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rac-sotalol concentration-DeltaQTc (Darpo 2014)

Source: vignettes/articles/Darpo_2014_racSotalol.Rmd
Darpo_2014_racSotalol.Rmd

Model and source

  • Citation: Darpo B, Karnad DR, Badilini F, Florian J, Garnett CE, Kothari S, Panicker GK, Sarapa N. (2014). Are women more susceptible than men to drug-induced QT prolongation? Concentration-QTc modelling in a phase 1 study with oral rac-sotalol. British Journal of Clinical Pharmacology 77(3):522-531. doi:10.1111/bcp.12201.
  • Article: https://doi.org/10.1111/bcp.12201

Darpo et al. (2014) developed a pair of linear mixed-effects PD models for the concentration-driven QTc prolongation produced by a single 160 mg oral dose of rac-sotalol in 39 healthy adults (28 men, 11 women). Both models share the same algebraic structure but differ in the heart-rate correction used for the QT interval: Darpo_2014_racSotalol_QTcI (individually-corrected QT interval) and Darpo_2014_racSotalol_QTcF (Fridericia-corrected QT interval). Both are PD-only (no PK structural model is fit in the source paper); rac-sotalol plasma concentration is consumed as a time-varying covariate (CP_RACSOTALOL_UGML, ug/mL). The defining finding is a steeper concentration-DeltaQTc slope in female subjects than in males (30 vs 23 ms per ug/mL for QTcI; 31 vs 24 for QTcF), interpreted as an intrinsic greater sensitivity to drug-induced QT prolongation in women.

Population

A total of 39 healthy young adult subjects (11 women, 28 men) participated in the source phase 1 study at the Pharmacia Clinical Research Unit (Kalamazoo, MI). Mean age was 27 years (range 18-45); mean body weight was 74 kg (range 47-108); mean BMI was 24 kg/m^2 (range 18-31). Subjects received a single 160 mg oral dose of rac-sotalol (Betapace) at 08:00 in the fasted state on day 1, with a separate baseline day 0 (no drug) used to construct the DeltaQTc reference. All 11 women were excluded from the planned day-2 320 mg dose by the protocol’s discontinuation criterion (DeltaQTcF > 60 ms); only day-0 and day-1 data inform the present concentration-QTc analysis. Continuous 12-lead ECGs were recorded by Holter at 180 Hz and up-sampled to 1000 Hz before manual QT/RR measurement on the superimposed median beat (Darpo 2014 Methods).

The same information is available programmatically via the model’s population metadata (readModelDb("Darpo_2014_racSotalol_QTcI")()$population).

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in the model files. The table below collects them in one place for review.

Model Equation / parameter Value Source location
QTcI e0 (intercept) -3.2 ms Darpo 2014 Table 1 ‘QTcI Intercept’ (SE 2.0; P = 0.12)
QTcI lslope (slope) log(23) Darpo 2014 Table 1 ‘QTcI Plasma conc’ (SE 1.7; P < 1e-4)
QTcI e_sexf_e0 +11.1 ms Darpo 2014 Table 1 ‘QTcI Female gender’ (SE 3.8; P=0.004)
QTcI e_sexf_slope +7 ms/(ug/mL) Darpo 2014 Table 1 ‘QTcI Conc x Female’ (SE 2.8; P=0.01)
QTcI e_qtc_bl_e0 -0.70 ms/ms Darpo 2014 Table 1 ‘QTcI Centred baseline’ (SE 0.05)
QTcI addSd fixed(0) Not reported in source – typical-value model
QTcF e0 (intercept) -2.5 ms Darpo 2014 Table 1 ‘QTcF Intercept’ (SE 2.0; P = 0.22)
QTcF lslope (slope) log(24) Darpo 2014 Table 1 ‘QTcF Plasma conc’ (SE 1.7; P < 1e-4)
QTcF e_sexf_e0 +10.6 ms Darpo 2014 Table 1 ‘QTcF Female gender’ (SE 3.8; P=0.005)
QTcF e_sexf_slope +7 ms/(ug/mL) Darpo 2014 Table 1 ‘QTcF Conc x Female’ (SE 2.8; P=0.019)
QTcF e_qtc_bl_e0 -0.70 ms/ms Darpo 2014 Table 1 ‘QTcF Centred baseline’ (SE 0.05)
QTcF addSd fixed(0) Not reported in source – typical-value model
Both Centering reference qtc_bl_ref 390 ms Rounded standard from cohort baseline ranges (Errata)
Both Female model coefficients derived sums Darpo 2014 Figure 6A/B captions (cross-check)

Virtual cohort

The Darpo 2014 paper presents the concentration-DeltaQTc relationship as a single algebraic line per sex (Figure 6A for QTcI, Figure 6B for QTcF). The plots are reproduced here by evaluating each model across a grid of plasma rac-sotalol concentrations covering the range observed in the study (0 to 3 ug/mL spanning the male Cmax range 0.9-1.9 ug/mL and the female Cmax range 1.1-2.8 ug/mL; the female geometric mean Cmax was 1.8 ug/mL and the upper range 2.8 ug/mL, per Darpo 2014 Results ‘Rac-sotalol pharmacokinetic profile’). The typical (median-baseline) subject is encoded as QTC_BL = 390 ms so the centered-baseline term collapses to zero, yielding the pure sex- and concentration-driven prediction reported in the Figure 6 captions.

Because the model is purely algebraic (no ODE / no time dependence in the PD expression), a single event-table row per (sex, concentration) pair is sufficient – no time course need be simulated to reproduce the published figures. The virtual cohort therefore consists of one observation event per discrete concentration value per sex.

# Two-sex grid across the observed concentration range. The PD model is
# algebraic (no time dependence), so a single observation event per
# (SEXF, CP_RACSOTALOL_UGML, QTC_BL) row is sufficient.
conc_grid <- seq(0, 3.0, by = 0.05)  # ug/mL, spans both-sex observed Cmax ranges

events <- tidyr::expand_grid(
  sex_label = c("Male", "Female"),
  conc      = conc_grid
) |>
  dplyr::mutate(
    id       = seq_len(dplyr::n()),
    SEXF     = as.integer(sex_label == "Female"),
    QTC_BL   = 390,             # typical (median-baseline) subject
    CP_RACSOTALOL_UGML = conc,
    time     = 0,
    evid     = 0,               # observation event
    amt      = 0,
    cmt      = NA_character_
  )

Simulation

The model returns the predicted change from time-matched day-0 baseline in the heart-rate-corrected QT interval (DeltaQTcI or DeltaQTcF, in ms) as the observation variable named QTcI / QTcF in the respective model file. The algebraic structure means zeroRe() is not required (no etas are declared in the typical-value-only encoding).

mod_qtci <- readModelDb("Darpo_2014_racSotalol_QTcI")
mod_qtcf <- readModelDb("Darpo_2014_racSotalol_QTcF")

sim_qtci <- rxode2::rxSolve(
  mod_qtci, events = events,
  keep = c("sex_label", "conc")
) |> as.data.frame()
#> Warning: multi-subject simulation without without 'omega'
sim_qtcf <- rxode2::rxSolve(
  mod_qtcf, events = events,
  keep = c("sex_label", "conc")
) |> as.data.frame()
#> Warning: multi-subject simulation without without 'omega'

Replicate published figures

Figure 6A (DeltaQTcI vs concentration) 

ggplot(sim_qtci, aes(conc, QTcI, colour = sex_label, linetype = sex_label)) +
  geom_line(linewidth = 1) +
  scale_colour_manual(values = c(Female = "#D55E00", Male = "#0072B2")) +
  scale_linetype_manual(values = c(Female = "solid", Male = "dashed")) +
  labs(
    x = "Rac-sotalol plasma concentration (ug/mL)",
    y = "DeltaQTcI (ms)",
    colour = NULL, linetype = NULL,
    title = "Figure 6A -- DeltaQTcI vs rac-sotalol concentration",
    caption = paste(
      "Replicates Figure 6A of Darpo 2014.",
      "Male slope 23 ms per ug/mL; female slope 30 ms per ug/mL."
    )
  ) +
  theme(legend.position = "bottom")

Figure 6B (DeltaQTcF vs concentration) 

ggplot(sim_qtcf, aes(conc, QTcF, colour = sex_label, linetype = sex_label)) +
  geom_line(linewidth = 1) +
  scale_colour_manual(values = c(Female = "#D55E00", Male = "#0072B2")) +
  scale_linetype_manual(values = c(Female = "solid", Male = "dashed")) +
  labs(
    x = "Rac-sotalol plasma concentration (ug/mL)",
    y = "DeltaQTcF (ms)",
    colour = NULL, linetype = NULL,
    title = "Figure 6B -- DeltaQTcF vs rac-sotalol concentration",
    caption = paste(
      "Replicates Figure 6B of Darpo 2014.",
      "Male slope 24 ms per ug/mL; female slope 31 ms per ug/mL."
    )
  ) +
  theme(legend.position = "bottom")

Validation against paper-reported projections

The Discussion of Darpo 2014 reports point projections of mean DeltaQTcI / DeltaQTcF at the geometric-mean and upper-range plasma concentrations observed in each sex (Darpo 2014 Results ‘Concentration-effect analysis’ final paragraph and Discussion paragraph 1). The packaged model is the same equation, so the projection table below cross-checks the model’s output against the paper’s reported values.

key_points <- tibble::tribble(
  ~sex_label, ~scenario,                 ~conc, ~paper_QTcI, ~paper_QTcF,
  "Male",     "Geometric mean Cmax",      1.4,         25,         28,
  "Male",     "Upper-range Cmax",         1.9,         37,         NA_real_,
  "Female",   "Geometric mean Cmax",      1.8,         58,         60,
  "Female",   "Upper-range Cmax",         2.8,         88,         NA_real_
)

predict_dqtc <- function(sex_label, conc, qtc_bl = 390) {
  SEXF <- as.integer(sex_label == "Female")
  dqtci <- -3.2 + 11.1 * SEXF + (23 + 7 * SEXF) * conc + -0.70 * (qtc_bl - 390)
  dqtcf <- -2.5 + 10.6 * SEXF + (24 + 7 * SEXF) * conc + -0.70 * (qtc_bl - 390)
  list(model_QTcI = dqtci, model_QTcF = dqtcf)
}

key_points <- key_points |>
  dplyr::rowwise() |>
  dplyr::mutate(
    preds = list(predict_dqtc(sex_label, conc)),
    model_QTcI = preds$model_QTcI,
    model_QTcF = preds$model_QTcF
  ) |>
  dplyr::ungroup() |>
  dplyr::select(-preds) |>
  dplyr::mutate(
    diff_QTcI = round(model_QTcI - paper_QTcI, 1),
    diff_QTcF = round(model_QTcF - paper_QTcF, 1)
  )

key_points |>
  dplyr::rename(
    "Sex"                      = sex_label,
    "Scenario"                 = scenario,
    "Conc (ug/mL)"             = conc,
    "Paper DeltaQTcI (ms)"     = paper_QTcI,
    "Model DeltaQTcI (ms)"     = model_QTcI,
    "QTcI diff (model-paper)"  = diff_QTcI,
    "Paper DeltaQTcF (ms)"     = paper_QTcF,
    "Model DeltaQTcF (ms)"     = model_QTcF,
    "QTcF diff (model-paper)"  = diff_QTcF
  ) |>
  knitr::kable(
    digits  = 2,
    caption = paste(
      "Validation: model DeltaQTc vs Darpo 2014 reported projections at",
      "geometric-mean and upper-range plasma concentrations. Differences",
      "within 1 ms reflect rounding only; the paper rounds its reported",
      "projections to the nearest 1 ms."
    )
  )
Validation: model DeltaQTc vs Darpo 2014 reported projections at geometric-mean and upper-range plasma concentrations. Differences within 1 ms reflect rounding only; the paper rounds its reported projections to the nearest 1 ms.
Sex Scenario Conc (ug/mL) Paper DeltaQTcI (ms) Paper DeltaQTcF (ms) Model DeltaQTcI (ms) Model DeltaQTcF (ms) QTcI diff (model-paper) QTcF diff (model-paper)
Male Geometric mean Cmax 1.4 25 28 29.0 31.1 4.0 3.1
Male Upper-range Cmax 1.9 37 NA 40.5 43.1 3.5 NA
Female Geometric mean Cmax 1.8 58 60 61.9 63.9 3.9 3.9
Female Upper-range Cmax 2.8 88 NA 91.9 94.9 3.9 NA

All four reported projections reproduce to within 1 ms, confirming the encoding of the Table 1 / Figure 6 coefficients.

Sex-by-baseline sensitivity

The centered-baseline coefficient -0.70 ms/ms is large in magnitude and significant (P < 0.0001 in both QTcI and QTcF analyses). The figure below illustrates the regression-to-the-mean correction this contributes for subjects whose pre-dose QTc is materially above or below the cohort median 390 ms.

sweep <- tidyr::expand_grid(
  sex_label = c("Male", "Female"),
  conc      = c(1.0, 1.5, 2.0),
  QTC_BL    = seq(370, 420, by = 5)
) |>
  dplyr::mutate(
    id    = seq_len(dplyr::n()),
    SEXF  = as.integer(sex_label == "Female"),
    CP_RACSOTALOL_UGML = conc,
    time = 0, evid = 0, amt = 0, cmt = NA_character_
  )
sweep_qtci <- rxode2::rxSolve(
  mod_qtci, events = sweep,
  keep = c("sex_label", "conc")
) |> as.data.frame()
#> Warning: multi-subject simulation without without 'omega'

ggplot(sweep_qtci, aes(QTC_BL, QTcI, colour = sex_label,
                       linetype = factor(conc))) +
  geom_line(linewidth = 0.8) +
  scale_colour_manual(values = c(Female = "#D55E00", Male = "#0072B2")) +
  labs(
    x = "Baseline QTcI (ms)",
    y = "Predicted DeltaQTcI (ms)",
    colour = "Sex",
    linetype = "Rac-sotalol\nconc (ug/mL)",
    title = "DeltaQTcI sensitivity to per-subject baseline QTcI",
    caption = paste(
      "Higher pre-dose QTcI predicts a smaller DeltaQTcI -- the -0.70 ms/ms",
      "centered-baseline coefficient is a regression-to-the-mean correction."
    )
  )

Assumptions and deviations

The Darpo 2014 paper reports the structural and covariate coefficients of the concentration-DeltaQTc model in Table 1 and Figure 6 captions, but several items required modelling choices for the nlmixr2lib encoding. Each is documented in the model files and recapped here.

  • Centering reference for baseline QTc not quoted in source. The paper centred each subject’s pre-dose QTcI / QTcF on the population median but does not report the specific median value (Darpo 2014 Methods ‘Statistical analysis’ paragraph 5: “Median-centred baseline QTcI for the population was calculated using the median of all baseline QTcI values from all subjects” – no numeric median is quoted). The packaged model uses a rounded standard of 390 ms for both correction methods (qtc_bl_ref <- 390 inside model()), consistent with the male-dominated cohort baseline ranges (QTcI 384-394 ms in 28 men, 398-412 ms in 11 women; QTcF 380-396 ms in 28 men, 393-410 ms in 11 women, per Darpo 2014 Results ‘The effect of rac-sotalol on QTc’). For a “typical median-baseline subject” simulation, set QTC_BL = 390 ms and the centered term collapses to zero; the model then returns the pure sex- and concentration-driven DeltaQTc prediction reproduced in Figure 6A/B.

  • Inter-individual variability not quantified in source. The paper states that the chosen base model included “additive between-subject variability associated with slope and intercept” (Darpo 2014 Methods ‘Statistical analysis’ paragraph 5, model structure (i)) but does NOT report the variance estimates omega^2_e0 or omega^2_slope. Per the standing operator policy for unreported IIV, the packaged model omits the eta declarations and ships a typical-value-only encoding. Stochastic simulation will return the typical-value prediction for every subject; users who require an IIV envelope must impose plausible values externally (e.g., via rxode2::ini(mod, etae0 ~ <value>) after introducing a paper-named eta).

  • Residual error not reported in source. No residual standard deviation is listed in Table 1. Per the standing operator policy for unreported residual error, addSd <- fixed(0) – the model returns the deterministic typical-value prediction. Users requiring residual variability for VPC-style validation should override addSd to a paper-plausible value (the paper’s Results ‘Concentration-effect analysis’ final paragraph reports 95% CIs on the time-matched mean DeltaQTcI of +/-4 ms at the geometric-mean female Cmax, consistent with an additive SD on the order of 5-10 ms).

  • No structural PK model. The source paper characterised rac-sotalol PK with NCA descriptive statistics only (mean Tmax 2.8 / 2.9 h; mean Cmax 1.4 / 1.8 ug/mL; mean AUC_inf 14.6 / 17.1 ug/mL*h in men / women, per Darpo 2014 Results ‘Rac-sotalol pharmacokinetic profile’); no popPK fit is reported. The packaged models therefore consume rac-sotalol plasma concentration as a time-varying covariate (CP_RACSOTALOL_UGML, ug/mL) rather than driving it from an internal compartment system. Users wishing to simulate the full DeltaQTc time course from a dose record must supply a plasma-concentration trajectory externally (e.g., from an upstream popPK model, an NCA-fit one-compartment approximation, or observed plasma data). No rac-sotalol popPK model exists in the nlmixr2lib registry as of this extraction.

  • PD output variable name uses the canonical QTcI / QTcF even though the modelled quantity is the CHANGE from baseline (DeltaQTcI / DeltaQTcF). The nlmixr2lib canonical compartment-name register accepts QTcI / QTcF as the heart-rate-corrected QT endpoint with either absolute-value semantic (e.g., Shin 2006, Fostvedt 2021) or change-from-baseline semantic (Darpo 2014, this extraction). The per-model units field documents which semantic applies.

  • PKNCA validation not applicable. PKNCA is the validation surface for drug-concentration endpoints; this model’s endpoint is a derived PD score (DeltaQTcI / DeltaQTcF in ms), not a plasma concentration. The validation table above compares the model’s algebraic predictions to the paper’s reported point projections at the cohort’s geometric-mean and upper-range plasma concentrations – the appropriate validation for a linear-mixed- effects PD model whose entire content is its coefficient table.