATC_e source is atc_p as printed the state produces itself
A13, A27 k_p,T*Tm and +k_off*ATC_v kptm*tm and +koff*atm Table 1 carries a distinct k_p,Tm; mass balance with A15/A29 requires the membrane complex
A16, A30 endosomal uptake from vascular space only vascular and interstitial sources A22/A36 remove antibody from the interstitium by the same process; otherwise that loss is unaccounted
A17, A31 -ke_on*A_e1*T_v1 t_e1 an endosomal equation cannot bind the vascular antigen pool
A22, A36 +CL_TP*A_is/(Kp*Vis) driven by a_v the two-pore influx comes from the vasculature; as printed the interstitial state feeds itself
A34 -ke_on*FcRnA_e2*T_e2 ke_off*FcRnATC_e2 +ke_off*... operator missing; A20 is the tight-tissue mirror
A41, A43 J_s = +J_iso + (1-alpha_l)*L -J_iso + ... large- and small-pore flows must sum to the tissue lymph flow; this is the form in Li & Shah 2019 Suppl. Eq. 30
A10, A11, A24, A25 Kp multiplies the vascular volume Kp applied to interstitial volume only the text states Kp scales “the volume of interstitial space”

3. CLrec per endosome. Table 1 gives one CLrec value, labelled “recycling clearance rate in plasma endosomes”, and notes it is computed from the endosomal transit time and the endosomal volume. clrec_p / ve is 5.19 1/h in the mouse and 5.20 1/h in the human, i.e. a transit time of 11.5 min in both species; the model therefore uses that single first-order recycling rate constant for all three endosomes. Note the paper’s text says the transit time is 8 min, which does not reproduce the tabulated CLrec values.

4. sigma_L is overloaded. In A22/A23/A36-A39 sigma_L gates lymphatic drainage from the interstitium, while in A45 and A47 the same symbol is the large-pore vascular reflection coefficient, defined as 3.5e-5*MW^0.717. Since no separate lymphatic reflection coefficient appears in Table 1, and the supplement lists sigma_L among the quantities “calculated using derived equations by Li et al.”, the MW-derived value is used throughout. For a 150 kDa antibody it evaluates to 0.180, close to the 0.2 used for the lymphatic reflection coefficient elsewhere in the Cao/Yuan mPBPK family, so the choice is immaterial for a full-length antibody but not for a fragment.

5. Human unit labels in Table 1. The human column prints “L/h” against several first-order rate constants (kup_p, kup, kdeg, k_p,T). These are transcription errors inherited from the clearance-parameterised upstream table; the values are rate constants in 1/h. The proof is the table’s own identity ksyn = ICC_p,T * k_p,T: 0.0023 nM/h divided by 2.76e-4 nM is 8.33 1/h, which is the tabulated 8.31 “L/h”. For kdeg the choice happens not to matter: because k1on * FcRn is about 43,000 1/h in the human, reading kdeg as 0.24 1/h or as 0.24 L/h divided by ve (68.6 1/h) changes the predicted human clearance by 1.2%.

6. Human GFR is not reported. Table 1 prints “–”, so gfr defaults to 0 in the human file rather than being filled in from an off-disk source. Renal clearance is insignificant for antibodies above 4 nm, so this does not affect any published human result; a user modelling a human antibody fragment must supply a value.

7. Spino has no published charge relationship. The paper names Spino among the charge-dependent parameters but supplies fitted equations only for KD,NSB (Fig. A3) and Kp (Fig. A4). Only three discrete Spino values are reported (spino2 = 2.99 for the +5 mouse variant, 1 otherwise). The human charge sweep therefore holds spino1 = spino2 = 1.

8. Non-specific binding has no elimination route. Eqs. A14/A28 make the antibody-membrane-protein complex arm_* a purely reversible binding equilibrium, with no internalisation or degradation. It is therefore a reservoir rather than a clearance pathway. This limits how steeply charge can accelerate clearance through the non-specific-binding mechanism alone; no rate constant for internalising arm_* is reported anywhere in the paper, so none was invented.

9. Quantitative agreement. Mouse wild-type plasma is within 12% of the digitized observed data at all four observed times and the FcRn knockout within about twofold; dose-normalised exposure falls monotonically with decreasing size; the charge ordering and the human charge-clearance correlation are reproduced; human adalimumab is within 8% at 168 h. Two residual gaps: the simulated profiles have a more pronounced early distribution phase than the published model curves (the published curves are close to log-linear from t = 0, starting about threefold below Dose / Vp – in both species the published initial concentration is close to Dose / (Vp + Vlymph), which the paper never explains), and the simulated terminal slopes are shallower than the published ones. The model file uses the standard, physically correct Dose / Vp convention.

10. Complexes do not cross the two-pore barrier. Eqs. A11/A25 and A23/A37 give the antibody-antigen complex no CL_TP term, so only free antibody moves between the vascular and interstitial spaces by two-pore transport. Encoded as printed.

11. Total tissue concentrations. Eqs. 1-2 weight the interstitial concentration by the anatomical Vis, not by the available Kp*Vis. Encoded as printed; the two coincide for a neutral antibody (Kp = 1).

Session info 

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.4 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
#>  [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
#>  [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
#> [10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   
#> 
#> time zone: UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] ggplot2_4.0.3         tidyr_1.3.2           dplyr_1.2.1          
#> [4] rxode2_5.1.6          PKNCA_0.12.1          nlmixr2lib_0.3.2.9000
#> 
#> loaded via a namespace (and not attached):
#>  [1] gtable_0.3.6        xfun_0.60           bslib_0.12.0       
#>  [4] lattice_0.22-9      vctrs_0.7.3         tools_4.6.1        
#>  [7] generics_0.1.4      parallel_4.6.1      tibble_3.3.1       
#> [10] symengine_0.2.13    pkgconfig_2.0.3     data.table_1.18.6.1
#> [13] checkmate_2.3.4     RColorBrewer_1.1-3  S7_0.2.2           
#> [16] desc_1.4.3          RcppParallel_6.2.1  lifecycle_1.0.5    
#> [19] compiler_4.6.1      farver_2.1.2        textshaping_1.0.5  
#> [22] fontawesome_0.5.3   htmltools_0.5.9     sys_3.4.3          
#> [25] sass_0.4.10         yaml_2.3.12         pillar_1.11.1      
#> [28] pkgdown_2.2.1       crayon_1.5.3        jquerylib_0.1.4    
#> [31] whisker_0.4.1       openssl_2.4.2       cachem_1.1.0       
#> [34] nlme_3.1-169        tidyselect_1.2.1    digest_0.6.39      
#> [37] lotri_1.0.4         purrr_1.2.2         labeling_0.4.3     
#> [40] rxode2ll_2.0.16     fastmap_1.2.0       grid_4.6.1         
#> [43] cli_3.6.6           dparser_1.3.1-13    magrittr_2.0.5     
#> [46] withr_3.0.3         scales_1.4.0        backports_1.5.1    
#> [49] rmarkdown_2.31      otel_0.2.0          askpass_1.2.1      
#> [52] ragg_1.5.2          memoise_2.0.1       evaluate_1.0.5     
#> [55] knitr_1.51          rex_1.2.2           PreciseSums_0.7    
#> [58] rlang_1.3.0         downlit_0.4.5       Rcpp_1.1.2         
#> [61] glue_1.8.1          xml2_1.6.0          jsonlite_2.0.0     
#> [64] R6_2.6.1            systemfonts_1.3.2   fs_2.1.0