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Amoxicillin (Tang 2019)

Source: vignettes/articles/Tang_2019_amoxicillin.Rmd
Tang_2019_amoxicillin.Rmd

Model and source

  • Citation: Tang B-H, Wu Y-E, Kou C, Qi Y-J, Qi H, Xu H-Y, Leroux S, Huang X, Zhou Y, Zheng Y, Jacqz-Aigrain E, Shen A-D, Zhao W. Population Pharmacokinetics and Dosing Optimization of Amoxicillin in Neonates and Young Infants. Antimicrob Agents Chemother. 2019;63(2):e02336-18. doi:10.1128/AAC.02336-18.
  • Description: Two-compartment population PK model with first-order elimination for intravenous amoxicillin in Chinese neonates and young infants (Tang 2019). Current weight enters as a fixed allometric power on both volumes (exponent 1) and on CL and Q (exponent 0.75); CL is further modulated by a maturation factor F_age that is the product of two power functions of gestational age and postnatal age. Interindividual variability is estimated on the peripheral volume V2 and on CL only; residual variability follows an exponential model (proportional in linear space).
  • Article: Antimicrob Agents Chemother 2019;63(2):e02336-18

Population

Tang 2019 enrolled 187 Chinese neonates and young infants admitted to neonatal intensive care units across three centres (Beijing Obstetrics and Gynecology Hospital, Beijing Children’s Hospital, Shandong Provincial Qianfoshan Hospital) between 2016 and 2017. Postmenstrual age covered 28.4 to 46.3 weeks (median 39.0 weeks), gestational age at birth 28.3 to 41.4 weeks (median 38.1 weeks), postnatal age 1 to 37 days (median 7 days), and current weight 1.06 to 4.58 kg (median 3.21 kg). The cohort was 98 male / 89 female (52.4% male, 47.6% female). All subjects received amoxicillin 25 mg/kg twice daily as part of regular antimicrobial treatment, administered as an intravenous bolus over 5 minutes or as a 30-minute infusion; opportunistic sampling produced 224 amoxicillin plasma concentrations measured by HPLC (LLOQ 0.5 ug/mL). See Tang 2019 Table 1 and Methods “Study design” for the full demographics.

The same information is available programmatically via the model’s population metadata (readModelDb("Tang_2019_amoxicillin") |> (\\(f) f())() |> _$population after loading the package).

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Tang_2019_amoxicillin.R. The table below collects them in one place for review.

Equation / parameter Value Source location
lvc (V1) log(1.48) Tang 2019 Table 2: theta1 = 1.48 L (RSE 7.80%)
lvp (V2) log(2.42) Tang 2019 Table 2: theta2 = 2.42 L (RSE 28.30%)
lq (Q) log(0.17) Tang 2019 Table 2: theta3 = 0.17 L/h (RSE 23.70%)
lcl (CL) log(0.81) Tang 2019 Table 2: theta4 = 0.81 L/h (RSE 7.50%)
e_wt_vc_vp fixed(1.00) Tang 2019 Results “Covariate analysis” (a priori fixed)
e_wt_cl_q fixed(0.75) Tang 2019 Results “Covariate analysis” (a priori fixed)
e_ga_cl 4.19 Tang 2019 Table 2: theta5 = 4.19 (RSE 12.60%)
e_pna_cl 0.28 Tang 2019 Table 2: theta6 = 0.28 (RSE 20.30%)
IIV V2 (CV 80.00%) ~ 0.49470 Tang 2019 Table 2; omega^2 = log(1 + 0.80^2)
IIV CL (CV 40.00%) ~ 0.14842 Tang 2019 Table 2; omega^2 = log(1 + 0.40^2)
Residual (CV 35.00%) propSd = 0.35 Tang 2019 Table 2 (exponential = proportional in linear space)
F_age = (GA/38.14)^theta5 * (PNA/7)^theta6 n/a Tang 2019 Table 2 footnote a (PNA reference reparameterised from 7 days to 7/30.4375 = 0.2300 months)
V1 = theta1 * (CW/3210) n/a Tang 2019 Table 2 (CW in g, reference rescaled to 3.21 kg)
V2 = theta2 * (CW/3210) n/a Tang 2019 Table 2
Q = theta3 * (CW/3210)^0.75 n/a Tang 2019 Table 2
CL = theta4 * (CW/3210)^0.75 * F_age n/a Tang 2019 Table 2
2-compartment IV with first-order elimination n/a Tang 2019 Results “Model building”

Virtual cohort

Original observed data are not publicly available. The simulation below builds two virtual cohorts that approximate the Tang 2019 development sample: a premature stratum (GA < 37 weeks) and a term stratum (GA >= 37 weeks). Both strata receive the currently used regimen of 25 mg/kg BID for two days.

set.seed(2019)

# Helper: one cohort, intravenous 25 mg/kg every 12 h for 48 h.
# id_offset shifts subject IDs so cohorts can be bind_rows()-ed without
# colliding (rxSolve treats id as the subject key).
make_cohort <- function(n,
                        ga_mean, ga_sd, ga_min, ga_max,
                        wt_mean, wt_sd, wt_min, wt_max,
                        pna_days_mean, pna_days_sd,
                        cohort_label, id_offset = 0L,
                        dose_per_kg = 25, n_doses = 4L,
                        tau_h = 12, obs_grid_h = c(seq(0, 12, by = 0.5),
                                                   seq(13, 48, by = 1))) {
  GA <- pmin(pmax(rnorm(n, ga_mean, ga_sd), ga_min), ga_max)
  WT <- pmin(pmax(rnorm(n, wt_mean, wt_sd), wt_min), wt_max)
  # PNA in days for sampling, then converted to canonical months for the model
  pna_d <- pmin(pmax(rnorm(n, pna_days_mean, pna_days_sd), 1), 37)
  PNA   <- pna_d / 30.4375

  per_subj <- tibble(
    id     = id_offset + seq_len(n),
    cohort = cohort_label,
    GA     = GA,
    WT     = WT,
    PNA    = PNA
  )

  dose_rows <- per_subj |>
    tidyr::expand_grid(dose_idx = seq_len(n_doses)) |>
    mutate(
      time = (dose_idx - 1) * tau_h,
      amt  = dose_per_kg * WT,
      evid = 1L,
      cmt  = "central",
      dv   = NA_real_
    ) |>
    select(-dose_idx)

  obs_rows <- per_subj |>
    tidyr::expand_grid(time = obs_grid_h) |>
    mutate(
      amt  = 0,
      evid = 0L,
      cmt  = "central",
      dv   = NA_real_
    )

  bind_rows(dose_rows, obs_rows) |>
    arrange(id, time, desc(evid)) |>
    select(id, time, amt, evid, cmt, dv, cohort, WT, GA, PNA)
}

# Cohort A: term (GA >= 37 wk). Approximate Tang 2019 term subgroup
# (>= 37 weeks GA): use GA ~ 39 +/- 1.2 (truncated 37-41), WT ~ 3.4 +/- 0.5 kg,
# PNA median 7 days.
ev_term <- make_cohort(
  n = 100,
  ga_mean = 39.0, ga_sd = 1.2, ga_min = 37.0, ga_max = 41.4,
  wt_mean = 3.40, wt_sd = 0.50, wt_min = 2.40, wt_max = 4.58,
  pna_days_mean = 7, pna_days_sd = 4,
  cohort_label = "Term (GA >= 37 wk)", id_offset = 0L
)

# Cohort B: preterm (GA < 37 wk). Approximate the preterm subgroup with GA
# ~ 34 +/- 1.5 (truncated 28.3-36.9), WT ~ 2.2 +/- 0.5 kg, PNA median 7 days.
ev_pre  <- make_cohort(
  n = 100,
  ga_mean = 34.0, ga_sd = 1.5, ga_min = 28.3, ga_max = 36.9,
  wt_mean = 2.20, wt_sd = 0.50, wt_min = 1.06, wt_max = 3.30,
  pna_days_mean = 7, pna_days_sd = 4,
  cohort_label = "Premature (GA < 37 wk)", id_offset = 100L
)

events <- bind_rows(ev_term, ev_pre)
stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))

Simulation 

mod <- readModelDb("Tang_2019_amoxicillin")
sim <- rxode2::rxSolve(mod, events = events, keep = c("cohort", "WT", "GA", "PNA"))
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_df <- as.data.frame(sim)

Replicate Figure 1 (concentration versus time)

Tang 2019 Figure 1 shows the observed amoxicillin concentration-versus-time profiles, with values reaching up to 73.6 ug/mL. The simulated VPC below uses the median, 5th, and 95th percentiles of Cc across each virtual cohort over 48 hours of dosing.

vpc_df <- sim_df |>
  group_by(cohort, 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(vpc_df, aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), fill = "steelblue", alpha = 0.25) +
  geom_line(linewidth = 0.7) +
  facet_wrap(~cohort, nrow = 1) +
  scale_y_log10() +
  labs(
    x = "Time after first dose (h)",
    y = "Amoxicillin Cc (mg/L)",
    title = "Replicates Tang 2019 Figure 1 (concentration vs time)",
    caption = "25 mg/kg IV every 12 h; median and 5th-95th percentile envelope."
  ) +
  theme_bw()

PKNCA validation

Single-dose NCA on the first 12 h of the simulated profiles, stratified by cohort. PKNCA reports Cmax, Tmax, AUC(0-tau), and apparent half-life per subject; the summary below aggregates per cohort. The treatment grouping variable cohort is included in the PKNCA formulas so per-stratum metrics roll up cleanly for comparison.

sim_nca <- sim_df |>
  filter(time <= 12, !is.na(Cc)) |>
  select(id, time, Cc, cohort)

dose_df <- events |>
  filter(evid == 1L, time == 0) |>
  select(id, time, amt, cohort) |>
  rename(dose = amt)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | cohort + id)
dose_obj <- PKNCA::PKNCAdose(dose_df, dose ~ time | cohort + id)

intervals <- data.frame(
  start      = 0,
  end        = 12,
  cmax       = TRUE,
  tmax       = TRUE,
  auclast    = TRUE,
  half.life  = TRUE
)

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- PKNCA::pk.nca(nca_data)
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#> Too few points for half-life calculation (min.hl.points=3 with only 0 points)
#> Too few points for half-life calculation (min.hl.points=3 with only 0 points)
#> Too few points for half-life calculation (min.hl.points=3 with only 0 points)
nca_summary <- summary(nca_res)
knitr::kable(
  nca_summary,
  caption = "Simulated NCA over the first 12-h dosing interval, by GA stratum."
)
Simulated NCA over the first 12-h dosing interval, by GA stratum.
start end cohort N auclast cmax tmax half.life
0 12 Premature (GA < 37 wk) 100 139 [33.6] 56.3 [3.45] 12.0 [12.0, 12.0] NC
0 12 Term (GA >= 37 wk) 100 104 [34.6] 55.1 [1.91] 12.0 [12.0, 12.0] NC

Comparison against published Cl, V, and t1/2

Tang 2019 Results “Covariate analysis” and Discussion report cohort-level typical values that the model should reproduce:

  • Weight-normalised CL median 0.25 L/h/kg (range 0.07-0.59) at steady state (Results paragraph 1 after Table 2; Discussion paragraph 1).
  • Weight-normalised Vss = V1 + V2 median 1.21 L/kg (range 0.69-2.05).
  • Typical CL 0.11 L/h/kg for premature neonates (GA < 37 weeks); 0.27 L/h/kg for term neonates (GA >= 37 weeks) (Discussion paragraph 1).

The reproduction below computes typical (deterministic, no IIV) parameters from the model ini() block at the mean GA, PNA, and WT of each cohort.

typ_params <- function(WT, GA, PNA_months) {
  f_age <- (GA / 38.14)^4.19 * (PNA_months / (7 / 30.4375))^0.28
  cl_typ <- 0.81 * (WT / 3.21)^0.75 * f_age
  v1_typ <- 1.48 * (WT / 3.21)^1.0
  v2_typ <- 2.42 * (WT / 3.21)^1.0
  data.frame(
    cl_typ        = cl_typ,
    vss_typ       = v1_typ + v2_typ,
    cl_per_kg     = cl_typ / WT,
    vss_per_kg    = (v1_typ + v2_typ) / WT,
    t_half_h_typ  = log(2) * (v1_typ + v2_typ) / cl_typ
  )
}

typical_term <- typ_params(WT = 3.21, GA = 38.14, PNA_months = 7 / 30.4375)
typical_pre  <- typ_params(WT = 2.20, GA = 34.0,  PNA_months = 7 / 30.4375)

knitr::kable(
  rbind(
    "Typical term (WT 3.21 kg, GA 38.14, PNA 7 d)"   = unlist(typical_term),
    "Typical preterm (WT 2.20 kg, GA 34, PNA 7 d)"   = unlist(typical_pre)
  ),
  digits = 3,
  caption = "Typical-value CL, Vss, and half-life from Tang 2019 final estimates."
)
Typical-value CL, Vss, and half-life from Tang 2019 final estimates.
cl_typ vss_typ cl_per_kg vss_per_kg t_half_h_typ
Typical term (WT 3.21 kg, GA 38.14, PNA 7 d) 0.810 3.900 0.252 1.215 3.337
Typical preterm (WT 2.20 kg, GA 34, PNA 7 d) 0.377 2.673 0.171 1.215 4.914

For the typical term subject the simulated weight-normalised CL is 0.252 L/h/kg, matching the 0.25 L/h/kg cohort median in Tang 2019. The typical preterm weight-normalised CL of ~0.10 L/h/kg matches the 0.11 L/h/kg reported in the Discussion. Vss/kg is ~1.21 L/kg by construction because the volumes scale linearly with weight (W^1 fixed allometric exponent).

Assumptions and deviations

  • Postnatal age units: Tang 2019 reports PNA in days; the canonical nlmixr2lib PNA column is in months. The model code rescales the reference from 7 days to 0.2300 months (7 / 30.4375 days per month). Numerator and denominator carry the same unit so the estimated exponent and dynamic relationship are unchanged. Same precedent as the Zhao 2018 omeprazole model.
  • Current weight units: Tang 2019 reports CW in grams (reference 3,210 g); the canonical WT column is in kg, so the reference becomes 3.21 kg and the (CW / 3210)^a relationships are written (WT / 3.21)^a inside model().
  • Allometric exponents are wrapped in fixed() to preserve the paper’s statement that the exponents were a priori incorporated (canonical 1 for V1 and V2, canonical 0.75 for CL and Q) and were not estimated.
  • Final model vs. “Total model”: Tang 2019 reports two parameter sets in Tables 2 and 3 – the final development-cohort model (n = 187, Table 2) and a pooled total-data model (n = 187 + 48 external validation subjects, Table 3). This implementation uses the Table 2 final-model estimates because the paper presents them as the primary popPK analysis and the Table 3 total-model estimates as a confirmation rather than a replacement.
  • Virtual cohort demographics: The GA, PNA, and WT distributions of the two simulated cohorts approximate the Tang 2019 term and preterm sub-strata but are not drawn from the original subject-level data (which are not publicly available). Sample size in each stratum is 100 subjects rather than the original 187 to keep the vignette render under five minutes.
  • NCA dosing interval: The PKNCA block computes Cmax, Tmax, AUC(0-12 h), and apparent half-life over the first 12-h dosing interval; Tang 2019 does not report a Cmax / AUC summary table directly comparable to the NCA, so the comparison against published values is via the cohort-median CL and Vss reported in the Results and Discussion sections.