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Tacrolimus (Chen 2020a)

Source: vignettes/articles/Chen_2020a_tacrolimus.Rmd
Chen_2020a_tacrolimus.Rmd

Model and source

  • Citation: Chen X, Wang DD, Xu H, Li ZP. Population pharmacokinetics model and initial dose optimization of tacrolimus in children and adolescents with lupus nephritis based on real-world data. Exp Ther Med. 2020;20(2):1423-1430. doi:10.3892/etm.2020.8821.
  • Description: One-compartment population PK model with first-order absorption for oral tacrolimus whole-blood concentrations in Chinese children and adolescents with lupus nephritis (Chen 2020). The absorption rate constant ka is fixed at 4.48 1/h from the literature. Apparent oral clearance CL/F is allometrically scaled by body weight (fixed exponent 0.75, reference 70 kg) and reduced by 29% with concomitant Wuzhi capsule; apparent volume V/F is scaled linearly by body weight. Exponential IIV on CL/F only; proportional residual error.
  • Article: https://doi.org/10.3892/etm.2020.8821 (open access, PMC7388563)

Chen et al. (2020) fitted a one-compartment model with first-order absorption to routine therapeutic-drug-monitoring whole-blood concentrations of oral tacrolimus in children and adolescents with lupus nephritis (LN), and used Monte Carlo simulation to recommend weight-banded initial doses with and without the CYP3A-inhibiting Chinese patent medicine Wuzhi capsule. The absorption rate constant was fixed at 4.48 1/h from earlier pediatric tacrolimus models (the paper’s references 13 and 16-18).

Population

Thirty-two Chinese pediatric and adolescent LN patients (5 male, 27 female) treated at the Children’s Hospital of Fudan University between August 2014 and September 2019. From Table I: age median 13.87 (range 2.86-17.99) years, body weight median 47.00 (17.00-66.50) kg. All patients received a glucocorticoid and 12 of 32 received Wuzhi capsule, the only co-medication retained on CL/F.

Source trace

Element Value Source
Structure: 1-compartment, first-order absorption and elimination – Methods, Population pharmacokinetics modeling
lka 4.48 1/h (fixed) Table II; Methods
lcl 15.5 L/h Table II; Results equation vi
lvc 174 L Table II; Results equation vii
e_wt_cl 0.75 (fixed), reference 70 kg Methods equation iii; equation vi
e_wt_vc 1 (fixed), reference 70 kg Methods equation iii; equation vii
e_conmed_wuzhi_cl -0.290 Table II (theta WZ); equation vi
Wuzhi form (1 + WZ * theta) – Methods equation v; equation vi
etalcl 0.172^2 = 0.029584 Table II (omega CL/F = 0.172, read as SD; see below)
Exponential IIV A = T(A) * exp(eta) – Methods equation i
propSd 0.281 Table II (sigma1, proportional); Methods equation ii

Apparent clearance by weight (Figure 3)

Figure 3 plots the typical weight-normalised CL/F with and without Wuzhi capsule at the six simulated body weights. The values below were digitised by the maintainers from Figure 3.

mod <- readModelDb("Chen_2020a_tacrolimus")
mod_typ <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'

fig3 <- tibble::tribble(
  ~WT, ~CONMED_WUZHI, ~pub_cl_kg,
  10, 0, 0.360, 20, 0, 0.302, 30, 0, 0.272,
  40, 0, 0.253, 50, 0, 0.240, 60, 0, 0.230,
  10, 1, 0.254, 20, 1, 0.214, 30, 1, 0.193,
  40, 1, 0.180, 50, 1, 0.170, 60, 1, 0.163
)

ev3 <- fig3 |>
  mutate(id = row_number(), time = 0, amt = 0, evid = 0L, cmt = "central") |>
  select(id, time, amt, evid, cmt, WT, CONMED_WUZHI)
cl3 <- rxode2::rxSolve(mod_typ, events = ev3, keep = c("WT", "CONMED_WUZHI")) |>
  as.data.frame() |>
  select(WT, CONMED_WUZHI, cl)
#> ℹ omega/sigma items treated as zero: 'etalcl'
#> Warning: multi-subject simulation without without 'omega'

cmp3 <- inner_join(fig3, cl3, by = c("WT", "CONMED_WUZHI")) |>
  mutate(sim_cl_kg = cl / WT, pct_diff = 100 * (sim_cl_kg - pub_cl_kg) / pub_cl_kg)

cmp3 |>
  select(WT, CONMED_WUZHI, pub_cl_kg, sim_cl_kg, pct_diff) |>
  dplyr::rename(
    "Weight (kg)" = WT, "Wuzhi capsule" = CONMED_WUZHI,
    "Figure 3 CL/F (L/h/kg)" = pub_cl_kg, "Model CL/F (L/h/kg)" = sim_cl_kg,
    "% diff" = pct_diff
  ) |>
  knitr::kable(digits = 3, caption = "Replicates Figure 3 of Chen 2020.")
Replicates Figure 3 of Chen 2020.
Weight (kg) Wuzhi capsule Figure 3 CL/F (L/h/kg) Model CL/F (L/h/kg) % diff
10 0 0.360 0.360 0.047
20 0 0.302 0.303 0.287
30 0 0.272 0.274 0.614
40 0 0.253 0.255 0.664
50 0 0.240 0.241 0.359
60 0 0.230 0.230 0.056
10 1 0.254 0.256 0.678
20 1 0.214 0.215 0.484
30 1 0.193 0.194 0.677
40 1 0.180 0.181 0.457
50 1 0.170 0.171 0.595
60 1 0.163 0.163 0.240 

# Typical values against a digitised curve: the only difference is reading
# error off the figure.
stopifnot(all(abs(cmp3$pct_diff) < 3))

Probability of target attainment (Figure 4)

The paper simulates six body weights (10-60 kg) and seven initial regimens (0.01-0.30 mg/kg/day split into two doses) with and without Wuzhi capsule, and plots the probability that the concentration falls in the 5-15 ng/mL window. We replicate it with 200 subjects per weight-by-dose-by-Wuzhi arm (the package cap), simulating 7 days of twice-daily dosing and reading the steady-state trough at the end of the last dosing interval. The model’s Cc output is the individual prediction (no residual error); this is what reproduces Figure 4 (see Assumptions). The published probabilities below were digitised by the maintainers from Figure 4 at the six simulated weights.

weights <- c(10, 20, 30, 40, 50, 60)
doses_kg <- c(0.01, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30)
nsub <- 200
ndays <- 7
tobs <- ndays * 24

arms <- tidyr::expand_grid(CONMED_WUZHI = 0:1, dose_kg = doses_kg, WT = weights) |>
  mutate(arm = row_number())

subj <- arms |>
  tidyr::uncount(nsub) |>
  mutate(id = row_number())
dtimes <- seq(0, by = 12, length.out = 2 * ndays)

ev_dose <- subj |>
  tidyr::expand_grid(time = dtimes) |>
  mutate(amt = dose_kg * WT / 2, evid = 1L, cmt = "depot")
ev_obs <- subj |>
  mutate(time = tobs, amt = 0, evid = 0L, cmt = "central")
ev4 <- bind_rows(ev_dose, ev_obs) |>
  arrange(id, time, desc(evid)) |>
  select(id, time, amt, evid, cmt, WT, CONMED_WUZHI, arm)

rxode2::rxSetSeed(2020)
sim4 <- rxode2::rxSolve(mod, events = ev4, keep = "arm") |>
  as.data.frame() |>
  filter(time == tobs)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> [====|====|====|====|====|====|====|====|====|====] 0:00:21

pta <- sim4 |>
  group_by(arm) |>
  summarise(sim_pta = 100 * mean(Cc >= 5 & Cc <= 15), .groups = "drop") |>
  left_join(arms, by = "arm")
pub4 <- tibble::tribble(
  ~CONMED_WUZHI, ~dose_kg, ~p10, ~p20, ~p30, ~p40, ~p50, ~p60,
  0, 0.01, 0, 0, 0, 0, 0, 0,
  0, 0.05, 1, 9, 22, 36, 47, 56,
  0, 0.10, 45, 82, 91, 93, 93, 91,
  0, 0.15, 84, 88, 78, 65, 53, 44,
  0, 0.20, 85, 65, 44, 26, 17, 12,
  0, 0.25, 73, 37, 17, 10, 6, 4,
  0, 0.30, 55, 18, 9, 4, 2, 0,
  1, 0.01, 0, 0, 0, 0, 0, 0,
  1, 0.05, 34, 72, 88, 93, 96, 98,
  1, 0.10, 93, 85, 67, 53, 41, 31,
  1, 0.15, 66, 28, 12, 6, 4, 2,
  1, 0.20, 28, 6, 1, 0, 0, 0,
  1, 0.25, 11, 1, 0, 0, 0, 0,
  1, 0.30, 5, 0, 0, 0, 0, 0
) |>
  tidyr::pivot_longer(p10:p60, names_to = "WT", values_to = "pub_pta") |>
  mutate(WT = as.numeric(sub("p", "", WT)))

cmp4 <- inner_join(pub4, pta, by = c("CONMED_WUZHI", "dose_kg", "WT")) |>
  mutate(diff_pts = sim_pta - pub_pta)

cmp4 |>
  filter(dose_kg != 0.01) |>
  mutate(cell = sprintf("%.0f / %.0f", pub_pta, sim_pta)) |>
  select(CONMED_WUZHI, dose_kg, WT, cell) |>
  tidyr::pivot_wider(names_from = WT, values_from = cell, names_prefix = "wt_") |>
  dplyr::rename(
    "Wuzhi capsule" = CONMED_WUZHI, "Dose (mg/kg/day)" = dose_kg,
    "10 kg" = wt_10, "20 kg" = wt_20, "30 kg" = wt_30,
    "40 kg" = wt_40, "50 kg" = wt_50, "60 kg" = wt_60
  ) |>
  knitr::kable(caption = "Replicates Figure 4 of Chen 2020: probability (%) of a trough in 5-15 ng/mL, published / simulated.")
Replicates Figure 4 of Chen 2020: probability (%) of a trough in 5-15 ng/mL, published / simulated.
Wuzhi capsule Dose (mg/kg/day) 10 kg 20 kg 30 kg 40 kg 50 kg 60 kg
0 0.05 1 / 0 9 / 10 22 / 16 36 / 30 47 / 38 56 / 56
0 0.10 45 / 34 82 / 72 91 / 88 93 / 93 93 / 96 91 / 94
0 0.15 84 / 76 88 / 94 78 / 85 65 / 74 53 / 62 44 / 44
0 0.20 85 / 88 65 / 73 44 / 48 26 / 32 17 / 24 12 / 16
0 0.25 73 / 72 37 / 46 17 / 20 10 / 16 6 / 6 4 / 2
0 0.30 55 / 61 18 / 25 9 / 10 4 / 3 2 / 2 0 / 2
1 0.05 34 / 26 72 / 67 88 / 82 93 / 93 96 / 97 98 / 98
1 0.10 93 / 94 85 / 87 67 / 72 53 / 54 41 / 40 31 / 25
1 0.15 66 / 72 28 / 32 12 / 14 6 / 8 4 / 4 2 / 1
1 0.20 28 / 34 6 / 12 1 / 2 0 / 3 0 / 0 0 / 0
1 0.25 11 / 14 1 / 3 0 / 0 0 / 0 0 / 0 0 / 0
1 0.30 5 / 4 0 / 0 0 / 0 0 / 0 0 / 0 0 / 0 
# The probabilities depend jointly on the typical values, the Wuzhi effect and
# the IIV scale; a wrong omega reading (variance instead of SD) flattens every
# curve to 40-60% and fails both bounds by a wide margin. Bounds are on the
# centre and the 90th percentile of the absolute difference (percentage
# points), robust to Monte Carlo noise at n = 200 per arm and to digitisation.
stopifnot(
  median(abs(cmp4$diff_pts)) < 5,
  quantile(abs(cmp4$diff_pts), 0.9) < 15
)
cmp4 |>
  mutate(
    dose = factor(sprintf("%.2f mg/kg/day", dose_kg)),
    panel = ifelse(CONMED_WUZHI == 1, "(B) with Wuzhi capsule", "(A) without Wuzhi capsule")
  ) |>
  ggplot(aes(WT, colour = dose)) +
  geom_line(aes(y = sim_pta)) +
  geom_point(aes(y = pub_pta)) +
  facet_wrap(~panel) +
  labs(x = "Body weight (kg)", y = "Probability of target attainment (%)", colour = "Dose")
Simulated probability of a 5-15 ng/mL trough (lines) against values digitised from Figure 4 of Chen 2020 (points). Replicates Figure 4 of Chen 2020.

Simulated probability of a 5-15 ng/mL trough (lines) against values digitised from Figure 4 of Chen 2020 (points). Replicates Figure 4 of Chen 2020.

The recommended regimens of Table III follow from these curves: without Wuzhi capsule 0.15 mg/kg/day below 23 kg and 0.10 mg/kg/day above; with Wuzhi capsule 0.10 and 0.05 mg/kg/day respectively.

PKNCA validation

The paper reports no NCA. As a structural check, typical-value subjects (zeroRe()) at each Figure 4 weight, with and without Wuzhi capsule, receive 0.10 mg/kg/day split into two doses for 7 days. PKNCA computes the steady-state AUC over the last dosing interval, which must equal the closed form Dose / (CL/F).

tlast <- tobs - 12
ev_nca <- tidyr::expand_grid(CONMED_WUZHI = 0:1, WT = weights) |>
  mutate(id = row_number())
nca_dose <- ev_nca |>
  tidyr::expand_grid(time = dtimes) |>
  mutate(amt = 0.10 * WT / 2, evid = 1L, cmt = "depot")
nca_obs <- ev_nca |>
  tidyr::expand_grid(time = tlast + c(0, 0.25, 0.5, 1, 1.5, 2, 3, 4, 6, 8, 10, 12)) |>
  mutate(amt = 0, evid = 0L, cmt = "central")
ev_nca_all <- bind_rows(nca_dose, nca_obs) |>
  arrange(id, time, desc(evid)) |>
  mutate(treatment = paste0(WT, " kg, WZ=", CONMED_WUZHI))

sim_nca <- rxode2::rxSolve(mod_typ, events = ev_nca_all, keep = c("WT", "CONMED_WUZHI", "treatment")) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalcl'
#> Warning: multi-subject simulation without without 'omega'

conc <- sim_nca |>
  filter(!is.na(Cc), time >= tlast) |>
  select(id, time, Cc, treatment)
doses <- ev_nca_all |>
  filter(evid == 1, time == tlast) |>
  select(id, time, amt, treatment)

o_conc <- PKNCAconc(conc, Cc ~ time | treatment + id)
o_dose <- PKNCAdose(doses, amt ~ time | treatment + id)
intervals <- data.frame(start = tlast, end = tlast + 12, auclast = TRUE, cmin = TRUE, cmax = TRUE)
nca_res <- pk.nca(PKNCAdata(o_conc, o_dose, intervals = intervals))

nca_wide <- as.data.frame(nca_res) |>
  select(treatment, PPTESTCD, PPORRES) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES) |>
  left_join(distinct(ev_nca_all, treatment, WT, CONMED_WUZHI), by = "treatment") |>
  mutate(auc_closed = 1000 * 0.10 * WT / 2 / (15.5 * (WT / 70)^0.75 * (1 - 0.290 * CONMED_WUZHI))) |>
  arrange(CONMED_WUZHI, WT)

nca_wide |>
  select(WT, CONMED_WUZHI, auclast, auc_closed, cmax, cmin) |>
  dplyr::rename(
    "Weight (kg)" = WT, "Wuzhi capsule" = CONMED_WUZHI,
    "AUCtau (PKNCA, ng*h/mL)" = auclast, "Dose/(CL/F) (ng*h/mL)" = auc_closed,
    "Cmax (ng/mL)" = cmax, "Cmin (ng/mL)" = cmin
  ) |>
  knitr::kable(digits = 2, caption = "Typical-value steady-state NCA, 0.10 mg/kg/day in two doses.")
Typical-value steady-state NCA, 0.10 mg/kg/day in two doses.
Weight (kg) Wuzhi capsule AUCtau (PKNCA, ng*h/mL) Dose/(CL/F) (ng*h/mL) Cmax (ng/mL) Cmin (ng/mL)
10 0 138.21 138.82 21.58 4.43
20 0 164.48 165.09 23.59 6.24
30 0 182.10 182.70 24.96 7.50
40 0 195.72 196.33 26.03 8.51
50 0 206.98 207.59 26.91 9.35
60 0 216.67 217.27 27.68 10.08
10 1 194.92 195.53 25.96 8.45
20 1 231.92 232.52 28.89 11.25
30 1 256.72 257.33 30.88 13.17
40 1 275.91 276.51 32.42 14.67
50 1 291.77 292.38 33.70 15.92
60 1 305.41 306.01 34.80 17.00 

# Same typical parameters on both sides: the only difference is trapezoidal
# error over a smooth 12-point interval, so a tight bound applies.
stopifnot(all(abs(nca_wide$auclast / nca_wide$auc_closed - 1) < 0.02))

Assumptions and deviations

  • IIV scale. Table II reports “omega CL/F = 0.172” without stating whether it is a variance or an SD. Simulating Figure 4 under both readings settles it: with 0.172 as the SD of eta (variance 0.029584) the simulated probabilities follow the published curves (e.g. 40 kg, 0.10 mg/kg/day, no Wuzhi: about 93% simulated vs 93% published; 60 kg, 0.05 mg/kg/day with Wuzhi: about 97% vs 98%), whereas treating 0.172 as the variance (SD 0.415) flattens every curve to roughly 40-60% (about 56% and 64% at those two points). The SD reading is used, the same convention the sister paper from this centre (Wang 2019, SOJIA; Wang_2019b_tacrolimus) resolves to.
  • Residual error scale and Figure 4. sigma1 = 0.281 is taken as the proportional SD (28.1%), consistent with the omega reading. Figure 4 is reproduced by individual predictions without residual error; adding the proportional error lowers the 40 kg, 0.10 mg/kg/day probability to about 81%, well below the published 93%.
  • Sampling time. The paper does not state when the TDM samples were drawn. The fixed ka and the Figure 4 replication (steady-state trough before the next dose) are consistent with trough sampling.
  • Figure 4 regimen. “Split into two doses” is read as equal doses every 12 h. The simulation runs 7 days (more than 10 half-lives at every weight) in place of an explicit steady-state solve.
  • Digitisation. Figure 3 and Figure 4 values were read off the published figures at the six simulated weights; Excel-style smoothing between those points is ignored. The largest Figure 4 discrepancy (about 8 percentage points at 20 kg without Wuzhi) is at the steepest part of the curves.
  • Wuzhi capsule indicator. The paper does not state whether it was time-varying; it is accepted per record. The Wuzhi dose is not reported.
  • Fixed ka is inherited from earlier pediatric models and is not identifiable from TDM data.
  • The body-weight reference (70 kg) lies above the observed median (47 kg); the typical CL/F of 15.5 L/h is an adult-equivalent value.