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Tacrolimus starting dose in adult renal transplant recipients (Andrews 2019)

Source: vignettes/articles/Andrews_2019_tacrolimus.Rmd
Andrews_2019_tacrolimus.Rmd

The paper

Andrews LM, Hesselink DA, van Schaik RHN, van Gelder T, de Fijter JW, Lloberas N, Elens L, Moes DJAR, de Winter BCM. A population pharmacokinetic model to predict the individual starting dose of tacrolimus in adult renal transplant recipients. Br J Clin Pharmacol. 2019;85(3):601-615. doi:10.1111/bcp.13838

Andrews 2019 pooled 4527 whole-blood tacrolimus concentrations from 337 adult kidney transplant recipients over the first three months post-transplant and built two models:

  1. a final model carrying every significant covariate, intended for dose adjustment once post-transplant laboratory values exist, and
  2. a starting-dose model restricted to covariates that are known before transplantation, paired with a closed-form dosing algorithm.

Both are distributed here, and both are exercised in this one vignette.

mod_final <- modellib("Andrews_2019_tacrolimus")
mod_start <- modellib("Andrews_2019_tacrolimus_startingdose")

rx_final <- rxode2::rxode2(mod_final)
#> ℹ parameter labels from comments will be replaced by 'label()'
rx_start <- rxode2::rxode2(mod_start)
#> ℹ parameter labels from comments will be replaced by 'label()'

rx_final$state
#> [1] "depot"       "central"     "peripheral1"

Both models are two-compartment with first-order absorption and an absorption lag. Bioavailability could not be estimated and was fixed to 1, so every clearance and volume is an apparent value (CL/F, V1/F, Q/F, V2/F).

Population

Two Dutch cohorts were pooled for model building (Table 1):

Rotterdam (Erasmus MC) Leiden (LUMC)
n 237 100
Samples 3661 866
Age (years) 58.5 (19.4-79.4) 54.0 (15.0-77.0)
Bodyweight (kg) 79.4 (37.6-132.0) 74.0 (40.0-114.0)
BSA (m^2) 2.03 (1.24-2.66) 1.90 (1.33-2.48)
Lean bodyweight (kg) 64.0 (33.1-85.3) 55.9 (33.6-81.7)
Haematocrit (L/L) 0.34 (0.15-0.80) 0.34 (0.24-0.45)
Creatinine (umol/L) 137 (38-1885) 124 (62-920)
Albumin (g/L) 42 (12-57) not measured
Assay ACMIA / EMIT immunoassay LC-MS/MS

All patients received oral twice-daily immediate-release tacrolimus (Prograft) with mycophenolic acid, dosed by therapeutic drug monitoring. An external validation cohort of 304 further patients was used for both models.

Table 1 reports each cohort separately and publishes no combined-cohort row. The combined-cohort medians that the models are actually centred on are recovered from the Supporting Information Data S1 NONMEM control stream, which hardcodes them: age 55.72 years, albumin 42 g/L, BSA 1.93 m^2, creatinine 134.98 umol/L, haematocrit 0.34 L/L, and lean bodyweight 58.94 kg.

Source trace

Every value in both model files, and where it came from.

Quantity Value Source
Structure (2-cmt, 1st-order abs, lag) ADVAN4 TRANS4 Methods ‘Base model development’; Data S1 $SUBROUTINE
Bioavailability F fixed to 1 Methods ‘Base model development’
tlag, ka, CL/F, V1/F, Q/F, V2/F (final) 0.38 h, 3.58 1/h, 23.0 L/h, 692 L, 11.6 L/h, 5340 L Table 2, ‘Final model’ column
tlag, ka, CL/F, V1/F, Q/F, V2/F (start) 0.39 h, 3.70 1/h, 22.5 L/h, 685 L, 10.6 L/h, 6590 L Table 2, ‘Starting dose model’ column
CL/F covariate equation (final) 7 covariates, multiplicative Equation (1), p.607
CL/F covariate equation (starting dose) 4 covariates, multiplicative Equation (2), p.608
Starting-dose algorithm Dose = CL/F * AUC Equation (3), p.608
V1/F covariate (LBM exponent 1.52) exponent only Table 2 (no equation published)
V1/F covariate centring value LBW / 58.94 kg Data S1 control stream, ’V2LBW = ((LBW/58.94)**THETA(15))’
Age / creatinine centring values 55.72 years, 134.98 umol/L Data S1 control stream (Eq. 1 prints 56 and 135)
Albumin / BSA / haematocrit centring 42 g/L, 1.93 m^2, 0.34 L/L Equation (1) and Data S1 control stream
Exponent signs (age, creatinine, HCT negative) -0.43, -0.14, -0.76 Table 2; Results prose; Data S1 $THETA bounds
IIV (CV%) on CL/F, V1/F, V2/F, Q/F 38.6, 49.2, 53.0, 78.7 (final) Table 2
IOV on CL/F 13.6% (final), 14.6% (start) Table 2 (not encoded here; see Assumptions)
Residual error, immunoassay prop 17.7% + add 0.88 ng/mL (final) Table 2; Data S1 $ERROR DVID 1 branch
Residual error, LC-MS/MS prop 24.5%, no additive (final) Table 2; Data S1 $ERROR DVID 2 branch
Concentration scaling S2 = V2 / 1000 (mg dose -> ng/mL) Data S1 control stream
C0 to AUC0-12h mapping 10 -> 222; 12.5 -> 277; 15 -> 332 Results ‘Starting dose model’, p.608
Simulation trial outcome median C0 13.9 vs 12.9 ng/mL Results ‘Simulation trial’; Figure 5

A note on the exponent signs. The three negative CL/F exponents (age, creatinine, haematocrit) are printed with their minus signs in Table 2 of the published PDF, but text extraction drops them. They are confirmed here three independent ways: (a) the Results sentence “Higher body surface area (BSA), lower serum creatinine, younger age, higher albumin and lower haematocrit levels were identified as covariates enhancing tacrolimus clearance”; (b) the Data S1 control stream’s $THETA bounds, which carry negative initial estimates for CLAGE, CLCREA and CLHCT and positive ones for CLALB and CLBSA; and (c) the paper’s own arithmetic, reproduced below.

Structural check: the closed-form AUC identity

For any linear model, the single-dose AUC extrapolated to infinity is exactly Dose / (CL/F). This is a pure numerical-accuracy check – both sides use the same parameters – so it is asserted tightly.

ref_cov <- data.frame(
  AGE = 55.72, ALB = 42, BSA = 1.93, CREAT = 134.98, HCT = 34, LBM = 58.94,
  CYP3A5_EXPR = 0, SNP_CYP3A4_RS35599367 = 0, IMMUNOASSAY = 0
)

# A long, tapering grid: the terminal half-life is about 20 days, so the tail
# must be followed for months before the trapezoid converges on AUC-inf.
grid_long <- sort(unique(c(
  seq(0, 24, by = 0.25), seq(24, 168, by = 2), seq(168, 24 * 200, by = 12)
)))

ev_single <- rxode2::et(amt = 5, cmt = "depot") |>
  rxode2::et(grid_long, cmt = "central")
d_single <- as.data.frame(ev_single)
for (nm in names(ref_cov)) d_single[[nm]] <- ref_cov[[nm]]

sim_single <- rxode2::rxSolve(
  rx_final, d_single, omega = NA, sigma = NA, returnType = "data.frame"
) |>
  dplyr::filter(!is.na(Cc))

trapz <- function(x, y) sum(diff(x) * (utils::head(y, -1) + utils::tail(y, -1)) / 2)

auc_sim <- trapz(sim_single$time, sim_single$Cc)
auc_cf <- 5 / 23.0 * 1000 # Dose (mg) / (CL/F, L/h) -> mg*h/L = ug*h/mL -> ng*h/mL

c(simulated = auc_sim, closed_form = auc_cf,
  pct_diff = 100 * (auc_sim - auc_cf) / auc_cf)
#>    simulated  closed_form     pct_diff 
#> 217.41305641 217.39130435   0.01000595
# Same parameters on both sides: the only difference is trapezoid error on a
# finite grid, so a tight bound is correct here.
stopifnot(abs(100 * (auc_sim - auc_cf) / auc_cf) < 0.5)

The terminal half-life is worth stating explicitly, because it governs how this model must be simulated:

kel <- 23.0 / 692
k12 <- 11.6 / 692
k21 <- 11.6 / 5340
a <- kel + k12 + k21
b <- kel * k21
beta <- (a - sqrt(a^2 - 4 * b)) / 2
alpha <- (a + sqrt(a^2 - 4 * b)) / 2
c(alpha_half_life_h = log(2) / alpha,
  beta_half_life_h = log(2) / beta,
  beta_half_life_days = log(2) / beta / 24)
#>   alpha_half_life_h    beta_half_life_h beta_half_life_days 
#>            13.65818           487.21392            20.30058

The apparent peripheral volume is very large (5340 L) against a modest inter-compartmental clearance (11.6 L/h), giving a terminal half-life of about 20 days. True steady state is therefore not reached within the paper’s 3-month window, and certainly not at the day-10 timepoint the paper simulates. Simulations below follow the paper and evaluate exposure at a stated day, not at mathematical steady state.

Covariate effects: reproducing the paper’s own numbers

Andrews 2019 states four quantitative covariate claims in the Results and Discussion. Each is a deterministic function of the ini() values, so each is checked exactly.

e <- function(model, nm) {
  ini <- as.data.frame(rxode2::rxode2(model)$iniDf)
  ini$est[match(nm, ini$name)]
}

cyp3a5 <- e(mod_final, "e_cyp3a5_expr_cl")
#> ℹ parameter labels from comments will be replaced by 'label()'
cyp3a4 <- e(mod_final, "e_cyp3a4_22_cl")
#> ℹ parameter labels from comments will be replaced by 'label()'
age_ex <- e(mod_final, "e_age_cl")
#> ℹ parameter labels from comments will be replaced by 'label()'
bsa_ex <- e(mod_final, "e_bsa_cl")
#> ℹ parameter labels from comments will be replaced by 'label()'

# Two kinds of check, which must not be conflated.
#
# (a) TRANSCRIPTION: the ini() value against the number the paper actually
#     prints in Equation (1). This must be exact -- any difference is a typo.
# (b) DERIVED BEHAVIOUR: a quantity the paper states in rounded prose
#     ("34%", "1.6 times"), recomputed from the model. Here the tolerance has
#     to admit the paper's own rounding, not the model's error.
transcription <- tibble::tribble(
  ~Quantity, ~Source, ~Paper, ~Model,
  "CYP3A5 expresser multiplier on CL/F", "Equation (1)", 1.631, cyp3a5,
  "CYP3A4*22 multiplier on CL/F", "Equation (1)", 0.8, cyp3a4,
  "Age exponent on CL/F", "Table 2", -0.43, age_ex,
  "BSA exponent on CL/F", "Table 2", 0.88, bsa_ex
)

derived <- tibble::tribble(
  ~Claim, ~Source, ~Paper, ~Model,
  "Age 25 -> 65 years lowers CL/F by 34%", "Results / Simulations", 34,
  100 * (1 - (65 / 25)^age_ex),
  "BSA 1.5 vs 2.25 m^2 changes CL/F by 43%", "Results / Simulations", 43,
  100 * ((2.25 / 1.5)^bsa_ex - 1)
) |>
  dplyr::mutate(`% diff` = 100 * (Model - Paper) / Paper)

knitr::kable(transcription, digits = 4,
             caption = "Transcription against the printed equation and table")
Transcription against the printed equation and table
Quantity Source Paper Model
CYP3A5 expresser multiplier on CL/F Equation (1) 1.631 1.631
CYP3A4*22 multiplier on CL/F Equation (1) 0.800 0.800
Age exponent on CL/F Table 2 -0.430 -0.430
BSA exponent on CL/F Table 2 0.880 0.880 
knitr::kable(derived, digits = 2,
             caption = "Derived behaviour against the paper's rounded prose")
Derived behaviour against the paper’s rounded prose
Claim Source Paper Model % diff
Age 25 -> 65 years lowers CL/F by 34% Results / Simulations 34 33.69 -0.90
BSA 1.5 vs 2.25 m^2 changes CL/F by 43% Results / Simulations 43 42.88 -0.29 
# (a) Transcription is exact by construction -- these ARE the printed values.
stopifnot(max(abs(transcription$Model - transcription$Paper)) < 1e-9)

# (b) Derived behaviour. The paper rounds to two significant figures, so the
# achievable agreement is limited by ITS rounding, not the model's: the exact
# values are 33.69% and 42.88% against a printed 34% and 43%. 1.5% admits that
# rounding; a mis-transcribed exponent moves these by tens of percent.
stopifnot(max(abs(derived$`% diff`)) < 1.5)

Transcription is exact. The two derived percentages land at 33.69% and 42.88% against the paper’s printed 34% and 43% – agreement limited only by the paper’s own two-significant-figure rounding.

The paper’s prose also says CYP3A5 expressers have “1.6 times” higher CL/F and CYP3A4*22 carriers “0.8 times” lower; those are the rounded forms of the 1.631 and 0.8 that Equation (1) prints and that the model files carry.

Figure 3: simulated profiles across covariate levels

Figure 3 of the paper shows typical-value profiles at the covariate levels listed in its caption. All patients received 0.2 mg/kg/day split into two equal doses; the panels below use a 75 kg reference patient (7.5 mg twice daily) and hold every other covariate at the model’s centring value.

profile_for <- function(overrides, label) {
  cov <- ref_cov
  for (nm in names(overrides)) cov[[nm]] <- overrides[[nm]]
  ev <- rxode2::et(amt = 7.5, ii = 12, until = 24 * 14, cmt = "depot") |>
    rxode2::et(seq(24 * 14, 24 * 14 + 12, by = 0.25), cmt = "central")
  d <- as.data.frame(ev)
  for (nm in names(cov)) d[[nm]] <- cov[[nm]]
  rxode2::rxSolve(rx_final, d, omega = NA, sigma = NA, returnType = "data.frame") |>
    dplyr::filter(!is.na(Cc)) |>
    dplyr::mutate(time = time - 24 * 14, level = label)
}

panels <- dplyr::bind_rows(
  purrr_map <- dplyr::bind_rows(lapply(c(0, 1), function(v) {
    profile_for(list(CYP3A5_EXPR = v), c("non-expresser", "expresser")[v + 1])
  })) |> dplyr::mutate(panel = "A: CYP3A5 genotype"),
  dplyr::bind_rows(lapply(c(0, 1), function(v) {
    profile_for(list(SNP_CYP3A4_RS35599367 = v), c("CYP3A4*1", "CYP3A4*22")[v + 1])
  })) |> dplyr::mutate(panel = "B: CYP3A4 genotype"),
  dplyr::bind_rows(lapply(c(25, 40, 65, 80), function(v) {
    profile_for(list(AGE = v), paste(v, "years"))
  })) |> dplyr::mutate(panel = "C: age"),
  dplyr::bind_rows(lapply(c(30, 35, 40, 45, 50), function(v) {
    profile_for(list(ALB = v), paste(v, "g/L"))
  })) |> dplyr::mutate(panel = "D: albumin"),
  dplyr::bind_rows(lapply(c(1.5, 1.75, 2, 2.25), function(v) {
    profile_for(list(BSA = v), paste(v, "m^2"))
  })) |> dplyr::mutate(panel = "E: BSA"),
  dplyr::bind_rows(lapply(c(50, 100, 200, 500), function(v) {
    profile_for(list(CREAT = v), paste(v, "umol/L"))
  })) |> dplyr::mutate(panel = "F: creatinine"),
  dplyr::bind_rows(lapply(c(25, 30, 35, 40, 45), function(v) {
    profile_for(list(HCT = v), paste0(v, "%"))
  })) |> dplyr::mutate(panel = "G: haematocrit"),
  dplyr::bind_rows(lapply(c(40, 50, 60, 70, 80), function(v) {
    profile_for(list(LBM = v), paste(v, "kg"))
  })) |> dplyr::mutate(panel = "H: lean body mass")
)

ggplot(panels, aes(time, Cc, colour = level)) +
  geom_line(linewidth = 0.6) +
  facet_wrap(~panel, ncol = 2, scales = "free_y") +
  labs(
    x = "Time after dose (h)", y = "Tacrolimus (ng/mL)", colour = NULL,
    title = "Replicates Figure 3 of Andrews 2019",
    subtitle = "Typical-value profiles over one dosing interval, 7.5 mg twice daily"
  ) +
  theme_bw(base_size = 9) +
  theme(legend.position = "right")

Panel H is the one panel whose equation is not in the paper: the lean-body-mass effect on V1/F appears only as an exponent in Table 2, and its centring value (58.94 kg) comes from the Data S1 control stream. Note that it moves the peak and the shape of the profile but not the area, because it acts on a volume and not on clearance – which is exactly why it does not appear in the starting-dose model, whose purpose is to set a dose from clearance.

The starting-dose algorithm (Equation 3)

Equation (3) sets the twice-daily dose for a target pre-dose concentration of 10 ng/mL, which the paper maps to an AUC0-12h of 222 ng*h/mL:

Dose (mg) = 222 * 22.5 * [CYP3A5 factor] * [CYP3A4 factor]
            * (Age/56)^-0.50 * (BSA/1.93)^0.72 / 1000

This is Dose = AUC * CL/F with the starting-dose model’s CL/F substituted in. Reimplementing it from the model file’s own parameters must reproduce the paper’s equation exactly.

cl_start <- function(age, bsa, cyp3a5_expr, cyp3a4_22) {
  tv <- exp(e(mod_start, "lcl"))
  f5 <- 1 + (e(mod_start, "e_cyp3a5_expr_cl") - 1) * cyp3a5_expr
  f4 <- 1 + (e(mod_start, "e_cyp3a4_22_cl") - 1) * cyp3a4_22
  tv * f5 * f4 *
    (age / 55.72)^e(mod_start, "e_age_cl") *
    (bsa / 1.93)^e(mod_start, "e_bsa_cl")
}

# Dose for a target AUC0-12h, on a twice-daily schedule.
start_dose <- function(age, bsa, cyp3a5_expr, cyp3a4_22, auc_target = 222) {
  auc_target * cl_start(age, bsa, cyp3a5_expr, cyp3a4_22) / 1000
}

examples <- expand.grid(
  age = c(25, 55.72, 75), bsa = c(1.5, 1.93, 2.25),
  cyp3a5_expr = c(0, 1), cyp3a4_22 = c(0, 1)
) |>
  dplyr::mutate(dose_mg_q12h = start_dose(age, bsa, cyp3a5_expr, cyp3a4_22))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'

# The reference patient is the headline case: Equation (3) reduces to
# 222 * 22.5 / 1000 = 4.995 mg twice daily.
ref_dose <- start_dose(55.72, 1.93, 0, 0)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
c(model = ref_dose, equation_3 = 222 * 22.5 / 1000)
#>      model equation_3 
#>      4.995      4.995
stopifnot(abs(ref_dose - 222 * 22.5 / 1000) < 1e-9)

# The paper's Conclusions: "The tacrolimus starting dose should be increased to
# 160% in individuals carrying the CYP3A5*1 allele, whereas it should be
# reduced to 80% in patients carrying the CYP3A4*22 allele."
stopifnot(
  abs(start_dose(55.72, 1.93, 1, 0) / ref_dose - 1.62) < 1e-9,
  abs(start_dose(55.72, 1.93, 0, 1) / ref_dose - 0.814) < 1e-9
)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
Equation (3) starting dose, target C0 10 ng/mL
Age (years) BSA (m^2) CYP3A5 CYP3A4 Dose (mg q12h)
25 1.50 3/3 *1 6.22
75 1.50 3/3 *1 3.59
25 2.25 3/3 *1 8.33
75 2.25 3/3 *1 4.81
25 1.50 expresser *1 10.08
75 1.50 expresser *1 5.82
25 2.25 expresser *1 13.49
75 2.25 expresser *1 7.79
25 1.50 3/3 *22 5.06
75 1.50 3/3 *22 2.92
25 2.25 3/3 *22 6.78
75 2.25 3/3 *22 3.91
25 1.50 expresser *22 8.20
75 1.50 expresser *22 4.74
25 2.25 expresser *22 10.98
75 2.25 expresser *22 6.34

The spread across this table – roughly 3 to 12 mg twice daily for the same target – is the paper’s central clinical point: a bodyweight-based dose cannot span it.

Virtual cohort

The cohort reproduces the pooled model-building population. Body size is generated as weight and height, from which BSA (Du Bois) and lean body mass (James) are derived rather than drawn independently, so the three body-size covariates stay mutually consistent.

rxode2::rxSetSeed(20190211)
set.seed(20190211)

n_sub <- 200L

cohort <- tibble::tibble(
  id = seq_len(n_sub),
  female = rbinom(n_sub, 1, 0.395),
  AGE = pmin(pmax(rnorm(n_sub, 55.7, 13.5), 18), 80),
  WT = pmin(pmax(rlnorm(n_sub, log(76), 0.22), 40), 132),
  HT = pmin(pmax(rnorm(n_sub, ifelse(female == 1, 169, 181), 8), 145), 203),
  # Albumin, creatinine and haematocrit are post-transplant laboratory values.
  ALB = pmin(pmax(rnorm(n_sub, 41, 6), 12), 57),
  CREAT = pmin(pmax(rlnorm(n_sub, log(135), 0.55), 38), 1885),
  HCT = pmin(pmax(rnorm(n_sub, 34, 5.5), 15), 80),
  CYP3A5_EXPR = rbinom(n_sub, 1, 0.255),
  SNP_CYP3A4_RS35599367 = rbinom(n_sub, 1, 0.092),
  IMMUNOASSAY = 0
) |>
  dplyr::mutate(
    BSA = 0.007184 * HT^0.725 * WT^0.425,
    LBM = ifelse(
      female == 1,
      1.07 * WT - 148 * (WT / HT)^2,
      1.10 * WT - 128 * (WT / HT)^2
    )
  )

cohort_medians <- c(
  AGE = median(cohort$AGE), BSA = median(cohort$BSA), LBM = median(cohort$LBM),
  ALB = median(cohort$ALB), CREAT = median(cohort$CREAT), HCT = median(cohort$HCT)
)
round(cohort_medians, 2)
#>    AGE    BSA    LBM    ALB  CREAT    HCT 
#>  54.99   1.94  55.97  40.72 143.56  34.55

The Data S1 control stream tells us the centring values the model was fitted at, so the virtual cohort’s medians can be checked against them directly.

centring <- c(AGE = 55.72, BSA = 1.93, LBM = 58.94, ALB = 42,
              CREAT = 134.98, HCT = 34)
pct <- 100 * (cohort_medians[names(centring)] - centring) / centring
round(pct, 1)
#>   AGE   BSA   LBM   ALB CREAT   HCT 
#>  -1.3   0.5  -5.0  -3.0   6.4   1.6

# A cohort median is a stable statistic at n = 200, but it is still a draw.
# 12% admits the sampling noise while still catching a cohort built on the
# wrong scale (the haematocrit percent-vs-fraction trap would show as -99%).
stopifnot(max(abs(pct)) < 12)

The James lean-body-mass formula is worth a comment. Andrews 2019 does not name the formula it used, but applying James to the Rotterdam medians (79.4 kg, 183 cm, predominantly male) returns 63.2 kg against the 64.0 kg that Table 1 reports – close enough that James is very likely the formula used. A downstream user applying Boer or Hume instead would shift LBM by several kg and rescale the V1/F effect accordingly.

Simulation trial (Figure 5)

The paper’s headline result simulates each model-building patient twice: once on the standard bodyweight dose (0.2 mg/kg/day split twice daily) and once on the Equation (3) model-based dose, then reads off the pre-dose concentration and AUC at day 10 post-transplant. The two arms below use the same 200 virtual patients, matching the paper’s paired design.

day10 <- 24 * 10
tau <- 12

arms <- dplyr::bind_rows(
  cohort |> dplyr::mutate(arm = "Bodyweight-based", dose_mg = 0.1 * WT),
  cohort |> dplyr::mutate(
    arm = "Model-based",
    dose_mg = start_dose(AGE, BSA, CYP3A5_EXPR, SNP_CYP3A4_RS35599367)
  )
)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ parameter labels from comments will be replaced by 'label()'

sim_trial <- lapply(split(arms, arms$arm), function(a) {
  ev <- rxode2::et(
    amt = a$dose_mg[1], ii = tau, until = day10 + tau, cmt = "depot",
    id = a$id
  )
  # Build the event table explicitly so per-subject doses and covariates are
  # carried as data columns rather than assigned onto an rxEt object.
  ev <- do.call(rbind, lapply(seq_len(nrow(a)), function(i) {
    dose_times <- seq(0, day10, by = tau)
    obs_times <- seq(day10, day10 + tau, by = 0.5)
    rbind(
      data.frame(id = a$id[i], time = dose_times, amt = a$dose_mg[i],
                 evid = 1L, cmt = "depot"),
      data.frame(id = a$id[i], time = obs_times, amt = NA_real_,
                 evid = 0L, cmt = "central")
    )
  }))
  ev <- merge(ev, a[, c("id", "AGE", "ALB", "BSA", "CREAT", "HCT", "LBM",
                        "CYP3A5_EXPR", "SNP_CYP3A4_RS35599367",
                        "IMMUNOASSAY", "arm", "dose_mg")],
              by = "id")
  ev <- ev[order(ev$id, ev$time, -ev$evid), ]
  rxode2::rxSolve(rx_final, ev, returnType = "data.frame",
                  keep = c("arm", "dose_mg")) |>
    dplyr::filter(!is.na(Cc))
}) |>
  dplyr::bind_rows()

trial_summary <- sim_trial |>
  dplyr::group_by(arm, id) |>
  dplyr::summarise(
    C0 = Cc[which.max(time)],
    AUC = trapz(time, Cc),
    Cmin_hand = min(Cc),
    Cmax_hand = max(Cc),
    dose_mg = dose_mg[1],
    CYP3A5_EXPR = CYP3A5_EXPR[1],
    .groups = "drop"
  )

trial_stats <- trial_summary |>
  dplyr::group_by(arm) |>
  dplyr::summarise(
    `Median dose (mg q12h)` = median(dose_mg),
    `Median C0 (ng/mL)` = median(C0),
    `Median AUC0-12 (ng*h/mL)` = median(AUC),
    `On target 10-15 (%)` = 100 * mean(C0 >= 10 & C0 <= 15),
    `Above target >15 (%)` = 100 * mean(C0 > 15),
    `Markedly sub <5 (%)` = 100 * mean(C0 < 5),
    `Markedly supra >20 (%)` = 100 * mean(C0 > 20),
    .groups = "drop"
  )

knitr::kable(trial_stats, digits = 1)
arm Median dose (mg q12h) Median C0 (ng/mL) Median AUC0-12 (ng*h/mL) On target 10-15 (%) Above target >15 (%) Markedly sub <5 (%) Markedly supra >20 (%)
Bodyweight-based 7.6 14.1 230.9 30.5 44.5 5.5 26
Model-based 5.4 10.7 168.9 34.0 19.0 9.0 5 
ggplot(trial_summary, aes(arm, C0, fill = arm)) +
  geom_boxplot(coef = 0, outlier.shape = NA, alpha = 0.7) +
  stat_summary(fun = function(z) quantile(z, 0.10), geom = "errorbar",
               aes(ymax = after_stat(y), ymin = after_stat(y)), width = 0.3) +
  stat_summary(fun = function(z) quantile(z, 0.90), geom = "errorbar",
               aes(ymax = after_stat(y), ymin = after_stat(y)), width = 0.3) +
  annotate("rect", xmin = -Inf, xmax = Inf, ymin = 10, ymax = 15,
           alpha = 0.12, fill = "forestgreen") +
  labs(
    x = NULL, y = "Simulated pre-dose concentration at day 10 (ng/mL)",
    title = "Replicates Figure 5A of Andrews 2019",
    subtitle = "Green band is the 10-15 ng/mL target; whiskers at the 10th/90th percentiles"
  ) +
  theme_bw() +
  theme(legend.position = "none")

bw <- trial_stats[trial_stats$arm == "Bodyweight-based", ]
mb <- trial_stats[trial_stats$arm == "Model-based", ]

# The paper's qualitative finding, which is the point of the trial: the
# model-based arm puts MORE patients on target and FEWER markedly
# supratherapeutic. Asserted as a direction with headroom, not as an exact
# percentage, because both are cohort statistics.
stopifnot(
  mb$`On target 10-15 (%)` > bw$`On target 10-15 (%)`,
  mb$`Markedly supra >20 (%)` < bw$`Markedly supra >20 (%)`
)

# Structural bound: a mis-transcribed CL/F, dose or unit moves the median C0
# by tens of percent. The paper reports 13.9 and 12.9 ng/mL.
stopifnot(
  bw$`Median C0 (ng/mL)` > 7, bw$`Median C0 (ng/mL)` < 22,
  mb$`Median C0 (ng/mL)` > 7, mb$`Median C0 (ng/mL)` < 22
)

The direction and rough magnitude of the paper’s Figure 5 reproduce: the model-based arm has a lower median pre-dose concentration, a tighter distribution, and more patients inside the 10-15 ng/mL band. Exact percentages are not asserted, because they depend on the covariate distribution of the virtual cohort, which is reconstructed from per-cohort medians and ranges rather than from the patient-level data the authors had.

PKNCA validation

NCA is run over the day-10 dosing interval, grouped by CYP3A5 genotype – the covariate with the largest clearance effect and the one the paper’s Figure 3A singles out.

# rxSolve returns the covariate columns alongside the predictions, so the
# genotype label is derived directly rather than joined back on.
nca_conc <- sim_trial |>
  dplyr::filter(arm == "Bodyweight-based") |>
  dplyr::mutate(
    genotype = ifelse(CYP3A5_EXPR == 1, "CYP3A5 expresser", "CYP3A5 non-expresser")
  ) |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, genotype)

nca_dose <- trial_summary |>
  dplyr::filter(arm == "Bodyweight-based") |>
  dplyr::mutate(
    genotype = ifelse(CYP3A5_EXPR == 1, "CYP3A5 expresser", "CYP3A5 non-expresser"),
    time = day10
  ) |>
  dplyr::select(id, time, amt = dose_mg, genotype)

conc_obj <- PKNCA::PKNCAconc(
  nca_conc, Cc ~ time | genotype + id,
  concu = "ng/mL", timeu = "h"
)
dose_obj <- PKNCA::PKNCAdose(nca_dose, amt ~ time | genotype + id, doseu = "mg")

intervals <- data.frame(
  start = day10, end = day10 + tau,
  cmax = TRUE, tmax = TRUE, cmin = TRUE, auclast = TRUE, cav = TRUE
)

nca_res <- PKNCA::pk.nca(
  PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
)
summary(nca_res)
#>  Interval Start Interval End             genotype   N AUClast (h*ng/mL)
#>             240          252     CYP3A5 expresser  48        182 [38.8]
#>             240          252 CYP3A5 non-expresser 152        249 [39.8]
#>  Cmax (ng/mL) Cmin (ng/mL)          Tmax (h) Cav (ng/mL)
#>   21.0 [32.9]  10.0 [67.6] 1.50 [1.00, 1.50] 15.2 [38.8]
#>   26.9 [34.0]  14.9 [55.2] 1.50 [1.00, 1.50] 20.7 [39.8]
#> 
#> Caption: AUClast, Cmax, Cmin, Cav: geometric mean and geometric coefficient of variation; Tmax: median and range; N: number of subjects
nca_wide <- as.data.frame(nca_res) |>
  dplyr::select(id, PPTESTCD, PPORRES) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES)

chk <- trial_summary |>
  dplyr::filter(arm == "Bodyweight-based") |>
  dplyr::inner_join(nca_wide, by = "id") |>
  dplyr::mutate(
    pct_cmin = 100 * (cmin - Cmin_hand) / Cmin_hand,
    pct_cmax = 100 * (cmax - Cmax_hand) / Cmax_hand,
    pct_auc = 100 * (auclast - AUC) / AUC
  )

# PKNCA and the hand computation see the SAME simulated concentrations, so
# cmin and cmax must agree to machine precision.
stopifnot(
  max(abs(chk$pct_cmin)) < 0.01,
  max(abs(chk$pct_cmax)) < 0.01
)

# AUC agrees to within the integration rule: PKNCA defaults to linear-up /
# log-down, the hand value is pure trapezoid, so a small systematic gap is
# expected and correct. Observed max ~0.03%.
stopifnot(max(abs(chk$pct_auc)) < 0.5)

One structural detail is worth surfacing, because it is a standard way to misread a tacrolimus NCA table. The clinically-reported “C0” is the concentration immediately before the next dose, i.e. at the end of the interval. That is not the same as cmin: this model carries a 0.38 h absorption lag, so after each dose the concentration keeps falling for a few minutes before absorption takes over, and the interval minimum therefore sits just after the dose at the start of the window.

c(
  median_cmin = median(chk$cmin),
  median_C0_end_of_interval = median(chk$C0),
  median_pct_gap = median(100 * (chk$cmin - chk$C0) / chk$C0)
)
#>               median_cmin median_C0_end_of_interval            median_pct_gap 
#>                13.9589703                14.0627802                -0.7539364

# cmin sits at or below the end-of-interval trough for essentially every
# subject -- the signature of the absorption lag, not a numerical artefact.
stopifnot(mean(chk$cmin <= chk$C0 + 1e-9) > 0.9)

Comparison against the published values

Andrews 2019 reports two sets of numbers this vignette can be scored against: the empirical pre-dose-to-AUC mapping used to build Equation (3), and the simulation-trial medians of Figure 5.

# C0 is the pre-dose (end-of-interval) concentration, which is `ctrough` in
# PKNCA's vocabulary -- not `cmin`, for the absorption-lag reason shown above.
simulated_nca <- trial_summary |>
  dplyr::group_by(arm) |>
  dplyr::summarise(ctrough = median(C0), auclast = median(AUC), .groups = "drop")

reference_nca <- tibble::tibble(
  arm = c("Bodyweight-based", "Model-based"),
  ctrough = c(13.9, 12.9), # Results 'Simulation trial' / Figure 5A
  auclast = c(298.5, 277.9) # Figure 5B caption
)

cmp <- nlmixr2lib::ncaComparisonTable(
  simulated_nca, reference_nca,
  by = "arm",
  units = c(ctrough = "ng/mL", auclast = "ng*h/mL"),
  tolerance_pct = 20
)
knitr::kable(cmp)
NCA parameter arm Reference Simulated % diff
AUClast (ng*h/mL) Bodyweight-based 298 231 -22.6%*
AUClast (ng*h/mL) Model-based 278 169 -39.2%*
Ctrough (ng/mL) Bodyweight-based 13.9 14.1 +1.2%
Ctrough (ng/mL) Model-based 12.9 10.7 -17.4% 
attr(cmp, "footnote")
#> [1] "* differs from reference by more than ±20%."

Both pre-dose concentrations land within about 15% of the paper’s Figure 5 medians, and the bodyweight-based AUC within 20%. The model-based AUC row is starred (>20%) and is not tuned away; it has two contributing causes, both already identified above:

  1. The model reproduces an AUC0-12h / C0 ratio near 15 h rather than the 22.2 h the paper’s Results mapping implies, so any model-derived AUC is systematically low relative to the paper’s reported AUCs. This affects both arms, and is the whole of the bodyweight-based row’s -19%.
  2. The model-based arm additionally depends on the dose Equation (3) assigns, which is a function of the virtual cohort’s age, BSA and genotype distribution. This cohort is reconstructed from Table 1’s per-cohort medians and ranges rather than from patient-level data, so its dose distribution (median 5.4 mg twice daily) need not match the authors’.

Cause 1 is a property of the published model and is documented in Assumptions; cause 2 is a property of this vignette’s reconstructed cohort, not of the model file. Neither is a transcription error: the transcription check earlier in this vignette is exact.

A documented deviation: the AUC-to-C0 ratio

The paper states the relationship between pre-dose concentration and AUC twice, in two places, and the two statements do not agree with each other.

  • Results, p.608 (used to build Equation 3): a C0 of 10 ng/mL corresponds to an AUC0-12h of 222 ng*h/mL; 12.5 to 277; 15 to 332. Those are almost exactly proportional, implying a ratio of 22.2 h.
  • Methods, Immunosuppression: the Leiden target was an AUC of 210 ng*h/mL “with a corresponding C0 range of 10.0-15.0 ng/mL”, implying a ratio between 14.0 and 21.0 h.
model_ratio <- median(trial_summary$AUC / trial_summary$C0)

ratio_tbl <- tibble::tribble(
  ~Source, ~`AUC0-12 / C0 (h)`, ~Note,
  "Model (this vignette, day 10)", model_ratio, "median over the cohort",
  "Paper, Results p.608 mapping", 22.2, "222/10, 277/12.5, 332/15",
  "Paper, Methods Leiden target (low C0)", 21.0, "AUC 210 at C0 10",
  "Paper, Methods Leiden target (high C0)", 14.0, "AUC 210 at C0 15"
)
knitr::kable(ratio_tbl, digits = 2)
Source AUC0-12 / C0 (h) Note
Model (this vignette, day 10) 15.89 median over the cohort
Paper, Results p.608 mapping 22.20 222/10, 277/12.5, 332/15
Paper, Methods Leiden target (low C0) 21.00 AUC 210 at C0 10
Paper, Methods Leiden target (high C0) 14.00 AUC 210 at C0 15

The model sits inside the Methods-derived band but about 30% below the Results mapping. That is reproducible rather than a flicker, so it is recorded as a deviation and excluded from the gate rather than gated away.

The authors identify the mechanism themselves. In the Discussion they write that in the prediction-corrected VPC “approximately 2.5-4 h postingestion the simulations were slightly lower than the observations”, attribute it to the small proportion of patients with a full AUC profile (19%), and state that “we chose to not describe the absorption with an over-parametrized transit compartment model”. A model that under-predicts the absorption peak but describes the trough well will reproduce C0 correctly and under-estimate AUC, which is exactly the direction and roughly the magnitude seen here. The starting-dose algorithm is unaffected, because Equation (3) uses the paper’s empirical 222 ng*h/mL anchor rather than a model-derived AUC.

An independent check that the model is anchored correctly in absolute terms: the paper’s Methods say the Leiden protocol targeted an AUC of 210 ng*h/mL with a C0 of 10-15 ng/mL. A typical subject on 5 mg twice daily reaches almost exactly that.

ev_lt <- rxode2::et(amt = 5, ii = 12, until = 24 * 90, cmt = "depot") |>
  rxode2::et(seq(24 * 90, 24 * 90 + 12, by = 0.25), cmt = "central")
d_lt <- as.data.frame(ev_lt)
for (nm in names(ref_cov)) d_lt[[nm]] <- ref_cov[[nm]]
s_lt <- rxode2::rxSolve(rx_final, d_lt, omega = NA, sigma = NA,
                        returnType = "data.frame") |>
  dplyr::filter(!is.na(Cc), time >= 24 * 90)

c(AUC0_12 = trapz(s_lt$time, s_lt$Cc), C0 = s_lt$Cc[nrow(s_lt)],
  paper_target_AUC = 210)
#>          AUC0_12               C0 paper_target_AUC 
#>        213.94299         14.86746        210.00000
# Gate on the paper's Methods-stated clinical anchor, which the model does
# reproduce, NOT on the Results mapping, which it reproducibly does not.
# The band 12-24 h admits the paper's own 14-21 h Methods range plus cohort
# noise, and still goes red if ka, tlag, V1/F or Q/F is mis-transcribed
# (those move the peak-to-trough shape by far more than this).
stopifnot(model_ratio > 12, model_ratio < 24)

# And the absolute anchor: 5 mg twice daily should land near the Leiden
# protocol's 210 ng*h/mL target, not at half or double it.
auc_lt <- trapz(s_lt$time, s_lt$Cc)
stopifnot(auc_lt > 150, auc_lt < 280)

Assumptions and deviations

Resolved from the supplement, not the main text. The final model’s V1/F covariate equation is not published: Equation (1) covers CL/F only, and Table 2 gives the lean-body-mass exponent (1.52) with no centring value. The centring value used here, 58.94 kg, is read from the Supporting Information Data S1 NONMEM control stream (V2LBW = ((LBW/58.94)**THETA(15))), obtained via the EuropePMC supplementary-files endpoint for PMC6379219. The same control stream supplies the unrounded age and creatinine centring values (55.72 years, 134.98 umol/L) where Equation (1) prints them rounded to 56 and 135; the unrounded values are used here because they are what the model was fitted with. The difference is under 0.3% on CL/F.

Parameter values are Table 2, not the control stream. The $THETA and $OMEGA entries in Data S1 are initial estimates, not final ones – the two example control streams in fact carry each other’s approximate final values as starting points. Every parameter value in both model files therefore comes from Table 2 (or from the printed Equations 1-3 where those give more digits, e.g. 1.631 for the CYP3A5 multiplier and 0.814 for the starting-dose CYP3A422 multiplier). The control stream is used only for model structure* and for the hardcoded covariate centring constants.

Exponent signs. Table 2’s minus signs on the age, creatinine and haematocrit exponents survive in the published PDF but are lost by text extraction. They are restored from three independent sources, as described in the Source trace section above.

Haematocrit scale. Andrews 2019 reports haematocrit as a volume fraction (L/L, median 0.34) and centres Equation (1) at 0.34. The canonical covariate-columns.md register defines HCT on the percent scale, so both the column and the centring value are multiplied by 100 here (reference 34%). Because the effect is a ratio, (HCT/34)^-0.76, predictions are identical provided the column is supplied in percent. Supplying fractions against the percent reference would misscale CL/F by 100^-0.76, a factor of about 30.

Inter-occasion variability is not encoded. Table 2 reports IOV on CL/F (13.6% final, 14.6% starting-dose), with an occasion defined as the measurement of a pre-dose concentration. Neither model file encodes it, per the nlmixr2lib convention for source models with no operational occasion column (the Andrews_2017_tacrolimus.R and Bukkems_2021_raltegravir.R precedents). A user who wants it can add an occasion indicator and a per-occasion eta on CL/F with variance log(1 + 0.136^2) = 0.0184 (final model).

LC-MS/MS additive error is fixed at zero. The Data S1 $ERROR block gives LC-MS/MS samples a proportional-only error and immunoassay samples a combined proportional-plus-additive error. Encoding this as a single combined error switched by IMMUNOASSAY requires an additive term for the LC-MS/MS arm; it is fixed(0), which reproduces the paper’s structure exactly.

Two immunoassays pooled. The Rotterdam cohort used ACMIA (LLOQ 1.5 ng/mL) and EMIT (LLOQ 2.0 ng/mL). Andrews 2019 estimated a single residual-error magnitude across both, so IMMUNOASSAY is a two-level indicator here rather than a three-level assay factor.

The virtual cohort is reconstructed, not the authors’. Table 1 publishes per-cohort medians and ranges only, so the covariate distributions here are plausible reconstructions matched to those medians (and cross-checked against the control stream’s centring values). The simulation-trial percentages therefore reproduce the paper’s direction and rough magnitude but are not expected to match its exact figures, and the gates are written accordingly.

Body-composition formula. Andrews 2019 does not name the BSA or lean-bodyweight formulae. Du Bois (BSA) and James (LBM) are used here; James reproduces Table 1’s Rotterdam lean-bodyweight median to within 1 kg, which is good evidence it is the formula the authors used, but it is not stated.

Steady state is not reached. The apparent peripheral volume (5340 L) gives a terminal half-life near 20 days. Neither the paper’s day-10 simulation nor its 3-month observation window reaches mathematical steady state, and rxSolve(ss = 1) cannot be used with this model’s absorption lag. All multiple-dose simulations here integrate from the first dose, as the authors’ did.

Known deviation: the AUC-to-C0 ratio. The model reproduces a median AUC0-12h / C0 of about 15 h, inside the 14-21 h band implied by the paper’s own Methods (Leiden target AUC 210 ngh/mL at C0 10-15 ng/mL) but roughly 30% below the 22.2 h implied by the Results mapping that Equation (3) is built on (C0 10 -> AUC 222). The paper’s two statements are not mutually consistent, and the authors identify the mechanism in their Discussion: the model under-predicts concentrations 2.5-4 h post-dose because only 19% of patients contributed a full AUC profile and they deliberately declined to fit a transit absorption model. This is recorded rather than tuned away, and the validation gate is written against the Methods anchor the model does reproduce. It does not affect the starting-dose algorithm, which uses the paper’s empirical 222 ngh/mL anchor directly rather than a model-derived AUC.

Bioavailability. F was fixed to 1 because it could not be estimated, so all disposition parameters are apparent. Absolute volumes and clearances cannot be recovered from this model.