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TMDD with two targets and with FcRn recycling (Dua 2015)

Source: vignettes/articles/Dua_2015_tmdd_tutorial.Rmd
Dua_2015_tmdd_tutorial.Rmd

Model and source

  • Article: Dua P, Hawkins E, van der Graaf PH. A Tutorial on Target-Mediated Drug Disposition (TMDD) Models. CPT Pharmacometrics Syst Pharmacol. 2015;4(6):324-337. https://doi.org/10.1002/psp4.41 (PMC4505827, open access).
  • Supplement: “Berkeley Madonna TMDD model library”, a Word document holding the authors’ runnable Berkeley Madonna code for eight TMDD models, each with a complete illustrative parameter set.

Dua 2015 is a tutorial. Its body walks through the TMDD model family, from the one-compartment Mager and Jusko model through the QE, QSS and Michaelis-Menten approximations to extensions for multiple targets, competing drugs, receptor-mediated endocytosis, FcRn recycling and binding outside the central compartment. It fits no data, and every structure in it is attributed to an earlier paper.

The library takes two of the eight supplement models, the two structures it did not already carry:

Supplement model Structure In this library
1 One-compartment full TMDD archetype PK_1cmt_tmdd_full
2 Two-compartment full TMDD structure covered by the archetypes
3 Two-compartment QE approximation structure covered by the archetypes
4 Two-compartment QSS approximation archetype PK_2cmt_tmdd_qss
5 Two-compartment MM approximation archetype PK_2cmt_tmdd_mm
6 Two targets: soluble (QSS) + membrane (MM), after Gibiansky & Gibiansky 2010 Dua_2015_tmdd_multitarget_qss
7 Full TMDD with FcRn endosomal recycling, after Xiao 2010 Dua_2015_tmdd_fcrn_recycling
8 Two-compartment QSS with denosumab values from Gibiansky 2012 not taken; the values belong to the Gibiansky 2012 denosumab analysis, which carries the variability and covariate model this transcription drops 
mods <- c("Dua_2015_tmdd_multitarget_qss", "Dua_2015_tmdd_fcrn_recycling")
m6 <- rxode2::rxode2(readModelDb(mods[1]))
m7 <- rxode2::rxode2(readModelDb(mods[2]))

tibble::tibble(
  Model = mods,
  `Supplement model` = c(6L, 7L),
  `ODE states` = c(length(m6$state), length(m7$state)),
  Parameters = c(nrow(m6$iniDf), nrow(m7$iniDf))
) |>
  knitr::kable()
Model Supplement model ODE states Parameters
Dua_2015_tmdd_multitarget_qss 6 3 14
Dua_2015_tmdd_fcrn_recycling 7 9 15 

# Both are explicit ODE systems. A cl/vc pair would make rxode2 solve them
# analytically and discard the TMDD terms, so confirm that did not happen.
stopifnot(is.null(m6$linCmt), is.null(m7$linCmt))

Population

There is no population. Both parameter sets are illustrative values for a generic monoclonal antibody, chosen by the authors for teaching. They are not fitted to data, and the source has no inter-individual variability and no residual error. Every ini() value is therefore fixed(), and the residual error is fixed(0) so the models simulate exactly as written.

pop <- readModelDb(mods[1])() |>
  environment() |>
  (\(e) e$population)()
tibble::tibble(Field = names(pop), Value = vapply(pop, \(x) paste(as.character(x), collapse = " "), "")) |>
  knitr::kable()
Value

Units and the dose conversion

Time is days and every concentration is nM.

  • Model 6 works per kg body weight. The supplement doses 10 mg/kg and converts it with Dose = (Dosemg * 1e-3 / MWlig) * 1e9 nmol/kg using MWlig = 150000 Da, then sets Lc0 = Dose / Vc with Vc = 0.04 L/kg. The model file is molar (dosing in nmol/kg), so the conversion is applied here.
  • Model 7 has no volume. The supplement gives the dose as the initial central concentration Lc0 = 200 nM, and every transfer between compartments is a symmetric first-order rate constant, so all nine states are concentrations referenced to the central volume. A dose of amt = 200 into central reproduces it.
MW <- 150000 # Supplement Model 6: MWlig = 150000 Da
VC <- 0.04 # Supplement Model 6: Vc = 0.04 L/kg
mgkg_to_nmolkg <- function(mgkg) mgkg * 1e-3 / MW * 1e9

dose6 <- mgkg_to_nmolkg(10)
tibble::tibble(
  Quantity = c("Model 6 dose (nmol/kg)", "Model 6 Lc0 = Dose/Vc (nM)", "Model 7 Lc0 (nM)"),
  Value = c(dose6, dose6 / VC, 200)
) |>
  knitr::kable(digits = 2)
Quantity Value
Model 6 dose (nmol/kg) 66.67
Model 6 Lc0 = Dose/Vc (nM) 1666.67
Model 7 Lc0 (nM) 200.00 
stopifnot(abs(dose6 - 66.6667) < 1e-3)

Source trace

Equations are the main text’s Eqs V.1-V.6 for Model 6. Model 7 is described only in prose in the main text (“FcRn recycling”), so its equations come from the supplement alone. All parameter values come from the supplement.

tibble::tribble(
  ~Model, ~Quantity, ~`Source location`,
  "6", "d/dt Ltot (central), d/dt LT (peripheral1)", "Eqs V.2-V.3; Supplement Model 6 Lctot', Lt'",
  "6", "d/dt Rtot_S (total_target)", "Eq V.4; Supplement Model 6 Rstot'",
  "6", "Free drug LC from the soluble-target QSS quadratic", "Eq V.5; Supplement Model 6 Lc",
  "6", "Rtot_S(0) = R0_S = kin_S/kout_S; LT(0) = 0", "Eq V.6; Supplement Model 6 INIT and kins",
  "6", "Kss_S, Kss_M, Vmax_M = R0_M * kint_M", "Main text 'Multiple targets'; Supplement Model 6",
  "6", "Soluble complex Ps and free soluble target Rs", "Supplement Model 6 'Finding Amount of Free Receptor'",
  "7", "d/dt Lc, Lt, R, Pc, Pt, Le, FcLe, Pe, FcPe", "Supplement Model 7 'Ordinary Differential Equations'",
  "7", "kin = kout * R0; R(0) = R0; all other states 0 except Lc(0) = Lc0", "Supplement Model 7 INIT and rate constants"
) |>
  knitr::kable()
Model Quantity Source location
6 d/dt Ltot (central), d/dt LT (peripheral1) Eqs V.2-V.3; Supplement Model 6 Lctot’, Lt’
6 d/dt Rtot_S (total_target) Eq V.4; Supplement Model 6 Rstot’
6 Free drug LC from the soluble-target QSS quadratic Eq V.5; Supplement Model 6 Lc
6 Rtot_S(0) = R0_S = kin_S/kout_S; LT(0) = 0 Eq V.6; Supplement Model 6 INIT and kins
6 Kss_S, Kss_M, Vmax_M = R0_M * kint_M Main text ‘Multiple targets’; Supplement Model 6
6 Soluble complex Ps and free soluble target Rs Supplement Model 6 ‘Finding Amount of Free Receptor’
7 d/dt Lc, Lt, R, Pc, Pt, Le, FcLe, Pe, FcPe Supplement Model 7 ‘Ordinary Differential Equations’
7 kin = kout * R0; R(0) = R0; all other states 0 except Lc(0) = Lc0 Supplement Model 7 INIT and rate constants 
trace_params <- function(ui) {
  ui$iniDf |>
    dplyr::filter(!is.na(ntheta)) |>
    dplyr::transmute(Parameter = name, Value = signif(ifelse(grepl("^l", name), exp(est), est), 4), Label = label)
}
knitr::kable(trace_params(m6), caption = "Model 6 parameters (all from Supplement Model 6)")
Model 6 parameters (all from Supplement Model 6)
Parameter Value Label
lvc 4.00e-02 Central volume of distribution (L/kg)
lkel 4.60e-02 Linear elimination rate constant of free drug from central (1/day)
lk12 9.02e-01 Central-to-tissue transfer rate constant of free drug (1/day)
lk21 7.25e-01 Tissue-to-central transfer rate constant of free drug (1/day)
lrbase_total_target 6.57e-03 Baseline soluble target concentration Rs0 (nM)
lkdeg_sol 1.73e+01 First-order degradation rate constant of free soluble target (1/day)
lkint_sol 1.70e-01 Elimination rate constant of the drug-soluble-target complex (1/day)
lkon_sol 3.02e+01 Association rate constant, drug to soluble target (1/(nM*day))
lkoff_sol 1.69e+02 Dissociation rate constant of the drug-soluble-target complex (1/day)
lrtot_mem 2.00e-03 Total membrane-target concentration, held constant Rm0 (nM)
lkint_mem 1.70e-01 Elimination rate constant of the drug-membrane-target complex (1/day)
lkon_mem 3.02e+01 Association rate constant, drug to membrane target (1/(nM*day))
lkoff_mem 1.69e+02 Dissociation rate constant of the drug-membrane-target complex (1/day)
propSd 0.00e+00 Proportional residual error (fraction; ZERO - none in the source) 
knitr::kable(trace_params(m7), caption = "Model 7 parameters (all from Supplement Model 7)")
Model 7 parameters (all from Supplement Model 7)
Parameter Value Label
lkel 0.024 Elimination rate constant of free drug from central (1/day)
lkel_endo 0.024 Elimination rate constant of non-FcRn-bound drug and complex from the endosome (1/day)
lk12 0.902 Central-to-tissue transfer rate constant of drug and complex (1/day)
lk21 0.725 Tissue-to-central transfer rate constant of drug and complex (1/day)
lkup 0.360 Central-to-endosome uptake rate constant of drug and complex (1/day)
lrbase_target 0.030 Baseline free target concentration R0 (nM)
lkdeg 0.823 First-order degradation rate constant of free target (1/day)
lkint 0.201 Elimination rate constant of the drug-target complex from central (1/day)
lkon 0.592 Association rate constant, drug to target (1/(nM*day))
lkoff 0.900 Dissociation rate constant of the drug-target complex (1/day)
lrtot_fcrn 0.050 Endosomal FcRn concentration, held constant Fc0 (nM)
lkon_fcrn 4.830 Association rate constant, drug or complex to FcRn (1/(nM*day))
lkoff_fcrn 2.000 Dissociation rate constant of the FcRn complexes (1/day)
lkrec_fcrn 0.400 Recycling rate constant of FcRn complexes back to central (1/day)
propSd 0.000 Proportional residual error (fraction; ZERO - none in the source)

Every value in both tables appears verbatim in the supplement; the in-file comment on each ini() line names the supplement variable it came from (kpt is the library’s k12, ktp its k21, kout its kdeg, and kep its kint).

Units of every ODE term

Model State Units Every term reduces to
6 central (total drug amount) nmol/kg (L/kg) x (1/day) x nM = nmol/kg/day
6 peripheral1 (tissue drug amount, = Lt x Vc) nmol/kg (1/day) x (L/kg) x nM or (1/day) x nmol/kg
6 total_target nM nM/day (ksyn) or (1/day) x nM
7 all nine states nM (1/day) x nM, or (1/(nM day)) x nM x nM for binding

The Model 6 Michaelis-Menten term Vmax_M * Lc / (Kss_M + Lc) has Vmax_M = R0_M * kint_M in nM/day, so it is a concentration rate and is multiplied by Vc along with the other central terms.

Model 6: soluble plus membrane target

Drug-free steady state

With no drug, the soluble target must hold at its baseline Rs0 = 0.00657 nM indefinitely, because kins = kouts * Rs0.

tight <- list(rtol = 1e-10, atol = 1e-12, maxsteps = 1e6)
ev0 <- rxode2::et(seq(0, 100, by = 1))
s0 <- do.call(rxode2::rxSolve, c(list(m6, ev0, returnType = "data.frame"), tight))
stopifnot(max(abs(s0$total_target / 0.00657 - 1)) < 1e-8, max(abs(s0$central)) == 0)

Single 10 mg/kg dose (Supplement Model 6 regimen)

The supplement simulates one 10 mg/kg i.v. bolus for 360 days.

grid6 <- sort(unique(c(0, exp(seq(log(1e-4), log(360), length.out = 2500)))))
ev6 <- rxode2::et(amt = dose6, cmt = "central") |> rxode2::et(grid6)
s6 <- do.call(rxode2::rxSolve, c(list(m6, ev6, returnType = "data.frame"), tight))

s6 |>
  dplyr::select(time, `Free drug Lc` = Cc, `Total drug Ltot` = ctot, `Total soluble target` = total_target) |>
  tidyr::pivot_longer(-time) |>
  ggplot(aes(time, value)) +
  geom_line() +
  facet_wrap(~name, scales = "free_y", ncol = 1) +
  scale_y_log10() +
  labs(x = "Time (day)", y = "Concentration (nM)", title = "Supplement Model 6, 10 mg/kg i.v.")

Removing the membrane target reduces Model 6 to the library’s QSS archetype

With the membrane target set to (numerically) zero, Model 6 is exactly the two-compartment QSS TMDD model (Supplement Model 4). The library already ships that structure as PK_2cmt_tmdd_qss, written independently with a CL/V parameterisation and an amount-based peripheral compartment. Mapping the Model 6 constants onto it (CL = kel * Vc, Q = kpt * Vc, Vp = Vc * kpt/ktp) must give the same total-drug profile. This checks the central, peripheral and soluble-target equations of Model 6 against a second, separately written implementation.

s6_nomem <- do.call(
  rxode2::rxSolve,
  c(list(m6, ev6, params = c(lrtot_mem = log(1e-15)), returnType = "data.frame"), tight)
)

kss <- (169 + 0.170) / 30.2 # Supplement Model 4/6: Kss = (koff + kep)/kon
arch <- readModelDb("PK_2cmt_tmdd_qss") |>
  rxode2::zeroRe() |>
  rxode2::ini(
    lcl = log(0.046 * VC), lvc = log(VC), lvp = log(VC * 0.902 / 0.725), lq = log(0.902 * VC),
    lT0 = log(0.00657), lkdeg = log(17.3), lkint = log(0.170), lkss = log(kss)
  )
s_arch <- do.call(rxode2::rxSolve, c(list(arch, ev6, returnType = "data.frame"), tight))

rel_arch <- max(abs(s_arch$Cc / s6_nomem$ctot - 1))
rel_arch
#> [1] 8.920642e-13
stopifnot(nrow(s_arch) == nrow(s6_nomem), rel_arch < 1e-6)

Drug mass balance

Drug leaves the body only through linear elimination of free drug, elimination of the soluble complex and the membrane-target Michaelis-Menten term, so the dose must equal the drug still in the body plus the cumulative eliminated flux. The flux is integrated with the trapezoidal rule on the dense log grid, which limits the closure to about 1e-5 (measured 6e-6).

flux6 <- with(s6, VC * (0.046 * Cc + total_target * 0.170 * Cc / (kss + Cc) +
  0.002 * 0.170 * Cc / ((169 + 0.170) / 30.2 + Cc)))
cum6 <- c(0, cumsum(diff(s6$time) * (head(flux6, -1) + tail(flux6, -1)) / 2))
closure6 <- (s6$central + s6$peripheral1 + cum6) / dose6 - 1
max(abs(closure6))
#> [1] 5.939772e-06
stopifnot(max(abs(closure6)) < 1e-4)

How much the membrane target contributes

The membrane term is a saturable extra elimination pathway, so removing it can only raise exposure, and its share of elimination grows as the dose falls and free drug approaches Kss_M (5.6 nM). With the tutorial’s illustrative values the pathway is small at every dose: the total membrane target is only Rm0 = 0.002 nM, against free-drug concentrations of tens to thousands of nM.

auc_free <- function(sim) sum(diff(sim$time) * (head(sim$Cc, -1) + tail(sim$Cc, -1)) / 2)
dose_tab <- lapply(c(0.01, 0.1, 1, 10), function(d) {
  ev <- rxode2::et(amt = mgkg_to_nmolkg(d), cmt = "central") |> rxode2::et(grid6)
  with_m <- do.call(rxode2::rxSolve, c(list(m6, ev, returnType = "data.frame"), tight))
  no_m <- do.call(rxode2::rxSolve, c(list(m6, ev, params = c(lrtot_mem = log(1e-15)), returnType = "data.frame"), tight))
  tibble::tibble(
    `Dose (mg/kg)` = d,
    `Free-drug AUC with membrane target (nM day)` = auc_free(with_m),
    `Free-drug AUC without (nM day)` = auc_free(no_m)
  )
}) |>
  dplyr::bind_rows() |>
  dplyr::mutate(`Exposure lost to membrane target (%)` = 100 * (1 - `Free-drug AUC with membrane target (nM day)` / `Free-drug AUC without (nM day)`))
knitr::kable(dose_tab, digits = 3)
Dose (mg/kg) Free-drug AUC with membrane target (nM day) Free-drug AUC without (nM day) Exposure lost to membrane target (%)
0.01 36.008 36.052 0.122
0.10 360.232 360.530 0.083
1.00 3605.363 3606.300 0.026
10.00 36108.590 36110.250 0.005 
lost <- dose_tab$`Exposure lost to membrane target (%)`
stopifnot(
  all(dose_tab$`Free-drug AUC with membrane target (nM day)` < dose_tab$`Free-drug AUC without (nM day)`),
  lost[1] > lost[4]
)

The membrane target removes 0.12% of free-drug exposure at 0.01 mg/kg and 0.0046% at 10 mg/kg. Free-drug AUC is close to dose-proportional across the whole range. With these values, neither target is abundant enough to give the nonlinear profile TMDD is known for; raise Rm0 or Rs0 to see it.

The supplement’s soluble-complex formula

The supplement computes the soluble complex as Ps = Lctot - Lc - Rm0 * Lc / (Kssm + Lc), subtracting the membrane-bound drug from total drug. But Lc comes from the soluble-target quadratic, which solves Lctot = Lc + Rstot * Lc / (Ksss + Lc) and so does not include membrane-bound drug in Lctot. The two definitions disagree by exactly the membrane-bound term, Rm0 * Lc / (Kssm + Lc). The model file keeps the printed form (complex_sol, target_sol) and adds the QSS-consistent form (complex_sol_qss, target_sol_qss). The identity below also confirms that the quadratic is solved correctly at every time point.

bias <- with(s6, (complex_sol_qss - complex_sol) - 0.002 * Cc / ((169 + 0.170) / 30.2 + Cc))
max(abs(bias))
#> [1] 3.72387e-13
stopifnot(max(abs(bias)) < 1e-9)

s6 |>
  dplyr::select(time, `As printed (target_sol)` = target_sol, `QSS-consistent (target_sol_qss)` = target_sol_qss) |>
  tidyr::pivot_longer(-time) |>
  ggplot(aes(time, value, colour = name)) +
  geom_line() +
  labs(x = "Time (day)", y = "Free soluble target (nM)", colour = NULL) +
  theme(legend.position = "bottom")

While drug is in excess the printed form understates the bound soluble target by about Rm0 = 0.002 nM. The baseline is only 0.00657 nM, so the printed free soluble target sits near 0.002 nM where the QSS relation puts it close to zero. Use target_sol_qss for target-suppression questions.

Model 7: full TMDD with FcRn recycling

Drug-free steady state 

s7_0 <- do.call(rxode2::rxSolve, c(list(m7, ev0, returnType = "data.frame"), tight))
stopifnot(max(abs(s7_0$target / 0.03 - 1)) < 1e-8, max(abs(s7_0$central)) == 0)

Single dose, Lc0 = 200 nM (Supplement Model 7 regimen)

The supplement simulates 28 days.

grid7 <- sort(unique(c(0, seq(0, 28, by = 0.005))))
ev7 <- rxode2::et(amt = 200, cmt = "central") |> rxode2::et(grid7)
s7 <- do.call(rxode2::rxSolve, c(list(m7, ev7, returnType = "data.frame"), tight))

s7 |>
  dplyr::select(
    time,
    `Free drug, central` = central, `Free drug, tissue` = peripheral1,
    `Free target` = target, `Complex, central` = complex,
    `Endosomal drug, free` = endosome, `Endosomal drug, FcRn-bound` = endosome_fcrn,
    `Endosomal complex, free` = complex_endosome, `Endosomal complex, FcRn-bound` = complex_endosome_fcrn
  ) |>
  tidyr::pivot_longer(-time) |>
  ggplot(aes(time, value)) +
  geom_line() +
  facet_wrap(~name, scales = "free_y", ncol = 2) +
  labs(x = "Time (day)", y = "Concentration (nM)", title = "Supplement Model 7, Lc0 = 200 nM")

Drug mass balance

Summing the eight drug-containing equations, every binding, transfer, uptake and recycling term cancels and only four elimination terms remain: kelc * Lc + kepc * Pc + kele * Le + kele * Pe. So the 200 nM dose must equal the drug left in the eight states plus the integrated elimination.

drug7 <- with(s7, central + peripheral1 + complex + complex_peripheral1 +
  endosome + endosome_fcrn + complex_endosome + complex_endosome_fcrn)
flux7 <- with(s7, 0.024 * central + 0.201 * complex + 0.024 * endosome + 0.024 * complex_endosome)
cum7 <- c(0, cumsum(diff(s7$time) * (head(flux7, -1) + tail(flux7, -1)) / 2))
closure7 <- (drug7 + cum7) / 200 - 1
max(abs(closure7))
#> [1] 4.089254e-08
stopifnot(max(abs(closure7)) < 1e-6) # measured 4e-8

Switching off distribution and endosomal uptake reduces Model 7 to the one-compartment full TMDD archetype

With kup and kpt set to (numerically) zero, no drug reaches the tissue or the endosome, and Model 7 is exactly the one-compartment full TMDD model (Supplement Model 1, whose binding constants Model 7 reuses). The library ships that structure as PK_1cmt_tmdd_full. With Vc = 1 L and the dose as 200 mg its central amount equals the Model 7 concentration, so the two free-drug profiles must coincide.

s7_red <- do.call(
  rxode2::rxSolve,
  c(list(m7, ev7, params = c(lkup = log(1e-15), lk12 = log(1e-15)), returnType = "data.frame"), tight)
)
arch1 <- readModelDb("PK_1cmt_tmdd_full") |>
  rxode2::zeroRe() |>
  rxode2::ini(
    lcl = log(0.024), lvc = log(1), lT0 = log(0.03), lkdeg = log(0.823),
    lkint = log(0.201), lkon = log(0.592), lkoff = log(0.9)
  )
s_arch1 <- do.call(rxode2::rxSolve, c(list(arch1, ev7, returnType = "data.frame"), tight))

rel_arch1 <- max(abs(s_arch1$Cc / s7_red$Cc - 1))
rel_arch1
#> [1] 3.528289e-13
stopifnot(nrow(s_arch1) == nrow(s7_red), rel_arch1 < 1e-6)

What FcRn recycling does to exposure

Without FcRn, drug taken up into the endosome is lost: it is eliminated there at kele and never returns. With FcRn, the bound fraction is recycled to the central compartment, so central exposure can only be higher.

s7_nofc <- do.call(
  rxode2::rxSolve,
  c(list(m7, ev7, params = c(lrtot_fcrn = log(1e-15)), returnType = "data.frame"), tight)
)
cmp7 <- dplyr::bind_rows(
  dplyr::mutate(s7, scenario = "FcRn 0.05 nM (Supplement Model 7)"),
  dplyr::mutate(s7_nofc, scenario = "No FcRn")
)
ggplot(cmp7, aes(time, Cc, colour = scenario)) +
  geom_line() +
  scale_y_log10() +
  labs(x = "Time (day)", y = "Free drug, central (nM)", colour = NULL) +
  theme(legend.position = "bottom")


auc7 <- c(fcrn = auc_free(s7), nofcrn = auc_free(s7_nofc))
auc7
#>     fcrn   nofcrn 
#> 722.1582 512.9660
stopifnot(auc7[["fcrn"]] > auc7[["nofcrn"]])

PKNCA on the free-drug profiles

The paper reports no NCA, so there is no published table to compare against. PKNCA is run here on the deterministic free-drug profiles as a descriptive summary, grouped by model and scenario. Model 7 is simulated over the supplement’s 28-day horizon, so a large part of its aucinf.obs is extrapolated.

conc_nca <- dplyr::bind_rows(
  dplyr::transmute(s6, id = 1L, treatment = "Model 6, 10 mg/kg", time, Cc),
  dplyr::transmute(s7, id = 2L, treatment = "Model 7, FcRn", time, Cc),
  dplyr::transmute(s7_nofc, id = 3L, treatment = "Model 7, no FcRn", time, Cc)
) |>
  dplyr::filter(!is.na(Cc))
dose_nca <- tibble::tibble(
  id = 1:3, time = 0,
  treatment = c("Model 6, 10 mg/kg", "Model 7, FcRn", "Model 7, no FcRn"),
  amt = c(dose6, 200, 200)
)
o_conc <- PKNCA::PKNCAconc(conc_nca, Cc ~ time | treatment + id)
o_dose <- PKNCA::PKNCAdose(dose_nca, amt ~ time | treatment + id, route = "intravascular")
intervals <- data.frame(start = 0, end = Inf, cmax = TRUE, aucinf.obs = TRUE, half.life = TRUE)
nca <- PKNCA::pk.nca(PKNCA::PKNCAdata(o_conc, o_dose, intervals = intervals))
summary(nca) |> knitr::kable()
start end treatment N cmax half.life aucinf.obs
0 Inf Model 6, 10 mg/kg 1 1670 34.1 36100
0 Inf Model 7, FcRn 1 200 28.0 1150
0 Inf Model 7, no FcRn 1 200 4.59 520

Assumptions and deviations

  • Only two of the eight supplement models are in the library. Models 1-5 repeat structures the library already carries as TMDD archetypes, and Model 8 transcribes the Gibiansky 2012 denosumab estimates without that analysis’s variability and covariate model, so it is best taken from the primary paper.
  • Illustrative parameters, not estimates. Neither model is fitted to data. Use them to explore model behaviour, not to predict any real antibody.
  • Model 6 is the i.v. form. Main-text Eqs V.1 and V.3 also carry a subcutaneous depot (dLD/dt = -ka*LD, LD(0) = D1). Supplement Model 6 is i.v.-only and no absorption rate constant is printed anywhere, so no depot is included. (The input term of Eq V.3 is typeset kf LD; with the depot of Eq V.1 it is the absorption term.)
  • Model 6 soluble-complex formula. The printed Ps is inconsistent with the quadratic that defines Lc, by exactly the membrane-bound drug. Both forms are reported; see the section above.
  • Model 7 kepe is unused. The supplement lists kepe = 0.201 1/day (“elimination of complex from endosome compartment”), but its Pe equation eliminates endosomal complex with kele = 0.024 1/day, the same rate as endosomal free drug. The model file follows the equation. The main text describes the Xiao model as having equal elimination rates for drug and complex (ke(LC) = ke(PC)); the supplement applies that equality only in the endosome, while in the central compartment it keeps kelc = 0.024 and kepc = 0.201 1/day. kepe is not carried in ini() because no equation uses it.
  • Model 7 FcRn is constant, and with these values that assumption is stretched. Following Xiao 2010, the tutorial holds FcRn fixed at Fc0 = 0.05 nM rather than tracking free FcRn. With the illustrative dose, the FcRn-bound drug pool reaches 10.7 nM, well above the stated FcRn concentration. A model that conserved FcRn would saturate the recycling pathway; this one does not. This is a property of the published model, reproduced as written.
  • Model 7 has no volume. Doses are concentrations (nM). To dose an amount, divide it by a central volume of your choosing; none is printed for Model 7.
  • No variability or residual error. propSd is fixed(0) in both files.
  • Integrator. The supplement integrates with fourth-order Runge-Kutta at DT = 0.01 day. Here the models are solved with LSODA at tight tolerances.