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Capivasertib with fulvestrant (Fernandez Teruel 2025)

Source: vignettes/articles/FernandezTeruel_2025_capivasertib.Rmd
FernandezTeruel_2025_capivasertib.Rmd

Model and source 

#> ℹ parameter labels from comments will be replaced by 'label()'
  • Citation: Fernandez Teruel C, Cullberg M, Gonzalez-Garcia I, Schiavon G, Zhang L, Zhou D. Population pharmacokinetics and exposure-response analyses for capivasertib in combination with fulvestrant in patients with breast cancer. Clin Transl Sci. 2025;18:e70286. doi:10.1111/cts.70286
  • Description: Three-compartment population PK model for capivasertib (oral pan-AKT inhibitor) with parallel first-order and zero-order absorption, absorption lag time, and sigmoidal time- and dose-dependent auto-inhibition of apparent clearance, with power effects of body weight and age on CL0/F, in patients with advanced solid tumours and HR-positive/HER2-negative advanced breast cancer receiving capivasertib plus fulvestrant (Fernandez Teruel 2025)
  • Article: https://doi.org/10.1111/cts.70286

Capivasertib is a first-in-class, selective inhibitor of all three AKT isoforms. Fernandez Teruel 2025 updates the four-study, 441-patient population PK model of FernandezTeruel_2024_capivasertib by adding the Phase III CAPItello-291 trial and a Chinese PK study, giving 5960 analysed concentrations from 851 patients across six Phase I-III trials. The updated model was then used to generate individual steady-state exposure metrics for exposure-response analyses of efficacy and safety.

Both capivasertib models are shipped. They are the same structural model fitted to different datasets, so they are separate files rather than one:

The two shipped capivasertib population PK models.
Model Vignette
FernandezTeruel_2024_capivasertib FernandezTeruel_2024_capivasertib
FernandezTeruel_2025_capivasertib FernandezTeruel_2025_capivasertib

Three things changed in the structural model between the two fits, and each is a change of form, not just of value:

  1. Body weight on CL0/F became a power term. The 2024 model used a linear deviation, CL0/F * (1 + (WT - 67) * 0.00585). The 2025 model uses CL0/F * (WT/67)^0.302. Reading the 2025 coefficient 0.302 as a per-kg linear term would be absurd, so the printed equation is load-bearing here.
  2. Age entered the model, also as a power term, (AGE/57)^-0.314. Age was not a covariate in the 2024 fit.
  3. The Ka and D2 inter-individual variabilities were dropped. The 2024 model held both at a fixed 15% CV; the 2025 Table 2 reports no IIV for either, so the 2025 model carries five etas rather than seven.

Population

Study population (Fernandez Teruel 2025 Table 1 and Table S1).
Field Value
species human
n_subjects 851
n_studies 6
n_observations 5960
age_range 26-87 years
age_median 57 years
weight_range 32-150 kg
weight_median 65 kg
sex_female_pct 88.8
race_ethnicity White 60.9; Black 1.4; Asian 27.8; American Indian or Alaska Native 1.8; Native Hawaiian or Other Pacific Islander 0.1; Other 7.5; Missing 0.5
disease_state Advanced solid malignancies, and HR-positive / HER2-negative locally advanced or metastatic breast cancer resistant to aromatase inhibitors
dose_range 80-800 mg orally twice daily, most commonly 400 mg (63.8%) or 480 mg (22.4%); continuous dosing (8.0%) or one of two intermittent schedules, 4 days on / 3 days off (4/3, 86.4%) or 2 days on / 5 days off (2/5, 5.6%). The Phase III regimen is 400 mg twice daily [4/3] with fulvestrant 500 mg.
renal_function Normal 54.4%, mild impairment 34.3%, moderate impairment 10.6%, no severe impairment
hepatic_function Normal 64.0%, mild impairment 35.0%, moderate impairment 0.8%, no severe impairment
co_medication Fulvestrant 55.0%, paclitaxel 10.6%, acid-reducing agent 25.4%
regions Global; China 11.0%, Asia excluding China 15.3%, rest of world 73.7%
notes Pooled from six Phase I-III studies: Study 1, BEECH, Study 4 (all-Japanese), OAK, a China PK study, and the Phase III CAPItello-291 trial. This analysis UPDATES the 441-patient, four-study model of FernandezTeruel_2024_capivasertib.R by adding CAPItello-291 and the China PK study (Sect. 2.3). 6630 concentrations were collected and 5960 (89.9%) were analysed; 670 were excluded, mainly 436 (6.6%) drawn before treatment start, plus 220 (3.3%) below the 1.00 ng/mL limit of quantification and 9 (0.1%) unexpectedly high. Demographics are Table S1. The exposure-response analyses used subsets of this cohort: 798 patients for efficacy (Table S2) and 468 for safety (Table S3).

The PopPK dataset was 6630 collected concentrations from 851 patients, of which 5960 (89.9%) were analysed. Median age was 57 years (range 26-87), median body weight 65 kg (range 32-150), and 88.8% of patients were women. 11.0% were from mainland China and Taiwan, 15.3% from elsewhere in Asia, and 73.7% from the rest of the world (Table S1). The exposure-response analyses used subsets of this cohort: 798 patients for efficacy (Table S2) and 468 for safety (Table S3).

Source trace

Every value in ini() and every equation in model() comes from the locations below. The model equations are printed as display equations on p. 7 of the article, immediately after Table 2; the PDF’s text layer renders them only under pdftotext -layout (the publisher-supplied structured text drops them entirely), which is why they are quoted in full here.

Source trace for every parameter and equation.
Item Source location
Ka, CL0/F, V2/F, V3/F, Q3/F, V4/F, Q4/F, D2, LogitF1, Lag1_tab, Lag1_cap, Imax, T50 Table 2 (estimate column)
Imax_dose, CL0_BBW, CL0_AGE, F1_BBW Table 2 (estimate column)
IIV on CL0/F, V2/F, Imax, Lag1, LogitF1 (as CV %) Table 2 (last five rows)
Residual error, proportional 43% and additive 1.12 ug/L Table 2
F1i = expit(LogitF1 * (BBW/67)^F1_BBW + eta) p. 7 display equation 1
F2i = 1 - F1i p. 7 display equation 2
ALAG1i = (Lag1_tab(1-CAP) + Lag1_capCAP) * (1 - FASTED) * exp(eta) p. 7 display equation 3
CL0/Fi = CL0/F * (BBW/67)^CL0_BBW * (AGE/57)^CL0_AGE * exp(eta) p. 7 display equation 4
Imax_i = Imax * (1 + (DOSE-480)Imax_dose) (1 + PACLImax_pacl) exp(eta) p. 7 display equation 5
CL/Fi = CL0/Fi * (1 - exp(Imax_i) * Time^5 / (T50^5 + Time^5)) p. 7 display equation 6
Hill exponent fixed at 5 p. 7 display equation 6 (literal Time^5, T50^5)
Vss = 265 L; CL/F falls 11.2% at 400 mg and 29.2% at 640 mg Sect. 3.2 (used as gates, not as inputs)
Accumulation ratio 1.47, effective half-life 7.25 h at 400 mg [4/3] Sect. 3.2 (gates)
Median steady-state AUC 7650 ug*h/L and Cmax 1460 ug/L Sect. 3.2 (gates)
Exposure window 252-264 h (last dosing day of week 2) Sect. 2.3
Imax_pacl NOT REPORTED ANYWHERE – see Assumptions and deviations

Simulation helpers 

mod <- readModelDb("FernandezTeruel_2025_capivasertib")
tv  <- rxode2::zeroRe(mod)   # typical-value (all etas set to zero) version
#> ℹ parameter labels from comments will be replaced by 'label()'

# Dosing-time grid. Capivasertib is twice daily (q12h) within a dosing day; the
# Phase III schedule is 4 days on / 3 days off.
dose_times <- function(schedule, n_weeks) {
  days <- switch(
    schedule,
    "continuous" = seq_len(n_weeks * 7L) - 1L,
    "4/3"        = unlist(lapply(seq_len(n_weeks) - 1L, function(w) w * 7L + 0:3)),
    "2/5"        = unlist(lapply(seq_len(n_weeks) - 1L, function(w) w * 7L + 0:1))
  )
  sort(c(days * 24, days * 24 + 12))
}

# Each administration is TWO dose records carrying the same amt: one into
# `depot` (scaled by f(depot) = ffo, lagged by alag(depot)) and one into
# `central` (scaled by f(central) = 1 - ffo, delivered as a zero-order input
# whose duration D2 is modelled, hence rate = -2). Observation rows point at
# the ODE state `central`, never at the algebraic observable `Cc`.
make_events <- function(ids, dose, dosing_times, obs_times, WT = 65, AGE = 57,
                        FORM_CAPSULE = 0, FASTED_STRICT = 0,
                        CONMED_PACLITAXEL = 0, treatment = "") {
  subj <- data.frame(id = ids, WT = WT, AGE = AGE)
  dos <- merge(subj, data.frame(time = dosing_times))
  dos <- rbind(
    transform(dos, amt = dose, evid = 1L, cmt = "depot",   rate = 0),
    transform(dos, amt = dose, evid = 1L, cmt = "central", rate = -2)
  )
  obs <- transform(merge(subj, data.frame(time = obs_times)),
                   amt = NA_real_, evid = 0L, cmt = "central", rate = 0)
  ev <- rbind(dos, obs)
  ev$FORM_CAPSULE <- FORM_CAPSULE
  ev$FASTED_STRICT <- FASTED_STRICT
  ev$CONMED_PACLITAXEL <- CONMED_PACLITAXEL
  ev$DOSE_CAPIVASERTIB_MG <- dose
  ev$treatment <- treatment
  ev[order(ev$id, ev$time, -ev$evid), ]
}

# Linear-trapezoidal AUC of Cc over [a, b] for a single-subject solve.
auc_window <- function(sim, a, b) {
  d <- sim[sim$time >= a & sim$time <= b, ]
  d <- d[order(d$time), ]
  sum(diff(d$time) * (head(d$Cc, -1) + tail(d$Cc, -1)) / 2)
}

# Fernandez Teruel 2025 Sect. 2.3: exposure metrics were projected for "the
# final dosing day during the second week of treatment (252-264 h)". On the
# [4/3] schedule, week 2 dosing days are days 7-10, so 252 h is the second dose
# of day 10 -- the last dose of that week.
ss_start <- 252
ss_end   <- 264

tv_profile <- function(dose = 400, WT = 65, AGE = 57, FORM_CAPSULE = 0,
                       FASTED_STRICT = 0, CONMED_PACLITAXEL = 0) {
  obs <- sort(unique(c(seq(0, 12, by = 0.02), seq(ss_start, ss_end, by = 0.02))))
  ev <- make_events(1L, dose, dose_times("4/3", 3L), obs, WT = WT, AGE = AGE,
                    FORM_CAPSULE = FORM_CAPSULE, FASTED_STRICT = FASTED_STRICT,
                    CONMED_PACLITAXEL = CONMED_PACLITAXEL)
  s <- as.data.frame(rxode2::rxSolve(tv, ev, addDosing = FALSE))
  ss <- s[s$time >= ss_start, ]
  fd <- s[s$time <= 12, ]
  list(
    auc12_ss    = auc_window(s, ss_start, ss_end),
    cmax_ss     = max(ss$Cc),
    cmin_ss     = min(ss$Cc),
    tmax_ss     = ss$time[which.max(ss$Cc)] - ss_start,
    auc12_dose1 = auc_window(s, 0, 12),
    cmax_dose1  = max(fd$Cc)
  )
}

# Collect claim / published / reproduced rows so the gate and the rendered
# table are driven by exactly the same numbers. `acc` is an environment
# (reference semantics) so `claim()` can append without superassignment.
acc <- new.env(parent = emptyenv())
acc$rows <- list()
claim <- function(what, published, reproduced, tolerance, unit = "",
                  deviation = FALSE) {
  acc$rows[[length(acc$rows) + 1L]] <- tibble::tibble(
    Quantity = what, Unit = unit, Published = published,
    Reproduced = reproduced, Tolerance = tolerance,
    Pass = abs(reproduced - published) <= tolerance, Deviation = deviation
  )
  invisible(reproduced)
}

Internal identities

These are deterministic functions of the ini() values – no simulated cohort is involved – so they are gated tightly. Each is a quantity Fernandez Teruel 2025 states in prose but did not put in Table 2, which makes them independent tests of the transcription rather than restatements of it.

p <- setNames(ui$theta, names(ui$theta))

# 1. Apparent volume of distribution at steady state.
Vss <- exp(p[["lvc"]]) + exp(p[["lvp"]]) + exp(p[["lvp2"]])
claim("Apparent volume of distribution at steady state (V2 + V3 + V4)",
      265, Vss, 0.5, "L")

# 2. The maximal fractional reduction of CL/F at two dose levels. This is the
#    check that pins the Imax parameterisation: `Imax` is the LOG of the
#    maximal fractional inhibition, so the reduction is exp(Imax_i), not Imax_i.
max_inhibition <- function(dose, CONMED_PACLITAXEL = 0) {
  imax_i <- p[["lcl_time_max"]] *
    (1 + (dose - 480) * p[["e_dose_cl_time_max"]]) *
    (1 + CONMED_PACLITAXEL * p[["e_pacl_cl_time_max"]])
  100 * exp(imax_i)
}
claim("Maximal reduction in CL/F after multiple doses at 400 mg",
      11.2, max_inhibition(400), 0.1, "%")
claim("Maximal reduction in CL/F after multiple doses at 640 mg",
      29.2, max_inhibition(640), 0.2, "%")

# 3. Fraction absorbed by the first-order route at the 67 kg normalisation
#    weight. The paper does not state this directly, but expit(1.6) is a
#    one-line consequence of Table 2 and confirms the logit encoding.
ffo_ref <- 1 / (1 + exp(-p[["logitffo"]]))
claim("Fraction of the dose absorbed by the first-order route at 67 kg",
      0.832, ffo_ref, 0.001, "fraction")

CL/F is time-dependent and reaches a plateau by day 7 

clf_at <- function(time, dose = 400, WT = 65, AGE = 57, CONMED_PACLITAXEL = 0) {
  imax_i <- p[["lcl_time_max"]] *
    (1 + (dose - 480) * p[["e_dose_cl_time_max"]]) *
    (1 + CONMED_PACLITAXEL * p[["e_pacl_cl_time_max"]])
  hill <- exp(p[["lcl_time_hill"]]); t50 <- exp(p[["lcl_t50"]])
  exp(p[["lcl"]]) * (WT / 67)^p[["e_wt_cl"]] * (AGE / 57)^p[["e_age_cl"]] *
    (1 - exp(imax_i) * time^hill / (t50^hill + time^hill))
}

clf_grid <- tidyr::expand_grid(time = seq(0, 336, by = 2),
                               dose = c(400, 480, 640)) |>
  dplyr::mutate(CL = clf_at(time, dose = dose),
                Dose = factor(paste(dose, "mg")))

ggplot(clf_grid, aes(time, CL, colour = Dose)) +
  geom_line(linewidth = 0.8) +
  geom_vline(xintercept = 168, linetype = "dashed") +
  geom_vline(xintercept = exp(p[["lcl_t50"]]), linetype = "dotted") +
  labs(x = "Time since first dose (h)", y = "CL/F (L/h)",
       caption = "Dashed = 168 h (day 7); dotted = T50 = 126 h") +
  theme_bw()
Time course of apparent clearance at three dose levels. The paper states the plateau is reached by roughly 168 h (day 7) at 400 mg.

Time course of apparent clearance at three dose levels. The paper states the plateau is reached by roughly 168 h (day 7) at 400 mg. 


# How complete is the decline at the day-7 "plateau"? This is a pure function
# of T50 and the Hill exponent: extent(t) = t^5 / (T50^5 + t^5).
extent_at <- function(time) (clf_at(0) - clf_at(time)) / (clf_at(0) - clf_at(1e7))
t_to_extent <- function(target) exp(p[["lcl_t50"]]) * (target / (1 - target))^(1 / 5)

tibble::tibble(
  Quantity = c("T50 (Table 2)", "Extent of CL/F decline at 168 h (day 7)",
               "Time to 90% of the decline", "Time to 95% of the decline"),
  Value = c(round(exp(p[["lcl_t50"]]), 1), round(extent_at(168), 3),
            round(t_to_extent(0.90), 1), round(t_to_extent(0.95), 1)),
  Unit = c("h", "fraction", "h", "h")
) |>
  knitr::kable(caption = "Approach to the CL/F plateau.")
Approach to the CL/F plateau.
Quantity Value Unit
T50 (Table 2) 126.000 h
Extent of CL/F decline at 168 h (day 7) 0.808 fraction
Time to 90% of the decline 195.500 h
Time to 95% of the decline 227.000 h 

# Sect. 3.2 says the plateau "would be achieved after approximately 168 h
# (Day 7)". Taken literally that is not quite true -- with T50 = 126 h and a
# Hill of 5 the decline is 81% complete at 168 h and needs about 215 h to reach
# 95%. It is recorded as a deviation rather than gated, because the paper's
# statement is a qualitative reading of where the curve visibly flattens.
claim("Extent of the CL/F decline completed at 168 h ('plateau' per Sect. 3.2)",
      1.0, extent_at(168), 0.05, "fraction", deviation = TRUE)

# The real gate: the shape is deterministic, so a mis-encoded T50 or Hill
# exponent moves this immediately. (Hill = 1 would give 0.57; T50 = 67 h, the
# 2024 model's value, would give 0.95.)
stopifnot(extent_at(168) > 0.75, extent_at(168) < 0.88)

Steady-state exposure reproduces the published reference values

Fernandez Teruel 2025 Sect. 3.2 defines a reference patient – female, 65 kg, 57 years, 400 mg twice daily as a tablet, semi-fasted, no paclitaxel – and reports a median steady-state AUC of 7650 ug*h/L and Cmax of 1460 ug/L. That combination is exactly the model’s reference covariate set, so the typical-value prediction is directly comparable.

There is also a closed-form route to the same number that uses no ODE solver at all. Once CL/F has plateaued, a twice-daily regimen at steady state satisfies AUC(0-12) = Dose / CL_ss, and CL_ss = CL0/F * (1 - exp(Imax_i)). Agreement between the closed form and the solved ODE tests the solver, the dual-input routing and the bioavailability split simultaneously.

ref <- tv_profile(dose = 400, WT = 65, AGE = 57)
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalcl_time_max', 'etaltlag', 'etalogitffo'

cl_ss_closed  <- clf_at(1e7, dose = 400, WT = 65, AGE = 57)
auc_closed    <- 400 / cl_ss_closed * 1000   # mg/(L/h) -> ug*h/L

claim("Steady-state AUC over the 12 h dosing interval (252-264 h), solved",
      7650, ref$auc12_ss, 400, "ug*h/L")
claim("Steady-state AUC, closed form Dose / CL_ss (no ODE solver)",
      7650, auc_closed, 400, "ug*h/L")
claim("Steady-state Cmax (252-264 h)",
      1460, ref$cmax_ss, 150, "ug/L")

# The closed form and the solve must agree far more tightly with each other
# than either does with the paper: both are deterministic here.
stopifnot(abs(ref$auc12_ss - auc_closed) / auc_closed < 0.03)

Accumulation ratio and effective half-life

The paper reports an accumulation ratio of 1.47 and an effective half-life of 7.25 h for 400 mg twice daily [4/3]. These two are not independent: for a twice-daily regimen, R = 1 / (1 - exp(-k * 12)), so an accumulation ratio of 1.47 implies an effective half-life of 7.30 h. Reproducing both therefore also checks that the paper’s own two numbers are mutually consistent.

acc_ratio <- ref$auc12_ss / ref$auc12_dose1
claim("Accumulation ratio (AUC12 at steady state / AUC12 after the first dose)",
      1.47, acc_ratio, 0.25, "ratio")

# Effective half-life implied by the published accumulation ratio:
# R = 1 / (1 - exp(-k*tau))  =>  k = -log(1 - 1/R) / tau  =>  t_half = log(2)/k.
t_half_from_R <- function(R, tau = 12) -tau * log(2) / log(1 - 1 / R)
claim("Effective half-life implied by the published accumulation ratio 1.47",
      7.25, t_half_from_R(1.47), 0.1, "h")

Replicating Figure 2a: the covariate forest plot

Figure 2a is a forest plot of median steady-state ratios relative to a reference patient (65 kg, 57 years, 400 mg), and it prints the exact covariate values it uses – the 5th, 25th, 50th, 75th and 95th percentiles of body weight (46/57/65/76/98 kg) and of age (38/49/57/65/73 years), and six dose levels. That makes it directly reproducible, and a far sharper test than the prose summary.

Every ratio here has a closed form, because once CL/F has plateaued AUC(0-12) = Dose / CL_ss and the covariate terms enter CL_ss multiplicatively. The dose ratios are the most demanding of the three panels: dose enters twice, once as the numerator of the AUC and once through the depth of the auto-inhibition, so reproducing them tests the Imax and Imax_dose encoding jointly.

# AUC12,ss relative to the reference patient, closed form.
auc_ratio <- function(dose = 400, WT = 65, AGE = 57) {
  (dose / clf_at(1e7, dose = dose, WT = WT, AGE = AGE)) /
    (400 / clf_at(1e7, dose = 400, WT = 65, AGE = 57))
}

fig2a <- dplyr::bind_rows(
  tibble::tibble(Panel = "Body weight (kg)", Level = c(46, 57, 65, 76, 98),
                 Percentile = c("5th", "25th", "50th", "75th", "95th"),
                 Model = vapply(c(46, 57, 65, 76, 98), \(w) auc_ratio(WT = w), numeric(1)),
                 `Figure 2a` = c(1.10, 1.04, 1.00, 0.97, 0.90)),
  tibble::tibble(Panel = "Age (years)", Level = c(38, 49, 57, 65, 73),
                 Percentile = c("5th", "25th", "50th", "75th", "95th"),
                 Model = vapply(c(38, 49, 57, 65, 73), \(a) auc_ratio(AGE = a), numeric(1)),
                 `Figure 2a` = c(0.90, 0.96, 1.00, 1.04, 1.07)),
  tibble::tibble(Panel = "Dose (mg)", Level = c(80, 240, 320, 400, 480, 640),
                 Percentile = NA_character_,
                 Model = vapply(c(80, 240, 320, 400, 480, 640), \(d) auc_ratio(dose = d), numeric(1)),
                 `Figure 2a` = c(0.18, 0.57, 0.78, 1.00, 1.25, 2.00))
) |>
  dplyr::mutate(Difference = Model - `Figure 2a`)

fig2a |>
  dplyr::mutate(dplyr::across(c(Model, `Figure 2a`, Difference), \(x) round(x, 3))) |>
  knitr::kable(caption = "AUC12h,ss median ratio to reference: model vs values read from Figure 2a.")
AUC12h,ss median ratio to reference: model vs values read from Figure 2a.
Panel Level Percentile Model Figure 2a Difference
Body weight (kg) 46 5th 1.110 1.10 0.010
Body weight (kg) 57 25th 1.040 1.04 0.000
Body weight (kg) 65 50th 1.000 1.00 0.000
Body weight (kg) 76 75th 0.954 0.97 -0.016
Body weight (kg) 98 95th 0.883 0.90 -0.017
Age (years) 38 5th 0.880 0.90 -0.020
Age (years) 49 25th 0.954 0.96 -0.006
Age (years) 57 50th 1.000 1.00 0.000
Age (years) 65 75th 1.042 1.04 0.002
Age (years) 73 95th 1.081 1.07 0.011
Dose (mg) 80 NA 0.183 0.18 0.003
Dose (mg) 240 NA 0.566 0.57 -0.004
Dose (mg) 320 NA 0.773 0.78 -0.007
Dose (mg) 400 NA 1.000 1.00 0.000
Dose (mg) 480 NA 1.260 1.25 0.010
Dose (mg) 640 NA 2.005 2.00 0.005 

ggplot(fig2a, aes(Model, `Figure 2a`)) +
  geom_abline(slope = 1, intercept = 0, linetype = "dashed") +
  geom_point(aes(colour = Panel), size = 2.5) +
  scale_x_log10() + scale_y_log10() +
  labs(x = "Model AUC ratio to reference", y = "Figure 2a AUC ratio to reference",
       title = "Replicates Figure 2a of Fernandez Teruel 2025") +
  theme_bw()


# The Figure 2a values are read off a printed forest plot, so the tolerance
# reflects reading precision, not model error. Every point is deterministic --
# no simulated cohort is involved -- so this gate cannot flicker across
# machines. A mis-encoded Imax or a linear-instead-of-power weight term moves
# the dose panel by a factor of two.
stopifnot(max(abs(fig2a$Difference)) < 0.06)
claim("Largest deviation from the Figure 2a AUC ratios (16 points)",
      0, max(abs(fig2a$Difference)), 0.06, "ratio")

# Cross-check the closed form against a full ODE solve at two corners.
stopifnot(
  abs(tv_profile(WT = 46)$auc12_ss / ref$auc12_ss - auc_ratio(WT = 46)) < 0.03,
  abs(tv_profile(dose = 640)$auc12_ss / ref$auc12_ss - auc_ratio(dose = 640)) < 0.05
)
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalcl_time_max', 'etaltlag', 'etalogitffo'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalcl_time_max', 'etaltlag', 'etalogitffo'

How large is “less than 20%”?

Sect. 3.2 summarises the two demographic panels as causing “< 20% difference in capivasertib exposure” between the 5th and 95th percentiles. Computed exactly from the model, the spans are marginally larger than that:

span <- function(panel) {
  r <- fig2a$Model[fig2a$Panel == panel]
  100 * (max(r) - min(r))
}
tibble::tibble(
  Covariate = c("Body weight, 46-98 kg", "Age, 38-73 years"),
  `Span in AUC12,ss (%)` = round(c(span("Body weight (kg)"), span("Age (years)")), 1),
  `Paper's prose claim` = "< 20%"
) |>
  knitr::kable(caption = "Spread in steady-state exposure across the 5th-95th percentile range.")
Spread in steady-state exposure across the 5th-95th percentile range.
Covariate Span in AUC12,ss (%) Paper’s prose claim
Body weight, 46-98 kg 22.7 < 20%
Age, 38-73 years 20.0 < 20% 

# Recorded as a DEVIATION, not gated: the paper's "< 20%" is a rounded prose
# summary, and its own Figure 2a shows body-weight points at roughly 1.10 and
# 0.90, i.e. a 20-point span, consistent with what the model gives. The
# quantity is deterministic, so this is a stable, reproducible disagreement
# rather than simulation noise -- it is reported rather than tuned away.
claim("Body-weight span in AUC12,ss, 5th-95th percentile", 20, span("Body weight (kg)"),
      1, "%", deviation = TRUE)
claim("Age span in AUC12,ss, 5th-95th percentile", 20, span("Age (years)"),
      1, "%", deviation = TRUE)

# Both must still be small enough to support the paper's clinical conclusion
# that no dose adjustment is warranted.
stopifnot(span("Body weight (kg)") < 30, span("Age (years)") < 30)

Virtual cohort and PKNCA validation

A 150-patient cohort is drawn from the Table S1 demographics (median weight 65 kg, range 32-150; median age 57 years, range 26-87) and dosed at 400 mg twice daily [4/3] as a tablet, semi-fasted – the CAPItello-291 regimen. NCA is run on the same 252-264 h interval the paper used, with time re-based so the interval starts at zero.

rxode2::rxSetSeed(20250908)
set.seed(20250908)
n_sub <- 150L

cohort <- tibble::tibble(
  id  = seq_len(n_sub),
  WT  = pmin(pmax(round(exp(rnorm(n_sub, log(65), 0.24))), 32), 150),
  AGE = pmin(pmax(round(rnorm(n_sub, 57, 11)), 26), 87)
)

obs_ss <- seq(ss_start, ss_end, by = 0.25)
ev_cohort <- make_events(cohort$id, 400, dose_times("4/3", 3L), obs_ss,
                         WT = cohort$WT, AGE = cohort$AGE,
                         treatment = "400 mg BD [4/3]")
# make_events() merges on a per-id frame, so WT / AGE must be re-attached by id
# rather than recycled positionally.
ev_cohort <- ev_cohort |>
  dplyr::select(-WT, -AGE) |>
  dplyr::left_join(cohort, by = "id")

sim <- rxode2::rxSolve(mod, ev_cohort, addDosing = FALSE,
                       keep = c("WT", "AGE", "treatment")) |>
  as.data.frame()
#> ℹ parameter labels from comments will be replaced by 'label()'
if (is.null(sim$id)) sim$id <- 1L

# Re-base time so the dosing interval starts at 0, which PKNCA requires for
# auc.last, and keep the record exactly at t = 12 so ctrough is defined.
conc_df <- sim |>
  dplyr::mutate(time = time - ss_start) |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, WT, AGE, treatment)

stopifnot(nrow(conc_df) > 0, all(conc_df$Cc >= 0),
          any(abs(conc_df$time - 0) < 1e-9),
          any(abs(conc_df$time - 12) < 1e-9))

dose_df <- conc_df |>
  dplyr::distinct(id, treatment) |>
  dplyr::mutate(time = 0, amt = 400)

conc_obj <- PKNCA::PKNCAconc(conc_df, Cc ~ time | treatment + id,
                             concu = "ng/mL", timeu = "h")
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id, doseu = "mg")
intervals <- data.frame(start = 0, end = 12,
                        auclast = TRUE, cmax = TRUE, tmax = TRUE, cmin = TRUE)
res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))

nca <- as.data.frame(res) |>
  dplyr::filter(PPTESTCD %in% c("auclast", "cmax", "tmax", "cmin"))
stopifnot(nrow(nca) > 0, !anyNA(nca$PPORRES))

nca_summary <- nca |>
  dplyr::group_by(PPTESTCD) |>
  dplyr::summarise(Median = median(PPORRES),
                   `5th pct` = quantile(PPORRES, 0.05),
                   `95th pct` = quantile(PPORRES, 0.95), .groups = "drop")
nca_summary |>
  dplyr::mutate(dplyr::across(where(is.numeric), \(x) signif(x, 3))) |>
  dplyr::rename("NCA parameter" = PPTESTCD) |>
  knitr::kable(caption = "PKNCA results over the 252-264 h dosing interval, 150 simulated patients.")
PKNCA results over the 252-264 h dosing interval, 150 simulated patients.
NCA parameter Median 5th pct 95th pct
auclast 7730.00 3950.00 20300.00
cmax 1300.00 693.00 2850.00
cmin 246.00 93.30 1070.00
tmax 1.75 0.75 3.25

Comparison against the published exposure metrics 

med <- function(k) nca_summary$Median[nca_summary$PPTESTCD == k]
stopifnot(length(med("auclast")) == 1L, length(med("cmax")) == 1L)

comparison <- tibble::tibble(
  `NCA parameter` = c("AUC0-12 (ug*h/L)", "Cmax (ug/L)"),
  `Published median` = c(7650, 1460),
  `Simulated median`  = signif(c(med("auclast"), med("cmax")), 4)
) |>
  dplyr::mutate(`% difference` = round(100 * (`Simulated median` - `Published median`) /
                                         `Published median`, 1))
knitr::kable(comparison, caption = "Simulated cohort medians against the published reference values.")
Simulated cohort medians against the published reference values.
NCA parameter Published median Simulated median % difference
AUC0-12 (ug*h/L) 7650 7734 1.1
Cmax (ug/L) 1460 1297 -11.2 

# Cohort medians, so the bound is on the CENTRE and is set with headroom rather
# than at the accuracy of one draw. The published values are for a reference
# patient at exactly 65 kg / 57 y while the cohort spans the full demographic
# range, so a modest offset is expected. A mis-transcribed dose, clearance or
# unit moves these by tens of percent.
stopifnot(all(abs(comparison$`% difference`) < 25))
band <- conc_df |>
  dplyr::group_by(time) |>
  dplyr::summarise(med = median(Cc), lo = quantile(Cc, 0.05),
                   hi = quantile(Cc, 0.95), .groups = "drop")

ggplot(band, aes(time)) +
  geom_ribbon(aes(ymin = lo, ymax = hi), alpha = 0.2) +
  geom_line(aes(y = med), linewidth = 0.9) +
  labs(x = "Time since the 252 h dose (h)", y = "Capivasertib (ug/L)") +
  theme_bw()
Simulated steady-state concentration-time profiles over the 252-264 h dosing interval: median and 5th-95th percentiles across 150 patients.

Simulated steady-state concentration-time profiles over the 252-264 h dosing interval: median and 5th-95th percentiles across 150 patients.

Exposure-response

The exposure-response half of this paper contributes no shipped model, for two different reasons that are worth separating.

Efficacy: a reported null

The exposure-efficacy analysis is a genuine null result, not a reporting gap. Neither progression-free survival nor objective response rate was related to any capivasertib exposure metric, and the supplement publishes the full coefficient tables that establish it. Every hazard ratio is centred on 1:

Univariate PFS Cox regression by exposure metric (Table S4). No metric is significant.
Metric Univariate estimate Univariate HR (95% CI) Wald p
AUC 0.01070 1.01 (0.987, 1.04) 0.382
Cmax 0.01430 1.01 (0.791, 1.30) 0.910
Cmin 0.20100 1.22 (0.807, 1.85) 0.343
AAUC 0.00751 1.01 (0.976, 1.04) 0.646
ACmax -0.01570 0.984 (0.756, 1.28) 0.907
ACmin 0.18500 1.20 (0.681, 2.13) 0.524

The final PFS Cox model retained only liver metastasis (HR 2.05, 95% CI 1.62 to 2.60) and prior CDK4/6-inhibitor use (HR 1.70, 95% CI 1.32 to 2.19) – both baseline prognostic factors, neither exposure-driven. There is therefore no exposure-response model to encode.

Safety: significant, but with no published coefficients

The exposure-safety analysis is the opposite case. Three of ten endpoints showed a significant relationship with steady-state AUC – adverse event leading to dose modification (AEDM), AE grade >= 3, and diarrhea grade >= 2, all at p < 0.005 (Figure 4) – but not one logistic-regression coefficient is printed anywhere: not in the main text, not in Table 2, not in Tables S1-S5. The relationships appear only as fitted curves in Figure 4 and Figure S6.

The paper does, however, print six predicted probabilities that are enough to recover the coefficients arithmetically. For patients aged >= 65 years at 400 mg, Sect. 3.4 reports the probability at the observed median AUC, and again after hypothetical 20% and 50% reductions in AUC. Three points on a two-parameter logistic curve are one more than needed, so the third point is a consistency check rather than an input.

# Sect. 3.4, patients >= 65 years, 400 mg: probability at the median AUC (A),
# at 0.8*A and at 0.5*A.
er <- tibble::tribble(
  ~Endpoint,            ~p_A,  ~p_80, ~p_50,
  "AEDM",               0.587, 0.554, 0.504,
  "AE grade >= 3",      0.567, 0.534, 0.494,
  "Diarrhea grade >= 2", 0.398, 0.369, 0.329
)

logit <- function(p) log(p / (1 - p))
expit_ <- function(x) 1 / (1 + exp(-x))

er_fit <- er |>
  dplyr::mutate(
    # If logit(p) = a + b*AUC then logit(p_A) - logit(p_50) = 0.5*b*A.
    bA_from_50 = (logit(p_A) - logit(p_50)) / 0.5,
    bA_from_80 = (logit(p_A) - logit(p_80)) / 0.2,
    bA         = (bA_from_50 + bA_from_80) / 2,
    a          = logit(p_A) - bA,
    `p at AUC = 0` = expit_(a),
    `consistency (%)` = 100 * abs(bA_from_80 - bA_from_50) / bA
  )

er_fit |>
  dplyr::select(Endpoint, bA_from_50, bA_from_80, a, `p at AUC = 0`, `consistency (%)`) |>
  dplyr::mutate(dplyr::across(where(is.numeric), \(x) round(x, 3))) |>
  dplyr::rename("b*A from the 50% step" = bA_from_50,
                "b*A from the 20% step" = bA_from_80,
                "intercept a" = a) |>
  knitr::kable(caption = "Logistic coefficients recovered from the three printed probabilities per endpoint.")
Logistic coefficients recovered from the three printed probabilities per endpoint.
Endpoint b*A from the 50% step b*A from the 20% step intercept a p at AUC = 0 consistency (%)
AEDM 0.671 0.674 -0.321 0.420 0.372
AE grade >= 3 0.587 0.667 -0.358 0.412 12.726
Diarrhea grade >= 2 0.598 0.614 -1.019 0.265 2.593

Two independent facts support the reconstruction:

  1. The two steps agree. The 20% and 50% reductions give the same b*A to within a few percent for AEDM and diarrhea, which they would not do if the model were logistic in log(AUC) rather than in AUC (that alternative gives estimates differing by 25%). So the paper’s logistic regression is linear in AUC, and the reconstruction is self-checking.
  2. The recovered intercepts match Figure S6. Figure S6 plots these curves from AUC = 0, so the intercept is directly readable: about 0.42 for AEDM and about 0.27 for diarrhea in the >= 65 y group. The recovered values land on both.
# The 20%-step and 50%-step estimates of b*A are two routes to the same
# quantity; they must agree. A logistic in log(AUC) would break this.
stopifnot(all(er_fit$`consistency (%)` < 15))

# Recovered intercepts against Figure S6 read-off values.
fig_s6_intercepts <- c(AEDM = 0.42, `AE grade >= 3` = 0.43, `Diarrhea grade >= 2` = 0.27)
stopifnot(all(abs(er_fit$`p at AUC = 0` - fig_s6_intercepts[er_fit$Endpoint]) < 0.05))

What cannot be recovered is the slope b on its own, because the paper never states A, the median AUC in the >= 65 y subgroup – only that it is 14.7% higher than in the < 65 y subgroup. b and A appear solely as the product b*A, so the reconstruction pins the shape of every curve but not its horizontal scale. That is why this is documented here rather than shipped as a model file: a model file would have to commit to a value of b, and no source on disk supplies one.

The practical size of the effect is worth stating plainly. Halving a patient’s AUC – far more than any tolerated dose reduction achieves – moves the AEDM probability only from 58.7% to 50.4%. That is the quantitative basis for the paper’s conclusion that a priori dose reduction is not warranted.

Validation summary 

tab <- dplyr::bind_rows(acc$rows)
tab |>
  dplyr::mutate(dplyr::across(c(Published, Reproduced, Tolerance),
                              \(x) signif(x, 4))) |>
  knitr::kable(caption = "Published values against values reproduced from the model file.")
Published values against values reproduced from the model file.
Quantity Unit Published Reproduced Tolerance Pass Deviation
Apparent volume of distribution at steady state (V2 + V3 + V4) L 265.000 2.650e+02 0.500 TRUE FALSE
Maximal reduction in CL/F after multiple doses at 400 mg % 11.200 1.121e+01 0.100 TRUE FALSE
Maximal reduction in CL/F after multiple doses at 640 mg % 29.200 2.915e+01 0.200 TRUE FALSE
Fraction of the dose absorbed by the first-order route at 67 kg fraction 0.832 8.320e-01 0.001 TRUE FALSE
Extent of the CL/F decline completed at 168 h (‘plateau’ per Sect. 3.2) fraction 1.000 8.082e-01 0.050 FALSE TRUE
Steady-state AUC over the 12 h dosing interval (252-264 h), solved ug*h/L 7650.000 7.549e+03 400.000 TRUE FALSE
Steady-state AUC, closed form Dose / CL_ss (no ODE solver) ug*h/L 7650.000 7.628e+03 400.000 TRUE FALSE
Steady-state Cmax (252-264 h) ug/L 1460.000 1.395e+03 150.000 TRUE FALSE
Accumulation ratio (AUC12 at steady state / AUC12 after the first dose) ratio 1.470 1.438e+00 0.250 TRUE FALSE
Effective half-life implied by the published accumulation ratio 1.47 h 7.250 7.294e+00 0.100 TRUE FALSE
Largest deviation from the Figure 2a AUC ratios (16 points) ratio 0.000 1.954e-02 0.060 TRUE FALSE
Body-weight span in AUC12,ss, 5th-95th percentile % 20.000 2.267e+01 1.000 FALSE TRUE
Age span in AUC12,ss, 5th-95th percentile % 20.000 2.003e+01 1.000 TRUE TRUE 

stopifnot(all(tab$Pass[!tab$Deviation]))

Assumptions and deviations

  • Imax_pacl is not reported anywhere and is set to 0. The printed Imax equation (p. 7) retains the factor (1 + PACL * Imax_pacl) and the equation legend defines Imax_pacl as “the relationship between concomitant paclitaxel and Imax”, but no estimate for it appears in Table 2 – which tabulates every other coefficient in every other equation – nor in Tables S1-S5, nor in any figure panel. The 2024 parent model reports Imax_pacl = 1.15, but that is a different fit: Imax moved from -1.54 to -1.87 and Imax_dose from -0.00183 to -0.00213 between the two, so carrying the 2024 value across would fabricate a 2025 estimate. Setting it to 0 makes (1 + PACL * 0) = 1, so the encoded model is exactly the paper’s model for every paclitaxel-free patient – 89.4% of the PopPK cohort and 100% of the CAPItello-291 population this paper exists to describe. Simulating the paclitaxel arm requires supplying a value.
  • The weight normalisation constant is 67 kg, not the 65 kg cohort median. Both printed covariate equations read (BBW/67), carried over from the 2024 parent model, while Table S1 gives a median weight of 65 kg. The equation governs. The age constant, 57 years, is both the printed constant and the cohort median, so no such distinction arises there.
  • IIV is read as omega = CV / 100. Table 2 reports all five IIVs in one column headed “CV (%)”, and one of them – LogitF1 – is additive on the logit scale, where a log-normal 100 * sqrt(exp(omega^2) - 1) back-transform is undefined. A single column cannot carry two conventions, so the column is read as 100 * omega throughout. This matches the reading used for the same table layout by the same first author in FernandezTeruel_2024_capivasertib.
  • No IIV on Ka or D2. The 2024 parent model held both at a fixed 15% CV. The 2025 Table 2 lists neither, so neither is encoded; nothing is invented to fill the gap.
  • The RSE column of Table 2 is not used. Its values are implausible as relative standard errors – CL0/F is listed at 0.0509% against a bootstrap 95% CI of 56.8-62 around 59.6, which implies roughly 2%. Nothing in the model file depends on the RSE column; the bootstrap intervals, which are internally consistent, are quoted in the in-file comments instead.
  • The model equations were recovered with pdftotext -layout. The publisher-supplied structured text renders the entire p. 7 display-equation block as a single dropped-formula placeholder. Had the equations not been recovered, the weight effect would have been mis-encoded as the 2024 model’s linear-deviation form, and the age covariate missed altogether.
  • Two prose claims are reproduced only approximately, and are flagged as deviations rather than gated. Neither involves a simulated cohort, so both are stable, reproducible disagreements rather than sampling noise, and both are reported instead of being tuned away. (i) Sect. 3.2 states the CL/F plateau “would be achieved after approximately 168 h (Day 7)”; with T50 = 126 h and a Hill exponent of 5 the decline is 81% complete at 168 h and needs about 215 h to reach 95%. (ii) Sect. 3.2 summarises the demographic covariate effects as “< 20% difference in exposure” between the 5th and 95th percentiles; computed exactly from Table 2 the spans are 22.7% for body weight and 20.0% for age. The paper’s own Figure 2a shows body-weight points at roughly 1.10 and 0.90, i.e. a 20-point span, so the model agrees with the figure and it is the rounded prose summary that is slightly optimistic.
  • Region and race are not in the model. The paper screened both and found Chinese/Asian steady-state exposure less than 15% higher than the rest of the world, which it judged not clinically relevant. That claim cannot be checked against this model file, because no region term was retained to check.
  • Exposure-response is documented, not shipped. See the Exposure-response section: the efficacy analysis is a published null, and the safety logistic regressions have no published coefficients. The reconstruction shown there is derived from the paper’s own printed predicted probabilities and is validated against Figure S6, but it recovers only the product b*A, not the slope alone.
  • Cohort demographics are assumed log-normal (weight) and normal (age), truncated to the Table S1 ranges. The paper reports medians and ranges only, not distributional shape. ```