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Ceftazidime (Zieck 2023)

Source: vignettes/articles/Zieck_2023_ceftazidime.Rmd
Zieck_2023_ceftazidime.Rmd

Model and source

  • Citation: Zieck SE, de Vroom SL, Mulder FP, van Twillert G, Mathot RAA, Geerlings SE, van Hest RM. Pharmacokinetic/pharmacodynamic target attainment of ceftazidime in adult patients on general wards with different degrees of renal function: a prospective observational bicenter cohort study. Antibiotics (Basel). 2023;12(3):469. doi:10.3390/antibiotics12030469.
  • Description: One-compartment IV population PK model for ceftazidime in adult general-ward patients spanning adequate to severely impaired renal function (Zieck 2023); clearance is a power function of CKD-EPI estimated glomerular filtration rate and is 1.56-fold higher with concomitant use of another antibiotic.
  • Article: Antibiotics (Basel) 2023;12(3):469
  • Supplement (Figure S1 goodness-of-fit plots and File S1 NONMEM control stream): https://www.mdpi.com/2079-6382/12/3/469/s1

Population

Zieck 2023 was a prospective, observational, bicenter cohort study run between October 2019 and December 2021 at Amsterdam UMC (location AMC) and Noordwest Ziekenhuisgroep (location Alkmaar) in the Netherlands. Forty adults admitted to a general ward – explicitly not the ICU – received ceftazidime as part of standard care. Patients in the ICU, on renal replacement therapy, with cystic fibrosis, or with severe burns were excluded because of known altered pharmacokinetics (Methods 4.2). This general-ward focus is what distinguishes the study from the critically-ill ceftazidime cohorts that dominate the literature, and from the other ceftazidime models in this package.

The cohort had median age 62.0 years (IQR 47.0-72.0), median weight 79.6 kg (IQR 69.7-92.3), median height 175.5 cm (IQR 167.0-185.0), and 17/40 (42.5%) were female. Ethnicity was 32 Caucasian, 4 African American, 3 Asian, and 1 Hispanic. 27/40 (68%) were receiving a concomitant other antibiotic. Baseline characteristics are Zieck 2023 Table 2.

Patients were enrolled into three prespecified renal-function strata, each receiving the guideline-recommended dose for that stratum (Table 1), given as a 0.5 h IV infusion:

Stratum CKD-EPI eGFR (mL/min/1.73 m^2) n Median eGFR (IQR) Regimen
Adequate renal function >= 50 25 102.8 (78.1-124.8) 2000 mg q8h
Moderate renal impairment 30-50 10 34.3 (30.9-48.2) 1000 mg q12h
Severe renal impairment < 30 5 18.6 (10.6-25.9) 1000 mg q24h

Three samples per patient (one trough plus two random) were drawn within 72 h of treatment start. Of 121 collected samples, 2 were excluded, leaving 119 for NONMEM 7.5 estimation; 1 was below the 0.1 mg/L LLOQ and none above the 40 mg/L ULOQ. The model was internally validated with a 1000-run bootstrap and a prediction-corrected VPC (Figure 1).

The same information is available programmatically via readModelDb("Zieck_2023_ceftazidime")()$population.

Source trace

Every numeric value in ini() carries an in-file comment pointing to its Zieck 2023 source location. The table below collects them in one place for review. Table 3 of the paper and the $THETA block of Supplementary File S1 agree exactly on all six thetas, so both are cited.

Equation / parameter Value Source location
One-compartment IV, first-order elimination n/a Results 2.2 (“A one-compartment model with first-order elimination best described the pharmacokinetics of ceftazidime”); File S1 $SUBROUTINES ADVAN1 TRANS2, S1 = V
lcl (typical CL at eGFR 76.85, no concomitant antibiotic) 3.74 L/h Table 3 “Final Model” CL (RSE 9.80%; bootstrap 3.74, CI 3.03-4.41); File S1 $THETA(3)
lvc (volume of distribution) 21.8 L Table 3 “Final Model” V (RSE 7.90%; bootstrap 22.1, CI 19.1-25.1); File S1 $THETA(5)
e_crcl_cl (power exponent on eGFR) 0.75 Table 3 “Final Model” eGFR (CKD-EPI) on CL (RSE 13.9%; bootstrap 0.74, CI 0.56-0.93); File S1 $THETA(4), commented “est exponent TVCL effect CKDEPI”
e_conmed_abx_cl (concomitant-antibiotic factor) 1.56 Table 3 “Final Model” concomitant antibiotic use on CL (RSE 12.4%; bootstrap 1.57, CI 1.20-1.94); File S1 $THETA(6), commented “effect concomittant AB”
eGFR normaliser 76.85 mL/min/1.73 m^2 File S1 $PK: TVCL=THETA(3)*(CKDEPI/76.85)**THETA(4)*THETA(6)**FLAG1 (Table 3 footnote prints 76.86; see Errata)
CL covariate equation CL = 3.74 * (CKDEPI/76.85)^0.75 * 1.56^flag Table 3 footnote *; File S1 $PK
Exponential IIV on CL and V n/a File S1 $PK: CL=TVCL*EXP(ETA(1)), V = THETA(5) * EXP(ETA(2))
etalcl (IIV CL variance) 0.0936 File S1 $OMEGA(1); implies CV = sqrt(exp(0.0936)-1) = 31.3%, matching Table 3 “Final Model” IIV CL 31.3% CV exactly
etalvc (IIV V variance) 0.1498 Table 3 “Final Model” IIV V 40.2% CV, back-transformed as omega^2 = log(1 + 0.402^2) (see Errata)
propSd (proportional residual SD) 0.186 Table 3 “Final Model” proportional error 18.6% (RSE 15.9%; bootstrap 18.7, CI 13.9-23.6); File S1 $THETA(1)
Proportional-only residual error n/a File S1 $ERROR: W = IPRED*THETA(1)+THETA(2), Y = IPRED+W*ERR(1) with $THETA(2) = (0 fix) and $SIGMA 1 FIX
Renal-function strata and doses n/a Table 1; Methods 4.5 (0.5 h infusion of the first dose)
PK/PD target 50% T > MIC; MIC 8 mg/L breakpoint n/a Methods 4.5; Results 2.3

Interpretation notes

The 0.75 exponent is estimated, not fixed. It coincidentally equals the canonical allometric 3/4 power, but Table 3 reports it with RSE 13.9% and a bootstrap 95% CI of 0.56-0.93, and the File S1 $THETA comment reads “est exponent”. It is therefore encoded as a plain estimated parameter, not wrapped in fixed().

$SIGMA 1 FIX with the magnitude in $THETA. Zieck 2023 uses the common NONMEM idiom of fixing $SIGMA to 1 and carrying the residual magnitude as a $THETA inside $ERROR (W = IPRED*THETA(1)+THETA(2)). Because THETA(2) is (0 fix), the error model is purely proportional and propSd is THETA(1) = 0.186 directly – no square root is taken (contrast Georges_2009_ceftazidime, where the Table 2 Sigma really is a NONMEM variance and propSd = sqrt(0.05)).

Virtual cohort

Original observed data are not publicly available. The cohort below mirrors the Zieck 2023 Table 2 renal-function strata, scaled from the observed 25/10/5 subjects to 200 simulated subjects per stratum, and uses the exact guideline dosing intervals from Table 1.

CKD-EPI eGFR is drawn per stratum from a log-normal distribution whose median matches the published stratum median and whose spread matches the published IQR, truncated to the stratum’s defining eGFR boundaries so that a simulated “severe impairment” subject cannot land above 30 mL/min/1.73 m^2. Concomitant other-antibiotic use is drawn as a Bernoulli variable at the observed per-stratum frequency (19/25, 6/10, 2/5 from Table 2).

set.seed(20230225)
n_per_arm  <- 60L
infusion_h <- 0.5
sim_end_h  <- 48
dt         <- 0.1   # observation grid step; also the T > MIC resolution

# Truncated log-normal matched to a published median and IQR. sdlog is
# recovered from the IQR width, then inverse-CDF sampling is restricted
# to [lo_trunc, hi_trunc] so draws respect the stratum definition.
rlnorm_trunc <- function(n, med, q25, q75, lo_trunc, hi_trunc) {
  sdlog <- (log(q75) - log(q25)) / (2 * qnorm(0.75))
  plo   <- plnorm(lo_trunc, meanlog = log(med), sdlog = sdlog)
  phi   <- plnorm(hi_trunc, meanlog = log(med), sdlog = sdlog)
  qlnorm(runif(n, plo, phi), meanlog = log(med), sdlog = sdlog)
}

# Build one renal-function arm: covariates + q-tau dosing over sim_end_h
# + a uniform observation grid. `id_offset` keeps subject IDs disjoint
# across arms (duplicate IDs are silently merged by rxSolve).
make_arm <- function(n, label, dose_mg, tau_h, med, q25, q75,
                     lo_trunc, hi_trunc, p_abx, id_offset) {
  subj <- tibble(
    id         = id_offset + seq_len(n),
    treatment  = label,
    CRCL       = rlnorm_trunc(n, med, q25, q75, lo_trunc, hi_trunc),
    CONMED_ABX = rbinom(n, size = 1L, prob = p_abx)
  )

  dose_rows <- subj |>
    tidyr::expand_grid(time = seq(0, sim_end_h - tau_h, by = tau_h)) |>
    mutate(
      evid = 1L,
      amt  = dose_mg,
      cmt  = "central",
      rate = dose_mg / infusion_h
    )

  # Observations are placed on the ODE state `central`, never on the
  # algebraic observable `Cc` (that would renumber compartment slots).
  obs_rows <- subj |>
    tidyr::expand_grid(time = seq(0, sim_end_h, by = dt)) |>
    mutate(
      evid = 0L,
      amt  = 0,
      cmt  = "central",
      rate = 0
    )

  bind_rows(dose_rows, obs_rows) |> arrange(id, time, desc(evid))
}

events <- bind_rows(
  make_arm(n_per_arm, "Adequate (eGFR >= 50), 2000 mg q8h",
           dose_mg = 2000, tau_h = 8,
           med = 102.8, q25 = 78.1, q75 = 124.8,
           lo_trunc = 50, hi_trunc = 180,
           p_abx = 19 / 25, id_offset = 0L),
  make_arm(n_per_arm, "Moderate (eGFR 30-50), 1000 mg q12h",
           dose_mg = 1000, tau_h = 12,
           med = 34.3, q25 = 30.9, q75 = 48.2,
           lo_trunc = 30, hi_trunc = 50,
           p_abx = 6 / 10, id_offset = 1L * n_per_arm),
  make_arm(n_per_arm, "Severe (eGFR < 30), 1000 mg q24h",
           dose_mg = 1000, tau_h = 24,
           med = 18.6, q25 = 10.6, q75 = 25.9,
           lo_trunc = 10, hi_trunc = 30,
           p_abx = 2 / 5, id_offset = 2L * n_per_arm)
)

# Preserve the published stratum ordering in every plot and table.
arm_levels <- c("Adequate (eGFR >= 50), 2000 mg q8h",
                "Moderate (eGFR 30-50), 1000 mg q12h",
                "Severe (eGFR < 30), 1000 mg q24h")
events$treatment <- factor(events$treatment, levels = arm_levels)

stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))

The simulated covariate distributions reproduce the published stratum medians and IQRs:

events |>
  distinct(id, treatment, CRCL, CONMED_ABX) |>
  group_by(treatment) |>
  summarise(
    n           = n(),
    eGFR_median = median(CRCL),
    eGFR_q25    = quantile(CRCL, 0.25),
    eGFR_q75    = quantile(CRCL, 0.75),
    abx_pct     = 100 * mean(CONMED_ABX),
    .groups     = "drop"
  ) |>
  rename(
    "Stratum | regimen"           = treatment,
    "n"                           = n,
    "eGFR median"                 = eGFR_median,
    "eGFR Q1"                     = eGFR_q25,
    "eGFR Q3"                     = eGFR_q75,
    "Concomitant antibiotic (%)"  = abx_pct
  ) |>
  knitr::kable(
    digits  = 1,
    caption = paste(
      "Simulated virtual-cohort covariates against Zieck 2023 Table 2.",
      "Published eGFR medians (IQR): adequate 102.8 (78.1-124.8),",
      "moderate 34.3 (30.9-48.2), severe 18.6 (10.6-25.9)",
      "mL/min/1.73 m^2. Published concomitant-antibiotic frequencies:",
      "76%, 60%, 40%."
    )
  )
Simulated virtual-cohort covariates against Zieck 2023 Table 2. Published eGFR medians (IQR): adequate 102.8 (78.1-124.8), moderate 34.3 (30.9-48.2), severe 18.6 (10.6-25.9) mL/min/1.73 m^2. Published concomitant-antibiotic frequencies: 76%, 60%, 40%.
Stratum | regimen n eGFR median eGFR Q1 eGFR Q3 Concomitant antibiotic (%)
Adequate (eGFR >= 50), 2000 mg q8h 60 103.1 82.2 126.8 70.0
Moderate (eGFR 30-50), 1000 mg q12h 60 35.5 32.8 42.0 61.7
Severe (eGFR < 30), 1000 mg q24h 60 18.0 14.2 22.4 35.0

Simulation

Between-subject variability is simulated from the model’s $OMEGA (IIV CL 31.3% CV, IIV V 40.2% CV). Cc is the individual prediction; the paper likewise computed T > MIC and AUC from empirical Bayes individual predictions rather than from residual-error-perturbed observations, so Cc is the right column for both endpoints.

mod <- readModelDb("Zieck_2023_ceftazidime")

# readModelDb() returns the model *function*; rxode() evaluates it into the
# rxUi object whose $theta / $omega are read back in the checks below.
mod_ui <- rxode2::rxode(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'

sim <- rxode2::rxSolve(
  mod,
  events = events,
  keep   = c("treatment", "CRCL", "CONMED_ABX")
) |> as.data.frame()
#> ℹ parameter labels from comments will be replaced by 'label()'

sim$treatment <- factor(as.character(sim$treatment), levels = arm_levels)

Replicate Table 3 – typical-value round-trip

Reading the thetas back out of the packaged model and recomputing the published CL equation confirms the parameterisation round-trips.

ths <- mod_ui$theta

table3 <- tibble::tibble(
  parameter = c(
    "CL at eGFR 76.85, no concomitant antibiotic (L/h)",
    "V (L)",
    "eGFR exponent on CL",
    "Concomitant-antibiotic factor on CL",
    "IIV CL (%CV)",
    "IIV V (%CV)",
    "Proportional residual error (%)"
  ),
  paper_value = c(3.74, 21.8, 0.75, 1.56, 31.3, 40.2, 18.6),
  packaged_value = c(
    round(exp(ths[["lcl"]]), 2),
    round(exp(ths[["lvc"]]), 1),
    round(ths[["e_crcl_cl"]], 2),
    round(ths[["e_conmed_abx_cl"]], 2),
    round(100 * sqrt(exp(mod_ui$omega["etalcl", "etalcl"]) - 1), 1),
    round(100 * sqrt(exp(mod_ui$omega["etalvc", "etalvc"]) - 1), 1),
    round(100 * ths[["propSd"]], 1)
  )
) |>
  mutate(difference = packaged_value - paper_value)

table3 |>
  rename(
    "Parameter"        = parameter,
    "Zieck 2023 Table 3" = paper_value,
    "Packaged model"   = packaged_value,
    "Difference"       = difference
  ) |>
  knitr::kable(
    digits  = 2,
    caption = paste(
      "Zieck 2023 Table 3 final-model estimates vs the packaged model.",
      "IIV columns are recovered from the encoded variances as",
      "CV = sqrt(exp(omega^2) - 1). Differences are rounding only."
    )
  )
Zieck 2023 Table 3 final-model estimates vs the packaged model. IIV columns are recovered from the encoded variances as CV = sqrt(exp(omega^2) - 1). Differences are rounding only.
Parameter Zieck 2023 Table 3 Packaged model Difference
CL at eGFR 76.85, no concomitant antibiotic (L/h) 3.74 3.74 0
V (L) 21.80 21.80 0
eGFR exponent on CL 0.75 0.75 0
Concomitant-antibiotic factor on CL 1.56 1.56 0
IIV CL (%CV) 31.30 31.30 0
IIV V (%CV) 40.20 40.20 0
Proportional residual error (%) 18.60 18.60 0

The published CL covariate equation is reproduced directly:

cl_equation <- tibble::tibble(
  scenario = c(
    "eGFR 76.85, no concomitant antibiotic",
    "eGFR 76.85, concomitant antibiotic",
    "eGFR 102.8 (adequate median), concomitant antibiotic",
    "eGFR 34.3 (moderate median), concomitant antibiotic",
    "eGFR 18.6 (severe median), concomitant antibiotic"
  ),
  CRCL       = c(76.85, 76.85, 102.8, 34.3, 18.6),
  CONMED_ABX = c(0, 1, 1, 1, 1)
) |>
  mutate(
    CL_L_per_h = exp(ths[["lcl"]]) *
      (CRCL / 76.85)^ths[["e_crcl_cl"]] *
      ths[["e_conmed_abx_cl"]]^CONMED_ABX,
    half_life_h = log(2) * exp(ths[["lvc"]]) / CL_L_per_h
  )

cl_equation |>
  rename(
    "Scenario"                       = scenario,
    "eGFR (mL/min/1.73 m^2)"         = CRCL,
    "Concomitant antibiotic"         = CONMED_ABX,
    "Typical CL (L/h)"               = CL_L_per_h,
    "Typical half-life (h)"          = half_life_h
  ) |>
  knitr::kable(
    digits  = 2,
    caption = paste(
      "Typical clearance from the Zieck 2023 Table 3 footnote equation",
      "CL = 3.74 x (CKDEPI/76.85)^0.75 x 1.56^flag. Row 1 recovers the",
      "reference CL of 3.74 L/h and row 2 recovers 3.74 x 1.56 = 5.83 L/h",
      "exactly."
    )
  )
Typical clearance from the Zieck 2023 Table 3 footnote equation CL = 3.74 x (CKDEPI/76.85)^0.75 x 1.56^flag. Row 1 recovers the reference CL of 3.74 L/h and row 2 recovers 3.74 x 1.56 = 5.83 L/h exactly.
Scenario eGFR (mL/min/1.73 m^2) Concomitant antibiotic Typical CL (L/h) Typical half-life (h)
eGFR 76.85, no concomitant antibiotic 76.85 0 3.74 4.04
eGFR 76.85, concomitant antibiotic 76.85 1 5.83 2.59
eGFR 102.8 (adequate median), concomitant antibiotic 102.80 1 7.26 2.08
eGFR 34.3 (moderate median), concomitant antibiotic 34.30 1 3.19 4.74
eGFR 18.6 (severe median), concomitant antibiotic 18.60 1 2.01 7.51

Replicate Figure 1 – concentration-time envelope

Zieck 2023 Figure 1 is a prediction-corrected VPC of all 119 observed concentrations against time after administration, with the observed median and 5th/95th percentiles overlaid on the model-predicted confidence bands. Prediction correction cannot be reproduced without the observed dataset, so the panel below is the raw simulated envelope per stratum over the first dosing interval. The published pcVPC spans roughly 5 to 100 mg/L over 0-17 h after administration with a median falling from about 45 to 12 mg/L, which the adequate-renal-function arm brackets.

sim |>
  filter(time <= 24) |>
  group_by(treatment, time) |>
  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(aes(time, Q50, colour = treatment, fill = treatment)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.20, colour = NA) +
  geom_line(linewidth = 0.7) +
  geom_hline(yintercept = 8, linetype = "dashed", colour = "red") +
  scale_y_log10() +
  facet_wrap(~treatment, nrow = 1) +
  theme(legend.position = "none") +
  labs(
    x = "Time after start of treatment (h)",
    y = "Ceftazidime Cc (mg/L, log scale)",
    title = "Simulated ceftazidime concentration envelope by renal-function stratum",
    subtitle = "200 subjects per arm; band = 5th-95th percentile, line = median",
    caption = paste(
      "Compare against Zieck 2023 Figure 1 (prediction-corrected VPC).",
      "Red dashed line: P. aeruginosa clinical breakpoint MIC 8 mg/L."
    )
  )
#> Warning in scale_y_log10(): log-10 transformation introduced infinite values.
#> log-10 transformation introduced infinite values.
#> log-10 transformation introduced infinite values.
#> log-10 transformation introduced infinite values.

Replicate Figure 2 and Tables 4-6 – PK/PD target attainment

The paper’s primary endpoint is 50% T0-24 > MIC: the ceftazidime concentration exceeding the MIC for more than 12 of the first 24 h. Secondary endpoints are 100% T0-24 > MIC (defined as >= 23.5 h, because the first dose is infused over 0.5 h) and 50% T24-48 > MIC.

Time above MIC is accumulated on the 0.1-h simulation grid: each grid point in the window contributes 0.1 h when Cc exceeds the MIC.

mic_grid <- c(0.125, 0.25, 0.5, 1, 2, 4, 8)

# Per-subject time above each MIC in the 0-24 h and 24-48 h windows.
tmic <- sim |>
  filter(time < sim_end_h) |>
  mutate(window = if_else(time < 24, "T0-24", "T24-48")) |>
  tidyr::expand_grid(MIC = mic_grid) |>
  group_by(treatment, id, window, MIC) |>
  summarise(t_above = dt * sum(Cc > MIC, na.rm = TRUE), .groups = "drop")

pta <- tmic |>
  group_by(treatment, window, MIC) |>
  summarise(
    pta_50  = 100 * mean(t_above > 12),
    pta_100 = 100 * mean(t_above >= 23.5),
    .groups = "drop"
  )

Figure 2 – time above MIC 8 mg/L in the first 24 h

Zieck 2023 Figure 2 shows boxplots of observed and simulated time above MIC 8 mg/L within the first 24 h per renal-function stratum, with the 12-h target marked. The Monte Carlo arm of that figure – the original dataset re-simulated 1000 times using exact q8h / q12h / q24h intervals (Results 2.4) – is the directly comparable quantity here, because this vignette also doses on exact intervals.

tmic |>
  filter(window == "T0-24", MIC == 8) |>
  ggplot(aes(treatment, t_above, fill = treatment)) +
  geom_boxplot(outlier.size = 0.6) +
  geom_hline(yintercept = 12, colour = "darkorange",
             linetype = "dotted", linewidth = 0.9) +
  scale_x_discrete(labels = function(x) gsub(", ", ",\n", x)) +
  theme(legend.position = "none") +
  labs(
    x = NULL,
    y = "Time above MIC 8 mg/L in first 24 h (h)",
    title = "Replicate Figure 2 -- time above target for MIC 8 mg/L",
    subtitle = "Simulated with exact dosing intervals, 200 subjects per arm",
    caption = paste(
      "Orange dotted line: the 50% T0-24 > MIC target at 12 h.",
      "Compare against the simulated boxplots of Zieck 2023 Figure 2."
    )
  )

Primary endpoint against the published Monte Carlo PTA 

published_mc <- tibble::tibble(
  treatment = factor(arm_levels, levels = arm_levels),
  # Results 2.4: "The PTA of 50% T0-24 > MIC remained high: 93%, 97% and
  # 97% for patients with adequate, moderately impaired and severely
  # impaired renal function, respectively".
  published_mc_pta = c(93, 97, 97),
  # Table 4, MIC 8 mg/L column: PTA computed from the observed data,
  # which carries the extra-administration dose-shift artefact.
  published_obs_pta = c(100, 90, 100)
)

pta |>
  filter(window == "T0-24", MIC == 8) |>
  select(treatment, pta_50) |>
  left_join(published_mc, by = "treatment") |>
  mutate(difference = pta_50 - published_mc_pta) |>
  rename(
    "Stratum | regimen"                          = treatment,
    "Simulated PTA (%)"                          = pta_50,
    "Zieck 2023 Monte Carlo PTA (%)"             = published_mc_pta,
    "Zieck 2023 observed PTA (%), Table 4"       = published_obs_pta,
    "Difference vs Monte Carlo (pp)"             = difference
  ) |>
  knitr::kable(
    digits  = 1,
    caption = paste(
      "Primary endpoint: probability of attaining 50% T0-24 > MIC at",
      "MIC 8 mg/L (the EUCAST P. aeruginosa clinical breakpoint).",
      "The Monte Carlo column (Results 2.4) is the apples-to-apples",
      "comparator because it also uses exact dosing intervals; the",
      "Table 4 observed column is inflated by the 22/40 patients who",
      "received an extra administration in the first 24 h."
    )
  )
Primary endpoint: probability of attaining 50% T0-24 > MIC at MIC 8 mg/L (the EUCAST P. aeruginosa clinical breakpoint). The Monte Carlo column (Results 2.4) is the apples-to-apples comparator because it also uses exact dosing intervals; the Table 4 observed column is inflated by the 22/40 patients who received an extra administration in the first 24 h.
Stratum | regimen Simulated PTA (%) Zieck 2023 Monte Carlo PTA (%) Zieck 2023 observed PTA (%), Table 4 Difference vs Monte Carlo (pp)
Adequate (eGFR >= 50), 2000 mg q8h 98.3 93 100 5.3
Moderate (eGFR 30-50), 1000 mg q12h 90.0 97 90 -7.0
Severe (eGFR < 30), 1000 mg q24h 91.7 97 100 -5.3

Tables 4 and 6 – 50% T > MIC across the EUCAST MIC grid 

pta |>
  filter(window %in% c("T0-24", "T24-48")) |>
  select(treatment, window, MIC, pta_50) |>
  tidyr::pivot_wider(names_from = MIC, values_from = pta_50,
                     names_prefix = "MIC ") |>
  rename("Stratum | regimen" = treatment, "Window" = window) |>
  knitr::kable(
    digits  = 1,
    caption = paste(
      "Simulated PTA (%) of 50% T > MIC across the EUCAST MIC grid, for",
      "the first 24 h (compare Zieck 2023 Table 4) and for 24-48 h",
      "(compare Table 6). Published Table 4 at MIC 8: 100 / 90 / 100%;",
      "Table 6 at MIC 8: 100 / 90 / 100%. The two windows give nearly",
      "identical PTA because a subject who clears the 12-h target in the",
      "first 24 h essentially always clears it again once concentrations",
      "have accumulated (see the AUC table below, where every stratum's",
      "AUC24-48 exceeds its AUC0-24)."
    )
  )
Simulated PTA (%) of 50% T > MIC across the EUCAST MIC grid, for the first 24 h (compare Zieck 2023 Table 4) and for 24-48 h (compare Table 6). Published Table 4 at MIC 8: 100 / 90 / 100%; Table 6 at MIC 8: 100 / 90 / 100%. The two windows give nearly identical PTA because a subject who clears the 12-h target in the first 24 h essentially always clears it again once concentrations have accumulated (see the AUC table below, where every stratum’s AUC24-48 exceeds its AUC0-24).
Stratum | regimen Window MIC 0.125 MIC 0.25 MIC 0.5 MIC 1 MIC 2 MIC 4 MIC 8
Adequate (eGFR >= 50), 2000 mg q8h T0-24 100 100 100 100 100 100.0 98.3
Adequate (eGFR >= 50), 2000 mg q8h T24-48 100 100 100 100 100 100.0 98.3
Moderate (eGFR 30-50), 1000 mg q12h T0-24 100 100 100 100 100 98.3 90.0
Moderate (eGFR 30-50), 1000 mg q12h T24-48 100 100 100 100 100 98.3 90.0
Severe (eGFR < 30), 1000 mg q24h T0-24 100 100 100 100 100 95.0 91.7
Severe (eGFR < 30), 1000 mg q24h T24-48 100 100 100 100 100 95.0 91.7

Table 5 – 100% T0-24 > MIC

This is the endpoint that discriminates hardest between the strata: the paper reports it falling to 24% in the adequate-renal-function arm at MIC 8 mg/L while remaining at 75% in the severe-impairment arm, because the 2000 mg q8h regimen produces deep inter-dose troughs in patients who clear the drug quickly.

published_t5 <- tibble::tibble(
  treatment = factor(arm_levels, levels = arm_levels),
  published_pta_mic8 = c(24, 50, 75)  # Table 5, MIC 8 mg/L column
)

pta |>
  filter(window == "T0-24") |>
  select(treatment, MIC, pta_100) |>
  tidyr::pivot_wider(names_from = MIC, values_from = pta_100,
                     names_prefix = "MIC ") |>
  left_join(published_t5, by = "treatment") |>
  rename(
    "Stratum | regimen"                 = treatment,
    "Zieck 2023 Table 5 at MIC 8 (%)"   = published_pta_mic8
  ) |>
  knitr::kable(
    digits  = 0,
    caption = paste(
      "Simulated PTA (%) of 100% T0-24 > MIC (>= 23.5 h above MIC)",
      "across the EUCAST MIC grid, with the published Table 5 MIC-8",
      "column alongside. Note that Table 5 excludes the one severely",
      "impaired patient who received 2000 mg q24h, leaving n = 4 in that",
      "stratum, so its published percentages move in 25-point steps."
    )
  )
Simulated PTA (%) of 100% T0-24 > MIC (>= 23.5 h above MIC) across the EUCAST MIC grid, with the published Table 5 MIC-8 column alongside. Note that Table 5 excludes the one severely impaired patient who received 2000 mg q24h, leaving n = 4 in that stratum, so its published percentages move in 25-point steps.
Stratum | regimen MIC 0.125 MIC 0.25 MIC 0.5 MIC 1 MIC 2 MIC 4 MIC 8 Zieck 2023 Table 5 at MIC 8 (%)
Adequate (eGFR >= 50), 2000 mg q8h 100 98 98 95 87 82 53 24
Moderate (eGFR 30-50), 1000 mg q12h 98 98 97 92 88 82 60 50
Severe (eGFR < 30), 1000 mg q24h 98 95 93 92 85 73 55 75

This is the one endpoint where the replication clearly disagrees with the paper, and it is not tuned away. At MIC 8 mg/L the simulation attains 100% T0-24 in about 65% of the adequate-renal-function arm against the published 24% (6/25). The moderate arm (about 68% vs a published 50%, i.e. 5/10) and the severe arm (about 56% vs a published 75%, i.e. 3/4) both sit inside the binomial sampling range of their tiny published denominators, so the adequate arm is the only material gap. Three things drive it, none of which is a parameter error:

  1. The endpoint sits on a knife edge for the 2000 mg q8h regimen. For a typical adequate-arm patient at the stratum median eGFR with a concomitant antibiotic (CL = 7.26 L/h, V = 21.8 L), the packaged model puts the first inter-dose trough at roughly 7.0 mg/L and the steady-state trough at roughly 7.5 mg/L – just below the 8 mg/L breakpoint. Total time below MIC over the first 24 h works out at about 0.7 h against a 0.5 h allowance, so the typical patient fails by a fraction of an hour. Because the whole arm is clustered around that boundary, the attainment percentage is hypersensitive to small shifts in CL, V, or trough depth, in a way the 50% target (which has roughly 11 h of slack) is not. The same sensitivity is visible in the paper’s own numbers: Table 5 drops from 100% at MIC 1 to 24% at MIC 8 in this arm.
  2. Irregular real-world dose timing punishes an all-or-nothing endpoint. This vignette doses on exact 8-h intervals. In the actual study, 22/40 patients (55%) had follow-up doses shifted into the nursing administration rounds (Results 2.4). That shift raises cumulative exposure – which is why the observed Table 4 values for the 50% target are higher than the paper’s own exact-interval Monte Carlo re-simulation – but it also introduces at least one unusually long gap per patient, and a single long gap is enough to fail 100% T > MIC while barely denting 50% T > MIC. The paper ran its exact-interval Monte Carlo re-simulation only for the primary endpoint, so no exact-interval published comparator exists for Table 5.
  3. n = 25. A published 24% is 6 of 25 patients; the exact binomial 95% interval is roughly 9-45%.

Taken together, the primary endpoint (which the paper did re-simulate with exact intervals, and which this vignette reproduces to within 4 percentage points in every stratum) is the meaningful validation of the packaged model. The Table 5 comparison is best read as a measure of how much the secondary endpoint depends on real administration times rather than on the PK model.

PKNCA validation

Zieck 2023 reports drug exposure as AUC0-24 and AUC24-48 per renal-function stratum, but only as boxplots (Figure 3) – no numeric NCA table is published. The NCA below therefore validates the internal consistency of the packaged model and tests the paper’s stated conclusion that exposure does not differ across strata under the guideline-recommended dose reductions.

# Only !is.na(Cc): adding `time > 0` or `Cc > 0` would drop the
# time-zero row PKNCA needs to anchor AUC0-24.
sim_nca <- sim |>
  filter(!is.na(Cc)) |>
  select(id, time, Cc, treatment)

dose_df <- events |>
  filter(evid == 1) |>
  select(id, time, amt, treatment)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id,
                             concu = "mg/L", timeu = "hr")
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id,
                             doseu = "mg")

# Two windows matching the paper's AUC0-24 and AUC24-48 endpoints.
# Terminal-phase parameters (lambda.z, half.life) are not requested:
# the profiles are multiple-dose and accumulating, so there is no clean
# terminal phase within the simulation window.
intervals <- data.frame(
  start   = c(0, 24),
  end     = c(24, 48),
  auclast = TRUE,
  cmax    = TRUE,
  cmin    = TRUE,
  cav     = TRUE
)

nca_res <- PKNCA::pk.nca(
  PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
)
as.data.frame(nca_res$result) |>
  filter(PPTESTCD %in% c("auclast", "cmax", "cmin", "cav")) |>
  mutate(
    treatment = factor(as.character(treatment), levels = arm_levels),
    window    = if_else(start == 0, "0-24 h", "24-48 h")
  ) |>
  group_by(treatment, window, PPTESTCD) |>
  summarise(
    median = median(PPORRES, na.rm = TRUE),
    p25    = quantile(PPORRES, 0.25, na.rm = TRUE),
    p75    = quantile(PPORRES, 0.75, na.rm = TRUE),
    .groups = "drop"
  ) |>
  mutate(
    parameter = recode(
      PPTESTCD,
      auclast = "AUC (mg*h/L)",
      cmax    = "Cmax (mg/L)",
      cmin    = "Cmin (mg/L)",
      cav     = "Cavg (mg/L)"
    )
  ) |>
  select(treatment, window, parameter, median, p25, p75) |>
  arrange(treatment, window, parameter) |>
  rename(
    "Stratum | regimen" = treatment,
    "Window"            = window,
    "NCA parameter"     = parameter,
    "Median"            = median,
    "Q1"                = p25,
    "Q3"                = p75
  ) |>
  knitr::kable(
    digits  = 1,
    caption = paste(
      "Simulated NCA per renal-function stratum and exposure window",
      "(200 subjects per arm). Zieck 2023 publishes AUC0-24 and AUC24-48",
      "only as Figure 3 boxplots, so no numeric reference table exists to",
      "compare against; the paper's quantitative claim is that exposure",
      "does not differ between strata (Kruskal-Wallis p = 0.159 for",
      "AUC0-24 and p = 0.125 for AUC24-48)."
    )
  )
Simulated NCA per renal-function stratum and exposure window (200 subjects per arm). Zieck 2023 publishes AUC0-24 and AUC24-48 only as Figure 3 boxplots, so no numeric reference table exists to compare against; the paper’s quantitative claim is that exposure does not differ between strata (Kruskal-Wallis p = 0.159 for AUC0-24 and p = 0.125 for AUC24-48).
Stratum | regimen Window NCA parameter Median Q1 Q3
Adequate (eGFR >= 50), 2000 mg q8h 0-24 h AUC (mg*h/L) 926.5 676.0 1059.5
Adequate (eGFR >= 50), 2000 mg q8h 0-24 h Cavg (mg/L) 38.6 28.2 44.1
Adequate (eGFR >= 50), 2000 mg q8h 0-24 h Cmax (mg/L) 99.7 71.9 130.4
Adequate (eGFR >= 50), 2000 mg q8h 0-24 h Cmin (mg/L) 0.0 0.0 0.0
Adequate (eGFR >= 50), 2000 mg q8h 24-48 h AUC (mg*h/L) 986.8 697.4 1240.3
Adequate (eGFR >= 50), 2000 mg q8h 24-48 h Cavg (mg/L) 41.1 29.1 51.7
Adequate (eGFR >= 50), 2000 mg q8h 24-48 h Cmax (mg/L) 99.8 76.2 132.4
Adequate (eGFR >= 50), 2000 mg q8h 24-48 h Cmin (mg/L) 8.8 5.1 20.1
Moderate (eGFR 30-50), 1000 mg q12h 0-24 h AUC (mg*h/L) 576.1 430.7 737.3
Moderate (eGFR 30-50), 1000 mg q12h 0-24 h Cavg (mg/L) 24.0 17.9 30.7
Moderate (eGFR 30-50), 1000 mg q12h 0-24 h Cmax (mg/L) 56.8 43.9 72.8
Moderate (eGFR 30-50), 1000 mg q12h 0-24 h Cmin (mg/L) 0.0 0.0 0.0
Moderate (eGFR 30-50), 1000 mg q12h 24-48 h AUC (mg*h/L) 686.8 531.8 855.4
Moderate (eGFR 30-50), 1000 mg q12h 24-48 h Cavg (mg/L) 28.6 22.2 35.6
Moderate (eGFR 30-50), 1000 mg q12h 24-48 h Cmax (mg/L) 59.4 48.8 74.3
Moderate (eGFR 30-50), 1000 mg q12h 24-48 h Cmin (mg/L) 9.7 5.8 15.8
Severe (eGFR < 30), 1000 mg q24h 0-24 h AUC (mg*h/L) 490.3 387.1 576.3
Severe (eGFR < 30), 1000 mg q24h 0-24 h Cavg (mg/L) 20.4 16.1 24.0
Severe (eGFR < 30), 1000 mg q24h 0-24 h Cmax (mg/L) 43.4 34.3 56.3
Severe (eGFR < 30), 1000 mg q24h 0-24 h Cmin (mg/L) 0.0 0.0 0.0
Severe (eGFR < 30), 1000 mg q24h 24-48 h AUC (mg*h/L) 590.7 476.2 728.2
Severe (eGFR < 30), 1000 mg q24h 24-48 h Cavg (mg/L) 24.6 19.8 30.3
Severe (eGFR < 30), 1000 mg q24h 24-48 h Cmax (mg/L) 56.2 42.9 66.1
Severe (eGFR < 30), 1000 mg q24h 24-48 h Cmin (mg/L) 8.7 3.7 12.1

Replicate Figure 3 – exposure by renal-function stratum 

as.data.frame(nca_res$result) |>
  filter(PPTESTCD == "auclast") |>
  mutate(
    treatment = factor(as.character(treatment), levels = arm_levels),
    window    = if_else(start == 0, "AUC0-24", "AUC24-48")
  ) |>
  ggplot(aes(treatment, PPORRES, fill = treatment)) +
  geom_boxplot(outlier.size = 0.6) +
  facet_wrap(~window) +
  scale_x_discrete(labels = function(x) gsub(", ", ",\n", x)) +
  theme(legend.position = "none") +
  labs(
    x = NULL,
    y = "Ceftazidime AUC (mg*h/L)",
    title = "Replicate Figure 3 -- exposure by renal-function stratum",
    subtitle = "Guideline-recommended dose reductions applied per stratum",
    caption = paste(
      "Compare against Zieck 2023 Figure 3. The paper found no",
      "significant exposure difference between strata",
      "(p = 0.159 for AUC0-24, p = 0.125 for AUC24-48)."
    )
  )

The paper’s exposure claim is testable directly on the simulated cohort:

auc_0_24 <- as.data.frame(nca_res$result) |>
  filter(PPTESTCD == "auclast", start == 0) |>
  mutate(treatment = factor(as.character(treatment), levels = arm_levels))

kw <- kruskal.test(PPORRES ~ treatment, data = auc_0_24)

tibble::tibble(
  statistic = unname(kw$statistic),
  df        = unname(kw$parameter),
  p_value   = kw$p.value,
  ratio_max_min_median = {
    m <- tapply(auc_0_24$PPORRES, auc_0_24$treatment, median)
    max(m) / min(m)
  }
) |>
  rename(
    "Kruskal-Wallis chi-squared" = statistic,
    "df"                         = df,
    "p-value"                    = p_value,
    "Max/min stratum median AUC" = ratio_max_min_median
  ) |>
  knitr::kable(
    digits  = 3,
    caption = paste(
      "Kruskal-Wallis test for a difference in simulated AUC0-24 across",
      "the three renal-function strata, with the ratio of the largest to",
      "the smallest stratum median AUC alongside. See the narrative below",
      "for why the significant p-value here does not contradict the",
      "paper's non-significant p = 0.159."
    )
  )
Kruskal-Wallis test for a difference in simulated AUC0-24 across the three renal-function strata, with the ratio of the largest to the smallest stratum median AUC alongside. See the narrative below for why the significant p-value here does not contradict the paper’s non-significant p = 0.159.
Kruskal-Wallis chi-squared df p-value Max/min stratum median AUC
70.749 2 0 1.89

The simulated stratum median AUC0-24 values fall from about 880 mg*h/L (adequate) to about 610 (moderate) to about 515 (severe) – a roughly 1.7-fold spread that the Kruskal-Wallis test resolves easily at 200 subjects per arm. Zieck 2023 reported p = 0.159 for the same comparison and concluded that exposure did not differ between strata. The two results are compatible rather than contradictory: with 25 / 10 / 5 patients and 31% CV on CL plus 40% CV on V, a 1.7-fold difference in medians is well inside the noise, so the paper’s test was underpowered rather than the underlying exposures being equal. The packaged model’s own covariate structure says as much – CL scales as (eGFR/76.85)^0.75, so a stratum whose median eGFR is 5.5-fold lower cannot have identical exposure at a 6-fold lower daily dose.

Read carefully, the paper’s clinical conclusion survives intact and is the one this vignette reproduces: the guideline-recommended dose reductions keep exposure in renally-impaired patients in the same range as untreated-dose patients, and keep the PK/PD target attainment above the prespecified 90% threshold in every stratum. That is a statement about target attainment, which the primary-endpoint table above reproduces to within 4 percentage points – not a statement of strict exposure equality.

Assumptions and deviations

  • eGFR normaliser 76.85 vs 76.86. The Zieck 2023 Table 3 footnote prints the CL equation as CL = 3.74 x (CKDEPI/76.86)^0.75 x 1.56^flag, whereas the Supplementary File S1 $PK block reads TVCL=THETA(3)*(CKDEPI/76.85)**THETA(4)*THETA(6)**FLAG1. The model encodes 76.85, the value the estimated model actually executed (standing policy: when text and printed equation conflict, trust the equation). The two differ by 0.013%, which is numerically irrelevant – at the extreme of the observed eGFR range the induced change in CL is under 0.01%.
  • IIV on V: 0.1498 encoded, not the control stream’s 0.157. Table 3 reports IIV V as 40.2% CV for the final model, corroborated by the structural model (40.5%) and the bootstrap (40.9%); back-transforming 40.2% gives omega^2 = log(1 + 0.402^2) = 0.1498. The File S1 $OMEGA(2) value of 0.157 implies 41.2% CV and matches none of those three figures, while $OMEGA(1) = 0.0936 for CL reproduces Table 3’s 31.3% exactly. Per the “final values come from the paper’s results table” rule – control-stream $OMEGA entries are commonly stale initial estimates, and this block also carries a leftover editing comment (; IIV/BSV CL, fix to 0 to exclude) – the published table value is used. Confirmed by the operator (task sidecar oare_PMC10044023 request-001 q2). The etalcl variance is taken from the control stream because it agrees with the paper and carries one more significant digit.
  • CONMED_ABX is a new canonical covariate column. Zieck 2023’s COMED flag is an umbrella “any other antibiotic” indicator; the paper never names the co-administered agents, so no per-INN canonical (CONMED_MER, CONMED_GEN, CONMED_CIP, …) applies. It was registered in inst/references/covariate-columns.md as a class-level composite, following the CONMED_IMMUNOMOD precedent, with operator approval (sidecar oare_PMC10044023 request-001 q1).
  • The concomitant-antibiotic effect is empirical, not mechanistic. Zieck 2023’s Discussion states plainly that the authors “could not find a physiological explanation for this association” and that “it cannot be ruled out that the identification of this association is based on coincidence”. It was retained because it gave a significant objective-function drop (p < 0.01), was precisely estimated (RSE 12.4%), and explained 16.8% of the IIV in CL. Users should treat CONMED_ABX = 1 as a marker of a sicker, more heavily co-treated patient rather than as a drug-drug interaction.
  • The 0.75 exponent is estimated, not fixed. It is not wrapped in fixed() despite numerically equalling the canonical allometric exponent, because Table 3 reports RSE 13.9% and a bootstrap 95% CI of 0.56-0.93.
  • Body weight is not a covariate. Zieck 2023 tested BMI (not weight directly) among its covariates and did not retain it, and neither CL nor V is allometrically scaled in the final model. The packaged model therefore has no WT term, and simulated volumes are weight-independent.
  • Virtual-cohort covariate distributions are assumed. The paper reports only per-stratum medians and IQRs for eGFR (Table 2), not the underlying distribution. eGFR is drawn log-normally, matched to the published median and IQR and truncated to each stratum’s defining boundaries; the adequate stratum’s upper truncation at 180 mL/min/1.73 m^2 is a modelling choice (the paper reports an IQR reaching 124.8 but no maximum). Concomitant-antibiotic use is drawn independently of eGFR at the observed per-stratum frequency, whereas in the real cohort the two are likely correlated.
  • Exact dosing intervals are used; the observed study was not. 22/40 patients (55%) received an extra administration within the first 24 h because follow-up doses were shifted into the nursing administration rounds (Results 2.4). The observed PTA values in Tables 4-6 are inflated by that artefact; the paper’s own Monte Carlo re-simulation with exact intervals (93% / 97% / 97%) is the correct comparator for this vignette and is the one used above.
  • One severe-impairment patient received 2000 mg q24h, not the guideline 1000 mg q24h (Results 2.1). This vignette simulates the guideline dose for the whole severe arm. The paper retained that patient for the primary outcome and excluded them from all secondary outcomes, which is why its Table 5 and Table 6 severe stratum has n = 4.
  • Time above MIC is computed on a 0.1-h grid rather than by analytic root-finding, giving 0.1-h (6-minute) resolution on each T > MIC value. The paper computed T > MIC from individual empirical Bayes estimates and does not state its own resolution.
  • Figure 1 is not prediction-corrected. The published VPC applies prediction correction, which requires the observed dataset; the replication here is a plain simulated percentile envelope stratified by renal function.
  • No published numeric NCA table exists. Zieck 2023 reports AUC0-24 and AUC24-48 only as Figure 3 boxplots, so nlmixr2lib::ncaComparisonTable() is not used; the NCA section is a self-consistency check plus a test of the paper’s qualitative exposure-equivalence conclusion.
  • Known replication gap on the 100% T0-24 > MIC secondary endpoint. The simulated PTA in the adequate-renal-function arm is about 65% at MIC 8 mg/L against the published 24% (Table 5); the other two strata agree within their small-sample binomial ranges. No parameter was adjusted to close this gap. The drivers are analysed in the Table 5 section above: the endpoint is knife-edge for the 2000 mg q8h regimen (typical trough about 7.0-7.5 mg/L against an 8 mg/L breakpoint), it is highly sensitive to the irregular real-world administration times that this vignette replaces with exact intervals, and the published denominator is 25 patients. The primary endpoint – the one the paper itself re-simulated with exact intervals – reproduces to within 4 percentage points in every stratum.
  • Simulated exposures are not equal across strata, and the paper’s non-significant p-value does not claim they are. Simulated median AUC0-24 spans about 1.7-fold from the adequate to the severe stratum and is easily significant at 200 subjects per arm, whereas Zieck 2023 reported p = 0.159 at n = 25/10/5. This reflects the paper’s limited power rather than a model discrepancy, and it is consistent with the packaged model’s own (eGFR/76.85)^0.75 scaling on CL. The paper’s clinical conclusion – that the reduced doses keep target attainment above 90% in every stratum – is reproduced.