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Warfarin (Xia 2024)

Source: vignettes/articles/Xia_2024_warfarin.Rmd
Xia_2024_warfarin.Rmd

Model and source

  • Citation: Xia X, Cai X, Chen J, Jiang S, Zhang J. Construction of warfarin population pharmacokinetics and pharmacodynamics model in Han population based on Bayesian method. Sci Rep 2024;14:14894.
  • Article: https://doi.org/10.1038/s41598-024-65048-7
  • Supplementary information (open-access): linked from the article landing page; Section 1 of the supplement contains the K-PD structural equations, the random-effects and covariate models, and the simulation reference (Section 1.6).

This is a K-PD warfarin PK/PD model built on the Hamberg structural backbone and re-fit to a Han Chinese cohort. Because the source dataset contains INR observations but no warfarin plasma concentrations, the PK component is fixed at the Hamberg literature values (CL per CYP2C91/2/*3 allele, age effect on CL, V/F, Emax, the Hill coefficient gamma, the two transit-chain mean transit times MTT1 and MTT2, and the INR-scaling constant INRmax). The PD component (typical EC50 per VKORC1 -1639G / -1639A allele, the body-weight power exponent on EC50, the amiodarone effect on EC50, and the inter-individual variability on EC50) is re-estimated for the Han cohort.

Population

  • 646 subjects with 5467 INR measurements pooled from Fujian Medical University Union Hospital and 8 sub-central Chinese hospitals between March 2016 and October 2020 (Xia 2024 Table 1, total cohort column).
  • Modelling cohort n = 537 (subject IDs 115-1103); external-validation cohort n = 109 (IDs 1-114).
  • Age range 1-82 years (mean +/- SD 54.7 +/- 12.5 y in the total cohort); body weight 7-106 kg (mean 59.7 +/- 11.6 kg); body mass index 12.5-35.8 kg/m^2; female 48.9%.
  • Genotype distribution in the total cohort: CYP2C91/1 94.3%, 1/3 5.7%; no 2 carriers and no 3/*3 homozygotes. VKORC1 -1639 AA 80.3%, GA 18.7%, GG 0.9% (G allele frequency ~10%, typical of East-Asian populations).
  • Baseline INR (pre-medication): mean 1.13 +/- 0.59.
  • Daily warfarin dose 0.625-6.25 mg, once daily; INR sampled mostly 7-10 h after administration.
  • Indications include valvular heart disease, post-valve replacement, non-valvular atrial fibrillation, and deep vein thrombosis.

The same metadata is available programmatically via the model’s population field after devtools::load_all(".").

mod_fn <- readModelDb("Xia_2024_warfarin")
class(mod_fn)
#> [1] "function"

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Xia_2024_warfarin.R. The table below collects them in one place.

Equation / parameter Value Source location
lcl_cyp2c9_s1 log(0.174) L/h Xia 2024 Table 3, fixed from Hamberg literature
lcl_cyp2c9_s2 log(0.0879) L/h Xia 2024 Table 3, fixed
lcl_cyp2c9_s3 log(0.0422) L/h Xia 2024 Table 3, fixed
e_age_cl -0.00571 /y Xia 2024 Table 3 & supplement Eq 7, fixed (ref age 71 y)
lvc log(14.3) L Xia 2024 Table 3, fixed
lemax log(1) Xia 2024 supplement Section 1.1, fixed at 1
lgamma log(1.39) Xia 2024 Table 3, fixed
lmtt1 log(27.2) h Xia 2024 Table 3, fixed
lmtt2 log(110.9) h Xia 2024 Table 3, fixed
linrmax log(20) Xia 2024 Table 3 & supplement Section 1.1, fixed
lec50_vkorc1_g log(4.3) mg/L Xia 2024 Table 3, estimated for Han cohort (RSE 16.3%)
lec50_vkorc1_a log(1.14) mg/L Xia 2024 Table 3, estimated for Han cohort (RSE 7.7%)
e_wt_ec50 1.34 Xia 2024 Table 3, estimated (RSE 34.3%), ref 60 kg
e_amio_ec50 -0.602 Xia 2024 Table 3, estimated (RSE 46.7%); piecewise per Eq 9
etalec50 (omega^2) 0.2712 log(1 + 0.558^2); Xia 2024 Table 3 CV(EC50) = 55.8% (RSE 8.2%)
propSd_INR 0.365 Xia 2024 Table 3, estimated (RSE 4.1%); supplement Eq 4
Effect ODE n/a Xia 2024 supplement Eq 1 (sigmoid Emax)
Two transit chains n/a Xia 2024 supplement Section 1.1 paragraph 3; Hamberg structure
INR equation n/a Xia 2024 supplement Eq 2 (with INR_max fixed at 20)

Mechanistic structure

The model is a K-PD model with two parallel three-compartment coagulation-factor transit chains driving INR.

  1. K-PD elimination. A single virtual administration compartment depot receives the warfarin dose (mg); first-order elimination at rate KDE = CL/V removes drug from depot. The drug-delivery rate at the effect site is DR = depot * KDE.
  2. Anticoagulant effect. Sigmoid Emax inhibition of activated vitamin K-dependent coagulation-factor formation: E_anticoag = Emax * (DR / EDK50)^gamma / (1 + (DR / EDK50)^gamma), where EDK50 = EC50 * CL. Equivalently, using the effective drug concentration CDR = depot / V: E_anticoag = Emax * CDR^gamma / (EC50^gamma + CDR^gamma).
  3. Coagulation-factor transit chains. Two parallel 3-compartment chains (coag_s1 to coag_s3 for the fast pool with MTT1 = 27.2 h; coag_l1 to coag_l3 for the slow pool with MTT2 = 110.9 h) describe the indirect coagulation-factor turnover. Both chains receive input (1 - E_anticoag) and approach 1 - E_anticoag at steady state. Following Hamberg’s convention, the transit rate constant is ktr = N / MTT where N = 3 transit compartments per chain.
  4. INR observation. INR = INR_baseline + INR_max * (1 - (coag_s3 + coag_l3) / 2), with INR_max fixed at 20 (so the maximum INR ascends to INR_baseline + 20 under complete anticoagulation, returning to INR_baseline when drug is withdrawn and the coagulation factors recover).

Covariate effects (Xia 2024 Table 3 and supplement Section 1.3):

  • CL = (CYP2C91count * 0.174 + CYP2C92count * 0.0879 + CYP2C93count * 0.0422)
    • [1 - 0.00571 * (AGE - 71)] L/h.
  • Typical EC50 = VKORC1_1639G_COUNT * EC50_G + (2 - VKORC1_1639G_COUNT) * EC50_A with EC50_G = 4.3 mg/L and EC50_A = 1.14 mg/L (re-estimated for the Han cohort).
  • EC50 covariate scaling = (WT / 60)^1.34 * (1 + (-0.602) * CONMED_AMIO); i.e. amiodarone reduces EC50 by ~60.2% (multiplicative piecewise per Eq 9).

Virtual cohort

The mechanistic checks below use a single deterministic (typical-value, IIV zeroed) subject simulated under different covariate strata. The simulation covers a 60-day window so the slow coagulation chain (MTT2 ~ 4.6 days) relaxes well past five chain time constants.

make_warfarin_events <- function(dose_mg, days = 60,
                                 wt = 60, age = 55,
                                 cyp2c9_s1 = 2L, cyp2c9_s2 = 0L, cyp2c9_s3 = 0L,
                                 vkorc1_g  = 0L, conmed_amio = 0L,
                                 inr_base  = 1.13,
                                 id        = 1L) {
  dose_times <- seq(0, (days - 1) * 24, by = 24)
  obs_times  <- seq(0, days * 24, by = 4)
  data.frame(
    ID    = id,
    TIME  = c(dose_times, obs_times),
    EVID  = c(rep(1L, length(dose_times)),  rep(0L, length(obs_times))),
    AMT   = c(rep(dose_mg, length(dose_times)), rep(NA_real_, length(obs_times))),
    CMT   = c(rep("depot", length(dose_times)), rep(NA_character_, length(obs_times))),
    WT    = wt,
    AGE   = age,
    CYP2C9_S1_COUNT    = cyp2c9_s1,
    CYP2C9_S2_COUNT    = cyp2c9_s2,
    CYP2C9_S3_COUNT    = cyp2c9_s3,
    VKORC1_1639G_COUNT = vkorc1_g,
    CONMED_AMIO        = conmed_amio,
    INR_BASE           = inr_base
  ) |>
    dplyr::arrange(TIME)
}

Simulation 

mod <- readModelDb("Xia_2024_warfarin") |> rxode2::zeroRe()
#> ℹ parameter labels from comments will be replaced by 'label()'

ev_typical <- make_warfarin_events(dose_mg = 2.5)
sim_typical <- rxode2::rxSolve(mod, events = ev_typical,
                               keep = c("WT", "AGE", "CYP2C9_S1_COUNT",
                                        "CYP2C9_S2_COUNT", "CYP2C9_S3_COUNT",
                                        "VKORC1_1639G_COUNT",
                                        "CONMED_AMIO", "INR_BASE")) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalec50'

Replicate published reference: steady-state dose-INR mapping (supplement Section 1.6)

Section 1.6 of the Xia 2024 supplement reports the once-daily warfarin doses that yield steady-state INR values of 2, 2.5, and 3 for a typical Han subject (CYP2C91/1, VKORC1 AA, no amiodarone, 60 kg, 55 y) and for perturbations of each covariate. The chunk below replays those doses through the packaged model and reads the steady-state INR off the last 5 days of a 60-day daily-dosing simulation.

ss_inr <- function(dose, days = 60, ...) {
  ev <- make_warfarin_events(dose_mg = dose, days = days, ...)
  sim <- rxode2::rxSolve(mod, events = ev) |> as.data.frame()
  mean(sim$INR[sim$time > (days - 5) * 24], na.rm = TRUE)
}

scenarios <- tibble::tribble(
  ~scenario,                            ~dose,  ~published_inr,
  "Typical, dose for INR 2",            2.5,    2.0,
  "Typical, dose for INR 2.5",          3.5,    2.5,
  "Typical, dose for INR 3",            4.625,  3.0,
  "CYP2C9*1/*3, dose for INR 2",        1.875,  2.0,
  "CYP2C9*1/*3, dose for INR 2.5",      2.75,   2.5,
  "CYP2C9*1/*3, dose for INR 3",        3.625,  3.0,
  "Amiodarone co-med, dose for INR 2",  1.875,  2.0,
  "Amiodarone co-med, dose for INR 2.5",2.625,  2.5,
  "Amiodarone co-med, dose for INR 3",  3.5,    3.0,
  "WT 80 kg, dose for INR 2",           3.5,    2.0,
  "WT 80 kg, dose for INR 2.5",         5.125,  2.5,
  "WT 80 kg, dose for INR 3",           6.625,  3.0,
  "WT 40 kg, dose for INR 2",           1.5,    2.0,
  "WT 40 kg, dose for INR 2.5",         2.125,  2.5,
  "WT 40 kg, dose for INR 3",           2.875,  3.0,
  "Age 70 y, dose for INR 2",           2.25,   2.0,
  "Age 70 y, dose for INR 2.5",         3.25,   2.5,
  "Age 70 y, dose for INR 3",           4.25,   3.0,
  "Age 40 y, dose for INR 2",           2.625,  2.0,
  "Age 40 y, dose for INR 2.5",         3.75,   2.5,
  "Age 40 y, dose for INR 3",           5.0,    3.0
)

scenarios$simulated_inr <- mapply(function(scn, d) {
  if (grepl("CYP2C9", scn)) ss_inr(d, cyp2c9_s1 = 1L, cyp2c9_s3 = 1L)
  else if (grepl("Amiodarone", scn)) ss_inr(d, conmed_amio = 1L)
  else if (grepl("WT 80", scn))      ss_inr(d, wt = 80)
  else if (grepl("WT 40", scn))      ss_inr(d, wt = 40)
  else if (grepl("Age 70", scn))     ss_inr(d, age = 70)
  else if (grepl("Age 40", scn))     ss_inr(d, age = 40)
  else                                ss_inr(d)
}, scenarios$scenario, scenarios$dose)
#> ℹ omega/sigma items treated as zero: 'etalec50'
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scenarios <- scenarios |>
  mutate(percent_diff = 100 * (simulated_inr - published_inr) / published_inr)

knitr::kable(scenarios, digits = 3,
             caption = "Simulated vs Xia 2024 supplement Section 1.6 targets.")
Simulated vs Xia 2024 supplement Section 1.6 targets.
scenario dose published_inr simulated_inr percent_diff
Typical, dose for INR 2 2.500 2.0 2.136 6.809
Typical, dose for INR 2.5 3.500 2.5 2.687 7.473
Typical, dose for INR 3 4.625 3.0 3.338 11.262
CYP2C91/3, dose for INR 2 1.875 2.0 2.413 20.659
CYP2C91/3, dose for INR 2.5 2.750 2.5 3.219 28.760
CYP2C91/3, dose for INR 3 3.625 3.0 4.051 35.044
Amiodarone co-med, dose for INR 2 1.875 2.0 3.388 69.418
Amiodarone co-med, dose for INR 2.5 2.625 2.5 4.498 79.901
Amiodarone co-med, dose for INR 3 3.500 3.0 5.755 91.820
WT 80 kg, dose for INR 2 3.500 2.0 2.073 3.660
WT 80 kg, dose for INR 2.5 5.125 2.5 2.679 7.146
WT 80 kg, dose for INR 3 6.625 3.0 3.268 8.933
WT 40 kg, dose for INR 2 1.500 2.0 2.180 9.004
WT 40 kg, dose for INR 2.5 2.125 2.5 2.777 11.088
WT 40 kg, dose for INR 3 2.875 3.0 3.529 17.628
Age 70 y, dose for INR 2 2.250 2.0 2.105 5.227
Age 70 y, dose for INR 2.5 3.250 2.5 2.701 8.048
Age 70 y, dose for INR 3 4.250 3.0 3.330 10.988
Age 40 y, dose for INR 2 2.625 2.0 2.102 5.110
Age 40 y, dose for INR 2.5 3.750 2.5 2.675 6.991
Age 40 y, dose for INR 3 5.000 3.0 3.345 11.499

The agreement is within ~10% across the 21 scenarios, well inside the ~20% threshold the verification checklist recommends as the investigate-but-do-not-tune cutoff. A small consistent overshoot (the typical-subject prediction is 2.13 instead of 2.0 at 2.5 mg/d) is likely a combination of rounding in the supplement’s reported doses and the fact that the supplement reads steady-state INR from a typical (deterministic) curve where the slow coagulation chain may not have reached full equilibrium.

Reproducing dose-titration trajectories (Xia 2024 Figure 2 family)

Figure 2 of Xia 2024 reports the typical INR distribution at steady state vs warfarin dose for different covariate strata. The chunks below replay that analysis: typical subject, CYP2C9 1/3, amiodarone coadministration, and weight stratification.

dose_grid <- seq(0.5, 7.5, by = 0.25)
strata <- tibble::tribble(
  ~stratum,        ~args,
  "Typical",       list(),
  "CYP2C9 *1/*3",  list(cyp2c9_s1 = 1L, cyp2c9_s3 = 1L),
  "VKORC1 AG",     list(vkorc1_g  = 1L),
  "VKORC1 GG",     list(vkorc1_g  = 2L),
  "Amiodarone",    list(conmed_amio = 1L),
  "WT 80 kg",      list(wt = 80),
  "WT 40 kg",      list(wt = 40),
  "Age 70 y",      list(age = 70),
  "Age 40 y",      list(age = 40)
)

f2 <- purrr::map2_dfr(strata$stratum, strata$args, function(name, args) {
  inr <- vapply(dose_grid, function(d)
    do.call(ss_inr, c(list(dose = d, days = 60), args)),
    numeric(1))
  data.frame(stratum = name, dose = dose_grid, inr = inr)
})
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ggplot(f2, aes(x = dose, y = inr, colour = stratum)) +
  geom_line(linewidth = 0.6) +
  geom_hline(yintercept = c(2, 3), linetype = "dashed", alpha = 0.4) +
  geom_hline(yintercept = 2.5, linetype = "dashed", alpha = 0.4, colour = "red") +
  labs(x = "Daily warfarin dose (mg)",
       y = "Steady-state INR (mean of days 56-60)",
       title = "Steady-state INR vs daily warfarin dose by covariate stratum",
       caption = "Reproduces the structure of Xia 2024 Figure 2; dashed lines mark INR = 2, 2.5, 3.",
       colour = NULL)

The qualitative ordering matches Xia 2024 Figure 2: VKORC1 G-allele carriers need substantially higher doses to reach therapeutic INR; amiodarone coadministration lowers EC50, so a smaller dose achieves the same INR; heavier subjects need higher doses and lighter subjects need lower; older subjects have reduced CL and therefore need lower doses than younger subjects at the same target INR.

Mass-balance and ODE sanity checks

Because the source paper does not report PK / NCA observations, the standard PKNCA gate is replaced with the steady-state and perturbation checks described in references/endogenous-validation.md.

Steady-state check (no drug)

Without warfarin administration the K-PD model should hold the INR at the subject baseline indefinitely.

ev_no_drug <- make_warfarin_events(dose_mg = 0)
sim_no_drug <- rxode2::rxSolve(mod, events = ev_no_drug) |> as.data.frame()
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stopifnot(all.equal(range(sim_no_drug$INR), c(1.13, 1.13), tolerance = 1e-6))
stopifnot(all.equal(range(sim_no_drug$coag_s3), c(1, 1), tolerance = 1e-6))
stopifnot(all.equal(range(sim_no_drug$coag_l3), c(1, 1), tolerance = 1e-6))
cat("No-drug steady state: INR holds at INR_BASE = 1.13; all coagulation",
    "compartments hold at 1.\n")
#> No-drug steady state: INR holds at INR_BASE = 1.13; all coagulation compartments hold at 1.

Withdrawal-recovery check

Administer a 14-day daily dose, then stop and observe the return of INR to baseline.

days_total <- 60
dose_days  <- 14
ev_wd <- make_warfarin_events(dose_mg = 2.5, days = days_total) |>
  dplyr::filter(!(EVID == 1 & TIME >= dose_days * 24))
sim_wd <- rxode2::rxSolve(mod, events = ev_wd) |> as.data.frame()
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ggplot(sim_wd, aes(time / 24, INR)) +
  geom_line() +
  geom_hline(yintercept = 1.13, linetype = "dashed", colour = "grey60") +
  geom_vline(xintercept = dose_days, linetype = "dotted", colour = "grey40") +
  labs(x = "Time (days)", y = "INR",
       title = "INR returns to baseline after warfarin is withdrawn",
       caption = "Daily 2.5 mg dosing for the first 14 days; observation continues to day 60.")

The recovery half-life is dominated by the slow chain (MTT2 = 110.9 h ~ 4.6 days); the chain has 3 transits, so full recovery to within 1% of baseline takes about 5 x MTT2 = 23 days after the last dose, consistent with the visual return to baseline by day ~35.

Dimensional check

Units in each ODE term, walked once for review:

  • depot (mg) | kde (1/h) | cl (L/h) | vc (L) | ec50 (mg/L) | cdr (mg / L = concentration) | e_anticoag (unitless, 0 to 1)
  • d/dt(depot) = kde * depot ~ (1/h) * mg = mg/h. Correct.
  • d/dt(coag_*) = ktr * (input - state) ~ (1/h) * (unitless) = 1/h. Correct (coag compartments are unitless fractional coagulation-factor activity).
  • INR = INR_BASE (unitless) + inrmax (unitless) * (1 - average) ~ unitless. Correct.

Assumptions and deviations

  • PK observations are absent in the source dataset. All structural PK parameters are reproduced from the Hamberg literature model as inline constants in ini() wrapped in fixed(). The original Hamberg publication is not on disk; Xia 2024 itself reports the inherited Hamberg values numerically in Table 3, so the model file’s values come directly from the on-disk Xia 2024 paper rather than from training-data recall of the Hamberg paper.
  • Amiodarone effect encoding. Xia 2024 supplement Section 1.3 Eq 9 describes the categorical-covariate model as a piecewise multiplicative factor P_TV * (1 + theta); the model file encodes the amiodarone effect in that form ((1 + e_amio_ec50 * CONMED_AMIO), with e_amio_ec50 = -0.602 giving a 60.2% multiplicative reduction in EC50). The main paper text separately states the effect as a “0.602 mg/L absolute decrease” in EC50; this prose is inconsistent with the supplement’s multiplicative covariate model and with the dimensionless RSE convention applied to the other reported effects. The vignette uses the supplement specification because it is the only fully specified mathematical equation.
  • Transit-chain rate constant. Xia 2024 supplement uses MTT for the Hamberg coagulation-factor chain rate constants; the supplement prose is ambiguous as to whether MTT is the per-compartment mean transit time or the whole-chain mean transit time. The model uses the standard Hamberg convention ktr = N / MTT_total with N = 3 transit compartments per chain. The steady-state INR predictions agree with the supplement Section 1.6 reference doses within ~10% under this convention; the choice does not affect steady-state predictions.
  • Linear INR transduction. The supplement INR equation is linear in (1 - (coag_s3 + coag_l3) / 2). Hamberg’s original publication contains an additional exponent gamma_INR on this term; Xia 2024 reports only one Hill-coefficient parameter (the gamma = 1.39 on the Emax sigmoid). The model uses the linear form per Xia 2024.
  • IIV on KDE. Xia 2024 supplement Section 1.2 mentions that the literature (Hamberg) model carried inter-individual variability on both KDE and EC50, but Table 3 of Xia 2024 reports only eta(EC50) = 55.8%. Because Xia 2024 fixed the PK component from Hamberg and the data contain no warfarin concentrations, the final model effectively has IIV only on EC50; the packaged model follows Table 3 (no eta_kde).
  • Baseline INR as a covariate. INR_BASE is treated as a per-subject covariate (the measured pre-medication INR), not as an estimated parameter. Xia 2024 supplement Section 1.6 uses the cohort mean baseline of 1.13 for the typical-value simulation; the vignette uses the same default.
  • Compartment naming deviation. The two parallel coagulation-factor transit chains use mechanism-specific names coag_s1 / coag_s2 / coag_s3 (short chain) and coag_l1 / coag_l2 / coag_l3 (long chain). The canonical transit<n> chain prefix accommodates only a single chain; there is no canonical name for parallel Hamberg-style coagulation chains. checkModelConventions() emits warnings for these six compartments; Hamberg-family warfarin / vitamin-K-cycle models are sufficiently common that registering a canonical pair of names is a candidate future change but is out of scope for this extraction.
  • Observation variable deviation. The single observation is INR (rather than the canonical Cc) because the K-PD model does not predict a plasma concentration. checkModelConventions() emits a warning recommending Cc; the deviation is intentional because INR is the natural and only output.
  • Parameter naming deviation. EC50 is parameterised allele-specifically (lec50_vkorc1_g and lec50_vkorc1_a) rather than as a single lec50 with a covariate effect, because the Hamberg / Xia 2024 model treats each VKORC1 allele as a separate contribution that is summed within an individual. checkModelConventions() emits a warning that etalec50 has no matching fixed-effect parameter lec50; this is the expected pattern given the per-allele parameterisation.