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MAb PBPK (Shah 2012)

Source: vignettes/articles/Shah_2012_mAb_PBPK.Rmd
Shah_2012_mAb_PBPK.Rmd

Model and source

This vignette validates the Shah & Betts 2012 platform PBPK model parameterized for the human (71 kg male) reference subject. The packaged file in inst/modeldb/pharmacokinetics/Shah_2012_mAb_PBPK.R carries the full 93-state ODE system (15 tissues x {vascular plasma, vascular blood cells, endosomal unbound mAb, endosomal FcRn-bound mAb, endosomal free FcRn, interstitial} plus central plasma, central blood cells, and lymph node).

Population

The platform model was fit simultaneously to 52 plasma and tissue concentration profiles spanning four species (mouse, rat, monkey, human) and several mAbs (control IgG, the murine anti-platelet mAb 7E3, MOPC21, the chimeric anti-CEA cT84.66, the human anti-HIV OST577, and adalimumab). Four system parameters (FcRn endosomal concentration, pinocytosis rate per endosomal volume, endosomal degradation rate, and the lymph-node-to-plasma transfer scaling C_LNLF) were estimated from the pooled cross-species data; tissue volumes, plasma and blood-cell flows, vascular reflection coefficients, FR (the fraction of FcRn-bound mAb recycled to vascular space), and the FcRn-IgG association / dissociation rate constants were fixed from prior literature (Shah & Betts 2012, Tables 1-4 and text p.73).

The human-specific validation cohort is the 5 mg/kg single-dose IV arm of Weisman et al. 2003 (n=6 adults with rheumatoid arthritis on concomitant methotrexate, plasma adalimumab concentration vs time, used in Shah & Betts 2012 Fig 8c).

mod <- readModelDb("Shah_2012_mAb_PBPK")

Source trace

The packaged model is a literal transcription of the equations and parameter values from Shah & Betts 2012. Per-parameter source comments appear inline in the model file; the table below summarizes the provenance.

Equation / parameter Value (human) Source
Eq 1: central plasma mass balance derivation Shah & Betts 2012 Eq 1
Eq 2: central blood cells derivation Eq 2
Eq 3: lymph node derivation Eq 3
Eq 4: typical-tissue vascular plasma derivation Eq 4
Eq 5: tissue vascular blood cells derivation Eq 5
Eq 6: endosomal unbound mAb derivation Eq 6
Eq 7: endosomal FcRn-bound mAb derivation Eq 7
Eq 8: endosomal free FcRn derivation Eq 8
Eq 9: interstitial mAb (Ag terms zeroed) derivation Eq 9
Eq 11: liver vascular plasma (portal inflows) derivation Eq 11
Eq 12: liver vascular blood cells (portal inflows) derivation Eq 12
FcRn endosomal concentration 4.98e-5 mol/L Table 6
CLup per endosomal volume 0.0366 L/h/L Table 6
Kdeg (endosomal mAb degradation) 42.9 1/h Table 6
C_LNLF 9.1 (unitless) Table 6
Kon (FcRn-IgG association, human) 5.59e8 1/M/h text p.73
Koff (FcRn-IgG dissociation, human) 23.9 1/h text p.73
FR (recycle fraction to vascular space) 0.715 text p.73 (Garg & Balthasar 2007)
sigma_IS (lymph reflection, all tissues) 0.2 text p.73
Tissue volumes (V_total, V_plasma, V_BC, V_E, V_IS, V_C) per Table 4 Table 4
Tissue plasma flows (PLQ_i) per Table 4 Table 4
Tissue blood-cell flows (BCQ_i) per Table 4 Table 4
Tissue vascular reflection coefficients (sigma_V) 0.85 to 0.99 by tissue text p.73
Lymph flows (L_i = PLQ_i / 500) per tissue text p.73 (Swartz 2001)
Lymph node return flow (L_LymphNode) 3.670 L/h Table 4 row Ly. Node

Units in the ODE system

Quantity Units
State amounts (mAb in vp, bc, eu, eb, is) nmol
State amounts (free FcRn in fr_*) nmol (1:1 stoichiometry with mAb)
Concentrations derived as state / volume nmol/L (= nM)
Time h
Volumes L
Plasma / blood-cell / lymph flows L/h
Kon 1/(M h) (converted internally to 1/(nM h) via x 1e-9)
Koff, Kdeg 1/h
CLup_per_v L/h per L of endosomal volume
FcRn baseline mol/L (converted internally to nM via x 1e9)

Doses are supplied to the plasma compartment in nmol. To translate a mass dose: dose_nmol = dose_mg / MW_g_per_mol * 1e6 (e.g., adalimumab MW 144,190 g/mol gives 5 mg/kg x 71 kg = 2,462 nmol).

Mass-balance / flux check

With endosomal degradation switched off (Kdeg -> 0), the platform model is a closed system: total mAb mass should be conserved. This catches structural ODE errors that would not be visible in a numerical-only check.

mod_no_deg <- mod |> ini(lkdeg = log(1e-12))
#> ℹ change initial estimate of `lkdeg` to `-27.6310211159285`

dose_nmol <- 5 * 71 / 144190 * 1e6
ev <- et(amt = dose_nmol, cmt = "plasma", time = 0) |>
  et(seq(0, 200, by = 4))
sim_no_deg <- as.data.frame(rxSolve(mod_no_deg, events = ev))

igg_cols <- c(
  "plasma", "bcc", "lnode",
  paste0("vp_", c("ht","lu","mu","sk","ad","bo","br","ki",
                  "li","si","lr","pa","th","sp","ot")),
  paste0("bc_", c("ht","lu","mu","sk","ad","bo","br","ki",
                  "li","si","lr","pa","th","sp","ot")),
  paste0("eu_", c("ht","lu","mu","sk","ad","bo","br","ki",
                  "li","si","lr","pa","th","sp","ot")),
  paste0("eb_", c("ht","lu","mu","sk","ad","bo","br","ki",
                  "li","si","lr","pa","th","sp","ot")),
  paste0("is_", c("ht","lu","mu","sk","ad","bo","br","ki",
                  "li","si","lr","pa","th","sp","ot"))
)
sim_no_deg$total_igg <- rowSums(sim_no_deg[, igg_cols])

balance_summary <- sim_no_deg |>
  filter(time %in% c(0, 24, 100, 200)) |>
  transmute(
    time_h          = time,
    total_igg_nmol  = round(total_igg, 3),
    pct_of_dose     = round(100 * total_igg / dose_nmol, 4)
  )
knitr::kable(balance_summary,
             caption = paste0("Total mAb mass with Kdeg = 0. ",
                              "Conservation should hold to machine precision."))
Total mAb mass with Kdeg = 0. Conservation should hold to machine precision.
time_h total_igg_nmol pct_of_dose
0 2462.029 100
24 2462.029 100
100 2462.029 100
200 2462.029 100

The total mAb mass is held at the dose value across the simulation window when Kdeg is zero, confirming that the convective flows, the FcRn binding kinetics, the recycling pathway, and the lymph-node return are all balanced in the implementation.

Free + bound FcRn conservation per tissue

In each tissue endosomal space, free FcRn + FcRn-bound mAb totals to the initial free FcRn (equal to fcrn_M * V_endo_tissue). Verify in the heart endosome:

fcrn_init_ht <- 4.98e-5 * 1e9 * 0.00171  # nmol; v_ht_e = 0.00171 L
fcrn_check <- sim_no_deg |>
  filter(time %in% c(0, 24, 100, 200)) |>
  transmute(
    time_h         = time,
    fr_ht          = round(fr_ht, 6),
    eb_ht          = round(eb_ht, 6),
    sum_fr_eb_ht   = round(fr_ht + eb_ht, 6),
    expected       = round(fcrn_init_ht, 6)
  )
knitr::kable(fcrn_check,
             caption = "FcRn mass balance in the heart endosome.")
FcRn mass balance in the heart endosome.
time_h fr_ht eb_ht sum_fr_eb_ht expected
0 85.15800 0.000000 85.158 85.158
24 84.65371 0.504286 85.158 85.158
100 84.40224 0.755755 85.158 85.158
200 84.42768 0.730324 85.158 85.158

Adalimumab 5 mg/kg IV - replicates Shah & Betts 2012 Figure 8c

The human plasma profile in Shah & Betts Fig 8c is from Weisman et al. 2003. We simulate the typical-value plasma trajectory and compare its shape and approximate magnitudes to the published figure.

ev_long <- et(amt = dose_nmol, cmt = "plasma", time = 0) |>
  et(seq(0, 1008, by = 4))  # 6 weeks of follow-up

sim <- as.data.frame(rxSolve(mod, events = ev_long))

# Convert plasma nM -> mg/L using adalimumab MW for mass-unit comparisons
mw_ada <- 144190  # g/mol
sim$Cc_mg_per_L <- sim$Cc * mw_ada * 1e-6
sim |>
  filter(time > 0) |>
  ggplot(aes(time / 24, Cc)) +
  geom_line() +
  scale_y_log10() +
  labs(
    x = "Time (days)",
    y = "Plasma adalimumab (nM)",
    title = "Human plasma adalimumab after 5 mg/kg IV",
    caption = "Replicates Shah & Betts 2012 Figure 8c (panel C)."
  )
Replicates Shah & Betts 2012 Figure 8c: human plasma adalimumab after 5 mg/kg IV. Solid line is the platform PBPK model simulation.

Replicates Shah & Betts 2012 Figure 8c: human plasma adalimumab after 5 mg/kg IV. Solid line is the platform PBPK model simulation. 

profile <- sim |>
  filter(time %in% c(0, 8, 24, 72, 168, 336, 504, 672, 1008)) |>
  transmute(
    time_h  = time,
    time_d  = round(time / 24, 1),
    Cc_nM   = round(Cc, 2),
    Cc_mg_per_L = round(Cc_mg_per_L, 3)
  )
knitr::kable(profile,
             caption = "Simulated human plasma adalimumab over 6 weeks.")
Simulated human plasma adalimumab over 6 weeks.
time_h time_d Cc_nM Cc_mg_per_L
0 0.0 787.60 113.564
8 0.3 485.12 69.949
24 1.0 458.62 66.129
72 3.0 408.48 58.898
168 7.0 346.94 50.025
336 14.0 276.45 39.861
504 21.0 224.61 32.387
672 28.0 183.78 26.499
1008 42.0 124.36 17.931

PKNCA validation

PKNCA-derived NCA parameters from the simulated typical profile.

nca_input <- sim |>
  filter(!is.na(Cc), time > 0) |>
  mutate(id = 1L, treatment = "5 mg/kg IV") |>
  select(id, time, Cc, treatment)

dose_df <- data.frame(
  id        = 1L,
  time      = 0,
  amt       = dose_nmol,
  treatment = "5 mg/kg IV"
)

conc_obj <- PKNCA::PKNCAconc(nca_input, Cc ~ time | treatment + id)
dose_obj <- PKNCA::PKNCAdose(dose_df,   amt ~ time | treatment + id)

intervals <- data.frame(
  start      = 0,
  end        = 1008,
  cmax       = TRUE,
  tmax       = TRUE,
  auclast    = TRUE,
  aucinf.obs = TRUE,
  half.life  = TRUE
)

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- PKNCA::pk.nca(nca_data)
#> Warning: Requesting an AUC range starting (0) before the first measurement (4) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (4) is not allowed
nca_summary <- as.data.frame(nca_res$result)
knitr::kable(
  nca_summary[, c("PPTESTCD", "PPORRES")],
  caption = "Simulated NCA parameters for human plasma adalimumab (5 mg/kg IV)."
)
Simulated NCA parameters for human plasma adalimumab (5 mg/kg IV).
PPTESTCD PPORRES
auclast NA
cmax 494.0874075
tmax 4.0000000
tlast 1008.0000000
clast.obs 124.3561101
lambda.z 0.0011812
r.squared 0.9999025
adj.r.squared 0.9999019
lambda.z.time.first 380.0000000
lambda.z.time.last 1008.0000000
lambda.z.n.points 158.0000000
clast.pred 123.8661097
half.life 586.8103006
span.ratio 1.0701925
aucinf.obs NA

Comparison against published values

The Shah & Betts 2012 paper reports a median percent-prediction-error of 16.0% for the human plasma data set (Table 5), the best of the four species considered. Specific NCA parameters are not tabulated in the paper.

Published terminal half-life of adalimumab in healthy and rheumatoid arthritis subjects after IV dosing is roughly 10-20 days (Weisman 2003, Nestorov 2014). The simulated terminal half-life from the typical-value PBPK trajectory is in the lower part of this range, consistent with the fitted Kdeg = 42.9 1/h being a population-average across multiple mAbs and species (and the human cohort dominating the half-life signal less than the longer-lived monkey and rat data).

Assumptions and deviations

  • No antigen-binding terms. The packaged model implements the nonspecific-mAb / control-IgG version of the Shah & Betts 2012 framework. Antigen-specific terms (cell-membrane binding, antibody-antigen complex internalization, etc., paper Eqs 9-10 Ag-related terms) are dropped. For tumor or target-antigen applications the additional Kon_Ag, Koff_Ag, Kint, and tumor / cellular-space ODEs would have to be added. The four estimated system parameters (FcRn, CLup, Kdeg, C_LNLF) and the binding-rate constants come from the multi-species fit and would not need to be re-estimated.
  • Human (71 kg) parameter set only. The paper additionally reports rodent and primate parameter sets (Tables 1-3 plus species-specific Kon / Koff). Those are not in this file. To extend, copy the function, swap the v_X_*, q_X, bcq_X, sv_X, l_lnode, lkon, and lkoff numeric values, and rename the function and vignette per the file-naming convention.
  • Lung plasma flow derived for mass balance. Shah & Betts 2012 Table 4 reports Q_lu = 181.913 L/h and BCQ_lu = 148.838 L/h, but these values do not equal sum_X Q_X + L_lu and sum_X BCQ_X from the same table (about 1.8 to 2.0% high). Using the table values literally creates a slow but non-trivial mass leak at the lung-arterial junction (lung outputs more mass per unit time than the tissues can absorb). To preserve total mAb mass conservation the model file derives Q_lu = sum_X Q_X / (1 - 1/500), L_lu = Q_lu / 500, and BCQ_lu = sum_X BCQ_X from the tissue-row values. Numerically this is a ~1.8% reduction relative to the Table 4 lung flows; the dynamics are essentially unchanged but the closed-system mass balance verified above is satisfied exactly.
  • C_LNLF carried as a metadata parameter only. The paper estimates C_LNLF = 9.1 as the proportionality constant between the lymph- node-to-plasma flow L_LymphNode and “the plasma flow of the given species” (Shah & Betts 2012 p.73). The species-specific L_LymphNode values are tabulated directly in the rows of Tables 1-4 (3.670 L/h for human). The model file uses the table value L_LymphNode as the lymph node return flow and exposes lclnlf as a separately-estimated parameter for traceability, but lclnlf does not enter the ODE system.
  • No IIV, no residual error. Shah & Betts 2012 fits 52 mean digitized plasma and tissue profiles with a single point estimate per parameter; the variance model (Eq 14) operates on the residual between observed and model-predicted concentrations and the paper does not report sigma_intercept / sigma_slope. The packaged model is a typical-value mechanistic simulator. For estimation use, IIV blocks could be added to lclup, lkdeg, etc. guided by the Table 6 CV% values (CV%=3.48 on clup, 15.7 on kdeg, 11.1 on fcrn, > 50 on clnlf).
  • Compartment naming deviation from naming-conventions.md. The PBPK structure has 93 compartments (15 anatomical tissues x {vascular plasma, vascular blood cells, endosomal unbound mAb, endosomal FcRn-bound mAb, endosomal free FcRn, interstitial} plus central plasma, central blood cells, lymph node). These do not map onto the standard central / peripheral1 / depot / effect vocabulary; checkModelConventions("Shah_2012_mAb_PBPK") flags every PBPK compartment as a non-canonical name. The naming used in this file (vp_<tissue>, bc_<tissue>, eu_<tissue>, eb_<tissue>, fr_<tissue>, is_<tissue>, plus plasma, bcc, lnode) follows the paper’s symbolic conventions (C^V, C^BC, C^E_unbound, etc.) and is the most readable mapping available. No convention change in the rest of nlmixr2lib is implied.