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TMDD high vs low-affinity approximations: omalizumab, caplacizumab and linagliptin (Straube 2025)

Source: vignettes/articles/Straube_2025_TMDD_high_vs_low_affinity.Rmd
Straube_2025_TMDD_high_vs_low_affinity.Rmd

Model and source

Straube (2025) revisits the classical Mager & Jusko target-mediated drug disposition (TMDD) model and derives high-affinity (KM≪RTK_M \ll R_T) and low-affinity (KM≫RTK_M \gg R_T) approximations of it. Section 3.2 of the paper then fits the full TMDD model to published PK data for three molecules, and those fits are what this package carries: the approximations are analytical results that the vignette checks the fitted models against, not separate models.

Each drug was fitted twice: a one-compartment fit in the main text (Table 2, Figure 5) and a two-compartment refit in the supplement (Table S1, Figures S2-S3). Both are reported by the author with complete parameter tables and dedicated figures, so both are packaged.

mods <- c(
  "Straube_2025_omalizumab_1cmt",   "Straube_2025_omalizumab_2cmt",
  "Straube_2025_caplacizumab_1cmt", "Straube_2025_caplacizumab_2cmt",
  "Straube_2025_linagliptin_1cmt",  "Straube_2025_linagliptin_2cmt"
)
# readModelDb() returns the model FUNCTION; rxode2::rxode() resolves it to the
# ui exactly once so every accessor below works.
ui <- lapply(mods, function(n) rxode2::rxode(readModelDb(n)))
names(ui) <- mods

tibble::tibble(
  Model       = mods,
  Structure   = vapply(ui, function(u) paste(u$state, collapse = ", "), character(1)),
  `Source table` = c("Table 2", "Table S1", "Table 2", "Table S1", "Table 2", "Table S1")
) |>
  knitr::kable(caption = "The six packaged fits and their ODE states.")
The six packaged fits and their ODE states.
Model Structure Source table
Straube_2025_omalizumab_1cmt depot, central, target, complex Table 2
Straube_2025_omalizumab_2cmt depot, central, peripheral1, target, complex Table S1
Straube_2025_caplacizumab_1cmt central, target, complex Table 2
Straube_2025_caplacizumab_2cmt central, peripheral1, target, complex Table S1
Straube_2025_linagliptin_1cmt central, target, complex Table 2
Straube_2025_linagliptin_2cmt central, peripheral1, target, complex Table S1

Population

The three datasets are all previously published profiles that Straube digitised and re-fitted; no new subjects were studied.

  • Omalizumab (human, n = 2) – total drug, total target and free target for two patients in a phase I study after a single subcutaneous dose, digitised from Meno-Tetang & Lowe (2005). Vc, CL and Rb were estimated patient-specific; everything else is pooled across the two patients. Omalizumab is an anti-IgE monoclonal antibody; this is the paper’s low-affinity example.
  • Caplacizumab / ALX-0081 – a single-domain antibody against von Willebrand factor, dosed intravenously at 0.02, 0.4 and 8 mg/kg, digitised from Glassman & Muzykantov (2020). The paper’s high-affinity example with target-mediated exposure enhancement (TMEE).
  • Linagliptin – a small-molecule DPP-4 inhibitor dosed intravenously at 0.5, 2.5 and 10 mg, also from Glassman & Muzykantov (2020). A second high-affinity / TMEE example, but one where the paper reports the approximations work markedly less well.

These are deterministic individual / typical-value fits: Straube reports no between-subject variability and no residual error, so none is encoded. The full metadata is available programmatically:

str(ui[["Straube_2025_caplacizumab_1cmt"]]$population, max.level = 1)
#> List of 7
#>  $ species      : chr "not stated in the source"
#>  $ n_subjects   : int NA
#>  $ n_studies    : int 1
#>  $ disease_state: chr "ALX-0081 is a single-domain antibody against von Willebrand factor (vWF) developed to treat acquired thrombotic"| __truncated__
#>  $ dose_range   : chr "Single intravenous doses of 0.02, 0.4 and 8 mg/kg (Figure 5b)."
#>  $ regions      : chr NA
#>  $ notes        : chr "SPECIES AND BODY WEIGHT ARE NOT STATED. Straube 2025 gives only the mg/kg dose levels and reports Vc = 0.046 L,"| __truncated__

Source trace

Every ini() entry carries an in-file comment naming its source location. The table below collects them.

Symbol Parameter 1-cmt source 2-cmt source
VcV_c lvc Table 2 Table S1
VpV_p lvp n/a Table S1
CLCL lcl Table 2 Table S1
QQ lq n/a Table S1
kak_a lka (omalizumab) Table 2 Table S1
FF lfdepot (omalizumab) Table 2, footnote * (fixed) Table S1, footnote * (fixed)
koffk_{off} lk2 Table 2 Table S1
KdK_d lkd Table 2 Table S1
RbR_b lrbase Table 2 Table S1
keRk_{eR} lkdeg Table 2 Table S1
keDRk_{eDR} lkint Table 2 Table S1

Quantities the tables print but that are derived in model() rather than stored, exactly as the tables’ own row labels define them: keD=CL/Vck_{eD} = CL/V_c, kon=koff/Kdk_{on} = k_{off}/K_d (Equation 7), ksyn=keRRbk_{syn} = k_{eR}R_b (Figure 1), and the target accumulation ratio Tacc=keR/keDRT_{acc} = k_{eR}/k_{eDR} (Equation 14).

Equation Source
d/dt(central), d/dt(target), d/dt(complex) Equation (2)
target(0) <- rbase * vc Equation (3)
depot + f(depot) (omalizumab) Equation (S20)-(S21); see Errata
peripheral1 distribution (2-cmt) Equation (S18)
Cc <- (central + complex)/vc (DTD_T) Section 2.1, DT=D+DRD_T = D + DR
ftbr <- freeTarget/rbase Equation (28)

Table consistency

The packaged parameters must reproduce every derived quantity the paper prints. This is a closed-form check on the paper’s own tables and needs no simulation.

printed <- tibble::tribble(
  ~Model,                            ~keD,   ~ksyn,  ~kon,   ~Tacc,   ~KM_Kd,
  "Straube_2025_omalizumab_1cmt",    0.0025,  1.130,  1.430,  5.058,  1.051,
  "Straube_2025_omalizumab_2cmt",    0.13,    1.15,   1.01,   5.7,    1.08,
  "Straube_2025_caplacizumab_1cmt", 58.80,   10.394,  2.3e4,  0.334,  1.01,
  "Straube_2025_caplacizumab_2cmt", 65.6,    10.24,   2.3e4,  0.334,  1.01,
  "Straube_2025_linagliptin_1cmt",   8.94,   18.529, 22.73,  86.804,  1.069,
  "Straube_2025_linagliptin_2cmt",  11.37,    1.573, 22.73,   5.399,  1.061
)

encoded <- lapply(mods, function(n) {
  g <- function(p) unname(exp(ui[[n]]$theta[[p]]))
  tibble::tibble(
    Model = n,
    keD   = g("lcl") / g("lvc"),
    ksyn  = g("lkdeg") * g("lrbase"),
    kon   = g("lk2") / g("lkd"),
    Tacc  = g("lkdeg") / g("lkint"),
    KM_Kd = 1 + g("lkint") / g("lk2")
  )
}) |> dplyr::bind_rows()

consistency <- printed |>
  tidyr::pivot_longer(-Model, names_to = "Quantity", values_to = "Printed") |>
  dplyr::inner_join(
    encoded |> tidyr::pivot_longer(-Model, names_to = "Quantity", values_to = "Encoded"),
    by = c("Model", "Quantity")
  ) |>
  dplyr::mutate(`% diff` = 100 * (Encoded - Printed) / Printed)

# Deterministic: no simulation, no RNG, so a tight bound is the right bound.
# Every residual is a printed-rounding artefact -- the largest (omalizumab keD,
# 1.9%) comes from Table 2 printing keD to two significant figures (0.0025 for
# 0.002548). 2.5% still goes red on a mis-transcribed digit.
stopifnot(max(abs(consistency$`% diff`)) < 2.5)

consistency |>
  dplyr::arrange(dplyr::desc(abs(`% diff`))) |>
  head(8) |>
  knitr::kable(digits = c(0, 0, 5, 5, 2),
               caption = "Largest deviations between encoded and printed derived quantities.")
Largest deviations between encoded and printed derived quantities.
Model Quantity Printed Encoded % diff
Straube_2025_omalizumab_1cmt keD 2.500e-03 0.00255 1.91
Straube_2025_omalizumab_2cmt keD 1.300e-01 0.13235 1.81
Straube_2025_omalizumab_2cmt Tacc 5.700e+00 5.62500 -1.32
Straube_2025_caplacizumab_2cmt keD 6.560e+01 64.83333 -1.17
Straube_2025_caplacizumab_1cmt kon 2.300e+04 23266.66667 1.16
Straube_2025_caplacizumab_2cmt kon 2.300e+04 23266.66667 1.16
Straube_2025_caplacizumab_1cmt keD 5.880e+01 59.30435 0.86
Straube_2025_linagliptin_1cmt kon 2.273e+01 22.63514 -0.42

Dose levels

There is no virtual cohort here: the models are deterministic fits with no IIV, so the “cohort” is the set of dose levels the paper itself simulated. Doses are in nmol, the unit Equation (3) uses (D0D_0 is “the drug dose (in nanomole)”).

MW <- c(omalizumab = 149000, caplacizumab = 28000, linagliptin = 472.54)  # Table 2, MW row

# Omalizumab and linagliptin doses are absolute mass, so the nmol dose follows
# from the molecular weight alone: mg / (g/mol) = 1e-3 mol, i.e. 1e6 nmol.
mg_to_nmol <- function(mg, mw) mg / mw * 1e6

# Caplacizumab doses are reported only as mg/kg and the paper states neither the
# species nor the body weight (see Errata). The nmol doses are therefore
# back-computed from the paper's OWN Vc and the initial total-drug concentration
# read off Figure 5b (~133 ug/mL for 8 mg/kg), then scaled by the mg/kg ratio.
# ug/mL / (g/mol) * 1e6 gives nM; nM * L is already nmol.
cap_top_ugml <- 133
cap_top_nmol <- cap_top_ugml / MW[["caplacizumab"]] * 1e6 *
  exp(ui[["Straube_2025_caplacizumab_1cmt"]]$theta[["lvc"]])

doses <- dplyr::bind_rows(
  tibble::tibble(drug = "omalizumab",   treatment = c("90 mg SC", "270 mg SC"),
                 amt = mg_to_nmol(c(90, 270), MW[["omalizumab"]]), cmt = "depot"),
  tibble::tibble(drug = "caplacizumab", treatment = c("0.02 mg/kg", "0.4 mg/kg", "8 mg/kg"),
                 amt = cap_top_nmol * c(0.02, 0.4, 8) / 8, cmt = "central"),
  tibble::tibble(drug = "linagliptin",  treatment = c("0.5 mg", "2.5 mg", "10 mg"),
                 amt = mg_to_nmol(c(0.5, 2.5, 10), MW[["linagliptin"]]), cmt = "central")
)
knitr::kable(doses, digits = 3, caption = "Dose levels simulated, in nmol.")
Dose levels simulated, in nmol.
drug treatment amt cmt
omalizumab 90 mg SC 604.027 depot
omalizumab 270 mg SC 1812.081 depot
caplacizumab 0.02 mg/kg 0.546 central
caplacizumab 0.4 mg/kg 10.925 central
caplacizumab 8 mg/kg 218.500 central
linagliptin 0.5 mg 1058.111 central
linagliptin 2.5 mg 5290.557 central
linagliptin 10 mg 21162.230 central

The linagliptin conversion is self-checking: 10 mg / 472.54 g/mol / 104.854 L = 201.8 nM, matching the ~200 nM intercept of the 10 mg curve in Figure 5c.

Simulation 

# Solve one (model, dose) pair over a window scaled to that model's own
# complex-elimination time constant.
solve_one <- function(model, amt, cmt, treatment, tmax, n = 1500) {
  ev <- rxode2::et(amt = amt, cmt = cmt)
  ev <- rxode2::et(ev, seq(0, tmax, length.out = n))
  # useLinCmt = FALSE: rxode2's automatic ODE -> linCmt conversion must not be
  # allowed to reinterpret the cl/vc pair and discard the explicit binding ODEs.
  s <- rxode2::rxSolve(ui[[model]], ev, returnType = "data.frame", useLinCmt = FALSE)
  if (is.null(s$id)) s$id <- 1L      # rxSolve omits `id` for a single subject
  s$model <- model
  s$treatment <- treatment
  s
}

Replicate published figures

Figure 5a – omalizumab (low-affinity example) 

# Replicates Figure 5a of Straube 2025: total drug (upper) and total / free
# target (lower) after a single 90 mg subcutaneous dose.
oma <- solve_one("Straube_2025_omalizumab_1cmt",
                 doses$amt[doses$treatment == "90 mg SC"], "depot", "90 mg SC", tmax = 60)

oma |>
  dplyr::select(time, `Total drug (D_T)` = Cc, `Total target (R_T)` = totalTarget,
                `Free target (R)` = freeTarget) |>
  tidyr::pivot_longer(-time) |>
  ggplot(aes(time, value)) +
  geom_line(linewidth = 0.8) +
  facet_wrap(~name, scales = "free_y") +
  labs(x = "Time (days)", y = "Concentration (nM)",
       caption = "Replicates Figure 5a of Straube 2025 (90 mg SC omalizumab).") +
  theme_bw()

The published Figure 5a peaks at roughly 60 nM around day 4-6 and falls to about 20 nM by day 60; total target rises above its 1.34 nM baseline because Tacc≈5>1T_{acc} \approx 5 > 1, while free target is suppressed and then recovers.

peak <- max(oma$Cc)
rb_oma <- exp(ui[["Straube_2025_omalizumab_1cmt"]]$theta[["lrbase"]])
# Deterministic solve of a fixed model at a fixed dose -- no cohort, no RNG.
stopifnot(
  peak > 55 && peak < 70,                                  # Figure 5a peak ~60 nM
  oma$time[which.max(oma$Cc)] > 2 && oma$time[which.max(oma$Cc)] < 8,
  max(oma$totalTarget) > rb_oma,                           # Tacc > 1 -> RT rises
  min(oma$freeTarget) < 0.2 * rb_oma                       # free target suppressed
)

Figures 5b and 5c – caplacizumab and linagliptin (high-affinity examples) 

# Replicates Figures 5b and 5c of Straube 2025. Both panels are plotted on the
# paper's x-axis units (hours) although the models work in days.
hi <- dplyr::bind_rows(
  lapply(which(doses$drug == "caplacizumab"), function(i)
    solve_one("Straube_2025_caplacizumab_1cmt", doses$amt[i], "central",
              doses$treatment[i], tmax = 80 / 24)),
  lapply(which(doses$drug == "linagliptin"), function(i)
    solve_one("Straube_2025_linagliptin_1cmt", doses$amt[i], "central",
              doses$treatment[i], tmax = 100 / 24))
)
hi <- hi |> dplyr::mutate(
  drug = ifelse(grepl("caplacizumab", model), "ALX-0081 (caplacizumab)", "Linagliptin"),
  # Figure 5b plots ug/mL; Figure 5c plots nM.
  y = ifelse(grepl("caplacizumab", model), Cc * MW[["caplacizumab"]] / 1e6, Cc),
  treatment = factor(treatment,
                     levels = c("0.02 mg/kg", "0.4 mg/kg", "8 mg/kg",
                                "0.5 mg", "2.5 mg", "10 mg"))
)

ggplot(hi, aes(time * 24, y, colour = treatment)) +
  geom_line(linewidth = 0.8) +
  facet_wrap(~drug, scales = "free") +
  scale_y_log10() +
  labs(x = "Time (h)", y = "Total drug (ug/mL for ALX-0081; nM for linagliptin)",
       colour = NULL,
       caption = "Replicates Figures 5b and 5c of Straube 2025 (upper panels).") +
  theme_bw()

# Lower panels of Figures 5b and 5c: free target to baseline ratio (Equation 28).
ftbr <- dplyr::bind_rows(
  lapply(which(doses$drug == "caplacizumab"), function(i)
    solve_one("Straube_2025_caplacizumab_1cmt", doses$amt[i], "central",
              doses$treatment[i], tmax = 500 / 24)),
  lapply(which(doses$drug == "linagliptin"), function(i)
    solve_one("Straube_2025_linagliptin_1cmt", doses$amt[i], "central",
              doses$treatment[i], tmax = 25 / 24))
) |>
  dplyr::mutate(drug = ifelse(grepl("caplacizumab", model),
                              "ALX-0081 (caplacizumab)", "Linagliptin"))

ggplot(ftbr, aes(time * 24, ftbr, colour = treatment)) +
  geom_line(linewidth = 0.8) +
  facet_wrap(~drug, scales = "free_x") +
  labs(x = "Time (h)", y = "FTBR", colour = NULL,
       caption = "Replicates Figures 5b and 5c of Straube 2025 (lower panels).") +
  theme_bw()

Mechanistic validation

The paper’s analytical results give several closed-form predictions that the packaged full-TMDD models must satisfy. All are deterministic, so the bounds below are tight by design (see pattern 11 of the skill’s failure-pattern notes).

Drug-free steady state (Equation 3)

With no dose, the target must sit exactly at its basal level RbR_b for all time – this is what ties target(0) <- rbase * vc to ksyn <- kdeg * rbase.

ss <- vapply(mods, function(n) {
  s  <- rxode2::rxSolve(ui[[n]], rxode2::et(seq(0, 30, length.out = 200)),
                        returnType = "data.frame", useLinCmt = FALSE)
  max(abs(s$freeTarget - exp(ui[[n]]$theta[["lrbase"]])))
}, numeric(1))
stopifnot(max(ss) < 1e-8)
knitr::kable(tibble::tibble(Model = mods, `Max |R - Rb| (nM)` = ss),
             caption = "Drug-free steady-state hold. Exact by construction.")
Drug-free steady-state hold. Exact by construction.
Model Max |R - Rb| (nM)
Straube_2025_omalizumab_1cmt 0
Straube_2025_omalizumab_2cmt 0
Straube_2025_caplacizumab_1cmt 0
Straube_2025_caplacizumab_2cmt 0
Straube_2025_linagliptin_1cmt 0
Straube_2025_linagliptin_2cmt 0

Sub-saturating high-affinity kinetics (Equation 12)

For a high-affinity drug dosed below the target level (DT(0)≤RbD_T(0) \le R_b), Equation (12) predicts that total drug decays as a single exponential with half-life ln⁡(2)/keDR\ln(2)/k_{eDR}, and hence that AUC0−∞=(D0/Vc)/keDRAUC_{0-\infty} = (D_0/V_c)/k_{eDR} and Cmax=D0/VcC_{max} = D_0/V_c. The 0.02 mg/kg caplacizumab dose is exactly this case – and the paper singles it out in the Figure 5 caption (“for 0.02 mg/kg the cross-over time does not exist since the initial drug concentration is lower than the target concentration”).

u <- ui[["Straube_2025_caplacizumab_1cmt"]]
g <- function(p) unname(exp(u$theta[[p]]))
D0  <- doses$amt[doses$treatment == "0.02 mg/kg"]
sub <- solve_one("Straube_2025_caplacizumab_1cmt", D0, "central", "0.02 mg/kg",
                 tmax = 8 * log(2) / g("lkint"), n = 4000)

stopifnot(sub$Cc[1] < g("lrbase"))          # confirms Equation (12) applies at all
w   <- sub$time > 6 * log(2) / g("lkint")   # terminal window, clear of the noise floor
slope <- coef(lm(log(sub$Cc[w]) ~ sub$time[w]))[2]

eq12 <- tibble::tibble(
  Quantity  = c("DT(0) (nM)", "terminal t1/2 (d)"),
  Predicted = c(D0 / g("lvc"), log(2) / g("lkint")),
  Observed  = c(sub$Cc[1], -log(2) / slope)
) |> dplyr::mutate(`% diff` = 100 * (Observed - Predicted) / Predicted)

# Achieved -0.7%; 3 still goes red on a mis-transcribed keDR, Vc or dose.
stopifnot(max(abs(eq12$`% diff`)) < 3)
knitr::kable(eq12, digits = 4,
             caption = "Equation (12) closed form vs the full TMDD solve.")
Equation (12) closed form vs the full TMDD solve.
Quantity Predicted Observed % diff
DT(0) (nM) 11.8750 11.8750 0.0000
terminal t1/2 (d) 0.7312 0.7259 -0.7218

Cross-over time (Equation 15)

Above saturation, Equation (15) predicts the time tct_c at which DTD_T falls to meet RTR_T.

crossover <- lapply(
  list(c("Straube_2025_caplacizumab_1cmt", "8 mg/kg", "3"),
       c("Straube_2025_linagliptin_1cmt",  "10 mg",   "5")),
  function(x) {
    uu <- ui[[x[1]]]; gg <- function(p) unname(exp(uu$theta[[p]]))
    d0 <- doses$amt[doses$treatment == x[2]]
    keD <- gg("lcl") / gg("lvc"); ksyn <- gg("lkdeg") * gg("lrbase")
    tc <- (1 / keD) * log((ksyn / keD + d0 / gg("lvc") - gg("lrbase")) / (ksyn / keD))
    s  <- solve_one(x[1], d0, "central", x[2], tmax = as.numeric(x[3]), n = 20000)
    tibble::tibble(Model = x[1], Dose = x[2],
                   `Eq (15) tc (h)` = tc * 24,
                   `Simulated DT=RT (h)` = s$time[which(s$Cc <= s$totalTarget)[1]] * 24)
  }) |> dplyr::bind_rows() |>
  dplyr::mutate(`% diff` = 100 * (`Simulated DT=RT (h)` - `Eq (15) tc (h)`) / `Eq (15) tc (h)`)

# Eq (15) comes from the high-affinity APPROXIMATION (Eq 13), so a few percent
# of approximation error is expected and is itself the paper's subject; the
# larger caplacizumab residual is the keD >> keR regime the paper flags in
# Figure 3b. 10% still goes red on a transcription error.
stopifnot(max(abs(crossover$`% diff`)) < 10)
knitr::kable(crossover, digits = 3,
             caption = "Equation (15) cross-over time vs the simulated DT = RT crossing.")
Equation (15) cross-over time vs the simulated DT = RT crossing.
Model Dose Eq (15) tc (h) Simulated DT=RT (h) % diff
Straube_2025_caplacizumab_1cmt 8 mg/kg 4.128 3.856 -6.592
Straube_2025_linagliptin_1cmt 10 mg 12.295 12.367 0.584

Target suppression and recovery (Equations 29-31)

The FTBR must be strongly suppressed while the target is saturated and return to unity once drug has washed out. Suppression is measured over the paper’s own figure windows; recovery needs a longer window, because for linagliptin the free target does not approach baseline monotonically (below).

suppression <- ftbr |> dplyr::group_by(drug, treatment) |>
  dplyr::summarise(`FTBR min` = min(ftbr), .groups = "drop")

# Recovery over a full-washout window rather than the figure window.
washout <- dplyr::bind_rows(lapply(seq_len(nrow(doses))[doses$drug != "omalizumab"], function(i) {
  model <- if (doses$drug[i] == "caplacizumab") "Straube_2025_caplacizumab_1cmt"
           else "Straube_2025_linagliptin_1cmt"
  s <- solve_one(model, doses$amt[i], "central", doses$treatment[i], tmax = 30, n = 2000)
  tibble::tibble(treatment = doses$treatment[i],
                 `FTBR max (overshoot)` = max(s$ftbr),
                 `FTBR at 30 d` = dplyr::last(s$ftbr))
}))

rec <- suppression |> dplyr::inner_join(washout, by = "treatment")

# Deterministic solves. Achieved: suppression <= 0.64, |1 - FTBR(30 d)| <= 7.4e-5.
stopifnot(all(rec$`FTBR min` < 0.75),
          all(abs(rec$`FTBR at 30 d` - 1) < 1e-3))
knitr::kable(rec, digits = 5,
             caption = "Target suppression, transient overshoot, and recovery to baseline.")
Target suppression, transient overshoot, and recovery to baseline.
drug treatment FTBR min FTBR max (overshoot) FTBR at 30 d
ALX-0081 (caplacizumab) 0.02 mg/kg 0.63970 1.00000 0.99997
ALX-0081 (caplacizumab) 0.4 mg/kg 0.00004 1.00000 0.99993
ALX-0081 (caplacizumab) 8 mg/kg 0.00000 1.00000 0.99993
Linagliptin 0.5 mg 0.08030 1.04723 1.00000
Linagliptin 10 mg 0.00264 1.10839 1.00000
Linagliptin 2.5 mg 0.01145 1.08051 1.00000

For linagliptin the FTBR overshoots baseline by up to 11% before settling. This is the expected consequence of Tacc=keR/keDR≈87≫1T_{acc} = k_{eR}/k_{eDR} \approx 87 \gg 1: by Equation (32) the total target accumulates towards TaccRbT_{acc}R_b while drug is present, so as the complex is cleared the liberated free target transiently exceeds its basal level. The paper’s Figure 5c is drawn on a 0-25 h window with the y-axis capped at 1, so it shows only the rise. Caplacizumab, with Tacc=0.334<1T_{acc} = 0.334 < 1, shows no overshoot – its total target is depleted rather than accumulated, and the FTBR approaches 1 from below.

Affinity classification (Equation 33)

Equation (33) classifies a drug as high-affinity when KM≪min⁡(TaccRb,Rb)K_M \ll \min(T_{acc}R_b, R_b) and low-affinity when KM≫max⁡(TaccRb,Rb)K_M \gg \max(T_{acc}R_b, R_b), with KM=Kd(1+keDR/koff)K_M = K_d(1 + k_{eDR}/k_{off}) (Equation 7). This reproduces the paper’s own assignment of each molecule.

cls <- lapply(mods[c(1, 3, 5)], function(n) {
  gg <- function(p) unname(exp(ui[[n]]$theta[[p]]))
  KM <- gg("lkd") * (1 + gg("lkint") / gg("lk2"))
  Tacc <- gg("lkdeg") / gg("lkint"); Rb <- gg("lrbase")
  tibble::tibble(Model = n, `KM (nM)` = KM, `Tacc*Rb (nM)` = Tacc * Rb, `Rb (nM)` = Rb,
                 `KM / min(Tacc*Rb, Rb)` = KM / min(Tacc * Rb, Rb),
                 `KM / max(Tacc*Rb, Rb)` = KM / max(Tacc * Rb, Rb))
}) |> dplyr::bind_rows()

# The paper states KM/(Tacc*Rb) = 3.3e-4 for ALX-0081 and "about 100-fold
# larger" for linagliptin (section 3.2).
alx <- cls$`KM / min(Tacc*Rb, Rb)`[grepl("caplacizumab", cls$Model)]
lin <- cls$`KM / min(Tacc*Rb, Rb)`[grepl("linagliptin",  cls$Model)]
stopifnot(abs(alx - 3.3e-4) / 3.3e-4 < 0.1,   # paper: 3.3e-4
          lin / alx > 30, lin / alx < 300)     # paper: "about 100-fold larger"
knitr::kable(cls, digits = 5,
             caption = "Equation (33) affinity classification, one-compartment fits.")
Equation (33) affinity classification, one-compartment fits.
Model KM (nM) Tacc*Rb (nM) Rb (nM) KM / min(Tacc*Rb, Rb) KM / max(Tacc*Rb, Rb)
Straube_2025_omalizumab_1cmt 2.41682 6.76625 1.342 1.80091 0.35719
Straube_2025_caplacizumab_1cmt 0.00364 10.96793 32.800 0.00033 0.00011
Straube_2025_linagliptin_1cmt 0.07908 161.11078 1.855 0.04263 0.00049

Omalizumab sits between the two limits – the paper notes that its KMK_M “is in the same order of magnitude as TaccRbT_{acc}R_b”, which is why the low-affinity approximation reproduces it only approximately.

PKNCA validation

NCA on the simulated total-drug profiles for the two intravenously dosed molecules. The observation window is eight terminal half-lives, long enough to capture AUC0−∞AUC_{0-\infty} but short enough that the tail has not decayed into solver noise (which would corrupt the half-life fit).

nca_sims <- lapply(which(doses$drug %in% c("caplacizumab", "linagliptin")), function(i) {
  model <- if (doses$drug[i] == "caplacizumab") "Straube_2025_caplacizumab_1cmt"
           else "Straube_2025_linagliptin_1cmt"
  solve_one(model, doses$amt[i], "central", doses$treatment[i],
            tmax = 8 * log(2) / exp(ui[[model]]$theta[["lkint"]]), n = 4000)
}) |> dplyr::bind_rows()

# Only `!is.na(Cc)` -- a `time > 0` or `Cc > 0` filter would drop the time-zero
# row that anchors AUC0-*.
sim_nca <- nca_sims |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, treatment)
stopifnot(nrow(sim_nca) > 0, !anyNA(sim_nca$Cc), all(sim_nca$Cc >= 0))

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id)
dose_df  <- doses |>
  dplyr::filter(drug %in% c("caplacizumab", "linagliptin")) |>
  dplyr::transmute(id = 1L, time = 0, amt, treatment)
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id)

intervals <- data.frame(start = 0, end = Inf,
                        cmax = TRUE, aucinf.obs = TRUE, half.life = TRUE)
nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))

Comparison against the paper’s closed-form predictions

Straube reports no NCA table, but Equations (12) and (17) are closed-form NCA predictions: for a high-affinity drug the terminal half-life is ln⁡(2)/keDR\ln(2)/k_{eDR} at every dose, and Cmax=D0/VcC_{max} = D_0/V_c for an IV bolus.

reference <- doses |>
  dplyr::filter(drug %in% c("caplacizumab", "linagliptin")) |>
  dplyr::rowwise() |>
  dplyr::mutate(
    .m = if (drug == "caplacizumab") "Straube_2025_caplacizumab_1cmt"
         else "Straube_2025_linagliptin_1cmt",
    cmax      = amt / exp(ui[[.m]]$theta[["lvc"]]),
    half.life = log(2) / exp(ui[[.m]]$theta[["lkint"]])
  ) |>
  dplyr::ungroup() |>
  dplyr::select(treatment, cmax, half.life)

cmp <- nlmixr2lib::ncaComparisonTable(
  simulated = nca_res, reference = reference, by = "treatment",
  units = c(cmax = "nM", half.life = "day"), tolerance_pct = 20
)
knitr::kable(cmp, caption = paste(
  "Simulated NCA vs the paper's closed-form predictions.",
  "* differs from reference by >20%."))
Simulated NCA vs the paper’s closed-form predictions. * differs from reference by >20%.
NCA parameter treatment Reference Simulated % diff
Cmax (nM) 0.02 mg/kg 11.9 11.9 +0.0%
Cmax (nM) 0.4 mg/kg 238 238 +0.0%
Cmax (nM) 8 mg/kg 4750 4750 +0.0%
Cmax (nM) 0.5 mg 10.1 10.1 +0.0%
Cmax (nM) 2.5 mg 50.5 50.5 +0.0%
Cmax (nM) 10 mg 202 202 +0.0%
t½ (day) 0.02 mg/kg 0.731 0.725 -0.8%
t½ (day) 0.4 mg/kg 0.731 0.72 -1.5%
t½ (day) 8 mg/kg 0.731 0.72 -1.5%
t½ (day) 0.5 mg 6.03 1.74 -71.1%*
t½ (day) 2.5 mg 6.03 1.74 -71.1%*
t½ (day) 10 mg 6.03 1.74 -71.1%* 
# ncaComparisonTable's "% diff" column is FORMATTED CHARACTER, so gate on raw
# numbers computed here rather than parsing the rendered table.
obs <- as.data.frame(nca_res) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "half.life")) |>
  dplyr::select(treatment, PPTESTCD, PPORRES) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES) |>
  dplyr::inner_join(reference, by = "treatment", suffix = c("_obs", "_ref")) |>
  dplyr::mutate(cmax_pct = 100 * (cmax_obs - cmax_ref) / cmax_ref,
                thalf_pct = 100 * (half.life_obs - half.life_ref) / half.life_ref,
                drug = ifelse(grepl("mg/kg", treatment), "caplacizumab", "linagliptin"))

# Cmax = D0/Vc is exact for an IV bolus, for both drugs.
stopifnot(max(abs(obs$cmax_pct)) < 0.5)

# The half-life claim holds for caplacizumab (achieved -0.8%) but NOT for
# linagliptin -- see the deviation discussion below. Gate the case that holds
# and keep the other visible in the table above rather than widening to cover it.
stopifnot(max(abs(obs$thalf_pct[obs$drug == "caplacizumab"])) < 3)
knitr::kable(obs |> dplyr::select(treatment, cmax_pct, thalf_pct),
             digits = 2,
             caption = "Percent difference from the closed-form predictions.")
Percent difference from the closed-form predictions.
treatment cmax_pct thalf_pct
0.02 mg/kg 0 -0.80
0.4 mg/kg 0 -1.46
0.5 mg 0 -71.11
10 mg 0 -71.11
2.5 mg 0 -71.12
8 mg/kg 0 -1.48

Known deviation: the linagliptin terminal half-life

For linagliptin the simulated terminal half-life is about 1.74 days against the ln⁡(2)/keDR=6.03\ln(2)/k_{eDR} = 6.03 days that Equation (17) predicts – a -71% deviation, identical across all three doses.

This is not an encoding error; it reproduces a result the paper itself reports. Equation (17) is the high-affinity approximation, which requires binding to be fast relative to drug elimination. Linearising the full binding system (Equation 2) about the basal target level gives a slow eigenvalue that is the true terminal rate constant, and it equals keDRk_{eDR} only in that fast-binding limit:

slow_eigen <- function(n) {
  gg <- function(p) unname(exp(ui[[n]]$theta[[p]]))
  konRb <- (gg("lk2") / gg("lkd")) * gg("lrbase"); keD <- gg("lcl") / gg("lvc")
  A <- matrix(c(-(konRb + keD), gg("lk2"), konRb, -(gg("lk2") + gg("lkint"))), 2, 2, byrow = TRUE)
  max(Re(eigen(A)$values))
}
tibble::tibble(
  Model = mods[c(3, 5)],
  `koff / keD` = vapply(mods[c(3, 5)], function(n)
    exp(ui[[n]]$theta[["lk2"]]) / (exp(ui[[n]]$theta[["lcl"]]) / exp(ui[[n]]$theta[["lvc"]])),
    numeric(1)),
  `t1/2 from slow eigenvalue (d)` = vapply(mods[c(3, 5)],
    function(n) -log(2) / slow_eigen(n), numeric(1)),
  `ln(2)/keDR (d)` = vapply(mods[c(3, 5)],
    function(n) log(2) / exp(ui[[n]]$theta[["lkint"]]), numeric(1)),
  `Simulated (d)` = c(mean(obs$half.life_obs[obs$drug == "caplacizumab"]),
                      mean(obs$half.life_obs[obs$drug == "linagliptin"]))
) |>
  knitr::kable(digits = 4, caption = paste(
    "The simulated terminal half-life tracks the slow eigenvalue of the",
    "linearised binding system, not ln(2)/keDR, whenever koff is not large",
    "relative to keD."))
The simulated terminal half-life tracks the slow eigenvalue of the linearised binding system, not ln(2)/keDR, whenever koff is not large relative to keD.
Model koff / keD t1/2 from slow eigenvalue (d) ln(2)/keDR (d) Simulated (d)
Straube_2025_caplacizumab_1cmt 1.4124 0.7263 0.7312 0.7220
Straube_2025_linagliptin_1cmt 0.1874 1.7409 6.0274 1.7411

Caplacizumab has koff/keD≈1.4k_{off}/k_{eD} \approx 1.4 with an extremely fast on-rate, so the approximation holds; linagliptin has koff/keD≈0.19k_{off}/k_{eD} \approx 0.19, and the paper’s own supplement (Equations S11-S17, Figure S1) analyses exactly this koff≪keDk_{off} \ll k_{eD} regime as the case where “the quasi-steady state assumption leading to Equation (5) is not valid anymore”. Consistently, the main text states that “the approximation for DTD_T (especially the slow terminal elimination phase) is much better for ALX-0081 than for linagliptin” (section 3.2), which is what Figure 5c shows: the dashed Equation (17) curves lie well above the solid full-model curves at late times.

Assumptions and deviations

  • Subcutaneous bioavailability (omalizumab). Supplement Equation (S20) is printed as dDdep/dt=−kaFDdepdD_{dep}/dt = -k_a F D_{dep} with dD/dt=kaFDdep/VcdD/dt = k_a F D_{dep}/V_c. Taken literally, FF is then only a rescaling of the absorption rate constant and the entire dose reaches the central compartment. Simulating that form gives a peak of 143 nM at day 9, against roughly 60 nM at day 4-6 in Figure 5a; encoding FF as an ordinary bioavailable fraction (f(depot) <- exp(lfdepot)) gives 61.6 nM at day 4.25 and reproduces the figure. The models therefore use standard bioavailability, and this is the only place where the encoding departs from an equation as printed.
  • Caplacizumab doses are figure-derived. The paper reports the ALX-0081 doses only as mg/kg and states neither the species nor the body weight; its fitted VcV_c of 0.046 L is not consistent with an adult human, and the upstream data source (Glassman & Muzykantov 2020) is not open access. The nmol doses used here were back-computed from the paper’s own VcV_c and the ~133 ug/mL initial concentration of the 8 mg/kg profile in Figure 5b, then scaled by the mg/kg ratio. This is a figure-derived input, not a paper-tabulated one. It is self-consistent: the resulting 0.02 mg/kg profile starts at 11.9 nM, below the 32.8 nM target level, which is exactly the condition the Figure 5 caption gives for that dose having no cross-over time. No model parameter depends on it.
  • Printed rounding. Several Table 2 / Table S1 columns are rounded more coarsely than the quantities derived from them. keD = CL/Vc computed from the printed CL and Vc differs from the printed keD by 1.9% (omalizumab, Table 2, where keD is printed as 0.0025 for 0.002548), 0.9% (caplacizumab Table 2) and 1.2% (caplacizumab Table S1); kon = koff/Kd differs by up to 1.2%; and Tacc = keR/keDR differs by 1.3% for omalizumab Table S1, where keDR is printed to two significant figures. The primary estimates (CL, Vc, koff, Kd, keR, keDR) are encoded as printed and the derived quantities are computed in model(), so these residuals are visible in the table-consistency check rather than hidden.
  • Target accumulation ratio TaccT_{acc} is not a stored parameter. It is exactly keR/keDRk_{eR}/k_{eDR} (Equation 14) and is recovered as kdeg/kint, so storing it would duplicate an existing degree of freedom. Where the paper fixed TaccT_{acc} (caplacizumab, Table 2 footnote a) and estimated keRk_{eR}, the derived keDRk_{eDR} is encoded un-fixed() because it inherits keRk_{eR}’s uncertainty; the in-file comment records this.
  • Specimen is not stated. The paper does not say whether concentrations are plasma or serum, so every circulating compartmentData entry is marked verified = FALSE with specimen = "plasma" as the assumption.
  • No IIV and no residual error. These are individual (omalizumab) or typical-value (caplacizumab, linagliptin) deterministic fits; the paper reports no variance components, and none were invented.
  • Two patients, one encoded. The omalizumab files carry the Figure 5a / Figure S3a patient. The second patient’s patient-specific values are recorded in each file’s population$notes.