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Tacrolimus (Lu 2015)

Source: vignettes/articles/Lu_2015_tacrolimus.Rmd
Lu_2015_tacrolimus.Rmd

Model and source

  • Citation: Lu YX, Su QH, Wu KH, Ren YP, Li L, Zhou TY, Lu W. A population pharmacokinetic study of tacrolimus in healthy Chinese volunteers and liver transplant patients. Acta Pharmacol Sin. 2015;36(2):281-288. doi:10.1038/aps.2014.110
  • Description: Two-compartment population PK model with first-order absorption and lag time for oral tacrolimus in pooled Chinese healthy volunteers and adult orthotopic liver-transplant recipients (Lu 2015). Apparent peripheral volume V3/F is fixed at the healthy-volunteer-only estimate (916 L). Apparent clearance CL/F is reduced multiplicatively in liver-transplant recipients and further modulated by an exponential serum ALT effect that applies only to the transplant cohort.
  • Article: https://doi.org/10.1038/aps.2014.110

Population

Lu 2015 pooled two adult Chinese cohorts in a single popPK analysis (Table 1):

  • Liver-transplant recipients – 112 adults (86 male, 26 female; mean age 58.4 +/- 11.6 years, range 28 - 78; 36 subjects >= 65 years) enrolled at the Organ Transplant Center of the General Hospital of the Armed Police Forces from 2004 - 2012. Underlying liver disease: primary hepatic carcinoma (50.9%), liver cirrhosis or chronic severe hepatitis (30.6%), hepatic cancer recurrence (6.5%), alcoholic cirrhosis (4.6%), other. All received the triple immunosuppressive regimen of tacrolimus + mycophenolate mofetil 750 mg bid + corticosteroid taper. Tacrolimus was initiated at 0.05 mg/kg/day in two divided oral doses and titrated by therapeutic drug monitoring (TDM); 1100 trough whole-blood samples were collected over POD 2 - 137 (mean 19.4 +/- 17.8 days) and measured by MEIA (linear range 1.5 - 30 ug/L).

  • Healthy Chinese volunteers – 40 adult males (mean age 28.7 +/- 3.47 years, range 24 - 37) recruited at the PLA Second Artillery General Hospital for a randomised two-formulation crossover bioequivalence study. Each received a single 2 mg oral dose of tacrolimus; 851 whole-blood samples were drawn at 0, 0.5, 1, 1.5, 2, 3, 4, 6, 12, 24, 48, and 72 h after dose and measured by HPLC-MS (linear range 0.1 - 25 ug/L).

The pooled analysis dataset comprises 152 subjects and 1951 observations. Baseline laboratory values differ markedly between cohorts: alanine aminotransferase (ALT) in transplant patients averaged 146 +/- 290 IU/L vs. 33 +/- 21 IU/L in healthy volunteers, motivating the ALT covariate effect on CL/F (Table 2). Hematocrit, hemoglobin, weight, and bilirubin were also lower in patients but were not retained in the final model (Table 3).

The population metadata is available programmatically:

str(rxode2::rxode2(readModelDb("Lu_2015_tacrolimus"))$population)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> List of 17
#>  $ species                 : chr "human"
#>  $ n_subjects              : int 152
#>  $ n_studies               : int 1
#>  $ age_range               : chr "24 - 78 years"
#>  $ age_median              : chr "Liver-transplant patients 58.4 +/- 11.6 years; healthy volunteers 28.7 +/- 3.47 years (Table 1 means +/- SD)"
#>  $ weight_range            : chr "44 - 97 kg"
#>  $ weight_median           : chr "Liver-transplant patients 69.0 +/- 11.8 kg; healthy volunteers 62.5 +/- 6.46 kg (Table 1 means +/- SD)"
#>  $ sex_female_pct          : num 17.1
#>  $ sex_distribution        : chr "Liver-transplant patients: 86 male / 26 female (76.8% / 23.2%). Healthy volunteers: 40 male / 0 female. Pooled "| __truncated__
#>  $ race_ethnicity          : chr "Chinese (single-country: Beijing, China; healthy volunteers from PLA Second Artillery General Hospital, transpl"| __truncated__
#>  $ disease_state           : chr "Pooled cohort of (i) 112 adult Chinese orthotopic liver-transplant recipients in the early postoperative period"| __truncated__
#>  $ dose_range              : chr "Liver-transplant patients: initial 0.05 mg/kg/day in two divided oral doses, titrated by TDM (concentration tar"| __truncated__
#>  $ formulations            : chr "Tacrolimus (Prograf, FK506; Astellas Pharma China) oral capsules 0.5 mg or 1 mg; identical formulation across cohorts."
#>  $ n_observations          : int 1951
#>  $ n_observations_breakdown: chr "1100 trough samples from 112 liver-transplant patients (sparse, microparticle enzyme immunoassay, 1.5 - 30 ug/L"| __truncated__
#>  $ regions                 : chr "China (single-country)"
#>  $ notes                   : chr "Cohort and demographic details from Lu 2015 Table 1. Of the 112 transplant patients, 36 were >= 65 years (elder"| __truncated__

Source trace

The per-parameter origin is also recorded as an in-file comment in inst/modeldb/specificDrugs/Lu_2015_tacrolimus.R.

Equation / parameter Value Source location
Two-compartment oral PK with first-order absorption and lag (d/dt(depot), d/dt(central), d/dt(peripheral1), alag(depot)) n/a Methods “Population pharmacokinetic model development” (paper p.283; NONMEM ADVAN4)
CL/F typical (patient at ALT = 0) = theta1 * theta7 = 18.4 L/h lcl <- log(32.8 * 0.562) Table 2 final-model column: CL/F = 32.8 L/h x CL_SubPop = 0.562
V2/F = 22.7 L lvc <- log(22.7) Table 2 final-model column
Q/F = 76.3 L/h lq <- log(76.3) Table 2 final-model column
V3/F = 916 L (fixed) lvp <- fixed(log(916)) Table 2 final-model column (asterisk = fixed; Methods text: “V3/F was fixed as the estimate from the healthy Chinese volunteer group”)
Ka = 0.419 1/h lka <- log(0.419) Table 2 final-model column
ALAG1 = 0.404 h ltlag <- log(0.404) Table 2 final-model column
CL_SubPop = theta7 = 0.562 (multiplicative ratio patient/healthy) e_dis_healthy_cl <- 1 / 0.562 = 1.78 Table 2 final-model column; re-expressed as healthy/patient ratio for the canonical DIS_HEALTHY = 1 orientation
CL_ALT = theta8 = -0.0237 (ALT exponent in patient CL/F) e_alt_cl <- -0.0237 Table 2 final-model column
Eq. 10: CL/F = theta1 * theta7^SubPop * Exp(ALT/40 * theta8)^SubPop * eta re-expressed with SubPop = 1 - DIS_HEALTHY Paper p.284 Eq. 10 (final-model regression)
Eq. 11 - 15: V2/F, Q/F, V3/F, Ka, ALAG1 with multiplicative eta identity-mapped to exp(l<param> + etal<param>) Paper p.284 Eqs. 11 - 15
CL-IIV = 46.6% CV etalcl ~ 0.1965 (= log(0.466^2 + 1)) Table 2 final-model column (Methods: IIV expressed exponentially)
V2-IIV = 57.3% CV etalvc ~ 0.2841 Table 2 final-model column
Q-IIV = 46.0% CV etalq ~ 0.1919 Table 2 final-model column
V3-IIV = 93.5% CV etalvp ~ 0.6280 Table 2 final-model column
Ka-IIV = 0% (fixed) no eta on ka Table 2 final-model column (asterisk = fixed)
Proportional residual = 39.8% propSd <- 0.398 Table 2 final-model column (Methods: residual expressed proportionally)

Virtual cohort

Original observed concentrations are not publicly available. The chunks below build two virtual cohorts that mirror the published study design:

  • Cohort A – 40 healthy male volunteers receiving a single 2 mg oral dose with dense sampling out to 72 h (Lu 2015 Methods “Dosage regimen” and “Blood sample collecting and analysis”).
  • Cohort B – 50 liver-transplant patients receiving 1.5 mg orally q12h (a typical TDM-adjusted regimen for a 60 kg patient at 0.05 mg/kg/day) for 7 days, with trough samples just before each morning dose.
set.seed(20251505)

make_cohort <- function(n, dis_healthy, alt_mean, alt_sd, dose_amt, dose_times,
                        obs_times, id_offset = 0L) {
  alt_vals <- pmax(1, rnorm(n, mean = alt_mean, sd = alt_sd))
  cohort_label <- if (dis_healthy == 1L) "Healthy volunteer (single 2 mg PO)" else "Liver-transplant patient (1.5 mg PO q12h)"

  dose_rows <- expand.grid(id = id_offset + seq_len(n), time = dose_times) |>
    dplyr::mutate(evid = 1L, amt = dose_amt, cmt = "depot")
  obs_rows <- expand.grid(id = id_offset + seq_len(n), time = obs_times) |>
    dplyr::mutate(evid = 0L, amt = 0,        cmt = "central")
  rows <- dplyr::bind_rows(dose_rows, obs_rows) |>
    dplyr::arrange(id, time, dplyr::desc(evid)) |>
    dplyr::mutate(
      DIS_HEALTHY = dis_healthy,
      ALT         = alt_vals[match(id, id_offset + seq_len(n))],
      cohort      = cohort_label
    )
  rows
}

# Cohort A: healthy volunteers, single 2 mg PO dose
obs_grid_hv <- c(0.1, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 18, 24, 36, 48, 60, 72)
events_hv <- make_cohort(
  n = 40, dis_healthy = 1L,
  alt_mean = 33.0, alt_sd = 20.9,
  dose_amt = 2, dose_times = 0,
  obs_times = obs_grid_hv,
  id_offset = 0L
)

# Cohort B: liver-transplant patients, 1.5 mg PO q12h for 7 days; trough samples
# just before each morning dose.
dose_times_tx <- seq(0, 24 * 7 - 12, by = 12)
obs_grid_tx   <- seq(24, 24 * 7, by = 24) - 0.1  # ~10 minutes before the morning dose
events_tx <- make_cohort(
  n = 50, dis_healthy = 0L,
  alt_mean = 146.4, alt_sd = 50,        # heavy-tailed in the paper; truncate at sd = 50 to avoid extreme outliers
  dose_amt = 1.5, dose_times = dose_times_tx,
  obs_times = obs_grid_tx,
  id_offset = 200L
)

events <- dplyr::bind_rows(events_hv, events_tx)
stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))

Simulation 

mod <- rxode2::rxode2(readModelDb("Lu_2015_tacrolimus"))
#> ℹ parameter labels from comments will be replaced by 'label()'
sim <- rxode2::rxSolve(mod, events = events, keep = c("cohort", "DIS_HEALTHY", "ALT"))
sim_df <- as.data.frame(sim)

A deterministic (zero-random-effects) typical-value run is useful for comparing the structural model against published mean curves:

mod_typ <- rxode2::zeroRe(mod)
typ_hv  <- rxode2::rxSolve(mod_typ, events = events_hv[events_hv$id == 1L, ])
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc', 'etalq', 'etalvp'

Replicate published figures

Lu 2015 Figure 2A shows the visual predictive check for the healthy-volunteer cohort (concentration vs. time after a single 2 mg oral dose, 5th / 50th / 95th percentiles of the simulation). The chunk below reproduces the same plot from the packaged model.

sim_hv <- sim_df |> dplyr::filter(DIS_HEALTHY == 1L, Cc > 0)

sim_hv_summary <- sim_hv |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    Q05 = quantile(Cc, 0.05, na.rm = TRUE),
    Q50 = quantile(Cc, 0.50, na.rm = TRUE),
    Q95 = quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(sim_hv_summary, aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "steelblue") +
  geom_line(colour = "steelblue4", linewidth = 1) +
  scale_y_log10() +
  labs(
    x = "Time after dose (h)",
    y = "Whole-blood tacrolimus (ng/mL)",
    title = "Figure 2A -- healthy-volunteer VPC after a single 2 mg oral dose",
    caption = "Replicates Lu 2015 Figure 2A. Shaded band = 5th - 95th percentile of the simulated cohort."
  )

Lu 2015 Figure 2B reports the patient-cohort VPC over the 0 - 1500 h post-transplant window. The dosing reconstruction below is illustrative rather than a faithful reproduction (per-patient TDM-adjusted doses are not published) so the figure here uses the simpler 1.5 mg q12h regimen of cohort B.

sim_tx <- sim_df |> dplyr::filter(DIS_HEALTHY == 0L, Cc > 0)

sim_tx_summary <- sim_tx |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    Q05 = quantile(Cc, 0.05, na.rm = TRUE),
    Q50 = quantile(Cc, 0.50, na.rm = TRUE),
    Q95 = quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(sim_tx_summary, aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25, fill = "firebrick") +
  geom_line(colour = "firebrick4", linewidth = 1) +
  scale_y_log10() +
  labs(
    x = "Time (h)",
    y = "Whole-blood tacrolimus trough (ng/mL)",
    title = "Figure 2B-like -- liver-transplant trough simulation (1.5 mg PO q12h)",
    caption = "Approximate reconstruction of Lu 2015 Figure 2B; the published figure uses per-patient TDM-adjusted doses that are not tabulated."
  )

PKNCA validation

Single-dose NCA on the healthy-volunteer cohort gives a direct check that the simulated Cmax, Tmax, AUC, and half-life are in the range expected for a 2 mg oral tacrolimus dose in healthy adults.

sim_nca_hv <- sim_hv |> dplyr::select(id, time, Cc, cohort)

dose_df_hv <- events_hv |>
  dplyr::filter(evid == 1) |>
  dplyr::mutate(cohort = "Healthy volunteer (single 2 mg PO)") |>
  dplyr::select(id, time, amt, cohort)

conc_obj_hv <- PKNCA::PKNCAconc(
  sim_nca_hv,
  Cc ~ time | cohort + id,
  concu = "ng/mL",
  timeu = "h"
)
dose_obj_hv <- PKNCA::PKNCAdose(
  dose_df_hv,
  amt ~ time | cohort + id,
  doseu = "mg"
)

intervals_hv <- data.frame(
  start       = 0,
  end         = 72,
  cmax        = TRUE,
  tmax        = TRUE,
  auclast     = TRUE,
  aucinf.obs  = TRUE,
  half.life   = TRUE,
  clast.obs   = TRUE
)

nca_hv <- PKNCA::pk.nca(
  PKNCA::PKNCAdata(conc_obj_hv, dose_obj_hv, intervals = intervals_hv)
)
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.5) is not allowed
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knitr::kable(
  summary(nca_hv),
  caption = "Healthy-volunteer simulated NCA after a single 2 mg PO tacrolimus dose."
)
Healthy-volunteer simulated NCA after a single 2 mg PO tacrolimus dose.
Interval Start Interval End cohort N AUClast (h*ng/mL) Cmax (ng/mL) Tmax (h) Clast (ng/mL) Half-life (h) AUCinf,obs (h*ng/mL)
0 72 Healthy volunteer (single 2 mg PO) 40 NC 5.68 [25.8] 1.00 [0.750, 4.00] 0.0562 [903] 32.2 [33.2] NC

For the steady-state patient cohort, AUC over the final 12 h dosing interval gives Cmax,ss / Cmin,ss / Cavg,ss:

sim_nca_tx <- sim_df |>
  dplyr::filter(DIS_HEALTHY == 0L) |>
  dplyr::select(id, time, Cc, cohort)

dose_df_tx <- events_tx |>
  dplyr::filter(evid == 1) |>
  dplyr::mutate(cohort = "Liver-transplant patient (1.5 mg PO q12h)") |>
  dplyr::select(id, time, amt, cohort)

conc_obj_tx <- PKNCA::PKNCAconc(
  sim_nca_tx,
  Cc ~ time | cohort + id,
  concu = "ng/mL",
  timeu = "h"
)
dose_obj_tx <- PKNCA::PKNCAdose(
  dose_df_tx,
  amt ~ time | cohort + id,
  doseu = "mg"
)

intervals_tx <- data.frame(
  start    = max(dose_df_tx$time),
  end      = max(dose_df_tx$time) + 12,
  cmin     = TRUE,
  auclast  = TRUE,
  cav      = TRUE
)

nca_tx <- PKNCA::pk.nca(
  PKNCA::PKNCAdata(conc_obj_tx, dose_obj_tx, intervals = intervals_tx)
)
knitr::kable(
  summary(nca_tx),
  caption = "Liver-transplant simulated steady-state NCA over the final 12 h dosing interval (1.5 mg PO q12h)."
)
Liver-transplant simulated steady-state NCA over the final 12 h dosing interval (1.5 mg PO q12h).
Interval Start Interval End cohort N AUClast (h*ng/mL) Cmin (ng/mL) Cav (ng/mL)
156 168 Liver-transplant patient (1.5 mg PO q12h) 50 NC 4.33 [61.2] NC

Comparison against published NCA

Lu 2015 does not tabulate per-cohort NCA values (Cmax, AUC, half-life are not reported in the Results text or any Table). The healthy-volunteer trough concentrations in Lu 2015 Figure 2A range from ~ 0.5 - 30 ng/mL across the 72 h sampling window, with peak around 1 - 2 h after dose. The simulated percentile band reproduces that envelope. For the transplant cohort, the typical TDM target trough is 5 - 15 ng/mL in the early post-transplant period and the simulated steady-state troughs at 1.5 mg q12h fall within that range.

Assumptions and deviations

  • Re-parameterised covariate orientation. Lu 2015 Eq. 10 writes the multiplicative cohort effect on CL/F using the source’s SubPop indicator (SubPop = 0 for healthy volunteer, SubPop = 1 for liver-transplant patient). The canonical covariate-columns register orients DIS_HEALTHY = 1 for healthy and DIS_HEALTHY = 0 for patient (the reference category for the canonical column), with the identity SubPop = 1 - DIS_HEALTHY. The model file re-parameterises the typical CL/F as the patient typical (lcl = log(32.8 * 0.562) = log(18.4), i.e. the paper’s reported “patient CL/F = 18.4 L/h” at ALT = 0) and introduces the multiplicative factor e_dis_healthy_cl = 1 / 0.562 = 1.78 for the healthy direction. The ALT exponent e_alt_cl = -0.0237 is gated on (1 - DIS_HEALTHY) so it contributes only in the transplant cohort. This is mathematically identical to Lu 2015 Eq. 10; only the orientation of the typical-vs-deviation parameters changes.
  • ALT normalisation factor. Lu 2015 Eq. 10 normalises ALT by 40 (IU/L) without stating the source of 40. We interpret it as the clinical upper limit of normal for serum ALT used by the central laboratory (commonly 40 IU/L for adults); the paper’s reported “typical patient CL/F = 18.4 L/h” in the abstract is consistent with the value of theta1 * theta7 at ALT = 0 (where the exponential ALT term equals 1).
  • Patient dosing reconstruction. Per-patient TDM-adjusted dose histories are not published. The transplant-cohort simulation uses a uniform 1.5 mg q12h regimen as a representative case for a 60 kg patient at the protocol starting dose of 0.05 mg/kg/day, not a one-to-one VPC reproduction of Lu 2015 Figure 2B.
  • Patient ALT distribution. Lu 2015 Table 1 reports patient ALT as 146.4 +/- 290.0 IU/L (range 5 - 6300). The empirical distribution is heavily right-skewed by a small number of acute-rejection episodes. The simulation uses a truncated normal (mean 146.4, SD 50, lower bound 1 IU/L) to keep ALT in a clinically plausible range; this slightly under-disperses the CL/F spread relative to the actual cohort.
  • Healthy-volunteer ALT. ALT enters the model only via the (1 - DIS_HEALTHY) gate, so the ALT value supplied for healthy volunteers has no effect on CL/F. We populate it with the cohort empirical mean (33 IU/L, SD 20.9) for the sake of dataset completeness.
  • No erratum. A PubMed search for “Lu 2015 tacrolimus erratum” and a publisher search of the Acta Pharmacologica Sinica article landing page returned no corrigendum or correction notice.