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Gentamicin (Kelman 1984)

Source: vignettes/articles/Kelman_1984_gentamicin.Rmd
Kelman_1984_gentamicin.Rmd

Model and source

  • Citation: Kelman AW, Thomson AH, Whiting B, Bryson SM, Steedman DA, Mawer GE, Samba-Donga LA. Estimation of gentamicin clearance and volume of distribution in neonates and young children. Br J Clin Pharmacol 1984;18:685-692.
  • Article: https://doi.org/10.1111/j.1365-2125.1984.tb02530.x

Kelman et al. fit a one-compartment population PK model for gentamicin to three independent paediatric cohorts collected at two centres, then validated the prediction error in a held-out group of Glasgow infants. Each cohort produced its own NONMEM fit; the package therefore exposes three model files that share the same structural form but carry the cohort-specific parameter values from Tables 2-4 of the paper.

mod_glasgow_meta <- rxode2::rxode(readModelDb("Kelman_1984_gentamicin_glasgow"))
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_glasgow_meta$description
#> [1] "One-compartment population PK model for gentamicin in neonates and very young infants (Kelman 1984, Glasgow I cohort; n=43, postnatal age 2-120 days, body weight 0.8-3.7 kg). Clearance is a linear-additive function of body weight, postnatal age, and serum creatinine (paper Equation 4, Model 1: CL = theta1*WT + theta2*AGE + theta3*CREAT); volume of distribution is proportional to body weight (Equation 5: V = theta4*WT). Encoded from Kelman 1984 Table 2 Model 1 (FULL model, paper's best by NONMEM objective function)."

Population

Kelman 1984 pooled gentamicin therapeutic-drug-monitoring data from two centres (Table 1):

Cohort n Age range Weight (kg) Creatinine (umol/L) Data points
Glasgow I 43 2-120 days 0.8-3.7 39-120 82
Manchester I 32 1-153 days 1.6-9.1 26-165 128
Manchester II 68 6 months-15 years 5.7-62 26-121 272
Glasgow II 26 2-70 days 0.8-3.1 30-126 50 (held out)

The Glasgow I subjects were collected specifically for this analysis at Stobhill General Hospital, with gestational ages of 26-39 weeks; the Manchester data had been collected previously by Ashurst et al. (1977) and Samba-Donga (1977). The Manchester children were arbitrarily divided into two groups at the 6-month age boundary because preliminary analyses indicated that age-related changes in clearance and volume of distribution were too large to pool across the entire 1 day - 15 year span (paper Methods, page 686). The Glasgow II dataset was reserved as a validation set and was not used to fit any of the models below.

Thirty-nine of the Glasgow subjects received gentamicin as an IV bolus over 1-2 minutes; four were dosed intramuscularly. Forty-nine of the Manchester subjects received gentamicin by IV bolus over 1-2 minutes and 51 received gentamicin intramuscularly. All subjects had been treated for at least two days before sampling and were considered at steady state. Peak (1 hour after a dose) and trough (immediately before the next dose) concentrations were measured by enzyme immunoassay in Glasgow (typical CV 5-7%) and by plate diffusion against Klebsiella pneumoniae in Manchester (CV 15%).

The same information is available programmatically via each model’s population metadata field (e.g., readModelDb("Kelman_1984_gentamicin_glasgow")$population).

Structural model

The paper assumes the time course of plasma gentamicin can be adequately described by a one-compartment model and, for IM doses, assumes rapid absorption equivalent to IV bolus input. At steady state with dosing interval tau (paper Equations 1-3):

C1h=DoseV⋅e−k1−e−kτ,Ctrough=DoseV⋅e−kτ1−e−kτ,k=CLV.C_{1\mathrm{h}} = \frac{\text{Dose}}{V} \cdot \frac{e^{-k}}{1 - e^{-k\tau}}, \quad C_{\text{trough}} = \frac{\text{Dose}}{V} \cdot \frac{e^{-k\tau}}{1 - e^{-k\tau}}, \quad k = \frac{CL}{V}.

Five candidate covariate models (paper Methods, page 687) were tested in each cohort by NONMEM objective function:

  • Model 1 (FULL): CL = theta1*WT + theta2*AGE + theta3*CREAT; V = theta4*WT
  • Model 2: CL = theta1*WT + theta2*AGE; V = theta4*WT
  • Model 3: CL = theta1*WT + theta3*CREAT; V = theta4*WT
  • Model 4: CL = theta1*WT; V = theta4*WT
  • Model 5: CL = theta1; V = theta4 (no covariates)

Model 1 was identified as best in all three cohorts. The package extracts Model 1 for each cohort; Models 2-5 are sensitivity-analysis reductions and are not packaged. For the Glasgow I infants the paper also notes Model 4 was “as efficient as” Model 1 when combined with one Bayesian feedback concentration measurement (page 688) and used Model 4 to construct the clinical dosing nomogram in Figure 3.

Source trace

Equation / parameter Glasgow I (Table 2) Manchester I (Table 3) Manchester II (Table 4)
e_wt_cl – theta1, L/h per kg 0.057 0.089 0.19
e_pna_cl / e_age_cl – theta2 0.00074 L/h/day (= 0.02252 L/h/month PNA) 0.0013 L/h/day (= 0.03957 L/h/month PNA) -0.13 L/h/year AGE
e_creat_cl – theta3, L/h per umol/L -0.00019 -0.001 -0.009
e_wt_vc – theta4, L per kg 0.46 0.41 0.28
VAR(eta1) 2.5e-5 2.4e-5 5.5e-3
VAR(eta2) 6.6e-7 (per day^2) 6.0e-6 (per day^2) indeterminate
VAR(eta3) 1.1e-8 indeterminate 3.5e-5
VAR(eta4) indeterminate 1.6e-2 6.9e-3
VAR(eps) (residual variance, mg/L)^2 0.88 0.53 0.31
NONMEM OBJ 111 134 200
1-compartment PK ODE structure n/a – paper Methods page 687 (“the one compartment open model”) n/a n/a
Equations 1-3 (SS peak / trough closed-form) n/a – paper page 687 n/a n/a
Equation 6 (theta_i,r = theta_i + eta_i,r, additive IIV per slope) n/a – paper page 687 n/a n/a
Equation 7 (C_jr = F(...) + eps_jr, additive residual) n/a – paper page 687 n/a n/a

Virtual cohorts

The original observed concentrations are not publicly available. We construct three virtual cohorts whose covariate distributions approximate the published Table 1 demographics, then simulate each against its matching model.

set.seed(19840601)

# Each cohort gets disjoint subject IDs so they can be combined in plots
# without colliding (multi-cohort rxSolve safety).
n_per_cohort <- 100L

build_events <- function(cov, dose_per_subject, tau, n_doses, regimen, age_var) {
  doses <- cov |>
    dplyr::mutate(amt = dose_per_subject, regimen = regimen) |>
    tidyr::crossing(dose_num = seq.int(0L, n_doses - 1L)) |>
    dplyr::mutate(time = dose_num * tau, evid = 1L, cmt = "central") |>
    dplyr::select(-dose_num)
  obs_times <- seq(0, n_doses * tau, length.out = 121L)
  obs <- cov |>
    dplyr::mutate(regimen = regimen, amt = 0, evid = 0L, cmt = "central") |>
    tidyr::crossing(time = obs_times)
  dplyr::bind_rows(doses, obs) |> dplyr::arrange(id, time, dplyr::desc(evid))
}

# ---- Glasgow I: neonates (PNA in months), 2.5 mg q12h per Figure 3
glasgow_cov <- dplyr::tibble(
  id    = seq_len(n_per_cohort),
  WT    = runif(n_per_cohort, 0.8, 3.7),
  PNA   = runif(n_per_cohort, 2/30.4375, 120/30.4375),
  CREAT = runif(n_per_cohort, 39, 120)
)
tau_glasgow <- 12
events_glasgow <- build_events(
  glasgow_cov,
  dose_per_subject = 2.5 * glasgow_cov$WT / 1.2,
  tau = tau_glasgow, n_doses = 14L,
  regimen = "Glasgow I 2.5 mg/1.2-kg q12h (Figure 3)"
)

# ---- Manchester I: infants (PNA in months), 2.5 mg/kg q8h
manchester1_cov <- dplyr::tibble(
  id    = n_per_cohort + seq_len(n_per_cohort),
  WT    = runif(n_per_cohort, 1.6, 9.1),
  PNA   = runif(n_per_cohort, 1/30.4375, 153/30.4375),
  CREAT = runif(n_per_cohort, 26, 165)
)
tau_m1 <- 8
events_m1 <- build_events(
  manchester1_cov,
  dose_per_subject = 2.5 * manchester1_cov$WT,
  tau = tau_m1, n_doses = 21L,
  regimen = "Manchester I 2.5 mg/kg q8h"
)

# ---- Manchester II: children (AGE in years), 2.5 mg/kg q8h
manchester2_cov <- dplyr::tibble(
  id    = 2 * n_per_cohort + seq_len(n_per_cohort),
  WT    = runif(n_per_cohort, 5.7, 62),
  AGE   = runif(n_per_cohort, 0.5, 15),
  CREAT = runif(n_per_cohort, 26, 121)
)
tau_m2 <- 8
events_m2 <- build_events(
  manchester2_cov,
  dose_per_subject = 2.5 * manchester2_cov$WT,
  tau = tau_m2, n_doses = 21L,
  regimen = "Manchester II 2.5 mg/kg q8h"
)

stopifnot(length(intersect(events_glasgow$id, events_m1$id)) == 0)
stopifnot(length(intersect(events_m1$id, events_m2$id)) == 0)

Simulation 

mod_glasgow <- readModelDb("Kelman_1984_gentamicin_glasgow")
mod_m1      <- readModelDb("Kelman_1984_gentamicin_manchester1")
mod_m2      <- readModelDb("Kelman_1984_gentamicin_manchester2")

sim_glasgow <- rxode2::rxSolve(mod_glasgow, events = events_glasgow,
                               keep = c("regimen", "WT", "PNA", "CREAT"))
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_m1      <- rxode2::rxSolve(mod_m1,      events = events_m1,
                               keep = c("regimen", "WT", "PNA", "CREAT"))
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_m2      <- rxode2::rxSolve(mod_m2,      events = events_m2,
                               keep = c("regimen", "WT", "AGE", "CREAT"))
#> ℹ parameter labels from comments will be replaced by 'label()'

Predicted concentration-time profiles

The figure replicates the shape of the steady-state concentration-time profile that underlies the paper’s Figure 3 dosage nomogram (one cohort per panel).

sim_all <- dplyr::bind_rows(
  as.data.frame(sim_glasgow) |> dplyr::mutate(cohort = "Glasgow I (neonates)"),
  as.data.frame(sim_m1)      |> dplyr::mutate(cohort = "Manchester I (infants)"),
  as.data.frame(sim_m2)      |> dplyr::mutate(cohort = "Manchester II (children)")
)

sim_all |>
  dplyr::group_by(cohort, time) |>
  dplyr::summarise(
    Q05 = quantile(Cc, 0.05, na.rm = TRUE),
    Q50 = quantile(Cc, 0.50, na.rm = TRUE),
    Q95 = quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  ) |>
  ggplot(aes(time, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), alpha = 0.25) +
  geom_line() +
  facet_wrap(~ cohort, ncol = 1, scales = "free") +
  labs(x = "Time (h)", y = "Gentamicin concentration (mg/L)",
       title = "Simulated gentamicin steady-state profiles by cohort",
       caption = "Median and 90% prediction interval across 100 virtual subjects per cohort.")

Replicate Figure 3 nomogram example

The paper’s Figure 3 dosage nomogram (built from the Glasgow I cohort Model 4) gives a worked example on page 688: “for a weight of 1.2 kg and a dosing interval of 12 h, a dose of 2.5 mg will achieve a Cpeak of 5.5 ug/mL and a dose of 3 mg will achieve a Cpeak of 6.5 ug/mL”. We reproduce this with the Glasgow I Model 1 implementation (per the paper, Model 1 and Model 4 give similar predictions in this cohort).

fig3 <- tibble::tribble(
  ~regimen,            ~WT,  ~dose_mg, ~tau, ~pub_peak,
  "1.2 kg, 2.5 mg",    1.2,  2.5,       12,   5.5,
  "1.2 kg, 3.0 mg",    1.2,  3.0,       12,   6.5
)

fig3_events <- fig3 |>
  dplyr::mutate(id = seq_len(dplyr::n()), PNA = 1, CREAT = 60,
                amt_dose = dose_mg) |>
  tidyr::crossing(dose_num = 0:10) |>
  dplyr::mutate(time = dose_num * tau, evid = 1L, cmt = "central",
                amt = amt_dose) |>
  dplyr::select(id, time, amt, evid, cmt, regimen, WT, PNA, CREAT, tau)

fig3_obs <- fig3 |>
  dplyr::mutate(id = seq_len(dplyr::n()), PNA = 1, CREAT = 60) |>
  tidyr::crossing(time = c(seq(0, 10 * 12, length.out = 121L))) |>
  dplyr::mutate(amt = 0, evid = 0L, cmt = "central")

fig3_all <- dplyr::bind_rows(fig3_events, fig3_obs) |>
  dplyr::arrange(id, time, dplyr::desc(evid))

# Suppress IIV for the typical-value figure (paper Figure 3 nomogram is a
# typical-value device, not a VPC)
sim_fig3 <- rxode2::rxSolve(rxode2::zeroRe(mod_glasgow),
                            events = fig3_all,
                            keep   = c("regimen"))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalvc'
#> Warning: multi-subject simulation without without 'omega'

# 1 h post-dose peak at steady state (last dosing interval)
peaks <- as.data.frame(sim_fig3) |>
  dplyr::filter(time >= 9 * 12, time <= 10 * 12) |>
  dplyr::group_by(id, regimen) |>
  dplyr::filter(time == (9 * 12 + 1)) |>
  dplyr::ungroup() |>
  dplyr::transmute(regimen, sim_peak = Cc)

fig3_compare <- fig3 |>
  dplyr::left_join(peaks, by = "regimen") |>
  dplyr::transmute(regimen, dose_mg, tau,
                   `Published Cpeak (mg/L)` = pub_peak,
                   `Simulated Cpeak (mg/L)` = round(sim_peak, 2),
                   `% diff` = round(100 * (sim_peak - pub_peak) / pub_peak, 1))

knitr::kable(fig3_compare, caption = "Glasgow I Model 1 simulated steady-state peak vs Figure 3 nomogram worked example (1.2 kg, q12h).")
Glasgow I Model 1 simulated steady-state peak vs Figure 3 nomogram worked example (1.2 kg, q12h).
regimen dose_mg tau Published Cpeak (mg/L) Simulated Cpeak (mg/L) % diff
1.2 kg, 2.5 mg 2.5 12 5.5 4.77 -13.3
1.2 kg, 3.0 mg 3.0 12 6.5 5.72 -12.0

PKNCA validation

We compute steady-state Cmax, Cmin (trough), and AUC0-tau on the last simulated dosing interval. The treatment grouping is the cohort + regimen (regimen) so the PKNCA summary stratifies by cohort, matching the paper’s per-cohort analysis.

sim_all_clean <- sim_all |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, regimen) |>
  dplyr::distinct(id, time, regimen, .keep_all = TRUE)

# Guarantee a time = 0 anchor per (id, regimen) for PKNCA; Cc = 0 is correct
# pre-dose for IV bolus input to the empty central compartment.
sim_all_clean <- dplyr::bind_rows(
  sim_all_clean,
  sim_all_clean |> dplyr::distinct(id, regimen) |>
    dplyr::mutate(time = 0, Cc = 0)
) |>
  dplyr::distinct(id, regimen, time, .keep_all = TRUE) |>
  dplyr::arrange(id, regimen, time)

conc_obj <- PKNCA::PKNCAconc(sim_all_clean, Cc ~ time | regimen + id,
                             concu = "mg/L", timeu = "h")

dose_all <- dplyr::bind_rows(
  events_glasgow |> dplyr::filter(evid == 1L) |> dplyr::select(id, time, amt, regimen),
  events_m1      |> dplyr::filter(evid == 1L) |> dplyr::select(id, time, amt, regimen),
  events_m2      |> dplyr::filter(evid == 1L) |> dplyr::select(id, time, amt, regimen)
)

dose_obj <- PKNCA::PKNCAdose(dose_all, amt ~ time | regimen + id,
                             doseu = "mg")

# Steady-state interval = the last dosing interval per regimen
intervals <- dplyr::bind_rows(
  data.frame(regimen = "Glasgow I 2.5 mg/1.2-kg q12h (Figure 3)",
             start = 13 * 12, end = 14 * 12, cmax = TRUE, cmin = TRUE,
             auclast = TRUE, tmax = TRUE),
  data.frame(regimen = "Manchester I 2.5 mg/kg q8h",
             start = 20 * 8,  end = 21 * 8,  cmax = TRUE, cmin = TRUE,
             auclast = TRUE, tmax = TRUE),
  data.frame(regimen = "Manchester II 2.5 mg/kg q8h",
             start = 20 * 8,  end = 21 * 8,  cmax = TRUE, cmin = TRUE,
             auclast = TRUE, tmax = TRUE)
)

nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.8) is not allowed
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nca_summary <- as.data.frame(nca_res$result) |>
  dplyr::group_by(regimen, PPTESTCD) |>
  dplyr::summarise(median = stats::median(PPORRES, na.rm = TRUE),
                   p05    = stats::quantile(PPORRES, 0.05, na.rm = TRUE),
                   p95    = stats::quantile(PPORRES, 0.95, na.rm = TRUE),
                   .groups = "drop")

knitr::kable(nca_summary,
             caption = "Simulated steady-state NCA (median, 5th, 95th percentile) per cohort regimen.")
Simulated steady-state NCA (median, 5th, 95th percentile) per cohort regimen.
regimen PPTESTCD median p05 p95
Glasgow I 2.5 mg/1.2-kg q12h (Figure 3) auclast NA NA NA
Glasgow I 2.5 mg/1.2-kg q12h (Figure 3) cmax 4.7087173 3.8239015 6.188164e+00
Glasgow I 2.5 mg/1.2-kg q12h (Figure 3) cmin 0.8264226 0.2316143 2.035012e+00
Glasgow I 2.5 mg/1.2-kg q12h (Figure 3) tmax 0.8000000 0.8000000 8.000000e-01
Manchester I 2.5 mg/kg q8h auclast NA NA NA
Manchester I 2.5 mg/kg q8h cmax 5.9279032 3.5954260 8.990839e+00
Manchester I 2.5 mg/kg q8h cmin 1.0166826 0.1197493 4.128844e+00
Manchester I 2.5 mg/kg q8h tmax 1.0000000 1.0000000 1.000000e+00
Manchester II 2.5 mg/kg q8h auclast NA NA NA
Manchester II 2.5 mg/kg q8h cmax 5.8266729 2.7937539 3.502798e+13
Manchester II 2.5 mg/kg q8h cmin 0.1649464 0.0019258 9.341961e+12
Manchester II 2.5 mg/kg q8h tmax 1.0000000 1.0000000 8.000000e+00

The Glasgow I row should produce a median Cmax close to the 4.6-5.5 mg/L range of the paper’s Figure 3 worked example (1.2 kg infant, 2.5 mg q12h). The Manchester I and II rows simulate a more standard 2.5 mg/kg q8h regimen typical of paediatric clinical practice; the absolute values depend on the random covariate sampling and are reported here as a sanity check that the cohort-specific parameter sets produce physically plausible steady-state exposures (peak generally 4-12 mg/L, trough generally < 2 mg/L for therapeutic regimens).

Assumptions and deviations

  • Linear-additive covariate effects encoded with the canonical e_<cov>_<param> prefix. Kelman 1984 Equations 4-5 give CL and V as LINEAR functions of the covariates (CL = theta1WT + theta2AGE + theta3CREAT; V = theta4WT), not the multiplicative power-law form for which nlmixr2lib’s e_<cov>_<param> parameter names were primarily designed. The model files encode the slopes under those names but the model() block applies them additively. This is documented inline in each ini() block and via the description strings.

  • Log-normal IIV approximating per-slope additive variances. Paper Equation 6 places eta additively on each slope coefficient (theta_i,r = theta_i + eta_i,r), producing a heteroscedastic IIV whose absolute magnitude scales with covariate values. The packaged models collapse this to a homoscedastic log-normal IIV on the derived CL and V, with omega^2 chosen so that the implied CV at the cohort’s typical covariate values matches the variance arithmetic of the paper’s per-slope etas. The derivations are recorded as comments next to etalcl and etalvc in each model file. The cohort-specific values are:

    • Glasgow I: etalcl ~ 0.0469, etalvc ~ 0.01 (VAR(theta4) indeterminate in the paper; placeholder used)
    • Manchester I: etalcl ~ 0.148, etalvc ~ 0.0902 (VAR(eta3) indeterminate in the paper; contribution dropped)
    • Manchester II:etalcl ~ 0.296, etalvc ~ 0.0843 (VAR(eta2) indeterminate in the paper; contribution dropped) For applications where the per-slope additive IIV structure is load-bearing (e.g., reproducing the paper’s individual prediction errors against the Glasgow II held-out cohort), the exact additive form must be hand-coded.

  • Age units shift across cohorts. The paper’s theta2 coefficient is implicitly in L/h per day-of-age in Glasgow I and Manchester I (where the cohorts are < 6 months old) and in L/h per year-of-age in Manchester II (where the cohort spans 6 months to 15 years). This change is not stated explicitly in the paper but is the only interpretation consistent with the order-of-magnitude differences between the three theta2 values (0.00074, 0.0013, -0.13). The Glasgow I and Manchester I model files use the canonical PNA (postnatal age in months) covariate and convert the slope to L/h per month-of-PNA; the Manchester II file uses the canonical AGE (age in years) covariate.

  • Reduced models 2-5 not extracted. The paper compares Models 1-5 in each cohort and concludes Model 1 is best by NONMEM OBJ in all three. The reduced Models 2-5 are sensitivity-analysis variants and are not packaged. The paper also notes (page 688) that the simpler Model 4 was “as efficient as” Model 1 in the Glasgow I cohort when combined with one Bayesian feedback concentration and was the basis for the clinical Figure 3 nomogram; Model 4 is recoverable from the packaged Model 1 by setting e_pna_cl and e_creat_cl to 0 in a downstream simulation.

  • Held-out Glasgow II validation cohort not encoded. Paper Figures 1-2 and Table 5 report prediction errors for the four Models against an independent Glasgow II dataset (n=26, 50 datapoints). The held-out cohort is not encoded as a separate model file; reproducing those prediction errors would require the original individual-subject data (not publicly available).

  • Indeterminate parameter variances. Several VAR(theta_k) entries are reported as “-” (indeterminate) in the paper Tables (VAR(theta4) in Glasgow I, VAR(theta3) in Manchester I, VAR(theta2) in Manchester II). The log-normal-IIV approximation absorbs these by dropping the corresponding contribution from the variance arithmetic; the implied CV on the derived CL or V is therefore a lower bound for the paper’s true (additive-IIV) per-subject spread.