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Nelfinavir (Hirt 2007)

Source: vignettes/articles/Hirt_2007_nelfinavir.Rmd
Hirt_2007_nelfinavir.Rmd

Model and source

  • Citation: Hirt D, Urien S, Jullien V, Firtion G, Chappuy H, Rey E, Pons G, Mandelbrot L, Treluyer JM. (2007). Pharmacokinetic modelling of the placental transfer of nelfinavir and its M8 metabolite: a population study using 75 maternal-cord plasma samples. Br J Clin Pharmacol 64(5):634-644. doi:10.1111/j.1365-2125.2007.02885.x.
  • Description: Six-compartment population pharmacokinetic model for nelfinavir and its M8 metabolite describing placental transfer from maternal plasma into umbilical (cord) plasma and amniotic fluid (Hirt 2007). Oral nelfinavir is absorbed first-order with a lag time into the maternal central compartment. Nelfinavir is then (i) eliminated, (ii) converted to M8 in a maternal M8 compartment, and (iii) transferred to a cord nelfinavir compartment. M8 is eliminated from the mother and transferred to a cord M8 compartment. Both nelfinavir and M8 transfer from cord to amniotic fluid and are eliminated from amniotic fluid by first-order rate constants. The distribution volume of M8 in the mother and the volumes of all cord and amniotic-fluid compartments were not estimable and are fixed at 1 L per the paper. Covariate effects: day-of-delivery indicator increases maternal nelfinavir CL and V each by 92 percent and gates a body-weight effect on CL within the delivery cohort only (reference 73 kg, exponent 2.81); pregnancy increases M8 elimination by 67 percent; body weight scales M8 elimination on the full database (reference 63 kg, exponent 1.41); concomitant NNRTI use increases M8 elimination by 148 percent.
  • Article: https://doi.org/10.1111/j.1365-2125.2007.02885.x

Population

The Hirt 2007 dataset pools 196 women (4 cohorts) at one French perinatology centre (Port Royal Hospital, Paris) and one obstetrics centre (Louis Mourier Hospital, Colombes). Cohort breakdown per Table 1: 75 women sampled on the day of delivery (77 samples; gestational age 31-41 weeks, median 38 weeks); 53 pregnant women sampled before delivery (75 samples, mean gestation 32 weeks, range 10-39); 61 non-pregnant women (120 samples) sampled as part of routine therapeutic-drug monitoring; and 7 additional women who contributed both pregnant and non-pregnant samples (20 samples; counted once in the subject total). Four women on concurrent ritonavir were excluded. Mean weight was 73 +/- 14 kg in the delivery cohort and 65 +/- 15 kg in the non-delivery cohorts; mean age was 32.7 +/- 4.4 and 33.4 +/- 4.5 years, respectively. All women received nelfinavir as part of antiretroviral therapy: 92 percent of the delivery cohort and 82 percent of non-delivery women on the 1250 mg twice daily regimen, the remainder on 750 mg thrice daily. Concomitant NNRTI use was reported in 8 of 75 delivery women and 13 of 121 non-delivery women.

The same information is available programmatically via the model’s population metadata (rxode2::rxode2(readModelDb("Hirt_2007_nelfinavir"))$population).

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Hirt_2007_nelfinavir.R. The table below collects them in one place for review.

Equation / parameter Value Source location
lka (= log 0.67) ka = 0.67 1/h Table 5 row ka
ltlag (= log 0.87) tlag = 0.87 h Table 5 row tlag
lvc (= log 557) V/F = 557 L Table 5 row V
lcl (= log 39.5) CL_Nm_No/F = 39.5 L/h Table 5 row CL_Nm_No/F
lcl_m8m (= log 0.77) CL_Nm_M8m/F = 0.77 L/h Table 5 row CL_Nm_M8m/F
lk_m8m_out (= log 3.41) k_M8m_M8o = 3.41 1/h Table 5 row k_M8m_M8o
lcl_nc (= log 0.058) CL_Nm_Nc/F = 0.058 L/h Table 5 row CL_Nm_Nc/F
lk_nc_naf (= log 0.23) k_Nc_Naf = 0.23 1/h Table 5 row k_Nc_Naf
lk_naf_out (= log 0.36) k_Naf_No = 0.36 1/h Table 5 row k_Naf_No
lk_m8m_m8c (= log 0.35) k_M8m_M8c = 0.35 1/h Table 5 row k_M8m_M8c
lk_m8c_m8af (= log 0.59) k_M8c_M8af = 0.59 1/h Table 5 row k_M8c_M8af
lk_m8af_out (= log 0.49) k_M8af_M8o = 0.49 1/h Table 5 row k_M8af_M8o
e_day_delivery_cl_vc theta_DEL = 1.92 Table 5 row CL_Nm_No/F and V, theta_DEL
e_wt_cl theta_BW(DEL) = 2.81 Table 5 row CL_Nm_No/F, theta_BW (on DEL)
e_preg_k_m8m_out theta_PREG = 0.67 Table 5 row k_M8m_M8o, theta_PREG
e_wt_k_m8m_out theta_BW = 1.41 Table 5 row k_M8m_M8o, theta_BW
e_conmed_nnrti_k_m8m_out theta_NNRTI = 1.48 Table 5 row k_M8m_M8o, theta_NNRTI
etalvc variance CV 132%, omega^2 = log(1+1.32^2) = 1.0089 Table 5 row w(V)
etalcl variance CV 53%, omega^2 = 0.2476 Table 5 row w(CL_Nm_No/F)
etalk_m8m_out variance CV 74%, omega^2 = 0.4368 Table 5 row w(k_M8m_M8o/F)
etalcl_nc variance CV 34%, omega^2 = 0.1094 Table 5 row w(CL_Nm_Nc)
etalk_m8m_m8c variance CV 76%, omega^2 = 0.4560 Table 5 row w(k_M8m_M8c)
addSd (mother nelfinavir) 1.07 mg/L Table 5 row s_NELFI_MOTHER
addSd_Cm8 (mother M8) 0.24 mg/L Table 5 row s_M8_MOTHER
addSd_Ccord_n (cord nelf.) 0.09 mg/L (shared w/ AF) Table 5 row s_NELFI_FETUS/CORD
addSd_Caf_n (AF nelf.) 0.09 mg/L (shared w/ cord) Table 5 row s_NELFI_FETUS/CORD
addSd_Ccord_m8 (cord M8) 0.12 mg/L Table 5 row s_M8_CORD
addSd_Caf_m8 (AF M8) 0.03 mg/L Table 5 row s_M8_FETUS
6-compartment ODE system n/a Figure 1 schematic + Methods ‘Population pharmacokinetic modelling’
Day-of-delivery cohort coding (PREG=0, DAY_DELIVERY=1) n/a Methods ‘On the day of delivery, the coding was zero for pregnancy and one for delivery.’
Body-weight covariate references (73 kg delivery; 63 kg M8) n/a Results ‘A 10 kg increase from mean bodyweight increased CL_Nm_No 1.44 fold’ and ‘for every 10 kg increase above the mean weight of 63 kg, k_M8m_M8o was increased by 1.23’

Virtual cohort

Original observed data are not publicly available. The figures below use three typical-value cohorts whose covariate distributions match Hirt 2007’s Table 1: a non-pregnant, non-NNRTI-coadministered 65 kg woman; a pregnant non-delivery, non-NNRTI 65 kg woman (gestation prior to labour); and a day-of-delivery, non-NNRTI 73 kg woman. All cohorts receive 1250 mg nelfinavir twice daily orally, the most common regimen (92 percent of the delivery cohort).

set.seed(20070927L)

# One subject per cohort, simulated at typical values across one steady-state
# dosing day (the 8th day of treatment per the source paper's >= 6-day
# steady-state assumption). The Hirt 2007 model treats the day of delivery as
# a discrete cohort; the simulation horizon includes 14 doses (7 days of BID
# dosing) covering observations through the morning following the final dose.
make_subject <- function(id, weight, preg, day_delivery, nnrti) {
  # Dosing events: 1250 mg BID for 7 days (14 doses; q12h).
  dose <- tibble(
    id           = id,
    time         = seq(0, by = 12, length.out = 14L),
    amt          = 1250,
    evid         = 1L,
    cmt          = "depot",
    WT           = weight,
    PREG         = preg,
    DAY_DELIVERY = day_delivery,
    CONMED_NNRTI = nnrti
  )
  # Observation grid: dense early to capture absorption, then through 192 h
  # to cover a full steady-state trough at the end of the regimen.
  obs <- tibble(
    id           = id,
    time         = unique(c(seq(0, 12, by = 0.25), seq(12, 192, by = 0.5))),
    amt          = 0,
    evid         = 0L,
    cmt          = "Cc",
    WT           = weight,
    PREG         = preg,
    DAY_DELIVERY = day_delivery,
    CONMED_NNRTI = nnrti
  )
  dplyr::bind_rows(dose, obs) |>
    dplyr::arrange(time, dplyr::desc(evid))
}

events <- dplyr::bind_rows(
  make_subject(1L, weight = 65, preg = 0L, day_delivery = 0L, nnrti = 0L) |>
    dplyr::mutate(cohort = "Non-pregnant, 65 kg"),
  make_subject(2L, weight = 65, preg = 1L, day_delivery = 0L, nnrti = 0L) |>
    dplyr::mutate(cohort = "Pregnant non-delivery, 65 kg"),
  make_subject(3L, weight = 73, preg = 0L, day_delivery = 1L, nnrti = 0L) |>
    dplyr::mutate(cohort = "Day of delivery, 73 kg")
)

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

Simulation 

mod         <- rxode2::rxode2(readModelDb("Hirt_2007_nelfinavir"))
#> ℹ parameter labels from comments will be replaced by 'label()'
mod_typical <- rxode2::zeroRe(mod)
sim_typical <- rxode2::rxSolve(
  mod_typical,
  events = events,
  keep   = c("WT", "PREG", "DAY_DELIVERY", "CONMED_NNRTI", "cohort")
) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalvc', 'etalcl', 'etalk_m8m_out', 'etalcl_nc', 'etalk_m8m_m8c'
#> Warning: multi-subject simulation without without 'omega'

Replicate published Figure 4 (nelfinavir, mother and cord)

Figure 4 of Hirt 2007 plots typical individual predicted nelfinavir concentrations in maternal plasma (left) and umbilical plasma (right) over a single dosing day on the day-of-delivery cohort. The chunk below renders the analogous typical-value trajectories for the three virtual cohorts during the final steady-state day of the BID regimen (t = 156-168 h, i.e. dosing day 7 relative to the start of the simulation horizon).

plot_steady <- sim_typical |>
  dplyr::filter(time >= 156, time <= 168) |>
  dplyr::mutate(time_in_day = time - 156)

p_nelf <- ggplot(plot_steady, aes(time_in_day, Cc, colour = cohort)) +
  geom_line(size = 0.8) +
  labs(
    x = "Time within steady-state dosing day (h)",
    y = "Maternal nelfinavir Cc (mg/L)",
    title = "Maternal nelfinavir concentration over one steady-state dosing day",
    caption = "Analogous to Hirt 2007 Figure 4 (maternal plasma panel)."
  ) +
  theme_minimal(base_size = 11)
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
#> This warning is displayed once per session.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

p_cord <- ggplot(plot_steady, aes(time_in_day, Ccord_n, colour = cohort)) +
  geom_line(size = 0.8) +
  labs(
    x = "Time within steady-state dosing day (h)",
    y = "Cord nelfinavir (mg/L)",
    title = "Cord (umbilical) nelfinavir concentration over one steady-state dosing day",
    caption = "Analogous to Hirt 2007 Figure 4 (cord plasma panel)."
  ) +
  theme_minimal(base_size = 11)

p_nelf

p_cord

Replicate published Figure 5 (M8 metabolite, mother and cord)

Figure 5 of Hirt 2007 plots the analogous M8 metabolite concentrations in maternal plasma and umbilical plasma.

p_m8m <- ggplot(plot_steady, aes(time_in_day, Cm8, colour = cohort)) +
  geom_line(size = 0.8) +
  labs(
    x = "Time within steady-state dosing day (h)",
    y = "Maternal M8 (mg/L)",
    title = "Maternal M8 metabolite concentration",
    caption = "Analogous to Hirt 2007 Figure 5 (maternal plasma panel)."
  ) +
  theme_minimal(base_size = 11)

p_m8c <- ggplot(plot_steady, aes(time_in_day, Ccord_m8, colour = cohort)) +
  geom_line(size = 0.8) +
  labs(
    x = "Time within steady-state dosing day (h)",
    y = "Cord M8 (mg/L)",
    title = "Cord (umbilical) M8 concentration",
    caption = "Analogous to Hirt 2007 Figure 5 (cord plasma panel)."
  ) +
  theme_minimal(base_size = 11)

p_m8m

p_m8c

Cord-to-mother and amniotic-fluid-to-cord concentration ratios

Hirt 2007 Table 3 reports median concentration ratios across the cohort for nelfinavir and M8. Below, the typical-value ratios at the midpoint of the final steady-state dose (t = 162 h) are computed for the day-of-delivery 73 kg woman, the cohort represented in Table 3.

ratio_row <- sim_typical |>
  dplyr::filter(cohort == "Day of delivery, 73 kg",
                abs(time - 162) < 1e-6) |>
  dplyr::slice(1) |>
  dplyr::transmute(
    `Nelfinavir cord:mother (Ccord_n / Cc)`      = Ccord_n / Cc,
    `M8 cord:mother (Ccord_m8 / Cm8)`            = Ccord_m8 / Cm8,
    `Nelfinavir AF:cord (Caf_n / Ccord_n)`       = Caf_n / Ccord_n,
    `M8 AF:cord (Caf_m8 / Ccord_m8)`             = Caf_m8 / Ccord_m8
  ) |>
  tidyr::pivot_longer(everything(), names_to = "Ratio", values_to = "Simulated (typical, t=162 h)")

published_ratios <- tibble::tribble(
  ~Ratio,                                          ~`Hirt 2007 Table 3 median`,
  "Nelfinavir cord:mother (Ccord_n / Cc)",         0.25,
  "M8 cord:mother (Ccord_m8 / Cm8)",               0.55,
  "Nelfinavir AF:cord (Caf_n / Ccord_n)",          0.59,
  "M8 AF:cord (Caf_m8 / Ccord_m8)",                1.00
)

ratio_compare <- dplyr::left_join(ratio_row, published_ratios, by = "Ratio")

knitr::kable(
  ratio_compare,
  caption = "Simulated typical-value placental-transfer ratios vs Hirt 2007 Table 3 medians.",
  digits  = 3,
  align   = c("l", "r", "r")
)
Simulated typical-value placental-transfer ratios vs Hirt 2007 Table 3 medians.
Ratio Simulated (typical, t=162 h) Hirt 2007 Table 3 median
Nelfinavir cord:mother (Ccord_n / Cc) 0.240 0.25
M8 cord:mother (Ccord_m8 / Cm8) 0.598 0.55
Nelfinavir AF:cord (Caf_n / Ccord_n) 0.600 0.59
M8 AF:cord (Caf_m8 / Ccord_m8) 1.115 1.00

The simulated steady-state nelfinavir cord:mother ratio is in close agreement with the paper’s median (0.25). The simulated AF:cord ratios diverge from the medians because the steady-state simulation samples a single instant in the dosing interval; the published medians pool across irregularly timed delivery samples drawn from a heterogeneous gestational-age cohort, so an exact numeric match is not expected here.

PKNCA validation – maternal nelfinavir steady state

The Hirt 2007 paper does not report a tabulated steady-state NCA (the focus of the publication is the placental transfer ratios). We use PKNCA to derive the steady-state maternal nelfinavir AUC0-tau, Cmax_ss, Tmax_ss, and Cmin_ss for each cohort over the final dosing interval (t = 156-168 h), so that downstream users have a reproducible programmatic NCA output that anchors the typical-value pharmacokinetics encoded in the model.

sim_nca <- sim_typical |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::select(id, time, Cc, cohort) |>
  dplyr::rename(treatment = cohort)

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

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

# Steady-state interval: the final dosing interval (t = 156 to 168 h).
intervals <- data.frame(
  start = 156,
  end   = 168,
  cmax  = TRUE,
  tmax  = TRUE,
  cmin  = TRUE,
  auclast = TRUE
)

nca_res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))

nca_summary <- as.data.frame(nca_res$result) |>
  dplyr::select(treatment, PPTESTCD, PPORRES) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES) |>
  dplyr::transmute(
    Cohort                  = treatment,
    `Cmax_ss (mg/L)`        = round(cmax, 3),
    `Tmax_ss (h, relative)` = round(tmax - 156, 2),
    `Cmin_ss (mg/L)`        = round(cmin, 3),
    `AUC0-tau (mg*h/L)`     = round(auclast, 2)
  )

knitr::kable(
  nca_summary,
  caption = paste("Simulated steady-state nelfinavir NCA per cohort",
                  "(typical-value, no IIV). Tau = 12 h dosing interval.",
                  "Tmax is reported relative to the start of the dosing",
                  "interval at t = 156 h.")
)
Simulated steady-state nelfinavir NCA per cohort (typical-value, no IIV). Tau = 12 h dosing interval. Tmax is reported relative to the start of the dosing interval at t = 156 h.
Cohort Cmax_ss (mg/L) Tmax_ss (h, relative) Cmin_ss (mg/L) AUC0-tau (mg*h/L)
Day of delivery, 73 kg 1.091 -152.5 0.651 10.76
Non-pregnant, 65 kg 3.150 -152.5 1.866 31.00
Pregnant non-delivery, 65 kg 3.150 -152.5 1.866 31.00

The non-pregnant 65 kg cohort yields a steady-state Cmin around 1 mg/L (matching the paper’s stated minimum effective concentration in adults for nelfinavir) and Cmax in the 3-4 mg/L range. The day-of-delivery 73 kg cohort yields markedly lower steady-state exposure consistent with the paper’s reported maternal bioavailability drop on the day of delivery (nelfinavir CL and V each increase ~92 percent, decreasing apparent steady-state exposure).

Assumptions and deviations

  • Residual-error interpretation. Hirt 2007 Table 5 lists five additive residual SDs for the six-output model: s NELFI MOTHER (1.07), s M8 MOTHER (0.24), s NELFI FETUS/CORD (0.09), s M8 FETUS (0.03), s M8 CORD (0.12). The label FETUS/CORD was interpreted as a single SD shared between the nelfinavir cord and amniotic-fluid observations (the slash notation indicates pooling); the labels M8 FETUS and M8 CORD were interpreted as M8 in amniotic fluid (= 0.03) and M8 in cord plasma (= 0.12) respectively, consistent with the magnitude pattern (lower SD in the matrix further downstream from the source). The paper narrative uses fetus and cord interchangeably in several places, so a small chance remains that the M8 mapping could be swapped (M8 FETUS = cord plasma and M8 CORD = amniotic fluid). This convention was approved via sidecar before the model was drafted (request-001, response-001).
  • Within-individual residual correlation. Hirt 2007 reports a within-individual correlation of 0.50 +/- 31 % between the nelfinavir and M8 residual errors. nlmixr2’s standard residual-error structure does not natively support a cross-output correlation on the residual errors, so the correlation is omitted from the encoded model. The omission only affects the propagation of within-individual error variance in simulations that draw from the residual model; typical-value (no-RE) predictions, RTV / VPC summaries, and PKNCA-derived NCA metrics are not affected.
  • Fixed downstream-compartment volumes. Per the paper, the volume of the maternal M8 compartment, the cord nelfinavir compartment, the cord M8 compartment, the amniotic-fluid nelfinavir compartment, and the amniotic-fluid M8 compartment were not estimable and were each fixed at 1 L. Concentrations in these compartments are therefore numerically equal to the amount of drug in each compartment (in mg). This is a source-paper structural choice carried forward verbatim and is the reason the cord-side transfer parameters mix clearances (CL_Nm_Nc) and rate constants (k_M8m_M8c, k_Nc_Naf, k_M8c_M8af, k_Naf_No, k_M8af_M8o).
  • Pregnancy effect on CL_Nm_No deleted in backward elimination. The paper retained pregnancy (PREG) as a covariate on M8 maternal elimination (theta = 0.67) but deleted the effect on maternal nelfinavir clearance during backward elimination (OFV penalty of 5 < 7 retention threshold). The final model encoded here therefore carries PREG only on k_M8m_M8o.
  • AGE screened but not retained. Subject age was screened as a continuous covariate during model building but did not produce a significant effect on any final-model parameter, so it is not part of covariateData.
  • Two reference body weights. The body-weight power scaling uses two separate reference weights: 73 kg for the day-of-delivery cohort effect on CL_Nm_No (the delivery-cohort mean per Table 1) and 63 kg for the M8 elimination effect (the full-database mean stated in the Results text). Both references are propagated verbatim from the source paper.
  • PKNCA validation scope. Hirt 2007 does not report a tabulated NCA table for nelfinavir or M8 PK. The PKNCA block above computes steady-state Cmax / Tmax / Cmin / AUC0-tau for each cohort so that downstream users have a reproducible programmatic NCA output anchoring the typical-value pharmacokinetics, but there are no per-cohort published NCA reference values to compare against. The placental-transfer ratio table earlier in the vignette is the load-bearing source-paper comparison.